February 9, 2008 | 14 Comments
It’s already well known that the Remote Sensing Systems satellite-derived temperature data has released the January figures: the finding is that it’s colder this January than it has been for some time. I wanted to look more carefully at this data, mostly to show how to avoid some common pitfalls when analyzing time series data, but also to show you that temperatures are not linearly increasing. (Readers Steve Hempell and Joe Daleo helped me get the data.)
First, the global average. The RSS satellite actually divides up the earth in swaths, or transects, which are bands across the earth whose widths vary as a function of the instrument that remotely senses the temperature. The temperature measured at any transect is, of course, subject to many kinds of errors, which must be corrected for. Although this is not the main point of this article, it is important to keep in mind that the number you see released by RSS is only an estimate of the true temperature. It’s a good one, but it does have error (usually depending on the location of the transect), which most of us never see or few actually use. That error, however, is extremely important to take into account when making statements like “The RSS data shows there’s a 90% chance it’s getting warmer.” Well, it might be 90% before taking into account the temperature error: afterwards, the probability might go down to, say, 75% (this is just an illustration; but no matter what, the original probability estimate will always go down).
Most people show the global average data, which is interesting, but taking an average is, of course, assuming a certain kind of statistical model is valid: one that says averaging transects gives an unbiased, low-variance estimate of the global temperature. Does that model hold? Maybe; I actually don’t know, but I have my suspicions it does not, which I’ll outline below.
So let’s look at the transects themselves. The ones I used are not perfect, but they are reasonable. They are: “Antarctica” (-60 to -70 degrees of latitude; obviously not the whole south pole), “Southern Hemisphere Extratropics” (-70 to -20 degrees; there is a 10 degree overlap with “Antarctica”), “Tropics” (-20 to 20 degrees), “Northern Hemisphere Extratropics” (20 to 82.5 degrees, a slightly wider transect than in the SH), and “Arctic” (60 to 82.5 degrees; there is a 22.5 degree overlap with NH Extratropics). Ideally, there would be no overlap between transects, global coverage would have been complete, and I would have preferred more instead of fewer transects, which would have allowed us to see greater detail. But we’ll work with what we have.
Here is the thumbnail of the transects. Click it (preferably open it in a new window so you can follow the discussion) to open the full-sized version.
All the transects are in one place, making it easy to do some comparisons. The scale for each is identical, each has only been shifted up or down so that they all fit on one picture. This is not a very sexy or colorful graph, but it is useful. First, each transect is shown with respect to its mean (the small, dashed line). Vertical lines have been placed at the maximum temperature for each. The peak for NH-Extratropics and Tropics was back in 1998 (a strong El Nino year). For the Arctic, the peak was in 1995. For the Antarctic, it was 1990. Finally, for the SH-Extratropics it was 1981.
You also often see, what I have drawn on the plot, a simple regression line (dash-dotted line), whose intent is usually to show a trend. Here, it appears that there were upward trends for the Tropics to the north pole, no sort of trend for the SH-Extratropics, and a downward trend for the Antarctic (recall there is overlap between the last two transects). Supposing these trends are real, they have to be explained. The obvious story is to say man-made increases due to CO2, etc. But it might also be that the northern hemisphere is measured differently (more coverage), or because there is obviously more land mass in the NH, and—don’t pooh-pooh this—the change of the tilt of the earth: the north pole tipped closer to the sun and so was warmer, the south pole tipped farther and so was cooler. Well, it’s true that the earth’s tilt has changed, and will always do so no matter what political party holds office, but effects due to its change are thought to be trivial at this time scale. Of course, there are other possibilities such as natural variation (which is sort of a cop out; what does “natural” mean anyway?).
To the eye, for example, the trend-regression for the Arctic looks good: there is an increase. Some people showing this data night calculate a classical test of significance (don’t get me started on these), but this is where most analysis usually stops. It shouldn’t. We need to ask, what we always need to ask when we fit a statistical model: how well does it fit? The first thing we can do is to collect the residuals, which are the distances between the model’s predictions and the actual data. What we’d like to see is that there is no “signal” or structure in these residuals, meaning that the model did its job of finding all the signal that there was. The only thing left after the model should be noise. A cursory glance at the classical model diagnostics would even show you, in this case, that the model is doing OK. But let’s do more. Below is a thumb-nail picture of two diagnostics that should always be examined for time series models (click for larger).
The bottom plot is a time-series plot of the residuals (the regression line minus the observed temperatures). Something called a non-parametric (loess) smoothing line is over-plotted. It is showing that there is some kind of cyclicity, or semi-periodic signal, left in the residuals. This is backed up by examining the top plot: which is the auto-correlation function. Each time-series residual is correlated with the one before it (lag 1), with the one two before it (lag 2), and so on. The lag-one correlation is almost 40%, again meaning that the residuals are certainly correlated, and that some signal is left in the residuals that the model didn’t capture. (The “lag 0” is always 1; the horizontal-dashed lines indicated classical 95% significance; the correlations have to reach above these lines to be significant.)
The gist is that the ordinary regression line is inadequate and we have to search for something better. We might try the non-parametric smoothing line for each series, which would be OK, but it is still difficult to ask whether trends exist in the data. Some kind of smoothing would be good, however, to avoid the visual distraction of the noise. We could, as many do, use a running mean, but I hate them and here is why.
Show in black is a pseudo-temperature series with noise: the actual temperature is dashed blue. Suppose you wanted to get rid of the noise using a “9-year ” running mean: the result is the orange line, which you can see does poorly, and shifts the actual peaks and troughs to the right. Well, that is only the start of the troubles, but I don’t go over any more here except to say that this technique is often misused, especially in hurricanes (two weeks ago a paper in Nature did just this sort of thing).
So what do we use? Another thing to try is something called Fourier, or spectral analysis, which is perfect for periodic data. This would be just the thing if the periodicities in the data were regular. They do not appear to be. We can take one step higher and use something called wavelet analysis, which is like spectral analysis (which I realize I did not explain), but instead of analyzing the time series globally like Fourier analysis, it does so locally. Which means it tends to under-smooth the data, and even allows some of the noise to “sneak through” in spots. This will be clearer when we look at this picture (again, just a thumb-nail: click for larger).
You can see what I mean by some of the original noise “sneaking through”: these are the spikes left over after the smoothing; however, you can also see that the spikes line up with the data, so we are not introducing any noise. The somewhat jaggy nature of the “smoothed” series has to do with the technicalities of using wavelets (I’ll have to explain this a latter time: but for technical completeness, I used a Daubechies orthonormal compactly supported wavelet, with soft probability thresholding by level). Anyway, some things that were hidden before are now clearer.
It looks like there was an increasing trend for most of the series starting in 1996 to 1998, but ending in late 2004, after which the data begin trending down: for the tropics to north pole, anyway. The signal in the southern hemisphere is weaker, or even non-existent at Antarctica.
This analysis is much stronger than the regression shown earlier; nevertheless, it is still not wonderful. The residuals don’t look much better, and are even worse in some places (e.g. early on in the Tropics), than the regression. But wavelet analysis is tricky: there are lots of choices of the so-called wavelet basis (the “Daubechies” thing above) and choices for thresholding. (I used the, more or less, defaults in the
R wavethresh package.)
But the smoothing is only a first start. We need to model this data all at once, and not transect by transect, taking into account the different relationships between each transect (I’m still working on a multivariate Bayesian hierarchical time-series model: it ain’t easy!). Not surprisingly, these relationships are not constant (shown below for fun). The main point is that modeling data of this type is difficult, and it is far too tempting to make claims that do not hold up upon closer analysis. One thing is certain: the hypothesis that the temperature is linearly increasing everywhere across the globe is just not true.
APPENDIX: Just for fun, here is a scatter-plot matrix of the data (click for larger): You can see that there is little correlation between the two poles, and more, but less than you would have thought, between bordering transects.
How to read this plot: it’s a scatter plot of each variable (transect), with each other. Pick a variable name. In that row, that variable is the y-axis. Pick another variable. In that column, that variable is the x-axis. This is a convenient way to look at all the data at the same time.