This material is to introduce the statistical concept of stationarity. A variable that is the cumulative sum of random disturbance is a stationary variable. Another name for such a variable is a random walk. What is shown below is a simulation of such a variable. A sample of 2000 random numbers between -0.5 and +0.5 are selected. The cumulative running sums of this set of random numbers are computed and plotted.

(Click on REFRESH to get a new sample and new time series.)

Although for a given sample the series may show upward trends or downward trends there are no long term trends. The series is stationary.

Temperature statistics may have such a stochastic structure because the rate of change of temperature T for some region is given by

C(dT/dt) = v(t)

where T and t are temperature and time, respectively, v(t) is the net energy inflow and C is the heat capacity coefficient of the region.

The average global temperatures for the period 1855 to 2003 are shown below.

The series has its ups and downs but appears to have a long term upward trend. This could occur even for a stationary stochastic variable.

There do appear to be trends. From 1855 to about 1870 there is an upward trend, then from 1870 to 1910 a downward trend. Without any obvious explanation from 1910 there is an upward trend that continues until about 1945. After 1945 the the trend is downward until about 1975. Since 1975 the trend has been upward. As the climatologist Patrick J. Michaels has pointed out the slopes of the trends from 1910 to 1945 and from 1975 onward are about the same. Moreover the slopes of the downward trends from 1870 to 1910 and from 1945 to 1975 are also about the same. The initial upward trend from 1855 to 1870 could be perceived as having the about the same slope as the two later upward trends.

Statistical methods such as regression analysis are inappropriate for variables which are the cumulative sum of random disturbances. This also applies for the informal analysis that people apply just looking at the graphs. It is the annual changes that are the appropriate subject of analysis.

The changes in global average temperature from one year to another for 1856 to 2003 are shown below:

It is much less evident that there is some long term upward shift in the above year-to-year changes. Statistical analysis can be applied to these year-to-year differences. In particular regression analysis can be used to fit a trend line to the data.

The data do not show any obvious trends. A regression line for a trend in the changes is barely perceptible because it is so close to the horizontal axis. The t-ratio (regression coefficient divided by its standard deviation) for the regression slope is a miniscule 0.001, definitely not significantly different from zero at the 95 percent level of confidence.

Another way of examining the temperature change data is to construct a frequency distribution (histogram). Here the temperature changes are grouped into temperature change intervals of 0.05°C width.

The average temperature change is 0.0055°C per year and that is equivalent to 0.55°C per century. The t-ratio for that change is 0.53 and not significantly different from zero at the 95 percent level of confidence. It is notable that the distribution looks more or less like a normal distribution. This is as would be expected from the Central Limit Theorem which says that some quantity which is the sum of a large number of independent random influences will have a frequency distribution which is closer to a normal distribution the larger the number of independent influences. This lends credence to the notion that the year-to-year temperature changes are stochastic (random).

Statistically unsophisicated observers, and this includes many bona fide scientists, look at the time series on global average temperature and say, "Obviously there has been an increase in temperature." But sometimes appearances can be misleading. In this matter I am reminded of a personal incident. Many years ago a woman I had recently met came over to my apartment which was on the fifth floor of an apartment building. She brought along her two year old son Sam. When she got to the apartment she remembered that there was something in her car she had forgotten to bring. She left Sam with me and headed back to her car. Sam started to sob and I realized that he did not understand that she had only left temporarily. I took Sam to the window where I knew we would be able to see his mother in the parking lot below. I said, "What's the matter, Sam? Do think that your mommy has gone off and left you?" Sam looked down at the parking lot where his mother was heading towards her car and said, "Well LOOK!"

So sometimes obvious things do not reflect the underlying reality.

The above material indicates that there are good reasons to question whether there are really any long term trends in global average temperatures. However, as Patrick J. Michaels wisely notes this type of analysis should not be the focus of the debate on policy concerning humans' effect on climate. He advises the acceptance of the moderate degree of global warming that the data supports. It is only about 0.5°C or 1°F per century when corrected for the changes in the intensity of sunlight. Then the public debate can focus on whether there is any evidence of actual catastrophic climate change. It appears that catastrophic climate change appears only in the climate models and these models are unvalidated or sometimes invalidated. For material on the validation of climate models see Backcasting.

However while the global warming alarmists are true-believing hysterics rather than truth-seekers, the skeptics are truth-seekers and issue of whether the global average temperature is stationary or non-stationary is important.

John V. Galbraith and Christopher Green of MacGill University in Montreal, Canada applied the latest statistical methods in a paper entitled, "Trends and Stationarity in Global Temperature Data." They found no evidence of a stochastic trend but did conclude that the data supported the existence of a small linear trend. Their study was concluded in 1990 and so there is the justification for the repition of the analysis including the more recent data.

The more recent data show a flat segment with an indication of a downward trend. Shown below are the month-by-month global temperature data from the Hadley Climate Research Unit

The temperature record for the lower troposphere show a similar pattern. Thus there is good reason to insist upon a statistical analysis of the matter of the stationarity of the average global temperature. Thermodynamics indicates that the governing equation is

C(dT/dt) = v(t)
but
v(t) = u(t) − γT⁴

where T is absolute temperature, u(t) is a stochastic variable and γ is parameter.

This can be approximated by

C(dΔT/dt) = u(t) − γ4T³ΔT

where T is average temperature and ΔT is deviation from average temperature. The issue is whether the net heat flow u(t) has an unchanging stochastic structure that would lead to a quasi-stationary temperature variable.

(To be continued.)