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+<header>
+<h1 class="title">Module <code>szyfrow.support.norms</code></h1>
+</header>
+<section id="section-intro">
+<p>Various norms, for calcuating the distances between two frequency
+profiles.</p>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">&#34;&#34;&#34;Various norms, for calcuating the distances between two frequency
+profiles.
+&#34;&#34;&#34;
+
+import collections
+from math import log10
+
+def lp(v1, v2=None, p=2):
+    &#34;&#34;&#34;Find the L_p norm. If passed one vector, find the length of that vector.
+    If passed two vectors, find the length of the difference between them.
+    &#34;&#34;&#34;
+    if v2:
+        vec = {k: abs(v1[k] - v2[k]) for k in (v1.keys() | v2.keys())}
+    else:
+        vec = v1
+        return sum(v ** p for v in vec.values()) ** (1.0 / p)
+
+def l1(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; l1({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; l1(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}))
+    0.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &#34;&#34;&#34;    
+    return lp(v1, v2, 1)
+
+def l2(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+    
+    &gt;&gt;&gt; l2({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.73205080...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}))
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.732050807...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}), \
+           normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.81649658...
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &#34;&#34;&#34;
+    return lp(v1, v2, 2)
+
+def l3(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; l3({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l3({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.44224957...
+    &gt;&gt;&gt; l3({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.4422495703...
+    &gt;&gt;&gt; l3(normalise({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}), \
+           normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.718144896...
+    &gt;&gt;&gt; l3({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &gt;&gt;&gt; l3(normalise({&#39;a&#39;:0, &#39;b&#39;:1}), normalise({&#39;a&#39;:1, &#39;b&#39;:1})) # doctest: +ELLIPSIS
+    0.6299605249...
+    &#34;&#34;&#34;
+    return lp(v1, v2, 3)
+
+def linf(v1, v2=None):   
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2&#34;&#34;&#34;
+    if v2:
+        vec = {k: abs(v1[k] - v2[k]) for k in (v1.keys() | v2.keys())}
+    else:
+        vec = v1
+        return max(v for v in vec.values())
+
+
+def scale(frequencies, norm=l2):
+    length = norm(frequencies)
+    return collections.defaultdict(int, 
+        {k: v / length for k, v in frequencies.items()})
+
+def l2_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they have a unit euclidean length
+    
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.7071067...), (2, 0.7071067...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+    &#34;&#34;&#34;
+    return scale(f, l2)
+
+def l1_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they sum to one
+    
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1}).items())
+    [(1, 0.5), (2, 0.5)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+    [(1, 0.25), (2, 0.5), (3, 0.25)]
+    &#34;&#34;&#34;    
+    return scale(f, l1)
+
+normalise = l1_scale
+euclidean_distance = l2
+euclidean_scale = l2_scale
+
+
+def geometric_mean(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the geometric mean of the absolute differences between two frequency profiles, 
+    expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+    
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.01382140...
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.0
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:0})) # doctest: +ELLIPSIS
+    0.009259259...
+    &#34;&#34;&#34;
+    total = 1.0
+    for k in frequencies1:
+        total *= abs(frequencies1[k] - frequencies2[k])
+    return total
+
+def harmonic_mean(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the harmonic mean of the absolute differences between two frequency profiles, 
+    expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.285714285...
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.228571428571...
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.0
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:0})) # doctest: +ELLIPSIS
+    0.2
+    &#34;&#34;&#34;
+    total = 0.0
+    for k in frequencies1:
+        if abs(frequencies1[k] - frequencies2[k]) == 0:
+            return 0.0
+        total += 1.0 / abs(frequencies1[k] - frequencies2[k])
+    return len(frequencies1) / total
+
+
+def cosine_similarity(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.0000000000...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.0000000000...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    0.5773502691...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1}) # doctest: +ELLIPSIS
+    0.7071067811...
+    &#34;&#34;&#34;
+    numerator = 0
+    length1 = 0
+    length2 = 0
+    for k in frequencies1:
+        numerator += frequencies1[k] * frequencies2[k]
+        length1 += frequencies1[k]**2
+    for k in frequencies2:
+        length2 += frequencies2[k]**2
+    return numerator / (length1 ** 0.5 * length2 ** 0.5)
+
+
+
+if __name__ == &#34;__main__&#34;:
+    import doctest
+    doctest.testmod()</code></pre>
+</details>
+</section>
+<section>
+</section>
+<section>
+</section>
+<section>
+<h2 class="section-title" id="header-functions">Functions</h2>
+<dl>
+<dt id="szyfrow.support.norms.cosine_similarity"><code class="name flex">
+<span>def <span class="ident">cosine_similarity</span></span>(<span>frequencies1, frequencies2)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as dictionaries.
+Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; cosine_similarity({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.0000000000...
+&gt;&gt;&gt; cosine_similarity({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.0000000000...
+&gt;&gt;&gt; cosine_similarity({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+0.5773502691...
+&gt;&gt;&gt; cosine_similarity({'a':0, 'b':1}, {'a':1, 'b':1}) # doctest: +ELLIPSIS
+0.7071067811...
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def cosine_similarity(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.0000000000...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.0000000000...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    0.5773502691...
+    &gt;&gt;&gt; cosine_similarity({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1}) # doctest: +ELLIPSIS
+    0.7071067811...
+    &#34;&#34;&#34;
+    numerator = 0
+    length1 = 0
+    length2 = 0
+    for k in frequencies1:
+        numerator += frequencies1[k] * frequencies2[k]
+        length1 += frequencies1[k]**2
+    for k in frequencies2:
+        length2 += frequencies2[k]**2
+    return numerator / (length1 ** 0.5 * length2 ** 0.5)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.euclidean_distance"><code class="name flex">
+<span>def <span class="ident">euclidean_distance</span></span>(<span>v1, v2=None)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as dictionaries.
+Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+&gt;&gt;&gt; l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.73205080...
+&gt;&gt;&gt; l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+&gt;&gt;&gt; l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.732050807...
+&gt;&gt;&gt; l2(normalise({'a':0, 'b':2, 'c':0}),            normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.81649658...
+&gt;&gt;&gt; l2({'a':0, 'b':1}, {'a':1, 'b':1})
+1.0
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l2(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+    
+    &gt;&gt;&gt; l2({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.73205080...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}))
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.732050807...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}), \
+           normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.81649658...
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &#34;&#34;&#34;
+    return lp(v1, v2, 2)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.euclidean_scale"><code class="name flex">
+<span>def <span class="ident">euclidean_scale</span></span>(<span>f)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Scale a set of frequencies so they have a unit euclidean length</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.7071067...), (2, 0.7071067...)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l2_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they have a unit euclidean length
+    
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.7071067...), (2, 0.7071067...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+    &#34;&#34;&#34;
+    return scale(f, l2)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.geometric_mean"><code class="name flex">
+<span>def <span class="ident">geometric_mean</span></span>(<span>frequencies1, frequencies2)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the geometric mean of the absolute differences between two frequency profiles,
+expressed as dictionaries.
+Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+&gt;&gt;&gt; geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+&gt;&gt;&gt; geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1})
+3.0
+&gt;&gt;&gt; geometric_mean(normalise({'a':2, 'b':2, 'c':2}),                        normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+0.01382140...
+&gt;&gt;&gt; geometric_mean(normalise({'a':2, 'b':2, 'c':2}),                        normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.0
+&gt;&gt;&gt; geometric_mean(normalise({'a':2, 'b':2, 'c':2}),                        normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+0.009259259...
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def geometric_mean(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the geometric mean of the absolute differences between two frequency profiles, 
+    expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+    
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; geometric_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.01382140...
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.0
+    &gt;&gt;&gt; geometric_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                       normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:0})) # doctest: +ELLIPSIS
+    0.009259259...
+    &#34;&#34;&#34;
+    total = 1.0
+    for k in frequencies1:
+        total *= abs(frequencies1[k] - frequencies2[k])
+    return total</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.harmonic_mean"><code class="name flex">
+<span>def <span class="ident">harmonic_mean</span></span>(<span>frequencies1, frequencies2)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the harmonic mean of the absolute differences between two frequency profiles,
+expressed as dictionaries.
+Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+&gt;&gt;&gt; harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+&gt;&gt;&gt; harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1}) # doctest: +ELLIPSIS
+1.285714285...
+&gt;&gt;&gt; harmonic_mean(normalise({'a':2, 'b':2, 'c':2}),                       normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+0.228571428571...
+&gt;&gt;&gt; harmonic_mean(normalise({'a':2, 'b':2, 'c':2}),                       normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.0
+&gt;&gt;&gt; harmonic_mean(normalise({'a':2, 'b':2, 'c':2}),                       normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+0.2
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def harmonic_mean(frequencies1, frequencies2):
+    &#34;&#34;&#34;Finds the harmonic mean of the absolute differences between two frequency profiles, 
+    expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    1.0
+    &gt;&gt;&gt; harmonic_mean({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.285714285...
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:5, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.228571428571...
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.0
+    &gt;&gt;&gt; harmonic_mean(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), \
+                      normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:0})) # doctest: +ELLIPSIS
+    0.2
+    &#34;&#34;&#34;
+    total = 0.0
+    for k in frequencies1:
+        if abs(frequencies1[k] - frequencies2[k]) == 0:
+            return 0.0
+        total += 1.0 / abs(frequencies1[k] - frequencies2[k])
+    return len(frequencies1) / total</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.l1"><code class="name flex">
+<span>def <span class="ident">l1</span></span>(<span>v1, v2=None)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as
+dictionaries. Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; l1({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+&gt;&gt;&gt; l1({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+3.0
+&gt;&gt;&gt; l1(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+&gt;&gt;&gt; l1({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1})
+3.0
+&gt;&gt;&gt; l1({'a':0, 'b':1}, {'a':1, 'b':1})
+1.0
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l1(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; l1({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; l1(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}))
+    0.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    3.0
+    &gt;&gt;&gt; l1({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &#34;&#34;&#34;    
+    return lp(v1, v2, 1)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.l1_scale"><code class="name flex">
+<span>def <span class="ident">l1_scale</span></span>(<span>f)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Scale a set of frequencies so they sum to one</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; sorted(normalise({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 1}).items())
+[(1, 0.5), (2, 0.5)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+[(1, 0.25), (2, 0.5), (3, 0.25)]
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l1_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they sum to one
+    
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1}).items())
+    [(1, 0.5), (2, 0.5)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+    [(1, 0.25), (2, 0.5), (3, 0.25)]
+    &#34;&#34;&#34;    
+    return scale(f, l1)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.l2"><code class="name flex">
+<span>def <span class="ident">l2</span></span>(<span>v1, v2=None)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as dictionaries.
+Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+&gt;&gt;&gt; l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.73205080...
+&gt;&gt;&gt; l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+&gt;&gt;&gt; l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.732050807...
+&gt;&gt;&gt; l2(normalise({'a':0, 'b':2, 'c':0}),            normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.81649658...
+&gt;&gt;&gt; l2({'a':0, 'b':1}, {'a':1, 'b':1})
+1.0
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l2(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as dictionaries.
+    Assumes every key in frequencies1 is also in frequencies2
+    
+    &gt;&gt;&gt; l2({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.73205080...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}), normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}))
+    0.0
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.732050807...
+    &gt;&gt;&gt; l2(normalise({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}), \
+           normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.81649658...
+    &gt;&gt;&gt; l2({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &#34;&#34;&#34;
+    return lp(v1, v2, 2)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.l2_scale"><code class="name flex">
+<span>def <span class="ident">l2_scale</span></span>(<span>f)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Scale a set of frequencies so they have a unit euclidean length</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.7071067...), (2, 0.7071067...)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+&gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l2_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they have a unit euclidean length
+    
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.7071067...), (2, 0.7071067...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+    &gt;&gt;&gt; sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+    &#34;&#34;&#34;
+    return scale(f, l2)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.l3"><code class="name flex">
+<span>def <span class="ident">l3</span></span>(<span>v1, v2=None)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as
+dictionaries. Assumes every key in frequencies1 is also in frequencies2</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; l3({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+&gt;&gt;&gt; l3({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.44224957...
+&gt;&gt;&gt; l3({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.4422495703...
+&gt;&gt;&gt; l3(normalise({'a':0, 'b':2, 'c':0}),            normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.718144896...
+&gt;&gt;&gt; l3({'a':0, 'b':1}, {'a':1, 'b':1})
+1.0
+&gt;&gt;&gt; l3(normalise({'a':0, 'b':1}), normalise({'a':1, 'b':1})) # doctest: +ELLIPSIS
+0.6299605249...
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l3(v1, v2=None):
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+    &gt;&gt;&gt; l3({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})
+    0.0
+    &gt;&gt;&gt; l3({&#39;a&#39;:2, &#39;b&#39;:2, &#39;c&#39;:2}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.44224957...
+    &gt;&gt;&gt; l3({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}, {&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1}) # doctest: +ELLIPSIS
+    1.4422495703...
+    &gt;&gt;&gt; l3(normalise({&#39;a&#39;:0, &#39;b&#39;:2, &#39;c&#39;:0}), \
+           normalise({&#39;a&#39;:1, &#39;b&#39;:1, &#39;c&#39;:1})) # doctest: +ELLIPSIS
+    0.718144896...
+    &gt;&gt;&gt; l3({&#39;a&#39;:0, &#39;b&#39;:1}, {&#39;a&#39;:1, &#39;b&#39;:1})
+    1.0
+    &gt;&gt;&gt; l3(normalise({&#39;a&#39;:0, &#39;b&#39;:1}), normalise({&#39;a&#39;:1, &#39;b&#39;:1})) # doctest: +ELLIPSIS
+    0.6299605249...
+    &#34;&#34;&#34;
+    return lp(v1, v2, 3)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.linf"><code class="name flex">
+<span>def <span class="ident">linf</span></span>(<span>v1, v2=None)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Finds the distances between two frequency profiles, expressed as
+dictionaries. Assumes every key in frequencies1 is also in frequencies2</p></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def linf(v1, v2=None):   
+    &#34;&#34;&#34;Finds the distances between two frequency profiles, expressed as 
+    dictionaries. Assumes every key in frequencies1 is also in frequencies2&#34;&#34;&#34;
+    if v2:
+        vec = {k: abs(v1[k] - v2[k]) for k in (v1.keys() | v2.keys())}
+    else:
+        vec = v1
+        return max(v for v in vec.values())</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.lp"><code class="name flex">
+<span>def <span class="ident">lp</span></span>(<span>v1, v2=None, p=2)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Find the L_p norm. If passed one vector, find the length of that vector.
+If passed two vectors, find the length of the difference between them.</p></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def lp(v1, v2=None, p=2):
+    &#34;&#34;&#34;Find the L_p norm. If passed one vector, find the length of that vector.
+    If passed two vectors, find the length of the difference between them.
+    &#34;&#34;&#34;
+    if v2:
+        vec = {k: abs(v1[k] - v2[k]) for k in (v1.keys() | v2.keys())}
+    else:
+        vec = v1
+        return sum(v ** p for v in vec.values()) ** (1.0 / p)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.normalise"><code class="name flex">
+<span>def <span class="ident">normalise</span></span>(<span>f)</span>
+</code></dt>
+<dd>
+<div class="desc"><p>Scale a set of frequencies so they sum to one</p>
+<pre><code class="language-python-repl">&gt;&gt;&gt; sorted(normalise({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 1}).items())
+[(1, 0.5), (2, 0.5)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+&gt;&gt;&gt; sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+[(1, 0.25), (2, 0.5), (3, 0.25)]
+</code></pre></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def l1_scale(f):
+    &#34;&#34;&#34;Scale a set of frequencies so they sum to one
+    
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 0}).items())
+    [(1, 1.0), (2, 0.0)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1}).items())
+    [(1, 0.5), (2, 0.5)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+    [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+    &gt;&gt;&gt; sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+    [(1, 0.25), (2, 0.5), (3, 0.25)]
+    &#34;&#34;&#34;    
+    return scale(f, l1)</code></pre>
+</details>
+</dd>
+<dt id="szyfrow.support.norms.scale"><code class="name flex">
+<span>def <span class="ident">scale</span></span>(<span>frequencies, norm=&lt;function l2&gt;)</span>
+</code></dt>
+<dd>
+<div class="desc"></div>
+<details class="source">
+<summary>
+<span>Expand source code</span>
+</summary>
+<pre><code class="python">def scale(frequencies, norm=l2):
+    length = norm(frequencies)
+    return collections.defaultdict(int, 
+        {k: v / length for k, v in frequencies.items()})</code></pre>
+</details>
+</dd>
+</dl>
+</section>
+<section>
+</section>
+</article>
+<nav id="sidebar">
+<h1>Index</h1>
+<div class="toc">
+<ul></ul>
+</div>
+<ul id="index">
+<li><h3>Super-module</h3>
+<ul>
+<li><code><a title="szyfrow.support" href="index.html">szyfrow.support</a></code></li>
+</ul>
+</li>
+<li><h3><a href="#header-functions">Functions</a></h3>
+<ul class="two-column">
+<li><code><a title="szyfrow.support.norms.cosine_similarity" href="#szyfrow.support.norms.cosine_similarity">cosine_similarity</a></code></li>
+<li><code><a title="szyfrow.support.norms.euclidean_distance" href="#szyfrow.support.norms.euclidean_distance">euclidean_distance</a></code></li>
+<li><code><a title="szyfrow.support.norms.euclidean_scale" href="#szyfrow.support.norms.euclidean_scale">euclidean_scale</a></code></li>
+<li><code><a title="szyfrow.support.norms.geometric_mean" href="#szyfrow.support.norms.geometric_mean">geometric_mean</a></code></li>
+<li><code><a title="szyfrow.support.norms.harmonic_mean" href="#szyfrow.support.norms.harmonic_mean">harmonic_mean</a></code></li>
+<li><code><a title="szyfrow.support.norms.l1" href="#szyfrow.support.norms.l1">l1</a></code></li>
+<li><code><a title="szyfrow.support.norms.l1_scale" href="#szyfrow.support.norms.l1_scale">l1_scale</a></code></li>
+<li><code><a title="szyfrow.support.norms.l2" href="#szyfrow.support.norms.l2">l2</a></code></li>
+<li><code><a title="szyfrow.support.norms.l2_scale" href="#szyfrow.support.norms.l2_scale">l2_scale</a></code></li>
+<li><code><a title="szyfrow.support.norms.l3" href="#szyfrow.support.norms.l3">l3</a></code></li>
+<li><code><a title="szyfrow.support.norms.linf" href="#szyfrow.support.norms.linf">linf</a></code></li>
+<li><code><a title="szyfrow.support.norms.lp" href="#szyfrow.support.norms.lp">lp</a></code></li>
+<li><code><a title="szyfrow.support.norms.normalise" href="#szyfrow.support.norms.normalise">normalise</a></code></li>
+<li><code><a title="szyfrow.support.norms.scale" href="#szyfrow.support.norms.scale">scale</a></code></li>
+</ul>
+</li>
+</ul>
+</nav>
+</main>
+<footer id="footer">
+<p>Generated by <a href="https://pdoc3.github.io/pdoc"><cite>pdoc</cite> 0.9.2</a>.</p>
+</footer>
+</body>
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