在glm中包含线性相关特征

Question

我有一个训练集，x列代表一个特定的体育场，比赛正在进行。显然，在训练集中，列是线性依赖的，因为必须在至少一个体育场中进行匹配。

然而，我的问题是，如果我通过测试数据，它可能包括一个体育场没有看到的训练数据。因此，我希望在训练一个R glm时包括所有x列，这样每个体育场馆系数的平均值都是零。然后，如果看到一个新的体育场，它本质上将得到所有体育场系数的平均值。

我的问题是，them函数似乎检测到我的训练集中有线性相关的列，并将其中一个系数设置为NA，使其余的系数都是线性独立的。我如何：

停止R在glm函数中插入我的一个列的NA系数，并确保所有体育场系数之和为0？

一些示例代码

# Past observations
outcome   = c(1  ,0  ,0  ,1  ,0  ,1  ,0  ,0  ,1  ,0  ,1  )
skill     = c(0.1,0.5,0.6,0.3,0.1,0.3,0.9,0.6,0.5,0.1,0.4)
stadium_1 = c(1  ,1  ,0  ,0  ,0  ,0  ,0  ,0  ,0  ,0  ,0  )
stadium_2 = c(0  ,0  ,1  ,1  ,1  ,1  ,1  ,0  ,0  ,0  ,0  )
stadium_3 = c(0  ,0  ,0  ,0  ,0  ,0  ,0  ,1  ,1  ,1  ,1  )

train_glm_data = data.frame(outcome, skill, stadium_1, stadium_2,     stadium_3)
LR = glm(outcome ~ . - outcome, data = train_glm_data,  family=binomial(link='logit'))
print(predict(LR, type = 'response'))

# New observations (for a new stadium we have not seen before)
skill     = c(0.1)
stadium_1 = c(0  )
stadium_2 = c(0  )
stadium_3 = c(0  )

test_glm_data = data.frame(outcome, skill, stadium_1, stadium_2, stadium_3)
print(predict(LR, test_glm_data, type = 'response'))

# Note that in this case, the observation is simply the same as if we had observed stadium_3
# Instead I would like it to be an average of all the known stadiums coefficients
# If they all sum to 0 this is essentially already done for me
# However if not then the stadium_3 coefficient is buried somewhere in the intercept term

Answer 1

train_glm_data$stadium <- NA
train_glm_data$stadium[train_glm_data$stadium_1==1] <- "Stadium 1"
train_glm_data$stadium[train_glm_data$stadium_2==1] <- "Stadium 2"
train_glm_data$stadium[train_glm_data$stadium_3==1] <- "Stadium 3"
train_glm_data$stadium_1 <- NULL
train_glm_data$stadium_2 <- NULL
train_glm_data$stadium_3 <- NULL

train_glm_data$stadium         <- as.factor(train_glm_data$stadium)
levels(train_glm_data$stadium) <- c("Stadium 1", "Stadium 2", "Stadium 3", "Stadium 4")
train_glm_data                 <- rbind(train_glm_data, c(
                                      round(mean(outcome)), mean(skill),
                                      "Stadium 4"
                                    ))
train_glm_data$outcome <- as.numeric(train_glm_data$outcome)
train_glm_data$skill   <- as.numeric(train_glm_data$skill)
LR = glm(outcome ~ stadium + skill, data = train_glm_data,  family=binomial(link='logit'))
print(predict(LR, type = 'response'))

# New observations (for a new stadium we have not seen before)
skill     = c(0.1)
stadium   = "Stadium 4"

test_glm_data = data.frame(skill, stadium)
print(predict(LR, test_glm_data, type = 'response'))

关于如何包含所有级别的系数的问题-- 不这样做。它被称为虚拟变量陷阱。如果不排除引用级别，则数据矩阵将变为奇异。

唯一的例外是，如果您估计一个无截距模型。了解有关虚拟变量陷阱这里。的更多信息

Answer 2

要估计所有虚拟变量的系数，可以在公式中添加"-1“，这将删除截距：

LR = glm(outcome ~ . - outcome - 1, data = train_glm_data, family=binomial(link='logit'))

系数：

coef(LR)
#      skill  stadium_1  stadium_2  stadium_3 
# -2.8080177  0.8424053  0.7541226  1.1313135 

对于看不见的训练水平问题，@hack-r提出了一些好主意。另一个想法是为新观测的所有虚拟变量计算1/n (其中n是观察到的体育场的数量)。