What’s Inside:
- Linear Regression
- Key Concepts
- Working of Linear Regression
- Prediction Process
- Ridge Regression
- Overview of L2 Regularization
- Working of Ridge Regression
- When to Use Ridge Regression
- Lasso Regression
- Overview of L1 Regularization
- Working of Lasso Regression
- When to Use Lasso Regression
- Comparison: Linear vs. Ridge vs. Lasso
- When to Use Which Regression Technique?
- What Makes a Good Model?
- Conclusion
1. Linear Regression
Linear Regression is a fundamental algorithm in machine learning used to predict continuous values. It assumes a linear relationship between the independent (input) variables and the dependent (output) variable.
Key Concept:
Linear regression tries to model the relationship between a set of input variables (features) and a continuous target variable by fitting a straight line. The line is drawn in such a way that it minimizes the error (or loss) between the predicted values and the actual values.
Working of Linear Regression:
- Cost Function: During the training phase, a cost function is used to evaluate how well the model is performing. The cost function measures the difference between the predicted values and the actual values. The most common cost function in linear regression is the Mean Squared Error (MSE).
- $$MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y_i})^2$$
- Gradient Descent: To minimize this cost, we use an optimization technique called Gradient Descent. The algorithm adjusts the model’s parameters (coefficients) to reduce the error iteratively. Initially, the model’s parameters are set to random values, and then the parameters are updated in each iteration by moving in the direction that reduces the cost function.
Prediction Process:
After training the model using the data, the prediction of new data points is calculated using the following formula:
$$y^=θ0+θ1×1+θ2×2+⋯+θnxn\hat{y} = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \dots + \theta_n x_n$$
Where:
- y^\hat{y} is the predicted value,
- θ0\theta_0 is the bias (intercept),
- θ1,θ2,…,θn\theta_1, \theta_2, \dots, \theta_n are the coefficients (weights) for each feature x1,x2,…,xnx_1, x_2, \dots, x_n.
2. Ridge Regression
Ridge Regression is a variant of Linear Regression that incorporates L2 regularization to address issues like overfitting and multicollinearity (when features are highly correlated).
Working of Ridge Regression:
- Cost Function: In ridge regression, we modify the cost function of linear regression by adding a penalty term. The penalty term adds the squared sum of the coefficients multiplied by a regularization parameter λ\lambda. This regularization term discourages large coefficients, thereby reducing overfitting.
Cost Function:
$$Ridge = \text{MSE} + \lambda \sum_{i=1}^{n} \theta_i^2$$
Where:
- λ\lambda is the regularization parameter (also known as the ridge parameter),
- θi\theta_i are the coefficients.
Ridge regression is typically used when there is multicollinearity in the dataset, as it helps to stabilize the regression by shrinking the coefficients of less important features.
When to Use Ridge Regression:
- Use ridge regression when features are highly correlated with each other, and the model needs to assign less importance to correlated features while retaining their presence.
3. Lasso Regression
Lasso Regression, like ridge regression, adds a regularization term to the cost function but uses L1 regularization instead of L2. This leads to a more sparse solution where some of the coefficients can become exactly zero.
Working of Lasso Regression:
- Cost Function: In lasso regression, the penalty term involves the absolute sum of the coefficients rather than their squared sum (as in ridge regression). This results in some coefficients being shrunk to exactly zero, effectively performing feature selection.
Cost Function:
$$Lasso = \text{MSE} + \lambda \sum_{i=1}^{n} |\theta_i|$$
Where:
- λ\lambda is the regularization parameter (also called the lasso parameter),
- θi\theta_i are the coefficients.
When to Use Lasso Regression:
- Feature Selection: Lasso regression is particularly useful when we want to select a subset of features from the dataset by forcing some coefficients to zero.
- High Multicollinearity: When there is high multicollinearity, lasso regression can be useful in reducing the influence of irrelevant features, thereby preventing overfitting.
4. Comparison: Linear vs. Ridge vs. Lasso
| Technique | Cost Function | Penalty | Use Case |
|---|---|---|---|
| Linear Regression | MSE | None | Used when the features are not correlated and overfitting is not a concern. |
| Ridge Regression | MSE | L2 Regularization (sum of squared coefficients) | Best when features are correlated (multicollinearity) and we want to reduce overfitting. |
| Lasso Regression | MSE | L1 Regularization (sum of absolute values of coefficients) | Used for feature selection, especially when we expect only a few features to be important. |
5. When to Use Which Regression Technique?
The choice between linear, ridge, and lasso regression depends on the dataset and the problem at hand:
- Simple Linear Regression:
- Use this when the relationship between the features and target variable is linear, and there is little or no multicollinearity between features.
- Ridge Regression:
- Use ridge regression when the dataset has multicollinearity, where multiple features are highly correlated. Ridge is preferred if you do not want to completely eliminate features but still wish to reduce their impact.
- Lasso Regression:
- Use lasso regression when you want to perform feature selection and eliminate irrelevant or redundant features. It works well when only a subset of features are truly informative, and the rest can be ignored.
6. What Makes a Good Model?
In machine learning, we often talk about bias and variance to evaluate the performance of a model:
- Bias refers to the error introduced by the model’s assumptions (e.g., assuming a linear relationship when the true relationship is more complex).
- Variance refers to the error introduced by the model’s sensitivity to small fluctuations in the training data.
A good model should ideally have low bias (it should fit the data well) and low variance (it should generalize well to unseen data). This is called a well-generalized model, and it performs well both on the training data and the testing data.
- Underfitting occurs when the model has high bias (too simple, e.g., using linear regression for non-linear data).
- Overfitting occurs when the model has high variance (too complex, e.g., using a very flexible model that learns the noise in the data).
A balance between bias and variance is crucial, and techniques like regularization (ridge and lasso) help control this balance by penalizing overly complex models.
Conclusion
- Linear Regression is a simple and effective method when there is a linear relationship between the features and the target.
- Ridge Regression is preferred when features are highly correlated, helping to control multicollinearity.
- Lasso Regression is valuable when you need feature selection by forcing irrelevant features’ coefficients to zero.
- Good model performance is achieved by minimizing both bias (training error) and variance (testing error), ensuring the model generalizes well to unseen data.
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