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Confidence Limits on the Forecasts and
Backcast of Average Global Temperature
Based Upon Its Trend and Cycles Over
the Past 128 Years

The record of annual average global temperatures from 1880 to 2008 from the National Oceanic and Atmospheric Administration (NOAA). The values are given as anomalies. In meteorology and climatology anomaly just means the deviation from some base level, usually a long term average.

In the graph there appears to be several episodes of increasing or decreasing temperature. These episodes appear to be linear. Furthermore the slopes of the upswings appear to be more or less equal. Likewise the slopes of the downswings appear to be equal. Underlying the upswings and downswings there seems to a long term trend. The extent of this long term trend can be discerned by plotting the averages for the various episodes, as is shown below.

A bent line can be fitted to the data using regression analysis. Such a regression function so fitted explains 88.06 percent of the variation in the average global temperature over the period 1880 to 2008. In this regression the slopes of the lines for the various episodes can be of any values.

The possibility of an accelerating trend was tested for by including a quadratic term in the regression. The coefficient for the quadratic term was not significantly different from zero at the 95 percent level of confidence. (Its t-ratio was 0.8.)

The values of the slopes for the two upswings are close, as are the values for the slopes of the two downswings.

The Regression Coefficients
for the Various Episodes of
Global Average Temperature

The difference in the slopes for the upswings is not significantly different from zero and likewise for the slopes for the downswings. Since the differences are not significantly different from zero it is appropriate to do another regression in which the slopes are exactly equal for the upswings and for the downswings.

Thus the appropriate regression function is the one in which the slopes for all of the upswings have to be the same and likewise for the slopes of the downswings. As noted above, visually and numerically this appears to be the case. The graph of the data along with the regression estimates is shown below.

The coefficient of determination (R²) for this regression is 87.98 percent, nearly as high as the value for the unconstrained regression. This means the correlation between the regression estimate and the actual temperature anomaly is 0.938.

Note the downturn for the last few years. If the cyclic pattern of the last 128 years continues, and there is no evidence that it will not, there will be a period of about 32 years during which the average global temperature will decline. The decline will be about 0.12°C over that period. The data and the projection of the long term trend along with the cycle are shown below.

There is a long term trend. Its value is found by computing the slope of the line between two low points in the cycle or two high points. That value is 0.005°C per year. This is equivalent to 0.05°C per decade and 0.5°C per century.

The slopes for the two types of episodes are the sums of the cycle trend and the long term trend. Thus the slope during an upswing is 0.0195°C per year which is the sum of 0.005°C per year for the long term trend and 0.0145°C per year for the cycle trend. Thus the cycle trend slope is nearly three times as large the long term trend. The slope for a downswing is -0.004°C/yr which is the net result of a slope of -0.009°C/yr for the cycle trend and +0.005°C/yr for the long term trend.

The projection on the basis of the past cycle in global average temperatures is then a downswing from now until about 2038 when the global temperature will be 0.12°C below the 2008 level. From 2038 the temperature will rise until about 2070 when the temperature anomaly will be about 0.969°C, about 0.35°C above the 2005 level. From 2070 the temperature will decline and by 2100 the temperature will be only about 0.25°C above the 2005 level.

Confidence Limits

The computation of a margin of error for the projections would be complicated, but it can be said that the standard error σ of the estimate for the regression equation is 0.087°C, so ±2σ is ±0.174°C.

Consider a regression of a variable y based upon a set of explanatory variables x. (The red color is to denote that x is a vector.)

y = β·x + u

where β is the vector of coefficients and u is a random variable with variance σu.

The general formula for the standard deviation of the regression estimate of y when x=x0, y(x0), is

= σu[1 + x0TV-1x0]½

where V is the variance-covariance matrix for the explanatory variables.

For a derivation of this formula see Confidence Limits.

From the standard deviation σy(x0) the 95 percent level confidence limits are computed as

y(x0) ± 2σy(x0)

For a simple, one variable regression y=a+bx+u the confidence limits would look like the following graph.

The quadratic spread of the limits comes from the uncertainty of the estimates of the regression coefficients. The farther the projection variables are from their average values the wider the spread. The spread at the average values comes from the intrinsic unpredictability due to the random term u.

Confidence Limits for the Forecasts
of Global Temperature Anomalies

The method applied to the forecasts given above results in the following graph.

The complex curvature results from the year for the projection deviating from its average value and the cycle variable also deviating from its average value.

A more visually interesting version of the same results is given below.

The results indicate that there is a 95 percent probability that the average global temperature in the year 2100 will be between 1.13°C above the 2008 level and 0.37°C below that 2008 level with the best estimate being that it will be 0.38°C above the 2008 level. The year 2100 is part of a downswing in the cycle. The year 2070 or thereabouts would be the year of the maximum increase. The best estimate of the temperature increase for that year over the 2008 level is 0.49°C. That is due to about 0.3°C from the long term trend and 0.2° from the upswing of the cycle compared to 2008.


Backcasts for the model are easily constructed. The first turning point was 1916. Going back 32 years puts the previous turning point at 1884. The annual increment for an upswing episode is then deducted from the regression estimate for 1884. The results are as shown.

The confidence limits computed for the backcast are as follows:

There is a slight glitch in the graph due to the backcasts computed on the basis of a turning point in the year 1884 whereas the original regression did not involve a turning point at that time.

NOAA declined to publish data for the years prior to 1880, probably because of questions of accuracy. Other organizations have published that data. Thus a comparison can be made of the backcasts with the display below.

As can be seen the backcasts fit the temperature trend that existed before 1880. The backcast value for 1855 is about 0.1°C too low but the actual value is within the confidence limits of the backcast.

If the backcasting were extended it would show declining temperatures from 1820 to 1852, making the fears of another ice age at the time understandable.


There is a discernable cycle in average global temperatures that goes back to the begining of the data 153 years ago. That cycle along with a long term trend of 0.5°C per century explain almost 90 percent of the variation in average global temperature. Statistically justified confidence limits can be computed and they indicate that the future does not hold any catastrophic changes in global temperature. The validity of the procedure is validated by the backcasts.

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