# Linear Regression

## 1. Regression Model

In statistics, linear regression is a linear approach to modelling the relationship between a dependent variable and one or more independent variables. Let $x_1$ be the independent variable and $y$ be the dependent variable. We will define a linear relationship between these two variables as follows:

$y = \theta_0+\theta_1 x_1$

## 2. Define Loss Function

We will use the Mean Squared Error function.

$L = \frac{1}{n}\sum_{i=1}^n (y_{true}-y_{predicted})^2$

## 3. Utilize the Gradient Descent Algorithm

You might know that the partial derivative of a function at its minimum value is equal to 0. So gradient descent basically uses this concept to estimate the parameters or weights of our model by minimizing the loss function.

1. Initialize the weights, $\theta_0 = 0$and $\theta_1 =0$

2. Calculate the partial derivatives w.r.t. to $\theta_0$and $\theta_1$ $d_{\theta_0} = -\frac{2}{n} \sum_{i=1}^n(y_i - \bar{y_i}) \\ d_{\theta_1} = -\frac{2}{n} \sum_{i=1}^n(y_i - \bar{y_i}) \times x_i$

3. Update the weights $\theta_0 = \theta_0 - l \times d_{\theta_0} \\ \theta_1 = \theta_1 - l \times d_{\theta_1}$

## Python Implementation

# Importing librariesimport numpy as npimport pandas as pdfrom sklearn.model_selection import train_test_split​# Preparing the datasetdata = pd.DataFrame({'feature' : [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], 'label' : [2,4,6,8,10,12,14,16,18,20,22,24,26,28,30]})# Divide the data to training set and test setX_train, X_test, y_train, y_test = train_test_split(data['feature'], data['label'], test_size=0.30)​# Method to make predictionsdef predict(X, theta0, theta1):    # Here the predict function is: theta0+theta1*x    return np.array([(theta0 + theta1*x) for x in X])​def linear_regression(X, Y):    # Initializing variables    theta0 = 0    theta1 = 0    learning_rate = 0.001    epochs = 300    n = len(X)​    # Training iteration    for epoch in range(epochs):        y_pred = predict(X, theta0, theta1)​        ## Here the loss function is: 1/n*sum(y-y_pred)^2 a.k.a mean squared error (mse)        # Derivative of loss w.r.t. theta0        theta0_d = -(2/n) * sum(Y-y_pred)        # Derivative of loss w.r.t. theta1        theta1_d = -(2/n) * sum(X*(Y-y_pred))​        theta0 = theta0 - learning_rate * theta0_d        theta1 = theta1 - learning_rate * theta1_d   ​    return theta0, theta1​# Training the modeltheta0, theta1 = linear_regression(X_train, y_train)   ​# Making predictionsy_pred = predict(X_test, theta0, theta1)​# Evaluating the modelprint(list(y_test))print(y_pred)