Autocorrelation « Open Mind

Autocorrelation

March 22, 2008 · 169 Comments

Global temperature shows a lot of variations; it goes up and down in a manner which is partly predictable and partly unpredictable. One of the important issues of global warming is how temperature will change over the long run. For this issue, we’re less interested in the short-term, year-to-year or month-to-month, day-to-day or moment-to-moment fluctuations which go up and down but average out to zero, and more interested in the persistent changes that behave more steadily and consistently over many years, decades, or longer. We’re interested in the trend.

It’s especially useful to estimate the trend rate. That’s because, if the trend rate stays constant or reasonably so, then knowing its value enables us to estimate what future temperature will be. We still won’t know what the moment-to-moment or year-to-year rapid fluctuations will be, and in the short run they turn out to be much larger than the change due to trend. But in the long run, the change due to trend accumulates while the fluctuations average out to zero, so the effect of the trend ends up being larger than the rapid fluctuations. I’d like to discuss how that trend rate is often estimated, and the effect of autocorrelation on that estimate. If you already know all the ins and outs of “linear regression” the first part will be no more than review. But for most people, the second part will be something new.

The model behind linear regression is that at any moment of time t, the data (which we’ll call x) is a linear function of time (i.e., a straight line), plus a random process:

$x(t) = \alpha t + \beta + \epsilon(t)$ .

The quantities $\alpha$ and $\beta$ are constants which represent the trend in the data; $\alpha$ is the “slope,” which is the rate of change of the trend line, and $\beta$ is its intercept, which is the value of the trend line when the time is $t=0$ . The $\epsilon$ term represents the random process. It’s sometimes called the “error” term, but it’s not necessarily an error. It might be! Or it might be a physical process which is truly random. It might even be a deterministic process (not at all random) which we don’t know how to predict, but behaves enough like a random process that we can safely model it as one. We can turn this equation around to say that

$x(t) - \alpha t - \beta = \epsilon(t)$ .

Presumably this equation holds for every moment of time. Of course, we can never have real data for every moment of time; there are (far!) too many moments to have measurements for all of them. Instead, we have estimates for a finite number of moments of time. Let’s write the times for which we have measurements as $t_1, t_2, ..., t_N$ , where N is the number of data points we have. The measurements associated with those times we’ll write as $x_1, x_2, ..., x_N$ . Hence for our linear model of how the data vary, we have

$x_j - \alpha t_j - \beta = \epsilon_j$ .

On the left hand side we have the difference between what was observed ( $x_j$ ) and what it would have been if it followed the straight line exactly ( $\alpha t_j + \beta$ ). These differences are the residuals. In all the following analysis, we’ll assume that the measurements are equally spaced in time. This is the case for almost all temperature analysis (the alternative case, referred to as “irregular sampling,” is one of the most fascinating aspects of time series analysis — but it doesn’t concern us here).

If we knew the true values of the constants $\alpha$ and $\beta$ , then the residuals are just a random process. Then the sum of the squares of the residuals will be the sum of the squares of that random process

$\sum (x_j - \alpha t_j - \beta)^2 = \sum (\epsilon_j)^2$ .

However, if we used the wrong values for the constants $\alpha$ and $\beta$ , then we expect (in the mathematical sense) that the sum of the squares of the residuals will be greater than the sum of the squares of the random process. Hence we use the available data to find the choices for the constants which minimize the sum of the squares of the quantities $x_j - \alpha t_j - \beta$ , i.e., the sum of the squares of the residuals. That’s why it’s called the method of least squares. In a wider context, it’s called ordinary least squares, or OLS.

This gives us estimates of the values of the constants $\alpha$ and $\beta$ . It doesn’t give us the absolutely true correct values! That’s because the assertion that “we expect (in the mathematical sense)” is a mathematical expectation, i.e., it’s a probabilistic assertion, not a deterministic one. In actual practice the values we estimate for $\alpha$ and $\beta$ will be close to, but not exactly equal, the true values. Fortunately, we can also estimate how far off they’re likely to be, i.e., the probable error of our estimates for the constants.

Given a large enough data sample (i.e., for N large enough), we get a “good” estimate of the constants regardless of the nature of the random process $\epsilon_j$ (before I’m flooded with objections, this isn’t strictly true always — but the exceptions are truly exceptional and we can safely ignore them). By “good” I mean that the mathematical expectation values of our estimates for the constants will be the true values; another way of saying this is that the estimates we get from least-squares regression are unbiased.

However, the nature of the random process can have a profound affect on our estimate of the probable error of the constants. The most common assumption about the nature of the random process is that the values of $\epsilon_j$ are white noise. This means that they all have expectation value zero, that they all have the same variance, and most important for this post, they’re all independent of each other. If they behave that way (or very approximately so), then the “standard” calculation of the probable error from least-squares regression gives a good estimate. But an important case, which applies to real temperature data (and a lot of other climate data too), is that the random terms are not independent of each other. A given term might depend on the preceding value of that term, or several preceding values, of show some other more complicated dependence. When the random terms are not independent of each other, and especially when a given random term is correlated with nearby (in time) values of that random term, we say that the random process exhibits autocorrelation.

What sort of correlation might the random terms exhibit? One of the most useful (and most applicable) models of the random process is an autoregressive model, in which a given random term is a linear combination of preceding random terms, plus a truly independent random part all its own. Hence the j^th random term might behave as

$\epsilon_j = \rho_1 \epsilon_{j-1} + \rho_2 \epsilon_{j-2} + \rho_3 \epsilon_{j-3} + e_j$ .

In this case, the random term depends linearly on the three precedent random terms, and it also has its own independent random part $e_j$ . The independent parts are assumed to be truly a white-noise process; all of the quantities $e_j$ are independent of each other. This example shows a 3rd-order autoregressive process, also denoted as an AR(3) process, and the quantities $\rho_1,\rho_2,\rho_3$ are the autoregressive coefficients.

In practice, the most common AR model used is the simplest: the AR(1), 1st-order process. Then the random terms follow

$\epsilon_j = \rho_1 \epsilon_{j-1} + e_j$ ,

and again the quantities $e_j$ are a white-noise process.

If the random terms follow an AR(1) process (or in fact if they exhibit any autocorrelation), then as mentioned before, linear regression still gives us a good (i.e. unbiased) estimate of the constants $\alpha$ and $\beta$ . But also as mentioned before, the OLS estimates of the probable error in the constants will not be good ones. In particular, when the autoregression coefficient $\rho_1$ is positive, the OLS estimates of probable errors will be too small; the real probable errors will be bigger than our estimates. When $\rho_1$ is large (i.e., close to 1), the OLS estimates will be way too small; the actual probable errors will be much larger than the OLS estimates. This can be a real problem; there’s quite a difference between saying that the rate of global warming is $0.018 \pm 0.002$ deg.C/yr, and saying that the rate is $0.018 \pm 0.2$ deg.C/yr (note: the $\pm$ value is usually twice the probable error). In the first case, we know the rate with good precision; in the second case, we don’t even know with confidence whether it’s positive or negative!

So in order to get anything like a realistic estimate of the probable error, we have to compensate for the autoregression of the random terms. The most common way to do this in practice, is to assume that the random terms follow an AR(1) process. Even under that assumption, there are many ways to skin the cat. One of the simplest and most common to simply to compute a first approximation to the impact of the AR(1) process on the likely errors. This leads to a very simple formulation. When we compute the probable errors using OLS, the answer depends critically on the number N of data points in our sample. This is the number of “degrees of freedom” in our data. But when the random process is autocorrelated, it turns out that we don’t have as many degrees of freedom as we have data points. Instead we can estimate the degrees of freedom by using an effective number of data, given by

$N_{eff} = N (1 - \rho_1) / (1 + \rho_1)$ .

In the formulae for the probable errors, we can replace $N$ by $N_{eff}$ to get realistic estimates.

As $\rho_1$ gets close to 1, the impact can be large. If, for example, $\rho_1$ is only as big as $0.5$ , then the effective number is only 1/3rd the actual number. The number enters into the formulae for the probable errors by dividing a quantity by the square root of N. If $N_{eff}$ is only 1/3rd as big as N, then our probable errors will be $\sqrt{3}$ times their OLS values. Suppose that $\rho_1$ is even bigger, say $0.9$ . Then the effective number will be 19 times smaller than the true number, and the probable errors will be $\sqrt{19}$ times (about $4.4$ times) as large as their OLS values. That’s nothing to sneeze at!

The aforementioned procedure is hardly perfect; it doesn’t compute the exact impact of an AR(1) process on linear regression, and especially as the autoregression coefficient gets very close to 1, it’s necessary to compare the “characteristic time scale” which it defines to the total time span of the data; only if the total time span is longer than the characteristic time scale does this process give a reliable estimate. But in the case of global temperature, for the modern global warming era (from about 1975 to the present) we have enough data for this procedure to work well.

There are other ways to estimate the impact of AR(1) autoregression on trend analysis. One which seems to be making the rounds in climate-related blogs is Cochrane-Orcutt estimation, based on a very useful insight. Take the equation (which we’ve already seen):

$x_j - \alpha t_j - \beta = \epsilon_j$ .

Then take the same equation, but with the index j reduced by 1 (i.e., for the immediately precedent measurement):

$x_{j-1} - \alpha t_{j-1} - \beta = \epsilon_{j-1}$ .

Now we’ll subtract $\rho_1$ times the 2nd equation, from the 1st. This gives:

$x_j - \rho_1 x_{j-1} - \alpha (t_j - \rho_1 t_{j-1}) - \beta (1 - \rho_1) = \epsilon_j - \rho_1 \epsilon_{j-1}$ .

From the definition of an AR(1) process, we recognize the right-hand-side as $e_j$ (which is a white-noise process), so we can say that

$x_j - \rho_1 x_{j-1} - \alpha (t_j - \rho_1 t_{j-1}) - \beta (1 - \rho_1) = e_j$ .

Now let’s define new variables, a new data series

$X_j = x_j - \rho_1 x_{j-1}$ ,

and a new set of times

$T_j = t_j - \rho_1 t_{j-1}$ .

Let’s also define a new intercept

$b = \beta (1 - \rho_1)$ .

The our equation becomes

$X_j - \alpha T_j - b = e_j$ .

This is just like the equation we used to define linear regression in the first place, using our new variables (and a different intercept). But now, the random process sitting on the right-hand-side is a white-noise process! Hence it’s safe to compute both the constants $\alpha$ and $b$ , as well as their probable errors, using standard OLS. This gives us valid estimates of the constants and their probable errors, but we’ll have to transform the intercept back, using $\beta = b/(1 - \rho_1)$ .

These aren’t the only ways to skin the cat. It is possible to compute the exact impact of autocorrelation on OLS, although the formulae are rather complicated and I won’t go into them here. It’s also possible to taylor the least-squares process so that instead of minimizing the sum of the squares of the residuals, we minimize a bilinear transform of the residuals which is designed to compensate for the autoregression. This process is referred to as generalized least squares. Both methods give good estimates of the regression constants and their probable errors. Both methods also accomodate autocorrelation which does not follow the AR(1) model.

Generalized least squares, in particular, gives the best estimates. What this means is that the regression constants determined by generalized least squares are the least-variance unbiased estimates. Unbiased means we expect to get the right answer; least-variance means that the probable error is as small as we can ever make it.

All the aforementioned methods depend on knowing the autocorrelation structure. When using the AR(1) model, we have to estimate the autoregression coefficient $\rho_1$ . For more complicated autocorrelation structures, we’ll have to estimate more coefficients. The uncertainties in those estimates will introduce further uncertainty into our estimates of regression constants. Especially when using more complex models than AR(1), this can be a tricky issue. Fortunately, it’s usually possible to put some limits on what the autocorrelation behavior is, and for many of the most important analyses it’s close enough to AR(1) that we can rely on that model to give us realistic, if imperfect, estimates. Just using OLS, on the other hand, in many cases gives estimates of probable errors which are not only imperfect, they’re not even realistic.

Let’s (finally!) take a look at some actual data, and what the various choices indicate. First let’s look at temperature data from NASA GISS from 1975 to the present:

The trend line plotted is from OLS, with a warming rate of 0.01775 deg.C/yr. The probable error from OLS is 0.00078 deg.C/yr. Doubling the probable error to get an estimate of the error range, we’d say that the warming rate since 1975 is $0.01775 \pm 0.00156$ deg.C/yr. I’ve quoted more digits accuracy than the accuracy of the answer, so it’s probably better to say $0.0177 \pm 0.0016$ deg.C/yr.

But we suspect that the random part of the data is not white noise, so we can expect that the estimated error is incorrect. If we study the residuals, we estimate the autoregressive constant for an AR(1) process as $\rho_1 = 0.6$ . So, the effective number of degrees of freedom is almost exactly $(1-0.6)/(1+0.6) = 1/4$ th the number of data points, in which case the probable error is almost exactly twice its OLS estimate. That means we need to double the size of our error range; the warming rate since 1975 is actually $0.0177 \pm 0.0032$ deg.C/yr. That’s still precise enough to say that the trend from 1975 to the present is positive, and at least as big as $0.0145$ deg.C/yr, possibly as high as $0.0209$ deg.C/yr.

We can also apply Cochrane-Orcutt estimation. This generates a new series of times and data values, which I’ll plot here:

The trend line now gives a slightly different rate, $0.01756$ deg.C/yr. The probable error is $0.00156$ deg.C/yr, almost exactly the same we got from AR(1)-compensated OLS. We now estimate the range of the warming rate as $0.0176 \pm 0.0031$ deg.C/yr. That’s very (very!) close to the result we got from AR(1)-compensated OLS. In fact, they’re statistically indistinguishable.

I’ve seen an application of this analysis to temperature data from 2001 to the present. That seems a bit silly to me; the fluctuations in temperature data are too great to give meaningful results from such a short time span. But it makes an interesting exercise. Using AR(1)-compensated OLS, the estimated trend in GISS data from 2001 to the present is $0.0024 \pm 0.0222$ deg.C/yr. Note that the probable error is nearly 10 times as large as the estimate itself. Cochrance-Orcutt estimation indicates a trend rate of $-0.0066 \pm 0.0268$ deg.C/yr. Again the probable error is many times larger than the rate itself. Again, each estimate is well within the error range defined by the other; the results are statistically indistinguishable.

It’s also worth noting that the error range from each estimate includes the value indicated by analysis of the data form 1975 to the present. Here, in fact, are the estimated trends for data 1975-present, and 2001-present, using both AR(1)-compensated OLS and Cochrane-Orcutt (indicate by “C-O”):

I draw two main conclusions from this analysis. First, there’s no hard evidence that the trend rate since 2001 is any different than it has been since 1975. Second, it’s abundantly clear that the probable error in the rate since 2001 is so large that it’s at best premature, at worst folly, to consider it any kind of revelation.

Categories: Global Warming · climate change · mathematics

169 responses so far ↓

TCO // March 22, 2008 at 8:05 pm | Reply

Makes sense. I can just look at it and see that.

I also think that now that 20 years have gone by since Hansen, we should asssess his model performance (re-run it with the actually conditions of CO2 and vulcanism exhibited and seeing how that compares to actual performance). I would also like to see an out of sample performance test of the Mannian method (I realize that requires gathering a lot of new data.

[Response: Re-running the GISS model with real-world forcings is, I believe, an effort presently underway.]
David B. Benson // March 22, 2008 at 9:22 pm | Reply

Again, an extremely helpful presentation.

I hope you’ll soon do one on generalized least squares.
TCO // March 22, 2008 at 11:08 pm | Reply

Well what’s taking them so long? Can’t we just feed it into a computer and run it already? What’s the big deal? Endele, endele.
dhogaza // March 22, 2008 at 11:29 pm | Reply

Well what’s taking them so long? Can’t we just feed it into a computer and run it already? What’s the big deal? Endele, endele

Maybe they’re writing it up, or it’s in submission and being reviewed, or?
George // March 23, 2008 at 12:21 am | Reply

Well what’s taking them so long? Can’t we just feed it into a computer and run it already? What’s the big deal? Endele, endele

Maybe they’re writing it up, or it’s in submission and being reviewed, or?”

Hey, why should Hansen even bother with all that journal publication nonsense when he can just post it as a comment on Climate Audit?

If the past is any indication, people like McIntyre won’t read the paper anyway. He didn’t read the first (Hansen 88) one, as is all too evident from his (bogus) claim that Hansen never said scenario B was “most plausible” (Hansen’s very words in that paper).

So what reason is there to believe McIntyre would read a new paper?

Forget the science formalities!

Let’s just get that ball rolling downhill post hates!

[Response: If you want to criticize Steve McIntyre, I suggest you address what he's done rather than what he might do.]
George // March 23, 2008 at 12:32 am | Reply

By the way, HB.

Thanks for the most excellent explanation of the impacts of autocorrelation and the problems with trying to divine a trend from very short periods.

… and please forgive the above off topic rant, but it was just too hard to resist.
TCO // March 23, 2008 at 12:44 am | Reply

Very gentlemanly comment Tammy.

I share with George a lot of frustration with SM. For instance the lack of publishing stuff. However, I don’t criticize SM for not reading literature. He’s actually read a lot of literature.
steven mosher // March 23, 2008 at 1:41 am | Reply

IF I predicted in 2001 that Warming would follow a .2C per decade trend what would you conclude today about the probability of that prediction being correct, based on observed data since my prediction?

(FWIW, this cold spell will break, Phil Jones has predicted that 2008 will be in the top 10 warmest years, so that’s NOT my issue.)

My question is simple. If you predicted .2C warming per decade in 2001, and if we look at the monthly temperature data collected since then, how would we assess the accuracy of that projection?

Forget that its an IPCC projection. Imagine it’s me claiming in 2001 that temps would DECLINE by .2C per decade. 6 years after me making that projection how would characterize my success against the observational record?

The answer doesnt matter. What method would you use?

[Response: The analysis in this post has the answer. Using OLS (compensated for AR1) or Cochrane-Orcutt, the data since 2001 indicate a likely range for the slope, and that likely range includes the value 0.02 per year (0.2 per decade). So the data don't contradict the hypothesis that the slope is +0.02. Both ranges also include the value -0.02 (or certainly close enough, since if we want to get fussy we should widen the error bars to include the uncertainty introduced by estimating the AR1 coefficient), so the data also don't contradict the hypothesis that the slope is -0.02. Hence the post-2001 data are compatible with either hypothesis.

The fact that the post-2001 data fail to negate either +0.2 or -0.2 per decade, emphasizes how little we can learn from trend analysis of such a brief span of time.

There's another important, often neglected, factor. We don't just have data since 2001 in isolation from any other data, we've been monitoring temperature for a lot longer than that. When we look at all the data since, say, 1975, the chance of *some* 7-year time span within that data showing a slope outside expectation is even greater; in fact if we keep measuring long enough, sooner or later we'll observe an exceptional slope. It's like flipping a coin to count heads: it's unlikely that we'll get 10 heads in a row, but if we keep flipping it long enough then it's not only *possible* that sooner or later we'll find an instance of 10 heads in a row, it's *inevitable* -- even if the coin is perfectly fair.]
wflamme // March 23, 2008 at 1:47 am | Reply

Tamino,

I’m afraid your conclusions might be wrong.

These are bell-shaped distributions of trend probability and you need to show that the 2001-2007/8 trend ’sample’ might well stem from the 1975-2007/8 distributon.
E.g. you could apply the Welch two sample t-test to the residuals.

[Response: Think about it: the difference between the estimates scaled by the standard deviation of just the 2001-present estimate, is less than the critical value for a normal test. The standard deviation of the difference will be greater than that of the 2001-present estimate alone so the *properly* scaled difference will be smaller still, while applying a t-test rather than normal test will make the critical value larger. So there's no doubt the conclusion is confirmed.]
cce // March 23, 2008 at 9:02 am | Reply

Re: Hansen ‘88 with real-world forcings

Actual (aggregate) forcing tracked scenario B, so I suspect that re-running it would get results similar to . . . scenario B.

http://www.realclimate.org/images/Hansen88_forc.jpg
S2 // March 23, 2008 at 10:21 am | Reply

Thanks, these explanations are really helpful.

If we study the residuals, we estimate the autoregressive constant for an AR(1) process as rho_1 = 0.6

How do we do this? Should we be looking at the difference between successive residuals, or is this too simplistic?

[Response: Perhaps the conceptually simplest way is to notice the AR(1) equation is similar to the linear regression equation -- but now it's a residual value depending linearly on the previous residual value rather than linearly on time. So, if you do a linear regression of the residuals against their immediately preceding values, you'll get a good estimate. Possibly more common (maybe not) is the "Yule-Walker" estimate. Both of them give a biased estimate, the more so as the autoregressive constant gets close to 1 or the number of data points N is small; the expected result underestimates the true value but in most cases the bias is small and can safely be ignored. I used a method designed to compensate for the bias and got 0.60; both the Yule-Walker and linear regression methods give 0.59.

So do a linear regression of residual(j) against residual(j-1), and the slope of that line is your estimate of the AR(1) coefficient.]
adimore // March 23, 2008 at 11:43 am | Reply

Thanks for the most excellent explanation of the impacts of autocorrelation
Ellis // March 23, 2008 at 1:36 pm | Reply

Scenario C was described as
‘‘a more drastic curtailment of emissions than has generally been imagined,’’ specifically GHGs were assumed to stop increasing after 2000.

From Hansen 2006. So, Dr. Hansen thinks that scenerio B was the most plausible. It looks to me like even that overshoots the reality. Now, don’t be confused, like I was, in comparing the models, which are based on land/ocean temperatures, with the actual land/ocean temperatures index. That would be truly insane. Nope, lets follow this logical assesment from Dr. Hansen,

Temperature change from climate models, including that reported
in 1988 (12), usually refers to temperature of surface air over both land and ocean. Surface air temperature change in a warming climate is slightly larger than the SST change (4), especially in regions of sea ice. Therefore, the best temperature observation for comparison with climate models probably falls between the meteorological station surface air analysis and the land–ocean temperature index.

Of course, even if you follow this non-sequitor, that still puts observations squarely on scenerio C, you know, the completely unrealistic, GHG’s stopped increasing after 2000, scenerio. What makes this period of no trend, or if you must, slight trend after 2000 remarkable is that aerosal optical depth was also decreasing. Sorry, for a link to only the abstract, the full paper was there last week.
Gavin says the models are wrong, but believes they further our knowledge of climate change. I won’t argue with his assessment, at the same time, I believe that models should be taken with a grain of salt until they are right.
Hansen's Hamster // March 23, 2008 at 2:34 pm | Reply

“He didn’t read the first (Hansen 88) one”

Dear George,

do you ever read CA? Guess not, so try this link:

http://www.climateaudit.org/?p=2630

or this one:

http://www.climateaudit.org/?p=2665

or this one:

http://www.climateaudit.org/?p=2645

Cheers.
George // March 23, 2008 at 4:19 pm | Reply

I have a couple questions.

First, what level of confidence is to be attached to the “probable error” that you give above for the trend? Is it a 1-sigma error? 2-sigma?

Note; I understand that if the “2001-present trend plus a 1-sigma error” encompasses another trend (eg, 1975-present), then the “2001-present trend plus a 2-sigma error” will certainly also encompass that second trend. But I am nonetheless curious what confidence level is attached to your stated error.

Second:

You say this about the use of the AR(1) model:

Fortunately, it’s usually possible to put some limits on what the autocorrelation behavior is, and for many of the most important analyses it’s close enough to AR(1) that we can rely on that model to give us realistic, if imperfect, estimates.

How close is close enough?

Is is safe to assume that if a process is nowhere close to AR(1), OLS might actually give a better result than the result obtained taking autocorrelation into account and assuming AR(1)?

How does one make this decision?

The reason I ask is that in a previous post (related to the Schwartz paper), you analyzed global temperature and indicated that it does not follow “linear trend + AR(1)”

“This makes it abundantly clear that if temperature did follow the stated assumption, it would not give the results reported by Schwartz. The conclusion is inescapable, that global temperature cannot be adequately modeled as a linear trend plus AR(1) process.

Does the decision whether AR(1) is “good enough” depend on what one is trying to determine from the analysis?

[Response: I use the phrase "probable error" to mean 1 sigma, but when I give a "plus or minus" range for an estimate it's 2-sigma.

When the autocorrelation structure departs from AR(1), it does indeed affect different things differently. In most cases (this one included) the impact on the probable error from OLS is small enough that the simplified AR(1) estimate is, to borrow a phrase from the post, "realistic, if imperfect." This is especially true considering that the simplification introduced to get the easy-to-use formula will tend to overestimate the probable error (so it's a conservative estimate); using the exact impact (complicated formula) for AR(1) will lower the value, but for this data the departure from AR(1) will raise it, so those two factors partly cancel each other out. However, departure from AR(1) will have a much larger effect on the estimated "characteristic time scale" for the system, which is the quantity Schwartz sought to estimate.

It's also crucial to realize that this isn't the same time span used by Schwartz. For the data from 1975 to the present, global temperature actually *does* follow the linear regression model rather well: a straight line plus autocorrelated noise that has a rather short characteristic time scale. For the time span used by Schwartz (1880-2004) it's very different. Most of us are familiar with the early-20th-century temperature rise, followed by the mid-century levelling off, followed by the late-20th-century rise -- not following a straight line, but since 1975, the pattern *is* reasonably linear. Schwartz explores the possibility that the departure from linearity 1880-2004 is due to autocorrelated noise with a much longer time scale; he doesn't claim that's the case, he just explores the possibility. Under that assumption, departure from the AR(1) model used by Schwartz has a profound affect on the result.

All told, I'd say that the departure from AR(1) of the residuals from a linear fit to the post-1975 data will change the probable error by not more than 10%, but the departure from AR(1) of the residuals from a linear fit to the 1880-2004 data changes the estimated "characteristic time scale" by about a factor of 3.]
DocMartyn // March 23, 2008 at 4:26 pm | Reply

Why on Earth should an increase in ‘Heat” to a system give rise to a linear increase in the (Tmax-Tmin)/2 ?

What line-shape would one expect form the change in CO2, giving rise to IR recycling, increasing temperature; in the form of (Tmax-Tmin)/2 ?

Modeling a linear function might be easy, but it is also very stupid.
TCO // March 23, 2008 at 5:05 pm | Reply

If I took 22 series of noise with the 0.60 redness and then did a multiple regression (in the manner that you do, allowing a coefficient for each series) versus the GISS temp, how would it compare to the GISS temp? Would it also roughly model it?

[Response: I don't know, off the top of my head, but I'll say this with confidence: if we withheld some of the temperature data so we could subject the model to completely independent verification (the test applied by MBH98), the independent verification would indicate complete failure.

Allowing a coefficient for each series in a multiple regression isn't "my" manner of doing things, it's what's usually done by everybody -- kinda the definition of "multiple regression".]
Barton Paul Levenson // March 23, 2008 at 6:03 pm | Reply

DocMartyn says:

[[Modeling a linear function might be easy, but it is also very stupid.]]

In all circumstances? If I model consumption as a fixed fraction of GDP, is that “very stupid?” How about national income to GDP? Still “very stupid?” How about the relationship of logged mass to logged luminosity in main sequence stars? “Very stupid” again?

Forgive me, but your blanket statement strikes me as very stupid.
Cthulhu // March 23, 2008 at 6:59 pm | Reply

“From Hansen 2006. So, Dr. Hansen thinks that scenerio B was the most plausible. It looks to me like even that overshoots the reality. ”

As far as I am concerned Hansen’s model passed potential falsification and AGW theory is stronger as a result. Temp could have stayed flat or even fallen since ‘88, yet it rose very close to what Hansen’s model projected. Either that’s coincidence or it reflects Something Right with the theory.

“Of course, even if you follow this non-sequitor, that still puts observations squarely on scenerio C, you know, the completely unrealistic, GHG’s stopped increasing after 2000, scenerio.”

The scenario C act of stable GHGs after 2000 does not produce much if any divergance from scenario B in just 8 years. You can see that from the graph where scenario B and C barely diverge over 2000-2008:
http://data.giss.nasa.gov/gistemp/graphs/GTCh_Fig2.pdf

In 10 years time when scenario B and C are about 0.5C apart we’ll know where temps lie in relation to either scenario.

But unless temps flatten or dive between now and it’ll be pretty acedemic anyway. A 1988 model is hardly expected to be a perfect predictor of future temp trends. Even if 2010-2020 average temp lies between scenario B and C it would be quite uncanny how closely that was predicted about 30 years previously.
George // March 23, 2008 at 7:08 pm | Reply

To my “He didn’t read the first (Hansen 88) one” [paper]

Hansen’s hamster asks:

do you ever read CA?

Which version?

The later ones that you link to or the original one that Tim Lambert quotes from here(critical parts of which were purged from Climate Audit without a trace) in which McIntyre made a patently false claim that no one who actually read Hansen’s paper (at least not with any care whatsoever) would ever have made (because it would just make them look foolish).

Here’s the original (now purged) text (preserved thanks to Tim Lambert):

In the right panel, only Scenario A is taken through to 2050 and in both panels, Scenario A is plotted as a solid line, which could be taken as according at least graphic precedence to Scenario A. Hansen has subsequently said that Scenario B was said by him at the time (in his testimony) to have been the “most plausible”, although the article itself contained no such statement.

Tim Lambert continues:

Here’s what Hansen wrote about the scenarios in his paper:

These scenarios are designed to yield sensitivity experiments for a broad range of future greenhouse forcings. Scenario A, since it is exponential, must eventually be on the high side of reality in view of finite resource constraints and environmental concerns … Scenario C is a more drastic curtailment of emissions than has generally been imagined … Scenario B is perhaps the most plausible of the three cases.

See if you can spot the words that McIntyre claimed were not there.

No reasonable person can read Hansen’s paper and conclude that A was his preferred scenario.

[end Tim Lambert quote.]

Indeed, no reasonable person could have actually read that Hansen paper and said what McIntyre originally said.

I’m not sure what “purging one’s mistakes without a trace” comprises but I am quite certain of what it is not: “auditing”.
EliRabett // March 23, 2008 at 7:13 pm | Reply

Hansen, et al essentially did rerun the model in the late 90s with the observed forcings. The code for the 1988 paper is available, have at it, just don’t expect others to do your work for you.
DocMartyn // March 23, 2008 at 7:30 pm | Reply

“How about the relationship of logged mass to logged luminosity in main sequence stars? ”

My point exactly, you should always do a fit to a model that is rational and appropriate. What is the shape of the distribution of error, in the Y and X plane’s of a line on a log-log ploy? Using linear regression analysis on data that is plotted in Log-log form is stupid.

[Response: Your comments have made you look completely ridiculous. Maybe more so than any other comments I've ever seen here.

The *data* since 1975 follow a linear progression + autocorrelated noise, to such a high degree that the linear model is undeniably applicable. The underlying signal need not be *perfectly* linear for that model to give an accurate estimate of its rate of change, if the departure from linearity is small. And in this case: it's not even a close call. Linear is as linear does.

Before you respond, consider the sage advice: it's better to remain silent and be thought a fool, than to open one's mouth and remove all doubt.]
TCO // March 23, 2008 at 8:05 pm | Reply

Eli: I hope you weren’t talking to me when suggesting that I do work? I’m more of an ask interesting questions kind of guy. I like goofing off on the net. Actually doing work? And then if I did, I would want to publish it. Used to love the simple pleasure of having my byline in the archived literature.
TCO // March 23, 2008 at 8:23 pm | Reply

Tammy: I didn’t mean it in a “gotcha” sense. I’m really curious. And sure, training on half the data and testing on the other half is helpful to being meaningful in the overall Mannian warfare battle (so is true out of sample testing). But to start, I just honestly wanted to know if that beautiful match that we saw in your second figure (I really like it, in that it tracks about 3 wiggles….makes me feel good) is what one would expect from multiple regression. IOW how powerful is multiple regression at driving for that sort of a nice looking result? (Even if in that particular instance, other significance tests will show the series to be useful in climate prediction ala Mann.)

Also apologies for the disconnect on the communications. It’s not your fault that I am used to the term from a very knuckel dragger six sigma green belt (and a course in DOE in college) level of understanding and from non time series work. I realize now that you are used to using the term to mean one coefficient for each series. And I guess that this is what the term means all the time in time series work. I am used to modeling of response surfaces not in time series but in simple multiple factors for a single output. For instace composition variables and temp and speed for a product’s physical property. In such situations, I’m used to seeing substantial degrees of freedom left over as well as some manual operation and decisions on how many terms to include (perhaps omitting some of the factors entirely, perhaps including some interaction terms), but roughly adding factors into the equation until a lot of variance is accounted for, but also a lot of degrees freedom left (to prevent overfitting, which will not bear out in out of sample work). Here’s a reference on the net to how I’m used to thinking about multiple regression:

http://ordination.okstate.edu/MULTIPLE.htm (see the end section especially.)
TCO // March 23, 2008 at 8:27 pm | Reply

Before you respond, consider the sage advice: it’s better to remain silent and be thought a fool, than to open one’s mouth and remove all doubt.

But my teacher’s used to say, there’s no such thing as a stupid question! ;)

[Response: Well, a comment such as "that's just stupid" isn't really a question. And if it were phrased as such, it would contradict your teacher's claim.]
DocMartyn // March 23, 2008 at 8:34 pm | Reply

Just may I ask what is the underlying model that predicts that there will be a linear increase in (Tmax-Tmin)/2 with respect to year in response to non0linear changes in human efflux of CO2 into the atmosphere and the resulting increases in atmospheric CO2? If you have no theoretical model that predicts a linear vs year result, the plot means nothing.
Fitting to data only matters if the fit is tied to some hypothetical function. You have made no attempt to explain what the fit should actually be, and if the fit you get agrees with your hypothesis.
If you are impressed with your linear fit, I suggest that you go the whole hog and fit it to a polynomial. You will find the fit is better; and as you don’t appear to care what the relationship between (Tmax-Tmin)/2 and the change in the date actually means, it should make you happier.

[Response: You seem to think that without a theoretical derivation indicating a linear time progression of temperature, the fact that THE DATA SHOW A LINEAR PROGRESSION is meaningless. You ignore the fact that whatever the predicted response is, as long as it's continuous, over the short term it will be approximately linear. As for fitting polynomials, THE DATA don't support that idea.

You really should have heeded my advice.

I promised readers not to let the abusive and the stupid dominate discussion. Your comments clarify the value of that policy.]
cce // March 23, 2008 at 8:51 pm | Reply

Hansen et al scenario B overshot land+ocean observations (by 25%) because the model had a sensitivity of 4.2 degrees which is “high”.

Scenario B vs land-only (the original comparison) show virtually identical rates of warming.
TCO // March 23, 2008 at 9:25 pm | Reply

“Well, a comment such as “that’s just stupid” isn’t really a question. And if it were phrased as such, it would contradict your teacher’s claim.”

I’m just playing on words, man. I wasn’t trying to upbraid you. I actually agree that a lot of us skeptics show ourselves poorly as well as doing no benefit to driving understanding, when we swing for the fences without stopping to think.
steven mosher // March 23, 2008 at 9:35 pm | Reply

Thanks for your response tamino. I think That you And Lucia did two different things, so we cant actually compare it ( she actually averaged the four temp series and is adding NOAA to the mix)

I have some issues with that method, but it is beside the point. I think what Lucia is doing on her blackboard, is showing a class an approach to a problem. Key points:

1. How to avoid cherry picking. She picks a date that is determined by the person making the prediction. ( the IPCC).

2. How to calculate a trend using OLS and O-C

3. How to compare that with the IPCC projection.

IT’S an excercise. So, you do the same.
If the IPCC predicted in 2001 that temps would increase at .2C per decade, what would it take to say that they MOST LIKELY got it wrong.
Not that AGW was wrong, not that warming had stopped, but that the projection of .2C per decade was most likely wrong.

It’s really all totally divorced from AGW, it goes more to methodology. Could we reject IPCC trend estimates after 6 years? YES. imagine they estimated that the Planet would warm by 1C per decade, instead of .2C per decade. 6 years of data could put a hefty nail in that coffin.

(Here is my Projection. The trend of earth’s climate over the next 1 billion years will be ZERO. People should test this hypothesis every million years. Ha.)
TCO // March 23, 2008 at 9:50 pm | Reply

Mosh: behave your thuggish self.

Tammy: Mosh actually brings up an interesting point. I readily agree that a few years of time is not enough to declare GW is over, given that the system has so much noise and other cycles which can run for a few years and overwhelm the long term trend (think about the difference in how much your 401K in an index fund grows over 40 years, versus over a select 5 years).

However, it’s still of interest to me, how much significance the match versus prediction has. Not just when we’re about 95% probability on a hypothesis test, but what we’re at now in terms of probability (and doing that calc with inclusion of the effect of autocorrelation, which will lower that prob number). Do you capisce my stupid question?
wflamme // March 23, 2008 at 11:02 pm | Reply

Tamino,

I haven’t checked this with GISS data yet, so my conclusions with respect to your data are provisional (some say, GISS is off a little these days).

Anyway your conclusions have a strong bias: They assume a H0: ‘no change in trends’ unless proven wrong with a high level of confidence. Thus they very much disregard erroneously accepting your null hypothesis (likely resulting in a high beta error) .

What if we couldn’t be so confident that one of our hypothesis be true in advance? Given that all false conclusions are equally bad, should we rather vote for a difference in means or against to use as null hypothesis?
elspi // March 23, 2008 at 11:14 pm | Reply

“You ignore the fact that whatever the predicted response is, as long as it’s continuous, over the short term it will be approximately linear.”

I think you meant to say “differentiable” (or continuously differentiable) instead of “continuous” there.

[Response: Right you are.]
David B. Benson // March 23, 2008 at 11:16 pm | Reply

steven mosher // March 23, 2008 at 9:35 pm — I encourage using the naive Bayesian factor method to compare two hypotheses, given some evidence (data):

http://en.wikipedia.org/wiki/Bayes_factor

which does not relieve one of checking that there is sufficient data to form good estimates of the variance, etc., of the residuals.
dhogaza // March 23, 2008 at 11:22 pm | Reply

http://rankexploits.com/musings/wp-content/uploads/2008/03/gmt_testnoextra.jpg

So why does Lucia label a line running from 2001 to 2025 “now to present”?

Have I pulled a Rip Van Winkle without realizing it?
dhogaza // March 23, 2008 at 11:29 pm | Reply

Eh, I always hit “submit” prematurely.

The point, of course, is that 25 years of declining temps *would* be significant, and her graph shows that quite well as by then the trend with uncertainty bands and the IPCC projections with uncertainty bands have clearly diverged.

But cut back her purple line to “now” (as I understand the word) and the divergence is not so pronounced.

Also, rather than average the series, I think it would be more useful to do the computation for each individually.
Hansen's Bulldog // March 23, 2008 at 11:46 pm | Reply

Some of the comments to this post have been a real eye-opener.

Here’s data; here are the results of tried-and-true well-known analysis methods; here are the obvious conclusions. No non-centered PCA, no bristlecone pines, no speculation about solar cycle 24, no mention of Al Gore. I half expected a dearth of comments because there’s nothing even remotely controversial, nothing to argue about.

But it turns out that comments have given us some real clues about which doubters might be skeptics and which deserve the name “denialist.” TCO might be a genuine skeptic; he launches commentary by saying “Makes sense.” Steven Mosher might be a genuine skeptic; he asks a lot of questions, probes the issue of hypothesis testing, mentions different data sets, but doesn’t try to contest what’s really indisputable.

But wflamme and DocMartyn are seriously trying to discredit the analysis itself. It seems to me that they’ve only succeeded in making themselves look like idiots. All the more so, because when their initial objections are answered unambiguously they *still* insist on denying the obvious. That’s denialism.

It’s also the kind of nonsense that has made regular readers consider not returning, the kind of contentiousness that obstructs understanding, the kind of garbage for which a response is really a waste of time. It’s the reason that stronger moderation of comments is necessary, as a service to readers.
Ellis // March 24, 2008 at 1:21 am | Reply

cce, I suppose you know and are just pulling my leg, but climate sensitivity is not an input of a model, it is the output of a model. So your first sentence reads, the model is wrong because the model is wrong. I have to tell you, that is not a very convincing argument.
As to your second line,

Scenario B vs land-only (the original comparison) show virtually identical rates of warming.

it leaves me in a quandry, should I believe you, and only you since you do not reference “(the original comparison)” or do I believe the man who made the model, again,

Temperature change from climate models, including that reported in 1988 (12), usually refers to temperature of surface air over
both land and ocean. Surface air temperature change in a warming climate is slightly larger than the SST change (4), especially in regions of sea ice. Therefore, the best temperature observation for comparison with climate models probably falls between the meteorological station surface air analysis and the land–ocean temperature index.

I am a stickler for references, so I have to go with Hansen on this one. Although if you want some real fun, go to Hansen 2006 and look up the reference number 4 which I made bold. That will lead you to Hansen 2001. Now for the fun, find any part of the 2001 paper that would make it useful as a reference to the statement,”Surface air temperature change in a warming climate is slightly larger than the SST change.”
Maybe I should have said, I am a stickler for valid references. In which case I could still believe you, but alas, no, I still gotta go with the Man.
Lee // March 24, 2008 at 4:15 am | Reply

TCO:
“But my teacher’s used to say, there’s no such thing as a stupid question!”

The guy who taught me to teach well used to say that every good teacher tells his students one lie. You just quoted it.
cce // March 24, 2008 at 6:18 am | Reply

We expect the model to have over-stated warming based on what we know more clearly today. 4.2 degrees is “high.”

The expectation of skeptics seems to be that if a 24 year old model is not perfect, then it is worthless, and all models everywhere from now until eternity are equally worthless.

As for the original comparison, I suppose its inconsequential that the land+ocean index didn’t exist in 1988. But if you want a source, you can read Hansen et al 1998 which updated the observations to 1997 and describes them as “based on meteorological station measurements.” I would refer you to Hansen et al 1987 which documented the methods and is cited in the original ‘88 paper, but it is not available online.
http://pubs.giss.nasa.gov/docs/1998/1998_Hansen_etal_1.pdf

From 1984 to 2007, scenario B calculated 0.25 degrees per decade of warming, the land index shows 0.24, and the land+ocean index shows 0.20. Thus, scenario B overstated warming by 25% based on the land+ocean index or 3% based on the land index.

If you don’t believe me you can download the data and do the calculations yourself.
http://cce.890m.com/hansen-88-scenarios.txt

You are referred to Plate 3 and figure 8 of the 1999 paper for the differences in the Land+Ocean and Meteorological Stations (land) indexes.
http://pubs.giss.nasa.gov/docs/1999/1999_Hansen_etal.pdf

The difficulty with sea-ice is described in this paragraph from the 2001 paper:
“Note that the 100-year temperature change in the North Polar region and at high latitudes in the Southern Hemisphere is uncertain, and indeed, we suspect that our illustrated temperature change in those regions understates
the warming of surface air. The reason for this belief is the realization that mean temperature changes at those latitudes are predominantly associated with changes in sea ice area. If an area of sea ice is replaced by open water, the local change of surface air temperature is exceptionally large because of the loss of insulating effect of the sea ice. Such large surface air temperature changes are captured in climate models but not in empirical studies in which the temperature changes of ocean areas are based on either estimated SST changes or extrapolations from measurements on coastal land areas.”

Is there anything else I can do for you?
fred // March 24, 2008 at 7:25 am | Reply

With regard to Lucia’s recent material, it is quite restricted in what it shows. It does seem to show that the IPCC predictions of 2000 or so have been falsified by later events, but it does not show (and she doesn’t say it does) that the planet is not warming, nor how much it is warming, and of course it says nothing about whether CO2 is causing what warming there is. It just shows these particular projections were probably too extreme. Its interesting and educational series of essays, but quite restricted in scope.

Its a very nice blog, Lucia’s. Good tempered, educational, interesting, reasonable, technically solid. Recommended.
Barton Paul Levenson // March 24, 2008 at 11:39 am | Reply

DocMartyn writes:

[[Using linear regression analysis on data that is plotted in Log-log form is stupid.]]

That may be the stupidest comment yet made on this thread. The usual reason for plotting things on a log-log plot is to see whether the resulting line comes out straight, SO THAT if it does you can use linear regression on the logged data. What you’re saying essentially is that log-transforming data is illegitimate. Which is very stupid.
dhogaza // March 24, 2008 at 12:31 pm | Reply

It does seem to show that the IPCC predictions of 2000 or so have been falsified by later events

Not to mention showing that it’s been “falsified” by 18 years of future events.

No one’s answered – why is a graph labelled 2000-now actually covering 2000-2025, thus visually exaggerating her claim?
steven mosher // March 24, 2008 at 12:50 pm | Reply

TCO, i’m behaving.

the salient points.

1. lucia is very precise. she is not claiming the warming is over, she accepts agw and expects the warming to return.

2. she uses all temp series not just giss. so she doesnt cherry pick one time series, she uses giss, hadcru, rss and uah and is adding noaa.

3. all she is rejecting is the 2001 claim that the rate of warming is .2C decade.

4. finally she is very clear that the conclusion is potentially wrong
dhogaza // March 24, 2008 at 2:02 pm | Reply

lucia is very precise.

Plotting 2001-2025 based on 7 years data and calling it “2001-now” is “precise”?

C’mon, why’d she visually exaggerate the result of her analysis?

Same graph is at prometheus.

[Response: Perhaps the "now" is just a typo.

I'm curious about exactly how the IPCC TAR prediction is phrased. Does anyone know the exact quote from IPCC TAR on which this is based? Is there some as-yet unmentioned context for that prediction? Does it say "about" 0.2 or "around" 0.2? Does it predict 0.2 over the "next several decades"?]
dhogaza // March 24, 2008 at 3:37 pm | Reply

I’m curious about exactly how the IPCC TAR prediction is phrased. Does anyone know the exact quote from IPCC TAR on which this is based?

If the discussion at William Connelly’s blog is to believe, the statement is “2C per century”.

Lucia herself states it that way.

I don’t see how that becomes “0.2C increase each and every decade”, nor how a trend based on 7 years that ends on a strong La Niña disproves a predicted 2C/century rise …

If this is sensible, do I get to say “0.02C increase each and every year” and “prove” the model predictions are bogus every time we hit a cold one?

And, I don’t think it’s a typo, not at all.
fred // March 24, 2008 at 3:39 pm | Reply

The 0.2 prediction.

0.2 a decade is just derived from the total warming predicted over the long term. Lucia’s question is whether there is enough of a divergence in the data that has arrived since the date when this prediction was made to make it seem implausible. I think she’s assuming linear progressive warming is what was forecast.

If the forecast was not in fact for linear warming, but for (eg) cooling for two decades followed by more intense warming to reach the average of 0.2 per decade, then her test would not be valid. But it would also be rather harder to confirm or falsify.

On the subject of autocorrelation, there are some interesting posts now, quite detailed and technical, on CA at the moment. Toeplitz matrices and so on. They seem quite persuasive about methods of assessing the validity of proxy reconstructions. Very hard for a layman to assess however.
Thomas Huxley // March 24, 2008 at 4:01 pm | Reply

Re Lucia’s graph. In small print but in the same colour as the line (at right) :
“projected after Jan 2008″.
Hansen's Hamster // March 24, 2008 at 4:18 pm | Reply

“Its a very nice blog, Lucia’s. Good tempered, educational, interesting, reasonable, technically solid. Recommended.”

Maybe it’s the female thing; more based on consensus, less cock’s behaviour as you can see on so many (climate) blogs. Thinking of Dr Judith Curry too, respect to her, although I don’t agree with her on many things.
Raven // March 24, 2008 at 4:45 pm | Reply

The graph clearly states that she is projecting the trend after the 2008.
The graph also clearly states that the trend is a C-O fit to the data from 2001-now.
Nothing in her graph or description is wrong or misleading.
Yet Tamino has yet to acknowledge that apply his analysis to any dataset other than GISS would show that a 0.2/decade is not consistent with the actual data since 2001.
The IPCC projections are in the graph presented on Lucia’s blog. The 0.2/decade trend is the short term projection calculated from the graph. The graph also clearing indicates the 1 sigma error bands for their projections.
Layman // March 24, 2008 at 5:01 pm | Reply

Lets stipulate that a method can be found to measure global mean surface temperature (GMST) accurately. I think nearly everyone would agree that, over a long time horizon, GMST would be a complex periodic function composed of several, perhaps many component periodic functions.

The theory is that humans have now injected something (lets say it CO2) into the system for which no natural theromstat exists, force the normal periodicity off track.

Mathematically, of course, its quite acceptable to do linear regression on periodic functions. And when performing them over short durations relative to the generally much longer periodicity of the function, linear regression is quite practical.

As a skeptic, my genuine question is, how do you know that what you are observing in the linear regression is due to some assignable cause vs. the normal upswing or downswing of the periodic function?
george // March 24, 2008 at 5:04 pm | Reply

lucia is very precise.

Given the considerable uncertainty, how is it possible to be “precise” when it comes to drawing conclusions about a trend over such a short period?

Also, is averaging data sets together the most “precise” way to deal with the case under consideration, when individual measurement errors are not the greatest contribution to the uncertainty in the trend?

I would think that the greatest contribution to the uncertainty in the trend comes from “noise” like that associated with like El Nino which can be an order of magnitude (or more) greater than the error associated with individual measurements (or more precisely, the average thereof, since most of the values used to determine trends are actually the average of a large number of individual measurements).
steven mosher // March 24, 2008 at 5:46 pm | Reply

Tamino, Part of the issue is the vagueness of the IPCC “claims” and the lack of data to back up simple charts. Lucia Has been very circumspect. If the IPCC wanted to be clearer, it had that choice.

Now, even with AR4 you cannot get the data backing up the charts and people resort to digtizing graphs. Thankfully, the datasets are becoming available, but only after registration ( I tried to register and was denied ) and the documentation that accompanys the data is abysmal.

Lucia, has all the relevant documentation. Just google “lucia rank exploits” and read, she will answer all your questions and is a very nice person. One thing I liked about her approach was that she actually passed her work around to some professional statisticians to review her maths. That’s not an endorsement or suggestion, but it showed some care.
steven mosher // March 24, 2008 at 6:06 pm | Reply

Dhog,

Since the IPCC does not provide the data underlying its charts and graphs, the best you can do to “illustrate” a point is to copy their chart and “draw” on it. The precision I refer to is this.

1. Lucia does not cherry pick a start date. The IPCC made it’s projection in 2001. She picks THEIR start date to test THEIR projection.

2. She does not cherry pick an instrument series. She uses all the time series: GISS, HADCRU, RSS, UAH.

3. She tests the claim that temperatures will increase at .2C per decade using the available data.

She rejects that claim at 95% confidence.

Now people have made several spurious arguments WRT to this.

1. 6 years is too short. It’s not. If I predicted in 2001 that temperatures would increase at 10C
per decade, thats right 10C per decade, then I suspect that after 6 years of relatvely flat temps people would reject my claims. The same if I projected a DECREASE of 10C per decade. The point is this. It’s TOUGH to reject the claim of .2c per decade after 6 years BECAUSE the CI is so wide.

2. We shouldnt test the claim for 30 years. Like I said before, my GCM says the TREND of earth temps is ZERO over the next million years. But you can’t test this clima until a million years from now.

3. The IPCC was unclear, so it doesnt matter.

The last is probably the defense.

The easiest way for you to get clear on this is as follows. IF I PREDICTED .2C of cooling per decade, and if after 6 years the data showed .1C WARMING per decade, would you think it was fair to submit my claim to a test?

Bottom line: AGW is true, the IPCC probably made a minor error. Fix it and move on. Disavow the words, without condemning the organization or science, and do the Obama thing.
Marty Ringo // March 24, 2008 at 7:18 pm | Reply

First, the linear trend is 1) significant for almost all 30 years past of further back to today (and that is with tests that account for almost all serial correlation except possibly fractional differencing, long memory) [see T. Vogelsang, “Trend Function Hypothesis Testing in the Presence of Serial Correlation,” Econometrica, Vol 66, No 1, 1998], 2) the coefficients are fairly stable across time dimension (months or years or greater), 3) the coefficients are not stable for subperiods, and 4) error structure of a linear trend model is uncertain other than it has serious serial correlation and moderate heteroskedastic (varying across time) errors. Thus, the apparent primary message of topical post – the standard errors of an OLS linear trend regression can seriously understate the uncertainty of the coefficient – is both true and important, at least to the extent that someone is looking at regression trends.

Second (and of a technical nature though critical is not meant to take away from the primary message), the adjustment of the OLS standard error by the effective N calculation is a poor way to correct the estimate. The Cochrane-Orcutt standard error is better but still underestimates the uncertainty especially when the autocorrelation function shows miscellaneous higher order correlation, i.e. higher order correlations that are noticeable but well under the 2/sqrt(N) significance bar. Both annual and monthly temperature data seem to have a highly significant ARMA(1,1) (one autoregressive term and one moving average term) structure, but as the grand boys of time series analysis, Box and Jenkins, pointed out, there can be more than one ARMA structure consistent with a time series. Thus, a Cochrane-Orcutt estimator, or even a maximum likelihood AR(1) estimator, can mis-estimate the standard errors. (Note: the moving average term has a negative coefficient which tends to reduce the overall equation error. That is, one is less likely to get spurious significance with a AR1 = 0.9 and a MA1 = -0.7 that with a simple AR1 = 0.9 and a MA1 = -0.1.) (As a second aside, there are plenty of free software packages that do quite sophisticated regression analysis – much more so that when I first started working with such packages on mainframes. Thus, there should be no excuse for using OLS when there is a high degree of serial correlation… although I confess to having run more than a few spreadsheet regressions on just such data. But I know it is a bad habit.)

Finally (with the same caveat as above), yes, Generalized Least Squares estimators are unbiased, but other than card games, some particle physics, and the like, we mere human beings don’t know the variance-covariance matrix. We can only estimate it. When we do, we then do feasible GLS estimation, or feasible Aitken estimation as it used to be called. These estimators are not unbiased, but they are consistent – approach the parameter with greater and greater probability as the sample size grows toward infinity. But we live in a finite sample world and consistency, while better than nothing, doesn’t mean your correction for the autocorrelation doesn’t bias your estimate. And since OLS estimates are unbiased, even with serial correlation, don’t correct for serial correlation unless that correction makes a big difference in the standard errors. … You probably just forgot to add that to your piece.

If you want to go further, you might try doing a piece of the forecast error with (as of yet) unknown regressors since there seems to be an interest in forecasted values.
george // March 24, 2008 at 7:34 pm | Reply

After looking at the relevant graphics and text from the 2001 report, I can’t see what the claim that “IPCC projections overpredict ” is about.

Those envelopes on the IPCC graphics are simply not equivalent to 2-sigma bounds on the calculated temperature trends, as far as I can see.

It also seems to me that the missing context is that the IPCC projections were intended to be long (not short) term ones.

That is implied, among other things, by the way they represent the temperature range associated with the different scenarios; as a bar for the year 2100.
Boris // March 24, 2008 at 8:15 pm | Reply

The graph also clearing indicates the 1 sigma error bands for their projections.

Yes, but their projections are derived from a mean of 19 GCMs, which has the effect of (nearly) removing unforced variability, AKA weather noise/ENSO.
Boris // March 24, 2008 at 8:32 pm | Reply

1. 6 years is too short. It’s not. If I predicted in 2001 that temperatures would increase at 10C
per decade, thats right 10C per decade, then I suspect that after 6 years of relatvely flat temps people would reject my claims.

While technically true, this does not mean that you can say anything after seven years in the current situation. In your 10C/decade(!) example, the yearly increase would be far greater than the internal variability.

The whole point now is that internal variability is far greater than the expected warming over these short timescales. The IPCC projection should yield about 0.14`C warming from 2001-2008, yet the recent January 2007 to January 2008 swing was over 0.5`C. It’s folly to think you can identify the trend in such noise.

And if one were to see a trend outside of the 95% confidence interval, one would expect it to happen with a strong la nina event.
dhogaza // March 24, 2008 at 8:44 pm | Reply

1. 6 years is too short. It’s not.

So do you think the performance of the stock market during Bush’s presidency proves that the market doesn’t, long term, say the century time frame, rise steadily?

It also seems to me that the missing context is that the IPCC projections were intended to be long (not short) term ones.

This is why they’re saying “0.2C per decade” rather than “2C per century”, as the IPCC did.

Because saying “6 years disproves a 10 year prediction” sounds much better than “6 years disproves a 100 year prediction” …
Adam // March 24, 2008 at 9:24 pm | Reply

According to John Cross, the model runs were independent from 1990 (I haven’t read the TAR), so the comparison should start then.

See:
http://www.skepticalscience.com/Comparing-IPCC-projections-to-observations.html

[Response: Fascinating. The above link also links to the published research, from which it's clear that choosing 2001 as a starting point is an error.]
David B. Benson // March 24, 2008 at 9:50 pm | Reply

Layman — I doubt that it is helpful to think of the climate as having periodic components. Taking climate as a 33 year average of the globally averaged weather, the obvious periodic forcings of day/night and summer/winter are averaged out. Everything else is only quasi-periodic oscillations, if that, until one gets out to the tens of millennia for the periodic components of orbital forcing.

[Response: Even the orbital forcing is not *strictly* periodic, and the response to them even less so.]
John Cross // March 24, 2008 at 11:40 pm | Reply

Adam: Thanks for the kudos. However Skeptical Science is run by John Cook, not John Cross. However the site is excellent and if you wish to associate me with it then I am happy to accept the compliment.

John Cook, John Cross, John Mashey, John Hunter, John A., John S. – there sure are a lot of Johns around. You can add your own joke to that.

John
TCO // March 24, 2008 at 11:57 pm | Reply

It’s not clear to me what SM means when he talks about spatial autocorrelation and spatial correlation (seems to use the terms interchangeably). Also, I’m interested in what spatial PCA would look like on temperature and precipitation.

Also, I don’t know how spatial patterns in PCs would interfere with the Mannian method. That method has an assumption that by training on performance in recent times, we can qualify some signals more than others and that one can have teleconnections to the global climate field (I’m skeptical of that as it worries me that we may do data mining and may lose physical justifications and just fish for results…but that’s a separate issue…we need to at least give the due). Note, that my lack of understanding of the SM crit is not a disagreement, per se. I just want the logic train expanded to explain why finding these regional patterns is a concern.
TCO // March 25, 2008 at 12:06 am | Reply

Dear John….paddump, bump
TCO // March 25, 2008 at 12:09 am | Reply

Oh….and I like Steve’s pretty patterns, but it’s too bad they will never be published and that he only thinks about pubs in terms of winning a propoganda battle (and not enough even for that), rather than as a way of crystallizing insights and of sharing knowledge.
George // March 25, 2008 at 1:01 am | Reply

Steven Mosher says

Lucia does not cherry pick a start date. The IPCC made it’s projection in 2001. She picks THEIR start date to test THEIR projection.

The IPCC also had earlier projections going back to 1990.

Why not start with those, given the longer time period?

2. She does not cherry pick an instrument series. She uses all the time series: GISS, HADCRU, RSS, UAH.

There is more than one way to pick cherries. Selecting the particular way to combine those data could (in theory, at least) also amount to cherry picking.

What if one combines the data in an alternative (though acceptable) manner? Does one reach the same conclusion?

3. She tests the claim that temperatures will increase at .2C per decade using the available data.

When it comes to testing projections intended to cover multiple decades, “available data” (covering only 7 years) is not necessarily the same as “sufficient data”.

I am not claiming that Lucia has herself picked cherries, but only pointing out that there are many ways to pick cherries and that there are gaping holes in the claim by Mosher (and Lucia herself) that she “has not engaged in cherry picking because…”.
Raven // March 25, 2008 at 1:02 am | Reply

Adam says:
“According to John Cross, the model runs were independent from 1990 (I haven’t read the TAR), so the comparison should start then.”

That claim is not true. If you look at the SRES you will find that the model outputs were adjusted to match the average emission and temperature trends from 1990-2000 (i.e. even if the models themselves were not re-tuned their outputs were calibrated to fit the actual data).

This point is explained completely by Ian Castles on both Lucia’s and Peike Jr.’s blog.

I don’t understand why you are so adverse to accepting Lucia’s analysis. If you are right the temperatures will turn around soon enough. If you are wrong and the temperatures continue to fall/stay stable then it won’t make a difference what you think because no one will believe the models anymore. It is worth noting that Lucia has stated many times that she belives the temperatures will turn around soon enough.

[Response: This is contradicted by Rahmstorf et al., who state "Although published in 2001, these model projections are essentially independent from the observed climate data since 1990." Can you give an exact reference (chapter and verse) to support your claim?]
Raven // March 25, 2008 at 3:10 am | Reply

Tamino says:
“This is contradicted by Rahmstorf et al., who state “Although published in 2001, these model projections are essentially independent from the observed climate data since 1990.”

Rahmstorf made an error of omission rather than fact.

See Box 5.1 here: http://www.grida.no/climate/ipcc/emission/115.htm

“One of the primary reasons for developing emissions scenarios is to enable coordinated studies of climate change, climate impacts, and mitigation options and strategies. With the multi-model approach used in the SRES process, 1990 and 2000 emissions do not agree in scenarios developed using different models. In addition, even with agreed reference values, it is time consuming and often impractical to fine-tune most integrated assessment models to reproduce a particular desired result.

Nevertheless, differences in the base year and 2000 emissions may lead to confusion among the scenario users. Therefore, the 1990 and 2000 emission estimates were standardized in all the SRES scenarios, with emissions diverging after the year 2000. The procedure for selecting 1990 and 2000 emission values and the subsequent adjustments to scenario emissions are described in this box.”

In other words, the scenarios were tuned to ensure that 2000 could be the baseline for any comparisons. Claiming that 1990 was the baseline is misleading.

[Response: You seem to be the one commiting an error of omission. Box 5.1 to which you refer *also* states: "Emissions for the year 2000 are, of course, not yet known and 1990 emissions are also uncertain." What is stated in the link you give is that "The standardized scenarios share the same values for emissions in both 1990 and 2000." This is *not* that emissions or temperature projections have any dependence on observed data post-1990. As for your conclusion that it means "that 2000 could be the baseline for any comparisons," that strikes me as a total non-sequitur.

And of course the aforementioned discussion is about *emissions* rather than temperature estimates. I don't see any evidence that the IPCC TAR temperature projections (or emissions scenarios for that matter) have any dependence on post-1990 observations.]
George // March 25, 2008 at 3:23 am | Reply

Its a very nice blog, Lucia’s. Good tempered, educational, interesting, reasonable, technically solid. Recommended.”

Except for the fact that she has a nasty habit of accusing some others (including our host here) of cherry picking based on the flimsiest of evidence.

As far as the comment that

“I don’t understand why you are so adverse [sic] to accepting Lucia’s analysis”

it’s not simply a matter of “accepting her analysis” without question, particularly when she herself has admitted that one has to be careful when drawing conclusions based on such short time periods.

If there are problems with her analysis (eg, with how she has combined the data from the various sets or how she has taken into account uncertainty attached to published IPCC trends over the short term ) OR, if she has claimed that the IPCC projections indicate something that they were not intended to indicate, then these issues are all fair game for criticism.

It’s really hard to tell what the actual point of this whole exercise is.

The IPCC trend of 2C per century is based on the trend over the past few decades (in fact, it is simply an extension of it through about 2030 or so).

Lucia has claimed that she accepts that the past warming and AGW are real and that “falsification of the projection” over the short term (which she claims to have done) would not falsify the theory of AGW.

So what exactly is she trying to show?

That there is a time span at which trend analysis breaks down and essentially becomes swamped by noise?

That the IPCC document is a policy document that is intended primarily for non-scientists (who are not versed in the fine details of error bands) to allow them to assess the effects of various possible future emission scenarios ?

What?
Raven // March 25, 2008 at 3:56 am | Reply

Tamino says:
“And of course the aforementioned discussion is about *emissions* rather than temperature estimates. I don’t see any evidence that the IPCC TAR temperature projections have any dependence on post-1990 observations”

The GHG forcings used for each scenario diverge quite rapidy which means the temperatures should have diverged as well. Setting the scenarios to use the same GHG emissions meant that the same model would produce the same temp in 2000 for all scenarios.

Therefore the following claim by Rahmstorf is misleading because it implies that the trend from 1990-2000 is a relevant reference point for comparison:

“The global mean surface temperature increase
(land and ocean combined) in both the
NASA GISS data set and the Hadley Centre/
Climatic Research Unit data set is 0.33°C for
the 16 years since 1990, which is in the upper
part of the range projected by the IPCC.”

[Response: You *still* haven't provided a single shred of evidence that any of the IPCC projections have any dependence at all on post-1990 observations. I have yet to see anything even close to contradicting that the projections start in 1990, exactly as claimed in Rahmstorf et al.

As for the "range" mentioned, IPCC projections are based on the use of multiple runs of multiple models. As Rahmstorf et al. state, "the gray band surrounding the scenarios shows the effect of uncertainty in climate sensitivity spanning a range from 1.70 to 4.2°C." And indeed the gray band starts in 1990.

Using 2000 as a starting point may well apply to comparing different emissions scenarios, but if you want to compare IPCC TAR projections with observations, the projections start in 1990.]
Raven // March 25, 2008 at 4:50 am | Reply

Tamino says:
“Using 2000 as a starting point may well apply to comparing different emissions scenarios, but if you want to compare IPCC TAR projections with observations, the projections start in 1990.”
The only thing that starts in 1990 is an artificial scenario that has no connection to stated scenario parameters nor with the actual emissions observed. More importantly, this artificial scenario was created *after* the actual temperature record was known. This means we can be certain that the IPCC would have revised this artificial scenario as many times as necessary to ensure that the ensemble means approximately matched the temperature record.

What you don’t seem to understand is that any change to the model input which uses knowledge after a certain date counts as ‘tuning’ which means the period prior to the ‘tuning’ cannot be used as evidence of the models effectiveness.

[Response: You're being absurd. If I run a model which is completely independent of observed temperature after 1990, then it's not only legitimate, it's only sensible to compare its output from 1990 onward to check the model effectiveness. This is no different from witholding data from a training set to provide an independent verification set.

I also notice that you've made a lot of claims about what the IPCC did with their models, but you have yet to provide any *evidence* for anything except the fact that various scenarios had the same forcing imposed in 1990 and 2000, even though it's clearly stated that at the time they were run the actual 2000 forcing wasn't known.

The most absurd part of your comment is the claim that "we can be certain that the IPCC would have revised..." In other words, you have no evidence but you assert it as certain.

It's also absurd to take a composite of *many* projections of long-term temperature trends, which by its nature smooths out short-term fluctuations, and compare it to a time span so brief that the short-term fluctuations dwarf the long-term trend. Rahmstorf et al. complain that the time since the projections begin (only 16 years at the time of their writing) is itself so short as to make comparisons difficult; the claim to have falsified IPCC projections based on only 7 years, with such a low signal-to-noise ratio, only gives evidence of a desire to bolster a preconceived notion. Maybe that's why several commenters have insisted that Lucia herself states the 7-year time span is insufficient for real falsification.

Your attempts to discredit Rahmstorf et al. 2007 have only strengthened my belief that their analysis is completely correct, that the projections start in 1990, and that any claim to have falsified IPCC projections using only post-2001 data is a sham.]
fred // March 25, 2008 at 6:49 am | Reply

George, this criticism is perfectly extraordinary.

Lucia is making a very restricted but valid point, and is quite clear about its limited extent. She simply takes predictions made on a certain date, for events which occurred after it. Then she asks whether the events are happening as projected. She picked 2001 as the start only because that was when the predictions were made. She ends with the latest data available. One of her difficulties was that she insisted on using predictions made before the start of the series that was predicted. Some of the IPCC forecasts apparently include elements of the series which occurred before the date of the forecasts. This is not necessarily illegitimate in their context, hindcasting has its uses, but it does make it hard to assess them as predictions. The 2001 projections were actually made for a series all of which was in the future.

This does not tell you anything about the issues you are concerned about. It does not say anything about the competence or integrity of the IPCC, the reality of AGW or anything else. It does not tell you whether other projections were better or worse. It just says that at the moment, this particular projection seems not to fit very well with what happened after it.

If the IPCC does not want its projections tested against what happens, it should stop making any. As with MBH, the fact that the 2001 projections seem not be borne out by experience so far is not central to the AGW hypothesis. The IPCC could have got one set of projections wrong and still be right about AGW where it matters. It is not necessary to defend this. They are human and can make mistakes.

Once again the response we are seeing is more reminiscent of people toeing the party or dogmatic line than scientific enquiry: the felt need to defend the indefensible, when nothing much hangs on it.
Barton Paul Levenson // March 25, 2008 at 12:20 pm | Reply

Raven writes:

[[The graph also clearly states that the trend is a C-O fit to the data from 2001-now.
Nothing in her graph or description is wrong or misleading.]]

It’s misleading to imply that you can tell anything from seven years worth of data. Did I mention, repeatedly, that climate is defined as regional or global mean weather conditions for 30 years or longer?

[[Yet Tamino has yet to acknowledge that apply his analysis to any dataset other than GISS would show that a 0.2/decade is not consistent with the actual data since 2001.]]

He doesn’t have to. Seven years is not a significant data set.
kim // March 25, 2008 at 12:49 pm | Reply

You miss an important point; lucia herself says that seven years is inadequate for confidence at the 95% level. She is explicit that her analysis requires ten years. That would require three more years of stable or dropping temperatures.

She is not a denier, merely curious and rigorous. Read her more carefully.
=========================
Adam // March 25, 2008 at 1:35 pm | Reply

“Adam: Thanks for the kudos. However Skeptical Science is run by John Cook, not John Cross. However the site is excellent and if you wish to associate me with it then I am happy to accept the compliment.”

Ooops. Apologies to all.
george // March 25, 2008 at 1:58 pm | Reply

lucia herself says that seven years is inadequate for confidence at the 95% level. She is explicit that her analysis requires ten years. That would require three more years of stable or dropping temperatures.
… Read her more carefully.

I guess that’s why she claimed the following:

I applied the technique to test the predictions of warming communicated by the IPCC in the AR4. I found their recent short term projections for warming are falsified to 95% confidence Their best estimate for the near term trend in the earth’s surface temperature is not consistent with observations of the earth’s surface temperature since the time when the IPCC made its projections. The mean trend they projected was too high. The lowest trend they communicated to the public, in graphical form, was too high to be consistent with recent observation of he earth’s surface temperature.

“Precise”, “rigorous” people are not only very careful about their analysis, but they are very careful about the words they use…if for no other reason that the words can come back to bite them.

They don’t claim things like “IPCC Falsification” without first thoroughly understanding
1) How the IPCC actually did their projections (including when the runs actually started) and what those projections actually show and
2) what the grayed region actually represents on those IPCC projection graphs (hint: it is not what Lucia seems to believe it is)
3) the relevant statistical analysis (in this case cochrane orcut)

From a post labeled: “Correcting for serial autocorrelation: cochrane orcut” [PS: Did i do this right?]“
*”To those wondering: Yes, I have done this for another more interesting data set. I have also attached the Excel spread sheet. Obviously, you can modify to look at other data sets. But, before you fiddle, let’s wait to hear back from the econometricians to see if I understood the method correctly.” — Lucia
kim // March 25, 2008 at 3:18 pm | Reply

George, I think you are confusing a 95% confidence from the IPCC’s prediction with a 95% confidence from lucia’s analysis. She has claimed that three more years of stable or dropping temperature will falsify the IPCC’s claim of 0.2 degrees centigrade warming per decade, at the 95% confidence level.
==============================
J // March 25, 2008 at 3:41 pm | Reply

Lucia, Anthony Watts, etc. are basically engaged in a fancified exercise in reading chicken entrails.

The fundamental premise is silly — that seven years’ worth of noisy data can confirm or contradict a projected trend that is unfolding on a timescale of decades. If you plot the past seven years’ worth of temperature data alongside the preceding 25 years’ data, they don’t show any departure from the long-term trend.

Lucia can dot every i and cross every t, and turn out a lovely example of statistical analysis, but it’s physically meaningless. You just can’t pin down climate change using less than a single decade’s worth of data.

Until recently, many “skeptics” were happy to scoff at a 30-year trend, so I am a bit bemused by their sudden fascination with 7, 5, or even 1-year trends.

This La Nina has produced a bit of a feeding frenzy on the skeptic side. From time to time I peek over at Anthony Watts’s blog, and it seems like he’s had about a zillion posts about “COLD COLD COLD!” temperatures in the past few months.

This raises a few questions in my mind:

(1) Does this mean that the surface and satellite temperature records are now considered reliable? Or is there some threshold, such that monthly anomalies below, say, 0.2 are considered good, but anomalies above that threshold are suspect?

(2) When temperatures inevitably begin to rise again, will Watts et al. continue to post about every single new monthly temperature measurement? Or will everyone’s interest mysteriously fade once it becomes clear that there isn’t, in fact, a cooling trend?

J.
fred // March 25, 2008 at 3:44 pm | Reply

George, if she has got something wrong, please just tell us what exactly it is. It will advance the discussion. Cochrane or the grey areas, whatever. Just state what is wrong.

Looks like she has falsified one particular prediction to the 95% level. Clearly there may be other predictions not falsified. They may be more interesting ones. It may prove nothing important that this particular one is falsified. But, falsified it seems to have been, at least for now.

If not, please tell us why not.

[Response: This issue having taken on a life of its own, I'll have quite a lot to say about it in an upcoming post.]
steven mosher // March 25, 2008 at 4:06 pm | Reply

This is a fun discussion. Lets suppose that it’s 2008. Oh it is. Do you think that 7 years from now we could reject a projection or prediction of warming? Is seven years too short a time period? Well it depends on the noise, and depends on the magnitude of the trend. Could we reject a projection of warming in a period less than 7 years?

Good Question! Let’s ask somebody who understands this:

“By 2015, the expected temperature from the regression-line fit and that expected from the “no change” hypothesis will be far enough apart that we’ll probably be able to distinguish between them with statistical significance. In other words, by 2015 either we’ll know that global warming has changed (possibly stopping, possibly reversing), or there’ll be no more of this “global warming stopped in 1998” malarkey.

It’s entirely possible that the numbers may give us statistically significant evidence even before 2015. If so, I’ll report the result. If it turns out that global warming is not continuing (which I seriously doubt), then I’ll readily admit that I was wrong. In fact, I’ll be keeping a close eye on the future evolution of global temperature and actively looking for such results, so if we do get valid evidence that global warming has stopped, I just might be the *first* one to say so.”

Very simply, If I bet that the next 7 years would show say .017C of warming per year on average,
or .17C per decade
AND IF, for example, we saw a couple years of zero warming, or Cooling, then as our author above suggests we could reject the claim that warming is proceeding apace at .017C per year,
AND we could reject this claim in a period shorter than 7 YEARS!.

Now, Which series would we look at? Just GISS? I dont know, Lucia makes a point to look at them all. let’s ask for some opinions. How should we determine the outcome with the HIGHEST RELIABILITY?

What would tamino say: We know, he said it a while back

“[Response: None of the metrics — popular or not — is 100% correct. And correcting the GISS Y2K error led to a net change in global average temperature anomaly of 0.003 deg.C.

As I said, I’m not betting money I’m trying to establish conditions under which we can confirm or deny various hypotheses. It was framed as a bet because that seems to be popular for discussion, and it does force one to be explicit about exactly what conditions will lead to a declaration for one or another hypothesis. For a bet, I think it’s better to keep it simple and agree on a single source of data for decision.

But for determining the outcome with highest reliability it’s better to use multiple data sets. I intend to keep track of GISS, HadCRU, and NCDC, and I’ll probably keep my eye on satellite data from RSS, UAH, UMd, and UW as well. I’ll report any significant results, regardless of the nature of the result or the source of the data. I expect they’ll end up telling the same story.]”

So, there you have it. It’s theoretically possible to reject a claimed warming trend, or cooling trend with less than 7 years of data ( it would be one hell of a rare event) and the most reliable method is to look at several measures ( giss, hadcru etc etc)

I dont know what you guys are arguing about. I agree with tamino

Cross posted other places.

[Response: Please don't mischaracterize my test so.

The original proposition was a speculation which was later expanded upon in a full post, where I made the conditions precise. It's a *serious* mischaracterization to state that the time frame for probable confirmation/denial (end of 2015) is "less than 7 years." As a minor point, that post was written in late 2007, and the post refers to the *end* of 2015, so it's more than 8 years, not less than 7, since the writing of that post. But that's very minor indeed; the major point is that the hypothesis being tested is "that global temperature hasn’t changed since 2001," so the actual time span suggested as "likely but by no means certain" to decide is 15 years (start of 2001 to end of 2015). That's not less than 7. It's more than twice as long.

I also stated:

By the end of 2015, it is in fact likely but by no means certain that one or the other side will have won. Eventually, the two regions get far enough apart that it’s certain to happen. In fact, by 2028 we’re sure to have two years outside the limits of one or the other side, so the bet can’t take longer than 2028 to be decided. But this test isn’t based on a particular future year; it’s possible (but highly unlikely) that either side could win if 2008 and 2009 both fall into its winning region.

I stand by the statement that it's the progression of the data, not the passing of a fixed amount of time, that decides whether a conclusion can be made. So the real question is, is Lucia's analysis correct? As I mentioned in a response to fred, I'll have a lot more to say about that in the next post.]
kim // March 25, 2008 at 4:08 pm | Reply

fred, perhaps I am a little behind the 8 ball. Nonetheless, if a flipped PDO means 30 years of cooling, there’s gonna be a whole lot of falsifying going on.
====================
Phil. // March 25, 2008 at 4:15 pm | Reply

“Lucia, Anthony Watts, etc. are basically engaged in a fancified exercise in reading chicken entrails.

The fundamental premise is silly — that seven years’ worth of noisy data can confirm or contradict a projected trend that is unfolding on a timescale of decades. If you plot the past seven years’ worth of temperature data alongside the preceding 25 years’ data, they don’t show any departure from the long-term trend.

Lucia can dot every i and cross every t, and turn out a lovely example of statistical analysis, but it’s physically meaningless. You just can’t pin down climate change using less than a single decade’s worth of data.”

I suggest you read Lucia more carefully particularly the following:
http://rankexploits.com/musings/2008/falsifying-is-hard-to-do-β-error-and-climate-change/
Her analysis is ongoing and will be updated regularly as the data is updated.
J // March 25, 2008 at 4:15 pm | Reply

Fred: “Looks like she has falsified one particular prediction to the 95% level.”

Yeah, she falsified that IPCC prediction that temperatures would rise at a rate of 0.02 degrees C/year over a particular seven-year period.

Remind me again where that one was? I’m having trouble finding it.
null{} // March 25, 2008 at 4:35 pm | Reply

Seven years was long enough here: http://tamino.wordpress.com/2008/01/24/giss-ncdc-hadcru/

Which was repeated here: http://tamino.wordpress.com/2008/01/31/you-bet/

Is it, or is it not … that’s the question.

[Response: You too have chosen to mischaracterize my post. See the response to steven mosher.]
Barton Paul Levenson // March 25, 2008 at 4:52 pm | Reply

fred writes:

[[George, if she has got something wrong, please just tell us what exactly it is. It will advance the discussion. Cochrane or the grey areas, whatever. Just state what is wrong. ]]

Inadequate sample size.
J // March 25, 2008 at 4:59 pm | Reply

Steven Mosher wrote: “Cross posted other places.”

How odd. A moment’s reflection would have made it obvious that Tamino was referring to calculating a trend based on 15 years’ worth of data, not 7.

I don’t know where else you cross-posted that, SM, but you might want to follow it up with a correction at the same site or sites.
Ian // March 25, 2008 at 5:12 pm | Reply

J,
The IPCC AR4 does actually mention the “0.2C per decade” figure, in two places:

(1) on page 7 of the SPM, there is this paragraph: “For the next two decades a warming of about 0.2°C per decade is projected for a range of SRES emissions scenarios. Even if the concentrations of all GHGs and aerosols had been kept constant at year 2000 levels, a further warming of about 0.1°C per decade would be expected. Afterwards, temperature projections increasingly depend on specific emissions scenarios.”

(2) which seems to be condensed from WG1 Chapter 10, page 822, when discussing committed climate change to 2300 and SRESs: “The committed warming trend values show a rate of warming averaged over the first two decades of the 21st century of about 0.1°C per decade, due mainly to the slow response of the oceans. About twice as much warming (0.2°C per decade) would be expected if emissions are within the range of the SRES scenarios.”

Both quotes are embedded in longer sections. Reading the longer sections, it might be argued (plausibly, I think) that these quotes are summary statements conveying the impact of estimated forcings, rather than predictions meant for formal significance testing. Even if taken as formal predictions, they’re obviously not a single summary prediction from the AR4 – testing every temp figure mentioned or implied in the AR4 against 7 yrs of data would take weeks.
dhogaza // March 25, 2008 at 5:33 pm | Reply

I don’t know where else you cross-posted that, SM, but you might want to follow it up with a correction at the same site or sites.

Obviously, CA, which is where I’m sure null{} found the links that he posted so triumphantly above.

As for a correction, I wouldn’t hold my breath.

I remember the post, and the bet, and yes it was clearly for the 15 year interval. Unbelievable.

[Response: How about it, steven? Will you post a correction in the places you cross-posted your original comment?]
Ian // March 25, 2008 at 5:38 pm | Reply

and, I left off: testing every figure in the AR4 against 7 yrs would be silly, for reasons detailed at length above.
fred // March 25, 2008 at 6:19 pm | Reply

BPL, how big a sample size is needed to assess the 2000 predictions?
fred // March 25, 2008 at 6:22 pm | Reply

J, its not silly. It may be wrong, but its not silly. Someone makes a prediction for any number of years, 20, 30 or 50. It is perfectly reasonable to ask after 7 years, how is the prediction doing. In fact, it would be a bit odd not to, when its about such an important matter.
null{} // March 25, 2008 at 6:22 pm | Reply

I did not choose to misrepresent, I didn’t read it close enough.

However, it now seems that you are stating that 15 years, not 30 years, are sufficient.

How about a clarification?

The continued presumption of motive is getting to be very discouraging. bye bye tamino

[Response: I've repeatedly stated that analysis of the data, not the passage of a fixed time span, is what's needed.

If "bye bye" means you won't be returning -- well, you won't be missed.]
george // March 25, 2008 at 7:01 pm | Reply

Here, let me help Steven Mosher remember where he hid those nuts of his…

Here’s one with no correction yet…
steven mosher // March 25, 2008 at 8:41 pm | Reply

Its simple, 6 years is “enough” data if

1. your prediction “trend = X” is extreme AND/OR
2. The data goes in the opposite direction, in the right magnitude.
3. The CI cooperates. ( varience is low)

Just do the math. Now, the hard part is drawing the CONCLUSION from rejecting the conjecture

Conclusions you cannot draw.

1. AGW is false. (unwad your panties guys.)
2. The IPCC is a sham. They might have got this one a little bit wrong in the short run, be more careful next time round.
3. Tamino is wrong. Cant conclude that either. Lucia confirms his results, matched his results for GISS…… the other indexes of course show the opposite. nice, that she shows results from a variety of methods. ( typcial engineer training)

people should Just reject the .2C /decade short term projection. dont reject the science . do the obama thing. It’s easy.
Dont throw lucia under the bus, that’s like selling your grandmother to win an election.

Some other tidbits just for grins. If 6 years ( ok 74 months) is too short to REJECT a hypothesis, then is it also too early to talk about confirmation. beta error and all. so please spare me the stories about the ice polar bears and whatnot.

Interesting point, Like the TAR, AR4, which used observations up to 2000-2001, ALSO predicted a .2C increase per decade. until 2011.
so the TAR using data to 1990 predicted .2C per decade from 1990 to 2021 and AR4, using data to 2000, predicted .2C per decade for the first 2 decades of 2000. Hmm. Not sure whatto make of that, but hey.

Can an AR4 Projection from 2000 to 2010 of .2C trend be rejected? Can Any projection be rejected? any at all? How long do I have to wait?
as long as you say? or is there and objective test.?

Another way to put that. If in the year 2001, I made a forecast ( hey god told me)
That the trend for 2001-2011 would be MINUS .2C per decade
How long would you wait to test my claim?

Now suppose I used statistics and guessed A .16C rise. what you make of that? Would you say I was “right”

Now suppsoed I used a bunch of physics based models, would you expect me to me closer to the truth or farther away? more confident or less confident?

Other interesting question.

Should a GCM or collection of GCM be expected to Outperform a Naive forecast over a 10 year period? 20 year period?

hmm. Anyway, I am sticking with my GCM. It shows that over the next 1 billion years the global warming trend will be 0.000C per billion years. Clearly such a projection canot be tested or evaluated for a couple hundred million years.

Just wait, it will get colder. err first it will get warmer, then colder. then really hot, then like absolute zero cold. but the trend, if you stretch out x long enough is zero. every tends toward it.

The hilarious thing is you guys are making Bob Carter’s argument without even realizing it. In a weird way.

Sorry rambling. Throw fruit if you like.
Ian // March 25, 2008 at 8:58 pm | Reply

Fred wrote:

“J, its not silly. It may be wrong, but its not silly. Someone makes a prediction for any number of years, 20, 30 or 50. It is perfectly reasonable to ask after 7 years, how is the prediction doing. In fact, it would be a bit odd not to, when its about such an important matter.”

Fred, you’re right, it is an important matter – and if you prefer to call a 7-yr trend check “wrong” instead of “silly,” it’s fine with me. But it doesn’t seem “reasonable” when everyone acknowledges that the trend of interest can’t be seen in such short periods. If I’m driving to the store, to me it’s not that reasonable to back down the driveway and have everyone ask “Are we there yet?”
J // March 25, 2008 at 9:32 pm | Reply

Fred, the problem is that people (e.g., you) are reading Lucia’s website and concluding things like “It does seem to show that the IPCC predictions of 2000 or so have been falsified by later events”

When in fact no such thing has been shown. IPCC did not predict that the climate would warm by (0.2 X 7 / 10) degrees during the period 2001-2007.

Your confusion on this may be understandable, if George quoted Lucia correctly: “I found their recent short term projections for warming are falsified to 95% confidence”

I described that as “silly”, and I stick by that description. Essentially, Lucia is taking a real IPCC prediction (something like 0.4 C warming over two decades), and turning it into an imaginary prediction (0.14 C warming over seven years). If IPCC had actually made Lucia’s imaginary prediction, then yes, she would have falsified it at 95% (assuming her stats are correct, which I haven’t checked).

There is a difference between “statistically significant” and “physically meaningful”. Climate trends of just a few years may be statistically significant, but not have any physical meaning.

IMHO, the soi-disant “skeptic” side is expending a lot of mental energy this winter on what is basically just reading chicken entrails and trying to find the patterns that they want to see. Look at Watts’s blog, for an example. Frankly, it’s a joke.
dhogaza // March 25, 2008 at 9:53 pm | Reply

Some other tidbits just for grins. If 6 years ( ok 74 months) is too short to REJECT a hypothesis, then is it also too early to talk about confirmation. beta error and all. so please spare me the stories about the ice polar bears and whatnot.

Over on the open thread I predicted that we’d hear that ice trends are too recent to be meaningful, while hearing that Lucia’s analysis is sound.

Too easy.
dhogaza // March 25, 2008 at 9:56 pm | Reply

And, oh, just had to check, Mosher still hasn’t corrected his mistatement of HB’s “bet” over on the blog george linked to above.

people should Just reject the .2C /decade short term projection. dont reject the science . do the obama thing. It’s easy.
Dont throw lucia under the bus, that’s like selling your grandmother to win an election.

Classic denialist posing. Not just climate science denialists, but all science denialists.

If you don’t accept their assertion that they’re right, all the professionals wrong, then you’re not doing the science, you’re (essentially) acting on faith.

Sorry, Mosher, that’s not going to fly.
J // March 25, 2008 at 9:59 pm | Reply

Mr Mosher writes:
Its simple, 6 years is “enough” data if

1. your prediction “trend = X” is extreme AND/OR
2. The data goes in the opposite direction, in the right magnitude.
3. The CI cooperates. ( varience is low)

Just do the math.

Okay … if all that matters is the statistics, why not give this a try:

During the five months beginning with July 1997, the GISSTEMP Land+Ocean monthly temperature anomalies are 0.26, 0.37, 0.41, 0.50, 0.56.

Calculate the trend over that time, and the 95% confidence intervals around that trend.

The trend itself is +9 degrees C/decade. Ignoring autocorrelation, the 95% CI is 7-11 degrees C/decade. Assume autocorrelation of 0.6, and double the confidence intervals, and you still have a 95% range of +5 to +13 degrees C/decade.

Whoa! We’ve just proved that the IPCC short-term prediction for mid-1997 was underestimating warming by at least a factor of 25 (!), possibly much more. It’s statistically significant!

But the IPCC didn’t make a “short-term prediction” for five months in 1997. And it also didn’t make one for seven years in 2001-2007.

Contrary to Mr Mosher, it’s not enough to “Just do the math”.
TCO // March 26, 2008 at 12:16 am | Reply

I looked at the Lucia post and kept looking for where she talked about the structure of the noise and finally found that she had an assumption of iid. That’s insane. We have all these kvetching from McI and his acolytes about autocorrelation and then we get her running this study assuming iid? When she can look at the theory and even just the previous history and see El Nino cycles? It’s like someone calculating energy use of a car at high speed and ignoring air friction (which overcomes rolling friction at high speeds). I mean it’s just OFF, OFF, OFF. Sheezu mama mia! Let’s do better than that, people. We give ourselves a bad name when we wade in with crap like that.

[Response: I thought (although I haven't checked in detail) she was using Cochrane-Orcutt estimation. That assumes an AR(1) model for the random fluctuations. If that model is correct, then the random fluctuations in the *transformed* data (as defined in this post) will be iid. Is this perhaps the case?

Response 2: On checking, that does appear to be the case; she appears to use both C-O estimation and the same simplified AR(1)-compensation to OLS that I outline in this post.]
TCO // March 26, 2008 at 12:18 am | Reply

Come on my fellow mujahedin! Look at the title of this thread! What’s missing in Lucia is what is the TITLE!
TCO // March 26, 2008 at 12:42 am | Reply

Well…then she should explain stuff better. ;) Any hoo. What is her error structure? What’s AR1 term?
TCO // March 26, 2008 at 12:46 am | Reply

And how well does AR1 model PDO or the previous random walky looking history?
TCO // March 26, 2008 at 12:47 am | Reply

Well I guess AR1 would do well for random walk. :) But how well does a random walk match previous expressed history?

[Response: A true random walk is AR(1) with autoregression coefficient = 1. But it's not a realistic model for many physical processes, because it's unbounded.

It turns out that for global temperature data, AR(1) is a decent first approximation -- it gets you in the ballpark -- but as one reader pointed out it's not complete. The departure from AR(1) tends to increase the probable error in the estimated trend rate. I'll have more to say about that in the upcoming post.]
Greg // March 26, 2008 at 12:51 am | Reply

J, you fail to grasp the mathematical basis. Let’s suppose you and I start tossing a coin (and, in this alternate reality, you and I have no idea how coin tossing distributes). We speculate that the long term trend will be 50% heads, 50% tails. The first seven tosses come up as heads. At this point, our prediction looks very, very poor. It is falsified to the 98% level or so (I haven’t done the exact maths).

Our prediction may still be correct (we know in this case it is, since we know it should be 50/50 heads/tails). But at that point, there is only a two percent chance of the random samples having deviated so strongly from the true trend.

Lucia is right, there is statistically only a very small chance that the IPCC prediction of 0.2C/decade will be correct, given how the last few years have unfolded.

Any disagreement you have with that is based on knowledge you think you have, not statistics. You know that a coin, given it’s physical arrangement, should be randomly falling on each side, and you know that a new el nino will come along and boost the temperatures.

[Response: I urge readers to withold concluding whether or not "Lucia is right" until after reading the next post.]
dhogaza // March 26, 2008 at 12:58 am | Reply

I think she’s quite clear about recognizing autocorrelation in the actual data.

Her problem isn’t with the data per se, her problem is her belief that the model projections are meant to be predictions than include weather phenomena such as El Niño and La Niña, which they clearly don’t.

That’s the point of William Connelly’s continuously pointing out that these projections don’t include weather noise. And others pointing out that if they did, the confidence intervals over the short term would be huge for the projections, because we KNOW that in the short term weather variability swamps the signal.
TCO // March 26, 2008 at 1:01 am | Reply

Hey! Uh…don’t expect us to put our thinking on hold waiting for you to spell check your next post, man. We are tearing into this thing and we won’t always wait for you. You may need to join in almost like a hoi polloi. But of course, much higher, much higher…still a little bit in the same soup.
TCO // March 26, 2008 at 1:05 am | Reply

Ok…if she recognizes autocorrelation, how exactly does she model the noise? Saying AR1 is not sufficient. What is the term equal to? Also, is Ar1 really adequate? If we looked at the DJIA for the last century and saw that there were some rapid drops (crashes), but that CAPM theory seems to be upheld in the long term with appreciation over very long time frames, would we take a recent crash and say that CAPM was wrong…without looking back at 1987 and the like?

(I’m drinking, so please read what I mean to say, not what I say.)

[Response: You lost me with this one.]
TCO // March 26, 2008 at 1:14 am | Reply

One at a time. if the noise is AR1. is the term = to 0.2, o.4, 1.0 (random walk) or what!!!
TCO // March 26, 2008 at 1:18 am | Reply

Oh…and maybe drinking would help you hang with me, better.

[Response: I'm beginning to suspect that I might be too moderate in that regard, to keep up with you.]
dhogaza // March 26, 2008 at 1:28 am | Reply

One at a time. if the noise is AR1. is the term = to 0.2, o.4, 1.0 (random walk) or what!!!

I, at least, don’t see where she’s given the term.
TCO // March 26, 2008 at 1:35 am | Reply

(still intoxicated, but this is an on-topic or at least on title post): If you are really interested in autocorrelation, you should look into the noise structure that Steve M used for his “red noise” series in his GRL article. They were modeled with a function that gives a LOT of different terms and which HeAVILy matches them against the ACTUAL DATA. Worth checking to see if this was over fitting. Just saying ya know. SM got awful snippy when I held the proverbial blowtorch to the proverbial nutswack on this issue. Just saying.
Johan i Kanada // March 26, 2008 at 2:18 am | Reply

Tamino,

1) Why would there be linear trend at all, as opposed to some other sort of trend? What is the physical theory/model supporting such a claim? (”It’s linear because I assumed it was linear” does not seem to be a strong argument.)

2) Depending in the selection of start and end points you will of course get different trend values. What physical meaning do you assign to these different trend values? If there is a “true” underlying trend, which one is it?

3) Presumably the hypothesis is that the global temperature T is positively affected by an increasing CO2 level. Thus you will need to establish some sort of causal relationship between observed T and CO2. Have you done this? (I realize that this last topic is slightly off topic in this dicussion thread, but nevertheless quite essential.)

[Response: I'd say it's safe to assume that the trend is *not* strictly linear. But if the nonlinear part is small compared to the linear part, then its presence may not even be strong enough to detect (let alone quantify) above the noise. That appears to be the case for post-1975 data; what's left over after subtracting the linear trend is indistinguishable from autocorrelated noise. So the data themselves confirm that a linear model is not only appropriate, it appears to be the only part of the signal that rises above the noise.

Any two start and end points, within the time span for which the data are indistinguishable from linear + noise, will of course give different estimates of the trend rate. But the *error range* will almost always include the true value. I say "almost always" because it's very unlikely for random fluctuations to cause the estimated error range to exclude the true value, but it's not *impossible* -- just as it's very unlikely for a fair coin to turn up 10 heads in a row, but it's not impossible. The differences reflect the fact that the *noise* is different for different time spans, so of course its impact on the estimated trend will be different. But that difference will -- almost always -- fall within expected limits. Of course computing those expected limits is nontrivial when the noise is autocorrelated, and the simple models of autocorrelation presented here are sufficient to get us in the ballpark, but *not* sufficient to encompass all the information we can determine from the data. I'll have more to say about that in the next post.

It's certainly not necessary to know *anything* about the physics underlying the data, to determine statistically that the data indicate a linear signal + (autocorrelated) noise. Statistical analysis of data doesn't tell us *why* the indicated signal is what it is. We *can* use it to test whether or not the observed signal is compatible with various hypothesized physical causes. The physics of climate forcing (including CO2 and other greenhouse gases) is best estimated using computer models, which indicate that for the modern era temperature should be rising in a very nearly linear fashion; the nonlinear part of the signal should be small compared to the linear part, and compared to the noise too. And that's what we observe.]
J // March 26, 2008 at 3:21 am | Reply

Tamino, good luck getting us to stop speculating …

Greg writes: Lucia is right, there is statistically only a very small chance that the IPCC prediction of 0.2C/decade will be correct, given how the last few years have unfolded.

Any disagreement you have with that is based on knowledge you think you have, not statistics.

I don’t exactly agree with you, but I might concede the point anyway, since I’m not interested in wrangling over the epistemology of this.

I suppose I would say that my fundamental disagreement with this line of argument (not specifically Lucia, since I haven’t read through everything on her site) is that it’s naively focused on statistical significance. As I pointed out up-thread, it’s possible to find sequences of temperature data that yield statistically significant trends that also happen to be physically absurd (+9 degC/decade) when extrapolated beyond the bounds of the data.

So we might have three trends, one over a five-month period (showing hyper-extreme warming), one over a seven-year period (showing only marginal warming), and one over a thirty-year period (showing moderate warming roughly in line with that expected from climate models). All three are statistically significant, but that doesn’t mean that we naively accept all three as representations of the real climate system.

Clearly, the five-month trend is just not telling us anything useful about climate; the 30-year trend probably is. The question of whether the 7-year trend is climatically meaningful probably doesn’t have an easy answer. You need statistics plus science, to answer that question, not statistics alone. I think the word “naive” is appropriate here.

Of course, I could be completely wrong about this.

[Response: I don't ask readers to inhibit speculation -- just to withold decision until I've had more to say on the subject.]
elspi // March 26, 2008 at 3:44 am | Reply

Johan i Kanada

“1) Why would there be linear trend at all”

BECAUSE EVERY DIFFERENTIABLE FUNCTION IS LINEAR IN THE SMALL

(sorry to yell, but who the hell taught you calculus?)

“3) Presumably the hypothesis is that the global temperature T is positively affected by an increasing CO2 level.”

CO2 is transparent in the visible spectrum (the light coming if from the sun) but opaque in the infrared spectrum (light going out from the earth). More CO2 warms the earth in EXACTLY the same way that putting on an extra coat warms a person.

[Response: No need to yell or curse -- a simple question.]
kim // March 26, 2008 at 3:54 am | Reply

No, a coat warms by stopping convection. But, keep thinking and trying. Climate sensitivity to CO2 is an important question.
=====================
kim // March 26, 2008 at 4:05 am | Reply

I think that if you look at the beginnings of lucia’s exercise she ignores weather noise because the IPCC does in its predictions. I also think that if you look closely, the IPCC explicitly denies being in the prediction business.

But there are a couple of ironies here. Despite the statistical rigor demonstrated by all concerned, it is apparent to hoi polloi that a plateauing of temperature, with now perhaps a downturn, is something NOT predicted by greenhouse gas theory. Even Pachauri, head of the IPCC is wondering out loud, now, about natural variability of the climate. Who will be next to wonder out loud, and louder?
==========================

[Response: But plateauing such as we see now (if it deserved the name) IS predicted by probability theory, due to nothing more than the nature of random noise. In fact it's not just predicted, it's inevitable. That was the point of this post.]
MrPete // March 26, 2008 at 6:45 am | Reply

Seems to me like the key question here is “what does ’small’ mean?” I’m sure there’s a math-valid way of answering that… and suspect that’s part of where Tamino is heading.

[Response: There is indeed a precise mathematical definition of "small" in this context. For the data from, say, 1975 onward, the signal is plenty big enough to be detected, and we can even quantify its linear part. But when we subtract the linear part leaving residuals, what remains is nonlinear signal + noise, and the signal-to-noise ratio in that remainder is so small that the nonlinear signal isn't detectable, let alone quantifiable.]
fred // March 26, 2008 at 6:54 am | Reply

“More CO2 warms the earth in EXACTLY the same way that putting on an extra coat warms a person.”

Well, no, it doesn’t. A much better analogy would be some moderate feedback processes. It might be something like this, you put on a coat. This has two effects, one you are more insulated and the retained heat accounts for about half of the forecast warming. This warming, the forecast is, then leads you to work harder, which warms you still more.

People then say, we need to establish two things about these forecasts. The first is whether insulation actually does retain heat. It is absurd to deny that, its physics. The second is whether retained heat leads to working harder. This takes more and different observations.

People who point out there are two issues not one, and that its not just a matter of insulation physics, are then called deniers. Despite the fact that the underlying two stage phenomenon is part of every model of the subject… And other people, who are of course not deniers at all, keep saying its just a matter of how insulation works.
gp // March 26, 2008 at 7:16 am | Reply

J writes:
“(2) When temperatures inevitably begin to rise again, will Watts et al. continue to post about every single new monthly temperature measurement? Or will everyone’s interest mysteriously fade once it becomes clear that there isn’t, in fact, a cooling trend?”

J you have just to wait few days since march is going to be much warmer then jan/feb at least for surface data due to very strong positive anomalies in Asia (and low snowcover!).
Satellite data likely will be colder since they are more sensitive to change in SST rather then land surface(there’s a physical reason for this is not an UHI sign) and la nina even if declining in est equatorial pacific still maintain low SST.
Barton Paul Levenson // March 26, 2008 at 9:17 am | Reply

fred posts:

[[BPL, how big a sample size is needed to assess the 2000 predictions?]]

Generally you want a sample size of 30 or more to potentially achieve 95% significance (p < 0.05).

fred also posts:

[[J, its not silly. It may be wrong, but its not silly. Someone makes a prediction for any number of years, 20, 30 or 50. It is perfectly reasonable to ask after 7 years, how is the prediction doing. In fact, it would be a bit odd not to, when its about such an important matter.]]

It’s perfectly natural, but it’s statistically illiterate. Seven is too small a sample size. Try 30.
Barton Paul Levenson // March 26, 2008 at 9:19 am | Reply

Steven Mosher posts:

[[hmm. Anyway, I am sticking with my GCM. It shows that over the next 1 billion years the global warming trend will be 0.000C per billion years. ]]

Your GCM doesn’t take stellar evolution into account, I take it.
Timothy Chase // March 26, 2008 at 12:27 pm | Reply

John Canada,

I don’t know as if you’ve noticed, I have a strange-looking planet for my current avatar.

Actually its our planet, Earth. It is an image of the earth in infrared at a particular part of the spectra showing the amount of CO2 in the atmosphere. The more CO2, the more opaque the atmosphere gets in those parts of the spectra where CO2 absorbs infrared radiation.

With a given satellite, we can measure the spectral emissions that escape the earth on over two thousand different channels — and then measure the quantities of methane, water vapor, etc.. that exist at various pressures (or altitudes) and temperatures because we have detailed information on the absorption spectra of those atmospheric constituents.

Here is another image at 8 KM:

http://www-airs.jpl.nasa.gov/Multimedia/Images/index.cfm?fuseaction=ShowMultimedia_ShowImage&ImageID=260

You will notice the higher levels of carbon dioxide near the east and west coast of the United States – and other places. Major population centers – and a lot of cars.

Radiation enters the climate system more or less at a constant rate via sunlight, not as much gets out. Things heat up until we are emitting more thermal radiation — enough to compensate for an increasingly opaque atmosphere — so that the rate at which radiation leaves the top of the atmosphere equals the rate at which it enters.

*

Here are some more images which may be of general interest:

http://www-airs.jpl.nasa.gov/Multimedia/Images/
Timothy Chase // March 26, 2008 at 1:54 pm | Reply

Barton Paul Levenson wrote:

[[hmm. Anyway, I am sticking with my GCM. It shows that over the next 1 billion years the global warming trend will be 0.000C per billion years. ]]

Your GCM doesn’t take stellar evolution into account, I take it.

It is a very special GCM: it takes into account only those aspects of reality he wants it to take into account.
Timothy Chase // March 26, 2008 at 2:09 pm | Reply

Fred wrote:

“More CO2 warms the earth in EXACTLY the same way that putting on an extra coat warms a person.”

Well, no, it doesn’t. A much better analogy would be some moderate feedback processes. It might be something like this, you put on a coat. This has two effects, one you are more insulated and the retained heat accounts for about half of the forecast warming. This warming, the forecast is, then leads you to work harder, which warms you still more.

Fred, the water vapor content of the atmosphere has been going up just as predicted — due to the rise in temperature due to carbon dioxide. And that increases the opacity of the atmosphere. No, it isn’t just radiation transfer theory — there is also thermodynamics (e.g., increased evaporation) and atmospheric chemistry, the response of the biosphere, etc. but it all begins with the radiation physics.

Fred wrote:

People who point out there are two issues not one, and that its not just a matter of insulation physics, are then called deniers.

Are they? Where?
george // March 26, 2008 at 2:55 pm | Reply

Greg says:

Let’s suppose you and I start tossing a coin (and, in this alternate reality, you and I have no idea how coin tossing distributes). We speculate that the long term trend will be 50% heads, 50% tails. The first seven tosses come up as heads. At this point, our prediction looks very, very poor. It is falsified to the 98% level or so (I haven’t done the exact maths).

I find it interesting to think about that, though probably not for the reason the author did.

Over the long term, the “trend” would tend toward a 50/50 ratio, but even with a fair coin, over the short term it could include “runs” of heads (and of tails). In fact, if one tossed a fair coin enough times, it would include at least one such run of seven heads in a row.

Nonetheless, if I saw 7 heads in a row right off the bat, what I (and most others) would probably wonder is this:

“Is that coin fair? Or ” is the toss itself fair”? ( ie, “is there perhaps something influencing the coin that we don’t know about making it land heads every time?”)

In the case before us, we know ahead of time that the coin (or at least the toss) is clearly not fair.

Over the short term, the “coin toss” (and the resulting temperature trend) is strongly biased by non-random (ie, correlated) events (eg, El Nino) that “push” the coin (and the trend) toward one outcome (hotter or colder temperature).

But over the long term, the effect of El nino is both positive and negative and tends to wash out leaving the basic underlying linear trend.

If you look at the projection graphs from the IPCC and read the accompanying text, it’s fairly clear that they were talking about the long term (ie, multi-decade) trends.

The “envelope” around the IPCC projections does not incorporate the influence of things like El Nino. In fact, they state quite clearly in the text accompanying the graphic that shows the projections:

This section investigates the range of future global mean temperature changes resulting from the thirty-five final SRES emissions scenarios with complete greenhouse gas emissions (Nakic´enovic´ et al., 2000). This range is compared to the expected range of uncertainty due to the differences in the response of several AOGCMs. Forcing uncertainties are not considered in these calculations.

While one is entitled to one’s opinion that the IPCC should have shown a short term projection with the relevant error bands, that was not their goal — and criticizing what they have shown based on the assumption that it shows something that it does not is certainly not valid criticism.

It is critical to consider what the IPCC intended with their projections. As indicated by the text and graphics, that was to show temperature change over the long(multi-decade) term given a certain set of emissions and climate sensitivity assumptions.

Their purpose was not the same as a a projection run by a scientist to test a climate model and if one assesses the IPCC projections based on that, one is simply barking up the wrong tree.
Johan i Kanada // March 26, 2008 at 3:25 pm | Reply

elspi,
I’m sorry, but your response:

” “1) Why would there be linear trend at all”
BECAUSE EVERY DIFFERENTIABLE FUNCTION IS LINEAR IN THE SMALL”

is not even wrong.

Tamino,
“Any two start and end points, within the time span for which the data are indistinguishable from linear + noise, will of course give different estimates of the trend rate. But the *error range* will almost always include the true value.”

The point is that the true value depends on the time span selected, i.e. there is not one true trend value, valid for all possible time spans. Why would there be?

Compare e.g. the trends:
- from 1900 – 2008
- from 1935 – 2008
- from 1975 – 2008
- from 1998 – 2008
Are you suggesing that there is one true trend value valid for all these time spans?

[Response: No. The data from 1900 to the present include a lot more variation that that from 1975 to the present (which is what I was considering), variation which is not just noise but signal as well. For the 1900-present time span, the *nonlinear* part of the signal is big enough to emerge from the noise; hence using only linear regression to estimate the trend doesn't properly characterized the signal. So the 1975-present data don't show nonlinear signal, but the 1900-present data do.

Even for the 1900-present time span, one can still use a linear model to quantify the *overall* behavior (which some might even call a trend); temperature has increased, not decreased. But to characterize the temperature signal 1900-present as a linear progression is an error.

And elspi does have a point: it's a property of differentiable functions that as one examines smaller and smaller time spans, the *size* of the nonlinear part of the signal (as measured by its total variation) also gets smaller and smaller in proportion to the size of the linear part of the signal.]
kim // March 26, 2008 at 3:29 pm | Reply

Heh, George at 2:55, please don’t tell the politicians that these are just projections from a model. They’ve been led to believe that they are predictions with physical validity. Are you trying to knock over their plans for wrenching economic changes with a feather?
===============================
fred // March 26, 2008 at 4:22 pm | Reply

BPL, is this not a different question? The question of sample size required to get a representation of the population sampled is one thing, and the answer is indeed around 30. But this is not what Lucia is doing at all.

Imagine a 2000 forecast of cooling at the average rate of 1 degree per decade over the next 50 years. You could not seriously argue that we cannot tell anything about its validity until we have 30 annual observations. Nor that the number of observations we need is the same, whether the forecast is for a 1 degree per decade cooling or a 2 or 3 degree per decade cooling.

One would have thought that events since 2000 can be shown using Lucia’s method to show that a forecast of 2 or 3 degree per decade cooling is falsified at the 99 percent level (haven’t done the math). Well, similarly, what she is doing is statistically legitimate also.
J // March 26, 2008 at 4:56 pm | Reply

Fred:

Let’s imagine, for the sake of argument, that the monthly surface temp anomalies during the next five months happen to show a steady increase with little variance. This isn’t a far-fetched supposition; in fact, this exact thing happened in July-November 1997. At the time, we were going into an El Nino, now we’re coming out of a La Nina.

Simple statistical analysis of those five months’ data shows a trend that is absurdly high (+9 C/decade), and is statistically significant at 95% (actually, significant even at a higher confidence level…)

Assume that the error bars are calculated to compensate for autocorrelation, and that all the statistical i’s are dotted and t’s are crossed.

Would you conclude that the IPCC is radically underestimating warming?

I wouldn’t.
fred // March 26, 2008 at 6:34 pm | Reply

J, its the year 2000. I make my prediction of the warmth or coolness in 2050. Say it is average temp up or down by 10 degrees by then. Are you really saying that nothing that happens between 2000 and 2010 can tell us anything about how likely my prediction is to be true?
TCO // March 26, 2008 at 8:30 pm | Reply

Fred: You answered a question with a question. That is a habit of debate rather than a habit of analysis and engagement. Instead of “counting on what point you think your opponent is driving at”, just ANSWER THE QUESTION. Wait for him to make a false further inference (who knows, maybe he won’t!?) In any case, answering a question with a question is annoying.
dhogaza // March 26, 2008 at 8:55 pm | Reply

Still no correction by Mosher of his misrepresentation of HB’s “bet” over at lucia’s blog …

Glad I didn’t hold my breath.
Johan i Kanada // March 26, 2008 at 11:32 pm | Reply

Tamino,

1) Regarding “The data from 1900 to the present include a lot more variation that that from 1975 to the present (which is what I was considering), variation which is not just noise but signal as well. “:
- So, putting it slightly provocatively, you simply selected a time span in which the trend seemed linear, and showed statistically that it was indeed linear + noise? So what, exactly, have we learned?

[Response: I didn't pick 1975 out of a hat. There are mathematical analyses which can identify likely "turning points" in a time series, and they indicate 1975. What we learn is that from 1975 to the present, the signal is indistinguishable from linear + noise, and that it's increasing, not decreasing.]

- Do the same analysis for the time span 1940 to 1975. Very likely you can identify a completely flat or slightly negative linear trend there, with pretty much the same certainty as for the 1975-2008 numbers. So what does that tell us?

[Response: I've done so, and it's flat. That tells us that the net effect of climate forcings during that time period was temperature stability.]

- As an AGW proponent (presumably), what you need to do is to look at the temperature data from let’s say 1900 (when the CO2 was at the “natural” level) until now, which likely will contain several superimposed signals (linear and/or otherwise, including also noise), and try to identify a trend (of any type) that can be reasonably connected with the recorded CO2 levels (causality, not just correlation, albeit the latter would be a good start) .

[Response: Trends in global temperature should be related to *climate forcing*, not just CO2. That includes below-average volcanic activity in the early 20th century, changes in solar activity, high sulfate emissions mid-century, and of course the CO2 and other greenhouse gases accumulating over the entire 20th century.

There are lots of researchers who do exactly that. Their computer models include the physics of all the aforementioned climate forcings. Hindcasts from those models only match observations if the warming influence of greenhouse gases (including but not limited to CO2) is included. Forecasts from those models indicate 2 to 4.5 deg.C warming this century. You can read about it in the IPCC reports.]

2) Regarding “it’s a property of differentiable functions …”. The point is that we are not looking at a mathematical differentiable function, we are looking at a time series of a pseudo-physical property values (calculated global temperature values).
Further, there is absolutely no a priori reason to assume that these property values should experience a multi-decadal exclusively linear trend.

[Response: Global temperature includes the impact of both external and internal variations. The variations due to external causes are a mathematical differentiable function. And nobody ever claimed "a priori" that the time evolution would be linear. The *data* say that for the last 30 years or so, it actually has been, within the limits of our ability to isolate the long-term evolution. But as I said earlier, although there's every reason to believe it's *not* linear, the nonlinear part of the signal hasn't risen above the noise.]

Remember what Einstein said: “As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.”
Johan i Kanada // March 27, 2008 at 4:18 am | Reply

Tamino,
“nobody ever claimed “a priori” that the time evolution would be linear”:

Well, DocMartyn (and perhaps others) was ridiculed for suggesting that assuming a linear trend was stupid. And I was admonished for not knowing calculus. Finally, you stated adamantly that the data proved a linear trend. Ok, I understand now that your statement refered to data from 1975 to present only, but that was certainly not clear in the subsequent debate.

I think we have beaten this horse to death now… no more fruitful arguments seem likely.

Thanks,
/Johan
Barton Paul Levenson // March 27, 2008 at 1:08 pm | Reply

fred writes:

[[Imagine a 2000 forecast of cooling at the average rate of 1 degree per decade over the next 50 years. You could not seriously argue that we cannot tell anything about its validity until we have 30 annual observations. ]]

Yes I could. In fact, I would. I certainly wouldn’t say that I could falsify the prediction with the first 7 years of observations.
Johan i Kanada // March 27, 2008 at 1:33 pm | Reply

Tamino,

Sorry, but I have a couple of additional comments.

1) You write that the data since 1975 consists “a linear progression + autocorrelated noise”
Noise, by definition, contains no information. But we know that the signal in question contains e.g. clear information about El Nino.
Hence, your statement cannot be correct.
Instead, it seems to me that your calculations have shown that there is linear *component* from 1975 with a certain value and error interval, but surely you have *not* shown that there is no other information (signal components) in the signal. (That would be un-physical, wouldn’t it?)

[Response: You're really grasping at straws. El Nino/la Nina don't contribute to climate *trend*, just it's fluctuations. Hence they give zero information about the *trend*, but certainly tell us a lot of information about ocean temperature in the Pacific.

You've also, once again, mischaracterized what I said. I've repeatedly made it clear that the data since 1975 are *indistinguishable* from a linear progression + autocorrelated noise. This means that whatever nonlinear signal is present (which I've asserted more than once is overwhelmingly likely to exist) is too small to be detected -- let alone quantified -- above the noise.]

2) If you look at your last illustration, doesn’t it actually indicate, quite clearly, that it is reasonably probable that there has been a change in the linear trend value around 2001?
In the same way you agree that 1940-1975 has a different linear trend value than 1975 to present, doesn’t it seem reasonable that 1975-2001 has one trend value, which is different from 2001-2008?
(Btw, if you calculate the trend value from 1975-2001, I bet the value is slightly different and the error interval significantly smaller, as compared to the 1975-2008 calculation.)

[Response: No. Comparing 1940-1975 with 1975-present, the error ranges (even when done rigorously) don't overlap; there's no value which is plausible for both time frames, so we conclude that the trend is different.]

Finally, a methodology question: When estimating the error interval, is it then intrinsically the case that a shorter time series gets a wider error interval than a long?

[Response: Usually yes, although there are exceptions. If the intrinsic nature of the noise changes, then we may get greater precision form a shorter time span.]

Thanks,
/Johan

[Response: It's starting to look like whatever I say, however many objections I respond to, you'll come up with yet *another* erroneous objection to occupy my time. I'm beginning to doubt your sincerity. Don't expect that pattern to continue.]
Johan i Kanada // March 27, 2008 at 4:02 pm | Reply

Tamino,

Perhaps it is a matter of semantics. Your definition of “noise” is any signal component that is not a linear trend, whereas my definition of “noise” is data without meaning/information (and the signal, temperature in this case, can thus consist of a number signal components, linear or otherwise, plus random noise).
(So e.g. I would not characterize a real (and physical) temperature fluctuation as noise.)

[Response: No! My definition of "noise" is that which is random. It's also legitimate to treat processes which we don't know how to predict despite their deterministic nature (like volcanic eruptions) as noise, when their statistical behavior is sufficiently like noise. And I repeat -- yet again -- that nonlinear signal is almost *surely* present in long-term temperature evolution, but it's small enough over the time span in question that not only is it hard to detect, it's impact on the progression of temperature can safely be neglected. How many more times will I have to repeat it?]

Regarding the 2001-2008 trend: Sure, the calculated error range includes also the calculated 1975-2008 trend value (at the fringe), but the likelihood of it actually being the same seems small. I.e. I fully understand that the the trend values have not been proven to be different, but that doesn’t mean they are the same.

Come to think of it, perhaps the above is crux of argument, i.e. “not disproven” does not mean “proven”.

(Possibly you have also stated that somewhere, I am not sure. In any case, this is not a criticism of you or your article, rather a clarification (at least for myself).)

Thanks,
/Johan
Johan i Kanada // March 27, 2008 at 7:59 pm | Reply

Tamino,

Final (?) attempt to find peace…

We agree, I think, that the linear trend during 1975-2008 differs from the linear trend during 1940-1975.

That means that during the 1940-2008 time period the temperature trend can reasonably be described by two linear segments.

Haven’t we now in fact defined a non-linear trend curve for the time period in question (1940- present)?

Further, it does not make physical sense to assume that the trend is fixed at A deg/dec from 1940-1975, at which time it suddenly changes to B deg/dec. (There are very few discontinuities in nature, why would temperature trend be one?)

Therefore, the piece-wise linear trend, according to my thinking, *approximates* the “real” physical temperature trend curve, which may or may not be linear. (It could be exponential, logarithmic, polynomial, chaotic, linear, a combination thereof, or whatever.)

[Response: I agree.]
fred // March 28, 2008 at 7:03 am | Reply

BPL, the problem with your argument is, it ends up proving rather too much. It ends up proving that no matter what happens to global temps for the next 20 years, we can legitimately conclude nothing about whether there is a long term warming, cooling, or unchanging trend.

Paradoxically, its precisely the argument the people you call ‘deniers’ make: that we need to wait for more data. You really cannot rescue the prediction that there is going to be catastrophic warming, and the concomittant argument that we must act now, by adding that we cannot know whether this warming is happening until 2030.

We could equally well make the proposition that there is going to be catastrophic cooling (or catastrophic stasis), and must act differently now to avert it, but we will not be able to disprove the cooling or the stasis either till 2030.

I think something rather at right angles to this. Regardless of whether CO2 warms catastrophically, we should still take steps to eliminate pollution and environmental wreckage, the chief contributors to which are the use of the automobile for mass transport, and current farming practices. The CO2 thesis is very important, but excessive focus on this gets in the way of something equally important in the way of issues on quality of life for its own sake.
kim // March 28, 2008 at 12:56 pm | Reply

I like ‘catastrophic stasis’. Ironically, in fact, climate stasis would be problematic. Changing biological niches help drive evolution. We would not be here but for climate change.

Furthermore, mistaken focus on CO2 is going to be catastrophic for science if, in fact, CO2 has a much smaller warming component than generally predicted. Present use of fossil carbon is critical for the lives of all of us, not just the poor and powerless of the world.

Science itself is in danger. When Gore can call skeptics the equivalent of flat earthers, when he himself may be the fool, may lead to an antiscientific reaction among the general population, if and when they realize that they have been asked to make huge sacrifices in pursuit of a chimera.

There is no question we have to keep the earth clean. You have to keep your room clean, too.
===============================
dhogaza // March 28, 2008 at 1:58 pm | Reply

Fred misrepresents BPL thusly…

It ends up proving that no matter what happens to global temps for the next 20 years, we can legitimately conclude nothing about whether there is a long term warming, cooling, or unchanging trend.

No, fred, saying “7 years isn’t long enough to reliably differentiate signal from noise” does not lead to the conclusion that 20 years also isn’t long enough.

Good grief.
TCO // March 28, 2008 at 10:01 pm | Reply

Plus there are the previous 30 years. But sheesh. Why do I have to keep making arguments that help the alarmists. I really am a napalm-spewing, commie-hating, ANWAR-drilling-loving conservative.

I just want our side to use logic. And to embrace truth. And to embrace curiosity. Zorita is one of my hopes.
S2 // March 28, 2008 at 11:02 pm | Reply

[Response: Perhaps the conceptually simplest way is to notice the AR(1) equation is similar to the linear regression equation ....

Got it. Thanks.
Barton Paul Levenson // March 29, 2008 at 12:06 pm | Reply

fred writes:

[[ You really cannot rescue the prediction that there is going to be catastrophic warming, and the concomittant argument that we must act now, by adding that we cannot know whether this warming is happening until 2030.]]

Nor do I. Unlike you, I don’t start the count from 2001. There’s data going back to 1850.
kim // March 30, 2008 at 4:16 am | Reply

BPL, how well does temperature correlate with CO2 level ever since 1850?
==========================
Barton Paul Levenson // April 1, 2008 at 1:09 pm | Reply

kim writes:

[[BPL, how well does temperature correlate with CO2 level ever since 1850?]]

I don’t have the Hadley Centre figures offhand, which go back to 1850, but using the NASA GISS figures from 1880 to 2007, the Pearson’s product-moment correlation coefficient is r = 0.899 (coefficient of determination indicates 80.9% of variance is accounted for). I’d like to say from that that CO2 is driving temperature — I believe it is — but in fact a correlation isn’t enough evidence to prove that. I would need to check for unit roots, cointegration, etc., and maybe perform tests for Granger causality.
Gavin's Pussycat // April 1, 2008 at 3:10 pm | Reply

I just want our side to use logic. And to embrace truth. And to embrace curiosity.

TCO, so did Gorbachev :-)
Lazar // April 4, 2008 at 10:28 am | Reply

HB,

Suppose x and y are two time series. If they are fitted by a linear regression of y against x, then how would significance levels be adjusted if both series are autocorrelated?
Is this something related to pre whitening?

Thanks.

[Response: No, it's not related to pre-whitening. That refers to finding a signal component, then removing it in order to look from more signal.

There's more than one way to skin the autocorrelation cat. For example, one can try to estimate the number of "effective degrees of freedom" in the system, and adjust the statistics accordingly. I much prefer to compute directly the impact of the autocorrelation on the expected variance of the correlation coefficient. If the autocorrelation structure of the two series is similar, then this can be greatly affected. It would probably make a worthwhile topic for a post.

One very great danger is that when you smooth data, the smoothing process itself imposes autocorrelation. If you smooth two time series in exactly the same way, then you impose very similar autocorrelation structure on them. Hence computing the correlation between two time series which have been smoothed in the same way introduces sizeable spurious correlation (for an example see this).]
Phil Scadden // June 11, 2008 at 3:04 am | Reply

Old thread but interesting. What if err(t) is autocorrelated AND also a function Abs(x(t)). ie
magnitude of random variability increases with say the temperature. least squares models are going to be biased to minimise error at high end? I am postulating you MIGHT see greater random variability in T as T increases and wonder whether this can be modelled.
cshme // July 13, 2008 at 8:31 am | Reply

Tamino, when I reproduce your analysis I used an iterative scheme to calculate rho. Starting from the first value of rho and transforming variables, the iteration becomes unstable after 4 loops. What do you think that result could mean? When I set initial rho=0, the iteration converges nicely. Strange. BTW, your presentation on autocorrelation is best and most clear that I’ve seen after searching multiple search engines for the past few days. Thanks.
Adrian Kerton // March 23, 2009 at 9:54 am | Reply

I have correlated positions of the magnetic poles with temperature over the last 100 years and found a very close correlation. What statistical tests can I use to show I do not have chance correlation? I used the Spearman coefficient but is there a better method?

Climate Change and the Earth’s Magnetic Poles, A Possible Connection
Author: Kerton, Adrian K.
Source: Energy & Environment, Volume 20, Numbers 1-2, January 2009 , pp. 75-83(9)
Publisher: Multi-Science Publishing Co Ltd

Abstract:
Many natural mechanisms have been proposed for climate change during the past millennia, however, none of these appears to have accounted for the change in global temperature seen over the second half of the last century. As such the rise in temperature has been attributed to man made mechanisms. Analysis of the movement of the Earth’s magnetic poles over the last 105 years demonstrates strong correlations between the position of the north magnetic, and geomagnetic poles, and both northern hemisphere and global temperatures. Although these correlations are surprising, a statistical analysis shows there is a less than one percent chance they are random, but it is not clear how movements of the poles affect climate. Links between changes in the Earth’s magnetic field and climate change, have been proposed previously although the exact mechanism is disputed. These include: The Earth’s magnetic field affects the energy transfer rates from the solar wind to the Earth’s atmosphere which in turn affects the North Atlantic Oscillation. Movement of the poles changes the geographic distribution of galactic and solar cosmic rays, moving them to particularly climate sensitive areas. Changes in distribution of ultraviolet rays resulting from the movement of the magnetic field, may result in increases in the death rates of carbon sinking oceanic plant life such as phytoplankton.

[Response: Hahahahahahahahaha! Funny! But April Fool's day is more than a week away.]
Ray Ladbury // March 23, 2009 at 1:26 pm | Reply

Adrian Kerton, your assertion that there is less than a 1% probability of random match presumes a statistical model–what model are you using and why do you believe this is the correct statistical model. Second, none of your proposed mechanisms makes any sense–e.g. how would movement of the geomagnetic poles alter UV fluxes? Third, it’s not even clear what you mean by “movement of the pole” and how you would do a correlation. Fourth, have you looked at the paleoclimate for past correlations? Finally, you are completely wrong in your attribution of climate change. The greenhouse effect is well established physics. If it did not exist, we wouldn’t be here to write this. So we KNOW that greenhouse gasses warm the planet. This was posited back ~1850, long before evidence of warming was well established. It was a prediction of climate science that has been observed to be true.

Tamino is right. This is an idea that really doesn’t make sense from a physical standpoint. Even if you were to find a physical mechanism, you’d still have to explain why adding a greenhouse gas to the atmosphere would fail to warm the planet.
Hank Roberts // March 23, 2009 at 10:38 pm | Reply

Both changed. Oh, wow.
http://imgs.xkcd.com/comics/correlation.png
Adrian Kerton // March 24, 2009 at 2:18 pm | Reply

to Hahaha,

I’m not sure I understand your comment, the correlation is there, try it yourself from the published data on the web. Whether or not is it just chance of course is a different matter, however there are plenty of paleomagnetic studies linking changes in the Earth’s magnetic field with climate change, do you think they are all a joke as well?

[Response: It's also correlated to historical beer sales, and the decline in pirate activity.

It's your critical thinking skills that are the joke.]
Adrian Kerton // March 24, 2009 at 10:21 pm | Reply

I’ll just reply on one topic mentioned :

“Third, it’s not even clear what you mean by “movement of the pole” and how you would do a correlation.”

Movement of the magnetic poles, it’s been happening for a long time now, if you can’t find the references then I really don’t know what to say. Correlation, just use the references in my paper. They are all on the internet. How to correlate, quite simply put both sets of data into a spread sheet and use the correlation functions.

Greenhouse gases undoubtedly contribute to warming but how much? I could not find a correlation between CO2 and temperature that was credible, though I did find lots of assumptions.

Byeee!
dhogaza // March 25, 2009 at 12:09 am | Reply

Even if you were to find a physical mechanism, you’d still have to explain why adding a greenhouse gas to the atmosphere would fail to warm the planet.

Adrian has it backwards, climate change is the cause of the magnetic poles wandering around, not vice-versa! :)
adrian kerton // March 25, 2009 at 11:20 am | Reply

Interesting comments.
I originally came here asking a question about how to correlate and now I get comments like “correlated to historical beer sales” Doesn’t answer my original question and hardly helpful, nor I suggest indicative of an open mind. We know greenhouse gases warm the atmosphere but the question is how much? I haven’t seen anything that correlates the temperture changes with actual emissions, I see plenty of computer models based on assumptions.

As to”how would movement of the geomagnetic poles alter UV fluxes?” read the original paper that proposes it.

I’m really dissapointed in that I came asking for help but all I get is abuse, hardly scientific debate which seems to characterise the pro carbon lobby.

[Response: You are a liar. Your motive was to push your theory about correlation between magnetic pole movement and temperature. You even posted the abstract from your paper.

You also stated that you can't find a reliable correlation between CO2 and temperature. You must not be trying. You did manage to throw in a ridiculous and very insulting comments about not seeing anything but "computer models based on assumptions."

And now you pretend to be wounded because I laughed at your laughable theory. You protest that you were "only asking a question." Liar. You came here to assassinate real climate science and confuse the unknowing with your truly laughable garbage about movement of the geomagnetic pole.

Having an open mind does NOT mean lifting off your skull and throwing your brain in the garbage bin. It certainly does not mean giving credence to crackpots.]
Ray Ladbury // March 25, 2009 at 3:36 pm | Reply

Adrain, I hardly think you could classify my comments as abusive. Indeed, I was trying to be helpful. It seems to me you came here looking for abuse so that you could play the martyr elsewhere.
The problem is that you are looking for correlation without having a firm idea of the mechanism. Yes, it is true that large changes in the geomagnetic field would change the terrestrial radiation environment. However, it is difficult for me to understand what mechansim would have a substantive effect on radiation levels given the rather modest changes we’ve seen recently, and I can’t think of any mechanism whereby UV levels would change as a result of geomagnetism.

What is more, I have to agree with Tamino that your failure to familiarize yourself with the evidence for a greenhouse mechanism in the current warming epoch:
1)does not imply the nonexistence of said evidence
2)does not constitute a ringing endoesement of your competence or credibility since said evidence is readily apparent from even a cursory glance at the peer-reviewed literature.

So, if your goal is to be a martyr, fine. Congrats. On the other hand if you really want to understand climate and statistical analysis, many of us here would be happy to provide you with all the references you might require. Start with
http://www.aip.org/history/climate/
Deech56 // March 25, 2009 at 7:05 pm | Reply

This may be fairly simple; not yet complete:

http://moregrumbinescience.blogspot.com/2009/03/does-co2-correlate-with-temperature.html
Philippe Chantreau // March 25, 2009 at 9:48 pm | Reply

Adrian, how does the solar wind transfer energy to the Earth’ atmosphere? I’m very curious about that physical mechanism and the quantities involved.
TCO // March 25, 2009 at 11:26 pm | Reply

Adrian:

With all due respect.

1. Why did you “publish” in EE, when you have such a fundamental question?

2. Try a university and talk to some statisticians.

3. Post the question in the Open Thread.

P.s. Tammy will still rip on you, so at least follow 1 and 2…
Philippe Chantreau // March 26, 2009 at 5:03 am | Reply

Adrian, you should follow TCO’s advice. He’s a genuine skeptic, and as such banned on some “skeptic” sites, but not here.

I’m still curious about that solar wind thing, btw.
Ray Ladbury // March 26, 2009 at 12:47 pm | Reply

Philippe, The couplings between the solar wind and the atmosphere are rather indirect. Only for very large solar particle events (and at the poles–e.g. auroras) do you get actual particle fluxes into the atmosphere. The largest known solar particle event–the Carrington event in 1851–brought auroras visible as far south as Havana! As a result, those advocating such couplings have had to get creative–positing indirect effects that modulate larger forcers. The solar wind can affect Earth’s magnetic field, which can in turn affect the galactic cosmic ray flux–but these aren’t large fluxes to begin with. The idea is that the GCR flux then modulates cloudcover–an idea that is currently unsupported by any evidence. There are other proposed mechanisms, but they’re all equally vague and unsupported.
Hank Roberts // March 26, 2009 at 3:14 pm | Reply

> How to correlate, quite simply put both sets of
> data into a spread sheet and use the correlation
> functions.

A college-level introductory statistics course will get you past this confusion.

Look, you can “correlate” any two sets of data.
Look up the correlation of “full moon” with any number of different things, you’ll find it.

You clearly expect to find something specific.

That expectation changes the statistical test you need to use. Understand this?
David B. Benson // March 26, 2009 at 9:45 pm | Reply

Deech56 // March 25, 2009 at 7:05 pm — Nice, but you’ll do better using ln(CO2) ala Arrenhius:

http://www.geocities.com/bpl1960/Correlation.html
bluegrue // March 26, 2009 at 11:48 pm | Reply

@adrian kerton

You have posted your abstract and figure 1 on your homepage

The axis label claims that you use “normalized” values. Is there any specific reason, why “North Pole Lattitude” starts out 0, whereas “North Pole Longitude” and “Temp. Trend” (smoothed 1year/5year/10year/whatever?) start out at 0.2? Any specific reason, why they intersect at 0.95 instead of 1? You are aware, that normalizing the data the way you did you introduce a correlation? Should I wonder, why peer review in E&E did not catch these obvious flaws?
Philippe Chantreau // March 27, 2009 at 8:53 am | Reply

Thx Ray, it confirms what I thought. My question was kinda loaded, I couldn’t see any plausible mechanism.
tomek // June 27, 2009 at 10:43 pm | Reply

Hi,
I was wondering about other type of data series, namely CO2 concentration, yearly means. I took the 45 yearly menas of CO2 conc. at Mauan Loa. Made the lag-1 autocorrelation and I end up with very high value for lag-1 autocorrelation (range of 0.999). It was expected, as this trend is not-so-far from linear for recent 4-5 decades. Anyway, using the equation you provided for Neff I obtained the number of DOF = 0.07 (!). Such a low value of DOF extremally extends the ‘possible error’, so the trend is (in ppmv per year): 1.36 +- 6.69. This result (error) is unaccpetable for the considered recent CO2 conc. trend. My question to experts and hobbiests is: where is the mistake made? Many thanks for any advice.
Tomek

[Response: You must have made a computational error, because I only get a lag-1 autocorrelation of 0.94; using that value and the 50 annual means available gives a rate of 1.43 +/- 0.27. And since the trend is significant, the confidence limits should be based on the autocorrelation of the residuals, not the data, which is only 0.87, so the trend rate over the 50 years is 1.43 +/- 0.18.

Did you remember to subtract the mean value from all the data before estimating the autocorrelation? If you work with original data rather than residuals, this is essential.]
tomek // June 28, 2009 at 10:05 pm | Reply

Hi there,
Many thanks for the answer on my previous question. Managed to get rid of the bug. I have one more problem to be solved, and I am not sure how to deal with it.
Let’s consider I have two climate runs, 100 years each. For simplification let’s say that I am interested only in annual, global mean near-surface temperatures. These two runs differ by a single forcing (in eg. one run contains the aerosol scheme, second run does not). The problem to solve is: ‘does the aerosol significantly changes the temperature trend over last 100 years?’.
My approach would be:
Calculate autocorrelations for both runs, then Neff, DOF etc., all these to end up with trends and error bands for both runs. If these trends, extended by error bands do overlap – there is no significant difference between both runs.
Am I right?

[Response: Not really. The "null hypothesis" is that the runs are equivalent (except for noise); if so then they'll give equal trends (within confidence limits). But getting that result doesn't confirm the null hypothesis, it just fails to reject it. There may be differences between the runs that don't affect the trend, or at least not much, and those differences won't affect the comparison of trends.

Also, when comparing trends you're comparing two estimates with their uncertainties, but that's not equivalent to testing whether their error ranges overlap. For the confidence limits to just barely overlap by chance you have to get an extreme value for one run (maybe only a 5% chance) AND for the other run too (another 5% chance) and the probability of *both* accidents is 5% x 5% = only 0.25%. Look up "t test" on wikipedia, particularly the version for comparing two means, and you might be able to figure out how to do the comparison correctly.]

To finally complicate the case I have to add that these two types of climate-model runs are ensembles with several members each. How to detect if there is significant difference between these two runs?
In the other words, how to combine the autocorrelation of time-series test with multiple runs?

[Response: One simple way is to treat the ensemble average as a single time series.]

If anyone could point me the way/appropriate reference I would be grateful.
Thanks in advance,
Tomek

	Timothy Chase on Breaking Records
	Timothy Chase on Breaking Records
	jyyh on Vapor Lock
	Hank Roberts on Vapor Lock
	Hank Roberts on Vapor Lock
	Hank Roberts on Vapor Lock
	Timothy Chase on Vapor Lock
	Ian Forrester on Vapor Lock
	Ray Ladbury on Vapor Lock
	suricat on Vapor Lock
	suricat on Vapor Lock
	David B. Benson on Open Thread #14
	MikeN on Open Thread #14
	freemike on Open Thread #14
	chriscolose on Vapor Lock

Open Mind