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