October 2021
Intermediate to advanced
289 pages
8h 31m
Chinese
Content preview from 数据科学中的实用统计学(第2版)
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170
｜
第
5
章
5.3.1
逻辑响应函数和
logit
函数
逻辑回归的核心部分是
逻辑响应函数
和
logit
函数
，它们可以将一个概率（
0
和
1
之间）映
射到一个更加广阔的、适合线性建模的空间。
首先，不将结果变量看作一个二元标签，而是看作标签为“
1
”的概率
p
。我们可能想将
p
作为预测变量的一个线性函数进行建模：
0 11 2 2 qq
p xx x
ββ β β
= + + ++
但是，拟合这个模型不能保证
p
在
0
和
1
之间，而这是概率必须满足的条件。
所以，我们对预测变量使用一个
逻辑响应函数
（或者
逆
logit
函数
），以此对
p
进行建模：
0 11 2 2
()
1
1e
qq
xx x
p
ββ β β
− + + ++
=
+
这种变换可以保证
p
始终在
0
和
1
之间。
为了去掉分母中的指数表达式，我们不使用概率，而是使用
发生比
（
odds
）
。赌徒们非常熟
悉发生比，它是“成功”（
1
）与
“不成功”（
0
）的比率。用概率术语来说，发生比就是某
件事情发生的概率除以它不发生的概率。例如，如果一匹马获胜的概率为
0.5
，那么它不
会获胜的概率就是（
1–0.5
）
=
0.5
，那么发生比就是
1.0
：
Odds( 1)
1
p
Y
p
= =
−
我们可以根据发生比的逆函数由发生比得到概率：
Odds
1 Odds
p =
+
将这个公式与前面的逻辑响应函数结合起来，就可以得到：
0 11 2 2
Odds( 1) e
qq
xx x
Y
ββ β β
+ + ++
= =
最后，对两边取对数，就可以得到一个由预测变量的线性函数所组成的表达式：
0 11 2 2
ln(Odds( 1))
qq
Y xx ...
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I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.