Ordinal Regression

Start writing here...

Great choice! Ordinal Regression is a super useful concept when you're dealing with ordered categories — more informative than classification, but simpler than full regression. Here's a well-structured breakdown, perfect for notes, presentations, or deeper learning.

🧮 Ordinal Regression (a.k.a. Ordinal Classification)

🧠 What Is Ordinal Regression?

Ordinal Regression is a type of supervised learning where the target variable has a natural order, but the differences between levels are unknown or not meaningful.

Think of it as the middle ground between classification and regression.

🎯 Real-World Examples

Problem	Classes (Ordered)
Customer satisfaction	😠 "Very Dissatisfied" → 😀 "Very Satisfied"
Star ratings	⭐, ⭐⭐, ⭐⭐⭐, ⭐⭐⭐⭐, ⭐⭐⭐⭐⭐
Disease severity	Mild → Moderate → Severe
Credit risk	Low → Medium → High

🧩 How Is It Different?

Problem Type	Target Variable
Classification	Discrete classes, unordered (e.g., cat/dog)
Regression	Continuous values (e.g., income)
Ordinal Regression	Discrete, ordered labels (e.g., rating levels)

⚙️ Common Approaches

1. Threshold Models / Cumulative Link Models

Learn a latent score s=wTxs = w^T x
Learn thresholds θ1,θ2,…,θK−1\theta_1, \theta_2, \ldots, \theta_{K-1}
Predict class based on which interval the score falls into

Class y=kif θk−1<s≤θk\text{Class } y = k \quad \text{if } \theta_{k-1} < s \leq \theta_k

✅ Simple and interpretable

✅ Used in proportional odds models (in statistics)

2. Ordinal Logistic Regression

Also called proportional odds model
Models cumulative probability:

P(y≤k∣x)=11+exp⁡(−(θk−wTx))P(y \leq k \mid x) = \frac{1}{1 + \exp(-( \theta_k - w^T x ))}

Used in statsmodels and R’s polr() function

3. Decomposition into Binary Classification

Train K−1K-1 binary classifiers to predict:
- Is label > 1?
- Is label > 2?
- ...
Final prediction based on the outputs of these classifiers

✅ Works with existing classification models

❌ Can be inconsistent if classifiers disagree

4. Deep Learning Approaches

Use neural nets with:
- Custom ordinal loss functions
- Cumulative logits
- Soft-label smoothing to enforce order

Popular in NLP tasks like sentiment scoring or emotion intensity prediction.

🧪 Example (Using mord in Python)

from mord import LogisticIT
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_iris

# Example using iris (hacky but illustrative)
X, y = load_iris(return_X_y=True)
y = y.astype(int)  # Assume ordered classes

model = LogisticIT()
model.fit(X, y)
preds = model.predict(X)

mord is a Python package for ordinal regression (pip install mord)

📈 Evaluation Metrics

Metric	Description
Mean Absolute Error (MAE)	Penalizes predictions further from true class
Quadratic Weighted Kappa (QWK)	Measures agreement accounting for order
Accuracy	Works but ignores order
Spearman’s Rank Correlation	Measures monotonic relationship

✅ Pros & ❌ Cons

✅ Pros	❌ Cons
Takes ordering into account	Less widely supported in libraries
More informative than classification	Requires careful model design
Works well with small datasets	Interpretation can be tricky in NN models

🔬 Use Cases in the Wild

Medical diagnosis: Disease stages
E-commerce: Customer satisfaction ratings
NLP: Emotion intensity, sentiment levels
Education: Grading levels or proficiency levels

🧠 Summary Table

Aspect	Ordinal Regression
Label Type	Discrete, ordered
Compared To	Between classification and regression
Model Types	Logistic models, threshold models, binary chains
Evaluation Metrics	MAE, QWK, Spearman’s rank

Let me know if you'd like:

A visual showing threshold models in action
Code using deep learning (PyTorch or TensorFlow)
Quiz questions for review
Comparisons with multiclass classification

Happy to expand or simplify based on your goals!

in Machine Learning