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核心梯度下降算法：
import numpy as np from utils.features import prepare_for_training class
LinearRegression: def __init__(self,data,labels,polynomial_degree =
0,sinusoid_degree = 0,normalize_data=True): """ 1.对数据进行预处理操作 2.先得到所有的特征个数
3.初始化参数矩阵 """ (data_processed, #预处理完之后的数据（标准化之后的数据） features_mean,
#预处理完之后的平均值和标准差 features_deviation) = prepare_for_training(data,
polynomial_degree, sinusoid_degree,normalize_data=True) #
在数据预处理中，对数据进行标准化（normalize）时，通常会使用数据的均值和标准差。标准化是一种常见的数据预处理技术， #
它通过减去均值并除以标准差，将数据转换为具有零均值和单位方差的形式。这样做可以使得不同尺度的特征具有相似的重要性，有助于提高模型的性能和收敛速度。
self.data = data_processed self.labels = labels self.features_mean =
features_mean self.features_deviation = features_deviation
self.polynomial_degree = polynomial_degree self.sinusoid_degree =
sinusoid_degree self.normalize_data = normalize_data #所有特征个数 num_features =
self.data.shape[1] #最终求解的 theta 值，初始化theta参数矩阵 self.theta =
np.zeros((num_features,1)) #alpha为学习率，也就是步长，越小越好；num_iterations为迭代次数 def
train(self,alpha,num_iterations = 500): """ 训练模块，执行梯度下降 """ #cost_history记录损失变化
cost_history = self.gradient_descent(alpha,num_iterations) return
self.theta,cost_history #梯度下降 def gradient_descent(self,alpha,num_iterations):
""" 实际迭代模块，会迭代num_iterations次 """ #cost_history记录损失变化 cost_history = [] for _
in range(num_iterations): self.gradient_step(alpha)
cost_history.append(self.cost_function(self.data,self.labels)) return
cost_history #实际参数更新的时候 计算步骤，公式在这里进行计算，梯度下降的核心计算过程 def
gradient_step(self,alpha): """ 梯度下降参数更新计算方法，注意是矩阵运算 """ #样本个数 num_examples =
self.data.shape[0] #预测值 prediction = LinearRegression.hypothesis(self.data,
self.theta) #误差值delta = 预测值-真实值 delta = prediction - self.labels
#通过步长来，对theta参数进行迭代更新 theta = self.theta #使用矩阵可以避免for循环 theta = theta -
alpha*(1/num_examples)*(np.dot(delta.T,self.data)).T self.theta = theta
#损失函数计算方法 def cost_function(self,data,labels): """ 损失计算方法 """ num_examples =
data.shape[0] delta = LinearRegression.hypothesis(self.data,self.theta) -
labels cost = (1/2)*np.dot(delta.T,delta)/num_examples return cost[0][0] #预测值 =
theta * 数据， 返回矩阵点乘数据 y = theta1*x1 + theta2*x2 + …… @staticmethod def
hypothesis(data,theta): predictions = np.dot(data,theta) return predictions
#获取损失值 def get_cost(self,data,labels): data_processed =
prepare_for_training(data, self.polynomial_degree, self.sinusoid_degree,
self.normalize_data )[0] return self.cost_function(data_processed,labels)
#获取预测值 def predict(self,data): """ 用训练的参数模型，与预测得到回归值结果 """ data_processed =
prepare_for_training(data, self.polynomial_degree, self.sinusoid_degree,
self.normalize_data )[0] predictions =
LinearRegression.hypothesis(data_processed,self.theta) return predictions
"""Prepares the dataset for training""" import numpy as np from .normalize
import normalize from .generate_sinusoids import generate_sinusoids from
.generate_polynomials import generate_polynomials def
prepare_for_training(data, polynomial_degree=0, sinusoid_degree=0,
normalize_data=True): # 计算样本总数 num_examples = data.shape[0] data_processed =
np.copy(data) # 预处理 features_mean = 0 features_deviation = 0 data_normalized =
data_processed if normalize_data: ( data_normalized, features_mean,
features_deviation ) = normalize(data_processed) data_processed =
data_normalized # 特征变换sinusoidal if sinusoid_degree > 0: sinusoids =
generate_sinusoids(data_normalized, sinusoid_degree) data_processed =
np.concatenate((data_processed, sinusoids), axis=1) # 特征变换polynomial if
polynomial_degree > 0: polynomials = generate_polynomials(data_normalized,
polynomial_degree, normalize_data) data_processed =
np.concatenate((data_processed, polynomials), axis=1) # 加一列1 data_processed =
np.hstack((np.ones((num_examples, 1)), data_processed)) return data_processed,
features_mean, features_deviation
绘图：
import numpy as np import pandas as pd import matplotlib.pyplot as plt from
linear_regression import LinearRegression data =
pd.read_csv('../data/world-happiness-report-2017.csv') # 得到训练和测试数据 train_data =
data.sample(frac = 0.8) test_data = data.drop(train_data.index)
input_param_name = 'Economy..GDP.per.Capita.' output_param_name =
'Happiness.Score' x_train = train_data[[input_param_name]].values y_train =
train_data[[output_param_name]].values x_test =
test_data[input_param_name].values y_test = test_data[output_param_name].values
plt.scatter(x_train,y_train,label='Train data')
plt.scatter(x_test,y_test,label='test data') plt.xlabel(input_param_name)
plt.ylabel(output_param_name) plt.title('Happy') plt.legend() plt.show()
num_iterations = 500 learning_rate = 0.01 linear_regression =
LinearRegression(x_train,y_train) (theta,cost_history) =
linear_regression.train(learning_rate,num_iterations) print
('开始时的损失：',cost_history[0]) print ('训练后的损失：',cost_history[-1])
plt.plot(range(num_iterations),cost_history) plt.xlabel('Iter')
plt.ylabel('cost') plt.title('GD') plt.show() predictions_num = 100
x_predictions =
np.linspace(x_train.min(),x_train.max(),predictions_num).reshape(predictions_num,1)
y_predictions = linear_regression.predict(x_predictions)
plt.scatter(x_train,y_train,label='Train data')
plt.scatter(x_test,y_test,label='test data')
plt.plot(x_predictions,y_predictions,'r',label = 'Prediction')
plt.xlabel(input_param_name) plt.ylabel(output_param_name) plt.title('Happy')
plt.legend() plt.show()
两个变量的线性回归模型，建议使用plotly进行绘图
import numpy as np import pandas as pd import matplotlib.pyplot as plt import
plotly import plotly.graph_objs as go plotly.offline.init_notebook_mode() from
linear_regression import LinearRegression data =
pd.read_csv('../data/world-happiness-report-2017.csv') train_data =
data.sample(frac=0.8) test_data = data.drop(train_data.index)
input_param_name_1 = 'Economy..GDP.per.Capita.' input_param_name_2 = 'Freedom'
output_param_name = 'Happiness.Score' x_train = train_data[[input_param_name_1,
input_param_name_2]].values y_train = train_data[[output_param_name]].values
x_test = test_data[[input_param_name_1, input_param_name_2]].values y_test =
test_data[[output_param_name]].values # Configure the plot with training
dataset. plot_training_trace = go.Scatter3d( x=x_train[:, 0].flatten(),
y=x_train[:, 1].flatten(), z=y_train.flatten(), name='Training Set',
mode='markers', marker={ 'size': 10, 'opacity': 1, 'line': { 'color': 'rgb(255,
255, 255)', 'width': 1 }, } ) plot_test_trace = go.Scatter3d( x=x_test[:,
0].flatten(), y=x_test[:, 1].flatten(), z=y_test.flatten(), name='Test Set',
mode='markers', marker={ 'size': 10, 'opacity': 1, 'line': { 'color': 'rgb(255,
255, 255)', 'width': 1 }, } ) plot_layout = go.Layout( title='Date Sets',
scene={ 'xaxis': {'title': input_param_name_1}, 'yaxis': {'title':
input_param_name_2}, 'zaxis': {'title': output_param_name} }, margin={'l': 0,
'r': 0, 'b': 0, 't': 0} ) plot_data = [plot_training_trace, plot_test_trace]
plot_figure = go.Figure(data=plot_data, layout=plot_layout)
plotly.offline.plot(plot_figure) num_iterations = 500 learning_rate = 0.01
polynomial_degree = 0 sinusoid_degree = 0 linear_regression =
LinearRegression(x_train, y_train, polynomial_degree, sinusoid_degree) (theta,
cost_history) = linear_regression.train( learning_rate, num_iterations )
print('开始损失',cost_history[0]) print('结束损失',cost_history[-1])
plt.plot(range(num_iterations), cost_history) plt.xlabel('Iterations')
plt.ylabel('Cost') plt.title('Gradient Descent Progress') plt.show()
predictions_num = 10 x_min = x_train[:, 0].min(); x_max = x_train[:, 0].max();
y_min = x_train[:, 1].min(); y_max = x_train[:, 1].max(); x_axis =
np.linspace(x_min, x_max, predictions_num) y_axis = np.linspace(y_min, y_max,
predictions_num) x_predictions = np.zeros((predictions_num * predictions_num,
1)) y_predictions = np.zeros((predictions_num * predictions_num, 1)) x_y_index
= 0 for x_index, x_value in enumerate(x_axis): for y_index, y_value in
enumerate(y_axis): x_predictions[x_y_index] = x_value y_predictions[x_y_index]
= y_value x_y_index += 1 z_predictions =
linear_regression.predict(np.hstack((x_predictions, y_predictions)))
plot_predictions_trace = go.Scatter3d( x=x_predictions.flatten(),
y=y_predictions.flatten(), z=z_predictions.flatten(), name='Prediction Plane',
mode='markers', marker={ 'size': 1, }, opacity=0.8, surfaceaxis=2, ) plot_data
= [plot_training_trace, plot_test_trace, plot_predictions_trace] plot_figure =
go.Figure(data=plot_data, layout=plot_layout) plotly.offline.plot(plot_figure)