Global Warming Trend Line Based On Years 2005-2014

  A new global warming trend appears to be developing. If one computes a trend line starting with the 2014 data and going back in time one gets a line that intercepts the previous one around 2006. The new trend line has a much lower growth rate of 0.24°C/century. Large fluctuations in the anomaly add a lot of uncertainty to the estimates of these trend lines.

Will The Global Warming Trend Of The Past Few Decades Continue?

  The NOAA land-ocean temperature anomaly for the years 1970 through 2007 shows a fairly linear global warming trend with a slope of 0.0173°C per year. The standard deviation of the anomaly from the trend line in this interval of time is σ = 0.1210. So far the anomaly has remained within the 3σ bounds for the fluctuations of the anomaly but the more recent anomalies appear to be dropping below the trend line. One has to ask if the warming trend will continue?

  The trend can be treated as a theory of global warming that can be subjected to statistical testing. If the anomaly shows a large number of months outside the bounds for the trend then we have to reject this theory of global warming. We would also have to ask if enough has been done in terms of reducing greenhouse gases to explain an observed reduction in the global warming trend. The next few years may help us decide if what we have been hearing is true science of mere speculation.

Correction (Mar 4): The value of a = 0.1408 above is for the reference year 1980. From the plot one can see that Trend for 1970 is slightly less than 0. The correct value with 1970 for the reference year is a = -0.0328.

Plot of the Global Ocean Anomaly from 1880 to 2014

  I computed a similar forecast for the global ocean temperature anomaly using data from Jan 1880 to the Aug 2013 and got similar results. The monthly data used, 20 year averages, fit and error bounds are shown in red and the additional data to the end of 2014 is in blue.

The actual anomaly data looks more like a random walk in this case and there is a more pronounced decrease in the projected temperature anomaly. It remains to be seen how reliable the prediction is. The global ocean anomaly appears to be much more stable than the global land anomaly.

Comparison of the 2014 Global Land Anomalies with a 2013 Prediction

  The observed global land temperature anomalies agree quite well with the a prediction made in October 2013. The observed values are within the error bounds of the computed curve. The forecast was based on the data in red and the blue points are for the observed data after the prediction. The solid red curve is a smoothed version of the curve obtained by taking 20 year averages. Combining the deviations from the 20 year with the 20 year average produced a much smoother curve. The doted lines are the fit to smoothed curve and the error bounds.

Supplemental (Feb 11): One might ask why the "corrected" 20 year average works so well and the answer might be that there is a feedback mechanism involved. The global temperatures are somewhat volatile but there also appears to be some resistance to change which is what one would expect if the temperatures are near an equilibrium point. Canceling the deviations from the 20 year average is a mathematical way of adding some resistance to change which results in a stiffened curve.

Systematic Error

  The problem that we encountered fitting the distribution for the standard deviation is a  systematic error due to the fact that the Theory of Errors does not work out exactly because the estimates of the moments are off. Bessel's correction is needed for the standard deviation and we saw that a second correction was needed for the standard deviation's own distribution function. One can plot the residuals for the original plot, the corrected plot and plot with the standard deviation calculated using its known mean.

The original plot shows evidence of a shift of the standard deviation distribution to left while the correction has greater standard deviations towards the sides and lower standard deviations near the peak as already mentioned. The plot using the known mean of the standard deviation shows a non-uniform distribution with greater fluctuations at the center of the plot. The probability distribution is responsible for reducing the counts on the sides.

Error Distribution for the Standard Deviation 2

  The two extra degrees of freedom in the standard deviation discussed in the last blog appear to be due to the use of the average of the x_k in the datasets for the computation of the observed values. If one uses the known value, μ_x, instead the observed and calculated curves fit quite well with n as the number of degrees of freedom. The observed values shift to the left when the averages are used to estimate the distribution mean.

The Error Distribution for the Standard Deviation

  One can also look more closely at the error distribution for the standard deviation by generating m datasets of n random numbers for a known probability distribution and compute a standard deviation for each dataset.

The computed distribution assuming the standard deviation of the sum is reduced by the square root of n gives a calculated distribution that is slightly offset from the observed distribution.

Using Bessel's correction and assuming two additional degrees of freedom gives a better fit to the observed distribution. The observed distribution appears to have a slightly different shape with a small reduction in the peak value and slightly larger values at the sides sides indicating a broadening of the peak. The sum of the probabilities for histogram intervals in both cases is 1 as expected. One has to use a large number of datasets to notice the differences.