目录

 

最小二乘法

5-3 简单线性回归的实现

实现简单线性回归法

封装我们自己的SimpleLinearRegression

5-4 向量化

向量化实现SimpleLinearRegression

向量化实现的性能测试

小测试 y = 2*x +1

---

 

 

 

之前的分类问题横纵轴都是特征，这节的回归问题则一个轴是特征，另一个轴已经是标签了，注意区分 

y = ax+b

 

不加绝对值可能正负相加为零---》| |绝对值不是处处可导的函数，后续的计算时需要函数可导----》（---）^2

找到模型，使其最大拟合数据

近乎所有参数学习算法都是这样的套路，很多算法都是这样

 

最小二乘法

a,b是未知数， x,y是监督学习提供的已知的数据

对a变形，便于计算

5-3 简单线性回归的实现

实现简单线性回归法

封装我们自己的SimpleLinearRegression

SimpleLinearRegression.py

import numpy as np


class SimpleLinearRegression1:

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

    def fit(self, x_train, y_train):
        """根据训练数据集x_train,y_train训练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):
        """给定待预测数据集x_predict，返回表示x_predict的结果向量"""
        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_single):
        """给定单个待预测数据x，返回x的预测结果值"""
        return self.a_ * x_single + self.b_

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

模块化需要__init__.py脚本其是空的

5-4 向量化

向量运算的效率会很高

 

向量化实现SimpleLinearRegression

 

SimpleLinearRegression.py

import numpy as np


class SimpleLinearRegression1:

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

    def fit(self, x_train, y_train):
        """根据训练数据集x_train,y_train训练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):
        """给定待预测数据集x_predict，返回表示x_predict的结果向量"""
        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_single):
        """给定单个待预测数据x，返回x的预测结果值"""
        return self.a_ * x_single + self.b_

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


class SimpleLinearRegression2:

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

    def fit(self, x_train, y_train):
        """根据训练数据集x_train,y_train训练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)

        self.a_ = (x_train - x_mean).dot(y_train - y_mean) / (x_train - x_mean).dot(x_train - x_mean)
        self.b_ = y_mean - self.a_ * x_mean

        return self

    def predict(self, x_predict):
        """给定待预测数据集x_predict，返回表示x_predict的结果向量"""
        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_single):
        """给定单个待预测数据x_single，返回x_single的预测结果值"""
        return self.a_ * x_single + self.b_

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

向量化实现的性能测试

小测试 y = 2*x +1