penaltyLearning (version 2020.5.13)

modelSelectionC: Exact model selection function

Description

Given loss.vec L_i, model.complexity K_i, the model selection function i*(lambda) = argmin_i L_i + lambda*K_i, compute all of the solutions (i, min.lambda, max.lambda) with i being the solution for every lambda in (min.lambda, max.lambda). This function uses the linear time algorithm implemented in C code. This function is mostly meant for internal use -- it is instead recommended to use modelSelection.

Usage

modelSelectionC(loss.vec, 
    model.complexity, 
    model.id)

Arguments

loss.vec

numeric vector: loss L_i

model.complexity

numeric vector: model complexity K_i

model.id

vector: indices i

Value

data.frame with a row for each model that can be selected for at least one lambda value, and the following columns. (min.lambda, max.lambda) and (min.log.lambda, max.log.lambda) are intervals of optimal penalty constants, on the original and log scale; model.complexity are the K_i values; model.id are the model identifiers (also used for row names); and model.loss are the C_i values.

Examples

# NOT RUN {
loss.vec <- c(
  -9.9, -12.8, -19.2, -22.1, -24.5, -26.1, -28.5, -30.1, -32.2, 
  -33.7, -35.2, -36.8, -38.2, -39.5, -40.7, -41.8, -42.8, -43.9, 
  -44.9, -45.8)
seg.vec <- seq_along(loss.vec)
exact.df <- penaltyLearning::modelSelectionC(loss.vec, seg.vec, seg.vec)
## Solve the optimization using grid search.
L.grid <- with(exact.df,{
  seq(min(max.log.lambda)-1,
      max(min.log.lambda)+1,
      l=100)
})
lambda.grid <- exp(L.grid)
kstar.grid <- sapply(lambda.grid, function(lambda){
  crit <- with(exact.df, model.complexity * lambda + model.loss)
  picked <- which.min(crit)
  exact.df$model.id[picked]
})
grid.df <- data.frame(log.lambda=L.grid, segments=kstar.grid)
library(ggplot2)
## Compare the results.
ggplot()+
  ggtitle("grid search (red) agrees with exact path computation (black)")+
  geom_segment(aes(min.log.lambda, model.id,
                   xend=max.log.lambda, yend=model.id),
               data=exact.df)+
  geom_point(aes(log.lambda, segments),
             data=grid.df, color="red", pch=1)+
  ylab("optimal model complexity (segments)")+
  xlab("log(lambda)")

# }