By DOUGLAS J. KEENAN WSJ.com

For years, some researchers have argued that the evidence for global warming is not nearly as strong as has been officially claimed. The details of the arguments are often technical. As a result, policy makers and other people outside the debate have relied on the pronouncements of a group of climate scientists. I think that is unnecessary. I believe that what is arguably the most important reason to doubt global warming can be explained in terms that most people can understand.

Consider the graph of global temperatures in Figure 1, which uses data from NASA. At first, it might seem obvious that the graph shows an increase in temperatures. In fact the story is more involved.

Imagine tossing a coin ten times. If the coin came up Heads each time, we would have very significant evidence that the coin was not a fair coin. Suppose instead that the coin was tossed only three times. If the coin came up Heads each time, we would not have significant evidence that the coin was unfair: Getting Heads three times can reasonably occur just by chance.

View figures here

In Figures 2 and 3, each graph has three segments, one segment for each toss of a coin. If the coin came up Heads, then the segment slopes upward; if it came up Tails, then it slopes downward. In Figure 2, the graph on the far left illustrates tossing Heads, Tails, Heads; the middle graph illustrates Tails, Heads, Tails; and the last graph illustrates Heads, Tails, Tails. Figure 3 illustrates Heads, Heads, Heads.

Three Heads is not significant evidence for anything other than random chance occurring. A statistician would say that although the graph shows an increase, the increase is "not significant."

Suppose now that instead of tossing coins, we roll ordinary six-sided dice. We will roll each die three times. If a die comes up 1, we will draw a line segment downward; if it comes up 6, the segment is drawn upward; and if it comes up 2, 3, 4 or 5, the segment is drawn straight across. Figure 4 gives some examples of possible outcomes.

Now consider Figure 5, which corresponds to rolling 6 three times. This outcome will occur by chance just once out of 216 times, and so offers significant evidence that the die is not rolling randomly. That is, the increase shown in Figure 5 is significant.

Note that Figure 3 and Figure 5 look identical. In Figure 3, the increase is not significant; yet in Figure 5, the increase is significant. These examples illustrate that we cannot determine whether a line shows a significant increase just by looking at it. Rather, we must know something about the process that generated the line. But in practice, the process might be very complicated, which can make the determination difficult.

Consider again the graph of global temperatures in Figure 1. We cannot tell if global temperatures are significantly increasing just by looking at the graph. Moreover, the process that generates global temperatures—Earth's climate system—is extremely complicated. Hence determining whether there is a significant increase is likely to be difficult.

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This brings us to the statistical concept of a time series, which is any series of measurements taken at regular time intervals. Examples include prices on the New York Stock Exchange at the close of each business day, the maximum daily temperature in London, the total wheat harvest in Canada each year and the average global temperature each year.

In the analysis of time series, a basic question is how to determine whether a given series is significantly increasing (or decreasing). The mathematics of time-series analysis gives us some tools to do this, requiring us first to state what we believe we know about the series in question. For example, we might state that we believe the series goes up one step whenever a certain coin comes up Heads, and that the series in question comprises three upward steps, as in Figure 3. Next, we must complete some computations based on what we have stated. For example, we compute that the probability of a coin coming up Heads three times in a row is ½ × ½ × ½ = 1/8, or a 12.5% probability of occurring randomly. From that, we conclude that the three upward steps in the coin-toss time series can be reasonably attributed to chance, and thus that the increase shown in Figure 3 is not significant.

Likewise, in order to determine if the global temperature series is increasing significantly, we must first state what we know about what causes those temperature movements. Because our understanding of the dynamics of global temperature is incomplete, we must make some assumptions. As long as the assumptions are reasonable, we can at least be confident that the conclusions drawn from our time-series analysis are reasonable.

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This is standard practice, but is it always adhered to in the work of climate scientists? The latest report from the U.N.'s Intergovernmental Panel on Climate Change (IPCC) was published in 2007. Chapter 3 of Working Group I considers the global temperature series illustrated in Figure 1. The chapter's principal conclusion is that the increase in global temperatures is extremely significant.

To draw that conclusion, the IPCC makes an assumption about the global temperature series, known as the "AR1" assumption, for the statistical concept of "first-order autoregression." That assumption implies, among other things, that only the current value in a time series has a direct effect on the next value. For the global temperature series, it means that this year's temperature affects next year's, but that the temperature in previous years does not. Intuitively, that seems unrealistic.

There are standard checks to (partially) test whether a given time series conforms to a given statistical assumption; if it does not, then any conclusions based on that assumption must be considered unfounded. For example, if the significance of the increase in Figure 5 were computed assuming that the probability of a line segment sloping upward were one in two instead of one in six, then that would lead to an incorrect conclusion. The need for such checks is taught in all introductory courses in time series. The IPCC chapter, however, does not report doing any such checks.

That is a startling omission, one with consequences for how the IPCC's recommendations should be interpreted. A fairly elementary alternative assumption that some researchers and I have tested fits the actual temperature data better than the IPCC's AR1 assumption—so much better that we can conclude that the IPCC's assumption has no support. Under the alternative assumption, the data do not show a significant increase in global temperatures. We don't know whether the alternative assumption itself is reasonable—other assumptions might be even better—but the improved fit does tell us that until more research is done on the best assumptions to apply to global average temperature series, the IPCC's conclusions about the significance of the temperature changes are unfounded.

None of this is opinion. This is factual and indisputable. It applies to any warming—whether attributable to humans or to nature. This assumption problem is not unique to the IPCC, either. The U.S. Climate Change Science Program, which advises Congress, published its report on temperature increases in 2006, and relied on the same insupportable assumption.

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This is not the only instance of serious incompetence in climate science.

Over many millennia, the most important cycles in Earth's climate have been those of the ice ages, which are caused by natural variations in Earth's orbit around the sun. These variations alter the intensity of summertime sunlight. The relevant data are presented in Figure 6: One line represents the amount of ice globally and the other line represents the intensity of summertime sunlight in the Northern Hemisphere, where the effects are greatest. But notice that the similarity between the two lines is very weak.

To understand what's going on, we have to consider the changes in the amount of ice globally. For example, if the amount of ice at different times were 17, 15, 14, 19, . . . , then we must subtract adjacent amounts to obtain the changes: 2, 1, -5, . . . . One line in Figure 7 shows these changes, while the other, as before, shows the intensity of summertime sunlight. Now we see that the similarity between the two lines is strong: one excellent piece of evidence that ice ages are indeed caused by orbital variations.

Serbian astrophysicist Milutin Milankovitch first proposed a connection between ice ages and orbital variations in 1920, though data on the amount of ice present in past millennia didn't become available until 1976. But not until 2006 did scientists first study the changes in the amount of ice. That is, it took 30 years for scientists to think to do the subtraction needed to draw the second line in Figure 7. During these three decades, scientists analyzing Milankovitch's proposed link based their studies on graphs like Figure 6, and they considered a variety of assumptions to try and explain the weak similarity of the two lines.

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We have already seen that the authors of the IPCC report have made one fundamental mistake in how they analyze their data, drawing conclusions based on an insupportable basic assumption. But they commit another error as well—the same one, in fact, that hindered the scientists working to verify Milankovitch's hypothesis. Nowhere in the IPCC report is any testing done on the changes in global temperatures; only the temperatures themselves are considered. The alternative assumption I tested does make use of the changes in global temperatures and obtains a better fit with the data.

To be sure, there have been other studies that consider other alternative starting points and thereby reach different conclusions about the temperature data. The IPCC report nods toward such work, but without really acknowledging how crucially the soundness of its conclusions rests upon its choice of assumptions. Making the right choice, the one that best corresponds to physical reality, requires further, difficult research, and accepting conclusions based on shaky premises risks foreclosing upon such work. That would be gross negligence for a field claiming to be scientific to commit.

*Mr. Keenan previously did mathematical research and financial trading on Wall Street and in the City of London; since 1995, he has been studying independently. He supports environmentalism and energy security. Technical details of this essay can be found at http://www.informath.org/media/a41/b8.pdf.*

I can't believe the readers of the Wall Street Journal would think this article was in "plain English" (i.e., readily understandable). More relevant, for scientists like me, is that it is not logically compelling. Many others have more persuasively and clearly laid out the cherry-picking of data by the IPCC climate scientists (thus relying on only a few "coin tosses", in Mr. Keenan's analogy). And dragging in Milankovitch theory, which is as troubled and unproven as anything else in climate science, merely shows personal scientific bias; I guarantee the "strong similarity" between sunlight and purported global ice changes, over hundreds of thousands of years, in his Figure 7, involves circular reasoning ("Oh, but it's so pretty, don't you think? Surely he must be right."). This is bad science chasing bad science, and I wouldn't bother with it, because there is plenty of good science for a proper judgment against the climate "consensus" (such as the graph and quotes at the top of this site).

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