copyright © 2007 Paolo B. DePetrillo, MD
Full Model
T = + P1 x Cosine {(Year+ P2) *3.1415) / P3 "PDO" term
+ P4 x Cosine {(Year +P5) *3.1415) / P6 "ENSO" term
+ P7 x {Cosine {(Year +P8) *3.1415) / P9} Sunspot Cycle
* (LogSun,lag 0 years + 2*LogSun, lag 1 years + 2*LogSun, lag 2 years +2*LogSun, lag 3 years+2*LogSun, lag 4 years + LogSun, lag 5 years )"Sunspot number" term
+ P10 baseline temperature
value [standard error]
P1 = 0.40 [ 0.09]
P2 = -1520 [ 252 ]
P3 = 32 [ 3 ] "PDO" term cycle length
P4 = 0.16 [ 0.09 ]
P5 = -151[ 22]
P6 = 2.02 [ 0.02 ] "ENSO" term cycle length
P7 = -0.009 [ 0.002 ] * {Sunspot Number Term)
P8= 1372 [ 8 ]
P9 = 5.6 [ 0.08 ] Sunspot cycle length
P10 = 13.74 [ 0.06]
T = predicted mean annual temperature
SSE 43.02 DFE 103 MSE 0.41 RMSE 0.64
Comparing to linear fit
Linear Fit Model
T = P1 * Year
There is no intercept term since it did not add anything to the model.
P1 = 0.00550 [0.00004]
SSE 86.9 DFE 126 MSE 0.69 RMSE 0.83 which is clearly better than a "mean fit" with SSE 93.4
Compare models with the corrected Akaike's Information Criteria
Linear Fit Full Model
Sum-of-squares 86.9 43.02
Number of data points 113 113
Number of parameters 2 13
Akaike's Information Criteria (corrected, AICc) -23.46 -76.84
Probability model is correct 0.00% 100.00%
Difference in AICc 53.38
Information ratio 391004117144.12
Full Model has a lower AICc than Linear Fit so is more likely to be the correct model.
It is more than one million times more likely to be correct than Linear Fit.
Compare models with F test
Model SS DF
Linear Fit (null) 86.9 111
Full Model (alternative) 43.02 100
Difference 43.88 11
Percentage Difference 102.00% 11.00%
Ratio (F) 9.27
P value <0.0001
If Linear Fit (the null hypothesis) were true, there would be a 0.00% chance of obtaining results that fit Full Model (the alternative hypothesis) so well.
Since the P value is less than the traditional significance level of 5%, you can conclude that the data fit significantly better to Full Model than to Linear Fit.
Thanks to the nice folks at GraphPad
Now, let's try the following. We are going to take the Full Model above, and ask the question: is there a linear trend once the cycles are taken into account. This is the key question!
It looks about the same as the full model. However, I added an extra parameter (the slope of the trend, if any) and the SSE = 43.02. This is not different from the Full Model without the linear trend and you don't have to do anything except say that there is, in the Cyclical Full Model, NO LINEAR TREND.
T = P1 x Cosine {(Year+ P2) *3.1415) / P3 "PDO" term
+ P4 x Cosine {(Year +P5) *3.1415) / P6 "ENSO" term
+ P7 x {Cosine {(Year +P8) *3.1415) / P9} Sunspot Cycle
* (LogSun,lag 0 years + 2*LogSun, lag 1 years + 2*LogSun, lag 2 years +2*LogSun, lag 3 years+2*LogSun, lag 4 years + LogSun, lag 5 years )"Sunspot number" term
+P10*Year Linear trend parameter
+ P11 baseline temperature
value [standard error]
P1 = 0.40 [ 0.09]
P2 = -1524 [ 306 ]
P3 = 32 [ 3 ] "PDO" term cycle length
P4 = -0.16 [ 0.09 ]
P5 = -151[ 28]
P6 = 2.02 [ 0.03 ] "ENSO" term cycle length
P7 = 0.009 [ 0.003 ] * {Sunspot Number Term)
P8= 1372 [ 7 ]
P9 = 5.6 [ 0.07 ] Sunspot cycle length
P10=0.0001[0.002] <--------------------------------------The linear trend(slope) is 0.0001 +/- 0.002 C / year. Which is of course nonsense since the error term is an order of magnitude higher!
P11 = 13.5 [ 4]
T = predicted mean annual temperature
SSE 43.02 DFE 102 MSE 0.42 RMSE 0.65 <-------- The error here also increases! Suggesting that in this model, THERE IS NO LINEAR TREND IN TEMPERATURE SINCE THE 1880'S!
But you may ask if the sunspot cycle theory is an artifact, which is a reasonable question. So here is a nice summary table of all the different models, in order of complexity.
1) Model Parameters T = SSE = Error
2) Mean fit P1 62.67
3) Linear only P1*Year 64.18 (OOPS!)
4) Linear + intercept P1*Year + P2 61.83
5) Sunspot Only P1 x {Cosine {(Year+ P2) *3.1415) / P3}xST 59.57
6)"PDO" Cycle Only P1 x Cosine {(Year+ P2) *3.1415) / P3 54.65 (Good start!)
7) "PDO" + "ENSO" P1 x Cosine {(Year+ P2) *3.1415) / P3
+ P4 x Cosine {(Year +P5) *3.1415) / P6 50.09
8) Full Model P1 x Cosine {(Year+ P2) *3.1415) / P3
+ P4 x Cosine {(Year +P5) *3.1415) / P6
+ P7 x {Cosine {(Year +P8) *3.1415) / P9}x ST
P10 - baseline 43.02
Models 5 - 8 all have the P10 baseline term. Nothing works without it!
ST = (LogSun,lag 0 years + 2*LogSun, lag 1 years + 2*LogSun, lag 2 years +2*LogSun, lag 3 years+2*LogSun, lag 4 years + LogSun, lag 5 years )
or the "Area under the curve" of the sunspot vs time graph done by trapezoidal method.
Addendum
After staring at the Full model graph, I noticed that there was a fairly strong cycle with a period of 24 years (12.06 x 2). I do not know if this is an artifact, or a climactic cycle, but for those who are interested, I have repeated some of the analysis.
To keep it interesting I have also outlined the longest cycle, which may be evidence of the Pacific Decadal Oscillation (PDO) using the blue lines. Depending on where you start measuring trends, you can get quite different answers for the slope!
T = + P1 x Cosine {(Year+ P2) *3.1415) / P3 "PDO" term
+ P4 x Cosine {(Year +P5) *3.1415) / P6 "24-year cycle?"
+P5 x Cosine {(Year +P6) *3.1415) / P7 "ENSO" term
+ P8 x {Cosine {(Year +P9) *3.1415) / P10} Sunspot Cycle
* (LogSun,lag 0 years + 2*LogSun, lag 1 years + 2*LogSun, lag 2 years +2*LogSun, lag 3 years+2*LogSun, lag 4 years + LogSun, lag 5 years )"Sunspot number" term
+ P13 baseline temperature
value [standard error]
P1 = 0.41 [ 0.09] "PDO" term
P2 = -1419 [ 261 ]
P3 = 31 [ 2 ] "PDO" term cycle period years 31 x 2
P4 = 0.23 [ 0.09 ]
P5 = 1283[ 31]
P6 = 12.1 [ 0.6 ]
P7 = -0.16[ 0.08 ])
P8= -153 [ 27 ]
P9 =2.02 [ 0.03 ] "ENSO" term cycle period years 2.02 x 2
P10=0.009[0.002] * {Sunspot Number Term
P11=1374[8]
P12=5.59[0.07] Sunspot cycle period years 5.59 x 2
P10 = 13.74 [ 0.06]
T = predicted mean annual temperature
SSE 40.22 DFE 100 MSE 0.40 RMSE 0.63
I added three extra parameters, but got a better fit - I will not present the math again!
Conclusions
In Wichita, KS, since 1889, these may have influenced the mean annual temperature: A cycle of about 64 years that looks like the PDO, a cycle of about 4 years that looks like ENSO. A cycle of about 11 years (sunspot cycle) and the effect of average sunspot number lagged from 2 to 4 years, and possibly a cycle of 24 years which is not named at this time. I'll leave it to the experts to figure out if it is an artifact or a real cycle.
Sunspot cycles probably influencer temperature variations and should probably be tested in GCM models. OK, this is a stretch, but if other locations show the same pattern, it strengthens the argument.
According to a linear fit model, we can conclude that since 1889, in Wichita, KS there may be a warming trend of 0.00702 +/- 0.00004 +/- SE Celsius degrees/ Year. However, the caveat is that this when we remove cyclical influences with the Full Model, this trend is no longer seen.
This raises the question if the linear trend is an artifact of attempting a linear fit on a cyclical time series.
The Full Model predicts a gradual decrease in average annual temperature for the next 20-25 years in Wichita, KS, based on entering the "cold" half of the PDO cycle as well as the cold part of the sunspot cycle. I have simply continued the past 11 years of sunspots,so prediction of solar activity might help make this a better model. I have not included the "24 year" cycle in the predictive model since I don't know of any natural climactic temperature cycle, like PDO or ENSO, that fits the periodicity of 24 years. If you know of one, please let me know!
Thanks to the folks at the station, there is no missing data. Even through the Great Depression.
.
Limitations
Data is from 1889 to present. This limits the conclusions to this time period, namely the magnitude of many of the parameters.
I have not ruled out an effect of carbon dioxide on temperature. In stat speak, I do not have the power to rule it out at some reasonable confidence level. And I don't reach a reasonable level of significance to rule it in. Hence, the question is not answered. Like everything else, it would have been nice to have a lot more data.
There are obviously other factors influencing the temperature, and of course it is possible that carbon dioxide may itself influence the length of some of the cycles. That is an empiric question which needs expert study.
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