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<div class="tex2jax_ignore mathjax_ignore section" id="evaluation">
<h1>Evaluation<a class="headerlink" href="#evaluation" title="Permalink to this headline"></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>
<div class="section" id="error-measures">
<h2>Error Measures<a class="headerlink" href="#error-measures" title="Permalink to this headline"></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 ommit 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">ae_</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">ae_</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>
</div>
<div class="section" id="evaluation-protocols">
<h2>Evaluation Protocols<a class="headerlink" href="#evaluation-protocols" title="Permalink to this headline"></a></h2>
<p>QuaPy implements the so-called “artificial sampling protocol”,
according to which a test set is used to generate samplings at
desired prevalences of fixed size and covering the full spectrum
of prevalences. This protocol is called “artificial” in contrast
to the “natural prevalence sampling” protocol that,
despite introducing some variability during sampling, approximately
preserves the training class prevalence.</p>
<p>In the artificial sampling procol, the user specifies the number
of (equally distant) points to be generated from the interval [0,1].</p>
<p>For example, if n_prevpoints=11 then, for each class, the prevalences
[0., 0.1, 0.2, …, 1.] will be used. This means that, for two classes,
the number of different prevalences will be 11 (since, once the prevalence
of one class is determined, the other one is constrained). For 3 classes,
the number of valid combinations can be obtained as 11 + 10 + … + 1 = 66.
In general, the number of valid combinations that will be produced for a given
value of n_prevpoints can be consulted by invoking
quapy.functional.num_prevalence_combinations, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">quapy.functional</span> <span class="k">as</span> <span class="nn">F</span>
<span class="n">n_prevpoints</span> <span class="o">=</span> <span class="mi">21</span>
<span class="n">n_classes</span> <span class="o">=</span> <span class="mi">4</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">num_prevalence_combinations</span><span class="p">(</span><span class="n">n_prevpoints</span><span class="p">,</span> <span class="n">n_classes</span><span class="p">,</span> <span class="n">n_repeats</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>in this example, n=1771. Note the last argument, n_repeats, that
informs of the number of examples that will be generated for any
valid combination (typical values are, e.g., 1 for a single sample,
or 10 or higher for computing standard deviations of performing statistical
significance tests).</p>
<p>One can instead work the other way around, i.e., one could set a
maximum budged of evaluations and get the number of prevalence points that
will generate a number of evaluations close, but not higher, than
the fixed budget. This can be achieved with the function
quapy.functional.get_nprevpoints_approximation, e.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">budget</span> <span class="o">=</span> <span class="mi">5000</span>
<span class="n">n_prevpoints</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">get_nprevpoints_approximation</span><span class="p">(</span><span class="n">budget</span><span class="p">,</span> <span class="n">n_classes</span><span class="p">,</span> <span class="n">n_repeats</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">num_prevalence_combinations</span><span class="p">(</span><span class="n">n_prevpoints</span><span class="p">,</span> <span class="n">n_classes</span><span class="p">,</span> <span class="n">n_repeats</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;by setting n_prevpoints=</span><span class="si">{</span><span class="n">n_prevpoints</span><span class="si">}</span><span class="s1"> the number of evaluations for </span><span class="si">{</span><span class="n">n_classes</span><span class="si">}</span><span class="s1"> classes will be </span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>that will print:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">by</span> <span class="n">setting</span> <span class="n">n_prevpoints</span><span class="o">=</span><span class="mi">30</span> <span class="n">the</span> <span class="n">number</span> <span class="n">of</span> <span class="n">evaluations</span> <span class="k">for</span> <span class="mi">4</span> <span class="n">classes</span> <span class="n">will</span> <span class="n">be</span> <span class="mi">4960</span>
</pre></div>
</div>
<p>The cost of evaluation will depend on the values of <em>n_prevpoints</em>, <em>n_classes</em>,
and <em>n_repeats</em>. Since it might sometimes be cumbersome to control the overall
cost of an experiment having to do with the number of combinations that
will be generated for a particular setting of these arguments (particularly
when <em>n_classes&gt;2</em>), evaluation functions
typically allow the user to rather specify an <em>evaluation budget</em>, i.e., a maximum
number of samplings to generate. By specifying this argument, one could avoid
specifying <em>n_prevpoints</em>, and the value for it that would lead to a closer
number of evaluation budget, without surpassing it, will be automatically set.</p>
<p>The following script shows a full example in which a PACC model relying
on a Logistic Regressor classifier is
tested on the <em>kindle</em> dataset by means of the artificial prevalence
sampling protocol on samples of size 500, in terms of various
evaluation metrics.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">quapy</span> <span class="k">as</span> <span class="nn">qp</span>
<span class="kn">import</span> <span class="nn">quapy.functional</span> <span class="k">as</span> <span class="nn">F</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</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">500</span>
<span class="n">dataset</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">fetch_reviews</span><span class="p">(</span><span class="s1">&#39;kindle&#39;</span><span class="p">)</span>
<span class="n">qp</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">preprocessing</span><span class="o">.</span><span class="n">text2tfidf</span><span class="p">(</span><span class="n">dataset</span><span class="p">,</span> <span class="n">min_df</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">training</span> <span class="o">=</span> <span class="n">dataset</span><span class="o">.</span><span class="n">training</span>
<span class="n">test</span> <span class="o">=</span> <span class="n">dataset</span><span class="o">.</span><span class="n">test</span>
<span class="n">lr</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">()</span>
<span class="n">pacc</span> <span class="o">=</span> <span class="n">qp</span><span class="o">.</span><span class="n">method</span><span class="o">.</span><span class="n">aggregative</span><span class="o">.</span><span class="n">PACC</span><span class="p">(</span><span class="n">lr</span><span class="p">)</span>
<span class="n">pacc</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">training</span><span class="p">)</span>
<span class="n">df</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">artificial_sampling_report</span><span class="p">(</span>
<span class="n">pacc</span><span class="p">,</span> <span class="c1"># the quantification method</span>
<span class="n">test</span><span class="p">,</span> <span class="c1"># the test set on which the method will be evaluated</span>
<span class="n">sample_size</span><span class="o">=</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="c1">#indicates the size of samples to be drawn</span>
<span class="n">n_prevpoints</span><span class="o">=</span><span class="mi">11</span><span class="p">,</span> <span class="c1"># how many prevalence points will be extracted from the interval [0, 1] for each category</span>
<span class="n">n_repetitions</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="c1"># number of times each prevalence will be used to generate a test sample</span>
<span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="c1"># indicates the number of parallel workers (-1 indicates, as in sklearn, all CPUs)</span>
<span class="n">random_seed</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="c1"># setting a random seed allows to replicate the test samples across runs</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> <span class="c1"># specify the evaluation metrics</span>
<span class="n">verbose</span><span class="o">=</span><span class="kc">True</span> <span class="c1"># set to True to show some standard-line outputs</span>
<span class="p">)</span>
</pre></div>
</div>
<p>The resulting report is a pandas’ dataframe that can be directly printed.
Here, we set some display options from pandas just to make the output clearer;
note also that the estimated prevalences are shown as strings using the
function strprev function that simply converts a prevalence into a
string representing it, with a fixed decimal precision (default 3):</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</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">pd</span><span class="o">.</span><span class="n">set_option</span><span class="p">(</span><span class="s2">&quot;precision&quot;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">df</span><span class="p">[</span><span class="s1">&#39;estim-prev&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">df</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">df</span><span class="p">)</span>
</pre></div>
</div>
<p>The output should look 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">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.000</span><span class="p">,</span> <span class="mf">1.000</span><span class="p">]</span> <span class="mf">0.000</span> <span class="mf">0.000</span> <span class="mf">0.000e+00</span>
<span class="mi">1</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.091</span><span class="p">,</span> <span class="mf">0.909</span><span class="p">]</span> <span class="mf">0.009</span> <span class="mf">0.048</span> <span class="mf">4.426e-04</span>
<span class="mi">2</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.163</span><span class="p">,</span> <span class="mf">0.837</span><span class="p">]</span> <span class="mf">0.037</span> <span class="mf">0.114</span> <span class="mf">4.633e-03</span>
<span class="mi">3</span> <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.283</span><span class="p">,</span> <span class="mf">0.717</span><span class="p">]</span> <span class="mf">0.017</span> <span class="mf">0.041</span> <span class="mf">7.383e-04</span>
<span class="mi">4</span> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.366</span><span class="p">,</span> <span class="mf">0.634</span><span class="p">]</span> <span class="mf">0.034</span> <span class="mf">0.070</span> <span class="mf">2.412e-03</span>
<span class="mi">5</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.459</span><span class="p">,</span> <span class="mf">0.541</span><span class="p">]</span> <span class="mf">0.041</span> <span class="mf">0.082</span> <span class="mf">3.387e-03</span>
<span class="mi">6</span> <span class="p">[</span><span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.565</span><span class="p">,</span> <span class="mf">0.435</span><span class="p">]</span> <span class="mf">0.035</span> <span class="mf">0.073</span> <span class="mf">2.535e-03</span>
<span class="mi">7</span> <span class="p">[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.654</span><span class="p">,</span> <span class="mf">0.346</span><span class="p">]</span> <span class="mf">0.046</span> <span class="mf">0.108</span> <span class="mf">4.701e-03</span>
<span class="mi">8</span> <span class="p">[</span><span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.725</span><span class="p">,</span> <span class="mf">0.275</span><span class="p">]</span> <span class="mf">0.075</span> <span class="mf">0.235</span> <span class="mf">1.515e-02</span>
<span class="mi">9</span> <span class="p">[</span><span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.858</span><span class="p">,</span> <span class="mf">0.142</span><span class="p">]</span> <span class="mf">0.042</span> <span class="mf">0.229</span> <span class="mf">7.740e-03</span>
<span class="mi">10</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]</span> <span class="p">[</span><span class="mf">0.945</span><span class="p">,</span> <span class="mf">0.055</span><span class="p">]</span> <span class="mf">0.055</span> <span class="mf">27.357</span> <span class="mf">5.219e-02</span>
</pre></div>
</div>
<p>One can get the averaged scores using standard pandas’
functions, i.e.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
</pre></div>
</div>
<p>will produce the following output:</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="mf">0.500</span>
<span class="n">mae</span> <span class="mf">0.035</span>
<span class="n">mrae</span> <span class="mf">2.578</span>
<span class="n">mkld</span> <span class="mf">0.009</span>
<span class="n">dtype</span><span class="p">:</span> <span class="n">float64</span>
</pre></div>
</div>
<p>Other evaluation functions include:</p>
<ul class="simple">
<li><p><em>artificial_sampling_eval</em>: that computes the evaluation for a
given evaluation metric, returning the average instead of a dataframe.</p></li>
<li><p><em>artificial_sampling_prediction</em>: that returns two np.arrays containing the
true prevalences and the estimated prevalences.</p></li>
</ul>
<p>See the documentation for further details.</p>
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