Contributed by: Prashanth Ashok
What is Ridge regression?
Ridge regression is a model-tuning method that is used to analyze any data that suffers from multicollinearity. This method performs L2 regularization. When the issue of multicollinearity occurs, least-squares are unbiased, and variances are large, resulting in predicted values being far away from the actual values.
The cost function for ridge regression:
Min(||Y – X(theta)||^2 + λ||theta||^2)
Lambda is the penalty term. λ given here is denoted by an alpha parameter in the ridge function. By changing the values of alpha, we are controlling the penalty term. The higher the values of alpha, the bigger is the penalty and therefore the magnitude of coefficients is reduced. It shrinks the parameters. Therefore, it is used to prevent multicollinearity and reduces the model complexity by coefficient shrinkage.
Check out the free course on regression analysis. Ridge Regression Models
For any type of regression machine learning model, the usual regression equation forms the base which is written as:
Y = XB + e
Where Y is the dependent variable, X represents the independent variables, B is the regression coefficients to be estimated, and e represents the errors are residuals.
Once we add the lambda function to this equation, the variance that is not evaluated by the general model is considered. After the data is ready and identified to be part of L2 regularization, there are steps that one can undertake.
Standardization
In ridge regression, the first step is to standardize the variables (both dependent and independent) by subtracting their means and dividing by their standard deviations. This causes a challenge in notation since we must somehow indicate whether the variables in a particular formula are standardized or not. As far as standardization is concerned, all ridge regression calculations are based on standardized variables. When the final regression coefficients are displayed, they are adjusted back into their original scale. However, the ridge trace is on a standardized scale.
Bias and variance trade-off
Bias and variance trade-off is generally complicated when it comes to building ridge regression models on an actual dataset. However, following the general trend which one needs to remember is:
- The bias increases as λ increases.
- The variance decreases as λ increases.
Assumptions of Ridge Regressions
The assumptions of ridge regression are the same as those of linear regression: linearity, constant variance, and independence. However, as ridge regression does not provide confidence limits, the distribution of errors to be normal need not be assumed. Now, let’s take an example of a linear regression problem and see how ridge regression if implemented, helps us to reduce the error. We shall consider a data set on Food restaurants trying to find the best combination of food items to improve their sales in a particular region.
Upload Required Libraries
import numpy as np
import pandas as pd
import os
import seaborn as sns
from sklearn.linear_model import LinearRegression
import matplotlib.pyplot as plt
import matplotlib.style plt.style.use('classic')
import warnings
warnings.filterwarnings("ignore")
df = pd.read_excel("food.xlsx")
After conducting all the EDA on the data, and treatment of missing values, we shall now go ahead with creating dummy variables, as we cannot have categorical variables in the dataset.
df =pd.get_dummies(df, columns=cat,drop_first=True)
Where columns=cat is all the categorical variables in the data set.
After this, we need to standardize the data set for the Linear Regression method.
Scaling the variables
#Scales the data. Essentially returns the z-scores of every attribute
from sklearn.preprocessing import StandardScaler
std_scale = StandardScaler()
std_scale df['week'] = std_scale.fit_transform(df[['week']])
df['final_price'] = std_scale.fit_transform(df[['final_price']])
df['area_range'] = std_scale.fit_transform(df[['area_range']])
Train-Test Split
# Copy all the predictor variables into X dataframe
X = df.drop('orders', axis=1)
# Copy target into the y dataframe. Target variable is converted in to Log.
y = np.log(df[['orders']])
# Split X and y into training and test set in 75:25 ratio
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25 , random_state=1)
Linear Regression Model
# invoke the LinearRegression function and find the bestfit model on training data
regression_model = LinearRegression()
regression_model.fit(X_train, y_train)
# Let us explore the coefficients for each of the independent attributes
for idx, col_name in enumerate(X_train.columns):
print("The coefficient for {} is {}".format(col_name, regression_model.coef_[0][idx]))
The coefficient for week is -0.0041068045722690814
The coefficient for final_price is -0.40354286519747384
The coefficient for area_range is 0.16906454326841025
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The coefficient for home_delivery_1.0 is 1.026400462237632
The coefficient for night_service_1 is 0.0038398863634691582
#checking the magnitude of coefficients
from pandas import Series, DataFrame
predictors = X_train.columns
coef = Series(regression_model.coef_.flatten(), predictors).sort_values()
plt.figure(figsize=(10,8))
coef.plot(kind='bar', title="Model Coefficients")
plt.show()
Variables showing Positive effect on regression model are food_category_Rice Bowl, home_delivery_1.0, food_category_Desert, food_category_Pizza, website_homepage_mention_1.0, food_category_Sandwich, food_category_Salad and area_range – these factors highly influencing our model.
Difference Between Ridge Regression Vs Lasso Regression
| Aspect | Ridge Regression | Lasso Regression |
|---|---|---|
| Regularization Approach | Adds penalty term proportional to square of coefficients | Adds penalty term proportional to absolute value of coefficients |
| Coefficient Shrinkage | Coefficients shrink towards but never exactly to zero | Some coefficients can be reduced exactly to zero |
| Effect on Model Complexity | Reduces model complexity and multicollinearity | Results in simpler, more interpretable models |
| Handling Correlated Inputs | Handles correlated inputs effectively | Can be inconsistent with highly correlated features |
| Feature Selection Capability | Limited | Performs feature selection by reducing some coefficients to zero |
| Preferred Usage Scenarios | All features assumed relevant or dataset has multicollinearity | When parsimony is advantageous, especially in high-dimensional datasets |
Ridge Regression in Machine Learning
Ridge regression is a key technique in machine learning, indispensable for creating robust models in scenarios prone to overfitting and multicollinearity. This method modifies standard linear regression by introducing a penalty term proportional to the square of the coefficients, which proves particularly useful when dealing with highly correlated independent variables. Among its primary benefits, ridge regression effectively reduces overfitting through added complexity penalties, manages multicollinearity by balancing effects among correlated variables, and enhances model generalization to improve performance on unseen data.
The implementation of ridge regression in practical settings involves the crucial step of selecting the right regularization parameter, commonly known as lambda. This selection, typically done using cross-validation techniques, is vital for balancing the bias-variance tradeoff inherent in model training. Ridge regression enjoys widespread support across various machine learning libraries, with Python’s scikit-learn being a notable example. Here, implementation entails defining the model, setting the lambda value, and employing built-in functions for fitting and predictions. Its utility is particularly notable in sectors like finance and healthcare analytics, where precise predictions and robust model construction are paramount. Ultimately, ridge regression’s capacity to improve accuracy and handle complex datasets solidifies its ongoing importance in the dynamic field of machine learning.
The higher the value of the beta coefficient, the higher is the impact. Dishes like Rice Bowl, Pizza, Desert with a facility like home delivery and website_homepage_mention plays an important role in demand or number of orders being placed in high frequency.
Variables showing negative effect on the regression model for predicting restaurant orders: cuisine_Indian, food_category_Soup, food_category_Pasta, food_category_Other_Snacks. Final_price has a negative effect on the order – as expected. Dishes like Soup, Pasta, other_snacks, Indian food categories hurt model prediction on the number of orders being placed at restaurants, keeping all other predictors constant.
Some variables which are hardly affecting model prediction for order frequency are week and night_service. Through the model, we are able to see object types of variables or categorical variables are more significant than continuous variables.
Regularization
Value of alpha, which is a hyperparameter of Ridge, which means that they are not automatically learned by the model instead they have to be set manually. We run a grid search for optimum alpha values to find optimum alpha for…
