WGI AR5 Final Draft Now Online

  The final draft of the IPCC Working Group I 5th Assessment Report (WGI AR5) is now available online. It is still subject to editing and is expected to be published in January 2014. Chapter 2 Observations: Atmosphere and Surface has a subsection 2.4.1 Land-Surface Air Temperature which discusses the global average temperature change over several periods for the temperature anomaly record in Table 2.4. The periods near the end of the record indicate a higher rate of change.

  The damped random walk discussed in the last few posts of this blog have to be considered hypothetical at this time. It is debatable whether the fluctuations in the 5-year average are part of the natural variation in the temperature anomaly or part of the changes in the equilibrium temperature or both. This complicates the analysis of the record. If there is a damped random walk present then it would indicate that part of the changes observed in the 5-year average are inconsequential to long term climate change. It suggests that we may need to clarify what we mean by climate change by focusing on changes to the equilibrium temperature.

A More Sophisticated Random Walk

  How can we modify the damped random walk model to allow for the other changes seen in the temperature anomaly. We arrived at the random walk model by subtracting the anomaly from the five year average and found a standard deviation which is characteristic of the "random" variations present in the data. I wanted to compare a one-dimensional random walk with the anomaly data and tried a simple sum first but it produced permanent displacements which were not seen in the anomaly. Multiplying the random walk displacement by a damping factor produced deviations more like those found in the anomaly. We can modify the formula used by assuming the zero value represented an equilibrium value, T₀, for the damped random walk. A new monthly average temperature, T', can be broken down into the equilibrium value plus a random change to the temperature fitting the normal distribution plus the damped anomaly.

To get this random walk model to produce results similar to the we would have to allow the equilibrium value to change over time. Once we have matched the changes in equilibrium over time numerically they would still have to be explained either as some external or internal change to the environment. One would have to consider sunspot activity, the melting ice pack, oscillations in the weather, greenhouse gases, etc.

A Random Walk Comparison with the Temperature Anomaly

  In order to compare the land N hemisphere temperature anomaly with a random walk I generated 1560 random numbers that fit a normal distribution with σ = 0.36 and summed them using,

Σ ₀ = δ ₀    Σ _k = 0.5 Σ _k-1 + δ _k    k>0

Then the result was summed over a period of 60 "months" to produce a five "year" average. The result for the random walk is shown in the following plot.

There are fluctuations present in the random walk and short linear segments but they are not as pronounced as the temperature anomaly data. Clearly something other than a random walk is also happening in the temperature anomaly data.

Fit for the temperature anomaly deviations

  The histograms of a normal distribution are difficult to fit. First of all one has to work with the frequencies for the intervals chosen which can be computed if the standard deviation, σ, is known. Next the values of the central peak are much larger than the "wings" for the extreme deviations and so is their variation. The best fit for the temperature anomaly deviations of the last blog resulted in σ = 0.334 but the fit was better for the central peak than the wings. Setting σ = 0.36 gave a better fit overall for an assumed normal distribution.

Deviations of the land N hemisphere temperature anomaly from the 5 yr averages

  One can plot the deviation of the NOAA land northern hemisphere temperature anomalies from the 5 year averages. Since the number of averages is less than the number of years for which there is data the dataset for the anomaly has to be reduced so that we are working with the points corresponding to the years of the midpoints for the 5 year average.

The distribution about the 5 year averages appears to be a normal distribution and shows how much natural variation there is in the temperature anomalies. Twenty equal intervals were used to get the histogram data.

Global Warming Update

  The points in the following plot are NOAA's monthly temperature anomaly data for land in the northern hemisphere. The solid line is a smoothed curve produced by averaging the data over five years.

The smoothed data towards the end of the curve suggests a slowdown in global warming but there is still a lot of fluctuation present.

Are Global Temperature Decreasing?

  There have been some recent news articles about decreasing global temperatures but this has been predicted previously by Don Easterbrook. Skeptics of global warming have claimed that lower than expected temperatures in 2013 are evidence against global warming. The counter argument is that this is a statistical fluctuation. One month's data or one year's data alone is not enough to determine a long term trend. Easterbrook argues that the historical record indicates that there may be a 30 year period of decreasing temperatures. Predictions of global warming assume a model for the Earth's behavior and the environmentalists are likely to error on the side of warming. If their predictions prove too high they will be forced to modify the models to get a better fit to the observations. This may not be the best approach to modeling since false assumptions are likely to lead to erroneous results. Conclusions based on a dozen or so years of satellite data may be a naive approach to the problem. It calls into question what is claimed to the an unnatural temperature change. Perhaps if we biased funding in favor of accurate predictions we would get a better idea of what is actually happening with global warming. There is more scientific research than just collecting data. The biggest problem is how to interpret the data.

Albert F. Ravenshear

  By searching for "line of closest fit" in Google Books yesterday I came across a reference to Ravenshear's letter to the editor in Nature by Keesom. By making another search for "A. F. Ravenshear" I was able to piece together a consistent picture of him which is shown in the following chronology.

1882 Whitworth Scholar

1899 Her Majesty's Patent Office

1899 Testimony and Authority in Mind

1901 Use of Least Squares in Physics in Nature

1904 name mentioned in The English Patent System


Supplemental (Sep 23):

1908 The Industrial and Commercial Influence of the English Patent System

Rescaling Normal Error Fits

  What can we do in the case where different units are used for the measurements of each axis and the quantities differ considerably in magnitude? A simple solution is to change the scale of the axes by dividing the difference from the mean for each point by the maximum difference for the axis. This results in sets of dimensionless values that vary between -1 and 1 for each axis. We can test this procedure by generating the data points, r^<k>, of a line and add small random variations δr^<k>. The function rnorm below generates a set of n random numbers with a normal distribution about the mean μ=0 and with standard deviation σ=0.01 and the function runif generates a set of n random numbers with a uniform distribution between 0 and 2π. Notice that the slope of the line was arbitrarily chosen to be 30 degrees.

The next step is to determine the differences of the points from the mean value for each axis and rescale each axis by dividing it by the maximum difference.  The rescaled data points can be used to solve for the fit with minimum error.

Since the rescaled axes vary between the same limits the angle of the line is approximately 45 degrees. When one converts back to the original scales by multiplying by the maximum differences for each axis the fit is very close to the arbitrarily chosen 30 degrees. The rms deviation is slightly less than the standard deviation since only the normal deviations from the line are used.

We can plot the rescaled data points with the fitted line to check the fit.

Comparison of the 2nd Method for a Line Fit a Pearson Example

  In On Lines and Planes of Closest Fit Pearson worked an example problem to illustrate the use normal errors. We can use the eigenmatrix equation to find the direction for the best fit.

Comparing the mean of the data points, r₀ above, with Pearson's centroid and the two values for the slope we see that we get the same solution both ways.