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I'm preparing a presentation about parallel statistics. I plan to illustrate the formulas for distributed computation of the mean and variance with examples involving center of gravity and moment of inertia. I'm wondering if there is a physical interpretation of the third or higher central moments that I can use to help illustrate the general formula.

James Koppel
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  • That relates the properties of moments to graphs of PDFs, but is not useful for visualizing computations done on moments.

    I'd like to be able to, for example, write Welford's algorithm (http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#On-line_algorithm) on the board, and then draw a picture and say "This is $x_n-\overline{x_n}$" and have the equality become somewhat obvious.

    Also at that link is a streaming algorithm for the mean. It takes a few seconds to see why the formula works, but becomes immediately obvious if I draw it as a center of gravity problem.

     – James Koppel Jul 11 '11 at 22:00
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    If you must preserve the interpretation of $x$ as a length, then anything above third moments will require too many dimensions to illustrate! In spirit, your question and comment are like the Ancient Greek drive to cast all arithmetic in terms of geometry. That indeed provides insight, but it also limits the scope of what can be done to a small number of dimensions (0 through 3). Creative use of graphics can help us visualize higher dimensions (and therefore, perhaps with higher moments). Thus I think you shouldn't dismiss graph-based illustrations out of hand. – whuber Jul 11 '11 at 22:16
  • I'm not sure what you mean by physical interpretation. (Perhaps it's that I don't think about mean or variance in terms of physical quantities).

    If you're just trying to to reprise the Welford algorithm, is it mathematically anything more than the insight that to compute statistics of the form $T = \frac{1}{N} \sum_{i=1}^n f(x_i)$, you can always compute them recursively via:

    $;T_1 = f(x_1)$, and

    $T_i = ((N-1)\cdot T_{i-1} + x_i) / N ;;$

    ? Raw moments can always be computed this way, and formulas for centered moments can (presumably) be derived from these?

     – jpillow Jul 12 '11 at 05:23
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    Closely related: http://stats.stackexchange.com/questions/132914, http://stats.stackexchange.com/questions/17595. – whuber May 08 '15 at 18:45
  • I show pictures on https://stats.stackexchange.com/a/324197/99274 higher moments also explained. – Carl Feb 16 '18 at 18:28
  • We used 3rd and 4th moments for input features in this image classification paper using Hermite and Laguerre neural networks ( http://bol.egr.uh.edu/sites/bol/files/files/publications/IJKESDP010304-PETERSON.pdf ). –  Apr 11 '21 at 22:10
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    For what purposes would the distributed computation of higher moments be useful? I’d say better to show those applications than to show physical analogies. (I would see applications easily enough for paralllel computation of quantiles, but there may not be so many applications with theee higher moments.) – Matt F. Feb 07 '22 at 09:52
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    Wow, this is a truly old question.

    I asked this question over a decade ago when I was a software-engineering intern at a hedge fund. I know there was demand from the traders to compute the kurtosis of large datasets; for what purpose, I was not privy. The audience of the talk would have known better than I did what the applications were.

     – James Koppel Mar 19 '22 at 23:46

1 Answers1

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If by graphical representation you meant histograms, I gather this is the best approach to provide a visual display of the moments. Please be aware we cannot specify any value for kurtosis. But there is a formula to provide kurtosis under random simulations.

Being k = kurtosis, sk = skewness, the formula is: k > sk^2 +1.

Below, a histogram with the specifications.

enter image description here

Marcos
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    This is being automatically flagged as low quality, probably because it is so short. At present it is more of a comment than an answer by our standards. Can you expand on it? We can also turn it into a comment. – Sycorax Apr 12 '21 at 00:53