A few years ago I carried out some analyses on the time series on the average annual temperature data provided by the National Oceanographic and Atmospheric Administration (NOAA). The graph of that data is shown below,

The temperatures are deviations from the long term average. Such deviatios are called anomalies.

Clearly there is a visible cyclic pattern invoving upswings of about 30 years and downswings of about 30 years. The cycle is on top of an ongoing trend of about 0.5° Celsius (C) per century.

Regression analysys showed tha the slopes of the two upswings are not significantly different statistically at the 95 percent level of confidence. Likewise the slopes of the two downswings are not statistically different at the 95 percent level of confidence. This was taken as definite evidence of a cycle in the data. However this is only required as evidence of a cycle if the trend in global temperatures is constant.

About the year 2000 global temperatures appeared to level off. There was a spike in AGT for the year 1998 that was not exceded for about 15 year. The 1998 spike in AGT occurred because the El Niño Southern Oscillation (ENSO). There was another ENSO event in 2010 and one is occurring in the current year of 2016. An ENSO event starts affecting AGT more than 12 months before the peak temperature month. It continues to affect AGT for more than 12 months after the peak. The pattern of the effect of an ENSO event on AGT is remarkably regular as shown below.

The net result of the previous analysis is that a cycle was identified. The continuation of that would mean AGT would peak in 2005-2010 and would start to go down, perhaps slowly at first. The AGT went up in 2015 and 2016 but only because of the ENSO peak that occurred in February of 2016. But AGT might have failed to go down in rececent because of an increase in the trend in AGT.

To examine this question it is desirable to subtract from AGT the cyclic componet and also the effects of ENSO events. This noncyclic component of AGT can be examined for possible acceleration of the trend.

The AGT series from NOAA ended in 2008 so an updated series was sought from the NOAA site. Here is the graph of temperature anomaly data found.

Lo and behold, the evidence for a cycle in terms of the equality of the slopes of the upswings and the equality of the slopes of the downswings has disappeared from the data. There is still evidence of a cycle it is not so clear cut. The remaining evidence is simply in terms of peaks and troughs.

The first regression equation used is

AGT*(anomaly) = c₀ + c₁Y + c₂(Y-Pk) + c₃(Y-Tr)

where Y is the years since 1880. The variable (Y-Pk) is the years since the most recent peak such that no trough has occurred. If a trough has occurred then (Y-Pk)=0. So if the year is 1950 then Y=70. A peak occurred in 1944 and no trough occurred between 1944 and 1950 so (Y-Pk)=6. For the year 1980 Y=100. But a trough occurred in 1975 so for 1980 (Y-Pk)=0. The variable (Y-Tr) is determined analogously.

The graph of the values of (Y-Pk) and (Y-Tr) are shown below:

Before any regressionn analysis was undertaken the effects of ENSO events were eliminated. The graph of the data used in the analysis is as follows.

The results of the regression analysis are:

AGT*(anomaly) = −0.39858 + 0.00601Y −0.0042(Y-Pk) + 0.00300(Y-Tr)
          [-12.3]          [15.3]          [-2.2]          [2.1]         

The coefficient of determination for this equation is 0.74368, good but not great.

The problem with this regression equation is in the nature of the pattern of the estimates as shown below:

The discontinuities at the points of the changes in the slope are incompatible with the nature of a cycle. Those discontinuities arise from the nature of the variables (Y-Pk) and (Y-Tr). Different variables with no such discontinuities must be defined. These are shown below.

The regression results using these variables are

AGT* = −0.31696 −0.00283ZPk + 0.01313ZTr
          [-9.8]          [-1.7]          [11.4]         

The time variable Y could not be included in the regression because ZPk + ZTr exactly equals Y.

The coefficient of determination for this equation is 0.76130.

The regression estimates based upon this equation are:

The statistical fit could obviously be improved if the slopes of the upswings could be different, but that would be incompatible with the notion of a natural unvarying cycle. NOAA's revisions of the AGT destroyed the evidence of such a cycle.

The magnitudes of the noncyclic component of AGT, in principle, can be found by subtracting the regression estimates of AGT* from the values of AGT*. Those values are shown below.

The values indicate an acceleration of the noncyclic AGT, but whether this is real or an artifact of NOAA's revision of the data is uncertain. `