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I'd like some history on the notion of cardinality before Cantor, and I'm coming up empty. Epp's Discrete Mathematics (text for the course I'm teaching) says:
I don't see anything specific in those articles about what exactly these guys said that we now recognize as being in the spirit of Cantor. I've heard that there is material in the Archimedes Palimpsest that suggests Cantor's approach, but couldn't find anything specific on that either. Any specifics would be greatly appreciated (not just limited to the folks I linked). I'm especially interested in Chinese, Hindu, and Islamic contributions. Thanks- Staecker (talk) 02:05, 29 November 2011 (UTC)
Oops- now that I reread that bit from Epp, it looks like maybe that quote is about the idea of real numbers as decimal expansions rather than cardinality via bijections. That would make more sense considering what little I know of Stevin. Anyway, I'm still interested in any Cantor-like ideas on cardinality before Cantor. I know (I assume) that nobody else had demonstrated that R was uncountable, but are there earlier sources who discuss cardinality in terms of bijections? Staecker (talk) 02:09, 29 November 2011 (UTC)
Here is the background. I have a bunch of data segments of different lengths each but they are all measurements from a single underlying signal and I want to use these data segments to estimate the power spectrum density of the underlying signal. Each data segment is uniformly spaced in time. I am not familiar with signal processing and Fourier analysis is notorious for being known as black magic. I ask two people and I get three different opinions on what to do and how to do it. And then no one can really convince me with their "reasoning" for why they did what they did so I appeal to wikipedia mathematicians. I have three specific questions and I ask for your opinions.
1. I am thinking of zero padding all of the data segments (to match the length of the longest segment) so that they can all be the same length. This way when I get their PSD, each segment will give me the same frequency resolution and then I can average them to get a single PSD. How does this sound? Should I add zeros at the end of the segment or should I do pad on both sides to keep the nonzero values in the middle so that the padded segment is (continuously) periodic? Is there any advantage of one over the other? Zero padding at the end is easier to program. And also, does it really matter nowadays if I pad to the next power of two? I mean the algorithms nowadays are fast enough either way, right?
2. To clarify, I want to get the PSD of each segment separately and then take their average to get a single PSD of the underlying signal. When I do take the average, I am also thinking about weighing it according to how many nonzero elements were there in the original segment. If I have three segments of length 100, 1900, and 2000, I would pad the first with 1900 zeros and the second with 100 to make them all the same length. Compute the PSD for each segment individually and then do
PSDaverage=PSD1*(100/4000)+PSD2*(1900/4000)+PSD3*(2000/4000).
Does this sound reasonable? Because obviously zero padding adds no new information and the 2000-segment contains much more information than 100-segment so it should be taken more seriously.
3. Lastly to get a single PSD of the underlying signal, I am thinking of just taking the (weighted?) arithmetic mean of all of the zero-padded PSDs. But one of the previous papers trying to do the same thing I am doing, took the log (base 10) of the power and then took their average. That doesn't make much sense to me. Is there a reason for it or any particular advantage someone knows of? The only thing I can think of is that the power might range over a few orders of magnitude but doing a log plot makes sense for plotting. But taking the log and then averaging weights different things differently. Is it better to do
PSDaverage=PSD1*(100/4000)+PSD2*(1900/4000)+PSD3*(2000/4000).
or should I do
PSDaverage=10^(log(PSD1)*(100/4000)+log(PSD2)*(1900/4000)+log(PSD3)*(2000/4000)).
Any comment/constructive criticism is welcome. If it is relevant, I am thinking of using the multitaper method with eight slepian sequences. Thank you in advance! - Looking for Wisdom and Insight! (talk) 05:01, 29 November 2011 (UTC)
Thanks for the reply but how do I measure how long the data segments are in relation to the signal frequencies I am interested in (what does that mean)? As for zero padding, I forgot to mention I will detrend each segment. Lastly, could the usage of geometric mean have something to do with various orders of magnitudes, especially if they were working in single precision or something (the paper is two decades old and I think they just wrote all of their own routines)? I suspect (I don't know for sure) the power might range from 10^-4 to 10^6 or something of the sort. So they were perhaps afraid of numerical error (like if they were using single precision) so taking the log makes the small numbers larger (in magnitude) and large numbers smaller. The results in their paper were over several orders of magnitudes. I am working in MATLAB so everything is double and I am not writing my own FFT or anything so I am not crazy about powers of two either. Another reason could be that because of the exponential decay in the power vs. frequency domain (that is what the PSD looks like in their paper), they thought that the geometric mean would be more appropriate than the arithmetic mean. What do you think? - Looking for Wisdom and Insight! (talk) 00:53, 30 November 2011 (UTC)