.. 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 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 30-39
.. code-block:: default
print(__doc__)
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 52-55
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 55-58
.. 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 68-71
.. code-block:: default
y = survey.target.values.ravel()
survey.target.head()
.. rst-class:: sphx-glr-script-out
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 72-77
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 77-84
.. 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 85-90
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 90-95
.. 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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_001.png
:alt: plot linear model coefficient interpretation
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 96-115
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 115-118
.. code-block:: default
survey.data.info()
.. rst-class:: sphx-glr-script-out
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 119-130
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;
- 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 130-143
.. 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'
)
.. GENERATED FROM PYTHON SOURCE LINES 144-146
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 146-161
.. 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 162-166
Processing the dataset
----------------------
First, we fit the model.
.. GENERATED FROM PYTHON SOURCE LINES 166-169
.. code-block:: default
_ = model.fit(X_train, y_train)
.. GENERATED FROM PYTHON SOURCE LINES 170-173
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 173-193
.. 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'
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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_002.png
:alt: Ridge model, small regularization
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 194-208
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 208-222
.. code-block:: default
feature_names = (model.named_steps['columntransformer']
.named_transformers_['onehotencoder']
.get_feature_names(input_features=categorical_columns))
feature_names = np.concatenate(
[feature_names, numerical_columns])
coefs = pd.DataFrame(
model.named_steps['transformedtargetregressor'].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>-0.013564</td>
</tr>
<tr>
<th>RACE_Other</th>
<td>-0.009120</td>
</tr>
<tr>
<th>RACE_White</th>
<td>0.022549</td>
</tr>
<tr>
<th>OCCUPATION_Clerical</th>
<td>0.000048</td>
</tr>
<tr>
<th>OCCUPATION_Management</th>
<td>0.090531</td>
</tr>
<tr>
<th>OCCUPATION_Other</th>
<td>-0.025098</td>
</tr>
<tr>
<th>OCCUPATION_Professional</th>
<td>0.071967</td>
</tr>
<tr>
<th>OCCUPATION_Sales</th>
<td>-0.046633</td>
</tr>
<tr>
<th>OCCUPATION_Service</th>
<td>-0.091050</td>
</tr>
<tr>
<th>SECTOR_Construction</th>
<td>-0.000180</td>
</tr>
<tr>
<th>SECTOR_Manufacturing</th>
<td>0.031273</td>
</tr>
<tr>
<th>SECTOR_Other</th>
<td>-0.031008</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 223-235
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 235-241
.. code-block:: default
coefs.plot(kind='barh', figsize=(9, 7))
plt.title('Ridge model, small regularization')
plt.axvline(x=0, color='.5')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_003.png
:alt: Ridge model, small regularization
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 242-253
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 253-263
.. code-block:: default
X_train_preprocessed = pd.DataFrame(
model.named_steps['columntransformer'].transform(X_train),
columns=feature_names
)
X_train_preprocessed.std(axis=0).plot(kind='barh', figsize=(9, 7))
plt.title('Features std. dev.')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_004.png
:alt: Features std. dev.
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 264-274
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 274-285
.. code-block:: default
coefs = pd.DataFrame(
model.named_steps['transformedtargetregressor'].regressor_.coef_ *
X_train_preprocessed.std(axis=0),
columns=['Coefficient importance'], index=feature_names
)
coefs.plot(kind='barh', figsize=(9, 7))
plt.title('Ridge model, small regularization')
plt.axvline(x=0, color='.5')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_005.png
:alt: Ridge model, small regularization
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 286-313
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 313-335
.. code-block:: default
from sklearn.model_selection import cross_validate
from sklearn.model_selection import RepeatedKFold
cv_model = cross_validate(
model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5),
return_estimator=True, n_jobs=-1
)
coefs = pd.DataFrame(
[est.named_steps['transformedtargetregressor'].regressor_.coef_ *
X_train_preprocessed.std(axis=0)
for est in cv_model['estimator']],
columns=feature_names
)
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.xlabel('Coefficient importance')
plt.title('Coefficient importance and its variability')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_006.png
:alt: Coefficient importance and its variability
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 336-348
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 348-358
.. 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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_007.png
:alt: Co-variations of coefficients for AGE and EXPERIENCE across folds
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 359-364
Two regions are populated: when the EXPERIENCE coefficient is
positive the AGE one is negative and viceversa.
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 364-386
.. code-block:: default
column_to_drop = ['AGE']
cv_model = cross_validate(
model, X.drop(columns=column_to_drop), y,
cv=RepeatedKFold(n_splits=5, n_repeats=5),
return_estimator=True, n_jobs=-1
)
coefs = pd.DataFrame(
[est.named_steps['transformedtargetregressor'].regressor_.coef_ *
X_train_preprocessed.drop(columns=column_to_drop).std(axis=0)
for est in cv_model['estimator']],
columns=feature_names[:-1]
)
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.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_008.png
:alt: Coefficient importance and its variability
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 387-401
The estimation of the EXPERIENCE coefficient is now less variable and
remain 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 to apply a similar amount 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 401-410
.. code-block:: default
from sklearn.preprocessing import StandardScaler
preprocessor = make_column_transformer(
(OneHotEncoder(drop='if_binary'), categorical_columns),
(StandardScaler(), numerical_columns),
remainder='passthrough'
)
.. GENERATED FROM PYTHON SOURCE LINES 411-412
The model will stay unchanged.
.. GENERATED FROM PYTHON SOURCE LINES 412-424
.. 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)
.. GENERATED FROM PYTHON SOURCE LINES 425-428
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 428-447
.. 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'
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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_009.png
:alt: Ridge model, small regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 448-449
For the coefficient analysis, scaling is not needed this time.
.. GENERATED FROM PYTHON SOURCE LINES 449-459
.. code-block:: default
coefs = pd.DataFrame(
model.named_steps['transformedtargetregressor'].regressor_.coef_,
columns=['Coefficients'], index=feature_names
)
coefs.plot(kind='barh', figsize=(9, 7))
plt.title('Ridge model, small regularization, normalized variables')
plt.axvline(x=0, color='.5')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_010.png
:alt: Ridge model, small regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 460-461
We now inspect the coefficients across several cross-validation folds.
.. GENERATED FROM PYTHON SOURCE LINES 461-478
.. code-block:: default
cv_model = cross_validate(
model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5),
return_estimator=True, n_jobs=-1
)
coefs = pd.DataFrame(
[est.named_steps['transformedtargetregressor'].regressor_.coef_
for est in cv_model['estimator']],
columns=feature_names
)
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 variability')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_011.png
:alt: Coefficient variability
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 479-492
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. 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 492-506
.. code-block:: default
from sklearn.linear_model import RidgeCV
model = make_pipeline(
preprocessor,
TransformedTargetRegressor(
regressor=RidgeCV(alphas=np.logspace(-10, 10, 21)),
func=np.log10,
inverse_func=sp.special.exp10
)
)
_ = model.fit(X_train, y_train)
.. GENERATED FROM PYTHON SOURCE LINES 507-508
First we check which value of :math:`\alpha` has been selected.
.. GENERATED FROM PYTHON SOURCE LINES 508-511
.. code-block:: default
model[-1].regressor_.alpha_
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
10.0
.. GENERATED FROM PYTHON SOURCE LINES 512-513
Then we check the quality of the predictions.
.. GENERATED FROM PYTHON SOURCE LINES 513-533
.. 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'
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, regularization, normalized variables')
plt.ylabel('Model predictions')
plt.xlabel('Truths')
plt.xlim([0, 27])
_ = plt.ylim([0, 27])
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_012.png
:alt: Ridge model, regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 534-536
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 536-546
.. code-block:: default
coefs = pd.DataFrame(
model.named_steps['transformedtargetregressor'].regressor_.coef_,
columns=['Coefficients'], index=feature_names
)
coefs.plot(kind='barh', figsize=(9, 7))
plt.title('Ridge model, regularization, normalized variables')
plt.axvline(x=0, color='.5')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_013.png
:alt: Ridge model, regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 547-560
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 560-581
.. code-block:: default
cv_model = cross_validate(
model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5),
return_estimator=True, n_jobs=-1
)
coefs = pd.DataFrame(
[est.named_steps['transformedtargetregressor'].regressor_.coef_ *
X_train_preprocessed.std(axis=0)
for est in cv_model['estimator']],
columns=feature_names
)
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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_014.png
:alt: Co-variations of coefficients for AGE and EXPERIENCE across folds
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 582-593
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. 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 593-607
.. code-block:: default
from sklearn.linear_model import LassoCV
model = make_pipeline(
preprocessor,
TransformedTargetRegressor(
regressor=LassoCV(alphas=np.logspace(-10, 10, 21), max_iter=100000),
func=np.log10,
inverse_func=sp.special.exp10
)
)
_ = model.fit(X_train, y_train)
.. GENERATED FROM PYTHON SOURCE LINES 608-609
First we verify which value of :math:`\alpha` has been selected.
.. GENERATED FROM PYTHON SOURCE LINES 609-612
.. code-block:: default
model[-1].regressor_.alpha_
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
0.001
.. GENERATED FROM PYTHON SOURCE LINES 613-614
Then we check the quality of the predictions.
.. GENERATED FROM PYTHON SOURCE LINES 614-634
.. 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'
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:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_015.png
:alt: Lasso model, regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 635-636
For our dataset, again the model is not very predictive.
.. GENERATED FROM PYTHON SOURCE LINES 636-646
.. code-block:: default
coefs = pd.DataFrame(
model.named_steps['transformedtargetregressor'].regressor_.coef_,
columns=['Coefficients'], index=feature_names
)
coefs.plot(kind='barh', figsize=(9, 7))
plt.title('Lasso model, regularization, normalized variables')
plt.axvline(x=0, color='.5')
plt.subplots_adjust(left=.3)
.. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_016.png
:alt: Lasso model, regularization, normalized variables
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 647-672
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. Additionnaly, this selection
is unstable for correlated features, and should be interpreted with
caution.
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 14.715 seconds)
.. _sphx_glr_download_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py:
.. only :: html
.. container:: sphx-glr-footer
:class: sphx-glr-footer-example
.. container:: binder-badge
.. image:: images/binder_badge_logo.svg
:target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.24.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>`_