prince 0.4.0

Statistical factor analysis in Python

Introduction

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 nice API.

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  # doctest: +SKIP

Via GitHub for the latest development version

>>> pip install git+https://github.com/MaxHalford/Prince  # doctest: +SKIP

Prince doesn't have any extra dependencies apart from the usual suspects (sklearn, pandas, matplotlib) which are included with Anaconda.

Usage

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 also 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:

• A Tutorial on Principal Component Analysis
• Theory of Correspondence Analysis
• Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
• Computation of Multiple Correspondence Analysis, with code in R
• Singular Value Decomposition Tutorial
• Multiple Factor Analysis

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', 'Sepal length'])
>>> y = pd.Series(y).map({0: 'Setosa', 1: 'Versicolor', 2: 'Virginica'})
>>> X.head()
   Sepal length  Sepal width  Petal length  Sepal length
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,
...     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 -2.264542  0.505704
1 -2.086426 -0.655405
2 -2.367950 -0.318477
3 -2.304197 -0.575368
4 -2.388777  0.674767

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.png')

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_  # doctest: +ELLIPSIS
[0.727704..., 0.230305...]

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_  # doctest: +ELLIPSIS
[436.622712..., 138.183139...]

>>> pca.total_inertia_
600.0

>>> pca.explained_inertia_
[0.727704..., 0.230305...]

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

>>> pca.column_correlations(X)
                     0         1
Sepal length  0.891224  0.357352
Sepal width  -0.449313  0.888351
Petal length  0.991684  0.020247
Sepal length  0.964996  0.062786

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  0.011745  0.001851
1  0.009970  0.003109
2  0.012842  0.000734
3  0.012160  0.002396
4  0.013069  0.003295

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,
...     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.png')

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_  # doctest: +ELLIPSIS
[0.199244..., 0.030086...]

>>> ca.total_inertia_  # doctest: +ELLIPSIS
0.230191...

>>> ca.explained_inertia_  # doctest: +ELLIPSIS
[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 ballons 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,
...     engine='auto',
...     random_state=42
... )
>>> mca = mca.fit(X)

As usual you can retrieve the row and column principal components via their respective methods.

>>> mca.row_coordinates(X).head()
          0         1
0  0.705387  0.000000
1 -0.386586  0.000000
2 -0.386586  0.000000
3 -0.852014  0.000000
4  0.783539 -0.633333

>>> mca.column_coordinates(X).head()
                     0         1
Color_PURPLE  0.117308  0.689202
Color_YELLOW -0.130342 -0.765780
Size_LARGE    0.117308 -0.689202
Size_SMALL   -0.130342  0.765780
Action_DIP   -0.853864 -0.000000

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.png')

The eigenvalues and inertia values are also accessible.

>>> mca.eigenvalues_  # doctest: +ELLIPSIS
[0.401656..., 0.211111...]

>>> mca.total_inertia_
1.0

>>> mca.explained_inertia_  # doctest: +ELLIPSIS
[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)]
... )

Next the authors decided to normalize the data so that the sum of squares of each column equals 1. This isn't always necessary but we'll still do it so as to exactly reproduce the paper's results.

>>> import numpy as np
>>> X = (X - X.mean()).apply(lambda x: x / np.sqrt((x ** 2).sum()), axis='rows')

Finally let's specify the type of oak used to store each wine.

>>> X['Oak type'] = [1, 2, 2, 2, 1, 1]

The groups are passed as a dictionary to the MFA class.

>>> import prince

>>> 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.PrettyPrinter().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`.

>>> mfa = prince.MFA(
...     groups=groups,
...     rescale_with_mean=False,
...     rescale_with_std=False,
...     n_components=2,
...     n_iter=3,
...     copy=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
0 -2.172155 -0.508596
1  0.557017 -0.197408
2  2.317663 -0.830259
3  1.832557  0.905046
4 -1.403787  0.054977
5 -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.png')

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
0 -2.764432 -1.104812 -2.213928 -0.863519 -1.538106  0.442545
1  0.773034  0.298919  0.284247 -0.132135  0.613771 -0.759009
2  1.991398  0.805893  2.111508  0.499718  2.850084 -3.796390
3  1.981456  0.927187  2.393009  1.227146  1.123206  0.560803
4 -1.292834 -0.620661 -1.492114 -0.488088 -1.426414  1.273679
5 -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.png')

As usual you have access to inertia information.

>>> mfa.eigenvalues_  # doctest: +ELLIPSIS
[2.834800..., 0.356859...]

>>> mfa.total_inertia_
3.353004...

>>> mfa.explained_inertia_  # doctest: +ELLIPSIS
[0.845450..., 0.106429...]

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()):  # doctest: +ELLIPSIS
...     print('{} eigenvalues: {}'.format(name, fa.eigenvalues_))
Expert #1 eigenvalues: [2.862595..., 0.119836...]
Expert #2 eigenvalues: [3.651083..., 0.194159...]
Expert #3 eigenvalues: [2.480488..., 0.441195...]

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.

Incoming features

I've got a lot on my hands aside from prince, so feel free to give me a hand!

• Factor Analysis of Mixed Data (FAMD)
• Generalized Procustean Analysis (GPA)
• Multiple Factorial Analysis (MFA)

License

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