我正在尝试使用 mgcv::gam 创建一个逻辑回归模型，我认为这是一个简单的决策边界，但我构建的模型表现很差。使用 locfit::locfit 在相同数据上构建的局部回归模型很容易找到边界。我想在我的真实模型中添加额外的参数回归器，所以我不想切换到纯粹的局部回归。
我想了解为什么 GAM 无法拟合数据，以及是否有方法可以指定可以更好地执行的平滑。
这是一个简化的、可重现的示例：
基本事实是 1 = 点位于单位圆内，如果在单位圆外则为 0
例如 z = 1 如果 sqrt(x^2 + y^2) <= 1，否则为 0
观察到的数据是嘈杂的，既有误报也有误报
根据点的笛卡尔坐标，构建逻辑回归以预测点是否在圆内。
局部回归可以很好地找到边界（50% 的概率轮廓非常接近单位圆），但逻辑 GAM 始终高估相同概率带的圆的大小。
library(ggplot2)
library(locfit)
library(mgcv)
library(plotrix)
set.seed(0)
radius <- 1 # actual boundary
n <- 10000 # data points
jit <- 0.5 # noise factor
# Simulate random data, add polar coordinates
df <- data.frame(x=runif(n,-3,3), y=runif(n,-3,3))
df$r <- with(df, sqrt(x^2+y^2))
df$theta <- with(df, atan(y/x))
# Noisy indicator for inside the boundary
df$inside <- with(df, ifelse(r < radius + runif(nrow(df),-jit,jit),1,0))
# Plot data, shows ragged edge
(ggplot(df, aes(x=x, y=y, color=inside)) + geom_point() + coord_fixed() + xlim(-4,4) + ylim(-4,4))
### Model boundary condition using x,y coordinates
### local regression finds the boundary pretty accurately
m.locfit <- locfit(inside ~ lp(x,y, nn=0.3), data=df, family="binomial")
plot(m.locfit, asp=1, xlim=c(-2,-2,2,2))
draw.circle(0,0,1, border="red")
### But GAM fits very poorly, also tried with fx=TRUE but didn't help
m.gam <- gam(inside ~ s(x,y), data=df, family=binomial)
plot(m.gam, trans=plogis, se=FALSE, rug=FALSE)
draw.circle(0,0,1, border="red")
### gam.check doesn't indicate a problem with the model itself
gam.check(m.gam)
Method: UBRE Optimizer: outer newton
full convergence after 8 iterations.
Gradient range [5.41668e-10,5.41668e-10]
(score -0.815746 & scale 1).
Hessian positive definite, eigenvalue range [0.0002169789,0.0002169789].
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.
k' edf k-index p-value
s(x,y) 29.000 13.795 0.973 0.08
#### Try using polar coordinates
### Again, locfit works well
m.locfit2 <- locfit(inside ~ lp(r, nn=0.3), data=df, family="binomial")
plot(m.locfit2)
abline(v=1, col="red")
### But GAM misses again
m.gam2 <- gam(inside ~ s(r, k=50), data=df, family=binomial)
plot(m.gam2, se=FALSE, rug=FALSE, trans=plogis)
abline(v=1, col="red")
### Can also plot gam on link scale for alternate view
plot(m.gam2, se=FALSE, rug=FALSE)
abline(v=1, col="red")
gam.check(m.gam2)
Method: UBRE Optimizer: outer newton
full convergence after 4 iterations.
Gradient range [-3.29203e-08,-3.29203e-08]
(score -0.8240065 & scale 1).
Hessian positive definite, eigenvalue range [7.290233e-05,7.290233e-05].
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.
k' edf k-index p-value
s(r) 49.000 10.537 0.979 0.06
