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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">"""Various norms, for calcuating the distances between two frequency
+profiles.
+"""
+
+import collections
+from math import log10
+
+def lp(v1, v2=None, p=2):
+ """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.
+ """
+ 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):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l1({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l1({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 3.0
+ >>> l1(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+ 0.0
+ >>> l1({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1})
+ 3.0
+ >>> l1({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ """
+ return lp(v1, v2, 1)
+
+def l2(v1, v2=None):
+ """Finds the distances between two frequency profiles, expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.73205080...
+ >>> l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+ 0.0
+ >>> l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.732050807...
+ >>> l2(normalise({'a':0, 'b':2, 'c':0}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.81649658...
+ >>> l2({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ """
+ return lp(v1, v2, 2)
+
+def l3(v1, v2=None):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l3({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l3({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.44224957...
+ >>> l3({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.4422495703...
+ >>> l3(normalise({'a':0, 'b':2, 'c':0}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.718144896...
+ >>> l3({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ >>> l3(normalise({'a':0, 'b':1}), normalise({'a':1, 'b':1})) # doctest: +ELLIPSIS
+ 0.6299605249...
+ """
+ return lp(v1, v2, 3)
+
+def linf(v1, v2=None):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2"""
+ 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):
+ """Scale a set of frequencies so they have a unit euclidean length
+
+ >>> sorted(euclidean_scale({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.7071067...), (2, 0.7071067...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+ """
+ return scale(f, l2)
+
+def l1_scale(f):
+ """Scale a set of frequencies so they sum to one
+
+ >>> sorted(normalise({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(normalise({1: 1, 2: 1}).items())
+ [(1, 0.5), (2, 0.5)]
+ >>> sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+ >>> sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+ [(1, 0.25), (2, 0.5), (3, 0.25)]
+ """
+ return scale(f, l1)
+
+normalise = l1_scale
+euclidean_distance = l2
+euclidean_scale = l2_scale
+
+
+def geometric_mean(frequencies1, frequencies2):
+ """Finds the geometric mean of the absolute differences between two frequency profiles,
+ expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1})
+ 3.0
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+ 0.01382140...
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.0
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+ 0.009259259...
+ """
+ total = 1.0
+ for k in frequencies1:
+ total *= abs(frequencies1[k] - frequencies2[k])
+ return total
+
+def harmonic_mean(frequencies1, frequencies2):
+ """Finds the harmonic mean of the absolute differences between two frequency profiles,
+ expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1}) # doctest: +ELLIPSIS
+ 1.285714285...
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+ 0.228571428571...
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.0
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+ 0.2
+ """
+ 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):
+ """Finds the distances between two frequency profiles, expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> cosine_similarity({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.0000000000...
+ >>> cosine_similarity({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.0000000000...
+ >>> cosine_similarity({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 0.5773502691...
+ >>> cosine_similarity({'a':0, 'b':1}, {'a':1, 'b':1}) # doctest: +ELLIPSIS
+ 0.7071067811...
+ """
+ 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__ == "__main__":
+ 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">>>> cosine_similarity({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.0000000000...
+>>> cosine_similarity({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.0000000000...
+>>> cosine_similarity({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+0.5773502691...
+>>> 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):
+ """Finds the distances between two frequency profiles, expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> cosine_similarity({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.0000000000...
+ >>> cosine_similarity({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.0000000000...
+ >>> cosine_similarity({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 0.5773502691...
+ >>> cosine_similarity({'a':0, 'b':1}, {'a':1, 'b':1}) # doctest: +ELLIPSIS
+ 0.7071067811...
+ """
+ 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">>>> l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+>>> l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.73205080...
+>>> l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+>>> l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.732050807...
+>>> l2(normalise({'a':0, 'b':2, 'c':0}), normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.81649658...
+>>> 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):
+ """Finds the distances between two frequency profiles, expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.73205080...
+ >>> l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+ 0.0
+ >>> l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.732050807...
+ >>> l2(normalise({'a':0, 'b':2, 'c':0}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.81649658...
+ >>> l2({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ """
+ 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">>>> sorted(euclidean_scale({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+>>> sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.7071067...), (2, 0.7071067...)]
+>>> sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+>>> 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):
+ """Scale a set of frequencies so they have a unit euclidean length
+
+ >>> sorted(euclidean_scale({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.7071067...), (2, 0.7071067...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+ """
+ 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">>>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+>>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+>>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1})
+3.0
+>>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+0.01382140...
+>>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.0
+>>> 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):
+ """Finds the geometric mean of the absolute differences between two frequency profiles,
+ expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> geometric_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1})
+ 3.0
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+ 0.01382140...
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.0
+ >>> geometric_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+ 0.009259259...
+ """
+ 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">>>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+>>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+1.0
+>>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1}) # doctest: +ELLIPSIS
+1.285714285...
+>>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+0.228571428571...
+>>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.0
+>>> 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):
+ """Finds the harmonic mean of the absolute differences between two frequency profiles,
+ expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 1.0
+ >>> harmonic_mean({'a':2, 'b':2, 'c':2}, {'a':1, 'b':5, 'c':1}) # doctest: +ELLIPSIS
+ 1.285714285...
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':5, 'c':1})) # doctest: +ELLIPSIS
+ 0.228571428571...
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.0
+ >>> harmonic_mean(normalise({'a':2, 'b':2, 'c':2}), \
+ normalise({'a':1, 'b':1, 'c':0})) # doctest: +ELLIPSIS
+ 0.2
+ """
+ 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">>>> l1({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+>>> l1({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+3.0
+>>> l1(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+>>> l1({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1})
+3.0
+>>> 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):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l1({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l1({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1})
+ 3.0
+ >>> l1(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+ 0.0
+ >>> l1({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1})
+ 3.0
+ >>> l1({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ """
+ 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">>>> sorted(normalise({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+>>> sorted(normalise({1: 1, 2: 1}).items())
+[(1, 0.5), (2, 0.5)]
+>>> sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+>>> 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):
+ """Scale a set of frequencies so they sum to one
+
+ >>> sorted(normalise({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(normalise({1: 1, 2: 1}).items())
+ [(1, 0.5), (2, 0.5)]
+ >>> sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+ >>> sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+ [(1, 0.25), (2, 0.5), (3, 0.25)]
+ """
+ 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">>>> l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+>>> l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.73205080...
+>>> l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+0.0
+>>> l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.732050807...
+>>> l2(normalise({'a':0, 'b':2, 'c':0}), normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.81649658...
+>>> 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):
+ """Finds the distances between two frequency profiles, expressed as dictionaries.
+ Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l2({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l2({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.73205080...
+ >>> l2(normalise({'a':2, 'b':2, 'c':2}), normalise({'a':1, 'b':1, 'c':1}))
+ 0.0
+ >>> l2({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.732050807...
+ >>> l2(normalise({'a':0, 'b':2, 'c':0}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.81649658...
+ >>> l2({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ """
+ 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">>>> sorted(euclidean_scale({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+>>> sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.7071067...), (2, 0.7071067...)]
+>>> sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+>>> 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):
+ """Scale a set of frequencies so they have a unit euclidean length
+
+ >>> sorted(euclidean_scale({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.7071067...), (2, 0.7071067...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.577350...), (2, 0.577350...), (3, 0.577350...)]
+ >>> sorted(euclidean_scale({1: 1, 2: 2, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.408248...), (2, 0.81649658...), (3, 0.408248...)]
+ """
+ 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">>>> l3({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+0.0
+>>> l3({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.44224957...
+>>> l3({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+1.4422495703...
+>>> l3(normalise({'a':0, 'b':2, 'c':0}), normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+0.718144896...
+>>> l3({'a':0, 'b':1}, {'a':1, 'b':1})
+1.0
+>>> 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):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2
+
+ >>> l3({'a':1, 'b':1, 'c':1}, {'a':1, 'b':1, 'c':1})
+ 0.0
+ >>> l3({'a':2, 'b':2, 'c':2}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.44224957...
+ >>> l3({'a':0, 'b':2, 'c':0}, {'a':1, 'b':1, 'c':1}) # doctest: +ELLIPSIS
+ 1.4422495703...
+ >>> l3(normalise({'a':0, 'b':2, 'c':0}), \
+ normalise({'a':1, 'b':1, 'c':1})) # doctest: +ELLIPSIS
+ 0.718144896...
+ >>> l3({'a':0, 'b':1}, {'a':1, 'b':1})
+ 1.0
+ >>> l3(normalise({'a':0, 'b':1}), normalise({'a':1, 'b':1})) # doctest: +ELLIPSIS
+ 0.6299605249...
+ """
+ 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):
+ """Finds the distances between two frequency profiles, expressed as
+ dictionaries. Assumes every key in frequencies1 is also in frequencies2"""
+ 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):
+ """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.
+ """
+ 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">>>> sorted(normalise({1: 1, 2: 0}).items())
+[(1, 1.0), (2, 0.0)]
+>>> sorted(normalise({1: 1, 2: 1}).items())
+[(1, 0.5), (2, 0.5)]
+>>> sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+[(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+>>> 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):
+ """Scale a set of frequencies so they sum to one
+
+ >>> sorted(normalise({1: 1, 2: 0}).items())
+ [(1, 1.0), (2, 0.0)]
+ >>> sorted(normalise({1: 1, 2: 1}).items())
+ [(1, 0.5), (2, 0.5)]
+ >>> sorted(normalise({1: 1, 2: 1, 3: 1}).items()) # doctest: +ELLIPSIS
+ [(1, 0.333...), (2, 0.333...), (3, 0.333...)]
+ >>> sorted(normalise({1: 1, 2: 2, 3: 1}).items())
+ [(1, 0.25), (2, 0.5), (3, 0.25)]
+ """
+ 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=<function l2>)</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>
+</html>
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