Would you have any better references for a layperson attempting to understand enough about wavelet analysis to make better sense of the Scafetta and West paper?

Can you link me to other climate studies where wavelet analysis was used?

Once a wavelet analysis is carried out what is the end game? Can you give examples? It appears from reviews that I have read that Scafetta and West were doing pattern matching of TSI and temperatures proxies. Is that often part of the analysis in wavelet analyses? Obviously I need to read more on wavelet analysis.

A Fifteen Minutes Introduction of Wavelet Transform and Applications, by Paul C. Liu:

http://www.iaa.ncku.edu.tw/~jjmiau/15Wavelet.pdf

A book review of “An introduction to wavelet analysis”, by David F. Walnut:

http://www.math.uiowa.edu/~jorgen/walnut.pdf

Wavelet analysis of chaotic time series by J.S. Murgu´ıa and E. Campos-Cant´on:

http://www.ejournal.unam.mx/rmf/no522/RMF52211.pdf

Even white noise will occasionally produce a match, further complicating the analysts life!

Which is how we get Allahfish and Mary on toast. And this:

What if one had a time series from a chaotic system and applied wavelet theory to it?

That’s what you would use it for, actually. As I hinted at, however, you have to be wary when you get “hits” that don’t repeat or are not otherwise regular. The same applies for Fourier analysis, too (which bender and I have discussed at length.) Even white noise will occasionally produce a match, further complicating the analysts life!

Is there something in the wavelet analysis that avoids or mitigates this problem?

Uh, not really, but each stage of the wavelet decomposition has a specific bandwidth, so you won’t “match” to a single pattern but a range of frequencies. This is not unlike a Fourier transform, btw, which has a range of frequencies in each “bin.” The difference, of course, is that the wavelet decomposition also provides time localization (as I described above) whereas a single Fourier Transform only has one time index: everywhere in the series.
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I think, btw, that you would note that even what you are doing would result in a range of frequencies/signals that provide a result, too. The correlator receiver, useful for CDMA communication systems, is similar. Consider all the effort that went into finding codes that do NOT have a range of signals that correlate well with your desired signal!
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Mark

Thanks again Mark for taking the time to reply to my questions. I think the point that you make in this comment is what I was looking for:

How do we know what to expect in TSI/temperature series? We don’t, but if similarities show up in both, I would not have a problem if someone claimed a connection. I’d expect them to describe the relationship, however, and would also be wary of any cause-effect attribution without some really, really good evidence.

From your question here I see that my original question was poorly worded:

I think “spurious” in the sense of a statistical correlation is an incorrect usage of the word here. Note, however, that filtering does represent a correlation of the data with the impulse response of the filter itself. What do you mean by “incorrect assumptions have been made about the generation of the signal?” Basically, if the data has energy at the frequencies represented by the particular wavelet stage, it will be resolved.

What if one had a time series from a chaotic system and applied wavelet theory to it?

What I sometimes see in pattern matching (not necessarily using wavelet analysis) is that the data set to be matched is cherry picked, or at least has the potential for such a selection, from several available series and then any discrepencies are more or less hand waved away. The end result can then a reasonably nice looking match with which I have trouble for the reason that you eluded to in the first excerpt above. Is there something in the wavelet analysis that avoids or mitigates this problem?

It would be nice to see Scafetta provide the frequency response function of each bandpass he employed. It is too painful, however, to watch this thread descend into an uncomprehending mangle of analytic concepts. Adios!

With VZ, MBH et al, the issues are totally different. They are not representing a time series in sinusoids. Instead they are calibrating by correlating instrumental readings with proxies. If you detrend you answer a different question – what is the correlation between the residuals after detrending? Which question is appropriate depends entirely on what you are looking for. If you believe the trends should be included in the correlation, then detrending weakens the test. I believe this was the Ammann et al view. As I understand, VZ et al believed that some of the trend in the proxies may have been unrelated to climate, so testing the correlation of residuals was safer. I have not looked into this thoroughly, and don’t have a view as to who is right. It’s just a totally different aspect of detrending.