import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# Parameters of the quadratic function
a = 400
b = 200
c = 900

# Generate 150 points between -100 and 100
np.random.seed(42)

X = np.random.uniform(0, 1, 150)

# Create Y values based on the quadratic function with Gaussian noise

Y = a * X**2 + b * X + c + np.random.normal(0, 15, 150)

# Reshape the data for modeling
X = X.reshape(-1, 1)
Y = Y.reshape(-1, 1)

# Perform linear regression
model = LinearRegression()
model.fit(X, Y)

# Display the data
plt.figure(figsize=(8, 6))
plt.scatter(X, Y, color='red',label='Data Points')
plt.plot(X, 500 * X +  c, color='blue', linewidth=2, label='Linear Regression Line')
#plt.plot(X, model.predict(X), color='blue', linewidth=2, label='Linear Regression Line')
plt.title('Linear Regression on Gold Price based on Purity')
plt.xlabel('Gold Purity')
plt.ylabel('Gold Price')
plt.legend()
plt.grid(ls='--')
plt.show()

import numpy as np
import matplotlib.pyplot as plt

# Parameters of the quadratic function
a = 400
b = 200
c = 900

# Generate random data
np.random.seed(42)
x = np.random.uniform(0, 1, 150)
y = a * x**2 + b * x + c + np.random.normal(0, 15, 150)
y = y.reshape(y.shape[0], 1)  # Reshaping y to fit the required shape

# Create the feature matrix X
X = np.vstack((x, np.ones(len(x)))).T

# Initialize the theta parameters for the linear model
theta = np.random.randn(2, 1)

# Define the linear model function
def model(X, theta):
    return X.dot(theta)

# Define the cost function
def cost_function(X, y, theta):
    m = len(y)
    return 1 / (2 * m) * np.sum((model(X, theta) - y)**2)

# Define the gradient calculation function
def grad(X, y, theta):
    m = len(y)
    return 1 / m * X.T.dot(model(X, theta) - y)

# Define the gradient descent function
def gradient_descent(X, y, theta, learning_rate, n_iterations):
    cost_history = np.zeros(n_iterations)
    for i in range(n_iterations):
        theta = theta - learning_rate * grad(X, y, theta)
        cost_history[i] = cost_function(X, y, theta)
    return theta, cost_history

# Define the learning rate and number of iterations
learning_rate = 0.2
n_iterations = 20

# Perform gradient descent to optimize the parameters
theta_final, cost_history = gradient_descent(X, y, theta, learning_rate, n_iterations)

# Make predictions with the optimized parameters
predictions = model(X, theta_final)

# Plot the data points and the linear model predictions
plt.scatter(x, y, c='r',label='Data Points')
plt.plot(x, predictions, c='blue',label='Linear Regression Line')
plt.xlabel('Gold Purity')
plt.ylabel('Gold Price')
plt.title('Linear Regression on Gold Price and Gold Purity')
plt.legend()
plt.grid(ls='--')
plt.show()

# Plot the cost history during gradient descent
plt.plot(range(n_iterations), cost_history)
plt.xlabel('Number of Steps')
plt.ylabel('Descent')
plt.title('Descent History during Gradient Descent')
plt.grid(ls='--')
plt.show()

# Calculate the coefficient of determination (R-squared)
def coef_determination(y, pred):
    u = ((y - pred)**2).sum()
    v = ((y - y.mean())**2).sum()
    return 1 - u / v

# Print Final theta

print(theta_final)

# Print the R-squared value
print(coef_determination(y, predictions))

# Display the first 5 samples of the data
print("First 5 samples of the data:")
for i in range(5):
    print(f"Sample {i + 1} - Gold Purity: {x[i]:.4f}, Gold Price: {y[i][0]:.4f}")

import numpy as np
import matplotlib.pyplot as plt

# Parameters of the quadratic function
a = 400
b = 200
c = 900

# Generate random data
np.random.seed(42)
x = np.random.uniform(0, 1, 150)
y = a * x**2 + b * x + c + np.random.normal(0, 15, 150)
y = y.reshape(y.shape[0], 1)  # Reshaping y to fit the required shape

# Create the feature matrix X

X = np.vstack((x**2,x, np.ones(len(x)))).T


print(X.shape)

# Initialize the theta parameters for the linear model
theta = np.random.randn(3, 1)

# Define the polynomial model function
def model(X, theta):
    return X.dot(theta)

# Define the cost function
def cost_function(X, y, theta):
    m = len(y)
    return 1 / (2 * m) * np.sum((model(X, theta) - y)**2)

# Define the gradient calculation function
def grad(X, y, theta):
    m = len(y)
    return 1 / m * X.T.dot(model(X, theta) - y)

# Define the gradient descent function
def gradient_descent(X, y, theta, learning_rate, n_iterations):
    cost_history = np.zeros(n_iterations)
    for i in range(n_iterations):
        theta = theta - learning_rate * grad(X, y, theta)
        cost_history[i] = cost_function(X, y, theta)
    return theta, cost_history

# Define the learning rate and number of iterations
learning_rate = 0.2
n_iterations = 20

# Perform gradient descent to optimize the parameters
theta_final, cost_history = gradient_descent(X, y, theta, learning_rate, n_iterations)

# Make predictions with the optimized parameters
predictions = model(X, theta_final)


# Plot the data points and the polynomial regression predictions
plt.scatter(x, y, c='r', label='Data Points')
plt.plot(x, predictions, c='blue', label='Polynomial Regression Line')
plt.xlabel('Gold Purity')
plt.ylabel('Gold Price')
plt.title('Polynomial Regression on Gold Price and Gold Purity')
plt.legend()
plt.grid(ls='--')
plt.show()

# Plot the cost history during gradient descent
plt.plot(range(n_iterations), cost_history)
plt.xlabel('Number of Steps')
plt.ylabel('Descent')
plt.title('Descent History during Gradient Descent')
plt.grid(ls='--')
plt.show()


# Calculate the coefficient of determination (R-squared)
def coef_determination(y, pred):
    u = ((y - pred)**2).sum()
    v = ((y - y.mean())**2).sum()
    return 1 - u / v

# Print Final theta
print(theta_final)

# Print the R-squared value
print(coef_determination(y, predictions))