Lære Polynomial Coding for Categorical Features | Advanced Categorical Coding Schemes

Polynomial coding is a specialized encoding technique designed for ordinal categorical variables—categories that have a clear, meaningful order but do not have consistent or measurable distances between them. These types of variables are common in real-world datasets, such as education level ("High School" < "Bachelor" < "Master"), product quality ratings, or survey responses ("Strongly Disagree" to "Strongly Agree").

What Makes Polynomial Coding Unique?

Captures Order, Not Distance: Unlike simple integer encoding, which assigns sequential numbers to categories, polynomial coding preserves the order of categories without misleading models into assuming equal spacing between them.
Cumulative Contrast: Each new feature created by polynomial coding represents a cumulative contrast between the current category and all previous ones. This helps models recognize the progression through ordered levels.
Improved Model Performance: Many machine learning algorithms, such as linear regression and tree-based models, can benefit from this encoding because it aligns with the natural ordering in the data, leading to more meaningful splits and coefficients.

How Polynomial Coding Works

For a variable with $k$ categories, polynomial coding generates $k-1$ new features.
Each feature encodes whether a sample's category is above a certain threshold in the order, resulting in a set of binary features that collectively describe the position of each category in the sequence.

Key Benefits

Preserves Ordinal Information: Retains the inherent order of categories, which is crucial for accurate modeling.
Avoids False Assumptions: Prevents models from interpreting differences between categories as equidistant, which can happen with simple integer encoding.
Enhances Interpretability: The resulting features are easier to interpret in the context of ordered data, making it clear how each category relates to the others.

Use polynomial coding when you want your machine learning models to leverage the order in ordinal data without imposing assumptions about the magnitude of differences between categories.


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import pandas as pd
import numpy as np

# Sample ordinal data
df = pd.DataFrame({
    "education_level": ["High School", "Associate", "Bachelor", "Master", "PhD"]
})

# Define the order of categories
categories = ["High School", "Associate", "Bachelor", "Master", "PhD"]

# Map categories to ordered values
df["education_level_ord"] = df["education_level"].map({cat: i for i, cat in enumerate(categories)})

# Polynomial coding: For k categories, create k-1 features
k = len(categories)
poly_codes = np.zeros((len(df), k - 1))

for i, val in enumerate(df["education_level_ord"]):
    for j in range(k - 1):
        if val > j:
            poly_codes[i, j] = 1
        else:
            poly_codes[i, j] = 0

# Add polynomial-coded columns to DataFrame
for j in range(k - 1):
    df[f"Poly_{j+1}"] = poly_codes[:, j]

print(df)

Step-by-Step Explanation of Polynomial Coding in the Code Sample

Follow these steps to understand how the code implements polynomial coding for the education_level feature:

Prepare the Data:
- A DataFrame is created with the education_level column containing ordered categories: "High School", "Associate", "Bachelor", "Master", and "PhD";
Define Category Order:
- The list categories explicitly sets the order of educational levels, ensuring the encoding respects the true sequence;
Map Categories to Ordinal Values:
- Each category is mapped to a unique integer based on its order using Python's enumerate, so "High School" becomes 0, "Associate" becomes 1, and so on;
- This creates a new column, education_level_ord, which holds these integer representations.
Initialize Polynomial Code Matrix:
- For $k$ categories, the code creates a matrix with $k-1$ columns (features);
- Each row will be filled with binary values that describe the cumulative position of its category.
Generate Polynomial Features:
- For each sample, the code checks, for every feature (from 1 to $k-1$ ): "Is this category above the current threshold?";
- If the ordinal value is greater than the feature index, it assigns 1; otherwise, 0;
- This produces a set of binary features where each additional 1 indicates the sample is at a higher level in the category order.
Add Features Back to the DataFrame:
- Each polynomial-coded column (Poly_1, Poly_2, etc.) is added to the DataFrame, making the transformed features ready for modeling.

How These Features Represent Cumulative Contrasts:

Each new feature marks whether a sample's category is above a certain threshold in the order. For example, only "Associate" and above get a 1 in Poly_1, "Bachelor" and above get a 1 in Poly_2, and so on;
The resulting pattern of 0s and 1s across the features clearly shows the progression through the ordered levels, preserving the ordinal structure without assuming equal spacing between categories.

Polynomial coding vs. Ordinal integer encoding

Polynomial coding encodes the ordinal relationship without assuming equal spacing between categories, while ordinal integer encoding simply assigns increasing integers, which can mislead some models into treating the difference between categories as uniform;

Polynomial coding vs. One-hot encoding

One-hot encoding ignores order and treats all categories as equally distinct, potentially losing valuable information when categories have a meaningful sequence;

Polynomial coding vs. Helmert/Backward Difference coding

While Helmert and Backward Difference coding also encode order, Polynomial coding specifically focuses on cumulative contrasts, making it more interpretable for strictly ordered features.

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