Multiple factor analysis

Table of contents

Resources

• Multiple Factor Analysis by Hervé Abdi
• Multiple Factor Analysis: main features and application to sensory data by Jérôme Pagès
• Wikipedia article

Data

Multiple factor analysis (MFA) is meant to be used when you have groups of variables. In practice, it builds a PCA on each group. It then fits a global PCA on the results of the so-called partial PCAs.

The following dataset contains end of season figures for Premier League football teams. It spans the 2021/22, 2022/23, and 2023/24 seasons. Only the 14 teams that have been in the Premier League for all three seasons are included.

import prince

dataset = prince.datasets.load_premier_league()
dataset

import pandas as pd

isinstance(dataset.columns, pd.MultiIndex)

True

Fitting

The groups are specified by the groups argument when calling fit.

groups = dataset.columns.levels[0].tolist()
groups

['2021-22', '2022-23', '2023-24']

mfa = prince.MFA(
    n_components=3,
    n_iter=3,
    copy=True,
    check_input=True,
    engine='sklearn',
    random_state=42
)
mfa = mfa.fit(
    dataset,
    groups=groups,
    supplementary_groups=None
)

There are several ways to specify the groups:

• If the columns of the dataframe are a MultiIndex:
  • By default the groups are all the columns in the first level.
  • You can also pass a list with a subset of the columns in the first level.
• You can also pass a dict that maps group names to the desired columns.

The supplementary_groups argument is expected to be a list with one more existing group names.

Eigenvalues

mfa.eigenvalues_summary

Coordinates

The MFA inherits from the PCA class, which means it provides access to the PCA methods and properties. For instance, the row_coordinates method will return the global coordinates of each wine.

mfa.row_coordinates(dataset)

There is also a partial_row_coordinates method that returns the coordinates projected onto each group.

mfa.partial_row_coordinates(dataset)

Visualization

mfa.plot(
    dataset,
    x_component=0,
    y_component=1
)

The first axis explains most of the difference between the wine ratings. This difference is actually due to the oak type of the barrels they were fermented in.

The show_partial_rows argument allows showing the global row coordinates together with the partial row coordinates. All the coordinates of each sample are connected with edges.

mfa.plot(
    dataset,
    show_partial_rows=True
)

Partial PCAs

An MFA is essentially a PCA applied to the outputs of partial PCA. Indeed, a PCA is first fitted to each group. A partial PCA can be accessed as so:

dataset['2022-23']

mfa['2022-23'].eigenvalues_summary