Linear Regression in Machine learning (original) (raw)

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It provides valuable insights for prediction and data analysis. This article will explore its types, assumptions, implementation, advantages and evaluation metrics.

**Understanding Linear Regression

**Linear regression is also a type of **supervised machine-learning algorithm that learns from the labelled datasets and maps the data points with most optimized linear functions which can be used for prediction on new datasets. It computes the linear relationship between the dependent variable and one or more independent features by fitting a linear equation with observed data. It predicts the continuous output variables based on the independent input variable.

For example if we want to predict house price we consider various factor such as house age, distance from the main road, location, area and number of room, linear regression uses all these parameter to predict house price as it consider a linear relation between all these features and price of house.

Why Linear Regression is Important?

The interpretability of linear regression is one of its greatest strengths. The model’s equation offers clear coefficients that illustrate the influence of each independent variable on the dependent variable, enhancing our understanding of the underlying relationships. Its simplicity is a significant advantage; linear regression is transparent, easy to implement, and serves as a foundational concept for more advanced algorithms.

Now that we have discussed why linear regression is important now we will discuss its working based on best fit line in regression.

What is the best Fit Line?

Our primary objective while using linear regression is to locate the best-fit line, which implies that the error between the predicted and actual values should be kept to a minimum. There will be the least error in the best-fit line.

The best Fit Line equation provides a straight line that represents the relationship between the dependent and independent variables. The slope of the line indicates how much the dependent variable changes for a unit change in the independent variable(s).

Linear Regression in Machine learning

Linear Regression

Here Y is called a dependent or target variable and X is called an independent variable also known as the predictor of Y. There are many types of functions or modules that can be used for regression. A linear function is the simplest type of function. Here, X may be a single feature or multiple features representing the problem.

Linear regression performs the task to predict a dependent variable value (y) based on a given independent variable (x)). Hence, the name is Linear Regression. In the figure above, X (input) is the work experience and Y (output) is the salary of a person. The regression line is the best-fit line for our model.

In linear regression some hypothesis are made to ensure reliability of the model’s results.

**Hypothesis function in Linear Regression

Assumptions are:

As we have discussed that our independent feature is the experience i.e X and the respective salary Y is the dependent variable. Let’s assume there is a linear relationship between X and Y then the salary can be predicted using:

[Tex]\hat{Y} = \theta_1 + \theta_2X [/Tex]

OR

[Tex]\hat{y}_i = \theta_1 + \theta_2x_i[/Tex]

Here,

The model gets the best regression fit line by finding the best θ1 and θ2 values.

Once we find the best θ1 and θ2 values, we get the best-fit line. So when we are finally using our model for prediction, it will predict the value of y for the input value of x.

**How to update θ 1 and θ 2 values to get the best-fit line?

To achieve the best-fit regression line, the model aims to predict the target value [Tex]\hat{Y} [/Tex] such that the error difference between the predicted value [Tex]\hat{Y} [/Tex] and the true value Y is minimum. So, it is very important to update the θ1 and θ2 values, to reach the best value that minimizes the error between the predicted y value (pred) and the true y value (y).

[Tex]minimize\frac{1}{n}\sum_{i=1}^{n}(\hat{y_i}-y_i)^2[/Tex]

Types of Linear Regression

When there is only one independent feature it is known as **Simple Linear Regression or Univariate Linear Regression and when there are more than one feature it is known as **Multiple Linear Regression orMultivariate Regression.

**1. Simple Linear Regression

Simple linear regression is the simplest form of linear regression and it involves only one independent variable and one dependent variable. The equation for simple linear regression is:
[Tex]y=\beta_{0}+\beta_{1}X[/Tex]
where:

Assumptions of Simple Linear Regression

Linear regression is a powerful tool for understanding and predicting the behavior of a variable, however, it needs to meet a few conditions in order to be accurate and dependable solutions.

  1. **Linearity: The independent and dependent variables have a linear relationship with one another. This implies that changes in the dependent variable follow those in the independent variable(s) in a linear fashion. This means that there should be a straight line that can be drawn through the data points. If the relationship is not linear, then linear regression will not be an accurate model.
  2. **Independence: The observations in the dataset are independent of each other. This means that the value of the dependent variable for one observation does not depend on the value of the dependent variable for another observation. If the observations are not independent, then linear regression will not be an accurate model.
  3. **Homoscedasticity: Across all levels of the independent variable(s), the variance of the errors is constant. This indicates that the amount of the independent variable(s) has no impact on the variance of the errors. If the variance of the residuals is not constant, then linear regression will not be an accurate model.

    Homoscedasticity in Linear Regression
  4. **Normality: The residuals should be normally distributed. This means that the residuals should follow a bell-shaped curve. If the residuals are not normally distributed, then linear regression will not be an accurate model.

Use Case of Simple Linear Regression

**2. Multiple Linear Regression

Multiple linear regression involves more than one independent variable and one dependent variable. The equation for multiple linear regression is:
[Tex]y=\beta_{0}+\beta_{1}X1+\beta_{2}X2+………\beta_{n}Xn[/Tex]
where:

The goal of the algorithm is to find the **best Fit Line equation that can predict the values based on the independent variables.

In regression set of records are present with X and Y values and these values are used to learn a function so if you want to predict Y from an unknown X this learned function can be used. In regression we have to find the value of Y, So, a function is required that predicts continuous Y in the case of regression given X as independent features.

Assumptions of Multiple Linear Regression

For Multiple Linear Regression, all four of the assumptions from Simple Linear Regression apply. In addition to this, below are few more:

  1. **No multicollinearity: There is no high correlation between the independent variables. This indicates that there is little or no correlation between the independent variables. Multicollinearity occurs when two or more independent variables are highly correlated with each other, which can make it difficult to determine the individual effect of each variable on the dependent variable. If there is multicollinearity, then multiple linear regression will not be an accurate model.
  2. **Additivity: The model assumes that the effect of changes in a predictor variable on the response variable is consistent regardless of the values of the other variables. This assumption implies that there is no interaction between variables in their effects on the dependent variable.
  3. **Feature Selection: In multiple linear regression, it is essential to carefully select the independent variables that will be included in the model. Including irrelevant or redundant variables may lead to overfitting and complicate the interpretation of the model.
  4. **Overfitting: Overfitting occurs when the model fits the training data too closely, capturing noise or random fluctuations that do not represent the true underlying relationship between variables. This can lead to poor generalization performance on new, unseen data.

Multiple linear regression sometimes faces issues like multicollinearity.

**Multicollinearity

Multicollinearity is a statistical phenomenon where two or more independent variables in a multiple regression model are highly correlated, making it difficult to assess the individual effects of each variable on the dependent variable.

**Detecting Multicollinearity includes two techniques:

Use Case of Multiple Linear Regression

Multiple linear regression allows us to analyze relationship between multiple independent variables and a single dependent variable. Here are some use cases:

Now that we have understood about linear regression, its assumption and its type now we will learn how to make a linear regression model.

Cost function for Linear Regression

As we have discussed earlier about best fit line in linear regression, its not easy to get it easily in real life cases so we need to calculate errors that affects it. These errors need to be calculated to mitigate them. The difference between the predicted value [Tex]\hat{Y} [/Tex]and the true value Y and it is called cost function or the loss function.

In Linear Regression, the **Mean Squared Error (MSE) cost function is employed, which calculates the average of the squared errors between the predicted values [Tex]\hat{y}_i[/Tex] and the actual values [Tex]{y}_i[/Tex]. The purpose is to determine the optimal values for the intercept [Tex]\theta_1[/Tex] and the coefficient of the input feature [Tex]\theta_2[/Tex] providing the best-fit line for the given data points. The linear equation expressing this relationship is [Tex]\hat{y}_i = \theta_1 + \theta_2x_i[/Tex].

MSE function can be calculated as:

[Tex]\text{Cost function}(J) = \frac{1}{n}\sum_{n}^{i}(\hat{y_i}-y_i)^2[/Tex]

Utilizing the MSE function, the iterative process of gradient descent is applied to update the values of \[Tex]\theta_1 \& \theta_2 [/Tex]. This ensures that the MSE value converges to the global minima, signifying the most accurate fit of the linear regression line to the dataset.

This process involves continuously adjusting the parameters \(\theta_1\) and \(\theta_2\) based on the gradients calculated from the MSE. The final result is a linear regression line that minimizes the overall squared differences between the predicted and actual values, providing an optimal representation of the underlying relationship in the data.

Now we have calculated loss function we need to optimize model to mtigate this error and it is done through gradient descent.

**Gradient Descent for Linear Regression

A linear regression model can be trained using the optimization algorithm gradient descent by iteratively modifying the model’s parameters to reduce the mean squared error (MSE) of the model on a training dataset. To update θ1 and θ2 values in order to reduce the Cost function (minimizing RMSE value) and achieve the best-fit line the model uses Gradient Descent. The idea is to start with random θ1 and θ2 values and then iteratively update the values, reaching minimum cost.

A gradient is nothing but a derivative that defines the effects on outputs of the function with a little bit of variation in inputs.

Let’s differentiate the cost function(J) with respect to [Tex]\theta_1 [/Tex]

[Tex]\begin {aligned} {J}’_{\theta_1} &=\frac{\partial J(\theta_1,\theta_2)}{\partial \theta_1} \\ &= \frac{\partial}{\partial \theta_1} \left[\frac{1}{n} \left(\sum_{i=1}^{n}(\hat{y}_i-y_i)^2 \right )\right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(\frac{\partial}{\partial \theta_1}(\hat{y}_i-y_i) \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(\frac{\partial}{\partial \theta_1}( \theta_1 + \theta_2x_i-y_i) \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(1+0-0 \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}(\hat{y}_i-y_i) \left(2 \right ) \right] \\ &= \frac{2}{n}\sum_{i=1}^{n}(\hat{y}_i-y_i) \end {aligned}[/Tex]

Let’s differentiate the cost function(J) with respect to [Tex]\theta_2[/Tex]

[Tex]\begin {aligned} {J}’_{\theta_2} &=\frac{\partial J(\theta_1,\theta_2)}{\partial \theta_2} \\ &= \frac{\partial}{\partial \theta_2} \left[\frac{1}{n} \left(\sum_{i=1}^{n}(\hat{y}_i-y_i)^2 \right )\right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(\frac{\partial}{\partial \theta_2}(\hat{y}_i-y_i) \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(\frac{\partial}{\partial \theta_2}( \theta_1 + \theta_2x_i-y_i) \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}2(\hat{y}_i-y_i) \left(0+x_i-0 \right ) \right] \\ &= \frac{1}{n}\left[\sum_{i=1}^{n}(\hat{y}_i-y_i) \left(2x_i \right ) \right] \\ &= \frac{2}{n}\sum_{i=1}^{n}(\hat{y}_i-y_i)\cdot x_i \end {aligned}[/Tex]

Finding the coefficients of a linear equation that best fits the training data is the objective of linear regression. By moving in the direction of the Mean Squared Error negative gradient with respect to the coefficients, the coefficients can be changed. And the respective intercept and coefficient of X will be if [Tex]\alpha [/Tex] is the learning rate.

Gradient Descent -Geeksforgeeks

Gradient Descent

[Tex]\begin{aligned} \theta_1 &= \theta_1 – \alpha \left( {J}’_{\theta_1}\right) \\&=\theta_1 -\alpha \left( \frac{2}{n}\sum_{i=1}^{n}(\hat{y}_i-y_i)\right) \end{aligned} \\ \begin{aligned} \theta_2 &= \theta_2 – \alpha \left({J}’_{\theta_2}\right) \\&=\theta_2 – \alpha \left(\frac{2}{n}\sum_{i=1}^{n}(\hat{y}_i-y_i)\cdot x_i\right) \end{aligned}[/Tex]

After optimizing our model, we evaluate our models accuracy to see how well it will perform in real world scenario.

Evaluation Metrics for Linear Regression

A variety of evaluation measures can be used to determine the strength of any linear regression model. These assessment metrics often give an indication of how well the model is producing the observed outputs.

The most common measurements are:

Mean Square Error (MSE)

Mean Squared Error (MSE) is an evaluation metric that calculates the average of the squared differences between the actual and predicted values for all the data points. The difference is squared to ensure that negative and positive differences don’t cancel each other out.

[Tex]MSE = \frac{1}{n}\sum_{i=1}^{n}\left ( y_i – \widehat{y_{i}} \right )^2[/Tex]

Here,

MSE is a way to quantify the accuracy of a model’s predictions. MSE is sensitive to outliers as large errors contribute significantly to the overall score.

Mean Absolute Error (MAE)

Mean Absolute Error is an evaluation metric used to calculate the accuracy of a regression model. MAE measures the average absolute difference between the predicted values and actual values.

Mathematically, MAE is expressed as:

[Tex]MAE =\frac{1}{n} \sum_{i=1}^{n}|Y_i – \widehat{Y_i}|[/Tex]

Here,

Lower MAE value indicates better model performance. It is not sensitive to the outliers as we consider absolute differences.

**Root Mean Squared Error (RMSE)

The square root of the residuals’ variance is the Root Mean Squared Error. It describes how well the observed data points match the expected values, or the model’s absolute fit to the data.

In mathematical notation, it can be expressed as:
[Tex]RMSE=\sqrt{\frac{RSS}{n}}=\sqrt\frac{{{\sum_{i=2}^{n}(y^{actual}_{i}}- y_{i}^{predicted})^2}}{n}[/Tex]
Rather than dividing the entire number of data points in the model by the number of degrees of freedom, one must divide the sum of the squared residuals to obtain an unbiased estimate. Then, this figure is referred to as the Residual Standard Error (RSE).

In mathematical notation, it can be expressed as:
[Tex]RMSE=\sqrt{\frac{RSS}{n}}=\sqrt\frac{{{\sum_{i=2}^{n}(y^{actual}_{i}}- y_{i}^{predicted})^2}}{(n-2)}[/Tex]

RSME is not as good of a metric as R-squared. Root Mean Squared Error can fluctuate when the units of the variables vary since its value is dependent on the variables’ units (it is not a normalized measure).

Coefficient of Determination (R-squared)

R-Squared is a statistic that indicates how much variation the developed model can explain or capture. It is always in the range of 0 to 1. In general, the better the model matches the data, the greater the R-squared number.
In mathematical notation, it can be expressed as:
[Tex]R^{2}=1-(^{\frac{RSS}{TSS}})[/Tex]

R squared metric is a measure of the proportion of variance in the dependent variable that is explained the independent variables in the model.

Adjusted R-Squared Error

Adjusted R2 measures the proportion of variance in the dependent variable that is explained by independent variables in a regression model. Adjusted R-square accounts the number of predictors in the model and penalizes the model for including irrelevant predictors that don’t contribute significantly to explain the variance in the dependent variables.

Mathematically, adjusted R2 is expressed as:

[Tex]Adjusted \, R^2 = 1 – (\frac{(1-R^2).(n-1)}{n-k-1})[/Tex]

Here,

Adjusted R-square helps to prevent overfitting. It penalizes the model with additional predictors that do not contribute significantly to explain the variance in the dependent variable.

While evaluation metrics help us measure the performance of a model, regularization helps in improving that performance by addressing overfitting and enhancing generalization.

Regularization Techniques for Linear Models

Lasso Regression (L1 Regularization)

Lasso Regression is a technique used for regularizing a linear regression model, it adds a penalty term to the linear regression objective function to prevent overfitting.

The objective function after applying lasso regression is:

[Tex]J(\theta) = \frac{1}{2m} \sum_{i=1}^{m}(\widehat{y_i} – y_i) ^2+ \lambda \sum_{j=1}^{n}|\theta_j|[/Tex]

Ridge Regression (L2 Regularization)

Ridge regression is a linear regression technique that adds a regularization term to the standard linear objective. Again, the goal is to prevent overfitting by penalizing large coefficient in linear regression equation. It useful when the dataset has multicollinearity where predictor variables are highly correlated.

The objective function after applying ridge regression is:

[Tex]J(\theta) = \frac{1}{2m} \sum_{i=1}^{m}(\widehat{y_i} – y_i)^2 + \lambda \sum_{j=1}^{n}\theta_{j}^{2}[/Tex]

Elastic Net Regression

Elastic Net Regression is a hybrid regularization technique that combines the power of both L1 and L2 regularization in linear regression objective.

[Tex]J(\theta) = \frac{1}{2m} \sum_{i=1}^{m}(\widehat{y_i} – y_i)^2 + \alpha \lambda \sum_{j=1}^{n}{|\theta_j|} + \frac{1}{2}(1- \alpha) \lambda \sum_{j=1}{n} \theta_{j}^{2}[/Tex]

Now that we have learned how to make a linear regression model, now we will implement it.

Python Implementation of Linear Regression

Import the necessary libraries:

Python `

import pandas as pd import numpy as np import matplotlib.pyplot as plt import matplotlib.axes as ax from matplotlib.animation import FuncAnimation

`

Load the dataset and separate input and Target variables

Here is the link for dataset: Dataset Link

Python `

url = 'https://media.geeksforgeeks.org/wp-content/uploads/20240320114716/data_for_lr.csv' data = pd.read_csv(url) data

Drop the missing values

data = data.dropna()

training dataset and labels

train_input = np.array(data.x[0:500]).reshape(500, 1) train_output = np.array(data.y[0:500]).reshape(500, 1)

valid dataset and labels

test_input = np.array(data.x[500:700]).reshape(199, 1) test_output = np.array(data.y[500:700]).reshape(199, 1)

`

Build the Linear Regression Model and Plot the regression line

Steps:

class LinearRegression: def init(self): self.parameters = {} 

def forward_propagation(self, train_input): 
    m = self.parameters['m'] 
    c = self.parameters['c'] 
    predictions = np.multiply(m, train_input) + c 
    return predictions 

def cost_function(self, predictions, train_output): 
    cost = np.mean((train_output - predictions) ** 2) 
    return cost 

def backward_propagation(self, train_input, train_output, predictions): 
    derivatives = {} 
    df = (predictions-train_output) 
    # dm= 2/n * mean of (predictions-actual) * input 
    dm = 2 * np.mean(np.multiply(train_input, df)) 
    # dc = 2/n * mean of (predictions-actual) 
    dc = 2 * np.mean(df) 
    derivatives['dm'] = dm 
    derivatives['dc'] = dc 
    return derivatives 

def update_parameters(self, derivatives, learning_rate): 
    self.parameters['m'] = self.parameters['m'] - learning_rate * derivatives['dm'] 
    self.parameters['c'] = self.parameters['c'] - learning_rate * derivatives['dc'] 

def train(self, train_input, train_output, learning_rate, iters): 
    # Initialize random parameters 
    self.parameters['m'] = np.random.uniform(0, 1) * -1
    self.parameters['c'] = np.random.uniform(0, 1) * -1

    # Initialize loss 
    self.loss = [] 

    # Initialize figure and axis for animation 
    fig, ax = plt.subplots() 
    x_vals = np.linspace(min(train_input), max(train_input), 100) 
    line, = ax.plot(x_vals, self.parameters['m'] * x_vals +
                    self.parameters['c'], color='red', label='Regression Line') 
    ax.scatter(train_input, train_output, marker='o', 
            color='green', label='Training Data') 

    # Set y-axis limits to exclude negative values 
    ax.set_ylim(0, max(train_output) + 1) 

    def update(frame): 
        # Forward propagation 
        predictions = self.forward_propagation(train_input) 

        # Cost function 
        cost = self.cost_function(predictions, train_output) 

        # Back propagation 
        derivatives = self.backward_propagation( 
            train_input, train_output, predictions) 

        # Update parameters 
        self.update_parameters(derivatives, learning_rate) 

        # Update the regression line 
        line.set_ydata(self.parameters['m'] 
                    * x_vals + self.parameters['c']) 

        # Append loss and print 
        self.loss.append(cost) 
        print("Iteration = {}, Loss = {}".format(frame + 1, cost)) 

        return line, 
    # Create animation 
    ani = FuncAnimation(fig, update, frames=iters, interval=200, blit=True) 

    # Save the animation as a video file (e.g., MP4) 
    ani.save('linear_regression_A.gif', writer='ffmpeg') 

    plt.xlabel('Input') 
    plt.ylabel('Output') 
    plt.title('Linear Regression') 
    plt.legend() 
    plt.show() 

    return self.parameters, self.loss

`

Trained the model and Final Prediction

Python `

#Example usage linear_reg = LinearRegression() parameters, loss = linear_reg.train(train_input, train_output, 0.0001, 20)

`

**Output:

Iteration = 1, Loss = 9130.407560462196
Iteration = 1, Loss = 1107.1996742908998
Iteration = 1, Loss = 140.31580932842422
Iteration = 1, Loss = 23.795780526084116
Iteration = 2, Loss = 9.753848205147605
Iteration = 3, Loss = 8.061641745006835
Iteration = 4, Loss = 7.8577116490914864
Iteration = 5, Loss = 7.8331350515579015
Iteration = 6, Loss = 7.830172502503967
Iteration = 7, Loss = 7.829814681591015
Iteration = 8, Loss = 7.829770758846183
Iteration = 9, Loss = 7.829764664327399
Iteration = 10, Loss = 7.829763128602258
Iteration = 11, Loss = 7.829762142342088
Iteration = 12, Loss = 7.829761222379141
Iteration = 13, Loss = 7.829760310486438
Iteration = 14, Loss = 7.829759399646989
Iteration = 15, Loss = 7.829758489015161
Iteration = 16, Loss = 7.829757578489033
Iteration = 17, Loss = 7.829756668056319
Iteration = 18, Loss = 7.829755757715535
Iteration = 19, Loss = 7.829754847466484
Iteration = 20, Loss = 7.829753937309139

ezgif-1-ba0c9540c5

The linear regression line provides valuable insights into the relationship between the two variables. It represents the best-fitting line that captures the overall trend of how a dependent variable (Y) changes in response to variations in an independent variable (X).

Applications of Linear Regression

Linear regression is used in many different fields including finance, economics and psychology to understand and predict the behavior of a particular variable.

For example linear regression is widely used in finance to analyze relationships and make predictions. It can model how a company’s earnings per share (EPS) influence its stock price. If the model shows that a 1increaseinEPSresultsina1 increase in EPS results in a 1increaseinEPSresultsina15 rise in stock price, investors gain insights into the company’s valuation. Similarly, linear regression can forecast currency values by analyzing historical exchange rates and economic indicators, helping financial professionals make informed decisions and manage risks effectively.

Also read – Linear Regression – In Simple Words, with real-life Examples

Advantages and Disadvantages of Linear Regression

Advantages of Linear Regression

Disadvantages of Linear Regression

Conclusion

Linear regression is a fundamental machine learning algorithm that has been widely used for many years due to its simplicity, interpretability, and efficiency. It is a valuable tool for understanding relationships between variables and making predictions in a variety of applications.

However, it is important to be aware of its limitations, such as its assumption of linearity and sensitivity to multicollinearity. When these limitations are carefully considered, linear regression can be a powerful tool for data analysis and prediction.