Segmented regression with quadratic polynomial and a strightline

Question

I am trying to implement segmented regression as per this example Segmented Regression, Breakpoint analysis.

Now, how can i implement it in such a way the second part will be quadratic polynomial and remaining other things same.

I tried the same by changing Z= ~poly(DistanceMeters, 2) however it didn't work.

Also, How can I get equations like

part 1: a1*x+b1
part 2: a2*x2**2 + b2*x + c1
part 3 :a3*x + b3

There are similar questions like this however they din't explain using segmented function.

Answer 1

I have two ideas, both with drawbacks. Maybe you can adjust one of them to your needs. Unfortunately cannot access Drive at the moment, so some artificial data used.

1. "Manually" fit polynomial models

Here you can specify whichever models you like, some segments can be lm's, some polynomials etc.

Code:

library(segmented)
library(ggplot2)
library(data.table)

# Data
set.seed(12)
xx <- 1:100
yy <- 2 + 1.5 * pmax(xx-35, 0) - 1.5 * pmax(xx-70, 0) + 15 * pmax(runif(100) - 0.5, 0) + rnorm(100, 0, 2)

dt <- data.table(x = xx, y = yy, type = 'act')
dt_all <- copy(dt)

# lm
lm_lin <- lm(y ~ x, data = dt)
summary(lm_lin)

# Find segments
lm_seg <- segmented(
  lm_lin, seg.Z = ~ x, psi = list(x = c(30, 80)))

# "Manual" lm's
breaks <- unname(lm_seg$psi[, 'Est.'])
lm_poly1 <- lm(y ~ poly(x, 4), data = dt[x < breaks[1], ])
lm_2 <- lm(y ~ x, data = dt[x > breaks[1] & x < breaks[2], ])
lm_poly3 <- lm(y ~ poly(x, 4), data = dt[x > breaks[2], ])

dt_all <- rbind(
  dt_all,
  data.table(x = xx, y = c(
    predict(lm_poly1),
    predict(lm_2),
    predict(lm_poly3)),
    type = 'lm_poly'
  )
)

2. Fit a gam model using breaks from segmented and some splines

Here you will get a smooth transition between segments, but you have much less control on what is happening.

# Using splines for smooth segments
library(mgcv)

spl <- gam(y ~ s(x, bs = "cc", k = 12), data = dt, knots = list(xx = breaks))

# Plot
dt_all <- rbind(dt_all, data.table(x = xx, y = predict(spl), type = 'spl'))
ggplot(dt_all, aes(x = x, y = y)) + geom_point(size = 1) +
  facet_grid(. ~ type) + theme_minimal()

Both can be done using e.g. list() and lapply() to automate a bit (for varying number of breaks etc.).

Edit:

By changing arguments of poly and s you can get slightly "better" fitting models, but for gam errors on the edges are quite big, see for degree = 6 and k = 30: