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<section id="evaluation">
<h1>Evaluation<a class="headerlink" href="#evaluation" title="Permalink to this heading"></a></h1>
<p>Quantification is an appealing tool in scenarios of dataset shift,
and particularly in scenarios of prior-probability shift.
That is, the interest in estimating the class prevalences arises
under the belief that those class prevalences might have changed
with respect to the ones observed during training.
In other words, one could simply return the training prevalence
as a predictor of the test prevalence if this change is assumed
to be unlikely (as is the case in general scenarios of
machine learning governed by the iid assumption).
In brief, quantification requires dedicated evaluation protocols,
which are implemented in QuaPy and explained here.</p>
<section id="error-measures">
<h2>Error Measures<a class="headerlink" href="#error-measures" title="Permalink to this heading"></a></h2>
<p>The module quapy.error implements the following error measures for quantification:</p>
<ul class="simple">
<li><p><em>mae</em>: mean absolute error</p></li>
<li><p><em>mrae</em>: mean relative absolute error</p></li>
<li><p><em>mse</em>: mean squared error</p></li>
<li><p><em>mkld</em>: mean Kullback-Leibler Divergence</p></li>
<li><p><em>mnkld</em>: mean normalized Kullback-Leibler Divergence</p></li>
</ul>
<p>Functions <em>ae</em>, <em>rae</em>, <em>se</em>, <em>kld</em>, and <em>nkld</em> are also available,
which return the individual errors (i.e., without averaging the whole).</p>
<p>Some errors of classification are also available:</p>
<ul class="simple">
<li><p><em>acce</em>: accuracy error (1-accuracy)</p></li>
<li><p><em>f1e</em>: F-1 score error (1-F1 score)</p></li>
</ul>
<p>The error functions implement the following interface, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span><span class="p">(</span><span class="n">true_prevs</span><span class="p">,</span> <span class="n">prevs_hat</span><span class="p">)</span>
</pre></div>
</div>
<p>in which the first argument is a ndarray containing the true
prevalences, and the second argument is another ndarray with
the estimations produced by some method.</p>
<p>Some error functions, e.g., <em>mrae</em>, <em>mkld</em>, and <em>mnkld</em>, are
smoothed for numerical stability. In those cases, there is a
third argument, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">mrae</span><span class="p">(</span><span class="n">true_prevs</span><span class="p">,</span> <span class="n">prevs_hat</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span> <span class="o">...</span>
</pre></div>
</div>
<p>indicating the value for the smoothing parameter epsilon.
Traditionally, this value is set to 1/(2T) in past literature,
with T the sampling size. One could either pass this value
to the function each time, or to set a QuaPy’s environment
variable <em>SAMPLE_SIZE</em> once, and omit this argument
thereafter (recommended);
e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">qp</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s1">&#39;SAMPLE_SIZE&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># once for all</span>
<span class="n">true_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span> <span class="c1"># let&#39;s assume 3 classes</span>
<span class="n">estim_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">])</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">error</span><span class="o">.</span><span class="n">mrae</span><span class="p">(</span><span class="n">true_prev</span><span class="p">,</span> <span class="n">estim_prev</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;mrae(</span><span class="si">{</span><span class="n">true_prev</span><span class="si">}</span><span class="s1">, </span><span class="si">{</span><span class="n">estim_prev</span><span class="si">}</span><span class="s1">) = </span><span class="si">{</span><span class="n">error</span><span class="si">:</span><span class="s1">.3f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>will print:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">mrae</span><span class="p">([</span><span class="mf">0.500</span><span class="p">,</span> <span class="mf">0.300</span><span class="p">,</span> <span class="mf">0.200</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.100</span><span class="p">,</span> <span class="mf">0.300</span><span class="p">,</span> <span class="mf">0.600</span><span class="p">])</span> <span class="o">=</span> <span class="mf">0.914</span>
</pre></div>
</div>
<p>Finally, it is possible to instantiate QuaPy’s quantification
error functions from strings using, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">error_function</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">error</span><span class="o">.</span><span class="n">from_name</span><span class="p">(</span><span class="s1">&#39;mse&#39;</span><span class="p">)</span>
<span class="n">error</span> <span class="o">=</span> <span class="n">error_function</span><span class="p">(</span><span class="n">true_prev</span><span class="p">,</span> <span class="n">estim_prev</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="evaluation-protocols">
<h2>Evaluation Protocols<a class="headerlink" href="#evaluation-protocols" title="Permalink to this heading"></a></h2>
<p>An <em>evaluation protocol</em> is an evaluation procedure that uses
one specific <em>sample generation procotol</em> to genereate many
samples, typically characterized by widely varying amounts of
<em>shift</em> with respect to the original distribution, that are then
used to evaluate the performance of a (trained) quantifier.
These protocols are explained in more detail in a dedicated <a class="reference internal" href="Protocols.html"><span class="doc std std-doc">entry
in the wiki</span></a>. For the moment being, let us assume we already have
chosen and instantiated one specific such protocol, that we here
simply call <em>prot</em>. Let also assume our model is called
<em>quantifier</em> and that our evaluatio measure of choice is
<em>mae</em>. The evaluation comes down to:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">mae</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">evaluation</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">quantifier</span><span class="p">,</span> <span class="n">protocol</span><span class="o">=</span><span class="n">prot</span><span class="p">,</span> <span class="n">error_metric</span><span class="o">=</span><span class="s1">&#39;mae&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;MAE = </span><span class="si">{</span><span class="n">mae</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>It is often desirable to evaluate our system using more than one
single evaluatio measure. In this case, it is convenient to generate
a <em>report</em>. A report in QuaPy is a dataframe accounting for all the
true prevalence values with their corresponding prevalence values
as estimated by the quantifier, along with the error each has given
rise.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">report</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">evaluation</span><span class="o">.</span><span class="n">evaluation_report</span><span class="p">(</span><span class="n">quantifier</span><span class="p">,</span> <span class="n">protocol</span><span class="o">=</span><span class="n">prot</span><span class="p">,</span> <span class="n">error_metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;mae&#39;</span><span class="p">,</span> <span class="s1">&#39;mrae&#39;</span><span class="p">,</span> <span class="s1">&#39;mkld&#39;</span><span class="p">])</span>
</pre></div>
</div>
<p>From a pandas’ dataframe, it is straightforward to visualize all the results,
and compute the averaged values, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">pd</span><span class="o">.</span><span class="n">set_option</span><span class="p">(</span><span class="s1">&#39;display.expand_frame_repr&#39;</span><span class="p">,</span> <span class="kc">False</span><span class="p">)</span>
<span class="n">report</span><span class="p">[</span><span class="s1">&#39;estim-prev&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">report</span><span class="p">[</span><span class="s1">&#39;estim-prev&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">strprev</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">report</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Averaged values:&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">report</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
</pre></div>
</div>
<p>This will produce an output like:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="n">true</span><span class="o">-</span><span class="n">prev</span> <span class="n">estim</span><span class="o">-</span><span class="n">prev</span> <span class="n">mae</span> <span class="n">mrae</span> <span class="n">mkld</span>
<span class="mi">0</span> <span class="p">[</span><span class="mf">0.308</span><span class="p">,</span> <span class="mf">0.692</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.314</span><span class="p">,</span> <span class="mf">0.686</span><span class="p">]</span> <span class="mf">0.005649</span> <span class="mf">0.013182</span> <span class="mf">0.000074</span>
<span class="mi">1</span> <span class="p">[</span><span class="mf">0.896</span><span class="p">,</span> <span class="mf">0.104</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.909</span><span class="p">,</span> <span class="mf">0.091</span><span class="p">]</span> <span class="mf">0.013145</span> <span class="mf">0.069323</span> <span class="mf">0.000985</span>
<span class="mi">2</span> <span class="p">[</span><span class="mf">0.848</span><span class="p">,</span> <span class="mf">0.152</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.809</span><span class="p">,</span> <span class="mf">0.191</span><span class="p">]</span> <span class="mf">0.039063</span> <span class="mf">0.149806</span> <span class="mf">0.005175</span>
<span class="mi">3</span> <span class="p">[</span><span class="mf">0.016</span><span class="p">,</span> <span class="mf">0.984</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.033</span><span class="p">,</span> <span class="mf">0.967</span><span class="p">]</span> <span class="mf">0.017236</span> <span class="mf">0.487529</span> <span class="mf">0.005298</span>
<span class="mi">4</span> <span class="p">[</span><span class="mf">0.728</span><span class="p">,</span> <span class="mf">0.272</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.751</span><span class="p">,</span> <span class="mf">0.249</span><span class="p">]</span> <span class="mf">0.022769</span> <span class="mf">0.057146</span> <span class="mf">0.001350</span>
<span class="o">...</span> <span class="o">...</span> <span class="o">...</span> <span class="o">...</span> <span class="o">...</span> <span class="o">...</span>
<span class="mi">4995</span> <span class="p">[</span><span class="mf">0.72</span><span class="p">,</span> <span class="mf">0.28</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.698</span><span class="p">,</span> <span class="mf">0.302</span><span class="p">]</span> <span class="mf">0.021752</span> <span class="mf">0.053631</span> <span class="mf">0.001133</span>
<span class="mi">4996</span> <span class="p">[</span><span class="mf">0.868</span><span class="p">,</span> <span class="mf">0.132</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.888</span><span class="p">,</span> <span class="mf">0.112</span><span class="p">]</span> <span class="mf">0.020490</span> <span class="mf">0.088230</span> <span class="mf">0.001985</span>
<span class="mi">4997</span> <span class="p">[</span><span class="mf">0.292</span><span class="p">,</span> <span class="mf">0.708</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.298</span><span class="p">,</span> <span class="mf">0.702</span><span class="p">]</span> <span class="mf">0.006149</span> <span class="mf">0.014788</span> <span class="mf">0.000090</span>
<span class="mi">4998</span> <span class="p">[</span><span class="mf">0.24</span><span class="p">,</span> <span class="mf">0.76</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.220</span><span class="p">,</span> <span class="mf">0.780</span><span class="p">]</span> <span class="mf">0.019950</span> <span class="mf">0.054309</span> <span class="mf">0.001127</span>
<span class="mi">4999</span> <span class="p">[</span><span class="mf">0.948</span><span class="p">,</span> <span class="mf">0.052</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.965</span><span class="p">,</span> <span class="mf">0.035</span><span class="p">]</span> <span class="mf">0.016941</span> <span class="mf">0.165776</span> <span class="mf">0.003538</span>
<span class="p">[</span><span class="mi">5000</span> <span class="n">rows</span> <span class="n">x</span> <span class="mi">5</span> <span class="n">columns</span><span class="p">]</span>
<span class="n">Averaged</span> <span class="n">values</span><span class="p">:</span>
<span class="n">mae</span> <span class="mf">0.023588</span>
<span class="n">mrae</span> <span class="mf">0.108779</span>
<span class="n">mkld</span> <span class="mf">0.003631</span>
<span class="n">dtype</span><span class="p">:</span> <span class="n">float64</span>
<span class="n">Process</span> <span class="n">finished</span> <span class="k">with</span> <span class="n">exit</span> <span class="n">code</span> <span class="mi">0</span>
</pre></div>
</div>
<p>Alternatively, we can simply generate all the predictions by:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">true_prevs</span><span class="p">,</span> <span class="n">estim_prevs</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">evaluation</span><span class="o">.</span><span class="n">prediction</span><span class="p">(</span><span class="n">quantifier</span><span class="p">,</span> <span class="n">protocol</span><span class="o">=</span><span class="n">prot</span><span class="p">)</span>
</pre></div>
</div>
<p>All the evaluation functions implement specific optimizations for speeding-up
the evaluation of aggregative quantifiers (i.e., of instances of <em>AggregativeQuantifier</em>).
The optimization comes down to generating classification predictions (either crisp or soft)
only once for the entire test set, and then applying the sampling procedure to the
predictions, instead of generating samples of instances and then computing the
classification predictions every time. This is only possible when the protocol
is an instance of <em>OnLabelledCollectionProtocol</em>. The optimization is only
carried out when the number of classification predictions thus generated would be
smaller than the number of predictions required for the entire protocol; e.g.,
if the original dataset contains 1M instances, but the protocol is such that it would
at most generate 20 samples of 100 instances, then it would be preferable to postpone the
classification for each sample. This behaviour is indicated by setting
<em>aggr_speedup=”auto”</em>. Conversely, when indicating <em>aggr_speedup=”force”</em> QuaPy will
precompute all the predictions irrespectively of the number of instances and number of samples.
Finally, this can be deactivated by setting <em>aggr_speedup=False</em>. Note that this optimization
is not only applied for the final evaluation, but also for the internal evaluations carried
out during <em>model selection</em>. Since these are typically many, the heuristic can help reduce the
execution time a lot.</p>
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