Harmonic entropy: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2017-12-27 19:46:10 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2017-12-27 19:57:22 UTC</tt>.<br>

: The original revision id was <tt>~~624270037~~</tt>.<br>

: The original revision id was <tt>624270115</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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Taking all of this, we can rewrite the original expression for Harmonic Renyi Entropy as follows:

[[math]]

H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)

H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)

[[math]]

where the expression $\left[S \ast K\right]^a(-d)$ represents the convolution of S and K, taken to the a'th power.

We have succeeded in representing Harmonic Renyi Entropy in simple terms of two convolution products, each of which can be computed in O(N log N) time.

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<!-- ws:start:WikiTextMathRule:33:

[[math]]&lt;br/&gt;

H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)&lt;br/&gt;[[math]]

H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)&lt;br/&gt;[[math]]

--><script type="math/tex">H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)</script><br />

--><script type="math/tex">H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)</script><br />

<br />

where the expression $\left[S \ast K\right]^a(-d)$ represents the convolution of S and K, taken to the a'th power.<br />

<br />

We have succeeded in representing Harmonic Renyi Entropy in simple terms of two convolution products, each of which can be computed in O(N log N) time.<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2017-12-27 19:46:10 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2017-12-27 19:57:22 UTC</tt>.<br>
-: The original revision id was <tt>624270037</tt>.<br>
+: The original revision id was <tt>624270115</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 299: / Line 299: @@
 Taking all of this, we can rewrite the original expression for Harmonic Renyi Entropy as follows:
 [[math]]
-H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)
+H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)
 [[math]]
+where the expression $\left[S \ast K\right]^a(-d)$ represents the convolution of S and K, taken to the a'th power.
 We have succeeded in representing Harmonic Renyi Entropy in simple terms of two convolution products, each of which can be computed in O(N log N) time.
@@ Line 647: / Line 649: @@
 &lt;!-- ws:start:WikiTextMathRule:33:
 [[math]]&amp;lt;br/&amp;gt;
-H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)&amp;lt;br/&amp;gt;[[math]]
+H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right](-d)} \right)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:33 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;H_a(d) = \frac{1}{1-a} \log_β \left( \frac{\left[S^a \ast K^a\right](-d)}{\left[S \ast K\right]^a(-d)} \right)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:33 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+where the expression $\left[S \ast K\right]^a(-d)$ represents the convolution of S and K, taken to the a'th power.&lt;br /&gt;
 &lt;br /&gt;
 We have succeeded in representing Harmonic Renyi Entropy in simple terms of two convolution products, each of which can be computed in O(N log N) time.&lt;br /&gt;