Ridge Regression

"Redoing Ridge Regression from scratch in Python"
08 February 2024

What is Ridge Regression?

Ridge Regression, also known as L2 Regularization or Tikhonov Regularization, is a type of linear regression that uses a regularization term to prevent overfitting just like Lasso Regression. The only difference is that Ridge Regression uses the sum of the squares of the weights instead of the sum of the absolute values of the weights.

Due to the square term, it minimizes the weights close to zero but not exactly zero. Thus, all the features will be used in the model. Moreover, Ridge Regression is good at when we have data in which the multicollinearity issue is not too severe.

Mathematics Behind Ridge Regression

In Ridge Regression, we are going to use the same linear function that Linear Regression uses:

y^=β0+β1x\hat{y} = \beta_0 + \beta_1 x

Similar to what we did in previous posts, we need to estimate the best β0\beta_0 and β1\beta_1 using the Gradient Descent algorithm. What the Gradient Descent algorithm does is to update the β0\beta_0 and β1\beta_1 values based on the cost function and the learning rate.

This example is just a simple linear model, we are going to use the following equations to update intercept and coefficient:

β0=β0αβ0J(β0,β1,,βp)β1=β1αβ1J(β0,β1,,βp)x1βp=βnαβnJ(β0,β1,,βp)xp\begin{gather*} \beta_0 = \beta_0 - \alpha \frac{\partial}{\partial \beta_0} J(\beta_0, \beta_1, \dots, \beta_p) \\ \beta_1 = \beta_1 - \alpha \frac{\partial}{\partial \beta_1} J(\beta_0, \beta_1, \dots, \beta_p)x_1 \\ \cdots \\ \beta_p = \beta_n - \alpha \frac{\partial}{\partial \beta_n} J(\beta_0, \beta_1, \dots, \beta_p)x_p \\ \end{gather*}

where α\alpha is the learning rate, βp\beta_p is the pp-th parameter, JJ is the cost function, and xpx_p is the pp-th feature.

Since we only have β0\beta_0 and β1\beta_1, we can simplify the equation above to:

β0=β0αβ0J(β0,β1)β1=β1αβ1J(β0,β1)x1\begin{gather*} \beta_0 = \beta_0 - \alpha \frac{\partial}{\partial \beta_0} J(\beta_0, \beta_1) \\ \beta_1 = \beta_1 - \alpha \frac{\partial}{\partial \beta_1} J(\beta_0, \beta_1)x_1 \\ \end{gather*}

However, the only difference in Ridge Regression is that we are going to add a penalty term to the cost function. This penalty term is the sum of the absolute values of the weights. This is also known as the L2 norm of the weights. The cost function for Ridge Regression that we have to minimize is given by:

minβ(1Ni=1n(y^iyi)2+λj=1pβj2)\min _\beta\left( \frac{1}{N} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2 + \lambda \sum_{j=1}^{p} |\beta_j^2| \right)

where

  1. 1Ni=1n(y^iyi)2\frac{1}{N} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2 is Mean Squared Error.
  2. λ\lambda is the regularization parameter.
  3. j=1pβj2\sum_{j=1}^{p} |\beta_j^2| is the L2 norm of the weights.

What's Wrong with the Data?

Unlike the Lasso Regression post, we are going to use the Diabetes dataset from the Scikit-Learn library. For more details, you can check the official documentation.

from sklearn.datasets import load_diabetes
 
data = load_diabetes()
feature_names = ['age', 'sex', 'body_mass_index', 'blood_pressure', 'serum_cholesterol', 'ldl', 'hdl', 'cholesterol_ratio', 'triglycerides', 'blood_sugar']

There are ten features in the dataset. Let's decipher what some features represent:

  1. body_mass_index: It's a measure of body fat based on height and weight.
  2. serum_cholesterol: The amount of cholesterol present in the blood.
  3. ldl: Low-density lipoprotein cholesterol, often referred to as "bad" cholesterol.
  4. hdl: High-density lipoprotein cholesterol, often referred to as "good" cholesterol.
  5. cholesterol_ratio: A ratio calculated by dividing the total cholesterol by the HDL cholesterol level.
  6. triglycerides: A type of fat found in the blood, expressed in logaritm.
  7. blood_sugar: The amount of glucose present in the blood.

Let's plot the heatmap to see the correlation between the features.

Diabetes Dataset Heatmap
Diabetes Dataset Heatmap

Here is a guide on how to interpret the values in the table above:

  1. Two or more features said to have a strong positive correlation if the correlation coefficient is close to 11.
  2. Two or more features said to have a moderate positive correlation if the correlation coefficient is close to 0.50.5.
  3. Two or more features said to have a weak positive correlation if the correlation coefficient is close to 00.
  4. Two or more features said to have a moderate negative correlation if the correlation coefficient is close to 0.5-0.5.
  5. Two or more features said to have a strong negative correlation if the correlation coefficient is close to 1-1.

Looking at the heatmap, we can determine easily what features are correlated with each other.

  1. ldl and serum_cholesterol have a strong positive correlation.
  2. ldl and cholesterol_ratio have a strong positive correlation.
  3. serum_cholesterol and cholesterol_ratio have a moderate positive correlation.
  4. serum_cholesterol and triglycerides have a moderate negative correlation.

These are the features that can be detected easily from the heatmap. With the following code, we can list out all the correlation values between the features.

df.corr()[df.corr() < 1]
  .unstack()
  .transpose()
  .sort_values(ascending=False)
  .drop_duplicates()
Variable 1Variable 2Correlation
ldlserum_cholesterol0.896663
cholesterol_ratio0.659817
cholesterol_ratiotriglycerides0.617859
serum_cholesterolcholesterol_ratio0.542207
triglycerides0.515503
.........
sexhdl-0.379090
triglycerideshdl-0.398577
hdlcholesterol_ratio-0.738493

We can also use Variance Inflation Factor (VIF) to determine the multicollinearity between the features.

from statsmodels.stats.outliers_influence import variance_inflation_factor
 
vif_data = pd.DataFrame()
vif_data["feature"] = feature_names
vif_data["VIF"] = [variance_inflation_factor(df.values, i) for i in range(len(data.feature_names))]
print(vif_data)
featureVIF
age1.217307
sex1.278071
body_mass_index1.509437
blood_pressure1.459428
serum_cholesterol59.202510
ldl39.193370
hdl15.402156
cholesterol_ratio8.890986
triglycerides10.075967
blood_sugar1.484623

Here is a guide on how to interpret VIF values:

  1. VIF=1\text{VIF} = 1: The feature is said to have no multicollinearity.
  2. 1<VIF<51 < \text{VIF} < 5: The feature is said to have a moderate multicollinearity.
  3. VIF>5\text{VIF} > 5: The feature is said to have a severe multicollinearity.

From the heatmap, the correlation table, as well as the VIF table, it's clear that ldl, serum_cholesterol, hdl, cholesterol_ratio, and triglycerides have a severe multicollinearity. Let's see how Ridge Regression can help us to solve this issue.

Implementation

Let's prepare the data for the Ridge Regression model by splitting the dataset into training and testing sets, and standardizing the feature values.

from sklearn.datasets import load_diabetes
 
data = load_diabetes()
X, y = data.data, data.target
 
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
 
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

Now out data is ready, we want pick a number of epoch, meaning how many times our model has to go through the dataset. In this example, we are going to use 100,000100,000 epochs, and it might take sometime. However, 50005_000 epochs should be enough to see the changes in the loss, intercept, and coefficients. Then we initialize the history of the loss, intercept, and coefficients so that we can visualize the changes in the values of these variables.

epochs = 100_000
loss_history = list()
intercept_history = list()
coefficients_history = np.zeros((scaled_X.shape[1], epochs))

Next, we would need two helper functions: predict and loss_function. Make sure to use vectorized operations to make the code faster. Remember, regularization_term is the λ\lambda in the cost function.

Unlike Lasso Regression, the loss function in Ridge Regression is the sum of the squares of the errors plus the sum of the squares of the weights.

def predict(intercept: float, coefficient: list, data: list) -> list:
    return intercept + np.dot(data, coefficient)
 
def loss_function(coefficients, errors, regularization_term):
    return np.mean(np.square(errors)) + regularization_term * np.sum(np.square(coefficients))

We also need a function called soft_threshold to update the coefficients. There are three conditions:

  1. If the coefficient is less than the negative of the regularization term, then we subtract the regularization term from the coefficient.
  2. If the coefficient is greater than the regularization term, then we subtract the regularization term from the coefficient.
  3. If the coefficient is between the negative and positive regularization term, then we set the coefficient to zero.
def soft_threshold(rho, lambda_):
    if rho < -lambda_:
        return (rho + lambda_)
    elif rho > lambda_:
        return (rho - lambda_)
    else:
        return 0
def ridge_regression(
    x, y,
    epochs,
    learning_rate = 0.1, 
    regularization_term = 0.001
):
    intercept, coefficients = 0, np.zeros(x.shape[1])
    length = x.shape[0]
 
    intercept_history.append(intercept)
    coefficients_history[:, 0] = coefficients
    loss_history.append(loss_function(coefficients, y, regularization_term))
 
    for i in range(1, epochs):
        predictions = predict(intercept, coefficients, x)
        errors = predictions - y
        intercept = intercept - learning_rate * np.sum(errors) / length
        intercept_history.append(intercept)
        
        for j in range(len(coefficients)):
            gradient = np.dot(x[:, j], errors) / length
            temp_coef = coefficients[j] - learning_rate * gradient
            coefficients[j] = soft_threshold(temp_coef, regularization_term)
            coefficients_history[j, i] = coefficients[j]
        
        loss_history.append(
            loss_function(
                coefficients, 
                errors, 
                regularization_term
            )
        )
 
    return intercept, coefficients
 
intercept, coefficients = ridge_regression(scaled_X, data.target, epochs)

Model Comparison

BaselineOurs
MSE2900.072878.51
age1.75-0.35
sex-11.51-11.29
body_mass_index25.6124.77
blood_pressure16.8315.31
serum_cholesterol-44.32-28.79
ldl24.5415.84
hdl7.620.56
cholesterol_ratio13.126.86
triglycerides35.1132.53
blood_sugar2.353.19

By comparing the baseline model and our model, you would be able to see the noticeable differences in the coefficients of the features. However, the Mean Squared Error values differ only by a small amount.

Changes in coefficients over time without regularization
Changes in coefficients over time without regularization

Changes in coefficients over time with regularization
Changes in coefficients over time with regularization

Conclusion

Here are the key takeaways from this post:

  1. Ridge Regression is a type of linear regression that uses a regularization term to prevent overfitting.
  2. It uses the sum of the squares of the weights multiplied by the regularization term to minimize coefficients.
  3. It will set coefficients close to zero but not exactly zero.
  4. It can be used to select important features in the dataset, just like Lasso Regression.
  5. It can increase the model's interpretability, just like Lasso Regression.

For the baseline model, you could see the code here. For own custom Lasso Regression model, you could see the code here.

Reference

  1. JMP. What is Multicollinearity? https://www.jmp.com/en_is/statistics-knowledge-portal/what-is-multiple-regression/multicollinearity.html
  2. Scikit-Learn. California Housing Dataset. https://scikit-learn.org/stable/datasets/real_world.html#california-housing-dataset
  3. StatsModel. Variance Inflation Factor. https://www.statsmodels.org/stable/generated/statsmodels.stats.outliers_influence.variance_inflation_factor.html
  4. NBSHARE. How to Use Pandas Correlation Matrix. https://www.nbshare.io/notebook/394171759/How-To-Use-Pandas-Correlation-Matrix/