Linear Regression

The formula: $y^{\wedge} = wx+b$

The cost function, $MSE = J(w,b) = 1/N \sum_{i=1}^{n}(y_i-(wx_i+b))^2$

The derivatives for Gradient Descent:

$df/dw = 1/N \sum_{i=1}^{n}-2x_i(y_i-(wx_i+b))$

$df/db = 1/N \sum_{i=1}^{n}-2(y_i-(wx_i+b))$

import numpy as np

# Calculate the R2 score
def r2_score(y_true, y_pred):
    corr_matrix = np.corrcoef(y_true, y_pred)
    corr = corr_matrix[0, 1]
    return corr ** 2


class LinearRegression:
    def __init__(self, learning_rate=0.001, n_iters=1000):
        self.lr = learning_rate
        self.n_iters = n_iters
        self.weights = None
        self.bias = None

    def fit(self, X, y):
        n_samples, n_features = X.shape

        # init parameters
        self.weights = np.zeros(n_features) # this can be random as well
        self.bias = 0

        # gradient descent
        for _ in range(self.n_iters):
            y_predicted = np.dot(X, self.weights) + self.bias
            # compute gradients
            dw = (1 / n_samples) * np.dot(X.T, (y_predicted - y))
            db = (1 / n_samples) * np.sum(y_predicted - y)

            # update parameters
            self.weights -= self.lr * dw
            self.bias -= self.lr * db

    def predict(self, X):
        y_approximated = np.dot(X, self.weights) + self.bias
        return y_approximated


# Testing
if __name__ == "__main__":
    # Imports
    import matplotlib.pyplot as plt
    from sklearn.model_selection import train_test_split
    from sklearn import datasets

    def mean_squared_error(y_true, y_pred):
        return np.mean((y_true - y_pred) ** 2)

    # We are making the dataset to test out algorithm
    X, y = datasets.make_regression(
        n_samples=100, n_features=1, noise=20, random_state=4
    )

    X_train, X_test, y_train, y_test = train_test_split(
        X, y, test_size=0.2, random_state=1234
    )

    regressor = LinearRegression(learning_rate=0.01, n_iters=1000)
    regressor.fit(X_train, y_train)
    predictions = regressor.predict(X_test)

    mse = mean_squared_error(y_test, predictions)
    print("MSE:", mse)

    accu = r2_score(y_test, predictions)
    print("Accuracy:", accu)

    y_pred_line = regressor.predict(X)
    cmap = plt.get_cmap("viridis")
    fig = plt.figure(figsize=(4, 3))
    m1 = plt.scatter(X_train, y_train, color=cmap(0.9), s=10)
    m2 = plt.scatter(X_test, y_test, color=cmap(0.5), s=10)
    plt.plot(X, y_pred_line, color="black", linewidth=2, label="Prediction")
    plt.show()