偏差与方差
- 题目
- 欠拟合
- 改进欠拟合
- 影响偏差和方差因素
- 训练集拟合情况
- 训练集和测试集代价函数
- 选择最优lamda
- 整体代码
训练集:训练模型
·验证集︰模型选择，模型的最终优化
·测试集:利用训练好的模型测试其泛化能力
#训练集
x_train,y_train = data['X'],data[ 'y']
#验证集
x_val,y_val = data['Xval'],data[ 'yval']
x_val.shape,y_val.shape
#测试集
x_test,y_test = data['Xtest'],data[ 'ytest']
x_test.shape,y_test.shape
题目
def reg_cost(theta,x,y,lamda):
cost=np.sum(np.power(x@theta-y.flatten(),2))
reg=theta[1:]@theta[1:]*lamda
return (cost+reg)/(2*len(x))
def reg_gradient(theta,x,y,lamda):
grad=(x@theta-y.flatten())@x
reg=lamda*theta
reg[0]=0
return (grad+reg)/(len(x))
def train_mode(x,y,lamda):
theta=np.ones(x.shape[1])
res=minimize(
fun=reg_cost,
x0=theta,
args=(x,y,lamda),
method='TNC',
jac=reg_gradient
)
return res.x
欠拟合
"""
训练样本从1开始递增进行训练
比较训练集和验证集上的损失函数的变化情况
"""
def plot_learning_curve(x_train,y_train,x_val,y_val,lamda):
x= range(1,len(x_train)+1)
training_cost =[]
cv_cost =[]
for i in x:
res = train_mode(x_train[:i,:],y_train[:i,:],lamda)
training_cost_i = reg_cost(res,x_train[:i,:],y_train[:i,:],lamda)
cv_cost_i = reg_cost(res,x_val,y_val,lamda)
training_cost.append(training_cost_i)
cv_cost.append(cv_cost_i)
plt.plot(x,training_cost,label = 'training cost')
plt.plot(x,cv_cost,label = 'cv cost')
plt.legend()
plt.xlabel("number of training examples")
plt.ylabel("error")
plt.show()
改进欠拟合
影响偏差和方差因素
"""
任务:构造多项式特征，进行多项式回归
"""
def poly_feature(x, power):
for i in range(2, power + 1):
x= np.insert(x, x.shape[1], np.power(x[:, 1], i), axis = 1)
return x
"""
归一化
"""
def get_means_stds(x):
means = np.mean(x, axis=0)
stds = np.std(x, axis=0)
return means, stds
def feature_normalize(x,means,stds):
x [:,1:]=(x[:,1:] - means[1:,])/stds[1:]
return x
power=6
x_train_poly=poly_feature(x_train,power)
x_val_poly=poly_feature(x_val,power)
x_test_poly=poly_feature(x_test,power)
train_means,train_stds=get_means_stds(x_train_poly)
x_train_norm=feature_normalize(x_train_poly,train_means,train_stds)
x_val_norm=feature_normalize(x_val_poly,train_means,train_stds)
x_test_norm=feature_normalize(x_test_poly,train_means,train_stds)
theta_fit=train_mode(x_train_norm,y_train,lamda =0)
训练集拟合情况
"""
训练集
绘制数据集和拟合函数
"""
def plot_poly_fit():
plot_data()
x = np.linspace(-60,60,100)
xx= x.reshape(100,1)
xx= np.insert(xx,0,1,axis=1)
xx= poly_feature(xx,power)
xx= feature_normalize(xx,train_means,train_stds)
plt.plot(x,xx@theta_fit,'r--')
训练集和测试集代价函数
plot_learning_curve(x_train_norm,y_train, x_val_norm, y_val, lamda=0)
此时lamda=0没有加入正则化
可以看出是高方差，过拟合了，此时lamda=0没有加入正则化
加入正则化如下
plot_learning_curve(x_train_norm,y_train, x_val_norm, y_val, lamda=1)
此时训练集误差增大，验证集误差减小了
但是lamda不能过大了，如下
plot_learning_curve(x_train_norm,y_train, x_val_norm, y_val, lamda=100)
选择最优lamda
lamdas=[0,0.001,0.003,0.01,0.03,0.1,0.3,1,2,3,10]
training_cost =[]
cv_cost =[]
for lamda in lamdas:
res = train_mode(x_train_norm,y_train,lamda)
tc = reg_cost(res,x_train_norm,y_train,lamda=0)
cv = reg_cost(res,x_val_norm,y_val,lamda=0)
training_cost.append(tc)
cv_cost.append(cv)
plt.plot(lamdas,training_cost,label="training cost")
plt.plot(lamdas,cv_cost,label="cv cos")
plt.legend()
plt.show()
l=lamdas[np.argmin(cv_cost)]#寻找最优lamda
print(l)
res = train_mode(x_train_norm,y_train,lamda =l)
test_cost = reg_cost(res,x_test_norm,y_test,lamda = 0)
print(test_cost)
整体代码
import numpy as np
import matplotlib.pyplot as plt
import scipy.io as sio
from scipy.optimize import minimize
def plot_data():
fig,ax = plt.subplots()
ax.scatter(x_train[:,1],y_train)
ax.set(xlabel = "change in water level(x)",
ylabel = 'water flowing out og the dam(y)')
def reg_cost(theta,x,y,lamda):
cost=np.sum(np.power(x@theta-y.flatten(),2))
reg=theta[1:]@theta[1:]*lamda
return (cost+reg)/(2*len(x))
def reg_gradient(theta,x,y,lamda):
grad=(x@theta-y.flatten())@x
reg=lamda*theta
reg[0]=0
return (grad+reg)/(len(x))
def train_mode(x,y,lamda):
theta=np.ones(x.shape[1])
res=minimize(
fun=reg_cost,
x0=theta,
args=(x,y,lamda),
method='TNC',
jac=reg_gradient
)
return res.x
"""
训练样本从1开始递增进行训练
比较训练集和验证集上的损失函数的变化情况
"""
def plot_learning_curve(x_train,y_train,x_val,y_val,lamda):
x= range(1,len(x_train)+1)
training_cost =[]
cv_cost =[]
for i in x:
res = train_mode(x_train[:i,:],y_train[:i,:],lamda)
training_cost_i = reg_cost(res,x_train[:i,:],y_train[:i,:],lamda)
cv_cost_i = reg_cost(res,x_val,y_val,lamda)
training_cost.append(training_cost_i)
cv_cost.append(cv_cost_i)
plt.plot(x,training_cost,label = 'training cost')
plt.plot(x,cv_cost,label = 'cv cost')
plt.legend()
plt.xlabel("number of training examples")
plt.ylabel("error")
plt.show()
"""
任务:构造多项式特征，进行多项式回归
"""
def poly_feature(x, power):
for i in range(2, power + 1):
x= np.insert(x, x.shape[1], np.power(x[:, 1], i), axis = 1)
return x
"""
归一化
"""
def get_means_stds(x):
means = np.mean(x, axis=0)
stds = np.std(x, axis=0)
return means, stds
def feature_normalize(x,means,stds):
x [:,1:]=(x[:,1:] - means[1:,])/stds[1:]
return x
"""
训练集
绘制数据集和拟合函数
"""
def plot_poly_fit():
plot_data()
x = np.linspace(-60,60,100)
xx= x.reshape(100,1)
xx= np.insert(xx,0,1,axis=1)
xx= poly_feature(xx,power)
xx= feature_normalize(xx,train_means,train_stds)
plt.plot(x,xx@theta_fit,'r--')
data=sio.loadmat("E:/学习/研究生阶段/python-learning/吴恩达机器学习课后作业/code/ex5-bias vs variance/ex5data1.mat")
#训练集
x_train,y_train = data['X'],data[ 'y']
#验证集
x_val,y_val = data['Xval'],data[ 'yval']
x_val.shape,y_val.shape
#测试集
x_test,y_test = data['Xtest'],data[ 'ytest']
x_test.shape,y_test.shape
#
x_train = np.insert(x_train,0,1,axis=1)
x_val = np.insert(x_val,0,1,axis=1)
x_test = np.insert(x_test,0,1,axis=1)
# plot_data()
theta=np.ones(x_train.shape[1])
lamda=1
# print(reg_cost(theta,x_train,y_train,lamda))
#print(reg_gradient(theta,x_train,y_train,lamda))
# theta_final=train_mode(x_train,y_train,lamda=0)
# plot_data()
# plt.plot(x_train[:,1],x_train@theta_final,c='r')
# plt.show()
#plot_learning_curve(x_train,y_train, x_val, y_val, lamda)
power=6
x_train_poly=poly_feature(x_train,power)
x_val_poly=poly_feature(x_val,power)
x_test_poly=poly_feature(x_test,power)
train_means,train_stds=get_means_stds(x_train_poly)
x_train_norm=feature_normalize(x_train_poly,train_means,train_stds)
x_val_norm=feature_normalize(x_val_poly,train_means,train_stds)
x_test_norm=feature_normalize(x_test_poly,train_means,train_stds)
theta_fit=train_mode(x_train_norm,y_train,lamda =0)
# plot_poly_fit()
#plot_learning_curve(x_train_norm,y_train, x_val_norm, y_val, lamda=100)
lamdas=[0,0.001,0.003,0.01,0.03,0.1,0.3,1,2,3,10]
training_cost =[]
cv_cost =[]
for lamda in lamdas:
res = train_mode(x_train_norm,y_train,lamda)
tc = reg_cost(res,x_train_norm,y_train,lamda=0)
cv = reg_cost(res,x_val_norm,y_val,lamda=0)
training_cost.append(tc)
cv_cost.append(cv)
plt.plot(lamdas,training_cost,label="training cost")
plt.plot(lamdas,cv_cost,label="cv cos")
plt.legend()
plt.show()
l=lamdas[np.argmin(cv_cost)]#寻找最优lamda
print(l)
res = train_mode(x_train_norm,y_train,lamda =l)
test_cost = reg_cost(res,x_test_norm,y_test,lamda = 0)
print(test_cost)
