5-3.简单的线性回归

import numpy as np
import matplotlib.pyplot as plt

x = np.array([1., 2., 3., 4., 5.])
y = np.array([1., 3., 2., 3., 5.])

plt.scatter(x, y)
plt.axis([0, 6, 0, 6])
plt.show()

​

​

用最小二乘法求出a， b

x_mean = np.mean(x)
y_mean = np.mean(y)

fenzi = 0.0
fenmu = 0.0
for x_i, y_i in zip(x, y):
    fenzi += (x_i - x_mean) * (y_i - y_mean)
    fenmu += (x_i - x_mean) ** 2

a = fenzi / fenmu
b = y_mean - a * x_mean

print(a)
print(b)

0.8
0.39999999999999947

#绘制直线
y_hat = a * x + b
plt.scatter(x, y)
plt.plot(x, y_hat, color = "red")
plt.axis([0, 6, 0, 6])
plt.show()

​

​

#新的样本
x_predict = 6
y_predict = a * x_predict + b
y_predict

5.2

使用我们自己封装的SimpleLinearRegression

在同级目录下使用pycharm新建工程play_ML

工程里新建一个python脚本命名为SimpleLinearRegression，写入以下代码

import numpy as np
from .metrics import r2_score


class SimpleLinearRegression1:
    def __init__(self):
        """初始化Simple Liner Regression模型"""
        self.a_ = None
        self.b_ = None

    def fit(self, x_train, y_train):
        """根据训练数据集X_train和y_train训练Simple Linear Regression模型"""
        assert x_train.ndim == 1, \
            "the Simple Linear Regression can only solve single fearture training data"
        assert len(x_train) == len(y_train), \
            "the size of x_train must be equal to the size of y_train"

        x_mean = np.mean(x_train)
        y_mean = np.mean(y_train)

        fenzi = 0.0
        fenmu = 0.0
        for x_i, y_i in zip(x_train, y_train):
            fenzi += (x_i - x_mean) * (y_i - y_mean)
            fenmu += (x_i - x_mean) ** 2

        self.a_ = fenzi / fenmu
        self.b_ = y_mean - self.a_ * x_mean

        return self

    def predict(self, x_predict):
        """对于给定的待预测数据集x_predict，返回预测结果向量"""
        assert x_predict.ndim == 1, \
            "the Simple Linear Regression can only solve single fearture training data"
        assert self.a_ is not None and self.b_ is not None, \
            "must be fitte before predict!"

        return np.array([self._predict(x) for x in x_predict])

    def _predict(self, x_single):
        """给定单个预测值x_single，返回预测结果"""

        return self.a_ * x_single + self.b_

    def __repr__(self):
        return "SimpleLinearRegression1()"


class SimpleLinearRegression2:
    def __init__(self):
        """初始化Simple Liner Regression模型"""
        self.a_ = None
        self.b_ = None

    def fit(self, x_train, y_train):
        """根据训练数据集X_train和y_train训练Simple Linear Regression模型"""
        assert x_train.ndim == 1, \
            "the Simple Linear Regression can only solve single feature training data"
        assert len(x_train) == len(y_train), \
            "the size of x_train must be equal to the size of y_train"

        x_mean = np.mean(x_train)
        y_mean = np.mean(y_train)

        fenzi = (x_train - x_mean).dot(y_train - y_mean)
        fenmu = (x_train - x_mean).dot(x_train - x_mean)

        self.a_ = fenzi / fenmu
        self.b_ = y_mean - self.a_ * x_mean

        return self

    def predict(self, x_predict):
        """对于给定的待预测数据集x_predict，返回预测结果向量"""
        assert x_predict.ndim == 1, \
            "the Simple Linear Regression can only solve single feature training data"
        assert self.a_ is not None and self.b_ is not None, \
            "must be fitted before predict!"

        return np.array([self._predict(x) for x in x_predict])

    def _predict(self, x_single):
        """给定单个预测值x_single，返回预测结果"""

        return self.a_ * x_single + self.b_

    def score(self, x_test, y_test):
        """根据测试数据集x_test和y_test确定当前模型的准确度"""
        y_predict = self.predict(x_test)

        return r2_score(y_test, y_predict)

    def __repr__(self):
        return "SimpleLinearRegression2()"

导入自定义的回归函数

from play_ML.SimpleLinearRegression import SimpleLinearRegression1

reg1 = SimpleLinearRegression1()

reg1.fit(x, y)

SimpleLinearRegression1()

reg1.predict(np.array([x_predict]))

array([5.2])

reg1.a_

0.8

reg1.b_

0.39999999999999947

#得到模型
y_hat1 = reg1.predict(x)
#绘图
plt.scatter(x, y)
plt.plot(x, y_hat1, color = "red")
plt.axis([0, 6, 0, 6])
plt.show()

​

​

向量化运算 实现SimpleLinearRegression

from play_ML.SimpleLinearRegression import SimpleLinearRegression2

reg2 = SimpleLinearRegression2()
reg2.fit(x, y)

SimpleLinearRegression2()

reg2.a_

0.8

reg2.b_

0.39999999999999947

#得到模型
y_hat2 = reg2.predict(x)
#绘图
plt.scatter(x, y)
plt.plot(x, y_hat1, color = "red")
plt.axis([0, 6, 0, 6])
plt.show()

​

​

向量化实现的性能测试

m = 1000000
big_x = np.random.random(size=m)
big_y = big_x * 2.0 + 3 + np.random.normal(size=m)

%timeit reg1.fit(big_x, big_y)
%timeit reg2.fit(big_x, big_y)

639 ms ± 11 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
7.76 ms ± 197 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

print(reg1.a_)
print(reg1.b_)

2.0045382114974615
2.9982778832590267

print(reg2.a_)
print(reg2.b_)

2.0045382114972243
2.9982778832591452