复习内容
Regularized linear regression cost function
对应matlab代码：
J = 1/(2*m)*sum((X*theta - y).^2) ;
regularize = lambda /(2*m)* (sum(theta.^2)-theta(1).^2);
J = J + regularize;
Regularized linear regression gradient
如果对这里有疑惑，这个部分也有相应的推导过程：
梯度对应代码：
n = size(theta);
for i=1:n
grad(i) = 1/m * sum((X*theta - y).*X(:,i)) + lambda/m *theta(i);
end
grad(1) = 1/m * sum((X*theta - y).^X(:,1));
Learning curves
Polynomial regression
Programming Exercise 5 - Regularized Linear Regression and Bias v.s. Variance
# %load ../../../standard_import.txt
import pandas as pd
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy.io import loadmat
from scipy.optimize import minimize
from sklearn.linear_model import LinearRegression, Ridge
from sklearn.preprocessing import PolynomialFeatures
pd.set_option('display.notebook_repr_html', False)
pd.set_option('display.max_columns', None)
pd.set_option('display.max_rows', 150)
pd.set_option('display.max_seq_items', None)
#%config InlineBackend.figure_formats = {'pdf',}
%matplotlib inline
import seaborn as sns
sns.set_context('notebook')
sns.set_style('white')
data = loadmat('data/ex5data1.mat')
data.keys()
dict_keys(['__header__', '__version__', '__globals__', 'X', 'y', 'Xtest', 'ytest', 'Xval', 'yval'])
y_train = data['y']
X_train = np.c_[np.ones_like(data['X']), data['X']]
yval = data['yval']
Xval = np.c_[np.ones_like(data['Xval']), data['Xval']]
print('X_train:', X_train.shape)
print('y_train:', y_train.shape)
print('Xval:', Xval.shape)
print('yval:', yval.shape)
X_train: (12, 2)
y_train: (12, 1)
Xval: (21, 2)
yval: (21, 1)
Regularized Linear Regression
plt.scatter(X_train[:,1], y_train, s=50, c='r', marker='x', linewidths=1)
plt.xlabel('Change in water level (x)')
plt.ylabel('Water flowing out of the dam (y)')
plt.ylim(ymin=0);
Regularized Cost function
def linearRegCostFunction(theta, X, y, reg):
m = y.size
h = X.dot(theta)
J = (1/(2*m))*np.sum(np.square(h-y)) + (reg/(2*m))*np.sum(np.square(theta[1:]))
return(J)
Gradient
def lrgradientReg(theta, X, y, reg):
m = y.size
h = X.dot(theta.reshape(-1,1))
grad = (1/m)*(X.T.dot(h-y))+ (reg/m)*np.r_[[[0]],theta[1:].reshape(-1,1)]
return(grad.flatten())
initial_theta = np.ones((X_train.shape[1],1))
cost = linearRegCostFunction(initial_theta, X_train, y_train, 0)
gradient = lrgradientReg(initial_theta, X_train, y_train, 0)
print(cost)
print(gradient)
303.951525554
[ -15.30301567 598.16741084]
def trainLinearReg(X, y, reg):
#initial_theta = np.zeros((X.shape[1],1))
initial_theta = np.array([[15],[15]])
# For some reason the minimize() function does not converge when using
# zeros as initial theta.
res = minimize(linearRegCostFunction, initial_theta, args=(X,y,reg), method=None, jac=lrgradientReg,
options={'maxiter':5000})
return(res)
fit = trainLinearReg(X_train, y_train, 0)
fit
fun: 1604.4002999186634
hess_inv: array([[ 1.03142187, 0.00617881],
[ 0.00617881, 0.001215 ]])
jac: array([ 3.42437190e-12, -5.70371898e-10])
message: 'Optimization terminated successfully.'
nfev: 6
nit: 4
njev: 6
status: 0
success: True
x: array([ 13.08790351, 0.36777923])
Comparison: coefficients and cost obtained with LinearRegression in Scikit-learn
regr = LinearRegression(fit_intercept=False)
regr.fit(X_train, y_train.ravel())
print(regr.coef_)
print(linearRegCostFunction(regr.coef_, X_train, y_train, 0))
[ 13.08790351 0.36777923]
1604.40029992
plt.plot(np.linspace(-50,40), (fit.x[0]+ (fit.x[1]*np.linspace(-50,40))), label='Scipy optimize')
#plt.plot(np.linspace(-50,40), (regr.coef_[0]+ (regr.coef_[1]*np.linspace(-50,40))), label='Scikit-learn')
plt.scatter(X_train[:,1], y_train, s=50, c='r', marker='x', linewidths=1)
plt.xlabel('Change in water level (x)')
plt.ylabel('Water flowing out of the dam (y)')
plt.ylim(ymin=-5)
plt.xlim(xmin=-50)
plt.legend(loc=4);
def learningCurve(X, y, Xval, yval, reg):
m = y.size
error_train = np.zeros((m, 1))
error_val = np.zeros((m, 1))
for i in np.arange(m):
res = trainLinearReg(X[:i+1], y[:i+1], reg)
error_train[i] = linearRegCostFunction(res.x, X[:i+1], y[:i+1], reg)
error_val[i] = linearRegCostFunction(res.x, Xval, yval, reg)
return(error_train, error_val)
t_error, v_error = learningCurve(X_train, y_train, Xval, yval, 0)
plt.plot(np.arange(1,13), t_error, label='Training error')
plt.plot(np.arange(1,13), v_error, label='Validation error')
plt.title('Learning curve for linear regression')
plt.xlabel('Number of training examples')
plt.ylabel('Error')
plt.legend();
Polynomial regression (Scikit-learn)
poly = PolynomialFeatures(degree=8)
X_train_poly = poly.fit_transform(X_train[:,1].reshape(-1,1))
regr2 = LinearRegression()
regr2.fit(X_train_poly, y_train)
regr3 = Ridge(alpha=20)
regr3.fit(X_train_poly, y_train)
# plot range for x
plot_x = np.linspace(-60,45)
# using coefficients to calculate y
plot_y = regr2.intercept_+ np.sum(regr2.coef_*poly.fit_transform(plot_x.reshape(-1,1)), axis=1)
plot_y2 = regr3.intercept_ + np.sum(regr3.coef_*poly.fit_transform(plot_x.reshape(-1,1)), axis=1)
plt.plot(plot_x, plot_y, label='Scikit-learn LinearRegression')
plt.plot(plot_x, plot_y2, label='Scikit-learn Ridge (alpha={})'.format(regr3.alpha))
plt.scatter(X_train[:,1], y_train, s=50, c='r', marker='x', linewidths=1)
plt.xlabel('Change in water level (x)')
plt.ylabel('Water flowing out of the dam (y)')
plt.title('Polynomial regression degree 8')
plt.legend(loc=4);
