.. 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

        Click :ref:`here <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 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.

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.

.. contents::
   :local:
   :depth: 1

.. GENERATED FROM PYTHON SOURCE LINES 33-39

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 40-46

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 46-51

.. code-block:: default


    from sklearn.datasets import fetch_openml

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








.. GENERATED FROM PYTHON SOURCE LINES 52-54

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 54-58

.. code-block:: default


    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;
        }

        .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>
          <th>OCCUPATION</th>
          <th>SECTOR</th>
          <th>MARR</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>count</th>
          <td>534.000000</td>
          <td>534</td>
          <td>534</td>
          <td>534.000000</td>
          <td>534</td>
          <td>534.000000</td>
          <td>534</td>
          <td>534</td>
          <td>534</td>
          <td>534</td>
        </tr>
        <tr>
          <th>unique</th>
          <td>NaN</td>
          <td>2</td>
          <td>2</td>
          <td>NaN</td>
          <td>2</td>
          <td>NaN</td>
          <td>3</td>
          <td>6</td>
          <td>3</td>
          <td>2</td>
        </tr>
        <tr>
          <th>top</th>
          <td>NaN</td>
          <td>no</td>
          <td>male</td>
          <td>NaN</td>
          <td>not_member</td>
          <td>NaN</td>
          <td>White</td>
          <td>Other</td>
          <td>Other</td>
          <td>Married</td>
        </tr>
        <tr>
          <th>freq</th>
          <td>NaN</td>
          <td>378</td>
          <td>289</td>
          <td>NaN</td>
          <td>438</td>
          <td>NaN</td>
          <td>440</td>
          <td>156</td>
          <td>411</td>
          <td>350</td>
        </tr>
        <tr>
          <th>mean</th>
          <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 59-62

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 62-65

.. code-block:: default


    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;
        }
    </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>
          <th>OCCUPATION</th>
          <th>SECTOR</th>
          <th>MARR</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>8.0</td>
          <td>no</td>
          <td>female</td>
          <td>21.0</td>
          <td>not_member</td>
          <td>35.0</td>
          <td>Hispanic</td>
          <td>Other</td>
          <td>Manufacturing</td>
          <td>Married</td>
        </tr>
        <tr>
          <th>1</th>
          <td>9.0</td>
          <td>no</td>
          <td>female</td>
          <td>42.0</td>
          <td>not_member</td>
          <td>57.0</td>
          <td>White</td>
          <td>Other</td>
          <td>Manufacturing</td>
          <td>Married</td>
        </tr>
        <tr>
          <th>2</th>
          <td>12.0</td>
          <td>no</td>
          <td>male</td>
          <td>1.0</td>
          <td>not_member</td>
          <td>19.0</td>
          <td>White</td>
          <td>Other</td>
          <td>Manufacturing</td>
          <td>Unmarried</td>
        </tr>
        <tr>
          <th>3</th>
          <td>12.0</td>
          <td>no</td>
          <td>male</td>
          <td>4.0</td>
          <td>not_member</td>
          <td>22.0</td>
          <td>White</td>
          <td>Other</td>
          <td>Other</td>
          <td>Unmarried</td>
        </tr>
        <tr>
          <th>4</th>
          <td>12.0</td>
          <td>no</td>
          <td>male</td>
          <td>17.0</td>
          <td>not_member</td>
          <td>35.0</td>
          <td>White</td>
          <td>Other</td>
          <td>Other</td>
          <td>Married</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 66-68

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

.. GENERATED FROM PYTHON SOURCE LINES 68-72

.. code-block:: default


    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 73-78

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 78-83

.. code-block:: default


    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 84-89

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 89-94

.. code-block:: default


    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 95-114

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 114-117

.. code-block:: default


    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    float64 
     1   SOUTH       534 non-null    category
     2   SEX         534 non-null    category
     3   EXPERIENCE  534 non-null    float64 
     4   UNION       534 non-null    category
     5   AGE         534 non-null    float64 
     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), float64(3)
    memory usage: 17.2 KB




.. GENERATED FROM PYTHON SOURCE LINES 118-129

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 129-142

.. code-block:: default


    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 143-145

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 145-157

.. code-block:: default


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

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








.. GENERATED FROM PYTHON SOURCE LINES 158-162

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

First, we fit the model.

.. GENERATED FROM PYTHON SOURCE LINES 162-165

.. code-block:: default


    model.fit(X_train, y_train)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <style>#sk-container-id-26 {color: black;background-color: white;}#sk-container-id-26 pre{padding: 0;}#sk-container-id-26 div.sk-toggleable {background-color: white;}#sk-container-id-26 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-26 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-26 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-26 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-26 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-26 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-26 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-26 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-26 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-26 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-26 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-26 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-26 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-26 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-26 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-26 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-26 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-26 div.sk-item {position: relative;z-index: 1;}#sk-container-id-26 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-26 div.sk-item::before, #sk-container-id-26 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-26 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-26 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-26 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-26 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-26 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-26 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-26 div.sk-label-container {text-align: center;}#sk-container-id-26 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-26 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-26" 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 sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-111" type="checkbox" ><label for="sk-estimator-id-111" class="sk-toggleable__label sk-toggleable__label-arrow">Pipeline</label><div class="sk-toggleable__content"><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 sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-112" type="checkbox" ><label for="sk-estimator-id-112" class="sk-toggleable__label sk-toggleable__label-arrow">columntransformer: ColumnTransformer</label><div class="sk-toggleable__content"><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-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-113" type="checkbox" ><label for="sk-estimator-id-113" class="sk-toggleable__label sk-toggleable__label-arrow">onehotencoder</label><div class="sk-toggleable__content"><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 sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-114" type="checkbox" ><label for="sk-estimator-id-114" class="sk-toggleable__label sk-toggleable__label-arrow">OneHotEncoder</label><div class="sk-toggleable__content"><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 sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-115" type="checkbox" ><label for="sk-estimator-id-115" class="sk-toggleable__label sk-toggleable__label-arrow">remainder</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">passthrough</label><div class="sk-toggleable__content"><pre>passthrough</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 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 sk-toggleable__label-arrow">transformedtargetregressor: TransformedTargetRegressor</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">regressor: Ridge</label><div class="sk-toggleable__content"><pre>Ridge(alpha=1e-10)</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator 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 sk-toggleable__label-arrow">Ridge</label><div class="sk-toggleable__content"><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 166-169

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 169-180

.. code-block:: default


    from sklearn.metrics import median_absolute_error

    y_pred = model.predict(X_train)

    mae = median_absolute_error(y_train, y_pred)
    string_score = f"MAE on training set: {mae:.2f} $/hour"
    y_pred = model.predict(X_test)
    mae = median_absolute_error(y_test, y_pred)
    string_score += f"\nMAE on testing set: {mae:.2f} $/hour"








.. GENERATED FROM PYTHON SOURCE LINES 181-191

.. code-block:: default

    fig, ax = plt.subplots(figsize=(5, 5))
    plt.scatter(y_test, y_pred)
    ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red")
    plt.text(3, 20, string_score)
    plt.title("Ridge model, small regularization")
    plt.ylabel("Model predictions")
    plt.xlabel("Truths")
    plt.xlim([0, 27])
    _ = plt.ylim([0, 27])




.. 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 192-206

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 red 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 206-216

.. code-block:: default

    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>-0.013531</td>
        </tr>
        <tr>
          <th>RACE_Other</th>
          <td>-0.009087</td>
        </tr>
        <tr>
          <th>RACE_White</th>
          <td>0.022582</td>
        </tr>
        <tr>
          <th>OCCUPATION_Clerical</th>
          <td>0.000061</td>
        </tr>
        <tr>
          <th>OCCUPATION_Management</th>
          <td>0.090544</td>
        </tr>
        <tr>
          <th>OCCUPATION_Other</th>
          <td>-0.025085</td>
        </tr>
        <tr>
          <th>OCCUPATION_Professional</th>
          <td>0.071980</td>
        </tr>
        <tr>
          <th>OCCUPATION_Sales</th>
          <td>-0.046620</td>
        </tr>
        <tr>
          <th>OCCUPATION_Service</th>
          <td>-0.091037</td>
        </tr>
        <tr>
          <th>SECTOR_Construction</th>
          <td>-0.000165</td>
        </tr>
        <tr>
          <th>SECTOR_Manufacturing</th>
          <td>0.031287</td>
        </tr>
        <tr>
          <th>SECTOR_Other</th>
          <td>-0.030993</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 217-229

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 229-236

.. code-block:: default


    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 237-248

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 248-258

.. code-block:: default


    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 259-269

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 269-281

.. code-block:: default


    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 282-309

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.

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 309-331

.. code-block:: default


    from sklearn.model_selection import cross_validate
    from sklearn.model_selection import RepeatedKFold

    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 332-341

.. code-block:: default

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", color="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 342-354

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 354-363

.. code-block:: default


    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 364-369

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 369-390

.. code-block:: default


    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 391-400

.. code-block:: default

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", color="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 401-416

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 416-424

.. code-block:: default


    from sklearn.preprocessing import StandardScaler

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








.. GENERATED FROM PYTHON SOURCE LINES 425-426

The model will stay unchanged.

.. GENERATED FROM PYTHON SOURCE LINES 426-435

.. code-block:: default


    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-27 {color: black;background-color: white;}#sk-container-id-27 pre{padding: 0;}#sk-container-id-27 div.sk-toggleable {background-color: white;}#sk-container-id-27 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-27 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-27 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-27 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-27 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-27 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-27 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-27 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-27 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-27 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-27 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-27 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-27 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-27 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-27 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-27 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-27 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-27 div.sk-item {position: relative;z-index: 1;}#sk-container-id-27 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-27 div.sk-item::before, #sk-container-id-27 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-27 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-27 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-27 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-27 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-27 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-27 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-27 div.sk-label-container {text-align: center;}#sk-container-id-27 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-27 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-27" 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 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 sk-toggleable__label-arrow">Pipeline</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">columntransformer: ColumnTransformer</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">onehotencoder</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">OneHotEncoder</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">standardscaler</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">StandardScaler</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">transformedtargetregressor: TransformedTargetRegressor</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">regressor: Ridge</label><div class="sk-toggleable__content"><pre>Ridge(alpha=1e-10)</pre></div></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator 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 sk-toggleable__label-arrow">Ridge</label><div class="sk-toggleable__content"><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 436-439

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 439-447

.. code-block:: default


    y_pred = model.predict(X_train)
    mae = median_absolute_error(y_train, y_pred)
    string_score = f"MAE on training set: {mae:.2f} $/hour"
    y_pred = model.predict(X_test)
    mae = median_absolute_error(y_test, y_pred)
    string_score += f"\nMAE on testing set: {mae:.2f} $/hour"








.. GENERATED FROM PYTHON SOURCE LINES 448-460

.. code-block:: default

    fig, ax = plt.subplots(figsize=(6, 6))
    plt.scatter(y_test, y_pred)
    ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red")

    plt.text(3, 20, string_score)

    plt.title("Ridge model, small regularization, normalized variables")
    plt.ylabel("Model predictions")
    plt.xlabel("Truths")
    plt.xlim([0, 27])
    _ = plt.ylim([0, 27])




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





.. GENERATED FROM PYTHON SOURCE LINES 461-463

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

.. GENERATED FROM PYTHON SOURCE LINES 463-475

.. code-block:: default


    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 476-480

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 480-493

.. code-block:: default


    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 494-501

.. code-block:: default

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", color="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 502-515

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 515-529

.. code-block:: default


    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-28 {color: black;background-color: white;}#sk-container-id-28 pre{padding: 0;}#sk-container-id-28 div.sk-toggleable {background-color: white;}#sk-container-id-28 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-28 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-28 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-28 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-28 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-28 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-28 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-28 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-28 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-28 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-28 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-28 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-28 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-28 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-28 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-28 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-28 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-28 div.sk-item {position: relative;z-index: 1;}#sk-container-id-28 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-28 div.sk-item::before, #sk-container-id-28 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-28 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-28 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-28 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-28 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-28 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-28 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-28 div.sk-label-container {text-align: center;}#sk-container-id-28 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-28 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-28" 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 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 sk-toggleable__label-arrow">Pipeline</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">columntransformer: ColumnTransformer</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">onehotencoder</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">OneHotEncoder</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">standardscaler</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">StandardScaler</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">transformedtargetregressor: TransformedTargetRegressor</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">regressor: RidgeCV</label><div class="sk-toggleable__content"><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 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 sk-toggleable__label-arrow">RidgeCV</label><div class="sk-toggleable__content"><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 530-531

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

.. GENERATED FROM PYTHON SOURCE LINES 531-534

.. code-block:: default


    model[-1].regressor_.alpha_





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

 .. code-block:: none


    10.0



.. GENERATED FROM PYTHON SOURCE LINES 535-536

Then we check the quality of the predictions.

.. GENERATED FROM PYTHON SOURCE LINES 536-544

.. code-block:: default


    y_pred = model.predict(X_train)
    mae = median_absolute_error(y_train, y_pred)
    string_score = f"MAE on training set: {mae:.2f} $/hour"
    y_pred = model.predict(X_test)
    mae = median_absolute_error(y_test, y_pred)
    string_score += f"\nMAE on testing set: {mae:.2f} $/hour"








.. GENERATED FROM PYTHON SOURCE LINES 545-557

.. code-block:: default

    fig, ax = plt.subplots(figsize=(6, 6))
    plt.scatter(y_test, y_pred)
    ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red")

    plt.text(3, 20, string_score)

    plt.title("Ridge model, optimum regularization, normalized variables")
    plt.ylabel("Model predictions")
    plt.xlabel("Truths")
    plt.xlim([0, 27])
    _ = plt.ylim([0, 27])




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





.. GENERATED FROM PYTHON SOURCE LINES 558-560

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 560-572

.. code-block:: default


    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 573-586

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 586-599

.. code-block:: default


    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 600-608

.. code-block:: default

    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 609-620

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 620-635

.. code-block:: default


    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)








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

.. GENERATED FROM PYTHON SOURCE LINES 637-640

.. code-block:: default


    model[-1].regressor_.alpha_





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

 .. code-block:: none


    0.001



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Then we check the quality of the predictions.

.. GENERATED FROM PYTHON SOURCE LINES 642-650

.. code-block:: default


    y_pred = model.predict(X_train)
    mae = median_absolute_error(y_train, y_pred)
    string_score = f"MAE on training set: {mae:.2f} $/hour"
    y_pred = model.predict(X_test)
    mae = median_absolute_error(y_test, y_pred)
    string_score += f"\nMAE on testing set: {mae:.2f} $/hour"








.. GENERATED FROM PYTHON SOURCE LINES 651-663

.. code-block:: default

    fig, ax = plt.subplots(figsize=(6, 6))
    plt.scatter(y_test, y_pred)
    ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red")

    plt.text(3, 20, string_score)

    plt.title("Lasso model, regularization, normalized variables")
    plt.ylabel("Model predictions")
    plt.xlabel("Truths")
    plt.xlim([0, 27])
    _ = plt.ylim([0, 27])




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





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

.. GENERATED FROM PYTHON SOURCE LINES 665-676

.. code-block:: default


    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 677-688

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 688-700

.. code-block:: default

    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 701-708

.. code-block:: default

    plt.figure(figsize=(9, 7))
    sns.stripplot(data=coefs, orient="h", color="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 709-727

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

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.
* 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.


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

   **Total running time of the script:** ( 0 minutes  16.491 seconds)


.. _sphx_glr_download_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.1.X?urlpath=lab/tree/notebooks/auto_examples/inspection/plot_linear_model_coefficient_interpretation.ipynb
        :alt: Launch binder
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_linear_model_coefficient_interpretation.py <plot_linear_model_coefficient_interpretation.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_linear_model_coefficient_interpretation.ipynb <plot_linear_model_coefficient_interpretation.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_