Segmentation

John Mount

2019-01-02

In this example we fit a piecewise constant function to example data.
Please see here for a discussion of the methodology.

library("RcppDynProg")

plot <- requireNamespace("ggplot2", quietly = TRUE)
if(plot) {
  library("ggplot2")
}

set.seed(2018)
g <- 50
d <- data.frame(
  x = 1:(3*g)) # ordered in x
d$y_ideal <- c(rep(0, g), rep(1, g), rep(-1, g))
d$y_observed <- d$y_ideal + rnorm(length(d$y_ideal))



if(plot) {
  plt1 <- ggplot(data= d, aes(x = x)) + 
    geom_line(aes(y = y_ideal), linetype=2) +
    geom_point(aes(y = y_observed)) +
    ylab("y") +
    ggtitle("raw data", 
            subtitle = "dots: observed values, dashed line: unobserved true values")
  print(plt1)
}

As a heuristic, we set our regularization penalty to a value that treats permuted data (no relation between x and y) as a single partition.

y_permuted <- d$y_ideal[sample.int(nrow(d), nrow(d), replace = FALSE)]


solve_with_penalty <- function(ycol, penalty) {
  n <- length(ycol)
  indices = seq_len(n)
  x <- const_costs(ycol, 1+numeric(n), 1, indices)
  x <- x + penalty
  solve_interval_partition(x, n)
}

lb <- 1
ub <- 10
while(length(solve_with_penalty(y_permuted, ub))>2) {
  ub <- ub*2
}
while(TRUE) {
  mid <- ceiling((ub+lb)/2)
  if(mid>=ub) {
    break
  }
  si <- solve_with_penalty(y_permuted, mid)
  if(length(si)<=2) {
    ub <- mid
  } else {
    lb <- mid
  }
}
print(ub)
## [1] 5

We now use this penalty to segment the data. Notice we recover the actual problem structure.

soln <- solve_with_penalty(d$y_observed, ub)
print(soln)
## [1]   1  50 101 151
d$group <- as.character(findInterval(d$x, soln))
group_means <- tapply(d$y_observed, d$group, mean)
d$group_mean <- group_means[d$group]

print(sum((d$y_observed - d$y_ideal)^2))
## [1] 151.876
print(sum((d$group_mean - d$y_ideal)^2))
## [1] 2.973557
if(plot) {
  plt2 <- ggplot(data= d, aes(x = x)) + 
    geom_line(aes(y = y_ideal), linetype=2) +
    geom_point(aes(y = y_observed, color = group)) +
    geom_line(aes(y = group_mean, color = group)) +
    ylab("y") +
    ggtitle("RcppDynProg piecewise constant estimate",
            subtitle = "dots: observed values, segments: observed group means, dashed line: unobserved true values") + 
    theme(legend.position = "none")
  print(plt2)
}

The same solution through the more succinct solve_for_partitionc() interface.

# x_cuts <- solve_for_partition(d$x, d$y_observed)
# sometimes a different penalty due to problem chunking
x_cuts <- solve_for_partitionc(d$x, d$y_observed, penalty = ub)
print(x_cuts)
##     x       pred group  what
## 1   1 -0.1147501     1  left
## 2  49 -0.1147501     1 right
## 3  50  1.1321951     2  left
## 4 100  1.1321951     2 right
## 5 101 -0.9412289     3  left
## 6 150 -0.9412289     3 right
d$estimate <- approx(x_cuts$x, x_cuts$pred, xout = d$x, method = "constant", rule = 2)$y
d$group <- as.character(findInterval(d$x, x_cuts[x_cuts$what=="left", "x"]))

print(sum((d$y_observed - d$y_ideal)^2))
## [1] 151.876
print(sum((d$estimate - d$y_ideal)^2))
## [1] 2.973557
print(sum((d$estimate - d$y_observed)^2))
## [1] 147.9564
if(plot) {
  plt3 <- ggplot(data= d, aes(x = x)) + 
    geom_line(aes(y = y_ideal), linetype=2) +
    geom_point(aes(y = y_observed, color = group)) +
    geom_line(aes(y = estimate, color = group)) +
    ylab("y") +
    ggtitle("RcppDynProg piecewise constant estimate",
            subtitle = "dots: observed values, segments: observed group means, dashed line: unobserved true values") + 
    theme(legend.position = "none")
  print(plt3)
}