Guest posts by Willis Eschenbach
Well, I’ve been thinking for a while about how to explain what I think is wrong with how climate trend uncertainty is calculated. Let me give it a shot.
Here, from an article at Carbon Site Summary, is an example of some of the stated trends and their associated uncertainties. The uncertainty (95% confidence interval in this case) is represented by the black “beard bars” extending below and above each data point.
Figure 1. Some observed and modeled temperature trends with their associated uncertainties.
To verify that I understand the graph, here is my own calculation of Berkeley Earth’s trend and uncertainty.
Figure 2. My own calculation of Berkeley Earth trend and uncertainty (95% confidence interval), from Berkeley Earth data. Model data is taken directly from the ClimateBrief graphic.
So far so good, I’ve reproduced their Berkeley Earth results.
And how is that trend and uncertainty calculated? It is done mathematically using a method called “linear regression”. Below are the results of linear regression, using the computer program R.
Figure 3. Berkeley Air temperature at Earth’s surface, seasonal anomalies removed. The black/yellow line is the linear regression trend.
The trend is expressed as an “Estimated” of the change in time listed as “time(tser)” by year and the uncertainty per year is “Std. Error” of the change in time.
This gives us an annual temperature trend of 0.18°C per decade (shown in the “Coefficient” as 1.809E-2 °C per year), with an uncertainty over the decade relative is ±0.004°C per decade (shown as 3.895E-4°C per year)
So… what’s not to like?
Well, the black line in Figure 3 is not a temperature record. It’s a temperature record that excludes seasonal changes. Here’s an example of how we eliminated seasonal variations, this time using the University of Alabama’s Huntsville Microwave Somatic Unit (UAH MSU) lower tropospheric temperature record. .
Figure 4. UAH MSU lower tropospheric temperature data (top panel), average seasonal component (middle panel) and remainder with seasonal component removed.
The seasonal component is calculated as the average temperature for each month. It iterates year after year for the duration of the original data set. The remaining component, shown in the bottom panel, is the original data (top panel) minus average seasonal variations (middle panel)
Now, this residual record (actual data minus seasonal variations) is very useful. It allows us to see slight variations from the average conditions for each month. For example, in the remaining data in the bottom panel, we can see the temperature peaks representing El Ninos 1998, 2011 and 2016.
In a nutshell: the rest is data minus seasonal variations.
Not only that, the remaining trend of 0.18°C per decade shown in Figure 3 above is the trend of the data itself minus the trend of seasonal variations. (Seasonal trends are close to but not exactly zero, since the final effects are based on exactly when the data starts and ends.)
So… what is the remaining trend uncertainty?
It is good Are not what is shown in Figure 3 above. According to the rules of uncertainty, the uncertainty of the difference of two values, each with an associated uncertainty, is the square root of the sum of the squares of the two uncertainties. But the uncertainty of the seasonal trend is quite small, usually at 1e-6 or so. (This very small uncertainty is due to the standard error of the mean of each monthly value.)
So the residual uncertainty is essentially equal to the uncertainty of the data itself.
And here is a much numbers are larger than what is normally calculated through linear regression.
How much bigger? Well, for the Berkeley Earth data, that’s eight times larger.
To see this graphically, here is Figure 2 again, but this time showing both the exact (red) and incorrect (black) uncertainties of Berkeley Earth.
Figure 5. As shown in Figure 2, but showing the actual uncertainty (95% confidence interval) for the Berkeley Earth data.
Here is another example. Much of it is generated from the difference in trends between the lower tropospheric temperature trends measured by the UAH MSU satellite and terrestrial trends such as the Berkeley Earth trend. Here are those two datasets, with related trends and an uncertainty (one standard deviation, also known as one sigma (1σ) uncertainty) calculated incorrectly through linear regression characterization of the data removed seasonal uncertainties.
Figure 6. UAH MSU’s lower tropospheric temperature and Earth’s surface air temperature in Berkeley, together with trends showing linear regression uncertainty.
Since the uncertainties (transparent red and blue triangles) do not overlap, it appears that the two datasets tend to be statistically different.
However, when we calculate the uncertainty correctly, we get a very different picture.
Figure 6. UAH MSU decreases in Berkeley Earth’s troposphere and surface air temperatures, along with trends showing correctly calculated uncertainties.
Since the one sigma (1σ) uncertainty essentially touches each other, we cannot say that the two trends are statistically different.
CODA: I’ve never taken a statistics class in my life. I am completely self-taught. So maybe my analysis is wrong. If you think it is, please quote exactly the words that you think are wrongand show (prove, not just claim) that they are wrong. I am always willing to learn more.
As always, my best wishes to everyone.
w.
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