Harmonic entropy: Difference between revisions

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However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.

It would be nice to show ~~that~~ the unnormalized entropy ~~converges~~ to the normalized entropy~~, up to a constant additive shift or multiplicative scaling,~~ in the limit of large <math>N</math>. However, we will leave this for future research, as well as the question of how to do an exact derivation of ~~the~~ normalized ~~entropy function~~.

Here are some examples for different values of <math>s</math>. All of these are Shannon HE (<math>a=1</math>), using <math>\sqrt{nd}</math> weights, with unreduced rationals (more on this below), with the bound that <math>nd < 1000000</math>, just with different values of <math>s</math>. All have been scaled so that the minimum entropy is 0, and the maximum entropy is 1:

[[File:HE vs UHE s=0.5%.png]]

[[File:HE vs UHE s=1%.png]]

[[File:HE vs UHE s=1.5%.png]]

As you can see, the unnormalized version is extremely close to a linear function of the normalized one. A similar situation holds for larger values of <math>a</math>. The Pearson correlation coefficient of "rho" is also given, and is typically very close to 1 - for example, for <math>s=1%</math>, it's equal to 0.99922. The correlation also seems to get better with increasing values of <math>N</math>, such that the correlation for N=1,000,000 (shown above) is much better than the one for N=10,000 (not pictured).

In the above examples, note that there are slightly adjusted values of <math>s</math> (usually by less than a cent) between the normalized and unnormalized comparisons for each plot. For example, in the plot for <math>s=1%</math>, corresponding to 17.2264 cents, we compare to a slightly adjusted UHE of 16.4764 cents. This is because, empirically, sometimes a very slight adjustment corresponds to a better correlation coefficient, suggesting that the UHE may be equivalent to the HE with a miniscule adjustment in the value of <math>s</math>.

It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large <math>N</math>, and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in <math>s</math> or <math>a</math>). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.

For now, we will start with a derivation of the unnormalized entropy for <math>N=\infty</math>, as an interesting function worthy of study in its own right - not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[The_Riemann_Zeta_Function_and_Tuning|Riemann Zeta function]].

@@ Line 357: / Line 357: @@
 However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.
-It would be nice to show that the unnormalized entropy converges to the normalized entropy, up to a constant additive shift or multiplicative scaling, in the limit of large <math>N</math>. However, we will leave this for future research, as well as the question of how to do an exact derivation of the normalized entropy function.
+Here are some examples for different values of <math>s</math>. All of these are Shannon HE (<math>a=1</math>), using <math>\sqrt{nd}</math> weights, with unreduced rationals (more on this below), with the bound that <math>nd < 1000000</math>, just with different values of <math>s</math>. All have been scaled so that the minimum entropy is 0, and the maximum entropy is 1:
+[[File:HE vs UHE s=0.5%.png]]
+[[File:HE vs UHE s=1%.png]]
+[[File:HE vs UHE s=1.5%.png]]
+As you can see, the unnormalized version is extremely close to a linear function of the normalized one. A similar situation holds for larger values of <math>a</math>. The Pearson correlation coefficient of "rho" is also given, and is typically very close to 1 - for example, for <math>s=1%</math>, it's equal to 0.99922. The correlation also seems to get better with increasing values of <math>N</math>, such that the correlation for N=1,000,000 (shown above) is much better than the one for N=10,000 (not pictured).
+In the above examples, note that there are slightly adjusted values of <math>s</math> (usually by less than a cent) between the normalized and unnormalized comparisons for each plot. For example, in the plot for <math>s=1%</math>, corresponding to 17.2264 cents, we compare to a slightly adjusted UHE of 16.4764 cents. This is because, empirically, sometimes a very slight adjustment corresponds to a better correlation coefficient, suggesting that the UHE may be equivalent to the HE with a miniscule adjustment in the value of <math>s</math>.
+It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large <math>N</math>, and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in <math>s</math> or <math>a</math>). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.
 For now, we will start with a derivation of the unnormalized entropy for <math>N=\infty</math>, as an interesting function worthy of study in its own right - not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[The_Riemann_Zeta_Function_and_Tuning|Riemann Zeta function]].