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\in\lbrace 0, 1\rbrace$,其中 0 表示**负向类(negative class)**,代表恶性肿瘤("-"),1 为**正向类(positive class)**,代表良性肿瘤("+")。如图,定义最右边的样本为**偏差项**。 在未加入偏差项时,线性回归算法给出了品红色的拟合直线,若规定 @@ -36,7 +36,25 @@ $h_\theta(x) \lt 0.5$ ,预测为 $y = 0$,即负向类。 ## 6.2 假设函数表示(Hypothesis Representation) +为了使 $h \in \left(0, 1\right)$,引入逻辑回归模型,定义假设函数 +$$ +h_\theta \left( x \right)=g\left(\theta^{T}X \right) +$$ +对比线性回归函数 $h_\theta \left( x \right)=\theta^{T}X$,$g$ 表示逻辑函数(logistic function),复合起来,则为线性回归函数。 +一个常用的逻辑函数是 S 形函数,叫做 **sigmoid 函数**(如下图),其公式为 $g\left( z \right)=\frac{1}{1+{{e}^{-z}}}$。 + +![sigmoid function](image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png) + +应用 sigmoid 函数,则逻辑回归模型:$$h_{\theta}(x)=g(\theta^Tx) =\frac{1}{1+e^{-\theta^Tx}}$$ + +$h_\theta \left( x \right)$ 的作用是,根据输入 $x$,参数 $\theta$ 计算得出”输出 $y=1$“的可能性(estimated probability),概率学中表示为: + +$\begin{align*}& h_\theta(x) = P(y=1 | x ; \theta) = 1 - P(y=0 | x ; \theta) \newline & P(y = 0 | x;\theta) + P(y = 1 | x ; \theta) = 1\end{align*}$ + + + +以肿瘤诊断为例,$h_\theta \left( x \right)=0.7$ 表示有 $70\%$ 的概率为恶性肿瘤。 ## 6.3 Decision Boundary