# Why does this logistic GAM fit so poorly?

I am trying to create a logistic regression model with mgcv::gam with what I think is a simple decision boundary, but the model I build performs very poorly. A local regression model built using locfit::locfit on the same data finds the boundary very easily. I want to add additional parametric regressors to my real-life model, so I do not want to switch to a purely local regression.

I want to understand why GAM is having trouble fitting the data, and whether there was ways of specifying the smooths that can perform better.

Here's a simplified, reproducible example:

Ground truth is 1 = point lies within the unit circle, 0 if outside

e.g. z = 1 if sqrt(x^2 + y^2) <= 1, 0 otherwise

The observed data is noisy, with both false positives and false negatives

Construct a logistic regression to predict whether a point is inside the circle or not, based on the point's Cartesian coordinates.

Local regression can find the boundary well (50% probability contour is very close to the unit circle), but a logistic GAM consistently overestimates the size of the circle for the same probability band.

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  ## 1 Answer You are ignoring the model intercept when evaluating the model fit. The plot method shows the fitted spline, but the model includes a parametric constant term, just like the intercept in a standard logistic regression model. Instead, predict from the fitted model using the predict() method for locations on a grid of locations over the interval. For example: m.gam <- gam(inside ~ te(x, y), data=df, family=binomial, method = "REML") locs <- with(df, data.frame(x = seq(min(x), max(x), length = 100), y = seq(min(y), max(y), length = 100))) pred <- expand.grid(locs) pred <- transform(pred, fitted = predict(m.gam, newdata = pred, type = "response")) contour(locs$x, locs$y, matrix(pred$fitted, ncol = 100))
draw.circle(0, 0, 1, border="red")


which gives

Using a te() smoother seems to do a bit better than s() and I used method = "REML" as this can help with situations where the objective function in GCV/UBRE-based selection can become flat (and hence these methods can undersmooth), in case that was the problem here.