fredag 21. september 2012

Solar Cycle Model fails after mid-1970s

Updated 17  December 2012.




Abstract.


Jan-Erik Solheim, Kjell Stordahl and Ole Humlum (hereafter SSH) have recently published two articles about the relationship between the mean temperature in a solar cycle and the length of the previous solar cycle [1, 2]. For the northern hemisphere, they found a negative correlation between those two variables. A long solar cycle is followed by a solar cycle with a low temperature, and a short solar cycle is followed by a solar cycle with a high temperature. SSH call this the Previous Solar Cycle Length Model. For simplicity, in this note I refer to it as the Solar Cycle Model.

When a solar cycle has ended, its length is known, and the model can predict the mean temperature in the next solar cycle.

The temperatures fitted well with the Solar Cycle Model until the mid-1970s, but not later. The mean temperatures during the last solar cycles have been much higher than predicted by the Solar Cycle Model. The mean temperature so far in the current solar cycle 24, which has lasted for about 3.5 years, is much higher than the Solar Cycle Model predicts.


1 Introduction

The solar intensity varies by approximately 0.1% over a solar cycle. Both the variations and the average value of the intensity differ a little from one cycle to the next. These variations are a solar radiative forcing that affects the energy balance of the Earth. A solar cycle lasts on average for just over 11 years. The previous solar cycle, number 23, ended in November 2008 after having lasted for an unusually long time, well over 12 years. The current solar cycle 24 will probably last for the rest of the decade. Due to the long solar cycle 23, the model predicts low temperatures in solar cycle 24.

SSH have demonstrated the negative correlation by studying long temperature series, most of them beginning in the late 1800s. I have analyzed the same temperature series. Although I mostly get the same statistical results in my analysis as do SSH in theirs, I believe that their conclusion about falling temperatures in the current decade is totally wrong.

SSH have analyzed many local temperature series. These are not independent of each other, because their locations are mostly exposed to the same climate changes. Local temperature series are noisy, and the noise can significantly affect the analysis. The noise in local temperatures tends to cancel out when they are used to calculate global temperatures, and global temperature series are therefore less noisy. We have global temperature series from about 1850, covering 14 solar cycles. Strictly speaking, SSH therefore start with only 14 observations, one in each solar cycle. Based on this small set of observations they make a model. Then, they apply statistical methods to check if the observations as a whole fit with the model. Mostly they fit. This is not surprising, because the model was made for that purpose. It does not at all prove that the model is correct and that it can be used to predict future temperatures.

I will show several plots of the predictions based on the HadCRUT3 NH temperature series. HadCRUT3 NH covers the northern hemisphere. I choose that temperature series because it is long (from 1850), it covers a large area, and it has less noise and is more reliable than the local series. SSH also analyze HadCRUT3 NH in [2], but they are wary of it, because those temperatures do not fit well with the model when they analyze the entire time interval from 1850 until the present. As I will show, however, the HadCRUT3 NH temperatures fit perfectly well with the model when we analyze the time interval from 1850 until the mid-1970s, but they fail to fit with the model for the more recent period.

SSH analyze many local temperature series in [2]. They concentrate on the Longyearbyen temperature series in [1]. The local temperature series vary considerably with respect to how well they fit with the Solar Cycle Model. Longyearbyen is a short series, starting in 1912. The other local series in [2] cover a longer period of time. I therefore think it is wrong to place special emphasis on Longyearbyen, and I show only one plot specific to this series (Figure 7). It is more correct to consider a median value of the local series, as I do in Figure 8. This median shows the same pattern as HadCRUT3 NH: The predictions of the Solar Cycle Model are wrong for the last solar cycles.

Figure 8 also shows that the predictions based on the global HadCRUT3 temperature series fail in the same way as the predictions based on HadCRUT3 NH. The predictions based on the global NASA GISS and NCDC temperature series also fail in the same way, and I therefore do not show any plots or numbers based on them.

I downloaded the temperature series from the Internet in May/June 2012, and I analyzed them with programs written in Scilab. The Internet sources are specified in [4]. All sources provide monthly temperatures. Some of the sources give the temperatures as anomalies. A monthly temperature anomaly is the difference between the absolute monthly temperature and the average temperature for that month. The other Internet sources provide absolute monthly temperatures, and I have converted these to anomalies. All calculations in this paper are based on monthly temperature anomalies.

In this paper I use the following notations:

  1. Observed mean temperature is the average of the observed temperature anomalies in a solar cycle. The observed mean temperature in the current solar cycle 24 is the average of the temperature anomalies observed so far in the cycle. At the time of writing, solar cycle 24 has lasted for approximately three and a half years.
  2. Predicted temperature is the prediction by the Solar Cycle Model for the mean temperature in a solar cycle. The prediction is calculated on the basis of both the observed mean temperatures in the previous solar cycles and the lengths of the previous solar cycles.

The observed and predicted temperatures are always for specific temperature series. A prediction derived from one temperature series is never compared to an observation from another temperature series.

SSH and I have used the same Internet sources for the solar cycles [3]. See details in Appendix A.

For those who might want to check my calculations, Appendix B describes the mathematical methods used, and Appendix C gives detailed statistics. It is not necessary for others to read those appendixes.

2 HadCRUT3 NH temperatures

The HadCRUT3 NH temperature series contains combined land and sea surface temperature anomalies [4].

Many temperature series, including HadCRUT3 NH, match the Solar Cycle Model well up to and including solar cycle 20, but not later. This chapter examines the match for the HadCRUT3 NH temperatures. The next chapter, chapter 3, examines it for some local temperature series.

2.1 Overview of mean temperatures and predictions


This paper plots mean temperatures either as a function of calendar year, or as a function of the length of the previous solar cycle. The first way, which gives the most general overview, is described in section 2.1.1. The second way, which gives the best overview for evaluation of the Solar Cycle Model, is described in section 2.1.2.

2.1.1 As a function of calendar year


Figure 1 shows the observed mean temperatures in solar cycles 10 to 23 as blue circles, and the mean temperature observed so far in solar cycle 24 as a blue star. It also shows the predicted temperatures in solar cycles 14 to 24 as red stars.
Figure 1.  The observed and predicted temperatures up to now as a function of calendar year.


The horizontal positions of the symbols in Figure 1 are in the middle of the solar cycle that they represent. Detailed information about the start, end and length of each solar cycle is given in Appendix A and in [3].

Figure 1 shows that the observed mean temperatures in the solar cycles have risen fairly steadily, and that the mean temperature observed so far in the current solar cycle 24 seems to follow the same pattern.

Figure 1 shows that the predictions for the solar cycles up to and including number 20 mostly fit the observed mean temperatures well. Solar cycle 20 ended in June 1976. The predictions for solar cycles 21 to 23 fit poorly with the observed mean temperatures; the observed mean temperatures are all much higher than their predictions. The current solar cycle 24 looks set to bring even more of the same. Given the failures of the predictions for cycles 21 to 23, and the high mean temperature observed so far in cycle 24, the model's prediction for cycle 24 is extremely unlikely to be right.

Figure 1 gives a fine overview. But it lacks information on solar cycle lengths, and on how a prediction is calculated. I therefore go on to show this information in Figure 2.


2.1.2 As a function of the length of the previous solar cycle


Figure 2 shows the same observed and predicted mean temperatures as Figure 1 does, but now they are drawn as a function of the length of the previous solar cycle.
Figure 2.  The observed and predicted temperatures up to now as a function of the length of the previous solar cycle.


Figure 2 clearly shows how the observed mean temperatures match the Solar Cycle Model. Given a perfect match, all the observed mean temperatures should lie on a straight line sloping down. The observed mean temperatures in solar cycles 10 to 20 do mostly lie close to a straight line, as we will see in Figure 4.

The blue trend line sloping downwards in Figure 2 is the best fit to the blue circles, which show the observed mean temperatures in solar cycles 10 to 23. The trend is calculated with linear regression analysis. The length of solar cycle 23 was 12.2 years, and the prediction for solar cycle 24 is therefore on the trend line at 12.2 years. The dotted red lines show the 95% confidence interval (uncertainty range) for that prediction. Strictly speaking, only the confidence interval at 12.2 years is interesting, because that is where the prediction is. But it is customary to draw the confidence interval for the entire area.

The predictions for solar cycles 14 to 23 do not lie on the trend line in Figure 2, because that line is made for the prediction for solar cycle 24. The trend line for the prediction for solar cycle 20 is shown in Figure 3, for solar cycle 21 in Figure 4, and so on.

Figure 2 shows that the observed mean temperatures in solar cycles 21 to 23 are much higher than predicted. The same holds, to an even greater extent, for the mean temperature observed so far in solar cycle 24. The model is obviously no longer able to predict future temperatures.


2.1.2.1 Autocorrelation



As we will see throughout, the observed mean temperatures are never exactly on the trend line. The deviations, called residuals, are the vertical distances between the observed mean temperatures and the trend line. When the residuals are small and random, the Solar Cycle Model fits well with the observed mean temperatures. There may be autocorrelation in the residuals, meaning that they are not random. Positive autocorrelation means that consecutive residuals tend to have the same sign. Negative autocorrelation means that consecutive residuals tend to have the opposite sign.

In Figure 2 we see that the observed mean temperatures in solar cycles 21 to 23 are far above the trend line, i.e. their residuals are large and positive. Statistical analysis shows that there is positive autocorrelation in the residuals, and that it is statistically significant. This suggests that the Solar Cycle Model does not reflect all factors that govern the mean temperatures in the solar cycles, and that the confidence interval is broader than shown in the figure. This is discussed in more detail in appendix C.


2.2 Predictions for solar cycles 20 to 24


This chapter shows how well the Solar Cycle Model has predicted the temperatures of solar cycles 20 to 23, and how well it seems to be doing for solar cycle 24. It is illustrated with four plots based on the HadCRUT3 NH temperature series. Table 3 in Appendix C contains the equivalent numerical information for 13 local temperature series and their median, for HadCRUT3 NH, and for the global HadCRUT3. The values for the local temperature series differ, and it would be misleading to single out one of them. Therefore HadCRUT3 NH has been chosen for the plots.

2.2.1 The prediction for solar cycle 20


Solar cycle 20 lasted from Desember 1964 till June 1976.
Figure 3.  The Solar Cycle Model, with the observed mean temperatures of solar cycles 10 to 19 as input, predicts the mean temperature for solar cycle 20. The observed mean temperatures (blue circles) are close to the blue trend line, and there is no significant autocorrelation in the residuals. The observed mean temperature in solar cycle 20 (blue star) is close to its prediction (red star)..

Figure 3 shows that the observed mean temperatures in the solar cycles 10 to 19 fit nicely with the Solar Cycle Model. There is virtually zero probability that random temperatures would fit so well with a trend line as steep as this one.

The prediction for solar cycle 20 matches the observed mean temperature of that cycle very well. In short, until June 1976 the Solar Cycle Model predicted the HadCRUT3 NH temperatures very well.

The next plots use the same scale as Figure 3 to make it easy to compare the uncertainties in the predictions. The uncertainty is shown as dashed red lines above and below the trend line, indicating the 95% confidence interval. The uncertainty in Figure 3 is small because the observed mean temperatures in solar cycles 10 to 19 fit so well with the trend line.

2.2.2 The prediction for solar cycle 21


Solar cycle 21 lasted from July 1976 till October 1986.
Figure 4.  The Solar Cycle Model, with the observed mean temperatures of solar cycles 10 to 20 as input, predicts the mean temperature for solar cycle 21. The observed mean temperatures (blue circles) are close to the blue trend line, and there is no significant autocorrelation in the residuals. But the observed mean temperature in solar cycle 21 (blue star) is much higher than its prediction (red star).
Figure 4 shows that the observed mean temperatures in solar cycles 10 to 20 fit nicely with the Solar Cycle Model. There is virtually zero probability that random temperatures would fit so well with a trend line as steep as this one. The 95% confidence interval around the prediction for solar cycle 21 is narrow.

The observed mean temperature in solar cycle 21 is far above the upper limit of the 95% confidence interval around its prediction. According to the model, there is only a 0.41% probability of measuring such a high mean temperature in solar cycle 21. The observed mean temperature in solar cycle 21 is not particularly high, the problem is that it does not fit with the Solar Cycle Model.

Figure 4 shows that the temperatures measured before the mid-1970s fit very well with the Solar Cycle Model, but that temperatures in the first solar cycle after the mid-1970s do not. Nor do those of the subsequent solar cycles. Figure 4 shows when the Solar Cycle Model collapsed with respect to the ability to predict future temperatures.

2.2.3 Predictions for solar cycles 22 and 23


Solar cycle 22

Solar cycle 22 lasted from November 1986 till November 1996. The observed mean temperature in the previous solar cycle 21 does not fit at all with the Solar Cycle Model, and the observed mean temperature in solar cycle 22 is much higher than the Solar Cycle Model predicted for it.

The case of the next solar cycle, number 23, is similar, see Figure 5; there is therefore no need for a plot showing the prediction for solar cycle 22.

Solar cycle 23

Solar cycle 23 lasted from December 1996 till November 2008.
Figure 5.  The Solar Cycle Model, with the observed mean temperatures of solar cycles 10 to 22 as input, predicts the mean temperature for solar cycle 23. The observed mean temperatures (blue circles) are NOT close to the blue trend line, and there is significant autocorrelation in the residuals. The observed mean temperature in solar cycle 23 (blue star) is much higher than its prediction (red star).
Figure 5 shows that the observed mean temperatures in the solar cycles 10 to 22 do not fit well with the Solar Cycle Model. The observed mean temperature in solar cycle 23 is much higher than predicted by the model.

Although the 95% confidence interval around the prediction for solar cycle 23 is much broader than it was for the previous predictions, the observed mean temperature in solar cycle 23 is far above the upper limit of the confidence interval. The model has collapsed, and it is almost pointless to calculate probabilities. For any practical purpose, there would be zero probability of measuring such a high mean temperature in solar cycle 23, if the model were correct.

2.2.4 The prediction for solar cycle 24


Solar cycle 24 started in December 2008, and it will probably last for the rest of the current decade.
Figure 6.  The Solar Cycle Model, with the observed mean temperatures of solar cycles 10 to 23 as input, predicts the mean temperature for solar cycle 24. The observed mean temperatures (blue circles) are NOT close to the blue trend line, and there is significant autocorrelation in the residuals. The mean temperature observed so far in solar cycle 24 (blue star) is much higher than its prediction (red star).

Figure 6 shows that the observed mean temperatures in solar cycles 10 to 23 do not fit with the Solar Cycle Model. The collapse of the model, as discussed in connection with the previous three solar cycles, is confirmed by Figure 6.

Note that the calculated 95% confidence interval is much broader in Figure 6 than in Figure 5. This is because the observed mean temperature in solar cycle 23 does not at all fit with the model, and it contributes to the confidence interval in Figure 6, but not in Figure 5. Despite the greatly expanded confidence interval, the mean temperature observed so far in solar cycle 24 is far above the upper limit of the confidence interval around its prediction.

3 Local temperatures

I downloaded the local temperature series from the Rimfrost and eKlima websites [4]. They contain monthly air temperatures. I calculated the temperature anomalies as described in the introduction, and used these anomalies to calculate the observed mean temperatures.

3.1 Longyearbyen


SSH analyzed the Longyearbyen temperature series in [1]. It is a short series, starting in 1912 and covering only 9 solar cycles. This is a small amount of data from which to draw statistical conclusions, and we should not place more emphasis on Longyearbyen than on the other, longer local temperature series. The Longyearbyen series does not have statistically significant autocorrelation in the residuals, and this may be the reason why SSH place so much emphasis on it.
Figure 7. Observed mean temperatures (blue circles) in solar cycles 15 to 23 for Longyearbyen in Svalbard as a function of calendar year, the mean temperature observed so far in cycle 24 (blue star), and the predictions for cycles 19 to 24 (red stars).


Figure 7 shows the Longyearbyen temperature series. It corresponds to Figure 1 for the HadCRUT3 NH temperature series. Figure 7 shows that the observed mean temperature in solar cycle 23 is much higher than predicted, and that the same seems to be the case for the current solar cycle 24.

The observed mean temperature in solar cycle 23 exceeds the upper limit of the 95% confidence interval around its prediction.

The mean temperature observed so far in solar cycle 24 exceeds the upper limit of the 99% confidence interval around its prediction. It is pointless to calculate probabilities for a model that obviously has collapsed. If we nevertheless do so, we see that there is only a 0.07% chance of observing a mean temperature as high as or higher than the one observed so far in solar cycle 24.

Many years remain of solar cycle 24, and the situation may of course change. But given the failure of the prediction for cycle 23, and the high mean temperature observed so far in cycle 24, the model's prediction for cycle 24 is extremely unlikely to be right.


3.2 The median of local temperature series



I have analyzed the same local temperature series as SSH did in [2]. I have examined how well the predictions match the observed mean temperatures in cycles 19 to 23, and so far in cycle 24. For each of these predictions I have examined how well the previously observed mean temperatures fit with the Solar Cycle Model. The detailed results from these examinations are given in two tables in appendix C. Figure 8 shows an overview of the results.

As expected, there is considerable variation among the local series, both with respect to how well the model fits the mean temperatures already observed, and with respect to how well the model predicts the mean temperature in the next solar cycle. To present the latter in a single plot would therefore have been confusing. Instead, Figure 8 shows the median of the values of the local temperature series. The figure also shows the same information for the two HadCRUT3 temperature series.

The vertical axis in Figure 8 is the probability of measuring mean temperatures as high as or higher than the ones observed, provided that the Solar Cycle Model is correct.

The probability is 50% when the observed mean temperature is exactly as predicted, because one would be just as likely to measure a mean temperature higher than predicted as to measure one lower than predicted.

The probability is less than 50% when the observed mean temperature is higher than predicted. For example, the predicted temperature for solar cycle 23 based on HadCRUT3 NH is 0.00°C. The upper limit of the 95% confidence interval around the prediction is +0.34°C. The observed mean temperature in solar cycle 23 is +0.51°C, which is much higher than the upper limit of the confidence interval. There would only be a 0.32% probability of a temperature this high or higher in solar cycle 23 if the Solar Cycle Model were correct. The blue circle in Figure 8 for HadCRUT3 NH for solar cycle 23 is therefore very close to the zero line.

The probability is greater than 50% when the observed mean temperature is lower than predicted. For example, the predicted temperature for solar cycle 20 based on the Arkangelsk temperature series is -0.08°C, with a 95% confidence interval between -1.38°C and +1.22°C. The observed mean temperature in Archangelsk in solar cycle 20 was -0.32 °C. This is lower than predicted, but well within the 95% confidence interval around the prediction. There is a 67.30% probability of measuring a mean temperature this high or higher in solar cycle 20, provided that the Solar Cycle Model is correct. This is the median probability among the local temperature series for solar cycle 20, and it is therefore plotted in Figure 8.


Figure 8.  The probability of measuring mean temperatures as high as or higher than those observed, provided that the Solar Cycle Model is correct. See explanation in the text.

Figure 8 shows a clear trend with respect to the ability of the Solar Cycle Model to predict the temperature in the next solar cycle. The median of the 13 local temperature series predicts the mean temperatures well (close to 50% probability) for predictions up to and including solar cycle 21. But thereafter the predictions start to fail. The probability drops to near zero for solar cycles 23 and 24, meaning that the observed mean temperatures in these solar cycles are much higher than predicted.

The two HadCRUT3 temperature series shown in Figure 8 fit well with the Solar Cycle Model up to and including solar cycle 20. But mean temperatures observed since are much higher than predicted.

Predictions based on the global NASA GISS and NCDC temperature series match the observed mean temperatures, or fail to do so, about the same as the predictions based on HadCRUT3. I therefore see no point in showing data and/or plots based on NASA GISS and NCDC.

4 SSH's robustness check


The agreement between SSH's calculations and mine is good when we predict the mean temperature for solar cycle 24. I therefore expect a similar good agreement for cycle 23. But there seems to be no agreement when we predict the mean temperature for solar cycle 23. In chapter 3.5 in [2] SSH write: "To check the robustness of the forecast with the PSCL-model we have for all 16 datasets removed the last entry, which is the SC23 temperature, and used the remaining observations for generating a forecast of t(SC23). The result is that the forecast for 11 of the 15 series with 95% confidence intervals are within this interval." In my calculations, t(SC23) is higher than the upper limit of the 95% confidence interval for 11 out of 15 temperature series. I present statistics and many plots supporting this, both here in this blog post and in another blog post. SSH do not present any plots or statistics that support their claim for solar cycle 23. I am therefore not able to comment further on the difference between their results and mine.

In my opinion the robustness test with the prediction for solar cycle 23 fails, also when we apply SSH's results. Statistically there is only 2.5% probability that an observed mean temperature would be higher than the upper limit of the 95% confidence interval around its prediction. I.e. it should happen only in 1 out of 40 cases. With SSH's results it happened in 4 out of 15 cases. With my results it happened in 11 out of 15 cases.

Let us assume that the 15 temperature series that SSH use in their robustness test are independent of each other. Then there is less than 0.1% probability that 4 or more observed mean temperatures would be higher than the upper limit of their 95% confidence intervals, if the model were correct. But the temperature series are not independent of each other, both because of geographical closeness and because they are measured in the same solar cycle. This is a major weakness in the test itself. It is better to check the predictions for several solar cycles based on a global temperature series, as has been done in this blog post. They show that the model has failed since the mid-1970s.

5 Conclusion


The temperatures fit well with the Solar Cycle Model until the mid-1970s. Since then, however, temperatures have been much higher than predicted by the model. The Solar Cycle Model therefore cannot be used to predict future temperatures.

If there is a real, physical reason why temperatures fitted so well with the Solar Cycle Model until the mid-1970s, that reason must be a solar radiative forcing. If so, as this forcing is still present after the mid-1970s, it can no longer dominate. Another forcing, or several, must have become dominant. My analysis says nothing about what the new dominant forcing(s) may be. But it is natural to think of the human-induced forcings, which many scientists claim became dominant in the 1970s.

Skeie et al [3] show that the sum of the various human-induced forcings first became positive in the 1970s. Before 1970, the sum was small and the sign varied. After 1970, the sum has increased steadily up to a substantial positive value in 2010. This is the probable explanation why there is an increasing gap between the Solar Cycle Model's predictions and the observed mean temperatures after the middle of the 1970s.

Acknowledgements

A special thank to Christian Moe for valuable comments and for language editing. Thanks also to the various agencies for providing the temperature series, especially to www.rimfrost.no, which is run on a voluntary basis.

Appendix A. Overview of the solar cycles

Different Internet sources show somewhat different solar cycle endpoints and lengths. I use the same NOAA source [3] as SSH.

In their calculations, SSH use a yearly resolution for the beginning and end of the solar cycle, although their NOAA source allows them to use a monthly resolution. SSH do not explain why. SSH use one tenth of a year as the resolution for solar cycle length, just as the NOAA does in [3].

In my calculations, I use one tenth of a year as the resolution for both the beginning, the end, and the length of the solar cycle, just as the NOAA does in [3]. I let the decimal number in the column Year of min in [3] distinguish between the last month in the previous solar cycle and the first month in the next cycle. In this way I get the start and end months shown in Table 1.

For test purposes, I have done many of the calculations with a yearly resolution for the beginning and end of the solar cycle, just as SSH do. This has little impact on the results. With yearly resolution, I also tried using absolute temperatures instead of temperature anomalies, something that has no impact on how well the temperatures fit with the Solar Cycle Model. None of the numerical results from these test calculations are used in this paper.
Table 1. Solar Cycle (SC) information. Solar cycle Minimum and Length are obtained from NOAA [3]. Start and End are the Minimum converted to calendar months, as explained in the text. The last column shows the difference between solar cycle lengths as provided by NOAA [3] and by Wikipedia.

Appendix B.  Mathematics used in the analysis

B.1 Correlation coefficient


The correlation between two variables x and y is positive when they tend to change in the same direction, such that when x increases, y does the same. The correlation is negative when they tend to change in opposite directions. A correlation may be due to a cause-effect relationship between the variables, but it may also be random, or controlled by something else.

The correlation coefficient r indicates the strength of the correlation. It is always between -1 and +1. It is +1 if all the points in the scatter plot of y against x lie on a straight line sloping upwards, and it is -1 if all points lie on a straight line sloping downwards. In both cases, there is a perfect linear relationship between the variables. r is close to zero when there is no correlation between the two variables.

I use Pearson's correlation formula for the coefficient. The formula can easily be looked up on the web, e.g. in Wikipedia [6].

SSH do not specify the formula that they use in [1, 2]. I get approximately the same correlation coefficients as they do, so I assume that they also use Pearson's formula.

B.2 Confidence interval for predictions


The results from a regression analysis can be used to predict future measurements, as I have explained in connection with Figure 2. There are of course uncertainties associated with predictions, and it is customary to express this uncertainty as a 95% confidence interval around the prediction. There is a 95% probability that the next measurement will fall inside this interval. When I calculate the confidence interval for the prediction, I take into account both the uncertainty of the next measurement and the uncertainty of the estimate itself. The difference between calculating the uncertainty of an estimate and of a prediction is explained in [7].

SSH get narrower confidence intervals than I do. They do not explain in detail how they calculate their confidence intervals. The difference between their values and mine does not matter for my conclusions.

B.3 The Durbin-Watson test for autocorrelation


SSH place great emphasis on the Durbin-Watson statistical test, so I have also used it in my analysis.

After a regression analysis, the measurements will usually deviate more or less from the regression curve. The deviations, the vertical distances between the measurements and the curve, are called residuals. The Durbin-Watson test calculates a value that is used to determine whether there is statistically significant autocorrelation in the residuals. Equation (1) shows how this value d is calculated. d is always between 0 and 4.

In Equation (1), N is the number of measurements and ei is the residual of measurement number i.

Positive autocorrelation means that consecutive residuals tend to have the same sign, causing d to be substantially less than 2. Negative autocorrelation means that consecutive residuals tend to have the opposite sign, causing d to be substantially larger than 2.

A positive autocorrelation is statistically significant when d is smaller than a critical value dc. The critical value depends on the significance level and on the number of observations. But dc also depends on the data. Therefore, statistical tables indicate a lower value dL and an upper value dU for dc. When d is smaller than dL, there is statistically significant positive autocorrelation. When d is greater than dU and less than (4-dU), there is no statistically significant autocorrelation, positive nor negative. When d is between dL and dU, we are not certain if the positive autocorrelation is statistically significant or not. The same holds for values of d greater than 2. When d is greater than (4-dL), there is statistically significant negative autocorrelation.
Table 2 Critical values for the Durbin-Watson test value d. The values are for α equal to 0.05 and when d is calculated using the residuals of a linear regression analysis. The values are taken from [8b].

Statistically significant autocorrelation is an indication that the regression curve is an incomplete model for the measurements. Autocorrelations are usually positive. We can compensate for positive autocorrelation when we calculate the uncertainties by setting the number of independent measurements Neff to less than the actual number of measurements N. This approach is often used when calculating uncertainties associated with trends based on monthly temperatures. For the Solar Cycle Model, on average, 11 years pass between each observation (mean temperature), and the number of observations is therefore only between 10 and 15. It feels wrong to reduce the number of independent measurements, and I have not done so. SSH touch on this in [1, 2], but they also opt not to reduce Neff. If we had reduced Neff, the 95% confidence interval would have broadened, making the prediction look both better and worse: better, because measurements would be more likely to fall within the 95% confidence interval around the prediction; worse, because the greater uncertainty of the predictions would make them less likely to be useful.

In Appendix C, the probabilities are written in red when they derive from calculations involving statistically significant autocorrelation and/or other statistical weaknesses.

The Durbin-Watson test is well described in Wikipedia [8a], with links to other sources.


Appendix C.  Statistics for the temperature series.

Table 3 shows the probabilities of measuring mean temperatures as high as or higher than observed in each of the solar cycles 19 to 24, provided that the Solar Cycle Model is correct. The table contains 13 local temperature series, the HadCRUT3 NH for the northern hemisphere and the global HadCRUT3. SSH analyzed the first 14 of these in [2].

I have color-coded the probabilities based on two statistical tests. Green means that the statistical tests are passed or almost passed. Red means that the tests are not passed. Green indicates that there is a good statistical foundation for saying the observed mean temperatures in the previous solar cycles fit well with the Solar Cycle Model. The results of these tests are only indicative; there are many examples of good data not passing such tests and of bad data passing.

Vardø is included twice in the table. For Vardø, the website eKlima has monthly temperatures from 1840, covering solar cycles 9 and onward. But SSH used only the temperatures from 1850 in [2], and they therefore omit solar cycle 9. The observed mean temperature in solar cycle 9 does not fit with the Solar Cycle Model. The observed mean temperatures in the next solar cycles fit well with the model. Therefore, all values for Vardø from 1840 are red, while five of the six values for Vardø from 1850 are green. There are more details in Table 4.

The observed mean temperature in a single solar cycle, number 9, greatly affects the statistics for Vardø. The observed mean temperatures for Torshavn in the Faroe Islands fit well with the model, and all six probabilities are coded green. But the observed mean temperatures for Armagh in Northern Ireland do not fit well with the model, and all six probabilities are coded red. This shows that the Solar Cycle Model is not robust with regard to local variations, and demonstrates that we cannot draw conclusions based on a few local temperature series.

Table 3. Probabilities [%] of measuring mean temperatures as high as or higher than observed in solar cycles 19 to 24, provided that the Solar Cycle Model is correct. The colors green and red indicate how well the observed mean temperatures in the previous solar cycles fit with the Solar Cycle Model. The line Median of locals contains the median of the probabilities for the local temperature series. Vardø from 1850 is not included in the calculation of the median, to avoid counting Vardø twice. HadCRUT3 NH is the HadCRUT3 temperature series for the northern hemisphere, and HadCRUT3 is the global HadCRUT3 temperature series.

The following example illustrates the probabilities in Table 3. The Longyearbyen probability for solar cycle 23 is 2.40%. It is coded green because the observed mean temperatures in the solar cycles up to and including number 22 fit well with the Solar Cycle Model, and they are therefore well suited to predicting the mean temperature of solar cycle 23, provided that the Solar Cycle Model is correct. But according to the model, there is only a 2.40% probability of measuring a mean temperature as high as or higher than the one measured, indicating that the model is not correct.

The median of the probabilities for the local series, and the probabilities for the two HadCRUT3 series, are shown graphically in Figure 8 in chapter 3.2.

I will now explain the statistics in Table 4 that are the basis for the color coding in table 3.

The Durbin-Watson test checks if there is autocorrelation in the residuals after a regression analysis. SSH use this test in [1, 2]. I use the same α (0.05) when testing for significance. The test classifies the result in one of three categories. They are, with my interpretation in parenthesis: no significant autocorrelation (passing the test), possibly significant autocorrelation (almost passing the test), or significant autocorrelation (failing the test).

As a second test, I use the confidence level of the calculated trend line between the observed mean temperatures and the lengths of the previous solar cycles. I classify a better than 95% confidence level as passing the test, between 95% and 90% as almost passing, and less than 90% as failing.

Green color in Table 3 shows either that both tests are passed, or that one of them is passed and the other is almost passed. Red color tells either that both tests are almost passed, or that one or both of them are failed.

The solar cycle numbers in the column headings in Table 4 are shifted left with regard to those in Table 3. A heading in Table 3 refers to the solar cycle for which the prediction is valid, whereas in Table 4, the heading refers to the solar cycles that are the basis for that prediction. As an example, the values in Table 3 under the heading 20 are the probabilities associated with the predictions for solar cycle 20. They are based on analysis of the observed mean temperatures in all the solar cycles up to and including solar cycle 19. Therefore the corresponding column heading in Table 4 is →19.

Table 4 also contains the correlation coefficients between the observed mean temperatures and the lengths of the previous solar cycles. All coefficients in table 4 are negative, and almost all of them are more negative than -0.5. The global HadCRUT3 shows a stable and strong negative correlation, at least as strong as the local temperature series. These facts are arguments both for analyzing HadCRUT3 NH as I did in Chapter 2, and for assuming that it is not coincidental that the temperatures up to the mid-1970s fit well with the Solar Cycle Model.


Table 4. Statistics that are the basis for the color coding in Table 3. p is the p-value of the regression analysis that calculated the best fit between the observed mean temperatures and the lengths of the previous solar cycles. r is the corresponding correlation coefficient. d is the Durbin-Watson test value. p and d determine the color green or red in Table 3, and they are written in the same color.


References

1.
Jan-Erik Solheim, Kjell Stordahl and Ole Humlum.

2.
Jan-Erik Solheim, Kjell Stordahl and Ole Humlum.

3.
NOAA data for the solar cycles. Both SSH and I use this NOAA reference.

4.
HadCRUT 3 temperatures were downloaded from UEA Climatic Research Unit 
Vardø temperatures were downloaded from eKlima
The other temperatures were downloaded from rimfrost.no

5.
Skeie et. al.

6.
Pearson product-moment correlation coefficient 

7.
Regression Analysis by Example, Fourth Edition. By Samprit Chatterjee and Ali S. Hadi.
Equation 2.38 calculates the Confidence interval for predictions, and equation 2.41 for estimations. 

8.
a) Durbin–Watson statistics. Wikipedia web page with further references.
b) Critical Values for the Durbin-Watson Test. Web page at Stanford University

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