.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/inspection/plot_linear_model_coefficient_interpretation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py>`
        to download the full example code. or to run this example in your browser via JupyterLite or Binder

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py:


======================================================================
Common pitfalls in the interpretation of coefficients of linear models
======================================================================

In linear models, the target value is modeled as a linear combination of the
features (see the :ref:`linear_model` User Guide section for a description of a
set of linear models available in scikit-learn). Coefficients in multiple linear
models represent the relationship between the given feature, :math:`X_i` and the
target, :math:`y`, assuming that all the other features remain constant
(`conditional dependence
<https://en.wikipedia.org/wiki/Conditional_dependence>`_). This is different
from plotting :math:`X_i` versus :math:`y` and fitting a linear relationship: in
that case all possible values of the other features are taken into account in
the estimation (marginal dependence).

This example will provide some hints in interpreting coefficient in linear
models, pointing at problems that arise when either the linear model is not
appropriate to describe the dataset, or when features are correlated.

.. note::

    Keep in mind that the features :math:`X` and the outcome :math:`y` are in
    general the result of a data generating process that is unknown to us.
    Machine learning models are trained to approximate the unobserved
    mathematical function that links :math:`X` to :math:`y` from sample data. As
    a result, any interpretation made about a model may not necessarily
    generalize to the true data generating process. This is especially true when
    the model is of bad quality or when the sample data is not representative of
    the population.

We will use data from the `"Current Population Survey"
<https://www.openml.org/d/534>`_ from 1985 to predict wage as a function of
various features such as experience, age, or education.

.. GENERATED FROM PYTHON SOURCE LINES 36-40

.. code-block:: Python


    # Authors: The scikit-learn developers
    # SPDX-License-Identifier: BSD-3-Clause








.. GENERATED FROM PYTHON SOURCE LINES 41-47

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import pandas as pd
    import scipy as sp
    import seaborn as sns








.. GENERATED FROM PYTHON SOURCE LINES 48-54

The dataset: wages
------------------

We fetch the data from `OpenML <http://openml.org/>`_.
Note that setting the parameter `as_frame` to True will retrieve the data
as a pandas dataframe.

.. GENERATED FROM PYTHON SOURCE LINES 54-58

.. code-block:: Python

    from sklearn.datasets import fetch_openml

    survey = fetch_openml(data_id=534, as_frame=True)








.. GENERATED FROM PYTHON SOURCE LINES 59-61

Then, we identify features `X` and targets `y`: the column WAGE is our
target variable (i.e., the variable which we want to predict).

.. GENERATED FROM PYTHON SOURCE LINES 61-65

.. code-block:: Python


    X = survey.data[survey.feature_names]
    X.describe(include="all")






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
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        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>EDUCATION</th>
          <th>SOUTH</th>
          <th>SEX</th>
          <th>EXPERIENCE</th>
          <th>UNION</th>
          <th>AGE</th>
          <th>RACE</th>
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          <td>534</td>
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          <td>534.000000</td>
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          <td>534</td>
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          <td>6</td>
          <td>3</td>
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        <tr>
          <th>top</th>
          <td>NaN</td>
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          <td>NaN</td>
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          <td>NaN</td>
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          <td>Other</td>
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        <tr>
          <th>freq</th>
          <td>NaN</td>
          <td>378</td>
          <td>289</td>
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          <td>NaN</td>
          <td>440</td>
          <td>156</td>
          <td>411</td>
          <td>350</td>
        </tr>
        <tr>
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          <td>13.018727</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>17.822097</td>
          <td>NaN</td>
          <td>36.833333</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>std</th>
          <td>2.615373</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>12.379710</td>
          <td>NaN</td>
          <td>11.726573</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>min</th>
          <td>2.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>0.000000</td>
          <td>NaN</td>
          <td>18.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>25%</th>
          <td>12.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>8.000000</td>
          <td>NaN</td>
          <td>28.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>50%</th>
          <td>12.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>15.000000</td>
          <td>NaN</td>
          <td>35.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>75%</th>
          <td>15.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>26.000000</td>
          <td>NaN</td>
          <td>44.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>max</th>
          <td>18.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>55.000000</td>
          <td>NaN</td>
          <td>64.000000</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
          <td>NaN</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 66-69

Note that the dataset contains categorical and numerical variables.
We will need to take this into account when preprocessing the dataset
thereafter.

.. GENERATED FROM PYTHON SOURCE LINES 69-72

.. code-block:: Python


    X.head()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
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    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>EDUCATION</th>
          <th>SOUTH</th>
          <th>SEX</th>
          <th>EXPERIENCE</th>
          <th>UNION</th>
          <th>AGE</th>
          <th>RACE</th>
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          <td>9</td>
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          <td>12</td>
          <td>no</td>
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          <td>17</td>
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          <td>35</td>
          <td>White</td>
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          <td>Other</td>
          <td>Married</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 73-75

Our target for prediction: the wage.
Wages are described as floating-point number in dollars per hour.

.. GENERATED FROM PYTHON SOURCE LINES 77-80

.. code-block:: Python

    y = survey.target.values.ravel()
    survey.target.head()





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    0    5.10
    1    4.95
    2    6.67
    3    4.00
    4    7.50
    Name: WAGE, dtype: float64



.. GENERATED FROM PYTHON SOURCE LINES 81-86

We split the sample into a train and a test dataset.
Only the train dataset will be used in the following exploratory analysis.
This is a way to emulate a real situation where predictions are performed on
an unknown target, and we don't want our analysis and decisions to be biased
by our knowledge of the test data.

.. GENERATED FROM PYTHON SOURCE LINES 86-91

.. code-block:: Python


    from sklearn.model_selection import train_test_split

    X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)








.. GENERATED FROM PYTHON SOURCE LINES 92-97

First, let's get some insights by looking at the variable distributions and
at the pairwise relationships between them. Only numerical
variables will be used. In the following plot, each dot represents a sample.

.. _marginal_dependencies:

.. GENERATED FROM PYTHON SOURCE LINES 97-102

.. code-block:: Python


    train_dataset = X_train.copy()
    train_dataset.insert(0, "WAGE", y_train)
    _ = sns.pairplot(train_dataset, kind="reg", diag_kind="kde")




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_001.png
   :alt: plot linear model coefficient interpretation
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 103-122

Looking closely at the WAGE distribution reveals that it has a
long tail. For this reason, we should take its logarithm
to turn it approximately into a normal distribution (linear models such
as ridge or lasso work best for a normal distribution of error).

The WAGE is increasing when EDUCATION is increasing.
Note that the dependence between WAGE and EDUCATION
represented here is a marginal dependence, i.e., it describes the behavior
of a specific variable without keeping the others fixed.

Also, the EXPERIENCE and AGE are strongly linearly correlated.

.. _the-pipeline:

The machine-learning pipeline
-----------------------------

To design our machine-learning pipeline, we first manually
check the type of data that we are dealing with:

.. GENERATED FROM PYTHON SOURCE LINES 122-125

.. code-block:: Python


    survey.data.info()





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    <class 'pandas.core.frame.DataFrame'>
    RangeIndex: 534 entries, 0 to 533
    Data columns (total 10 columns):
     #   Column      Non-Null Count  Dtype   
    ---  ------      --------------  -----   
     0   EDUCATION   534 non-null    int64   
     1   SOUTH       534 non-null    category
     2   SEX         534 non-null    category
     3   EXPERIENCE  534 non-null    int64   
     4   UNION       534 non-null    category
     5   AGE         534 non-null    int64   
     6   RACE        534 non-null    category
     7   OCCUPATION  534 non-null    category
     8   SECTOR      534 non-null    category
     9   MARR        534 non-null    category
    dtypes: category(7), int64(3)
    memory usage: 17.2 KB




.. GENERATED FROM PYTHON SOURCE LINES 126-137

As seen previously, the dataset contains columns with different data types
and we need to apply a specific preprocessing for each data types.
In particular categorical variables cannot be included in linear model if not
coded as integers first. In addition, to avoid categorical features to be
treated as ordered values, we need to one-hot-encode them.
Our pre-processor will

- one-hot encode (i.e., generate a column by category) the categorical
  columns, only for non-binary categorical variables;
- as a first approach (we will see after how the normalisation of numerical
  values will affect our discussion), keep numerical values as they are.

.. GENERATED FROM PYTHON SOURCE LINES 137-150

.. code-block:: Python


    from sklearn.compose import make_column_transformer
    from sklearn.preprocessing import OneHotEncoder

    categorical_columns = ["RACE", "OCCUPATION", "SECTOR", "MARR", "UNION", "SEX", "SOUTH"]
    numerical_columns = ["EDUCATION", "EXPERIENCE", "AGE"]

    preprocessor = make_column_transformer(
        (OneHotEncoder(drop="if_binary"), categorical_columns),
        remainder="passthrough",
        verbose_feature_names_out=False,  # avoid to prepend the preprocessor names
    )








.. GENERATED FROM PYTHON SOURCE LINES 151-153

To describe the dataset as a linear model we use a ridge regressor
with a very small regularization and to model the logarithm of the WAGE.

.. GENERATED FROM PYTHON SOURCE LINES 153-165

.. code-block:: Python


    from sklearn.compose import TransformedTargetRegressor
    from sklearn.linear_model import Ridge
    from sklearn.pipeline import make_pipeline

    model = make_pipeline(
        preprocessor,
        TransformedTargetRegressor(
            regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10
        ),
    )








.. GENERATED FROM PYTHON SOURCE LINES 166-170

Processing the dataset
----------------------

First, we fit the model.

.. GENERATED FROM PYTHON SOURCE LINES 170-173

.. code-block:: Python


    model.fit(X_train, y_train)






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      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-32 div.sk-label.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator-specific style */

    /* Colorize estimator box */
    #sk-container-id-32 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-32 div.sk-estimator.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    #sk-container-id-32 div.sk-label label.sk-toggleable__label,
    #sk-container-id-32 div.sk-label label {
      /* The background is the default theme color */
      color: var(--sklearn-color-text-on-default-background);
    }

    /* On hover, darken the color of the background */
    #sk-container-id-32 div.sk-label:hover label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    /* Label box, darken color on hover, fitted */
    #sk-container-id-32 div.sk-label.fitted:hover label.sk-toggleable__label.fitted {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator label */

    #sk-container-id-32 div.sk-label label {
      font-family: monospace;
      font-weight: bold;
      display: inline-block;
      line-height: 1.2em;
    }

    #sk-container-id-32 div.sk-label-container {
      text-align: center;
    }

    /* Estimator-specific */
    #sk-container-id-32 div.sk-estimator {
      font-family: monospace;
      border: 1px dotted var(--sklearn-color-border-box);
      border-radius: 0.25em;
      box-sizing: border-box;
      margin-bottom: 0.5em;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-32 div.sk-estimator.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    /* on hover */
    #sk-container-id-32 div.sk-estimator:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-32 div.sk-estimator.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Specification for estimator info (e.g. "i" and "?") */

    /* Common style for "i" and "?" */

    .sk-estimator-doc-link,
    a:link.sk-estimator-doc-link,
    a:visited.sk-estimator-doc-link {
      float: right;
      font-size: smaller;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1em;
      height: 1em;
      width: 1em;
      text-decoration: none !important;
      margin-left: 0.5em;
      text-align: center;
      /* unfitted */
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
      color: var(--sklearn-color-unfitted-level-1);
    }

    .sk-estimator-doc-link.fitted,
    a:link.sk-estimator-doc-link.fitted,
    a:visited.sk-estimator-doc-link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    div.sk-estimator:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover,
    div.sk-label-container:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    div.sk-estimator.fitted:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover,
    div.sk-label-container:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    /* Span, style for the box shown on hovering the info icon */
    .sk-estimator-doc-link span {
      display: none;
      z-index: 9999;
      position: relative;
      font-weight: normal;
      right: .2ex;
      padding: .5ex;
      margin: .5ex;
      width: min-content;
      min-width: 20ex;
      max-width: 50ex;
      color: var(--sklearn-color-text);
      box-shadow: 2pt 2pt 4pt #999;
      /* unfitted */
      background: var(--sklearn-color-unfitted-level-0);
      border: .5pt solid var(--sklearn-color-unfitted-level-3);
    }

    .sk-estimator-doc-link.fitted span {
      /* fitted */
      background: var(--sklearn-color-fitted-level-0);
      border: var(--sklearn-color-fitted-level-3);
    }

    .sk-estimator-doc-link:hover span {
      display: block;
    }

    /* "?"-specific style due to the `<a>` HTML tag */

    #sk-container-id-32 a.estimator_doc_link {
      float: right;
      font-size: 1rem;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1rem;
      height: 1rem;
      width: 1rem;
      text-decoration: none;
      /* unfitted */
      color: var(--sklearn-color-unfitted-level-1);
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
    }

    #sk-container-id-32 a.estimator_doc_link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    #sk-container-id-32 a.estimator_doc_link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    #sk-container-id-32 a.estimator_doc_link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
    }
    </style><div id="sk-container-id-32" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(remainder=&#x27;passthrough&#x27;,
                                       transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;])],
                                       verbose_feature_names_out=False)),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=Ridge(alpha=1e-10)))])</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-116" type="checkbox" ><label for="sk-estimator-id-116" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>Pipeline</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.pipeline.Pipeline.html">?<span>Documentation for Pipeline</span></a><span class="sk-estimator-doc-link fitted">i<span>Fitted</span></span></div></label><div class="sk-toggleable__content fitted"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(remainder=&#x27;passthrough&#x27;,
                                       transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;])],
                                       verbose_feature_names_out=False)),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=Ridge(alpha=1e-10)))])</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-117" type="checkbox" ><label for="sk-estimator-id-117" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>ColumnTransformer</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.ColumnTransformer.html">?<span>Documentation for ColumnTransformer</span></a></div></label><div class="sk-toggleable__content fitted"><pre>ColumnTransformer(remainder=&#x27;passthrough&#x27;,
                      transformers=[(&#x27;onehotencoder&#x27;,
                                     OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                     [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;, &#x27;SECTOR&#x27;, &#x27;MARR&#x27;,
                                      &#x27;UNION&#x27;, &#x27;SEX&#x27;, &#x27;SOUTH&#x27;])],
                      verbose_feature_names_out=False)</pre></div> </div></div><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-118" type="checkbox" ><label for="sk-estimator-id-118" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>transformedtargetregressor: TransformedTargetRegressor</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.TransformedTargetRegressor.html">?<span>Documentation for transformedtargetregressor: TransformedTargetRegressor</span></a></div></label><div class="sk-toggleable__content fitted"><pre>TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;, inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                               regressor=Ridge(alpha=1e-10))</pre></div> </div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-119" type="checkbox" ><label for="sk-estimator-id-119" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>regressor: Ridge</div></div></label><div class="sk-toggleable__content fitted"><pre>Ridge(alpha=1e-10)</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-120" type="checkbox" ><label for="sk-estimator-id-120" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>Ridge</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.linear_model.Ridge.html">?<span>Documentation for Ridge</span></a></div></label><div class="sk-toggleable__content fitted"><pre>Ridge(alpha=1e-10)</pre></div> </div></div></div></div></div></div></div></div></div></div></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 174-177

Then we check the performance of the computed model plotting its predictions
on the test set and computing,
for example, the median absolute error of the model.

.. GENERATED FROM PYTHON SOURCE LINES 177-188

.. code-block:: Python


    from sklearn.metrics import PredictionErrorDisplay, median_absolute_error

    mae_train = median_absolute_error(y_train, model.predict(X_train))
    y_pred = model.predict(X_test)
    mae_test = median_absolute_error(y_test, y_pred)
    scores = {
        "MedAE on training set": f"{mae_train:.2f} $/hour",
        "MedAE on testing set": f"{mae_test:.2f} $/hour",
    }








.. GENERATED FROM PYTHON SOURCE LINES 189-199

.. code-block:: Python

    _, ax = plt.subplots(figsize=(5, 5))
    display = PredictionErrorDisplay.from_predictions(
        y_test, y_pred, kind="actual_vs_predicted", ax=ax, scatter_kwargs={"alpha": 0.5}
    )
    ax.set_title("Ridge model, small regularization")
    for name, score in scores.items():
        ax.plot([], [], " ", label=f"{name}: {score}")
    ax.legend(loc="upper left")
    plt.tight_layout()




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_002.png
   :alt: Ridge model, small regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 200-214

The model learnt is far from being a good model making accurate predictions:
this is obvious when looking at the plot above, where good predictions
should lie on the black dashed line.

In the following section, we will interpret the coefficients of the model.
While we do so, we should keep in mind that any conclusion we draw is
about the model that we build, rather than about the true (real-world)
generative process of the data.

Interpreting coefficients: scale matters
----------------------------------------

First of all, we can take a look to the values of the coefficients of the
regressor we have fitted.

.. GENERATED FROM PYTHON SOURCE LINES 214-224

.. code-block:: Python

    feature_names = model[:-1].get_feature_names_out()

    coefs = pd.DataFrame(
        model[-1].regressor_.coef_,
        columns=["Coefficients"],
        index=feature_names,
    )

    coefs






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>Coefficients</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>RACE_Hispanic</th>
          <td>-0.013520</td>
        </tr>
        <tr>
          <th>RACE_Other</th>
          <td>-0.009077</td>
        </tr>
        <tr>
          <th>RACE_White</th>
          <td>0.022593</td>
        </tr>
        <tr>
          <th>OCCUPATION_Clerical</th>
          <td>0.000045</td>
        </tr>
        <tr>
          <th>OCCUPATION_Management</th>
          <td>0.090528</td>
        </tr>
        <tr>
          <th>OCCUPATION_Other</th>
          <td>-0.025102</td>
        </tr>
        <tr>
          <th>OCCUPATION_Professional</th>
          <td>0.071964</td>
        </tr>
        <tr>
          <th>OCCUPATION_Sales</th>
          <td>-0.046636</td>
        </tr>
        <tr>
          <th>OCCUPATION_Service</th>
          <td>-0.091053</td>
        </tr>
        <tr>
          <th>SECTOR_Construction</th>
          <td>-0.000198</td>
        </tr>
        <tr>
          <th>SECTOR_Manufacturing</th>
          <td>0.031255</td>
        </tr>
        <tr>
          <th>SECTOR_Other</th>
          <td>-0.031026</td>
        </tr>
        <tr>
          <th>MARR_Unmarried</th>
          <td>-0.032405</td>
        </tr>
        <tr>
          <th>UNION_not_member</th>
          <td>-0.117154</td>
        </tr>
        <tr>
          <th>SEX_male</th>
          <td>0.090808</td>
        </tr>
        <tr>
          <th>SOUTH_yes</th>
          <td>-0.033823</td>
        </tr>
        <tr>
          <th>EDUCATION</th>
          <td>0.054699</td>
        </tr>
        <tr>
          <th>EXPERIENCE</th>
          <td>0.035005</td>
        </tr>
        <tr>
          <th>AGE</th>
          <td>-0.030867</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 225-237

The AGE coefficient is expressed in "dollars/hour per living years" while the
EDUCATION one is expressed in "dollars/hour per years of education". This
representation of the coefficients has the benefit of making clear the
practical predictions of the model: an increase of :math:`1` year in AGE
means a decrease of :math:`0.030867` dollars/hour, while an increase of
:math:`1` year in EDUCATION means an increase of :math:`0.054699`
dollars/hour. On the other hand, categorical variables (as UNION or SEX) are
adimensional numbers taking either the value 0 or 1. Their coefficients
are expressed in dollars/hour. Then, we cannot compare the magnitude of
different coefficients since the features have different natural scales, and
hence value ranges, because of their different unit of measure. This is more
visible if we plot the coefficients.

.. GENERATED FROM PYTHON SOURCE LINES 237-244

.. code-block:: Python


    coefs.plot.barh(figsize=(9, 7))
    plt.title("Ridge model, small regularization")
    plt.axvline(x=0, color=".5")
    plt.xlabel("Raw coefficient values")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_003.png
   :alt: Ridge model, small regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 245-256

Indeed, from the plot above the most important factor in determining WAGE
appears to be the
variable UNION, even if our intuition might tell us that variables
like EXPERIENCE should have more impact.

Looking at the coefficient plot to gauge feature importance can be
misleading as some of them vary on a small scale, while others, like AGE,
varies a lot more, several decades.

This is visible if we compare the standard deviations of different
features.

.. GENERATED FROM PYTHON SOURCE LINES 256-266

.. code-block:: Python


    X_train_preprocessed = pd.DataFrame(
        model[:-1].transform(X_train), columns=feature_names
    )

    X_train_preprocessed.std(axis=0).plot.barh(figsize=(9, 7))
    plt.title("Feature ranges")
    plt.xlabel("Std. dev. of feature values")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_004.png
   :alt: Feature ranges
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 267-277

Multiplying the coefficients by the standard deviation of the related
feature would reduce all the coefficients to the same unit of measure.
As we will see :ref:`after<scaling_num>` this is equivalent to normalize
numerical variables to their standard deviation,
as :math:`y = \sum{coef_i \times X_i} =
\sum{(coef_i \times std_i) \times (X_i / std_i)}`.

In that way, we emphasize that the
greater the variance of a feature, the larger the weight of the corresponding
coefficient on the output, all else being equal.

.. GENERATED FROM PYTHON SOURCE LINES 277-289

.. code-block:: Python


    coefs = pd.DataFrame(
        model[-1].regressor_.coef_ * X_train_preprocessed.std(axis=0),
        columns=["Coefficient importance"],
        index=feature_names,
    )
    coefs.plot(kind="barh", figsize=(9, 7))
    plt.xlabel("Coefficient values corrected by the feature's std. dev.")
    plt.title("Ridge model, small regularization")
    plt.axvline(x=0, color=".5")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_005.png
   :alt: Ridge model, small regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 290-345

Now that the coefficients have been scaled, we can safely compare them.

.. warning::

  Why does the plot above suggest that an increase in age leads to a
  decrease in wage? Why the :ref:`initial pairplot
  <marginal_dependencies>` is telling the opposite?

The plot above tells us about dependencies between a specific feature and
the target when all other features remain constant, i.e., **conditional
dependencies**. An increase of the AGE will induce a decrease
of the WAGE when all other features remain constant. On the contrary, an
increase of the EXPERIENCE will induce an increase of the WAGE when all
other features remain constant.
Also, AGE, EXPERIENCE and EDUCATION are the three variables that most
influence the model.

Interpreting coefficients: being cautious about causality
---------------------------------------------------------

Linear models are a great tool for measuring statistical association, but we
should be cautious when making statements about causality, after all
correlation doesn't always imply causation. This is particularly difficult in
the social sciences because the variables we observe only function as proxies
for the underlying causal process.

In our particular case we can think of the EDUCATION of an individual as a
proxy for their professional aptitude, the real variable we're interested in
but can't observe. We'd certainly like to think that staying in school for
longer would increase technical competency, but it's also quite possible that
causality goes the other way too. That is, those who are technically
competent tend to stay in school for longer.

An employer is unlikely to care which case it is (or if it's a mix of both),
as long as they remain convinced that a person with more EDUCATION is better
suited for the job, they will be happy to pay out a higher WAGE.

This confounding of effects becomes problematic when thinking about some
form of intervention e.g. government subsidies of university degrees or
promotional material encouraging individuals to take up higher education.
The usefulness of these measures could end up being overstated, especially if
the degree of confounding is strong. Our model predicts a :math:`0.054699`
increase in hourly wage for each year of education. The actual causal effect
might be lower because of this confounding.

Checking the variability of the coefficients
--------------------------------------------

We can check the coefficient variability through cross-validation:
it is a form of data perturbation (related to
`resampling <https://en.wikipedia.org/wiki/Resampling_(statistics)>`_).

If coefficients vary significantly when changing the input dataset
their robustness is not guaranteed, and they should probably be interpreted
with caution.

.. GENERATED FROM PYTHON SOURCE LINES 345-366

.. code-block:: Python


    from sklearn.model_selection import RepeatedKFold, cross_validate

    cv = RepeatedKFold(n_splits=5, n_repeats=5, random_state=0)
    cv_model = cross_validate(
        model,
        X,
        y,
        cv=cv,
        return_estimator=True,
        n_jobs=2,
    )

    coefs = pd.DataFrame(
        [
            est[-1].regressor_.coef_ * est[:-1].transform(X.iloc[train_idx]).std(axis=0)
            for est, (train_idx, _) in zip(cv_model["estimator"], cv.split(X, y))
        ],
        columns=feature_names,
    )








.. GENERATED FROM PYTHON SOURCE LINES 367-376

.. code-block:: Python

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", palette="dark:k", alpha=0.5)
    sns.boxplot(data=coefs, orient="h", color="cyan", saturation=0.5, whis=10)
    plt.axvline(x=0, color=".5")
    plt.xlabel("Coefficient importance")
    plt.title("Coefficient importance and its variability")
    plt.suptitle("Ridge model, small regularization")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_006.png
   :alt: Ridge model, small regularization, Coefficient importance and its variability
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_006.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 377-389

The problem of correlated variables
-----------------------------------

The AGE and EXPERIENCE coefficients are affected by strong variability which
might be due to the collinearity between the 2 features: as AGE and
EXPERIENCE vary together in the data, their effect is difficult to tease
apart.

To verify this interpretation we plot the variability of the AGE and
EXPERIENCE coefficient.

.. _covariation:

.. GENERATED FROM PYTHON SOURCE LINES 389-398

.. code-block:: Python


    plt.ylabel("Age coefficient")
    plt.xlabel("Experience coefficient")
    plt.grid(True)
    plt.xlim(-0.4, 0.5)
    plt.ylim(-0.4, 0.5)
    plt.scatter(coefs["AGE"], coefs["EXPERIENCE"])
    _ = plt.title("Co-variations of coefficients for AGE and EXPERIENCE across folds")




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_007.png
   :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_007.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 399-404

Two regions are populated: when the EXPERIENCE coefficient is
positive the AGE one is negative and vice-versa.

To go further we remove one of the 2 features and check what is the impact
on the model stability.

.. GENERATED FROM PYTHON SOURCE LINES 404-425

.. code-block:: Python


    column_to_drop = ["AGE"]

    cv_model = cross_validate(
        model,
        X.drop(columns=column_to_drop),
        y,
        cv=cv,
        return_estimator=True,
        n_jobs=2,
    )

    coefs = pd.DataFrame(
        [
            est[-1].regressor_.coef_
            * est[:-1].transform(X.drop(columns=column_to_drop).iloc[train_idx]).std(axis=0)
            for est, (train_idx, _) in zip(cv_model["estimator"], cv.split(X, y))
        ],
        columns=feature_names[:-1],
    )








.. GENERATED FROM PYTHON SOURCE LINES 426-435

.. code-block:: Python

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", palette="dark:k", alpha=0.5)
    sns.boxplot(data=coefs, orient="h", color="cyan", saturation=0.5)
    plt.axvline(x=0, color=".5")
    plt.title("Coefficient importance and its variability")
    plt.xlabel("Coefficient importance")
    plt.suptitle("Ridge model, small regularization, AGE dropped")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_008.png
   :alt: Ridge model, small regularization, AGE dropped, Coefficient importance and its variability
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_008.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 436-451

The estimation of the EXPERIENCE coefficient now shows a much reduced
variability. EXPERIENCE remains important for all models trained during
cross-validation.

.. _scaling_num:

Preprocessing numerical variables
---------------------------------

As said above (see ":ref:`the-pipeline`"), we could also choose to scale
numerical values before training the model.
This can be useful when we apply a similar amount of regularization to all of them
in the ridge.
The preprocessor is redefined in order to subtract the mean and scale
variables to unit variance.

.. GENERATED FROM PYTHON SOURCE LINES 451-459

.. code-block:: Python


    from sklearn.preprocessing import StandardScaler

    preprocessor = make_column_transformer(
        (OneHotEncoder(drop="if_binary"), categorical_columns),
        (StandardScaler(), numerical_columns),
    )








.. GENERATED FROM PYTHON SOURCE LINES 460-461

The model will stay unchanged.

.. GENERATED FROM PYTHON SOURCE LINES 461-470

.. code-block:: Python


    model = make_pipeline(
        preprocessor,
        TransformedTargetRegressor(
            regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10
        ),
    )
    model.fit(X_train, y_train)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <style>#sk-container-id-33 {
      /* Definition of color scheme common for light and dark mode */
      --sklearn-color-text: #000;
      --sklearn-color-text-muted: #666;
      --sklearn-color-line: gray;
      /* Definition of color scheme for unfitted estimators */
      --sklearn-color-unfitted-level-0: #fff5e6;
      --sklearn-color-unfitted-level-1: #f6e4d2;
      --sklearn-color-unfitted-level-2: #ffe0b3;
      --sklearn-color-unfitted-level-3: chocolate;
      /* Definition of color scheme for fitted estimators */
      --sklearn-color-fitted-level-0: #f0f8ff;
      --sklearn-color-fitted-level-1: #d4ebff;
      --sklearn-color-fitted-level-2: #b3dbfd;
      --sklearn-color-fitted-level-3: cornflowerblue;

      /* Specific color for light theme */
      --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, white)));
      --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-icon: #696969;

      @media (prefers-color-scheme: dark) {
        /* Redefinition of color scheme for dark theme */
        --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, #111)));
        --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-icon: #878787;
      }
    }

    #sk-container-id-33 {
      color: var(--sklearn-color-text);
    }

    #sk-container-id-33 pre {
      padding: 0;
    }

    #sk-container-id-33 input.sk-hidden--visually {
      border: 0;
      clip: rect(1px 1px 1px 1px);
      clip: rect(1px, 1px, 1px, 1px);
      height: 1px;
      margin: -1px;
      overflow: hidden;
      padding: 0;
      position: absolute;
      width: 1px;
    }

    #sk-container-id-33 div.sk-dashed-wrapped {
      border: 1px dashed var(--sklearn-color-line);
      margin: 0 0.4em 0.5em 0.4em;
      box-sizing: border-box;
      padding-bottom: 0.4em;
      background-color: var(--sklearn-color-background);
    }

    #sk-container-id-33 div.sk-container {
      /* jupyter's `normalize.less` sets `[hidden] { display: none; }`
         but bootstrap.min.css set `[hidden] { display: none !important; }`
         so we also need the `!important` here to be able to override the
         default hidden behavior on the sphinx rendered scikit-learn.org.
         See: https://github.com/scikit-learn/scikit-learn/issues/21755 */
      display: inline-block !important;
      position: relative;
    }

    #sk-container-id-33 div.sk-text-repr-fallback {
      display: none;
    }

    div.sk-parallel-item,
    div.sk-serial,
    div.sk-item {
      /* draw centered vertical line to link estimators */
      background-image: linear-gradient(var(--sklearn-color-text-on-default-background), var(--sklearn-color-text-on-default-background));
      background-size: 2px 100%;
      background-repeat: no-repeat;
      background-position: center center;
    }

    /* Parallel-specific style estimator block */

    #sk-container-id-33 div.sk-parallel-item::after {
      content: "";
      width: 100%;
      border-bottom: 2px solid var(--sklearn-color-text-on-default-background);
      flex-grow: 1;
    }

    #sk-container-id-33 div.sk-parallel {
      display: flex;
      align-items: stretch;
      justify-content: center;
      background-color: var(--sklearn-color-background);
      position: relative;
    }

    #sk-container-id-33 div.sk-parallel-item {
      display: flex;
      flex-direction: column;
    }

    #sk-container-id-33 div.sk-parallel-item:first-child::after {
      align-self: flex-end;
      width: 50%;
    }

    #sk-container-id-33 div.sk-parallel-item:last-child::after {
      align-self: flex-start;
      width: 50%;
    }

    #sk-container-id-33 div.sk-parallel-item:only-child::after {
      width: 0;
    }

    /* Serial-specific style estimator block */

    #sk-container-id-33 div.sk-serial {
      display: flex;
      flex-direction: column;
      align-items: center;
      background-color: var(--sklearn-color-background);
      padding-right: 1em;
      padding-left: 1em;
    }


    /* Toggleable style: style used for estimator/Pipeline/ColumnTransformer box that is
    clickable and can be expanded/collapsed.
    - Pipeline and ColumnTransformer use this feature and define the default style
    - Estimators will overwrite some part of the style using the `sk-estimator` class
    */

    /* Pipeline and ColumnTransformer style (default) */

    #sk-container-id-33 div.sk-toggleable {
      /* Default theme specific background. It is overwritten whether we have a
      specific estimator or a Pipeline/ColumnTransformer */
      background-color: var(--sklearn-color-background);
    }

    /* Toggleable label */
    #sk-container-id-33 label.sk-toggleable__label {
      cursor: pointer;
      display: flex;
      width: 100%;
      margin-bottom: 0;
      padding: 0.5em;
      box-sizing: border-box;
      text-align: center;
      align-items: start;
      justify-content: space-between;
      gap: 0.5em;
    }

    #sk-container-id-33 label.sk-toggleable__label .caption {
      font-size: 0.6rem;
      font-weight: lighter;
      color: var(--sklearn-color-text-muted);
    }

    #sk-container-id-33 label.sk-toggleable__label-arrow:before {
      /* Arrow on the left of the label */
      content: "▸";
      float: left;
      margin-right: 0.25em;
      color: var(--sklearn-color-icon);
    }

    #sk-container-id-33 label.sk-toggleable__label-arrow:hover:before {
      color: var(--sklearn-color-text);
    }

    /* Toggleable content - dropdown */

    #sk-container-id-33 div.sk-toggleable__content {
      max-height: 0;
      max-width: 0;
      overflow: hidden;
      text-align: left;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-33 div.sk-toggleable__content.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-33 div.sk-toggleable__content pre {
      margin: 0.2em;
      border-radius: 0.25em;
      color: var(--sklearn-color-text);
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-33 div.sk-toggleable__content.fitted pre {
      /* unfitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-33 input.sk-toggleable__control:checked~div.sk-toggleable__content {
      /* Expand drop-down */
      max-height: 200px;
      max-width: 100%;
      overflow: auto;
    }

    #sk-container-id-33 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {
      content: "▾";
    }

    /* Pipeline/ColumnTransformer-specific style */

    #sk-container-id-33 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-33 div.sk-label.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator-specific style */

    /* Colorize estimator box */
    #sk-container-id-33 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-33 div.sk-estimator.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    #sk-container-id-33 div.sk-label label.sk-toggleable__label,
    #sk-container-id-33 div.sk-label label {
      /* The background is the default theme color */
      color: var(--sklearn-color-text-on-default-background);
    }

    /* On hover, darken the color of the background */
    #sk-container-id-33 div.sk-label:hover label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    /* Label box, darken color on hover, fitted */
    #sk-container-id-33 div.sk-label.fitted:hover label.sk-toggleable__label.fitted {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator label */

    #sk-container-id-33 div.sk-label label {
      font-family: monospace;
      font-weight: bold;
      display: inline-block;
      line-height: 1.2em;
    }

    #sk-container-id-33 div.sk-label-container {
      text-align: center;
    }

    /* Estimator-specific */
    #sk-container-id-33 div.sk-estimator {
      font-family: monospace;
      border: 1px dotted var(--sklearn-color-border-box);
      border-radius: 0.25em;
      box-sizing: border-box;
      margin-bottom: 0.5em;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-33 div.sk-estimator.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    /* on hover */
    #sk-container-id-33 div.sk-estimator:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-33 div.sk-estimator.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Specification for estimator info (e.g. "i" and "?") */

    /* Common style for "i" and "?" */

    .sk-estimator-doc-link,
    a:link.sk-estimator-doc-link,
    a:visited.sk-estimator-doc-link {
      float: right;
      font-size: smaller;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1em;
      height: 1em;
      width: 1em;
      text-decoration: none !important;
      margin-left: 0.5em;
      text-align: center;
      /* unfitted */
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
      color: var(--sklearn-color-unfitted-level-1);
    }

    .sk-estimator-doc-link.fitted,
    a:link.sk-estimator-doc-link.fitted,
    a:visited.sk-estimator-doc-link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    div.sk-estimator:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover,
    div.sk-label-container:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    div.sk-estimator.fitted:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover,
    div.sk-label-container:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    /* Span, style for the box shown on hovering the info icon */
    .sk-estimator-doc-link span {
      display: none;
      z-index: 9999;
      position: relative;
      font-weight: normal;
      right: .2ex;
      padding: .5ex;
      margin: .5ex;
      width: min-content;
      min-width: 20ex;
      max-width: 50ex;
      color: var(--sklearn-color-text);
      box-shadow: 2pt 2pt 4pt #999;
      /* unfitted */
      background: var(--sklearn-color-unfitted-level-0);
      border: .5pt solid var(--sklearn-color-unfitted-level-3);
    }

    .sk-estimator-doc-link.fitted span {
      /* fitted */
      background: var(--sklearn-color-fitted-level-0);
      border: var(--sklearn-color-fitted-level-3);
    }

    .sk-estimator-doc-link:hover span {
      display: block;
    }

    /* "?"-specific style due to the `<a>` HTML tag */

    #sk-container-id-33 a.estimator_doc_link {
      float: right;
      font-size: 1rem;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1rem;
      height: 1rem;
      width: 1rem;
      text-decoration: none;
      /* unfitted */
      color: var(--sklearn-color-unfitted-level-1);
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
    }

    #sk-container-id-33 a.estimator_doc_link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    #sk-container-id-33 a.estimator_doc_link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    #sk-container-id-33 a.estimator_doc_link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
    }
    </style><div id="sk-container-id-33" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                                     (&#x27;standardscaler&#x27;,
                                                      StandardScaler(),
                                                      [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;,
                                                       &#x27;AGE&#x27;])])),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=Ridge(alpha=1e-10)))])</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-121" type="checkbox" ><label for="sk-estimator-id-121" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>Pipeline</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.pipeline.Pipeline.html">?<span>Documentation for Pipeline</span></a><span class="sk-estimator-doc-link fitted">i<span>Fitted</span></span></div></label><div class="sk-toggleable__content fitted"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                                     (&#x27;standardscaler&#x27;,
                                                      StandardScaler(),
                                                      [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;,
                                                       &#x27;AGE&#x27;])])),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=Ridge(alpha=1e-10)))])</pre></div> </div></div><div class="sk-serial"><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-122" type="checkbox" ><label for="sk-estimator-id-122" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>columntransformer: ColumnTransformer</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.ColumnTransformer.html">?<span>Documentation for columntransformer: ColumnTransformer</span></a></div></label><div class="sk-toggleable__content fitted"><pre>ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                     OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                     [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;, &#x27;SECTOR&#x27;, &#x27;MARR&#x27;,
                                      &#x27;UNION&#x27;, &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                    (&#x27;standardscaler&#x27;, StandardScaler(),
                                     [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;, &#x27;AGE&#x27;])])</pre></div> </div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-123" type="checkbox" ><label for="sk-estimator-id-123" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>onehotencoder</div></div></label><div class="sk-toggleable__content fitted"><pre>[&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;, &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;, &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-124" type="checkbox" ><label for="sk-estimator-id-124" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>OneHotEncoder</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.preprocessing.OneHotEncoder.html">?<span>Documentation for OneHotEncoder</span></a></div></label><div class="sk-toggleable__content fitted"><pre>OneHotEncoder(drop=&#x27;if_binary&#x27;)</pre></div> </div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-125" type="checkbox" ><label for="sk-estimator-id-125" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>standardscaler</div></div></label><div class="sk-toggleable__content fitted"><pre>[&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;, &#x27;AGE&#x27;]</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-126" type="checkbox" ><label for="sk-estimator-id-126" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>StandardScaler</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.preprocessing.StandardScaler.html">?<span>Documentation for StandardScaler</span></a></div></label><div class="sk-toggleable__content fitted"><pre>StandardScaler()</pre></div> </div></div></div></div></div></div></div><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-127" type="checkbox" ><label for="sk-estimator-id-127" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>transformedtargetregressor: TransformedTargetRegressor</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.TransformedTargetRegressor.html">?<span>Documentation for transformedtargetregressor: TransformedTargetRegressor</span></a></div></label><div class="sk-toggleable__content fitted"><pre>TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;, inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                               regressor=Ridge(alpha=1e-10))</pre></div> </div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-128" type="checkbox" ><label for="sk-estimator-id-128" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>regressor: Ridge</div></div></label><div class="sk-toggleable__content fitted"><pre>Ridge(alpha=1e-10)</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-129" type="checkbox" ><label for="sk-estimator-id-129" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>Ridge</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.linear_model.Ridge.html">?<span>Documentation for Ridge</span></a></div></label><div class="sk-toggleable__content fitted"><pre>Ridge(alpha=1e-10)</pre></div> </div></div></div></div></div></div></div></div></div></div></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 471-474

Again, we check the performance of the computed
model using, for example, the median absolute error of the model and the R
squared coefficient.

.. GENERATED FROM PYTHON SOURCE LINES 474-493

.. code-block:: Python


    mae_train = median_absolute_error(y_train, model.predict(X_train))
    y_pred = model.predict(X_test)
    mae_test = median_absolute_error(y_test, y_pred)
    scores = {
        "MedAE on training set": f"{mae_train:.2f} $/hour",
        "MedAE on testing set": f"{mae_test:.2f} $/hour",
    }

    _, ax = plt.subplots(figsize=(5, 5))
    display = PredictionErrorDisplay.from_predictions(
        y_test, y_pred, kind="actual_vs_predicted", ax=ax, scatter_kwargs={"alpha": 0.5}
    )
    ax.set_title("Ridge model, small regularization")
    for name, score in scores.items():
        ax.plot([], [], " ", label=f"{name}: {score}")
    ax.legend(loc="upper left")
    plt.tight_layout()




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_009.png
   :alt: Ridge model, small regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_009.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 494-496

For the coefficient analysis, scaling is not needed this time because it
was performed during the preprocessing step.

.. GENERATED FROM PYTHON SOURCE LINES 496-508

.. code-block:: Python


    coefs = pd.DataFrame(
        model[-1].regressor_.coef_,
        columns=["Coefficients importance"],
        index=feature_names,
    )
    coefs.plot.barh(figsize=(9, 7))
    plt.title("Ridge model, small regularization, normalized variables")
    plt.xlabel("Raw coefficient values")
    plt.axvline(x=0, color=".5")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_010.png
   :alt: Ridge model, small regularization, normalized variables
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_010.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 509-513

We now inspect the coefficients across several cross-validation folds. As in
the above example, we do not need to scale the coefficients by the std. dev.
of the feature values since this scaling was already
done in the preprocessing step of the pipeline.

.. GENERATED FROM PYTHON SOURCE LINES 513-526

.. code-block:: Python


    cv_model = cross_validate(
        model,
        X,
        y,
        cv=cv,
        return_estimator=True,
        n_jobs=2,
    )
    coefs = pd.DataFrame(
        [est[-1].regressor_.coef_ for est in cv_model["estimator"]], columns=feature_names
    )








.. GENERATED FROM PYTHON SOURCE LINES 527-534

.. code-block:: Python

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", palette="dark:k", alpha=0.5)
    sns.boxplot(data=coefs, orient="h", color="cyan", saturation=0.5, whis=10)
    plt.axvline(x=0, color=".5")
    plt.title("Coefficient variability")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_011.png
   :alt: Coefficient variability
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_011.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 535-548

The result is quite similar to the non-normalized case.

Linear models with regularization
---------------------------------

In machine-learning practice, ridge regression is more often used with
non-negligible regularization.

Above, we limited this regularization to a very little amount. Regularization
improves the conditioning of the problem and reduces the variance of the
estimates. :class:`~sklearn.linear_model.RidgeCV` applies cross validation
in order to determine which value of the regularization parameter (`alpha`)
is best suited for prediction.

.. GENERATED FROM PYTHON SOURCE LINES 548-562

.. code-block:: Python


    from sklearn.linear_model import RidgeCV

    alphas = np.logspace(-10, 10, 21)  # alpha values to be chosen from by cross-validation
    model = make_pipeline(
        preprocessor,
        TransformedTargetRegressor(
            regressor=RidgeCV(alphas=alphas),
            func=np.log10,
            inverse_func=sp.special.exp10,
        ),
    )
    model.fit(X_train, y_train)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <style>#sk-container-id-34 {
      /* Definition of color scheme common for light and dark mode */
      --sklearn-color-text: #000;
      --sklearn-color-text-muted: #666;
      --sklearn-color-line: gray;
      /* Definition of color scheme for unfitted estimators */
      --sklearn-color-unfitted-level-0: #fff5e6;
      --sklearn-color-unfitted-level-1: #f6e4d2;
      --sklearn-color-unfitted-level-2: #ffe0b3;
      --sklearn-color-unfitted-level-3: chocolate;
      /* Definition of color scheme for fitted estimators */
      --sklearn-color-fitted-level-0: #f0f8ff;
      --sklearn-color-fitted-level-1: #d4ebff;
      --sklearn-color-fitted-level-2: #b3dbfd;
      --sklearn-color-fitted-level-3: cornflowerblue;

      /* Specific color for light theme */
      --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, white)));
      --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-icon: #696969;

      @media (prefers-color-scheme: dark) {
        /* Redefinition of color scheme for dark theme */
        --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, #111)));
        --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-icon: #878787;
      }
    }

    #sk-container-id-34 {
      color: var(--sklearn-color-text);
    }

    #sk-container-id-34 pre {
      padding: 0;
    }

    #sk-container-id-34 input.sk-hidden--visually {
      border: 0;
      clip: rect(1px 1px 1px 1px);
      clip: rect(1px, 1px, 1px, 1px);
      height: 1px;
      margin: -1px;
      overflow: hidden;
      padding: 0;
      position: absolute;
      width: 1px;
    }

    #sk-container-id-34 div.sk-dashed-wrapped {
      border: 1px dashed var(--sklearn-color-line);
      margin: 0 0.4em 0.5em 0.4em;
      box-sizing: border-box;
      padding-bottom: 0.4em;
      background-color: var(--sklearn-color-background);
    }

    #sk-container-id-34 div.sk-container {
      /* jupyter's `normalize.less` sets `[hidden] { display: none; }`
         but bootstrap.min.css set `[hidden] { display: none !important; }`
         so we also need the `!important` here to be able to override the
         default hidden behavior on the sphinx rendered scikit-learn.org.
         See: https://github.com/scikit-learn/scikit-learn/issues/21755 */
      display: inline-block !important;
      position: relative;
    }

    #sk-container-id-34 div.sk-text-repr-fallback {
      display: none;
    }

    div.sk-parallel-item,
    div.sk-serial,
    div.sk-item {
      /* draw centered vertical line to link estimators */
      background-image: linear-gradient(var(--sklearn-color-text-on-default-background), var(--sklearn-color-text-on-default-background));
      background-size: 2px 100%;
      background-repeat: no-repeat;
      background-position: center center;
    }

    /* Parallel-specific style estimator block */

    #sk-container-id-34 div.sk-parallel-item::after {
      content: "";
      width: 100%;
      border-bottom: 2px solid var(--sklearn-color-text-on-default-background);
      flex-grow: 1;
    }

    #sk-container-id-34 div.sk-parallel {
      display: flex;
      align-items: stretch;
      justify-content: center;
      background-color: var(--sklearn-color-background);
      position: relative;
    }

    #sk-container-id-34 div.sk-parallel-item {
      display: flex;
      flex-direction: column;
    }

    #sk-container-id-34 div.sk-parallel-item:first-child::after {
      align-self: flex-end;
      width: 50%;
    }

    #sk-container-id-34 div.sk-parallel-item:last-child::after {
      align-self: flex-start;
      width: 50%;
    }

    #sk-container-id-34 div.sk-parallel-item:only-child::after {
      width: 0;
    }

    /* Serial-specific style estimator block */

    #sk-container-id-34 div.sk-serial {
      display: flex;
      flex-direction: column;
      align-items: center;
      background-color: var(--sklearn-color-background);
      padding-right: 1em;
      padding-left: 1em;
    }


    /* Toggleable style: style used for estimator/Pipeline/ColumnTransformer box that is
    clickable and can be expanded/collapsed.
    - Pipeline and ColumnTransformer use this feature and define the default style
    - Estimators will overwrite some part of the style using the `sk-estimator` class
    */

    /* Pipeline and ColumnTransformer style (default) */

    #sk-container-id-34 div.sk-toggleable {
      /* Default theme specific background. It is overwritten whether we have a
      specific estimator or a Pipeline/ColumnTransformer */
      background-color: var(--sklearn-color-background);
    }

    /* Toggleable label */
    #sk-container-id-34 label.sk-toggleable__label {
      cursor: pointer;
      display: flex;
      width: 100%;
      margin-bottom: 0;
      padding: 0.5em;
      box-sizing: border-box;
      text-align: center;
      align-items: start;
      justify-content: space-between;
      gap: 0.5em;
    }

    #sk-container-id-34 label.sk-toggleable__label .caption {
      font-size: 0.6rem;
      font-weight: lighter;
      color: var(--sklearn-color-text-muted);
    }

    #sk-container-id-34 label.sk-toggleable__label-arrow:before {
      /* Arrow on the left of the label */
      content: "▸";
      float: left;
      margin-right: 0.25em;
      color: var(--sklearn-color-icon);
    }

    #sk-container-id-34 label.sk-toggleable__label-arrow:hover:before {
      color: var(--sklearn-color-text);
    }

    /* Toggleable content - dropdown */

    #sk-container-id-34 div.sk-toggleable__content {
      max-height: 0;
      max-width: 0;
      overflow: hidden;
      text-align: left;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-34 div.sk-toggleable__content.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-34 div.sk-toggleable__content pre {
      margin: 0.2em;
      border-radius: 0.25em;
      color: var(--sklearn-color-text);
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-34 div.sk-toggleable__content.fitted pre {
      /* unfitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-34 input.sk-toggleable__control:checked~div.sk-toggleable__content {
      /* Expand drop-down */
      max-height: 200px;
      max-width: 100%;
      overflow: auto;
    }

    #sk-container-id-34 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {
      content: "▾";
    }

    /* Pipeline/ColumnTransformer-specific style */

    #sk-container-id-34 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-34 div.sk-label.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator-specific style */

    /* Colorize estimator box */
    #sk-container-id-34 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-34 div.sk-estimator.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    #sk-container-id-34 div.sk-label label.sk-toggleable__label,
    #sk-container-id-34 div.sk-label label {
      /* The background is the default theme color */
      color: var(--sklearn-color-text-on-default-background);
    }

    /* On hover, darken the color of the background */
    #sk-container-id-34 div.sk-label:hover label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    /* Label box, darken color on hover, fitted */
    #sk-container-id-34 div.sk-label.fitted:hover label.sk-toggleable__label.fitted {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator label */

    #sk-container-id-34 div.sk-label label {
      font-family: monospace;
      font-weight: bold;
      display: inline-block;
      line-height: 1.2em;
    }

    #sk-container-id-34 div.sk-label-container {
      text-align: center;
    }

    /* Estimator-specific */
    #sk-container-id-34 div.sk-estimator {
      font-family: monospace;
      border: 1px dotted var(--sklearn-color-border-box);
      border-radius: 0.25em;
      box-sizing: border-box;
      margin-bottom: 0.5em;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-34 div.sk-estimator.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    /* on hover */
    #sk-container-id-34 div.sk-estimator:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-34 div.sk-estimator.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Specification for estimator info (e.g. "i" and "?") */

    /* Common style for "i" and "?" */

    .sk-estimator-doc-link,
    a:link.sk-estimator-doc-link,
    a:visited.sk-estimator-doc-link {
      float: right;
      font-size: smaller;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1em;
      height: 1em;
      width: 1em;
      text-decoration: none !important;
      margin-left: 0.5em;
      text-align: center;
      /* unfitted */
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
      color: var(--sklearn-color-unfitted-level-1);
    }

    .sk-estimator-doc-link.fitted,
    a:link.sk-estimator-doc-link.fitted,
    a:visited.sk-estimator-doc-link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    div.sk-estimator:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover,
    div.sk-label-container:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    div.sk-estimator.fitted:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover,
    div.sk-label-container:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    /* Span, style for the box shown on hovering the info icon */
    .sk-estimator-doc-link span {
      display: none;
      z-index: 9999;
      position: relative;
      font-weight: normal;
      right: .2ex;
      padding: .5ex;
      margin: .5ex;
      width: min-content;
      min-width: 20ex;
      max-width: 50ex;
      color: var(--sklearn-color-text);
      box-shadow: 2pt 2pt 4pt #999;
      /* unfitted */
      background: var(--sklearn-color-unfitted-level-0);
      border: .5pt solid var(--sklearn-color-unfitted-level-3);
    }

    .sk-estimator-doc-link.fitted span {
      /* fitted */
      background: var(--sklearn-color-fitted-level-0);
      border: var(--sklearn-color-fitted-level-3);
    }

    .sk-estimator-doc-link:hover span {
      display: block;
    }

    /* "?"-specific style due to the `<a>` HTML tag */

    #sk-container-id-34 a.estimator_doc_link {
      float: right;
      font-size: 1rem;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1rem;
      height: 1rem;
      width: 1rem;
      text-decoration: none;
      /* unfitted */
      color: var(--sklearn-color-unfitted-level-1);
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
    }

    #sk-container-id-34 a.estimator_doc_link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    #sk-container-id-34 a.estimator_doc_link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    #sk-container-id-34 a.estimator_doc_link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
    }
    </style><div id="sk-container-id-34" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                                     (&#x27;standardscaler&#x27;,
                                                      StandardScaler(),
                                                      [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;,
                                                       &#x27;AGE&#x27;])])),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=RidgeCV(alphas=array([1.e-10, 1.e-09, 1.e-08, 1.e-07, 1.e-06, 1.e-05, 1.e-04, 1.e-03,
           1.e-02, 1.e-01, 1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05,
           1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10]))))])</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-130" type="checkbox" ><label for="sk-estimator-id-130" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>Pipeline</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.pipeline.Pipeline.html">?<span>Documentation for Pipeline</span></a><span class="sk-estimator-doc-link fitted">i<span>Fitted</span></span></div></label><div class="sk-toggleable__content fitted"><pre>Pipeline(steps=[(&#x27;columntransformer&#x27;,
                     ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                                      OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                                      [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;,
                                                       &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;,
                                                       &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                                     (&#x27;standardscaler&#x27;,
                                                      StandardScaler(),
                                                      [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;,
                                                       &#x27;AGE&#x27;])])),
                    (&#x27;transformedtargetregressor&#x27;,
                     TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;,
                                                inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                                                regressor=RidgeCV(alphas=array([1.e-10, 1.e-09, 1.e-08, 1.e-07, 1.e-06, 1.e-05, 1.e-04, 1.e-03,
           1.e-02, 1.e-01, 1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05,
           1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10]))))])</pre></div> </div></div><div class="sk-serial"><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-131" type="checkbox" ><label for="sk-estimator-id-131" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>columntransformer: ColumnTransformer</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.ColumnTransformer.html">?<span>Documentation for columntransformer: ColumnTransformer</span></a></div></label><div class="sk-toggleable__content fitted"><pre>ColumnTransformer(transformers=[(&#x27;onehotencoder&#x27;,
                                     OneHotEncoder(drop=&#x27;if_binary&#x27;),
                                     [&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;, &#x27;SECTOR&#x27;, &#x27;MARR&#x27;,
                                      &#x27;UNION&#x27;, &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]),
                                    (&#x27;standardscaler&#x27;, StandardScaler(),
                                     [&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;, &#x27;AGE&#x27;])])</pre></div> </div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-132" type="checkbox" ><label for="sk-estimator-id-132" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>onehotencoder</div></div></label><div class="sk-toggleable__content fitted"><pre>[&#x27;RACE&#x27;, &#x27;OCCUPATION&#x27;, &#x27;SECTOR&#x27;, &#x27;MARR&#x27;, &#x27;UNION&#x27;, &#x27;SEX&#x27;, &#x27;SOUTH&#x27;]</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-133" type="checkbox" ><label for="sk-estimator-id-133" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>OneHotEncoder</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.preprocessing.OneHotEncoder.html">?<span>Documentation for OneHotEncoder</span></a></div></label><div class="sk-toggleable__content fitted"><pre>OneHotEncoder(drop=&#x27;if_binary&#x27;)</pre></div> </div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-134" type="checkbox" ><label for="sk-estimator-id-134" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>standardscaler</div></div></label><div class="sk-toggleable__content fitted"><pre>[&#x27;EDUCATION&#x27;, &#x27;EXPERIENCE&#x27;, &#x27;AGE&#x27;]</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-135" type="checkbox" ><label for="sk-estimator-id-135" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>StandardScaler</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.preprocessing.StandardScaler.html">?<span>Documentation for StandardScaler</span></a></div></label><div class="sk-toggleable__content fitted"><pre>StandardScaler()</pre></div> </div></div></div></div></div></div></div><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-136" type="checkbox" ><label for="sk-estimator-id-136" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>transformedtargetregressor: TransformedTargetRegressor</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.compose.TransformedTargetRegressor.html">?<span>Documentation for transformedtargetregressor: TransformedTargetRegressor</span></a></div></label><div class="sk-toggleable__content fitted"><pre>TransformedTargetRegressor(func=&lt;ufunc &#x27;log10&#x27;&gt;, inverse_func=&lt;ufunc &#x27;exp10&#x27;&gt;,
                               regressor=RidgeCV(alphas=array([1.e-10, 1.e-09, 1.e-08, 1.e-07, 1.e-06, 1.e-05, 1.e-04, 1.e-03,
           1.e-02, 1.e-01, 1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05,
           1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10])))</pre></div> </div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-137" type="checkbox" ><label for="sk-estimator-id-137" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>regressor: RidgeCV</div></div></label><div class="sk-toggleable__content fitted"><pre>RidgeCV(alphas=array([1.e-10, 1.e-09, 1.e-08, 1.e-07, 1.e-06, 1.e-05, 1.e-04, 1.e-03,
           1.e-02, 1.e-01, 1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05,
           1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10]))</pre></div> </div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-138" type="checkbox" ><label for="sk-estimator-id-138" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>RidgeCV</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.6/modules/generated/sklearn.linear_model.RidgeCV.html">?<span>Documentation for RidgeCV</span></a></div></label><div class="sk-toggleable__content fitted"><pre>RidgeCV(alphas=array([1.e-10, 1.e-09, 1.e-08, 1.e-07, 1.e-06, 1.e-05, 1.e-04, 1.e-03,
           1.e-02, 1.e-01, 1.e+00, 1.e+01, 1.e+02, 1.e+03, 1.e+04, 1.e+05,
           1.e+06, 1.e+07, 1.e+08, 1.e+09, 1.e+10]))</pre></div> </div></div></div></div></div></div></div></div></div></div></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 563-564

First we check which value of :math:`\alpha` has been selected.

.. GENERATED FROM PYTHON SOURCE LINES 564-567

.. code-block:: Python


    model[-1].regressor_.alpha_





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    np.float64(10.0)



.. GENERATED FROM PYTHON SOURCE LINES 568-569

Then we check the quality of the predictions.

.. GENERATED FROM PYTHON SOURCE LINES 569-587

.. code-block:: Python

    mae_train = median_absolute_error(y_train, model.predict(X_train))
    y_pred = model.predict(X_test)
    mae_test = median_absolute_error(y_test, y_pred)
    scores = {
        "MedAE on training set": f"{mae_train:.2f} $/hour",
        "MedAE on testing set": f"{mae_test:.2f} $/hour",
    }

    _, ax = plt.subplots(figsize=(5, 5))
    display = PredictionErrorDisplay.from_predictions(
        y_test, y_pred, kind="actual_vs_predicted", ax=ax, scatter_kwargs={"alpha": 0.5}
    )
    ax.set_title("Ridge model, optimum regularization")
    for name, score in scores.items():
        ax.plot([], [], " ", label=f"{name}: {score}")
    ax.legend(loc="upper left")
    plt.tight_layout()




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_012.png
   :alt: Ridge model, optimum regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_012.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 588-590

The ability to reproduce the data of the regularized model is similar to
the one of the non-regularized model.

.. GENERATED FROM PYTHON SOURCE LINES 590-602

.. code-block:: Python


    coefs = pd.DataFrame(
        model[-1].regressor_.coef_,
        columns=["Coefficients importance"],
        index=feature_names,
    )
    coefs.plot.barh(figsize=(9, 7))
    plt.title("Ridge model, with regularization, normalized variables")
    plt.xlabel("Raw coefficient values")
    plt.axvline(x=0, color=".5")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_013.png
   :alt: Ridge model, with regularization, normalized variables
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_013.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 603-616

The coefficients are significantly different.
AGE and EXPERIENCE coefficients are both positive but they now have less
influence on the prediction.

The regularization reduces the influence of correlated
variables on the model because the weight is shared between the two
predictive variables, so neither alone would have strong weights.

On the other hand, the weights obtained with regularization are more
stable (see the :ref:`ridge_regression` User Guide section). This
increased stability is visible from the plot, obtained from data
perturbations, in a cross-validation. This plot can be compared with
the :ref:`previous one<covariation>`.

.. GENERATED FROM PYTHON SOURCE LINES 616-629

.. code-block:: Python


    cv_model = cross_validate(
        model,
        X,
        y,
        cv=cv,
        return_estimator=True,
        n_jobs=2,
    )
    coefs = pd.DataFrame(
        [est[-1].regressor_.coef_ for est in cv_model["estimator"]], columns=feature_names
    )








.. GENERATED FROM PYTHON SOURCE LINES 630-638

.. code-block:: Python

    plt.ylabel("Age coefficient")
    plt.xlabel("Experience coefficient")
    plt.grid(True)
    plt.xlim(-0.4, 0.5)
    plt.ylim(-0.4, 0.5)
    plt.scatter(coefs["AGE"], coefs["EXPERIENCE"])
    _ = plt.title("Co-variations of coefficients for AGE and EXPERIENCE across folds")




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_014.png
   :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_014.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 639-650

Linear models with sparse coefficients
--------------------------------------

Another possibility to take into account correlated variables in the dataset,
is to estimate sparse coefficients. In some way we already did it manually
when we dropped the AGE column in a previous ridge estimation.

Lasso models (see the :ref:`lasso` User Guide section) estimates sparse
coefficients. :class:`~sklearn.linear_model.LassoCV` applies cross
validation in order to determine which value of the regularization parameter
(`alpha`) is best suited for the model estimation.

.. GENERATED FROM PYTHON SOURCE LINES 650-665

.. code-block:: Python


    from sklearn.linear_model import LassoCV

    alphas = np.logspace(-10, 10, 21)  # alpha values to be chosen from by cross-validation
    model = make_pipeline(
        preprocessor,
        TransformedTargetRegressor(
            regressor=LassoCV(alphas=alphas, max_iter=100_000),
            func=np.log10,
            inverse_func=sp.special.exp10,
        ),
    )

    _ = model.fit(X_train, y_train)








.. GENERATED FROM PYTHON SOURCE LINES 666-667

First we verify which value of :math:`\alpha` has been selected.

.. GENERATED FROM PYTHON SOURCE LINES 667-670

.. code-block:: Python


    model[-1].regressor_.alpha_





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    np.float64(0.001)



.. GENERATED FROM PYTHON SOURCE LINES 671-672

Then we check the quality of the predictions.

.. GENERATED FROM PYTHON SOURCE LINES 672-691

.. code-block:: Python


    mae_train = median_absolute_error(y_train, model.predict(X_train))
    y_pred = model.predict(X_test)
    mae_test = median_absolute_error(y_test, y_pred)
    scores = {
        "MedAE on training set": f"{mae_train:.2f} $/hour",
        "MedAE on testing set": f"{mae_test:.2f} $/hour",
    }

    _, ax = plt.subplots(figsize=(6, 6))
    display = PredictionErrorDisplay.from_predictions(
        y_test, y_pred, kind="actual_vs_predicted", ax=ax, scatter_kwargs={"alpha": 0.5}
    )
    ax.set_title("Lasso model, optimum regularization")
    for name, score in scores.items():
        ax.plot([], [], " ", label=f"{name}: {score}")
    ax.legend(loc="upper left")
    plt.tight_layout()




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_015.png
   :alt: Lasso model, optimum regularization
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_015.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 692-693

For our dataset, again the model is not very predictive.

.. GENERATED FROM PYTHON SOURCE LINES 693-704

.. code-block:: Python


    coefs = pd.DataFrame(
        model[-1].regressor_.coef_,
        columns=["Coefficients importance"],
        index=feature_names,
    )
    coefs.plot(kind="barh", figsize=(9, 7))
    plt.title("Lasso model, optimum regularization, normalized variables")
    plt.axvline(x=0, color=".5")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_016.png
   :alt: Lasso model, optimum regularization, normalized variables
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_016.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 705-716

A Lasso model identifies the correlation between
AGE and EXPERIENCE and suppresses one of them for the sake of the prediction.

It is important to keep in mind that the coefficients that have been
dropped may still be related to the outcome by themselves: the model
chose to suppress them because they bring little or no additional
information on top of the other features. Additionally, this selection
is unstable for correlated features, and should be interpreted with
caution.

Indeed, we can check the variability of the coefficients across folds.

.. GENERATED FROM PYTHON SOURCE LINES 716-728

.. code-block:: Python

    cv_model = cross_validate(
        model,
        X,
        y,
        cv=cv,
        return_estimator=True,
        n_jobs=2,
    )
    coefs = pd.DataFrame(
        [est[-1].regressor_.coef_ for est in cv_model["estimator"]], columns=feature_names
    )








.. GENERATED FROM PYTHON SOURCE LINES 729-736

.. code-block:: Python

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", palette="dark:k", alpha=0.5)
    sns.boxplot(data=coefs, orient="h", color="cyan", saturation=0.5, whis=100)
    plt.axvline(x=0, color=".5")
    plt.title("Coefficient variability")
    plt.subplots_adjust(left=0.3)




.. image-sg:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_017.png
   :alt: Coefficient variability
   :srcset: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_017.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 737-787

We observe that the AGE and EXPERIENCE coefficients are varying a lot
depending of the fold.

Wrong causal interpretation
---------------------------

Policy makers might want to know the effect of education on wage to assess
whether or not a certain policy designed to entice people to pursue more
education would make economic sense. While Machine Learning models are great
for measuring statistical associations, they are generally unable to infer
causal effects.

It might be tempting to look at the coefficient of education on wage from our
last model (or any model for that matter) and conclude that it captures the
true effect of a change in the standardized education variable on wages.

Unfortunately there are likely unobserved confounding variables that either
inflate or deflate that coefficient. A confounding variable is a variable that
causes both EDUCATION and WAGE. One example of such variable is ability.
Presumably, more able people are more likely to pursue education while at the
same time being more likely to earn a higher hourly wage at any level of
education. In this case, ability induces a positive `Omitted Variable Bias
<https://en.wikipedia.org/wiki/Omitted-variable_bias>`_ (OVB) on the EDUCATION
coefficient, thereby exaggerating the effect of education on wages.

See the :ref:`sphx_glr_auto_examples_inspection_plot_causal_interpretation.py`
for a simulated case of ability OVB.

Lessons learned
---------------

* Coefficients must be scaled to the same unit of measure to retrieve
  feature importance. Scaling them with the standard-deviation of the
  feature is a useful proxy.
* Interpreting causality is difficult when there are confounding effects. If
  the relationship between two variables is also affected by something
  unobserved, we should be careful when making conclusions about causality.
* Coefficients in multivariate linear models represent the dependency
  between a given feature and the target, **conditional** on the other
  features.
* Correlated features induce instabilities in the coefficients of linear
  models and their effects cannot be well teased apart.
* Different linear models respond differently to feature correlation and
  coefficients could significantly vary from one another.
* Inspecting coefficients across the folds of a cross-validation loop
  gives an idea of their stability.
* Coefficients are unlikely to have any causal meaning. They tend
  to be biased by unobserved confounders.
* Inspection tools may not necessarily provide insights on the true
  data generating process.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 12.364 seconds)


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