【机器学习实战】五、实现非线性回归模型案例

非线性回归模型案例

一、导入第三方库、读取显示数据

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from linear_regression import LinearRegression

data = pd.read_csv('./data/non-linear-regression-x-y.csv')

x = data['x'].values.reshape((data.shape[0],1)) # 拿到x
y = data['y'].values.reshape((data.shape[0],1)) # 拿到y
data.head(10)

plt.plot(x , y)
plt.show()

部分数据展示：

通过折线图展示数据，横轴为x，纵轴为y

二、设置参数

num_iterations ：模型迭代次数；
learning_rate ：学习率；
polynomial_degree : 特征变换
sinusoid_degree ： sin(x) 非线性变换
normalize_data ：标准化数据

num_iterations = 5000 # 设置模型迭代次数5000
learning_rate = 0.02 # 学习率
polynomial_degree = 15 #
sinusoid_degree = 15
normalize_data = True

三、多项式特征变换

"""Add polynomial features to the features set"""
import numpy as np
from .normalize import normalize

def generate_polynomials(dataset, polynomial_degree, normalize_data=False):
    """变换方法：
    x1, x2, x1^2, x2^2, x1*x2, x1*x2^2, etc.
    """
    features_split = np.array_split(dataset, 2, axis=1)
    dataset_1 = features_split[0]
    dataset_2 = features_split[1]

    (num_examples_1, num_features_1) = dataset_1.shape
    (num_examples_2, num_features_2) = dataset_2.shape

    if num_examples_1 != num_examples_2:
        raise ValueError('Can not generate polynomials for two sets with different number of rows')

    if num_features_1 == 0 and num_features_2 == 0:
        raise ValueError('Can not generate polynomials for two sets with no columns')

    if num_features_1 == 0:
        dataset_1 = dataset_2
    elif num_features_2 == 0:
        dataset_2 = dataset_1

    num_features = num_features_1 if num_features_1 < num_examples_2 else num_features_2
    dataset_1 = dataset_1[:, :num_features]
    dataset_2 = dataset_2[:, :num_features]
    polynomials = np.empty((num_examples_1, 0))

    for i in range(1, polynomial_degree + 1):
        for j in range(i + 1):
            polynomial_feature = (dataset_1 ** (i - j)) * (dataset_2 ** j)
            polynomials = np.concatenate((polynomials, polynomial_feature), axis=1)
    if normalize_data:
        polynomials = normalize(polynomials)[0]
    return polynomials

四、sin(x)非线性变换

import numpy as np

def generate_sinusoids(dataset, sinusoid_degree):
    """
    sin(x).
    """
    num_examples = dataset.shape[0]
    sinusoids = np.empty((num_examples, 0))

    for degree in range(1, sinusoid_degree + 1):
        sinusoid_features = np.sin(degree * dataset)
        sinusoids = np.concatenate((sinusoids, sinusoid_features), axis=1)
    return sinusoids

五、调用训练模型方法

linear_regression = LinearRegression(x , y, polynomial_degree ,sinusoid_degree , normalize_data)
(theta , cost_history) = linear_regression.train(
    learning_rate , num_iterations
)
print('开始损失 : {:.2f}'.format(cost_history[0])) # 第一个损失值
print('结束损失 : {:.2f}'.format(cost_history[-1])) # 最后一个损失值

六、绘制曲线图

绘制损失变化曲线图

theta_table = pd.DataFrame({'Model Parameters' : theta.flatten()})
plt.plot(range(num_iterations), cost_history)
plt.xlabel('Lterations')
plt.ylabel('Cost')
plt.title('Gradient Descent Progress')
plt.show()

绘制非线性回归预测模型折线图

predictions_num = 1000
x_predictions = np.linspace(x.min() , x.max() , predictions_num).reshape(predictions_num , 1)
y_predictions = linear_regression.predict(x_predictions)

plt.scatter(x, y, label = 'Training Dataset')
plt.plot(x_predictions , y_predictions , 'r' ,label = 'Prediction')
plt.show()