Ordinal Regression

When we deal with data in machine learning, we often try to predict something. Sometimes we want to predict a number (like a price) and sometimes we want to predict a category (like "cat" or "dog"). When the categories have a natural order, such as 'bad', 'average', and 'good', ordinal regression is the appropriate technique to use.

Ordinal Regression is a type of predictive modeling where the target variable is ordered but not measured numerically. The values have a specific sequence, but we don't know the exact gap between them.

For example:

• Rating a product as: 1 - Poor, 2 - Average, 3 - Good, 4 - Excellent
• Education level: High School, Bachelor’s, Master’s, PhD

In both examples, there’s a clear order, but the difference between the categories isn’t a fixed value.

Why Not Use Regular Classification?

The problem is that normal classification treats all categories as separate and equal, with no idea of order. It sees "Poor" and "Excellent" as just two different labels without knowing that "Excellent" is better than "Poor".

Ordinal regression respects the order of the categories. This helps improve accuracy and provides more meaningful predictions.

Real-Life Examples of Ordinal Regression

1. Movie Ratings – Predicting if someone will rate a movie 1 to 5 stars.
2. Customer Feedback – Predicting how satisfied a customer is: Very Unsatisfied, Unsatisfied, Neutral, Satisfied, Very Satisfied.
3. Education Levels – Predicting someone's education level based on their background.

How Does It Work?

• Ordered Labels: It starts with labels that are categorical but ordered. For example:
  • 1 = Poor
  • 2 = Fair
  • 3 = Good
  • 4 = Excellent
• Transforms the Problem: It doesn’t just treat this as a multi-class problem. Instead, it turns it into a series of binary problems. Example:
  • Is the rating > 1?
  • Is the rating > 2?
  • Is the rating > 3?
• Fits a Model to These Thresholds: The model tries to learn thresholds on a continuous scale. These thresholds split the data into ordered regions, matching your categories.
• Uses Logistic (or Probit) Link Function: Most ordinal regression models use a logistic function (like in logistic regression) to estimate probabilities.

Difference Between Nominal and Ordinal Variables

Common Algorithms for Ordinal Regression

• Proportional Odds Model (POM) (also called Ordered Logit Model)
• Ordinal SVM
• Cumulative Link Models
• Neural Networks for ordinal output (using special loss functions)

Mathematical Formulation (Proportional Odds Model)

The Proportional Odds Model (POM), also known as the Ordered Logit Model, is commonly used for ordinal regression. It models the cumulative probability that the response variable falls in or below a particular category.

Formula:

Where:

• Probability that the response variable Y is in category 𝑗 or lower.
• :Threshold (also called cutpoint or intercept) for category 𝑗 ; separates category 𝑗 from the next.
• β: Coefficient vector (weights for the model).
• X: Feature vector (input variables).
• :Dot product (linear combination) of features and coefficients.
• exp: Exponential function.

Implementing Ordinal Regression in Python

Output:

Accuracy: 0.9733333333333334

When implementing ordinal regression in Python, especially using libraries like statsmodels, mord, or custom PyTorch models, you’ll come across the following key outputs and terms:

• Accuracy: % of correct predictions (e.g., 97.33%).
• Thresholds (): Cut-off points between ordered classes.
• Coefficients (𝛽): Show how features influence the outcome order.
• Cumulative Probability: Model predicts the chance of being in or below each category.
• Predicted Class: Chosen based on the highest class probability.
• Loss Function: Penalizes bigger mistakes more (e.g., Poor → Excellent).

Popular Algorithms for Ordinal Regression

Here are some methods used for ordinal regression:

• Ordinal Logistic Regression
• Ordinal SVM (Support Vector Machine)
• Decision Trees for Ordered Categories
• Neural Networks with Ordered Output

These models are adjusted so that they know the order of the output labels.

Advantages of Ordinal Regression

• Makes use of the order in data
• More accurate than simple classification in ordered problems
• Easy to interpret results

Challenges

• Harder to implement than regular classification
• Needs careful handling of data and labels
• Less support in some libraries

Limitations

• Model assumptions (e.g., proportional odds) may not always hold.
• Interpretation of coefficients can be challenging.
• Not all libraries offer native support.