As I’ve previously mentioned, Climacograms with finite time series suspected of containing a deterministic trend to detect the magnitude of the Hurst parameter give results that are ambiguous. Even not considering the lack of statistical resolution of the final data points, the existence of a deterministic trend in the process can appear as evidence of Hurst-Kolmogorov long term persistence even when there is no persistence at all. When the process does exhibit long term persistence, the existence of a trend can biase the estimate of the Hurst parameter upwards. The result is one might tend to mistake short term persistence for long term persistence or obtain estimate of the Hurst parameter that are biased high.
An example of where the latter is can be seen by generating 2^14 data points, x, using a model that is the sum of fractionally differenced series with d=0.3 (H=0.8) and a trend, creating a climacogram and fitting to find the rate of decay of log(σx) with the log of the scale. I repeated this 5 times and plotted each of the series. The resulting climacograms are illustrated with open circles and results are provided in the legend on the lower right of the figure:
A series generated by fractional differencing does correspond to long term memory, so we do expect to detect this. Specifically, we expect σx~1/scale^(0.5-d); so for d=0.3, we expect m_first=-0.2 to pass through the points on the climacogram.
Instead, if the best fit value is computed based on the first few points at the lowest scales, the results for the decay rate range from -0.09 and -0.13. This suggests longer term memory corresponding to fractional differencing near -0.4 rather than -0.3. (That is a Hurst parameter of 0.9 ). If standard deviations at higher scales are included in the fit, the analysis suggests decay rates that approach zero, or even result in increasing standard deviations with scale. So, this example is one in which the Hurst parameter is biased high if one estimates the parameter based on the climacogram. (Some analysts might notice the concave upward behavior; others might attribute it’s appearance to lack of statistical resolution of the final points in the climacogram.)
Obviously, if someone is using climacograms to disagnose or reveal the presence of Hurst-Kolomogorov behavior, it would be nice to supplement the graphs with an additional trace that might help viewers distinguish between climacograms that suggests HK behavior incorrectly and those where the HK behavior is real.
In this particular case, it occurred to me that I could create a second from of the climacogram based on a single realization. In this second form of the climacogram, I computed trends over a particular scale and then computed the standard deviation of those trend. That is: If I had 16 data points, I can compute the trend based on points 1-8 and a second using 9-16. I can then find the sample standard deviation of these two trends. Then, to create a variable with the same decay rate as for the standard deviation of means I multiplied by the true standard deviation of the series of values from (1:scale). Note: this is a variant of the climacogram computed based on scales (and the method of computing this differs from the slope-o-grams I discussed for models.)
The values obtained this way were plotted using ‘*’ symbols, and the same fit applied to this second form of climacogram. Note in this case, the estimated value of the fractional difference parameter is closer to d=0.2; all five fall between 0.2 and 0.23 suggesting the method will estimate the Hurst parameter slightly low.
In fact this method can be shown to be unaffected by introduction of a deterministic trend. Unfortunately, if the time series includes a deterministic trend containing a quadratic term it suffers the same problems seen in the ordinary climacograms. So, basing the climacogram on the trend does not solve all problems of diagnosis– but it solves some.
What does seem promising is this however: If I systematically create bothtypes of climacograms, I find that I have a pretty powerful graphic method to diagnose many instances where the hypothesis the process can be modeled by assuming pure HK behavior is clearly incorrect; this could help prevent misdiagnosing the type of persistence contained in data. Features I tend to see:
When data are generated by a process with short term persistence but contains no deterministic component, the two climagograms will differ at short scales, but will converge to similar values at large scales. As example of climacograms for a pure AR1 process are shown to the right. This feature is useful to avoid misdiagnosing short term persistent processing as long term persistent processes.
- When the data does contain a trend, the climacogram based on the running means will be concave upward and eventually turn up at higher scales. The climacograms based on trends will continue declining at all scales. This deviation at large scales helps prevent misdiagnosing the existence of a determiniitic linear trend as evidence of long term persistence.
When data are created by taking the sum of a stochastic process and a periodic function, the climacogram based on the means, the standard deviation initially decreases very slowly (suggesting HK behavior when short data sets are available); the standard deviation undergoes a dramatic drop at a scale equal to the period of the deterministic function, and then declines morerapidly than expected of short term persistent processes. This appears as a form of anti-persistence.
In contrast, the climacograms based on trends initially show an increase in value at small scales, exhibit a peak at the period equal to that of the deterministic trend and then decline rapidly. Observing this contradictory behavior in both types of climacograms could give a tip-off to prevent an analyst from mis-diagnosing flat-slow decay at at small scales seen in the traditional climacogram for long term persistence.
Because the second sort of climacogram can often reveal features that are masked in the first type of climacogram, it seems to me that it would be beneficial to routinely show both types of climacograms when this method is being used to diagnose HK behavior. Ambiguities will still be possible for cases where the deterministic trend is non-linear or short term persistence (aka ‘markovian’ ) behavior does exhibit long integral time scales relative to length of time spanned by the data. But often, the difference in character between the two types of climacograms will reveal features that are important to identifying a useful model to describe the process that generated the time series. Plotting both may also help analyst have more confidence in their diagnosis of Hurst type long term persistence when this feature really is exhibited by the data. Certainly, showing both types of traces could help persuade readers who might doubt the presence of HK behavior in a series.
wow complete silence here on the biggest climate story happening on the cybernet
http://news.google.com.au/news/more?pz=1&cf=all&ned=au&cf=all&ncl=dC3i_C9v4lXT6fM5hc6ayHMqN1V7M
The first news story title at your link was “King’s School prep teacher among 11 men charged over child abuse images “
Lucia this is major maybe the last part was cut off try this apologies
http://news.google.com.au/news/more?pz=1&cf=all&ned=au&cf=all&ncl=dC3i_C9v4lXT6fM5hc6ayHMqN1V7M
Or if that doesn’t work got to “watts up with that” its all there. Happening real time enjoy!
You’ve done a serious lot of work, Lucia, in making this post, and I’m rather sad to see no comments about it.
It’s a fascinating way of studying the data in another light, using the standard deviation of trends. Be interesting to see how common GCMs look when your second evaluation method is applied, and what it tells us (if possible).
I know you’ve said previously that the observed temperature data, being a single run, can’t have these methods reliably applied to it to evaluate short and long term persistence? But, is it possible to use multiple runs from model -hindcasts-, which are meant to simulate the observed historical temperatures, and then apply these methods to see how and if short term and long term persistence appears backwards through history?
After all, looking at the models predicting into the future (where they can diverge from reality without us able to know) and what types of persistence they have doesn’t really tell us much about the real world right here and now till we catch up with the models in time — only the math the models were built on. But hindcasts… give us something we can evaluate and merge with historical data, so it seems to me.
I dunno, it’s just a random thought. I’m not a statistician, and maybe you’ve already thought of this or maybe it is not applicable, or maybe it won’t actually tell us anything interesting at all.
Because I have multiple runs of GCM’s I could do something better that what I did here. It’s sort of similar to the “trend” climcaograms, but involves multiple runs. Some of the GCM’s look like they show Hurst behavior, some don’t.
I can apply this method to individual model runs, but I don’t really know what it tells me about the models. It would tell me whether the climacogram for the model runs look like the data– and I will be showing somethign with the multi-model mean + “noise” to explain another ambiguity with looking at GISTemp.
You are correct. It doesn’t. The only thing detecting persistence in the models tells us is that models contain persistence. After that, it is worth while trying to figure out whether the earth data have LTP or STP and whether the type of persistence matches models. But that’s difficult because we only have 1 ‘run’ for the earth.
One of my points here is actually that climacograms give ambiguous results given the amount of data available and the types of explanations for features in the data we wish to investigate. In the simplest terms:
Suppose guy named Gavin suggest the reason the data look the way they do is “A”. A guy named Joe suggest reason B. Then Joe shows a climacogram, and shows the climacogram is consistent with reason B. Look good for “B” so far. But what if the climacogram would look exactly the same if Gavin’s theory A is right? Then the climacogram doesn’t help us to chose between A&B. It’s true it might rule out some theory “C” that no one believes in. But that’s not particularly useful.
But, it seem to me that in quite a few instances, at least adding the second trace does distinguish some features in the data. There is a third thing I need to discuss and it has to do with non-linearity in the forced function.
Ah-hah, I see what you are aiming for. I’ve been reading all of your posts as they’ve progressed, and I very much see the ambiguity you mention.
One potentially interesting thing in my mind about this method of evaluation for a single model run, is it could allow finding instances of divergence from the multi-run mean (which could average out finer details), that may give insights into the way the math and forcing parameters mix together in the model’s construction. Things noticeable not in theory, but only in practice. In a sense, with a sound enough theory, it could be used to do another level of quality control.
However, at that point, I think we’d be studying the science of modeling in general rather than the science of climate. What you are doing already is far superior when it comes to evaluating what all this means in the context of real world climate and what we can look for as the future unfolds to distinguish between which model, and it’s underlying scientific assumptions, is most closely related to reality.
Anyways, thanks for all the work and effort you put into this. It’s fun having another unique way to examine and try to test these hypotheses put forth in the models.
Ged–
I suspect comparing trends, absolute values and much more familiar things is probably easier. To some extent, if the trend is wrong, who cares if the model captures the Hurst parameter? Mind you, if the trend is right then one might be interested in learning if other features are also right. Also, there is also a difficulty that, to some extent, some people estimate the variability of the earths trends based on models. So, if we could show the HURST (or anything) about model variability was either right or wrong, that would affect statistical comparisons of mean trends.
In the end though: The scientists discussing Hurst and climacograms tend to be evaluating earth observations. I approve of evaluating earth obseravatins. But also, I think we need to be careful and recognize the limitations of what we can learn from a tool, and in particular how using at tool a certain way affects what we think we’ve learned.