Multiple correspondence analysis

Resources

Computation of Multiple Correspondence Analysis, with code in R

Data

Multiple correspondence analysis is an extension of correspondence analysis. It should be used when you have more than two categorical variables. The idea is to one-hot encode a dataset, before applying correspondence analysis to it.

As an example, we’re going to use the balloons dataset taken from the UCI datasets website.

import pandas as pd

dataset = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/balloons/adult+stretch.data')
dataset.columns = ['Color', 'Size', 'Action', 'Age', 'Inflated']
dataset.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

Fitting

import prince

mca = prince.MCA(
    n_components=3,
    n_iter=3,
    copy=True,
    check_input=True,
    engine='sklearn',
    random_state=42
)
mca = mca.fit(dataset)

The way MCA works is that it one-hot encodes the dataset, and then fits a correspondence analysis. In case your dataset is already one-hot encoded, you can specify one_hot=False to skip this step.

one_hot = pd.get_dummies(dataset)

mca_no_one_hot = prince.MCA(one_hot=False)
mca_no_one_hot = mca_no_one_hot.fit(one_hot)

Eigenvalues

mca.eigenvalues_summary

	eigenvalue	% of variance	% of variance (cumulative)
component
0	0.402	40.17%	40.17%
1	0.211	21.11%	61.28%
2	0.186	18.56%	79.84%

Coordinates

mca.row_coordinates(dataset).head()

	0	1	2
0	0.705387	5.369158e-15	0.758639
1	-0.386586	5.724889e-15	0.626063
2	-0.386586	4.807799e-15	0.626063
3	-0.852014	5.108782e-15	0.562447
4	0.783539	-6.333333e-01	0.130201

mca.column_coordinates(dataset).head()

	0	1	2
Color_PURPLE	0.117308	6.892024e-01	-0.641270
Color_YELLOW	-0.130342	-7.657805e-01	0.712523
Size_LARGE	0.117308	-6.892024e-01	-0.641270
Size_SMALL	-0.130342	7.657805e-01	0.712523
Action_DIP	-0.853864	-6.367615e-16	-0.079340

Visualization

mca.plot(
    dataset,
    x_component=0,
    y_component=1,
    show_column_markers=True,
    show_row_markers=True,
    show_column_labels=False,
    show_row_labels=False
)

Contributions

mca.row_contributions_.head().style.format('{:.0%}')

	0	1	2
0	7%	0%	16%
1	2%	0%	11%
2	2%	0%	11%
3	10%	0%	9%
4	8%	10%	0%

mca.column_contributions_.head().style.format('{:.0%}')

	0	1	2
Color_PURPLE	0%	24%	23%
Color_YELLOW	0%	26%	26%
Size_LARGE	0%	24%	23%
Size_SMALL	0%	26%	26%
Action_DIP	15%	0%	0%

Cosine similarities

mca.row_cosine_similarities(dataset).head()

	0	1	2
0	0.461478	2.673675e-29	0.533786
1	0.152256	3.338988e-29	0.399316
2	0.152256	2.354904e-29	0.399316
3	0.653335	2.348969e-29	0.284712
4	0.592606	3.871772e-01	0.016363

mca.column_cosine_similarities(dataset).head()

	0	1	2
Color_PURPLE	0.015290	5.277778e-01	0.456920
Color_YELLOW	0.015290	5.277778e-01	0.456920
Size_LARGE	0.015290	5.277778e-01	0.456920
Size_SMALL	0.015290	5.277778e-01	0.456920
Action_DIP	0.530243	2.948838e-31	0.004578