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Best-fit Line

We find the best-fit line, using gradient descent optimization, in order to make predictions. The core part, Gradient Descent, is an iterative optimization algorithm. It that starts with an initial guess for the slope and the intercept and updates them. The params are updated in the direction of steepest descent of the cost function. The slope or gradient of a function in (x,y) point is the derivative. The cost function measures the error between the predicted and actual values. By iteratively updating the m and b values in the direction of the negative gradient, the algorithm finds the values that minimize the mean squared error.

 
import numpy as np
import matplotlib.pyplot as plt

# Input training dataset
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])

# Initialize variables: slope (m) and y-intercept (b)
m = 0
b = 0

# Set the learning rate and the number of iterations for gradient descent
learning_rate = 0.01
num_iterations = 1000

# Perform gradient descent to find the best-fit line
for i in range(num_iterations):
    
    # Calculate the predicted values of y (y_pred) based on the current m and b
    y_pred = m*x + b

    # Calculate the error between the predicted values and the actual values
    error = y - y_pred 

    # Calculate the derivatives of the cost function with respect to m and b
    m_derivative = -(2/len(x)) * sum(x * error)
    b_derivative = -(2/len(x)) * sum(error)

    # Update the values of m and b using the gradient descent algorithm
    m = m - learning_rate * m_derivative
    b = b - learning_rate * b_derivative

# Output the equation of the best-fit line
print(f'Best fit line for given data: y = {m}x + {b}')

# Round the values of m and b for clarity
m = round(m, 1)
b = round(b, 1)


# Create a plot to visualize the data and the best-fit line
fig, ax = plt.subplots()
plt.ylim(0, 10)
plt.xlim(0, 10)

# Plot the data points as 'x' markers in green
ax.plot(x,  y,  'x', color='g', label='Training data')
ax.plot(x, m*x + b,  label=f'h(x) = {m} + {b}x')
plt.legend()
plt.show()

Model Class

Encapsulate algorithm into a class that fits training data and make predictions.

 
import numpy as np
import matplotlib.pyplot as plt

# Training datasets
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])

class LinearRegression:

    def __init__(self):
        self.coef_ = []
        self.intercept_ = 0

    def fit(self, X_train, Y_train, learning_rate=0.01, num_iterations=1000):
        
        x = X_train
        y = Y_train
        m = 0 
        b = 0
        for i in range(num_iterations):
            y_pred = m*x + b
            error = y - y_pred

            m_derivative = -(2/len(x)) * sum(x * error)
            b_derivative = -(2/len(x)) * sum(error)

            m -= learning_rate * m_derivative
            b -= learning_rate * b_derivative

        obj = LinearRegression()
        obj.coef_.append(m)
        obj.intercept_ = b
        
        return obj

# Learn a prediction function
r = LinearRegression().fit(x, y)
m = r.coef_[0].round(1)
b = r.intercept_.round(1)

# Prediction
x1 = 3
y1 = m*x1 + b

# Output
print('Best line:', f"y = {m}x + {b}")
print('Prediction for x=3:', f"y = {y1:.1f}")

m = round(m, 1)
b = round(b, 1)

fig, ax = plt.subplots()
plt.ylim(0, 10)
plt.xlim(0, 10)

ax.plot(x,  y,  'x', color='g', label='Training data')
ax.plot(x, m*x + b,  label=f'h(x) = {m} + {b}x')
ax.plot(x1, y1, 'o', color='r', label=f'h({x1}) = {y1}') # Draw unknown point
plt.legend()
plt.show()