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<li><a class="reference internal" href="#">4.1. Partial Dependence and Individual Conditional Expectation plots</a><ul>
<li><a class="reference internal" href="#partial-dependence-plots">4.1.1. Partial dependence plots</a></li>
<li><a class="reference internal" href="#individual-conditional-expectation-ice-plot">4.1.2. Individual conditional expectation (ICE) plot</a></li>
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<div class="section" id="partial-dependence-and-individual-conditional-expectation-plots">
<span id="partial-dependence"></span><h1><span class="section-number">4.1. </span>Partial Dependence and Individual Conditional Expectation plots<a class="headerlink" href="#partial-dependence-and-individual-conditional-expectation-plots" title="Permalink to this headline">¶</a></h1>
<p>Partial dependence plots (PDP) and individual conditional expectation (ICE)
plots can be used to visualize and analyze interaction between the target
response <a class="footnote-reference brackets" href="#id2" id="id1">1</a> and a set of input features of interest.</p>
<p>Both PDPs and ICEs assume that the input features of interest are independent
from the complement features, and this assumption is often violated in practice.
Thus, in the case of correlated features, we will create absurd data points to
compute the PDP/ICE.</p>
<div class="section" id="partial-dependence-plots">
<h2><span class="section-number">4.1.1. </span>Partial dependence plots<a class="headerlink" href="#partial-dependence-plots" title="Permalink to this headline">¶</a></h2>
<p>Partial dependence plots (PDP) show the dependence between the target response
and a set of input features of interest, marginalizing over the values
of all other input features (the ‘complement’ features). Intuitively, we can
interpret the partial dependence as the expected target response as a
function of the input features of interest.</p>
<p>Due to the limits of human perception the size of the set of input feature of
interest must be small (usually, one or two) thus the input features of interest
are usually chosen among the most important features.</p>
<p>The figure below shows two one-way and one two-way partial dependence plots for
the California housing dataset, with a <a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>:</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/inspection/plot_partial_dependence.html"><img alt="../_images/sphx_glr_plot_partial_dependence_0031.png" src="../_images/sphx_glr_plot_partial_dependence_0031.png" style="width: 630.0px; height: 280.0px;" /></a>
</div>
<p>One-way PDPs tell us about the interaction between the target response and an
input feature of interest feature (e.g. linear, non-linear). The left plot
in the above figure shows the effect of the average occupancy on the median
house price; we can clearly see a linear relationship among them when the
average occupancy is inferior to 3 persons. Similarly, we could analyze the
effect of the house age on the median house price (middle plot). Thus, these
interpretations are marginal, considering a feature at a time.</p>
<p>PDPs with two input features of interest show the interactions among the two
features. For example, the two-variable PDP in the above figure shows the
dependence of median house price on joint values of house age and average
occupants per household. We can clearly see an interaction between the two
features: for an average occupancy greater than two, the house price is nearly
independent of the house age, whereas for values less than 2 there is a strong
dependence on age.</p>
<p>The <a class="reference internal" href="classes.html#module-sklearn.inspection" title="sklearn.inspection"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.inspection</span></code></a> module provides a convenience function
<a class="reference internal" href="generated/sklearn.inspection.plot_partial_dependence.html#sklearn.inspection.plot_partial_dependence" title="sklearn.inspection.plot_partial_dependence"><code class="xref py py-func docutils literal notranslate"><span class="pre">plot_partial_dependence</span></code></a> to create one-way and two-way partial
dependence plots. In the below example we show how to create a grid of
partial dependence plots: two one-way PDPs for the features <code class="docutils literal notranslate"><span class="pre">0</span></code> and <code class="docutils literal notranslate"><span class="pre">1</span></code>
and a two-way PDP between the two features:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_hastie_10_2</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">plot_partial_dependence</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_hastie_10_2</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
<span class="gp">>>> </span><span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">features</span><span class="p">)</span>
</pre></div>
</div>
<p>You can access the newly created figure and Axes objects using <code class="docutils literal notranslate"><span class="pre">plt.gcf()</span></code>
and <code class="docutils literal notranslate"><span class="pre">plt.gca()</span></code>.</p>
<p>For multi-class classification, you need to set the class label for which
the PDPs should be created via the <code class="docutils literal notranslate"><span class="pre">target</span></code> argument:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">mc_clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]</span>
<span class="gp">>>> </span><span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">mc_clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">features</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
<p>The same parameter <code class="docutils literal notranslate"><span class="pre">target</span></code> is used to specify the target in multi-output
regression settings.</p>
<p>If you need the raw values of the partial dependence function rather than
the plots, you can use the
<a class="reference internal" href="generated/sklearn.inspection.partial_dependence.html#sklearn.inspection.partial_dependence" title="sklearn.inspection.partial_dependence"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.inspection.partial_dependence</span></code></a> function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">partial_dependence</span>
<span class="gp">>>> </span><span class="n">pdp</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">partial_dependence</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">pdp</span>
<span class="go">array([[ 2.466..., 2.466..., ...</span>
<span class="gp">>>> </span><span class="n">axes</span>
<span class="go">[array([-1.624..., -1.592..., ...</span>
</pre></div>
</div>
<p>The values at which the partial dependence should be evaluated are directly
generated from <code class="docutils literal notranslate"><span class="pre">X</span></code>. For 2-way partial dependence, a 2D-grid of values is
generated. The <code class="docutils literal notranslate"><span class="pre">values</span></code> field returned by
<a class="reference internal" href="generated/sklearn.inspection.partial_dependence.html#sklearn.inspection.partial_dependence" title="sklearn.inspection.partial_dependence"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.inspection.partial_dependence</span></code></a> gives the actual values
used in the grid for each input feature of interest. They also correspond to
the axis of the plots.</p>
</div>
<div class="section" id="individual-conditional-expectation-ice-plot">
<span id="individual-conditional"></span><h2><span class="section-number">4.1.2. </span>Individual conditional expectation (ICE) plot<a class="headerlink" href="#individual-conditional-expectation-ice-plot" title="Permalink to this headline">¶</a></h2>
<p>Similar to a PDP, an individual conditional expectation (ICE) plot
shows the dependence between the target function and an input feature of
interest. However, unlike a PDP, which shows the average effect of the input
feature, an ICE plot visualizes the dependence of the prediction on a
feature for each sample separately with one line per sample.
Due to the limits of human perception, only one input feature of interest is
supported for ICE plots.</p>
<p>The figures below show four ICE plots for the California housing dataset,
with a <a class="reference internal" href="generated/sklearn.ensemble.HistGradientBoostingRegressor.html#sklearn.ensemble.HistGradientBoostingRegressor" title="sklearn.ensemble.HistGradientBoostingRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">HistGradientBoostingRegressor</span></code></a>. The second figure plots
the corresponding PD line overlaid on ICE lines.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/inspection/plot_partial_dependence.html"><img alt="../_images/sphx_glr_plot_partial_dependence_0021.png" src="../_images/sphx_glr_plot_partial_dependence_0021.png" style="width: 448.0px; height: 336.0px;" /></a>
</div>
<p>While the PDPs are good at showing the average effect of the target features,
they can obscure a heterogeneous relationship created by interactions.
When interactions are present the ICE plot will provide many more insights.
For example, we could observe a linear relationship between the median income
and the house price in the PD line. However, the ICE lines show that there
are some exceptions, where the house price remains constant in some ranges of
the median income.</p>
<p>The <a class="reference internal" href="classes.html#module-sklearn.inspection" title="sklearn.inspection"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.inspection</span></code></a> module’s <a class="reference internal" href="generated/sklearn.inspection.plot_partial_dependence.html#sklearn.inspection.plot_partial_dependence" title="sklearn.inspection.plot_partial_dependence"><code class="xref py py-func docutils literal notranslate"><span class="pre">plot_partial_dependence</span></code></a>
convenience function can be used to create ICE plots by setting
<code class="docutils literal notranslate"><span class="pre">kind='individual'</span></code>. In the example below, we show how to create a grid of
ICE plots:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_hastie_10_2</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.inspection</span> <span class="kn">import</span> <span class="n">plot_partial_dependence</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_hastie_10_2</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">features</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">kind</span><span class="o">=</span><span class="s1">'individual'</span><span class="p">)</span>
</pre></div>
</div>
<p>In ICE plots it might not be easy to see the average effect of the input
feature of interest. Hence, it is recommended to use ICE plots alongside
PDPs. They can be plotted together with
<code class="docutils literal notranslate"><span class="pre">kind='both'</span></code>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">plot_partial_dependence</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">features</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">kind</span><span class="o">=</span><span class="s1">'both'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="mathematical-definition">
<h2><span class="section-number">4.1.3. </span>Mathematical Definition<a class="headerlink" href="#mathematical-definition" title="Permalink to this headline">¶</a></h2>
<p>Let <span class="math notranslate nohighlight">\(X_S\)</span> be the set of input features of interest (i.e. the <code class="docutils literal notranslate"><span class="pre">features</span></code>
parameter) and let <span class="math notranslate nohighlight">\(X_C\)</span> be its complement.</p>
<p>The partial dependence of the response <span class="math notranslate nohighlight">\(f\)</span> at a point <span class="math notranslate nohighlight">\(x_S\)</span> is
defined as:</p>
<div class="math notranslate nohighlight">
\[\begin{split}pd_{X_S}(x_S) &\overset{def}{=} \mathbb{E}_{X_C}\left[ f(x_S, X_C) \right]\\
&= \int f(x_S, x_C) p(x_C) dx_C,\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(f(x_S, x_C)\)</span> is the response function (<a class="reference internal" href="../glossary.html#term-predict"><span class="xref std std-term">predict</span></a>,
<a class="reference internal" href="../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> or <a class="reference internal" href="../glossary.html#term-decision_function"><span class="xref std std-term">decision_function</span></a>) for a given sample whose
values are defined by <span class="math notranslate nohighlight">\(x_S\)</span> for the features in <span class="math notranslate nohighlight">\(X_S\)</span>, and by
<span class="math notranslate nohighlight">\(x_C\)</span> for the features in <span class="math notranslate nohighlight">\(X_C\)</span>. Note that <span class="math notranslate nohighlight">\(x_S\)</span> and
<span class="math notranslate nohighlight">\(x_C\)</span> may be tuples.</p>
<p>Computing this integral for various values of <span class="math notranslate nohighlight">\(x_S\)</span> produces a PDP plot
as above. An ICE line is defined as a single <span class="math notranslate nohighlight">\(f(x_{S}, x_{C}^{(i)})\)</span>
evaluated at <span class="math notranslate nohighlight">\(x_{S}\)</span>.</p>
</div>
<div class="section" id="computation-methods">
<h2><span class="section-number">4.1.4. </span>Computation methods<a class="headerlink" href="#computation-methods" title="Permalink to this headline">¶</a></h2>
<p>There are two main methods to approximate the integral above, namely the
‘brute’ and ‘recursion’ methods. The <code class="docutils literal notranslate"><span class="pre">method</span></code> parameter controls which method
to use.</p>
<p>The ‘brute’ method is a generic method that works with any estimator. Note that
computing ICE plots is only supported with the ‘brute’ method. It
approximates the above integral by computing an average over the data <code class="docutils literal notranslate"><span class="pre">X</span></code>:</p>
<div class="math notranslate nohighlight">
\[pd_{X_S}(x_S) \approx \frac{1}{n_\text{samples}} \sum_{i=1}^n f(x_S, x_C^{(i)}),\]</div>
<p>where <span class="math notranslate nohighlight">\(x_C^{(i)}\)</span> is the value of the i-th sample for the features in
<span class="math notranslate nohighlight">\(X_C\)</span>. For each value of <span class="math notranslate nohighlight">\(x_S\)</span>, this method requires a full pass
over the dataset <code class="docutils literal notranslate"><span class="pre">X</span></code> which is computationally intensive.</p>
<p>Each of the <span class="math notranslate nohighlight">\(f(x_{S}, x_{C}^{(i)})\)</span> corresponds to one ICE line evaluated
at <span class="math notranslate nohighlight">\(x_{S}\)</span>. Computing this for multiple values of <span class="math notranslate nohighlight">\(x_{S}\)</span>, one
obtains a full ICE line. As one can see, the average of the ICE lines
correspond to the partial dependence line.</p>
<p>The ‘recursion’ method is faster than the ‘brute’ method, but it is only
supported for PDP plots by some tree-based estimators. It is computed as
follows. For a given point <span class="math notranslate nohighlight">\(x_S\)</span>, a weighted tree traversal is performed:
if a split node involves an input feature of interest, the corresponding left
or right branch is followed; otherwise both branches are followed, each branch
being weighted by the fraction of training samples that entered that branch.
Finally, the partial dependence is given by a weighted average of all the
visited leaves values.</p>
<p>With the ‘brute’ method, the parameter <code class="docutils literal notranslate"><span class="pre">X</span></code> is used both for generating the
grid of values <span class="math notranslate nohighlight">\(x_S\)</span> and the complement feature values <span class="math notranslate nohighlight">\(x_C\)</span>.
However with the ‘recursion’ method, <code class="docutils literal notranslate"><span class="pre">X</span></code> is only used for the grid values:
implicitly, the <span class="math notranslate nohighlight">\(x_C\)</span> values are those of the training data.</p>
<p>By default, the ‘recursion’ method is used for plotting PDPs on tree-based
estimators that support it, and ‘brute’ is used for the rest.</p>
<div class="admonition note" id="pdp-method-differences">
<p class="admonition-title">Note</p>
<p>While both methods should be close in general, they might differ in some
specific settings. The ‘brute’ method assumes the existence of the
data points <span class="math notranslate nohighlight">\((x_S, x_C^{(i)})\)</span>. When the features are correlated,
such artificial samples may have a very low probability mass. The ‘brute’
and ‘recursion’ methods will likely disagree regarding the value of the
partial dependence, because they will treat these unlikely
samples differently. Remember, however, that the primary assumption for
interpreting PDPs is that the features should be independent.</p>
</div>
<div class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_partial_dependence.html#sphx-glr-auto-examples-inspection-plot-partial-dependence-py"><span class="std std-ref">Partial Dependence and Individual Conditional Expectation Plots</span></a></p></li>
</ul>
</div>
<p class="rubric">Footnotes</p>
<dl class="footnote brackets">
<dt class="label" id="id2"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>For classification, the target response may be the probability of a
class (the positive class for binary classification), or the decision
function.</p>
</dd>
</dl>
<div class="topic">
<p class="topic-title">References</p>
<p>T. Hastie, R. Tibshirani and J. Friedman, <a class="reference external" href="https://web.stanford.edu/~hastie/ElemStatLearn//">The Elements of
Statistical Learning</a>,
Second Edition, Section 10.13.2, Springer, 2009.</p>
<p>C. Molnar, <a class="reference external" href="https://christophm.github.io/interpretable-ml-book/">Interpretable Machine Learning</a>, Section 5.1, 2019.</p>
<p>A. Goldstein, A. Kapelner, J. Bleich, and E. Pitkin, <a class="reference external" href="https://arxiv.org/abs/1309.6392">Peeking Inside the
Black Box: Visualizing Statistical Learning With Plots of Individual
Conditional Expectation</a>,
Journal of Computational and Graphical Statistics, 24(1): 44-65, Springer,
2015.</p>
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