patterns of a 2d binary matrix

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Feb 25, 2021 at 16:21	history	edited	whuber♦	CC BY-SA 4.0	added 453 characters in body
Feb 25, 2021 at 16:06	comment	added	whuber♦		@Our The same way you would compute the entropy of any dataset. For instance, $a_1$ has ten black cells and 15 white cells, making their relative frequencies 10/25 and 15/25. The entropy is $H(a_1) = -(10/25)\log_2(10/25) - (15/25)\log_2(15/25) = 0.97.$ I should have defined this explicitly in the post.
Feb 25, 2021 at 16:02	comment	added	Our		@whuber yes, I meant, how do you compute the entropy of a matrix
Feb 25, 2021 at 13:24	comment	added	whuber♦		@Our Are you asking what entropy means, or did you mean to ask how the entropy is computed?
Feb 25, 2021 at 8:22	comment	added	Our		ı still do not understand why do you measure the entropy of the resulting matrices in the first example $a_1$
Sep 19, 2018 at 15:29	comment	added	whuber♦		@subhacom 20 years ago, an employee of mine was working on related problems as a PhD student in the stats department of Penn State. That's what I was thinking of in stating this "has been used in the analysis of images." I don't know whether any of that work was published.
Sep 19, 2018 at 15:10	comment	added	subhacom		@whuber Excellent answer. While it makes intuitive sense, is there an article or textbook one can cite for the original derivation of this (I am assuming that if this is your original work you have published it formally in a journal)?
May 2, 2016 at 23:23	comment	added	r.e.s.		(+1) Here's a very general principle: With any multiset $M$, there's the naturally associated entropy of the probability distribution determined by the multiplicities $\mu(e)$ of its distinct elements $e$, namely $p(e) := \frac{\mu(e)}{\sum_{e\in S}\mu(e)}\ \ (e\in S)$, where $S$ is the set of distinct elements in $M$. Examples are multisets formed by size-$k$ neighborhoods of various shapes in objects of various dimensions. (I just posted a 1D application to length-$k$ substrings.)
Dec 14, 2014 at 10:03	comment	added	Cosmo Harrigan		I reproduced the results from this excellent answer by @whuber using NumPy and matplotlib in Python, available here: github.com/cosmoharrigan/matrix-entropy
Apr 18, 2014 at 0:17	history	edited	whuber♦	CC BY-SA 3.0	added 3 characters in body
Nov 7, 2011 at 13:57	vote	accept	Felix S
Oct 28, 2011 at 14:10	history	edited	whuber♦	CC BY-SA 3.0	Typos; clarified some phrases.
Oct 28, 2011 at 12:26	history	bounty ended	Felix S
Oct 27, 2011 at 16:42	comment	added	whuber♦		@ttnphns Here is a popular illustrated help page on the topic.
Oct 27, 2011 at 15:03	comment	added	ttnphns		I'm sorry, I could not understand how you produced your moving sums plots. Please, explain in more detail how to compute the moving sum.
Oct 26, 2011 at 5:12	history	edited	whuber♦	CC BY-SA 3.0	added 21 characters in body
Oct 26, 2011 at 5:02	history	answered	whuber♦	CC BY-SA 3.0

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