[Machine Learning]4.逻辑回归（logistic regression）

时间：2022-03-30 23:49:18

接下来我们将从回归问题切换至分类问题。分类问题的名字logistic regression是由于历史原因，但事实上讨论的并不是用于regression问题，而是classification问题。


分类

在这时，我们的输出值y再也不是连续量，而是离散值，不是0就是1.

y=0代表反向类别，y=1代表正向类别，当然我们也可以根据自身需要给y自己的类别定义。

现在我们只讨论只有两个类别的问题，称为“二元分类问题”。


假说函数表示

根据我们前面的定义，我们的假说函数应该满足

[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）

我们在这里的选择是sigmoid函数，也成为“逻辑函数”

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sigmoid函数图像如下：

[Machine Learning]4.逻辑回归（logistic regression）

可以看到，在Z=0,也即预估值
[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）=0的时候，假说函数等于0.5,因此，可以用假说函数大等于于0.5说明当前预测值是正向类别的，反之则是反向类别的。

假说函数事实上提供的是一种概率，即类别是正向类别的概率。


决策边界

区分两个类别的边界称为决策边界，其中在如果取
[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）=0作为决策边界，那么由满足
[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）=0的所有x所构成的线即使决策边界，当然，也可以根据需求采取其他的决策边界，比如取满足
[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）=1.0的x值构成的边界作为决策边界，那么此时的分类就会比较严格，在做一些需要安全性较高的预测可以采用这种决策边界。


开销函数

在分类问题中，由于具有不同的类别并且会带来大量的局部最优点，因此不能像回归问题那样简单的计算方差。

取而代之的，我们采用如下的开销函数形式：

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假说函数值越原离目标值，开销越大。

这个开销函数的图像是一个convex的图像，也就是说总能收敛到全局最优点。


简化的开销函数

我们将分类讨论的开销函数整合成一个函数
[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression）[Machine Learning]4.逻辑回归（logistic regression），可得

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向量化：

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梯度下降

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向量化：

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可以发现与回归问题的公式是一样的，但其实是不一样的，因为这里采用的是logistic函数


多类别分类问题：一对多

当问题存在多个类别时，我们依然可以利用我们上面所用的方法，技巧是将多个类别看作是多个二元分类的问题，对每一个类别都计算它们自身的假说函数值（其实二元分类也是具有两个假说函数，只是由于和为1,因此另一个是冗余的），然后取其中具有最大的假说函数值的作为当前输入的类别。


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参考：https://share.coursera.org/wiki/index.php/ML:Logistic_Regression


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