Linear Least Squares 的多项式表达和矩阵表达 与 Python 实现

import numpy as np
import random
import sklearn
from sklearn.datasets.samples_generator import make_regression 
import pylab
from scipy import stats
%matplotlib inline

def gradient_descent(alpha, x, y, ep=0.0001, max_iter=100):
    converged = False
    iter = 0
    m = x.shape[0] # number of samples

    # initial theta
    t0 = np.random.random(x.shape[1])
    t1 = np.random.random(x.shape[1])

    # total error, J(theta)
    J = sum([(t0 + t1*x[i] - y[i])**2 for i in range(m)])

    # Iterate Loop
    while not converged:
        # for each training sample, compute the gradient (d/d_theta j(theta))
        grad0 = 1.0/m * sum([(t0 + t1*x[i] - y[i]) for i in range(m)]) 
        grad1 = 1.0/m * sum([(t0 + t1*x[i] - y[i])*x[i] for i in range(m)])

        # update the theta_temp
        temp0 = t0 - alpha * grad0
        temp1 = t1 - alpha * grad1
    
        # update theta
        t0 = temp0
        t1 = temp1

        # mean squared error
        e = sum( [ (t0 + t1*x[i] - y[i])**2 for i in range(m)] ) 

        if abs(J-e) <= ep:
            print 'Converged, iterations: ', iter, '!!!'
            converged = True
    
        J = e   # update error 
        iter += 1  # update iter
    
        if iter == max_iter:
            print 'Max interactions exceeded!'
            converged = True

    return t0,t1

if __name__ == '__main__':

    x, y = make_regression(n_samples=100, n_features=1, n_informative=1, 
                        random_state=0, noise=35) 
    print 'x.shape = %s y.shape = %s' %(x.shape, y.shape)
 
    alpha = 0.01 # learning rate
    ep = 0.01 # convergence criteria

    # call gredient decent, and get intercept(=theta0) and slope(=theta1)
    theta0, theta1 = gradient_descent(alpha, x, y, ep, max_iter=1000)
    print ('theta0 = %s theta1 = %s') %(theta0, theta1) 

    # check with scipy linear regression 
    slope, intercept, r_value, p_value, slope_std_error = stats.linregress(x[:,0], y)
    print ('intercept = %s slope = %s') %(intercept, slope) 

    # plot
    for i in range(x.shape[0]):
        y_predict = theta0 + theta1*x 

    pylab.plot(x,y,'o')
    pylab.plot(x,y_predict,'k-')
    pylab.show()
    print "Done!"

import numpy as np
import random
from sklearn.datasets.samples_generator import make_regression 
import pylab
from scipy import stats

def gradient_descent_2(alpha, x, y, numIterations):
    m = x.shape[0] # number of samples
    theta = np.ones(2)
    x_transpose = x.transpose()
    for iter in range(0, numIterations):
        hypothesis = np.dot(x, theta)
        loss = hypothesis - y
        J = np.sum(loss ** 2) / (2 * m)  # cost
        print "iter %s | J: %.3f" % (iter, J)      
        gradient = np.dot(x_transpose, loss) / m         
        theta = theta - alpha * gradient  # update
    return theta

if __name__ == '__main__':

    x, y = make_regression(n_samples=100, n_features=1, n_informative=1, 
                        random_state=0, noise=35) 

    m, n = np.shape(x)
    x = np.c_[ np.ones(m), x] # insert column

    alpha = 0.01 # learning rate
    theta = gradient_descent_2(alpha, x, y, 1000)

    # plot
    for i in range(x.shape[1]):
        y_predict = theta[0] + theta[1]*x 
    pylab.plot(x[:,1],y,'o')
    pylab.plot(x,y_predict,'k-')
    pylab.show()
    print "Done!"