def main():
m = 3
n = 15
test_size = 40
k = int(1e6)
# 产生数据
ori_x, ori_y = generate_data(n)
test_x, test_y = generate_data(test_size)
# 确定梯度下降初始化内容，不带正则项
# w0 = best_answer_without_correct(ori_x, ori_y, m)
w0 = np.zeros((m, 1))
x = vandermonde(ori_x, m)
_x = vandermonde(test_x, m)
A = x.T @ x
t = np.mat(ori_y).T
b = x.T @ t
# 梯度下降求解w
w = grad_down(w0, A, b, k, x, t)
# 画图
# 计算E_RMS
e_rms = get_RMS(get_Ew(_x, w, np.mat(test_y).T), test_size)
print('without correct in test data:e_rms=', e_rms)
match_plot((ori_x, ori_y), m, n, w, 'grad_down')
# 梯度下降求解w,带正则项
la = np.exp(-9)
A = x.T @ x + la * np.mat(np.eye(A.shape[0], A.shape[1]))
# w0 = np.mat(best_answer_with_correct(ori_x, ori_y, m, la)).T
w = grad_down(w0, A, b, k, x, t)
# 计算E_RMS
e_rms = get_RMS(get_Ew(_x, w, np.mat(test_y).T), test_size)
print('with correct in test data:e_rms=', e_rms)
# 画图
match_plot((ori_x, ori_y), m, n, w, 'grad_down with correct')
def main():
m = 14
n = 50
test_size = 40
# 产生数据
ori_x, ori_y = generate_data(n)
test_x, test_y = generate_data(test_size)
# 确定梯度下降初始化内容，不带正则项
w0 = np.zeros((m, 1))
x = vandermonde(ori_x, m)
A = x.T @ x
b = x.T @ np.mat(ori_y).T
# 共轭梯度下降
w = conjugate_down(w0, A, b)
# 画图
match_plot((ori_x, ori_y), m, n, w)
# 计算E_RMS
_x = vandermonde(test_x, m)
e_rms = get_RMS(get_Ew(_x, w, np.mat(test_y).T), test_size)
print('without correct in test data:e_rms=', e_rms)
# 带正则项
la = np.exp(-9)
A = x.T @ x + la * np.mat(np.eye(A.shape[0], A.shape[1]))
w = conjugate_down(w0, A, b)
# 计算E_RMS
e_rms = get_RMS(get_Ew(_x, w, np.mat(test_y).T), test_size)
print('with correct in test data:e_rms=', e_rms)
# 画图
match_plot((ori_x, ori_y),
m,
n,
w,
line_name='conjugate_down_with_correct')
# 确定最好的lambda
la_list = np.array([np.exp(i) for i in range(-15, 0)])
e_rms_train = []
e_rms_test = []
for _la in la_list:
A = x.T @ x + _la * np.mat(np.eye(A.shape[0], A.shape[1]))
w = conjugate_down(w0, A, b)
# 计算E_RMS
e_rms_te = get_RMS(get_Ew(_x, w, np.mat(test_y).T), test_size)
e_rms_test.append(float(e_rms_te))
e_rms_tr = get_RMS(get_Ew(x, w, np.mat(ori_y).T), n)
e_rms_train.append(float(e_rms_tr))
# 画图 e_rms ---- lambda
# plt.xlabel('lambda($e^{-x}$)')
s = 'train_size=' + str(n) + ',test_size=' + str(test_size)
plt.title(s)
plt.xlabel('lambda')
plt.ylabel('$E_{RMS}$')
plt.xscale('log')
plt.xticks([np.exp(-i) for i in range(5, 15, 2)],
['$e^{' + str(-i) + '}$' for i in range(5, 15, 2)])
plt.plot(la_list, e_rms_train, color='r', label='train')
plt.plot(la_list, e_rms_test, color='b', label='test')
plt.legend()
plt.show()
def grad_down(w0, A, b, k, x, t):
w = np.mat(w0)
A = np.mat(A)
b = np.mat(b)
step_length = 0.001 # 步长
r = np.mat(b - A * w) # 梯度反方向
value1 = get_Ew(x, w, t) # 第二个
value0 = value1
precision = 1e-5
for _ in range(k):
w = w + step_length * r
value0 = value1 # 前一个
value1 = get_Ew(x, w, t) # 当前的
# 如果二次函数取值变大，则减小步长
if value1 - value0 > 0: step_length = 0.5 * step_length
# 跳出条件 当loss足够小时
if value1 < precision:
print(_)
break
r = np.mat(b - A * w) # 梯度反方向
return w
match_plot(ori, m, n, w, "solution with correct")
# 确定最好的lambda
# 通过画图来寻找近似的E(w)最低的点 绘制 train_RMS-lambda 和 test_RMS-lambda 图像
la_list = []
train_RMS = []
test_RMS = []
for _ in np.arange(-15, 0):
la_list.append(np.exp(_))
for lal in la_list:
_w = np.array(best_answer_with_correct(ori[0], ori[1], m, lal))
_w = np.reshape(_w, (m, 1))
_x = vandermonde(ori[0], m)
_t = np.array(ori[1]).reshape(ori[1].shape[0], 1)
train_RMS.append(get_RMS(float(get_Ew(_x, _w, _t)), n))
_x = vandermonde(test_data[0], m)
_t = np.array(test_data[1]).reshape(test_data[1].shape[0], 1)
test_RMS.append(get_RMS(float(get_Ew(_x, _w, _t)), test_size))
# RMS-lambda图像
plt.xscale('log')
plt.xlabel('lambda')
plt.ylabel('$E_{RMS}$')
plt.xticks([np.exp(-i) for i in range(5, 15, 2)], ['$e^{' + str(-i) + '}$' for i in range(5, 15, 2)])
plt.plot(la_list, train_RMS, color='r', label='train')
plt.plot(la_list, test_RMS, color='b', label='test')
plt.legend()
plt.show()
# 求解
w = best_answer_without_correct(ori[0], ori[1], m)
# 拟合曲线
match_plot(ori, m, n, w, 'solution')
# 绘制loss-m图像,解释过拟合现象 m-w-E(w)
title = 'train_size=' + str(n) + ',test_size=' + str(test_size)
plt.title(title)
plt.xlabel('m')
plt.ylabel('$E_{RMS}$')
plt.xlim(0, 11)
m_list = np.arange(0, 12)
RMS_test = []
RMS_train = []
test_data = generate_data(test_size)
test = np.array(test_data[1]).reshape(test_data[1].shape[0], 1)
train = np.array(ori[1]).reshape(ori[1].shape[0], 1)
for _m in m_list:
_w = best_answer_without_correct(ori[0], ori[1], _m + 1)
_w = np.reshape(_w, (_m + 1, 1))
_x = x = vandermonde(test_data[0], _m + 1)
RMS_test.append(get_RMS(float(get_Ew(_x, _w, test)), test_size))
_x = x = vandermonde(ori[0], _m + 1)
RMS_train.append(get_RMS(float(get_Ew(_x, _w, train)), n))
plt.plot(m_list, RMS_test, color='b', label='test set', marker='o')
plt.plot(m_list, RMS_train, color='r', label='train set', marker='o')
plt.legend()
plt.show()