Harmonic entropy: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 515669614 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 515669634 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2014-07-06 13: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:50:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515669634</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<span style="background-color: #ffffff;">This uses as a spreading function the Gaussian distribution with s=~17 cents (or a lin-frequency deviation of 1%). The basis set is all rationals of Tenney height less than 10000. This uses the complexity-normalization approach, and the complexity function is sqrt(n·d):</span> | <span style="background-color: #ffffff;">This uses as a spreading function the Gaussian distribution with s=~17 cents (or a lin-frequency deviation of 1%). The basis set is all rationals of Tenney height less than 10000. This uses the complexity-normalization approach, and the complexity function is sqrt(n·d):</span> | ||
[[image:http://i.imgur.com/tNg7z1P.png | [[image:http://i.imgur.com/tNg7z1P.png]] | ||
<span style="background-color: #ffffff;">This example uses the same spreading function and standard deviation, but this time the basis set is all rationals of Weil height less than 100. The complexity function here is max(n,d):</span> | <span style="background-color: #ffffff;">This example uses the same spreading function and standard deviation, but this time the basis set is all rationals of Weil height less than 100. The complexity function here is max(n,d):</span> | ||
[[image:http://i.imgur.com/TZdU6eD.png | [[image:http://i.imgur.com/TZdU6eD.png]] | ||
The following image compares the domain-integral and complexity-normalization approaches by overlaying the two curves on top of each other. In both cases, the spreading function is again a Gaussian with s=~17 cents, and the basis set is all those rationals with Tenney height ≤ 10000. It can be seen that the curves are extremely similar, and that the locations of the minima and maxima are largely preserved: | The following image compares the domain-integral and complexity-normalization approaches by overlaying the two curves on top of each other. In both cases, the spreading function is again a Gaussian with s=~17 cents, and the basis set is all those rationals with Tenney height ≤ 10000. It can be seen that the curves are extremely similar, and that the locations of the minima and maxima are largely preserved: | ||
[[image:http://i.imgur.com/5QPTsEP.png width="800" height="600"]] | [[image:http://i.imgur.com/5QPTsEP.png width="800" height="600"]] | ||
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<span style="background-color: #ffffff;">This uses as a spreading function the Gaussian distribution with s=~17 cents (or a lin-frequency deviation of 1%). The basis set is all rationals of Tenney height less than 10000. This uses the complexity-normalization approach, and the complexity function is sqrt(n·d):</span><br /> | <span style="background-color: #ffffff;">This uses as a spreading function the Gaussian distribution with s=~17 cents (or a lin-frequency deviation of 1%). The basis set is all rationals of Tenney height less than 10000. This uses the complexity-normalization approach, and the complexity function is sqrt(n·d):</span><br /> | ||
<!-- ws:start:WikiTextRemoteImageRule:71:&lt;img src=&quot;http://i.imgur.com/tNg7z1P.png&quot; alt=&quot; | <!-- ws:start:WikiTextRemoteImageRule:71:&lt;img src=&quot;http://i.imgur.com/tNg7z1P.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://i.imgur.com/tNg7z1P.png" alt="external image tNg7z1P.png" title="external image tNg7z1P.png" /><!-- ws:end:WikiTextRemoteImageRule:71 --><br /> | ||
<span style="background-color: #ffffff;">This example uses the same spreading function and standard deviation, but this time the basis set is all rationals of Weil height less than 100. The complexity function here is max(n,d):</span><br /> | <span style="background-color: #ffffff;">This example uses the same spreading function and standard deviation, but this time the basis set is all rationals of Weil height less than 100. The complexity function here is max(n,d):</span><br /> | ||
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<!-- ws:start:WikiTextRemoteImageRule:72:&lt;img src=&quot;http://i.imgur.com/TZdU6eD.png&quot; alt=&quot; | <!-- ws:start:WikiTextRemoteImageRule:72:&lt;img src=&quot;http://i.imgur.com/TZdU6eD.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://i.imgur.com/TZdU6eD.png" alt="external image TZdU6eD.png" title="external image TZdU6eD.png" /><!-- ws:end:WikiTextRemoteImageRule:72 --><br /> | ||
The following image compares the domain-integral and complexity-normalization approaches by overlaying the two curves on top of each other. In both cases, the spreading function is again a Gaussian with s=~17 cents, and the basis set is all those rationals with Tenney height ≤ 10000. It can be seen that the curves are extremely similar, and that the locations of the minima and maxima are largely preserved:<br /> | The following image compares the domain-integral and complexity-normalization approaches by overlaying the two curves on top of each other. In both cases, the spreading function is again a Gaussian with s=~17 cents, and the basis set is all those rationals with Tenney height ≤ 10000. It can be seen that the curves are extremely similar, and that the locations of the minima and maxima are largely preserved:<br /> | ||
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