May 15 2008
Using Auto-Regressive Time Series to Constrain Calculated Temperature Trends
In a not-so-bizarre twist, the bloghorrea continues about the consistency between the IPCC predictions projections and recent observations. This most recent round includes a newcomer to the fray, RealClimate. Dr. Gavin Schmidt wrote about What the IPCC models really say while introducing the term ‘blog-viating’, which I think should be used much more often.
Lucia chimed in and asked that question that we were all dying to know the answer to, Are Swedes Tall?. She also tried to answer whether the IPCC projections actually falsify, again.
Dr. Roger Pielke Jr. then asked, again, How to Make Two Decades of Cooling Consistent with Warming. Dr. James Annan replied with The consistently wrong chronicles, as well as comments at Pielke’s blog. And finally, there’s Comparing Distrubutions of Observations and Predictions: A Response to James Annan.
Autoregressive Moving Average Models
Instead of doing something useful, I decided to play around with an auto-regressive model to show that calculating trends over 7 years isn’t very useful. If you already know this, please skip ahead to the end of the post and leave a comment about whatever is on your mind.
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This plot shows the global average temperature as derived from several different sources. I took the temperature anomalies as reported by GISS, Hadley, UAH, RSS, and NCDC and put them on a common baseline. Actual values are available here. The average value of these 5 anomalies is considered to be the ‘true’ global average temperature anomaly.
The linear trend over the entire time series was calculated (around 0.166 C per decade). This was then removed, and the residual is the black curve above which represents the weather noise in the climate system. This noise is clearly not random, so I chose to represent it using an AR(2) model. Wikipedia has a nice page on Autoregressive moving average models if you’re interested.
First I generated a time series that looked like the temperature anomaly time series above.
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Like the actual temperature anomaly data, this time series has a linear long-term trend, which was actually added afterwards. The trend is exactly equal to the linear trend from the temperature anomalies. There are sharp peaks and valleys that look a lot like El Ninos and La Ninas.
We can also generate time series that are much longer.
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This AR(2) time series was generated using the same parameters as the one above. The only difference is that it is 10 times longer. The little wiggles on the above plot are the same magnitude as those on the plot above. However, it should be clear that as the time series becomes longer, the trend becomes easier to see and differentiate from the background noise.
Wavelets: Again?
Like in previous posts, we’ll use our wavelet black box to see the frequencies that are dominant. This was done for both of the AR(2) model data as well as the temperature anomaly data.
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The top panel shows the wavelet analysis when performed on the temperature anomaly noise series; that is, after the linear trend was removed. Most of the time, there is the signature of signals with 3 and 8 year periods. The 3 year period is most dominant around the large El Nino in 1998.
The middle panel shows the results from the short AR(2) model. There is again a 3 year periodicity associated with the peaks and troughs that look like ENSO. These significant around the time of the deviations but are insignificant in the intervening years. There is also a fairly strong 10 year cycle as in the actual temperature anomaly wavelet.
The bottom panel has the results from the long AR(2) model. It is different that the previous two in that there does not appear to be any regions in th time series where there is a large significance in the 3 year periodicities. When zoomed in, there still appear to be the spikes with around 3 year duration. There is also the 10-12 year periodicity that is significant throughout much of the time period.
Histogram of Trends
The usefulness of the AR(2) model is that it will allow us to compare temperature trends of these different time series and we can see how often there are negative trends. Note that this is purely a statistical exercise, and not a prediction of future temperatures.
All of the above allows us to calculate temperature trends on a data series other than the ‘real’ temperature anomaly data.
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The above plot shows the histogram of 7 year trends computed on the actual temperature anomaly data. If you use your imagination, it very roughly Gaussian. The peak occurs at the mean of the distribution, denoted by the vertical line in the ‘T’. The horizontal line in the ‘T’ shows the standard deviation of the trends. Note that there are many instances where the trend is less than zero.
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The above results were obtained when the same analysis was done on the long AR(2) model data. The upper-left panel shows when 7 year trends were calculated. As with the actual temperature, there is a wide spread about the mean. It is not uncommon to have negative temperature trends.
If the trends are instead calculated with 14 years, the number of extremely small and extremely large trends is reduced, as expected. In this panel, most of the trend lie withing 0.1-0.3 C/decade, but there are still a few negative trends.
As the number of years used when calculating the trend increases, the width of the distribution narrows. By the time 43 years is used, the width is negligible compared to the 7 year trends.
Conclusion
There is only one global temperature time series for the Earth. It has many short-term effects such volcanoes and ENSO that can influence calculations of short-term linear trends. This post used a second order auto-regressive model to generate a time series with similar short term characteristics in order to compile statistics about the linear trends.
The histogram analysis of the trends calculated using 4 different lengths shows, not surprisingly, that as more time is taken into account the number of extremely large and extremely small trends is reduced. As the number gets very large, such that all the ‘weather noise’ is averaged out, the width of the distribution becomes negligible.
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6 Responses to “Using Auto-Regressive Time Series to Constrain Calculated Temperature Trends”
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I too liked Gavin’s use of “blog-viating”, and think it should be used more often.
Yep. More time, smaller uncertainties whether or not Swedes are tall. My father-in-law is Swedish, but only 5′11″.
What period of time would you consider long enough to determine a trend?
I guess if you used 100 years since the dip of the little age age you would have predicted AGW temperatures too
Unfortunately we have nothing near as good as we have now since then end of the 1970’s. 30 years is still not very long to determine a trend. Fortunately I’m likely to still be alive when we have several more decades of data to work from.
According to the IPCC “predictions projections “, when will the AGW trend reach the tipping point and begin to trend down?
The lack of ability to differentiate type I and Type II errors by my side (and endless strained efforts to contaminate language by changing the meaning of falsification, etc.) is a hassle. These guys seem to never have heard of the fallacy of the excluded middle. Or if they have, they haven’t internalized it. So they return to efforts to avoid making clear statements. What a bunch of widget heads.
Plus, just LOOKING at the time series shows that it is prone to large 7 year deviations greater than the sum of a slow, small linear trend.
Maybe Lucia will next falsify CAPM theory by showing the stock market appreciation from 2000-2008? Call the committee! Another Nobel is waiting! Lucia killed beta!