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How to Perform Correspondence Analysis in Python

Step-by-step guide to implementing Correspondence Analysis (CA) in Python using the prince library, including data preparation, model fitting, biplot visualization, and result interpretation for categorical contingency tables.

Correspondence Analysis (CA) is a dimensionality reduction technique designed for categorical data organized in contingency tables. Similar to PCA for numerical data, CA visualizes relationships between row categories (e.g., regions) and column categories (e.g., products) on a two-dimensional map.

Setting Up the Environment

The prince library provides the standard implementation for CA and MCA in Python.

pip install prince pandas matplotlib

Preparing the Contingency Table

CA requires a cross-tabulation table where rows and columns represent categorical variables and cells contain frequency counts.

import pandas as pd

data = {
    'Coffee': [50, 30, 20],
    'Tea':    [30, 60, 10],
    'Soda':   [10, 20, 80]
}
df = pd.DataFrame(data, index=['North', 'South', 'East'])
print(df)

Output:

       Coffee  Tea  Soda
North      50   30    10
South      30   60    20
East       20   10    80

Fitting the CA Model

Initialize and fit the model by specifying the number of components to extract:

import prince

ca = prince.CA(n_components=2)
ca = ca.fit(df)

To examine the explained variance for each dimension:

print(ca.eigenvalues_summary)

Creating the Biplot Visualization

The biplot displays both row and column categories in the same coordinate space, revealing their associations.

import matplotlib.pyplot as plt

row_coords = ca.row_coordinates(df)
col_coords = ca.column_coordinates(df)

print("Row coordinates:\n", row_coords)
print("\nColumn coordinates:\n", col_coords)

Using the built-in plotting method:

ax = ca.plot(
    df,
    figsize=(8, 8),
    x_component=0,
    y_component=1,
    show_row_markers=True,
    show_col_markers=True,
    show_row_labels=True,
    show_col_labels=True
)
plt.title("Correspondence Analysis Biplot")
plt.axhline(y=0, color='gray', linestyle='--', linewidth=0.5)
plt.axvline(x=0, color='gray', linestyle='--', linewidth=0.5)
plt.savefig("ca_biplot.png", dpi=150, bbox_inches='tight')
plt.show()

For more control over the visualization:

fig, ax = plt.subplots(figsize=(10, 8))

ax.scatter(row_coords[0], row_coords[1], c='blue', s=100, label='Regions')
for idx, row in row_coords.iterrows():
    ax.annotate(idx, (row[0], row[1]), fontsize=12, color='blue')

ax.scatter(col_coords[0], col_coords[1], c='red', s=100, marker='^', label='Beverages')
for idx, row in col_coords.iterrows():
    ax.annotate(idx, (row[0], row[1]), fontsize=12, color='red')

ax.axhline(y=0, color='gray', linestyle='--', linewidth=0.5)
ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.5)
ax.set_xlabel('Dimension 1')
ax.set_ylabel('Dimension 2')
ax.legend()
ax.set_title('CA Biplot: Regional Beverage Preferences')
plt.savefig("ca_biplot_custom.png", dpi=150, bbox_inches='tight')
plt.show()

Interpreting the Results

Understanding the biplot requires attention to several key aspects:

Visual Cue Interpretation
Proximity Categories plotted close together have strong positive associations
Distance from origin Points far from center (0,0) contribute most to the overall variance
Opposite positions Categories on opposite sides of an axis are negatively correlated
Axis alignment Points aligned along an axis share similar profiles on that dimension

Extracting Additional Statistics

print(f"Total inertia: {ca.total_inertia_:.4f}")

print(f"Explained inertia: {ca.explained_inertia_}")

row_contrib = ca.row_contributions_
col_contrib = ca.column_contributions_
print("\nRow contributions:\n", row_contrib)
print("\nColumn contributions:\n", col_contrib)

When to Use CA vs MCA

Summary

Correspondence Analysis transforms complex categorical relationships into interpretable visual maps. By projecting row and column categories onto shared dimensions, CA reveals association patterns that would otherwise remain hidden in large contingency tables.

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