Harmonic entropy: Difference between revisions

Line 359:

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=0.5%.png|640px]]

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

[[File:HE vs UHE s=1%.png|640px]]

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

[[File:HE vs UHE s=1.5%.png|640px]]

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).

@@ Line 359: / Line 359: @@
 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=0.5%.png|640px]]
-[[File:HE vs UHE s=1%.png]]
+[[File:HE vs UHE s=1%.png|640px]]
-[[File:HE vs UHE s=1.5%.png]]
+[[File:HE vs UHE s=1.5%.png|640px]]
 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).