一、目標

              尋找一條直線，最大程度的“擬合”樣本特徵和樣本輸出標記之間的關係。在迴歸問題中我們預測的是一個具體的數值，這個具體的數值是在一個連續的空間裡的，如果想看兩個特徵的迴歸問題就需要在三維空間裡進行觀察。樣本特徵有多個的迴歸稱為多元線性迴歸

損失函式

對a求偏導數：

最後得到的結果：

求a、b的Python程式碼：

封裝SampleLinearRegression演算法的程式碼實現

"""coding:utf-8"""
import numpy as np
class SimpleLinearRegression(object):
    def __init__(self):
        """初始化Simple Linear Regression 模型"""
        self.a_ = None
        self.b_ = None
    def fit(self,x_train,y_train):
        """根據訓練資料集x_train,y_train訓練Simple Linear Regression模型"""
        assert x_train.ndim == 1, \
            "Simple Linear Regressor 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)
        num = 0.0
        d = 0.0
        for x,y in zip(x_train,y_train):
            num += (x-x_mean)*(y-y_mean)
            d += (x-x_mean)**2

        self.a_ = num/d
        self.b_ = y_mean-self.a_*x_mean
        return self

    def predict(self,x_predict):
        """給定待預測資料集x_predict,返回表示x_predict的結果向量"""
        assert x_predict.ndim == 1, \
            "Simple Linear Regressor can only solve single feature training data."
        assert self.a_ is not None and self.b_ is not None, \
            "must fit before predict!"
        return np.array([self._predict(x) for x in x_predict])

    def _predict(self,x):
        """給定單個待預測資料x，返回x的預測結果值"""
        return self.a_ * x +self.b_

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

檢驗封裝演算法的測試程式碼

"""coding:utf-8"""
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()
from play_ML.SimpleLinearRrgression import SimpleLinearRegression
slr = SimpleLinearRegression()
slr.fit(X,y)
y_hat = slr.predict(X)
plt.scatter(X,y)
plt.plot(X,y_hat,color="r")
plt.axis([0,6,0,6])
plt.show()
 

測試結果