Apprendre Helmert Coding Explained | Advanced Categorical Coding Schemes

Helmert coding is a way to encode categorical variables as numerical values for use in regression analysis and statistical modeling.

How Helmert Coding Works

Compares each level to the mean of subsequent levels;
Produces orthogonal contrasts, meaning the resulting columns are uncorrelated;
Generates k-1 columns for a variable with k levels, with each column representing the contrast between one category and the mean of the categories that follow it.

This approach is different from one-hot encoding, which compares each category to a single baseline. Helmert coding instead lets you interpret the effect of each category relative to the average of the remaining categories.

Why Use Helmert Coding?

Orthogonality: The columns are uncorrelated, providing unique information to the model;
Prevents multicollinearity, making linear regression coefficients easier to interpret;
Useful for group comparisons, especially when you want to understand how each category compares to the average of subsequent groups.

You will often use Helmert coding in experimental design, psychology, and other fields where comparing groups and maintaining orthogonality of predictors are important.


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import pandas as pd
import numpy as np

# Sample categorical data
data = pd.DataFrame({
    "Color": ["Red", "Blue", "Green", "Blue", "Green", "Red", "Green"]
})

# Get unique categories and sort them
categories = sorted(data["Color"].unique())

# Number of unique categories
k = len(categories)

# Create the Helmert coding matrix
def helmert_matrix(k):
    matrix = np.zeros((k, k-1))
    for i in range(k-1):
        matrix[:i+1, i] = 1
        matrix[i+1, i] = -(i+1)
    # Normalize each column
    matrix /= np.sqrt(np.sum(matrix**2, axis=0))
    return matrix

# Build a mapping from category to Helmert code
helmert_codes = helmert_matrix(k)
category_to_code = {cat: helmert_codes[i] for i, cat in enumerate(categories)}

# Apply the mapping to the data
helmert_encoded = data["Color"].map(category_to_code)
helmert_df = pd.DataFrame(helmert_encoded.tolist(),
                          columns=[f"Helmert_{i+1}" for i in range(k-1)])

# Concatenate with original data
result = pd.concat([data, helmert_df], axis=1)
print(result)

Helmert Coding Step by Step

This code sample demonstrates how to apply Helmert coding to a categorical variable using Python and pandas.

1. Creating the Helmert Coding Matrix

The function helmert_matrix(k) builds a Helmert matrix for k unique categories;
For each column i, the first i+1 rows are set to 1, the next row is set to -(i+1), and all others are 0;
Each column is normalized so that the sum of squares is 1, ensuring orthogonality.

2. Mapping Categories to Helmert Codes

The sorted unique categories from the Color column are assigned to the rows of the Helmert matrix;
A dictionary category_to_code maps each category name (like "Blue", "Green", "Red") to its Helmert code (a numerical vector from the matrix).

3. Encoding the Data

The original Color values are mapped to their corresponding Helmert code vectors using .map(category_to_code);
The resulting list of vectors is converted into a new DataFrame, where each column represents one Helmert contrast (e.g., Helmert_1, Helmert_2).

4. Building the Final DataFrame

The original data and the encoded columns are concatenated side by side using pd.concat;
The final output shows each color along with its Helmert-encoded values, ready for use in a regression model or analysis.

Definition

Orthogonal coding refers to a method of encoding categorical variables so that the resulting columns are uncorrelated (orthogonal) with each other. This is beneficial in regression analysis because it reduces multicollinearity, improves the interpretability of model coefficients, and can enhance numerical stability during model fitting.

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