I wanted to demonstrate the importance of knowing even some elementary applied mathematics, but of course for those not already convinced that they need any more math (and don’t want to learn more) I need an example that is sensational or inflammatory. So I’ve gone with a recent Science paper on global warming. It’s fairly widely known that temperature records have tended to show no significant increase in global temperatures over the past 15+ years (technically, simulations of temperature records because these records are never constructed using the raw data but rather changed to account for a large number of positive and negative biases). For many (few of whom are climate scientists), this is used as evidence that mainstream climate science is wrong about anthropogenic global warming (AGW), or “human-caused” global warming. For others (mostly climate scientists), this “hiatus” has been a frustrating mystery.
Recently, a paper published in Science by Karl et al. claimed that this “apparent slowdown” is likely due to biases in the surface temperature record, and the idea that “the global surface temperature ‘has shown a much smaller increasing linear trend over the past 15 years…’ is no longer valid”.
As is usual in Science papers (particularly Scienceexpress Reports like this one), there isn’t much new in this paper. In fact, the paper itself is more like a literature review than a research paper (they relegated their new analysis to “Supplementary Material”, distinct from the actual paper). But if one looks at the graph in their paper, it’s pretty clear that their new analysis shows there’s no “hiatus”:
(Fig. 2. Global (land and ocean) surface temperature anomaly time series with new analysis, old analysis, and with and without time-dependent bias corrections.)
As instructive as the graphs are, the number of “wiggles”, what “time-dependent bias” means, what the actual values of differences per year and/or per time period were between the new and old record, etc., make these graphs somewhat less than illuminating. Also, the only place that the new record temperature trends are compared with the old is in a table of values in the supplement:
(“Table S1. Trends of temperature (°C/decade) and the 90% confidence intervals for various time periods, subsets of the globe, and dataset versions. “)
The confidence intervals (CIs) are the numbers in parenthesis, and as you might guess, the “+/-” numbers are the error margins (i.e., the value given could be less than or greater the “real” value by that much). Here we have the opposite problem: exact numbers, but no good “visuals” to see what they mean. I’ve taken the liberty of making some EXTREMLY simple bar graphs using MATLAB and these data. Lets look at the values for the new vs. the old temperature trends per time period:
There’s a few important things to note already. First, all the changes were increases (more on that later). Second, the largest changes are BY FAR over the “hiatus” time periods. Before we analyze this further, it turns out that almost all the significant changes come from changes to the ocean temperature records:
Again, the changes in the trends are all positive (except for 1880-2014, which is unchanged), and the most significant changes are all during the “hiatus”. Only now the changes aren’t just significant they’re humongous. Note also that in order to compare new vs. old trends per time period, I’ve plotted them along the y-axis from 0 to 1, while the GLOBAL trends were plotted from 0 to 0.14. It’s not that important here, as the graphs are designed to compare the changes from old to new, and significance is scale dependent (that is, if e.g., the land temperature trends ranged from 100C to 10,000C, and the ocean temperature trends ranged from .0001 to .0009, using the same scale for both graphs would make it impossible to even SEE the temperature trends for the ocean, let alone the differences between old and new).
However, like all good scientists, the researchers didn’t just provide information about the temperature trend changes, but the margins of error. Here are the error margins per time period for global and ocean trends:
Again, there are a few items of note. First, notice that the the margins for error are much larger for the time period of interest. Second, note that here the range along the y-axes for both graphs don’t differ by much. Finally, note that the total time period is from 1880 onwards. This is important. Over the years, we’ve had more and more coverage (i.e., more places where surface temperatures have been measured), and more reliable coverage (the instruments have improved, the recording process has been more reliable, etc.). So why is it that the margins for error are greatest for the time periods over which our records are the most comprehensive and reliable?
That’s not all. If you look at the table I’ve copied from the Science paper’s supplement, you’ll note that the error margins for both new and old are close to if not greater than the changes. In other words, the margins for error are greater than the changes between the new and old records.
Finally, I mentioned I would address the fact that the changes were all positive (with one exception where there was no change). Technically, it’s in accurate to say that these are records of temperature data. They aren’t. They’re SIMULATIONS of temperature data. The raw data is adjusted in all sorts of ways EVEN BEFORE it is “averaged” into a yearly value. And this averaging isn’t simple: our coverage is horribly biased (e.g., there are and have been FAR, FAR more places recording surface temperatures on land than on the ocean), so any good yearly “average” has to use some fairly sophisticated “averaging” methods to give a good estimate. So we have a lot of biases and they change in different ways over time (for example, most of our temperature data comes from areas most subject to biases from development and other anthropogenic surface processes, so increased coverage doesn’t necessarily mean more accurate records).
Every measurement comes with a certain amount of error. When you are dealing with a lot of measurements all subject to different biases and over a long time period, these biases causing these errors are numerous, but more importantly they are both positive and negative. So why is it that this new and improved record involves only positive changes, and the most dramatic changes are during the period where we would expect there to be the least amount of uncertainty or error margins? It turns out that in general, corrections of climate records, from satellite to sea surface readings, have been in the direction that fits mainstream climate change theory. This is natural. When e.g., you expect satellite readings to match surface readings and they don’t, you look for reasons why they don’t (this was done), but you don’t just look for problems in general. After all, you have a record you suppose is good, so you look for reasons why the new record is off, so your efforts to “correct” the records are biased. Hence, even though this new “correction” tells us that we are more uncertain about records from time periods during which our coverage was better and records more reliable, and even though we find the smallest corrections for periods during which our records were the worst, we still end up correcting the record to reflect mainstream theory.
Now, does this mean that the corrections are flawed or that mainstream climate theory is wrong? Of course not. I personally think it is likely that the corrections are accurate in that they account for real biases, but far more likely that the corrections don’t account for a large number of errors the researchers didn’t bother to look for because there was no reason to. But whether that’s true or not is kind of unimportant. There are lots of studies more comprehensive than this one, there are differing accounts of sources for bias and their magnitude in the literature, and we will continue to see new records, new criticisms of them, new corrections, etc. The point here is just that the only applied math I’ve really used here is so basic I haven’t required any mathematics beyond arithmetic, elementary statistics/probability, and logic. Yet being able to think critically about reported values in terms of e.g., how they were obtained, what biases there could be just from knowing a bit about how the data were collected (for example, most studies that use human participants use college students to make claims about human beings in general, despite the fact that there is frequently good reason NOT to do this), read charts and graphs, and due some simple data analysis enables one to be that much more informed. And I haven’t just used this research as an example, but this post: I invite you to think about ways my data analysis is off by continuing the kind of elementary quantitative reasoning and the application of basic logic I’ve illustrated here.