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MaxHalford
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Description

:crown: Python factor analysis library (PCA, CA, MCA, MFA, FAMD)

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Prince is a library for doing factor analysis. This includes a variety of methods including principal component analysis (PCA) and correspondence analysis (CA). The goal is to provide an efficient implementation for each algorithm along with a scikit-learn API.

Table of contents

Installation

:warning: Prince is only compatible with Python 3.

:snake: Although it isn't a requirement, using Anaconda is highly recommended.

Via PyPI

$ pip install prince

Via GitHub for the latest development version

$ pip install git+https://github.com/MaxHalford/Prince

Prince doesn't have any extra dependencies apart from the usual suspects (

sklearn
,
pandas
,
matplotlib
) which are included with Anaconda.

Usage

import numpy as np; np.random.set_state(42)  # this is for doctests reproducibility

Guidelines

Each estimator provided by

prince
extends scikit-learn's
TransformerMixin
. This means that each estimator implements a
fit
and a
transform
method which makes them usable in a transformation pipeline. The
fit
method is actually an alias for the
row_principal_components
method which returns the row principal components. However you can also access the column principal components with the
column_principal_components
.

Under the hood Prince uses a randomised version of SVD. This is much faster than using the more commonly full approach. However the results may have a small inherent randomness. For most applications this doesn't matter and you shouldn't have to worry about it. However if you want reproducible results then you should set the

random_state
parameter.

The randomised version of SVD is an iterative method. Because each of Prince's algorithms use SVD, they all possess a

n_iter
parameter which controls the number of iterations used for computing the SVD. On the one hand the higher
n_iter
is the more precise the results will be. On the other hand increasing
n_iter
increases the computation time. In general the algorithm converges very quickly so using a low
n_iter
(which is the default behaviour) is recommended.

You are supposed to use each method depending on your situation:

  • All your variables are numeric: use principal component analysis (
    prince.PCA
    )
  • You have a contingency table: use correspondence analysis (
    prince.CA
    )
  • You have more than 2 variables and they are all categorical: use multiple correspondence analysis (
    prince.MCA
    )
  • You have groups of categorical or numerical variables: use multiple factor analysis (
    prince.MFA
    )
  • You have both categorical and numerical variables: use factor analysis of mixed data (
    prince.FAMD
    )

The next subsections give an overview of each method along with usage information. The following papers give a good overview of the field of factor analysis if you want to go deeper:

Principal component analysis (PCA)

If you're using PCA it is assumed you have a dataframe consisting of numerical continuous variables. In this example we're going to be using the Iris flower dataset.

>>> import pandas as pd
>>> import prince
>>> from sklearn import datasets

>>> X, y = datasets.load_iris(return_X_y=True) >>> X = pd.DataFrame(data=X, columns=['Sepal length', 'Sepal width', 'Petal length', 'Petal width']) >>> y = pd.Series(y).map({0: 'Setosa', 1: 'Versicolor', 2: 'Virginica'}) >>> X.head() Sepal length Sepal width Petal length Petal width 0 5.1 3.5 1.4 0.2 1 4.9 3.0 1.4 0.2 2 4.7 3.2 1.3 0.2 3 4.6 3.1 1.5 0.2 4 5.0 3.6 1.4 0.2

The

PCA
class implements scikit-learn's
fit
/
transform
API. It's parameters have to passed at initialisation before calling the
fit
method.
>>> pca = prince.PCA(
...     n_components=2,
...     n_iter=3,
...     rescale_with_mean=True,
...     rescale_with_std=True,
...     copy=True,
...     check_input=True,
...     engine='auto',
...     random_state=42
... )
>>> pca = pca.fit(X)

The available parameters are:

  • n_components
    : the number of components that are computed. You only need two if your intention is to make a chart.
  • n_iter
    : the number of iterations used for computing the SVD
  • rescale_with_mean
    : whether to substract each column's mean
  • rescale_with_std
    : whether to divide each column by it's standard deviation
  • copy
    : if
    False
    then the computations will be done inplace which can have possible side-effects on the input data
  • engine
    : what SVD engine to use (should be one of
    ['auto', 'fbpca', 'sklearn']
    )
  • random_state
    : controls the randomness of the SVD results.

Once the

PCA
has been fitted, it can be used to extract the row principal coordinates as so:
>>> pca.transform(X).head()  # same as pca.row_coordinates(X).head()
          0         1
0 -2.264703  0.480027
1 -2.080961 -0.674134
2 -2.364229 -0.341908
3 -2.299384 -0.597395
4 -2.389842  0.646835

Each column stands for a principal component whilst each row stands a row in the original dataset. You can display these projections with the

plot_row_coordinates
method:
>>> ax = pca.plot_row_coordinates(
...     X,
...     ax=None,
...     figsize=(6, 6),
...     x_component=0,
...     y_component=1,
...     labels=None,
...     color_labels=y,
...     ellipse_outline=False,
...     ellipse_fill=True,
...     show_points=True
... )
>>> ax.get_figure().savefig('images/pca_row_coordinates.svg')

Each principal component explains part of the underlying of the distribution. You can see by how much by using the accessing the

explained_inertia_
property:
>>> pca.explained_inertia_
array([0.72962445, 0.22850762])

The explained inertia represents the percentage of the inertia each principal component contributes. It sums up to 1 if the

n_components
property is equal to the number of columns in the original dataset. you The explained inertia is obtained by dividing the eigenvalues obtained with the SVD by the total inertia, both of which are also accessible.
>>> pca.eigenvalues_
array([2.91849782, 0.91403047])

>>> pca.total_inertia_ 4.000000...

>>> pca.explained_inertia_ array([0.72962445, 0.22850762])

You can also obtain the correlations between the original variables and the principal components.

>>> pca.column_correlations(X)
                     0         1
Petal length  0.991555  0.023415
Petal width   0.964979  0.064000
Sepal length  0.890169  0.360830
Sepal width  -0.460143  0.882716

You may also want to know how much each observation contributes to each principal component. This can be done with the

row_contributions
method.
>>> pca.row_contributions(X).head()
          0         1
0  1.757369  0.252098
1  1.483777  0.497200
2  1.915225  0.127896
3  1.811606  0.390447
4  1.956947  0.457748

You can also transform row projections back into their original space by using the

inverse_transform
method.
>>> pca.inverse_transform(pca.transform(X)).head()
          0         1         2         3
0  5.018949  3.514854  1.466013  0.251922
1  4.738463  3.030433  1.603913  0.272074
2  4.720130  3.196830  1.328961  0.167414
3  4.668436  3.086770  1.384170  0.182247
4  5.017093  3.596402  1.345411  0.206706

Correspondence analysis (CA)

You should be using correspondence analysis when you want to analyse a contingency table. In other words you want to analyse the dependencies between two categorical variables. The following example comes from section 17.2.3 of this textbook. It shows the number of occurrences between different hair and eye colors.

>>> import pandas as pd

>>> pd.set_option('display.float_format', lambda x: '{:.6f}'.format(x)) >>> X = pd.DataFrame( ... data=[ ... [326, 38, 241, 110, 3], ... [688, 116, 584, 188, 4], ... [343, 84, 909, 412, 26], ... [98, 48, 403, 681, 85] ... ], ... columns=pd.Series(['Fair', 'Red', 'Medium', 'Dark', 'Black']), ... index=pd.Series(['Blue', 'Light', 'Medium', 'Dark']) ... ) >>> X Fair Red Medium Dark Black Blue 326 38 241 110 3 Light 688 116 584 188 4 Medium 343 84 909 412 26 Dark 98 48 403 681 85

Unlike the

PCA
class, the
CA
only exposes scikit-learn's
fit
method.
>>> import prince
>>> ca = prince.CA(
...     n_components=2,
...     n_iter=3,
...     copy=True,
...     check_input=True,
...     engine='auto',
...     random_state=42
... )
>>> X.columns.rename('Hair color', inplace=True)
>>> X.index.rename('Eye color', inplace=True)
>>> ca = ca.fit(X)

The parameters and methods overlap with those proposed by the

PCA
class.
>>> ca.row_coordinates(X)
               0         1
Blue   -0.400300 -0.165411
Light  -0.440708 -0.088463
Medium  0.033614  0.245002
Dark    0.702739 -0.133914

>>> ca.column_coordinates(X) 0 1 Fair -0.543995 -0.173844 Red -0.233261 -0.048279 Medium -0.042024 0.208304 Dark 0.588709 -0.103950 Black 1.094388 -0.286437

You can plot both sets of principal coordinates with the

plot_coordinates
method.
>>> ax = ca.plot_coordinates(
...     X=X,
...     ax=None,
...     figsize=(6, 6),
...     x_component=0,
...     y_component=1,
...     show_row_labels=True,
...     show_col_labels=True
... )
>>> ax.get_figure().savefig('images/ca_coordinates.svg')

Like for the

PCA
you can access the inertia contribution of each principal component as well as the eigenvalues and the total inertia.
>>> ca.eigenvalues_
[0.199244..., 0.030086...]

>>> ca.total_inertia_ 0.230191...

>>> ca.explained_inertia_ [0.865562..., 0.130703...]

Multiple correspondence analysis (MCA)

Multiple correspondence analysis (MCA) is an extension of correspondence analysis (CA). It should be used when you have more than two categorical variables. The idea is simply to compute the one-hot encoded version of a dataset and apply CA on it. As an example we're going to use the balloons dataset taken from the UCI datasets website.

>>> import pandas as pd

>>> X = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/balloons/adult+stretch.data') >>> X.columns = ['Color', 'Size', 'Action', 'Age', 'Inflated'] >>> X.head() Color Size Action Age Inflated 0 YELLOW SMALL STRETCH ADULT T 1 YELLOW SMALL STRETCH CHILD F 2 YELLOW SMALL DIP ADULT F 3 YELLOW SMALL DIP CHILD F 4 YELLOW LARGE STRETCH ADULT T

The

MCA
also implements the
fit
and
transform
methods.
>>> import prince
>>> mca = prince.MCA(
...     n_components=2,
...     n_iter=3,
...     copy=True,
...     check_input=True,
...     engine='auto',
...     random_state=42
... )
>>> mca = mca.fit(X)

Like the

CA
class, the
MCA
class also has
plot_coordinates
method.
>>> ax = mca.plot_coordinates(
...     X=X,
...     ax=None,
...     figsize=(6, 6),
...     show_row_points=True,
...     row_points_size=10,
...     show_row_labels=False,
...     show_column_points=True,
...     column_points_size=30,
...     show_column_labels=False,
...     legend_n_cols=1
... )
>>> ax.get_figure().savefig('images/mca_coordinates.svg')

The eigenvalues and inertia values are also accessible.

>>> mca.eigenvalues_
[0.401656..., 0.211111...]

>>> mca.total_inertia_ 1.0

>>> mca.explained_inertia_ [0.401656..., 0.211111...]

Multiple factor analysis (MFA)

Multiple factor analysis (MFA) is meant to be used when you have groups of variables. In practice it builds a PCA on each group -- or an MCA, depending on the types of the group's variables. It then constructs a global PCA on the results of the so-called partial PCAs -- or MCAs. The dataset used in the following examples come from this paper. In the dataset, three experts give their opinion on six different wines. Each opinion for each wine is recorded as a variable. We thus want to consider the separate opinions of each expert whilst also having a global overview of each wine. MFA is the perfect fit for this kind of situation.

First of all let's copy the data used in the paper.

>>> import pandas as pd

>>> X = pd.DataFrame( ... data=[ ... [1, 6, 7, 2, 5, 7, 6, 3, 6, 7], ... [5, 3, 2, 4, 4, 4, 2, 4, 4, 3], ... [6, 1, 1, 5, 2, 1, 1, 7, 1, 1], ... [7, 1, 2, 7, 2, 1, 2, 2, 2, 2], ... [2, 5, 4, 3, 5, 6, 5, 2, 6, 6], ... [3, 4, 4, 3, 5, 4, 5, 1, 7, 5] ... ], ... columns=['E1 fruity', 'E1 woody', 'E1 coffee', ... 'E2 red fruit', 'E2 roasted', 'E2 vanillin', 'E2 woody', ... 'E3 fruity', 'E3 butter', 'E3 woody'], ... index=['Wine {}'.format(i+1) for i in range(6)] ... ) >>> X['Oak type'] = [1, 2, 2, 2, 1, 1]

The groups are passed as a dictionary to the

MFA
class.
>>> groups = {
...    'Expert #{}'.format(no+1): [c for c in X.columns if c.startswith('E{}'.format(no+1))]
...    for no in range(3)
... }
>>> import pprint
>>> pprint.pprint(groups)
{'Expert #1': ['E1 fruity', 'E1 woody', 'E1 coffee'],
 'Expert #2': ['E2 red fruit', 'E2 roasted', 'E2 vanillin', 'E2 woody'],
 'Expert #3': ['E3 fruity', 'E3 butter', 'E3 woody']}

Now we can fit an

MFA
.
>>> import prince
>>> mfa = prince.MFA(
...     groups=groups,
...     n_components=2,
...     n_iter=3,
...     copy=True,
...     check_input=True,
...     engine='auto',
...     random_state=42
... )
>>> mfa = mfa.fit(X)

The

MFA
inherits from the
PCA
class, which entails that you have access to all it's methods and properties. The
row_coordinates
method will return the global coordinates of each wine.
>>> mfa.row_coordinates(X)
               0         1
Wine 1 -2.172155 -0.508596
Wine 2  0.557017 -0.197408
Wine 3  2.317663 -0.830259
Wine 4  1.832557  0.905046
Wine 5 -1.403787  0.054977
Wine 6 -1.131296  0.576241

Just like for the

PCA
you can plot the row coordinates with the
plot_row_coordinates
method.
>>> ax = mfa.plot_row_coordinates(
...     X,
...     ax=None,
...     figsize=(6, 6),
...     x_component=0,
...     y_component=1,
...     labels=X.index,
...     color_labels=['Oak type {}'.format(t) for t in X['Oak type']],
...     ellipse_outline=False,
...     ellipse_fill=True,
...     show_points=True
... )
>>> ax.get_figure().savefig('images/mfa_row_coordinates.svg')

You can also obtain the row coordinates inside each group. The

partial_row_coordinates
method returns a
pandas.DataFrame
where the set of columns is a
pandas.MultiIndex
. The first level of indexing corresponds to each specified group whilst the nested level indicates the coordinates inside each group.
>>> mfa.partial_row_coordinates(X)  # doctest: +NORMALIZE_WHITESPACE
  Expert #1           Expert #2           Expert #3
               0         1         0         1         0         1
Wine 1 -2.764432 -1.104812 -2.213928 -0.863519 -1.538106  0.442545
Wine 2  0.773034  0.298919  0.284247 -0.132135  0.613771 -0.759009
Wine 3  1.991398  0.805893  2.111508  0.499718  2.850084 -3.796390
Wine 4  1.981456  0.927187  2.393009  1.227146  1.123206  0.560803
Wine 5 -1.292834 -0.620661 -1.492114 -0.488088 -1.426414  1.273679
Wine 6 -0.688623 -0.306527 -1.082723 -0.243122 -1.622541  2.278372

Likewhise you can visualize the partial row coordinates with the

plot_partial_row_coordinates
method.
>>> ax = mfa.plot_partial_row_coordinates(
...     X,
...     ax=None,
...     figsize=(6, 6),
...     x_component=0,
...     y_component=1,
...     color_labels=['Oak type {}'.format(t) for t in X['Oak type']]
... )
>>> ax.get_figure().savefig('images/mfa_partial_row_coordinates.svg')

As usual you have access to inertia information.

>>> mfa.eigenvalues_
array([0.47246678, 0.05947651])

>>> mfa.total_inertia_ 0.558834...

>>> mfa.explained_inertia_ array([0.84545097, 0.10642965])

You can also access information concerning each partial factor analysis via the

partial_factor_analysis_
attribute.
>>> for name, fa in sorted(mfa.partial_factor_analysis_.items()):
...     print('{} eigenvalues: {}'.format(name, fa.eigenvalues_))
Expert #1 eigenvalues: [0.47709918 0.01997272]
Expert #2 eigenvalues: [0.60851399 0.03235984]
Expert #3 eigenvalues: [0.41341481 0.07353257]

The

row_contributions
method will provide you with the inertia contribution of each row with respect to each component.
>>> mfa.row_contributions(X)
               0         1
Wine 1  9.986433  4.349104
Wine 2  0.656699  0.655218
Wine 3 11.369187 11.589968
Wine 4  7.107942 13.771950
Wine 5  4.170915  0.050817
Wine 6  2.708824  5.582943

The

column_correlations
method will return the correlation between the original variables and the components.
>>> mfa.column_correlations(X)
                     0         1
E1 coffee    -0.918449 -0.043444
E1 fruity     0.968449  0.192294
E1 woody     -0.984442 -0.120198
E2 red fruit  0.887263  0.357632
E2 roasted   -0.955795  0.026039
E2 vanillin  -0.950629 -0.177883
E2 woody     -0.974649  0.127239
E3 butter    -0.945767  0.221441
E3 fruity     0.594649 -0.820777
E3 woody     -0.992337  0.029747

Factor analysis of mixed data (FAMD)

A description is on it's way. This section is empty because I have to refactor the documentation a bit.

>>> import pandas as pd

>>> X = pd.DataFrame( ... data=[ ... ['A', 'A', 'A', 2, 5, 7, 6, 3, 6, 7], ... ['A', 'A', 'A', 4, 4, 4, 2, 4, 4, 3], ... ['B', 'A', 'B', 5, 2, 1, 1, 7, 1, 1], ... ['B', 'A', 'B', 7, 2, 1, 2, 2, 2, 2], ... ['B', 'B', 'B', 3, 5, 6, 5, 2, 6, 6], ... ['B', 'B', 'A', 3, 5, 4, 5, 1, 7, 5] ... ], ... columns=['E1 fruity', 'E1 woody', 'E1 coffee', ... 'E2 red fruit', 'E2 roasted', 'E2 vanillin', 'E2 woody', ... 'E3 fruity', 'E3 butter', 'E3 woody'], ... index=['Wine {}'.format(i+1) for i in range(6)] ... ) >>> X['Oak type'] = [1, 2, 2, 2, 1, 1]

Now we can fit an

FAMD
.
>>> import prince
>>> famd = prince.FAMD(
...     n_components=2,
...     n_iter=3,
...     copy=True,
...     check_input=True,
...     engine='auto',
...     random_state=42
... )
>>> famd = famd.fit(X.drop('Oak type', axis='columns'))

The

FAMD
inherits from the
MFA
class, which entails that you have access to all it's methods and properties. The
row_coordinates
method will return the global coordinates of each wine.
>>> famd.row_coordinates(X)
               0         1
Wine 1 -1.488689 -1.002711
Wine 2 -0.449783 -1.354847
Wine 3  1.774255 -0.258528
Wine 4  1.565402  0.016484
Wine 5 -0.349655  1.516425
Wine 6 -1.051531  1.083178

Just like for the

MFA
you can plot the row coordinates with the
plot_row_coordinates
method.
>>> ax = famd.plot_row_coordinates(
...     X,
...     ax=None,
...     figsize=(6, 6),
...     x_component=0,
...     y_component=1,
...     labels=X.index,
...     color_labels=['Oak type {}'.format(t) for t in X['Oak type']],
...     ellipse_outline=False,
...     ellipse_fill=True,
...     show_points=True
... )
>>> ax.get_figure().savefig('images/famd_row_coordinates.svg')

Going faster

By default

prince
uses
sklearn
's randomized SVD implementation (the one used under the hood for
TruncatedSVD
). One of the goals of Prince is to make it possible to use a different SVD backend. For the while the only other supported backend is Facebook's randomized SVD implementation called fbpca. You can use it by setting the
engine
parameter to
'fbpca'
:
>>> import prince
>>> pca = prince.PCA(engine='fbpca')

If you are using Anaconda then you should be able to install

fbpca
without any pain by running
pip install fbpca
.

License

The MIT License (MIT). Please see the license file for more information.

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