def plot_regularization_path(degree):
# 填充训练数据
feature = PolynomialFeature(degree)
X = feature.transform(x_train)
# 未经正则化操作的w_infinite
w_infinite = np.linalg.pinv(X) @ y_train
# 正则项系数（范围可调节，0.04-100是较好的范围）
alpha = np.linspace(0.04, 100, 10000)
# 注意report中要对w随lambda的变化进行分析以及横坐标的范围
w_lambda = None
# 依次计算不同正则项系数下的 w_lambda
for i in alpha:
w_lambda_i = np.linalg.solve(i * np.eye(np.size(X, 1)) + X.T @ X,
X.T @ y_train)
w_lambda = w_lambda_i if w_lambda is None else np.vstack(
(w_lambda, w_lambda_i))
# 横坐标 ||w_lambda||/||w_infinite||
path_x = [np.linalg.norm(i) / np.linalg.norm(w_infinite) for i in w_lambda]
# 绘制Regularization Path
plt.title('Regularization Path (degree={0})'.format(degree))
plt.xlabel('$||w_\lambda||/||w_∞||$')
plt.ylabel('$w_i$')
for i in range(0, degree + 1):
# 依次绘制wi的正则化路径
plt.plot(path_x,
w_lambda[:, i],
label='$w_{0}$'.format(i),
linewidth='2')
# plt.plot(path_x, abs(w_lambda[:, i]), label='$w_{0}$'.format(i), linewidth='2')
plt.legend()
plt.show()
pass
def rmse_measure_1():
training_errors = []
test_errors = []
for i in range(10):
feature = PolynomialFeature(i)
X_train = feature.transform(x_train)
X_test = feature.transform(x_test)
model = LinearRegression()
# model = RidgeRegression()
model.fit(X_train, y_train)
y = model.predict(X_test)
training_errors.append(rmse(model.predict(X_train), y_train))
# test_errors.append(rmse(model.predict(X_test), y_test + np.random.normal(scale=0.25, size=len(y_test))))
test_errors.append(rmse(model.predict(X_test), y_test))
plt.title('Curve Fitting Result with Training Size=%d' % len(X_train))
# plt.title('Curve Fitting Result with $ln(λ)$=%.2f' % 1.0)
plt.plot(training_errors,
'o-',
mfc="none",
mec="b",
ms=10,
c="b",
label="Training")
plt.plot(test_errors,
'o-',
mfc="none",
mec="r",
ms=10,
c="r",
label="Test")
plt.legend()
plt.xlabel("$degree$")
plt.ylabel("$RMSE$")
plt.show()
def linear_fit(x, y, degree, alpha):
# 填充数据
feature = PolynomialFeature(degree)
X = feature.transform(x)
# 引入正则化后，求解得到的参数w_lambda
w_lambda = np.linalg.solve(alpha * np.eye(np.size(X, 1)) + X.T @ X,
X.T @ y)
return X, w_lambda
def plot_rmse(degree):
# 填充训练数据
feature = PolynomialFeature(degree)
X = feature.transform(x_train)
# 未经正则化操作的w_infinite
w_infinite = np.linalg.pinv(X) @ y_train
# 正则项系数
alpha = np.linspace(0.04, 100, 10000)
# 注意report中要说明：lambda不能调过大，否则影响准确率（通过RMSE指标说明）
# 不同正则项系数下的 w_lambda
w_lambda = None
# 训练集和测试集RMSE误差
training_errors = []
test_errors = []
for i in alpha:
w_lambda_i = np.linalg.solve(i * np.eye(np.size(X, 1)) + X.T @ X,
X.T @ y_train)
w_lambda = w_lambda_i if w_lambda is None else np.vstack(
(w_lambda, w_lambda_i))
model = RidgeRegression(i)
model.fit(X, y_train)
training_errors.append(rmse(model.predict(X), y_train))
test_errors.append(
rmse(model.predict(feature.transform(x_test)), y_test))
# 横坐标 ||w_lambda||/||w_infinite||
path_x = [np.linalg.norm(i) / np.linalg.norm(w_infinite) for i in w_lambda]
# 绘制RMSE曲线
plt.title('RMSE (degree={0})'.format(degree))
plt.xlabel('$||w_\lambda||/||w_∞||$')
plt.ylabel("$RMSE$")
plt.plot(path_x,
training_errors,
'-',
mfc="none",
mec="b",
ms=10,
c="b",
label="Train")
plt.plot(path_x,
test_errors,
'-',
mfc="none",
mec="r",
ms=10,
c="r",
label="Test")
plt.legend()
plt.show()
pass
def find_best_degree_based_on_rmse():
row_training_errors = []
column_training_errors = []
degrees = np.arange(2, 20, 2)
for degree in degrees:
X = PolynomialFeature(degree).transform(index)
model_row = RidgeRegression(alpha_row)
model_column = RidgeRegression(alpha_column)
model_row.fit(X, row)
model_column.fit(X, column)
row_training_errors.append(rmse(model_row.predict(X), row))
column_training_errors.append(rmse(model_column.predict(X), column))
plt.title('Linear Fitting Result ($RMSE$)')
plt.plot(degrees,
row_training_errors,
'o-',
mfc="none",
mec="r",
ms=10,
c="r",
label="row-fitting-RMSE ($\lambda=%.3f$)" % alpha_row)
plt.plot(degrees,
column_training_errors,
'o-',
mfc="none",
mec="b",
ms=10,
c="b",
label="column-fitting-RMSE ($\lambda=%.3f$)" % alpha_column)
plt.xlabel("$degree$")
plt.ylabel("$RMSE$")
plt.legend()
plt.show()
def rmse_measure_2():
training_errors = []
test_errors = []
alpha_ln = np.linspace(-50, 0, 100)
alpha = np.exp(alpha_ln)
degree = 9
for i in alpha:
feature = PolynomialFeature(degree)
X_train = feature.transform(x_train)
X_test = feature.transform(x_test)
# model = LinearRegression()
model = RidgeRegression(i)
model.fit(X_train, y_train)
training_errors.append(rmse(model.predict(X_train), y_train))
test_errors.append(rmse(model.predict(X_test), y_test))
plt.title('Curve Fitting Result with $M$=%d' % degree)
plt.plot(alpha_ln,
training_errors,
'-',
mfc="none",
mec="b",
ms=10,
c="b",
label="Training")
plt.plot(alpha_ln,
test_errors,
'-',
mfc="none",
mec="r",
ms=10,
c="r",
label="Test")
plt.xlim((-40, 0))
plt.ylim((0, 1))
plt.legend()
plt.xlabel("$ln(λ)$")
plt.ylabel("$RMSE$")
plt.show()
def poly_fit_regularized():
for i, degree in enumerate([0, 1, 3, 6, 9]):
feature = PolynomialFeature(degree)
X_train = feature.transform(x_train)
X_test = feature.transform(x_test)
model = RidgeRegression(alpha=1e-3)
model.fit(X_train, y_train)
y = model.predict(X_test)
plt.title('Curve Fitting Result with Regularization, M=%d' % degree)
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.scatter(x_train,
y_train,
facecolor="none",
edgecolor="b",
s=50,
label="training data")
plt.plot(x_test, y_test, c="g", label="$\sin(2\pi x)$")
plt.plot(x_test, y, c="r", label="fitting")
plt.ylim(-1.5, 1.5)
plt.annotate("M={}".format(degree), xy=(-0.15, 1))
plt.legend()
plt.show()