Multiple factor analysis

Table of contents

Resources

• Multiple Factor Analysis

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 dataset used in the following example 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.

import prince 

dataset = prince.datasets.load_burgundy_wines()
dataset

Fitting

The groups are passed as a dictionary to the fit method.

groups = dataset.columns.levels[0].drop("Oak type").tolist()
groups

['Expert 1', 'Expert 2', 'Expert 3']

mfa = prince.MFA(
    n_components=2,
    n_iter=3,
    copy=True,
    check_input=True,
    engine='sklearn',
    random_state=42
)
mfa = mfa.fit(dataset, groups=groups)

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)

However, all the other methods are not implemented yet. They will raise a NotImplemented exception if you call them.

mfa.group_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.

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['Expert 1']

mfa['Expert 1'].eigenvalues_summary