一、广义线性模型概念

在讨论广义线性模型之前,先回顾一下基本线性模型,也就是线性回归

在线性回归模型中的假设中,有两点需要提出:

(1)假设因变量服从高斯分布:$Y={{\theta }^{T}}x+\xi $,其中误差项$\xi \sim N(0,{{\sigma }^{2}})$,那么因变量$Y\sim N({{\theta }^{T}}x,{{\sigma }^{2}})$。  

(2)模型预测的输出为$E[Y]$,根据$Y={{\theta }^{T}}x+\xi $,$E[Y]=E[{{\theta }^{T}}x+\xi ]={{\theta }^{T}}x$,记$\eta ={{\theta }^{T}}x$,则$\eta =E[Y]$

广义线性模型可以认为在以上两点假设做了扩展:

(1)因变量分布不一定是高斯分布,服从一个指数分布族(见下文)即可。

(2)模型预测输出仍然可以认为是$E[Y]$(实际上是$E[T(Y)]$,许多情况下$T(Y)=Y$),但是$Y$的分布不一定是高斯分布,$E[Y]$和$\eta ={{\theta }^{T}}x$也不一定是简单的相等关系,它们的关系用$\eta =g(E[Y])$描述,称为连接函数,其中$\eta $称为自然参数。

由于以上两点的扩展,广义线性模型的应用比基本线性模型广泛许多。对于广义线性这个术语,可以理解为广义体现在因变量的分布形式比较广,只要是一指数分布族即可,而线性则体现在自然参数$\eta ={{\theta }^{T}}x$是$\theta $的线性函数。

二、广义线性模型的构建

上文提到指数分布族,它是广义线性模型的基础,所以先简单了解一下指数分布族。

对于变量$y$,如果其分布可写成$p(y;\eta )=b(\eta )\exp ({{\eta }^{T}}T(y)-a(\eta ))$的形式,则称$y$服从一个指数分布族,自然参数$\eta $是分布的参数。为什么这样定义是牛逼的数学家弄的,咱就看看它在广义线性模型中怎么用的吧~

实际中的许多分布都是一个指数分布族,如高斯分布,二项分布,泊松分布,多项分布等等,所以之前写的线性回归逻辑回归实际上都是个广义线性模型。以逻辑回归为例来看看。

逻辑回归假设$y$服从参数为$\phi $伯努利分布,$p(y)={{\phi }^{y}}{{(1-\phi )}^{1-y}}$,$E[y]=\phi $。

下面将其写出指数分布族的形式:

$\begin{align}  p(y)&={{\phi }^{y}}{{(1-\phi )}^{1-y}} \\& =\exp (\log ({{\phi }^{y}}{{(1-\phi )}^{1-y}})) \\& =\exp (y\log (\phi )+(1-y)\log (1-\phi )) \\ & =\exp (y\log (\frac{\phi }{1-\phi })+\log (1-\phi )) \\\end{align}$

与指数分布族的一般形式对比可发现:

$\begin{align}& b(\eta )=1 \\& \eta =\log (\frac{\phi }{1-\phi })\Rightarrow \phi =\frac{1}{1+{{e}^{-\eta }}} \\& T(y)=y \\& a(\eta )=-\log (1-\phi ) \\\end{align}$

可见这是符合假设(1)的。根据假设(2),我们预测的是

$\begin{align}{{h}_{\theta }}(x)&=E[T(y)] \\& =E[y](=\phi ) \\& =\frac{1}{1+{{e}^{-\eta }}} \\& =\frac{1}{1+{{e}^{-{{\theta }^{T}}x}}} \\\end{align}$

这正是之前逻辑回归的模型。

可以看出以上推导过程中$E[y]=\frac{1}{1+{{e}^{-\eta }}}$这一步比较重要,起到了连接预测值$E[y]$和自然参数$\eta $的作用,这就是连接函数的作用。

回顾假设(1)(2)和以上逻辑回归推导过程,可以看出,构建一个广义线性模型需要两个步骤:

(1)确定预测变量$y$的分布是一个指数分布族

(2)确定连接函数。连接函数可以是任意的,但根据上文可以看出,一但步骤(1)中的分布形式给定,就可以推导出一个连接函数,这个根据分布推导出的分布称为标准连接函数,也是通常默认采用的。所以一般步骤(1)中分布的形式给定,步骤(2)也就默认确定了。

经过以上两个步骤,模型就建立好了,接下来就是写出似然函数,最大化似然函数估计模型参数。对于广义线性模型的参数估计,有个专门的算法IRWLS(iteratively weighted least squares),感兴趣的可以查阅相关文献。另外,关于模型的假设检验,也不写了。写不出来。。。看来几天数学课本,头炸了。。还是安静的做个程序员吧~

三、广义线性模型应用

广义线性模型的应用最广泛的的是逻辑回归和泊松回归。逻辑回归将因变量建模为伯努利分布,输出是二值的,通常用来做二分类。泊松回归将因变量的分布建模为泊松分布,一般用来预测类似顾客数目、一个时间段内给定事件发生数目的问题。

另外,对于多分类问题,将因变量建模为多项分布也是一个广义线性模型。

之前在逻辑回归中,没有提到广义线性模型,现在可以直接用R中提供的广义线性模型来拟合。

 x<-read.table("q1x.txt");
x<-as.matrix(x);
y<-scan("q1y.txt");
y<-matrix(y,ncol=1); gfit<-glm(y~x,family=binomial());
print(coef(gfit));

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" alt="" />

可以看出这和之前拟合的参数基本一致。

泊松回归和多项回归没弄得数据,不代码模拟了~

附:R中GLM相关

R中用glm(formula,family...)函数来做广义线性模型,并且提供了一下指数族分布:

binomial(link = "logit")
gaussian(link = "identity")
Gamma(link = "inverse")
inverse.gaussian(link = "1/mu^2")
poisson(link = "log")
quasi(link = "identity", variance = "constant")
quasibinomial(link = "logit")
quasipoisson(link = "log")

用上选项指定glm()中family参数,就可以得到不同的模型。

另外,robust包中的glmRob()函数可用来拟合稳健的广义线性模型,包括稳健Logistic回归,稳健泊松回归等。当拟合Logistic回归模型数据出现离群点和强影响点时,稳健Logistic回归便可派上用场;
多项分布回归:若响应变量包含两个以上的无序类别,便可使用mlogit包中的mlogit()函数拟合多项Logistic回归;
序数Logistic回归:若响应变量是一组有序的类别(比如,信用风险为差/良/好),便可使用rms包中的lrm()函数拟合序数Logistic回归

参考资料

[1]Andrew Ng 机器学习视频讲义:http://cs229.stanford.edu/

[2]统计之都:http://cos.name/2011/01/how-does-glm-generalize-lm-assumption/

[3]《R语言实战》 /(美)科巴科弗(Kabacoff,R.I.)著;高涛,肖楠,陈钢译.人民邮电出版社,2013.1

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