Structure metric

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[[toc]]

=Definition=
The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[constant structure]] [[periodic scale]] within the period, which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. If **s** is a periodic scale with quasiperiod **P**, and if **s**[i] with  0≤i<**P** is a note of **s** within the period **P**, then we may define the base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i], 0≤j<**P**}. These have the property that the interval between the base note **s**[j] and the note i steps away, **s**[j+i], is in class(i), the interval class to which **s**[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of **s**[i], and **P**-n which correspond to indicies of intervals other than **s**[i]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// ||**s**[i]|| of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |**s**[i] - **s**[j]| ||.

=Properties=
The structure metric has the following properties:

1. d(**s**[i], **s**[i]) = 0
|| **s**[i] - **s**[i] || = ||0|| which equals 0.

2. d(**s**[i], **s**[j]) ≥ 0
This is so since the cardinality n of the base point set is less than or equal to **P**.

3. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]
Suppose ||**s**[**I**]|| equals 0 with 0 < **I** < **P**. Then **s**[j+**I**] - **s**[j] = **s**[**I**], so that **s** is periodic with quasiperiod **I**. But by assumption, **P** is the least quasiperiod of **s**. Hence, ||**s**[**I**]|| equals 0 implies **I** equals 0. It follows that if d(**s**[i], **s**[j]) equals || **s**[i] - **s**[j] || equals 0, then **s**[i] - **s**[j] equals 0 and **s**[i] equals **s**[j]. 

4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
d(**s**[i], **s**[j]) equals || |**s**[i] - **s**[j]| || equals || |**s**[j] - **s**[i]| || equals d(**s**[j], **s**[i])

5. d(**s**[i], **s**[k]) ≤ d(**s**[i], **s**[j]) + d(**s**[j], **s**[k])
First, || **s**[i + j] mod **O** || ≤ ||**s**[i]|| + ||**s**[j]|| where **O** is the interval of equivalence. If an interval in the interval class of **s**[i] equals **s**[i] and an interval in the interval class of **s**[j] equals **s**[j], then their product, reduced modulo the interval of equivalence **O** equals **s**[**P**], will be **s**[i + j] mod **O**. Hence to get an interval in the class of **s**[i + j] mod **O** other than **s**[i + j] mod **O** as a product, either the interval in the class of **s**[i] must be other than **s**[i], or the interval in the class of **s**[j] must be other than **s**[j]. If always only one of the intervals is different than the defining interval for its class, then || **s**[i + j] mod **O** || equals ||**s**[i]|| + ||**s**[j]||. However, there may be overlap, so that the first interval is not in the class for **s**[i] and the second not in the class for **s**[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(**s**[i], **s**[j]) + d(**s**[j], **s**[k]) = || |**s**[i] - **s**[j]| || + || |**s**[j] - **s**[k]| || ≥ || |**s**[i] - **s**[k]| || = d(**s**[i], **s**[k]).

These properties mean that the structure metric defines a //finite metric space//. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.

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<html><head><title>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule:4:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>
<!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --><div style="margin-left: 1em;"><a href="#Properties">Properties</a></div>
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
The <em>structure metric</em> is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within the period, which give to it the property of being a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow">finite metric space</a>. If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if <strong>s</strong>[i] with  0≤i&lt;<strong>P</strong> is a note of <strong>s</strong> within the period <strong>P</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i], 0≤j&lt;<strong>P</strong>}. These have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. In other words, there are <strong>P</strong>-n intervals, counting multiplicities, in the class of <strong>s</strong>[i] other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| ||.<br />
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1>
The structure metric has the following properties:<br />
<br />
1. d(<strong>s</strong>[i], <strong>s</strong>[i]) = 0<br />
|| <strong>s</strong>[i] - <strong>s</strong>[i] || = ||0|| which equals 0.<br />
<br />
2. d(<strong>s</strong>[i], <strong>s</strong>[j]) ≥ 0<br />
This is so since the cardinality n of the base point set is less than or equal to <strong>P</strong>.<br />
<br />
3. d(<strong>s</strong>[i], <strong>s</strong>[j]) = 0 implies <strong>s</strong>[i] equals <strong>s</strong>[j]<br />
Suppose ||<strong>s</strong>[<strong>I</strong>]|| equals 0 with 0 &lt; <strong>I</strong> &lt; <strong>P</strong>. Then <strong>s</strong>[j+<strong>I</strong>] - <strong>s</strong>[j] = <strong>s</strong>[<strong>I</strong>], so that <strong>s</strong> is periodic with quasiperiod <strong>I</strong>. But by assumption, <strong>P</strong> is the least quasiperiod of <strong>s</strong>. Hence, ||<strong>s</strong>[<strong>I</strong>]|| equals 0 implies <strong>I</strong> equals 0. It follows that if d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || <strong>s</strong>[i] - <strong>s</strong>[j] || equals 0, then <strong>s</strong>[i] - <strong>s</strong>[j] equals 0 and <strong>s</strong>[i] equals <strong>s</strong>[j]. <br />
<br />
4. d(<strong>s</strong>[i], <strong>s</strong>[j]) = d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || |<strong>s</strong>[i] - <strong>s</strong>[j]| || equals || |<strong>s</strong>[j] - <strong>s</strong>[i]| || equals d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
<br />
5. d(<strong>s</strong>[i], <strong>s</strong>[k]) ≤ d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k])<br />
First, || <strong>s</strong>[i + j] mod <strong>O</strong> || ≤ ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]|| where <strong>O</strong> is the interval of equivalence. If an interval in the interval class of <strong>s</strong>[i] equals <strong>s</strong>[i] and an interval in the interval class of <strong>s</strong>[j] equals <strong>s</strong>[j], then their product, reduced modulo the interval of equivalence <strong>O</strong> equals <strong>s</strong>[<strong>P</strong>], will be <strong>s</strong>[i + j] mod <strong>O</strong>. Hence to get an interval in the class of <strong>s</strong>[i + j] mod <strong>O</strong> other than <strong>s</strong>[i + j] mod <strong>O</strong> as a product, either the interval in the class of <strong>s</strong>[i] must be other than <strong>s</strong>[i], or the interval in the class of <strong>s</strong>[j] must be other than <strong>s</strong>[j]. If always only one of the intervals is different than the defining interval for its class, then || <strong>s</strong>[i + j] mod <strong>O</strong> || equals ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]||. However, there may be overlap, so that the first interval is not in the class for <strong>s</strong>[i] and the second not in the class for <strong>s</strong>[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| || + || |<strong>s</strong>[j] - <strong>s</strong>[k]| || ≥ || |<strong>s</strong>[i] - <strong>s</strong>[k]| || = d(<strong>s</strong>[i], <strong>s</strong>[k]).<br />
<br />
These properties mean that the structure metric defines a <em>finite metric space</em>. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.</body></html>