Autocorrelation « Open Mind

Autocorrelation

March 22, 2008 · 148 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

148 responses so far ↓

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

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

Again, an extremely helpful presentation.

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

“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

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

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

“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

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

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

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

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

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

“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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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