<!--
%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{Older resamplers}
-->

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  fig.width=6,
  fig.height=6)
data.table::setDTthreads(1)
## output: rmarkdown::html_vignette above creates html where figures are limited to 700px wide.
## Above CSS from https://stackoverflow.com/questions/34906002/increase-width-of-entire-html-rmarkdown-output main-container is for html_document, body is for html_vignette
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

The goal of this vignette is to explain the older resamplers:
`ResamplingVariableSizeTrainCV` and `ResamplingSameOtherCV`, which
output some data which are useful for visualizing the train/test
splits. If you do not want to visualize the train/test splits, then it
is recommended to instead use the newere resampler,
`ResamplingSameOtherSizesCV` (see other vignette).

# Same/Other/All resampler

The goal of thie section is to explain how to quantify the extent to
which it is possible to train on one data subset, and predict on
another data subset. This kind of problem occurs frequently in many
different problem domains:

* geography: can we train on one region (say Europe) and accurately
  predict on another? (North America)
* time series: can we train on one time period (2000) and accurately
  predict on another? (2001)
* personalization: can we train on one person (Alice) and accurately
  predict on another? (Bob)

The ideas are similar to my previous blog posts about how to do this
in
[python](https://tdhock.github.io/blog/2022/generalization-to-new-subsets/)
and [R](https://tdhock.github.io/blog/2023/R-gen-new-subsets/). Below
we explain how to use `mlr3resampling` for this purpose, in simulated
regression and classification problems. To use this method in
real data, the important sections to read below are named "Benchmark:
computing test error," which show how to create these cross-validation
experiments using mlr3 code.

## Simulated regression problems

We begin by generating some data which can be used with regression algorithms.
Assume there is a data set with some rows from one person, some rows
from another,

```{r}
N <- 300
library(data.table)
set.seed(1)
abs.x <- 2
reg.dt <- data.table(
  x=runif(N, -abs.x, abs.x),
  person=rep(1:2, each=0.5*N))
reg.pattern.list <- list(
  easy=function(x, person)x^2,
  impossible=function(x, person)(x^2+person*3)*(-1)^person)
reg.task.list <- list()
for(task_id in names(reg.pattern.list)){
  f <- reg.pattern.list[[task_id]]
  yname <- paste0("y_",task_id)
  reg.dt[, (yname) := f(x,person)+rnorm(N)][]
  task.dt <- reg.dt[, c("x","person",yname), with=FALSE]
  reg.task <- mlr3::TaskRegr$new(
    task_id, task.dt, target=yname)
  reg.task$col_roles$subset <- "person"
  reg.task$col_roles$stratum <- "person"
  reg.task$col_roles$feature <- "x"
  reg.task.list[[task_id]] <- reg.task
}
reg.dt
```

The table above shows some simulated data for two regression problems:

* easy problem has the same pattern for each person, so it is possible/easy to train on one person, and accurately predict on another.
* impossible problem has a different pattern for each person, so it is impossible to train on one person, and accurately predict on another.
* when adapting the code above to real data, the important part is the
  `mlr3::TaskRegr` line which tells mlr3 what data set to use, what is
  the target column, and what is the subset/stratum column.

### Static visualization of simulated data

First we reshape the data using the code below,

```{r}
(reg.tall <- nc::capture_melt_single(
  reg.dt,
  task_id="easy|impossible",
  value.name="y"))
```

The table above is a more convenient form for the visualization which we create using the code below,

```{r}
if(require(animint2)){
  ggplot()+
    geom_point(aes(
      x, y),
      data=reg.tall)+
    facet_grid(
      task_id ~ person,
      labeller=label_both,
      space="free",
      scales="free")+
    scale_y_continuous(
      breaks=seq(-100, 100, by=2))
}
```

In the simulated data above, we can see that 

* for the easy pattern, it is the same for both people, so it should
  be possible/easy to train on one person, and accurately predict on
  another.
* for the impossible pattern, it is different for each person, so it
  should not be possible to train on one person, and accurately
  predict on another.
  
### Benchmark: computing test error

In the code below, we define a K-fold cross-validation experiment.

```{r}
(reg_same_other <- mlr3resampling::ResamplingSameOtherCV$new())
```

In the code below, we define two learners to compare,

```{r}
(reg.learner.list <- list(
  if(requireNamespace("rpart"))mlr3::LearnerRegrRpart$new(),
  mlr3::LearnerRegrFeatureless$new()))
```

In the code below, we define the benchmark grid, which is all
combinations of tasks (easy and impossible), learners (rpart and
featureless), and the one resampling method.

```{r}
(reg.bench.grid <- mlr3::benchmark_grid(
  reg.task.list,
  reg.learner.list,
  reg_same_other))
```

In the code below, we execute the benchmark experiment (in parallel
using the multisession future plan).

```{r}
if(FALSE){#for CRAN.
  if(require(future))plan("multisession")
}
if(require(lgr))get_logger("mlr3")$set_threshold("warn")
(reg.bench.result <- mlr3::benchmark(
  reg.bench.grid, store_models = TRUE))
```

The code below computes the test error for each split,

```{r}
reg.bench.score <- mlr3resampling::score(reg.bench.result)
reg.bench.score[1]
```

The code below visualizes the resulting test accuracy numbers.

```{r}
if(require(animint2)){
  ggplot()+
    scale_x_log10()+
    geom_point(aes(
      regr.mse, train.subsets, color=algorithm),
      shape=1,
      data=reg.bench.score)+
    facet_grid(
      task_id ~ person,
      labeller=label_both,
      scales="free")
}
```

It is clear from the plot above that 

* for the easy task, training on same is just as good as all or other
  subsets. rpart has much lower test error than featureless, in all
  three train subsets.
* for the impossible task, the least test error is using rpart with same train subsets; featureless with same train subsets is next best; training on all is substantially worse (for both featureless and rpart); training on other is even worse (patterns in the two people are completely different).
* in a real data task, training on other will most likely not be quite as bad as in the impossible task above, but also not as good as in the easy task.

### Interactive visualization of data, test error, and splits

The code below can be used to create an interactive data visualization
which allows exploring how different functions are learned during
different splits.

```{r SimulationsAnimintRegression}
inst <- reg.bench.score$resampling[[1]]$instance
rect.expand <- 0.2
grid.dt <- data.table(x=seq(-abs.x, abs.x, l=101), y=0)
grid.task <- mlr3::TaskRegr$new("grid", grid.dt, target="y")
pred.dt.list <- list()
point.dt.list <- list()
for(score.i in 1:nrow(reg.bench.score)){
  reg.bench.row <- reg.bench.score[score.i]
  task.dt <- data.table(
    reg.bench.row$task[[1]]$data(),
    reg.bench.row$resampling[[1]]$instance$id.dt)
  names(task.dt)[1] <- "y"
  set.ids <- data.table(
    set.name=c("test","train")
  )[
  , data.table(row_id=reg.bench.row[[set.name]][[1]])
  , by=set.name]
  i.points <- set.ids[
    task.dt, on="row_id"
  ][
    is.na(set.name), set.name := "unused"
  ]
  point.dt.list[[score.i]] <- data.table(
    reg.bench.row[, .(task_id, iteration)],
    i.points)
  i.learner <- reg.bench.row$learner[[1]]
  pred.dt.list[[score.i]] <- data.table(
    reg.bench.row[, .(
      task_id, iteration, algorithm
    )],
    as.data.table(
      i.learner$predict(grid.task)
    )[, .(x=grid.dt$x, y=response)]
  )
}
(pred.dt <- rbindlist(pred.dt.list))
(point.dt <- rbindlist(point.dt.list))
set.colors <- c(
  train="#1B9E77",
  test="#D95F02",
  unused="white")
algo.colors <- c(
  featureless="blue",
  rpart="red")
make_person_subset <- function(DT){
  DT[, "person/subset" := person]
}
make_person_subset(point.dt)
make_person_subset(reg.bench.score)

if(require(animint2)){
  viz <- animint(
    title="Train/predict on subsets, regression",
    pred=ggplot()+
      ggtitle("Predictions for selected train/test split")+
      theme_animint(height=400)+
      scale_fill_manual(values=set.colors)+
      geom_point(aes(
        x, y, fill=set.name),
        showSelected="iteration",
        size=3,
        shape=21,
        data=point.dt)+
      scale_color_manual(values=algo.colors)+
      geom_line(aes(
        x, y, color=algorithm, subset=paste(algorithm, iteration)),
        showSelected="iteration",
        data=pred.dt)+
      facet_grid(
        task_id ~ `person/subset`,
        labeller=label_both,
        space="free",
        scales="free")+
      scale_y_continuous(
        breaks=seq(-100, 100, by=2)),
    err=ggplot()+
      ggtitle("Test error for each split")+
      theme_animint(height=400)+
      scale_y_log10(
        "Mean squared error on test set")+
      scale_fill_manual(values=algo.colors)+
      scale_x_discrete(
        "People/subsets in train set")+
      geom_point(aes(
        train.subsets, regr.mse, fill=algorithm),
        shape=1,
        size=5,
        stroke=2,
        color="black",
        color_off=NA,
        clickSelects="iteration",
        data=reg.bench.score)+
      facet_grid(
        task_id ~ `person/subset`,
        labeller=label_both,
        scales="free"),
    diagram=ggplot()+
      ggtitle("Select train/test split")+
      theme_bw()+
      theme_animint(height=300)+
      facet_grid(
        . ~ train.subsets,
        scales="free",
        space="free")+
      scale_size_manual(values=c(subset=3, fold=1))+
      scale_color_manual(values=c(subset="orange", fold="grey50"))+
      geom_rect(aes(
        xmin=-Inf, xmax=Inf,
        color=rows,
        size=rows,
        ymin=display_row, ymax=display_end),
        fill=NA,
        data=inst$viz.rect.dt)+
      scale_fill_manual(values=set.colors)+
      geom_rect(aes(
        xmin=iteration-rect.expand, ymin=display_row,
        xmax=iteration+rect.expand, ymax=display_end,
        fill=set.name),
        clickSelects="iteration",
        data=inst$viz.set.dt)+
      geom_text(aes(
        ifelse(rows=="subset", Inf, -Inf),
        (display_row+display_end)/2,
        hjust=ifelse(rows=="subset", 1, 0),
        label=paste0(rows, "=", ifelse(rows=="subset", subset, fold))),
        data=data.table(train.name="same", inst$viz.rect.dt))+
      scale_x_continuous(
        "Split number / cross-validation iteration")+
      scale_y_continuous(
        "Row number"),
    source="https://github.com/tdhock/mlr3resampling/blob/main/vignettes/ResamplingSameOtherCV.Rmd")
  viz
}
if(FALSE){
  animint2pages(viz, "2023-12-13-train-predict-subsets-regression")
}
```

If you are viewing this in an installed package or on CRAN, 
then there will be no data viz on this page, 
but you can view it on:
<https://tdhock.github.io/2023-12-13-train-predict-subsets-regression/>

## Simulated classification problems

The previous section investigated a simulated regression problem, whereas in this section we simulate a binary classification problem.
Assume there is a data set with some rows from one person, some rows
from another,

```{r}
N <- 200
library(data.table)
(full.dt <- data.table(
  label=factor(rep(c("spam","not spam"), l=N)),
  person=rep(1:2, each=0.5*N)
)[, signal := ifelse(label=="not spam", 0, 3)][])
```

Above each row has an person ID between 1 and 2. 
We can imagine a spam filtering system, that has training data for multiple people (here just two).
Each row in the table above represents a message which has been labeled as spam or not, by one of the two people.
Can we train on one person, and accurately predict on the other person?
To do that we will need some features, which we generate/simulate below:

```{r}
set.seed(1)
n.people <- length(unique(full.dt$person))
for(person.i in 1:n.people){
  use.signal.vec <- list(
    easy=rep(if(person.i==1)TRUE else FALSE, N),
    impossible=full.dt$person==person.i)
  for(task_id in names(use.signal.vec)){
    use.signal <- use.signal.vec[[task_id]]
    full.dt[
    , paste0("x",person.i,"_",task_id) := ifelse(
      use.signal, signal, 0
    )+rnorm(N)][]
  }
}
full.dt
```

In the table above, there are two sets of two features:

* For easy features, one is correlated with the label (`x1_easy`), and
  one is random noise (`x2_easy`), so the algorithm just needs to
  learn to ignore the noise feature, and concentrate on the signal
  feature. That should be possible given data from either person (same
  signal in each person).
* Each impossible feature is correlated with the label (when feature
  number same as person number), or is just noise (when person number
  different from feature number). So if the algorithm has access to
  the correct person (same as test, say person 2), then it needs to
  learn to use the corresponding feature `x2_impossible`. But if
  the algorithm does not have access to that person, then the best it
  can do is same as featureless (predict most frequent class label in
  train data).
  
### Static visualization of simulated data

Below we reshape the data to a table which is more suitable for visualization:

```{r}
(scatter.dt <- nc::capture_melt_multiple(
  full.dt,
  column="x[12]",
  "_",
  task_id="easy|impossible"))
```

Below we visualize the pattern for each person and feature type:

```{r}
if(require(animint2)){
  ggplot()+
    geom_point(aes(
      x1, x2, color=label),
      shape=1,
      data=scatter.dt)+
    facet_grid(
      task_id ~ person,
      labeller=label_both)
}
```

In the plot above, it is apparent that 

* for easy features (left), the two label classes differ in x1 values
  for both people. So it should be possible/easy to train on person 1, and
  predict accurately on person 2.
* for impossible features (right), the two people have different label
  patterns. For person 1, the two label classes differ in x1 values,
  whereas for person 2, the two label classes differ in x2 values. So
  it should be impossible to train on person 1, and predict accurately
  on person 2.

### Benchmark: computing test error

We use the code below to create a list of classification tasks, for
use in the mlr3 framework.

```{r}
class.task.list <- list()
for(task_id in c("easy","impossible")){
  feature.names <- grep(task_id, names(full.dt), value=TRUE)
  task.col.names <- c(feature.names, "label", "person")
  task.dt <- full.dt[, task.col.names, with=FALSE]
  this.task <- mlr3::TaskClassif$new(
    task_id, task.dt, target="label")
  this.task$col_roles$subset <- "person"
  this.task$col_roles$stratum <- c("person","label")
  this.task$col_roles$feature <- setdiff(names(task.dt), this.task$col_roles$stratum)
  class.task.list[[task_id]] <- this.task
}
class.task.list
```

Note in the code above that person is assigned roles subset and
stratum, whereas label is assigned roles target and stratum.  When
adapting the code above to real data, the important part is the
`mlr3::TaskClassif` line which tells mlr3 what data set to use, and
what columns should be used for target/subset/stratum.

The code below is used to define a K-fold cross-validation experiment,

```{r}
(class_same_other <- mlr3resampling::ResamplingSameOtherCV$new())
```

The code below is used to define the learning algorithms to test,

```{r}
(class.learner.list <- list(
  if(requireNamespace("rpart"))mlr3::LearnerClassifRpart$new(),
  mlr3::LearnerClassifFeatureless$new()))
```

The code below defines the grid of tasks, learners, and resamplings.

```{r}
(class.bench.grid <- mlr3::benchmark_grid(
  class.task.list,
  class.learner.list,
  class_same_other))
```

The code below runs the benchmark experiment grid. Note that each
iteration can be parallelized by declaring a future plan.

```{r}
if(FALSE){
  if(require(future))plan("multisession")
}
if(require(lgr))get_logger("mlr3")$set_threshold("warn")
(class.bench.result <- mlr3::benchmark(
  class.bench.grid, store_models = TRUE))
```

Below we compute scores (test error) for each resampling iteration,
and show the first row of the result.

```{r}
class.bench.score <- mlr3resampling::score(class.bench.result)
class.bench.score[1]
```

Finally we plot the test error values below.

```{r}
if(require(animint2)){
  ggplot()+
    geom_point(aes(
      classif.ce, train.subsets, color=algorithm),
      shape=1,
      data=class.bench.score)+
    facet_grid(
      person ~ task_id,
      labeller=label_both,
      scales="free")
}
```

It is clear from the plot above that 

* for the easy task, training on same is just as good as all or other
  subsets.
* for the impossible task, we must train on same subset for minimal
  test error; training on all is almost as good, because the pattern
  in person 1 is orthogonal to person 2; training on other is just as
  bad as featureless, because patterns are different.
* in a real data task, training on other will most likely not be quite as bad as in the impossible task above, but also not as good as in the easy task.
  
### Interactive visualization of data, test error, and splits

The code below can be used to create an interactive data visualization
which allows exploring how different functions are learned during
different splits.

```{r SimulationsAnimintClassification}
inst <- class.bench.score$resampling[[1]]$instance
rect.expand <- 0.2
grid.value.dt <- scatter.dt[
, lapply(.SD, function(x)do.call(seq, c(as.list(range(x)), l=21)))
, .SDcols=c("x1","x2")]
grid.class.dt <- data.table(
  label=full.dt$label[1],
  do.call(
    CJ, grid.value.dt
  )
)
class.pred.dt.list <- list()
class.point.dt.list <- list()
for(score.i in 1:nrow(class.bench.score)){
  class.bench.row <- class.bench.score[score.i]
  task.dt <- data.table(
    class.bench.row$task[[1]]$data(),
    class.bench.row$resampling[[1]]$instance$id.dt)
  names(task.dt)[2:3] <- c("x1","x2")
  set.ids <- data.table(
    set.name=c("test","train")
  )[
  , data.table(row_id=class.bench.row[[set.name]][[1]])
  , by=set.name]
  i.points <- set.ids[
    task.dt, on="row_id"
  ][
    is.na(set.name), set.name := "unused"
  ][]
  class.point.dt.list[[score.i]] <- data.table(
    class.bench.row[, .(task_id, iteration)],
    i.points)
  if(class.bench.row$algorithm!="featureless"){
    i.learner <- class.bench.row$learner[[1]]
    i.learner$predict_type <- "prob"
    i.task <- class.bench.row$task[[1]]
    setnames(grid.class.dt, names(i.task$data()))
    grid.class.task <- mlr3::TaskClassif$new(
      "grid", grid.class.dt, target="label")
    pred.grid <- as.data.table(
      i.learner$predict(grid.class.task)
    )[, data.table(grid.class.dt, prob.spam)]
    names(pred.grid)[2:3] <- c("x1","x2")
    pred.wide <- dcast(pred.grid, x1 ~ x2, value.var="prob.spam")
    prob.mat <- as.matrix(pred.wide[,-1])
    contour.list <- contourLines(
      grid.value.dt$x1, grid.value.dt$x2, prob.mat, levels=0.5)
    class.pred.dt.list[[score.i]] <- data.table(
      class.bench.row[, .(
        task_id, iteration, algorithm
      )],
      data.table(contour.i=seq_along(contour.list))[, {
        do.call(data.table, contour.list[[contour.i]])[, .(level, x1=x, x2=y)]
      }, by=contour.i]
    )
  }
}
(class.pred.dt <- rbindlist(class.pred.dt.list))
(class.point.dt <- rbindlist(class.point.dt.list))

set.colors <- c(
  train="#1B9E77",
  test="#D95F02",
  unused="white")
algo.colors <- c(
  featureless="blue",
  rpart="red")
make_person_subset <- function(DT){
  DT[, "person/subset" := person]
}
make_person_subset(class.point.dt)
make_person_subset(class.bench.score)
if(require(animint2)){
  viz <- animint(
    title="Train/predict on subsets, classification",
    pred=ggplot()+
      ggtitle("Predictions for selected train/test split")+
      theme_animint(height=400)+
      scale_fill_manual(values=set.colors)+
      scale_color_manual(values=c(spam="black","not spam"="white"))+
      geom_point(aes(
        x1, x2, color=label, fill=set.name),
        showSelected="iteration",
        size=3,
        stroke=2,
        shape=21,
        data=class.point.dt)+
      geom_path(aes(
        x1, x2, 
        subset=paste(algorithm, iteration, contour.i)),
        showSelected=c("iteration","algorithm"),
        color=algo.colors[["rpart"]],
        data=class.pred.dt)+
      facet_grid(
        task_id ~ `person/subset`,
        labeller=label_both,
        space="free",
        scales="free")+
      scale_y_continuous(
        breaks=seq(-100, 100, by=2)),
    err=ggplot()+
      ggtitle("Test error for each split")+
      theme_animint(height=400)+
      theme(panel.margin=grid::unit(1, "lines"))+
      scale_y_continuous(
        "Classification error on test set",
        breaks=seq(0, 1, by=0.25))+
      scale_fill_manual(values=algo.colors)+
      scale_x_discrete(
        "People/subsets in train set")+
      geom_hline(aes(
        yintercept=yint),
        data=data.table(yint=0.5),
        color="grey50")+
      geom_point(aes(
        train.subsets, classif.ce, fill=algorithm),
        shape=1,
        size=5,
        stroke=2,
        color="black",
        color_off=NA,
        clickSelects="iteration",
        data=class.bench.score)+
      facet_grid(
        task_id ~ `person/subset`,
        labeller=label_both),
    diagram=ggplot()+
      ggtitle("Select train/test split")+
      theme_bw()+
      theme_animint(height=300)+
      facet_grid(
        . ~ train.subsets,
        scales="free",
        space="free")+
      scale_size_manual(values=c(subset=3, fold=1))+
      scale_color_manual(values=c(subset="orange", fold="grey50"))+
      geom_rect(aes(
        xmin=-Inf, xmax=Inf,
        color=rows,
        size=rows,
        ymin=display_row, ymax=display_end),
        fill=NA,
        data=inst$viz.rect.dt)+
      scale_fill_manual(values=set.colors)+
      geom_rect(aes(
        xmin=iteration-rect.expand, ymin=display_row,
        xmax=iteration+rect.expand, ymax=display_end,
        fill=set.name),
        clickSelects="iteration",
        data=inst$viz.set.dt)+
      geom_text(aes(
        ifelse(rows=="subset", Inf, -Inf),
        (display_row+display_end)/2,
        hjust=ifelse(rows=="subset", 1, 0),
        label=paste0(rows, "=", ifelse(rows=="subset", subset, fold))),
        data=data.table(train.name="same", inst$viz.rect.dt))+
      scale_x_continuous(
        "Split number / cross-validation iteration")+
      scale_y_continuous(
        "Row number"),
    source="https://github.com/tdhock/mlr3resampling/blob/main/vignettes/ResamplingSameOtherCV.Rmd")
  viz
}
if(FALSE){
  animint2pages(viz, "2023-12-13-train-predict-subsets-classification")
}
```

If you are viewing this in an installed package or on CRAN, 
then there will be no data viz on this page, 
but you can view it on:
<https://tdhock.github.io/2023-12-13-train-predict-subsets-classification/>

## Conclusion

In this section we have shown how to use mlr3resampling for comparing
test error of models trained on same/all/other subsets.

# Variable size train resampler

The goal of this section is to explain how to
`ResamplingVariableSizeTrainCV`, which can be used to determine how
many train data are necessary to provide accurate predictions on a
given test set. 

## Simulated regression problems

The code below creates data for simulated regression problems. First
we define a vector of input values,

```{r}
N <- 300
abs.x <- 10
set.seed(1)
x.vec <- runif(N, -abs.x, abs.x)
str(x.vec)
```

Below we define a list of two true regression functions (tasks in mlr3
terminology) for our simulated data,

```{r}
reg.pattern.list <- list(
  sin=sin,
  constant=function(x)0)
```

The constant function represents a regression problem which can be
solved by always predicting the mean value of outputs (featureless is
the best possible learning algorithm). The sin function will be used
to generate data with a non-linear pattern that will need to be
learned. Below we use a for loop over these two functions/tasks, to
simulate the data which will be used as input to the learning
algorithms:

```{r}
library(data.table)
reg.task.list <- list()
reg.data.list <- list()
for(task_id in names(reg.pattern.list)){
  f <- reg.pattern.list[[task_id]]
  task.dt <- data.table(
    x=x.vec,
    y = f(x.vec)+rnorm(N,sd=0.5))
  reg.data.list[[task_id]] <- data.table(task_id, task.dt)
  reg.task.list[[task_id]] <- mlr3::TaskRegr$new(
    task_id, task.dt, target="y"
  )
}
(reg.data <- rbindlist(reg.data.list))
```

In the table above, the input is x, and the output is y. Below we
visualize these data, with one task in each facet/panel:

```{r}
if(require(animint2)){
  ggplot()+
    geom_point(aes(
      x, y),
      data=reg.data)+
    facet_grid(task_id ~ ., labeller=label_both)
}
```

In the plot above we can see two different simulated data sets
(constant and sin).  Note that the code above used the `animint2`
package, which provides interactive extensions to the static graphics
of the `ggplot2` package (see below section Interactive data viz). 

### Visualizing instance table

In the code below, we define a K-fold cross-validation experiment,
with K=3 folds.

```{r}
reg_size_cv <- mlr3resampling::ResamplingVariableSizeTrainCV$new()
reg_size_cv$param_set$values$train_sizes <- 6
reg_size_cv
```

In the output above we can see the parameters of the resampling
object, all of which should be integer scalars:

* `folds` is the number of cross-validation folds.
* `min_train_data` is the minimum number of train data to consider.
* `random_seeds` is the number of random seeds, each of which
  determines a different random ordering of the train data. The random
  ordering determines which data are included in small train set
  sizes.
* `train_sizes` is the number of train set sizes, evenly spaced on a
  log scale, from `min_train_data` to the max number of train data
  (determined by `folds`).

Below we instantiate the resampling on one of the tasks:

```{r}
reg_size_cv$instantiate(reg.task.list[["sin"]])
reg_size_cv$instance
```

Above we see the instance, which need not be examined by the user, but
for informational purposes, it contains the following data:

* `iteration.dt` has one row for each train/test split,
* `id.dt` has one row for each data point.

### Benchmark: computing test error

In the code below, we define two learners to compare,

```{r}
(reg.learner.list <- list(
  if(requireNamespace("rpart"))mlr3::LearnerRegrRpart$new(),
  mlr3::LearnerRegrFeatureless$new()))
```

The code above defines 

* `regr.rpart`: Regression Tree learning algorithm, which should be
  able to learn the non-linear pattern in the sin data (if there are
  enough data in the train set).
* `regr.featureless`: Featureless Regression learning algorithm, which
  should be optimal for the constant data, and can be used as a
  baseline in the sin data. When the rpart learner gets smaller
  prediction error rates than featureless, then we know that it has
  learned some non-trivial relationship between inputs and outputs.

In the code below, we define the benchmark grid, which is all
combinations of tasks (constant and sin), learners (rpart and
featureless), and the one resampling method.

```{r}
(reg.bench.grid <- mlr3::benchmark_grid(
  reg.task.list,
  reg.learner.list,
  reg_size_cv))
```

In the code below, we execute the benchmark experiment (optionally in parallel
using the multisession future plan).

```{r}
if(FALSE){
  if(require(future))plan("multisession")
}
if(require(lgr))get_logger("mlr3")$set_threshold("warn")
(reg.bench.result <- mlr3::benchmark(
  reg.bench.grid, store_models = TRUE))
```

The code below computes the test error for each split, and visualizes
the information stored in the first row of the result:

```{r}
reg.bench.score <- mlr3resampling::score(reg.bench.result)
reg.bench.score[1]
```

The output above contains all of the results related to a particular
train/test split. In particular for our purposes, the interesting
columns are:

* `test.fold` is the cross-validation fold ID.
* `seed` is the random seed used to determine the train set order.
* `train_size` is the number of data in the train set.
* `train` and `test` are vectors of row numbers assigned to each set.
* `iteration` is an ID for the train/test split, for a particular
  learning algorithm and task. It is the row number of `iteration.dt`
  (see instance above), which has one row for each unique combination
  of `test.fold`, `seed`, and `train_size`.
* `learner` is the mlr3 learner object, which can be used to compute
  predictions on new data (including a grid of inputs, to show
  predictions in the visualization below).
* `regr.mse` is the mean squared error on the test set.
* `algorithm` is the name of the learning algorithm (same as
  `learner_id` but without `regr.` prefix).

The code below visualizes the resulting test accuracy numbers.

```{r}
train_size_vec <- unique(reg.bench.score$train_size)
if(require(animint2)){
  ggplot()+
    scale_x_log10(
      breaks=train_size_vec)+
    scale_y_log10()+
    geom_line(aes(
      train_size, regr.mse,
      group=paste(algorithm, seed),
      color=algorithm),
      shape=1,
      data=reg.bench.score)+
    geom_point(aes(
      train_size, regr.mse, color=algorithm),
      shape=1,
      data=reg.bench.score)+
    facet_grid(
      test.fold~task_id,
      labeller=label_both,
      scales="free")
}
```

Above we plot the test error for each fold and train set size. 
There is a different panel for each task and test fold.
Each line represents a random seed (ordering of data in train set), 
and each dot represents a specific train set size.
So the plot above shows that some variation in test error, for a given test fold, 
is due to the random ordering of the train data.

Below we summarize each train set size, by taking the mean and standard deviation over each random seed.

```{r}
reg.mean.dt <- dcast(
  reg.bench.score,
  task_id + train_size + test.fold + algorithm ~ .,
  list(mean, sd),
  value.var="regr.mse")
if(require(animint2)){
  ggplot()+
    scale_x_log10(
      breaks=train_size_vec)+
    scale_y_log10()+
    geom_ribbon(aes(
      train_size,
      ymin=regr.mse_mean-regr.mse_sd,
      ymax=regr.mse_mean+regr.mse_sd,
      fill=algorithm),
      alpha=0.5,
      data=reg.mean.dt)+
    geom_line(aes(
      train_size, regr.mse_mean, color=algorithm),
      shape=1,
      data=reg.mean.dt)+
    facet_grid(
      test.fold~task_id,
      labeller=label_both,
      scales="free")
}
```

The plot above shows a line for the mean, 
and a ribbon for the standard deviation, 
over the three random seeds.
It is clear from the plot above that 

* in constant task, the featureless always has smaller or equal
  prediction error rates than rpart, which indicates that rpart
  sometimes overfits for large sample sizes.
* in sin task, more than 30 samples are required for rpart to be more
  accurate than featureless, which indicates it has learned a
  non-trivial relationship between input and output.
  
### Interactive data viz

The code below can be used to create an interactive data visualization
which allows exploring how different functions are learned during
different splits.

```{r ResamplingVariableSizeTrainCVAnimintRegression}
grid.dt <- data.table(x=seq(-abs.x, abs.x, l=101), y=0)
grid.task <- mlr3::TaskRegr$new("grid", grid.dt, target="y")
pred.dt.list <- list()
point.dt.list <- list()
for(score.i in 1:nrow(reg.bench.score)){
  reg.bench.row <- reg.bench.score[score.i]
  task.dt <- data.table(
    reg.bench.row$task[[1]]$data(),
    reg.bench.row$resampling[[1]]$instance$id.dt)
  set.ids <- data.table(
    set.name=c("test","train")
  )[
  , data.table(row_id=reg.bench.row[[set.name]][[1]])
  , by=set.name]
  i.points <- set.ids[
    task.dt, on="row_id"
  ][
    is.na(set.name), set.name := "unused"
  ]
  point.dt.list[[score.i]] <- data.table(
    reg.bench.row[, .(task_id, iteration)],
    i.points)
  i.learner <- reg.bench.row$learner[[1]]
  pred.dt.list[[score.i]] <- data.table(
    reg.bench.row[, .(
      task_id, iteration, algorithm
    )],
    as.data.table(
      i.learner$predict(grid.task)
    )[, .(x=grid.dt$x, y=response)]
  )
}
(pred.dt <- rbindlist(pred.dt.list))
(point.dt <- rbindlist(point.dt.list))
set.colors <- c(
  train="#1B9E77",
  test="#D95F02",
  unused="white")
algo.colors <- c(
  featureless="blue",
  rpart="red")
if(require(animint2)){
  viz <- animint(
    title="Variable size train set, regression",
    pred=ggplot()+
      ggtitle("Predictions for selected train/test split")+
      theme_animint(height=400)+
      scale_fill_manual(values=set.colors)+
      geom_point(aes(
        x, y, fill=set.name),
        showSelected="iteration",
        size=3,
        shape=21,
        data=point.dt)+
      scale_size_manual(values=c(
        featureless=3,
        rpart=2))+
      scale_color_manual(values=algo.colors)+
      geom_line(aes(
        x, y,
        color=algorithm,
        size=algorithm,
        group=paste(algorithm, iteration)),
        showSelected="iteration",
        data=pred.dt)+
      facet_grid(
        task_id ~ .,
        labeller=label_both),
    err=ggplot()+
      ggtitle("Test error for each split")+
      theme_animint(width=500)+
      theme(
        panel.margin=grid::unit(1, "lines"),
        legend.position="none")+
      scale_y_log10(
        "Mean squared error on test set")+
      scale_color_manual(values=algo.colors)+
      scale_x_log10(
        "Train set size",
        breaks=train_size_vec)+
      geom_line(aes(
        train_size, regr.mse,
        group=paste(algorithm, seed),
        color=algorithm),
        clickSelects="seed",
        alpha_off=0.2,
        showSelected="algorithm",
        size=4,
        data=reg.bench.score)+
      facet_grid(
        test.fold~task_id,
        labeller=label_both,
        scales="free")+
      geom_point(aes(
        train_size, regr.mse,
        color=algorithm),
        size=5,
        stroke=3,
        fill="black",
        fill_off=NA,
        showSelected=c("algorithm","seed"),
        clickSelects="iteration",
        data=reg.bench.score),
    source="https://github.com/tdhock/mlr3resampling/blob/main/vignettes/Simulations.Rmd")
  viz
}
if(FALSE){
  animint2pages(viz, "2023-12-26-train-sizes-regression")
}
```

If you are viewing this in an installed package or on CRAN, 
then there will be no data viz on this page, 
but you can view it on:
<https://tdhock.github.io/2023-12-26-train-sizes-regression/>

The interactive data viz consists of two plots:

* The first plot shows the data, with each point colored according to
  the set it was assigned, in the currently selected
  split/iteration. The red/blue lines additionally show the learned
  prediction functions for the currently selected split/iteration.
* The second plot shows the test error rates, as a function of train
  set size. Clicking a line selects the corresponding random seed,
  which makes the corresponding points on that line appear. Clicking a
  point selects the corresponding iteration (seed, test fold, and train
  set size).

## Simulated classification problems

Whereas in the section above, we focused on regression (output is a real number),
in this section we simulate a binary classification problem (output if a factor with two levels).

```{r}
class.N <- 300
class.abs.x <- 1
rclass <- function(){
  runif(class.N, -class.abs.x, class.abs.x)
}
library(data.table)
set.seed(1)
class.x.dt <- data.table(x1=rclass(), x2=rclass())
class.fun.list <- list(
  constant=function(...)0.5,
  xor=function(x1, x2)xor(x1>0, x2>0))
class.data.list <- list()
class.task.list <- list()
for(task_id in names(class.fun.list)){
  class.fun <- class.fun.list[[task_id]]
  y <- factor(ifelse(
    class.x.dt[, class.fun(x1, x2)+rnorm(class.N, sd=0.5)]>0.5,
    "spam", "not"))
  task.dt <- data.table(class.x.dt, y)
  this.task <- mlr3::TaskClassif$new(
    task_id, task.dt, target="y")
  this.task$col_roles$stratum <- "y"
  class.task.list[[task_id]] <- this.task
  class.data.list[[task_id]] <- data.table(task_id, task.dt)
}
(class.data <- rbindlist(class.data.list))
```

The simulated data table above consists of two input features (`x1`
and `x2`) along with an output/label to predict (`y`). Below we count
the number of times each label appears in each task:

```{r}
class.data[, .(count=.N), by=.(task_id, y)]
```

The table above shows that the `spam` label is the minority class
(`not` is majority, so that will be the prediction of the featureless
baseline). Below we visualize the data in the feature space:

```{r}
if(require(animint2)){
  ggplot()+
    geom_point(aes(
      x1, x2, color=y),
      shape=1,
      data=class.data)+
    facet_grid(. ~ task_id, labeller=label_both)+
    coord_equal()
}
```

The plot above shows how the output `y` is related to the two inputs `x1` and
`x2`, for the two tasks.

* For the constant task, the two inputs are not related to the output.
* For the xor task, the spam label is associated with either `x1` or
  `x2` being negative (but not both).
  
In the mlr3 code below, we define a list of learners, our resampling
method, and a benchmark grid:

```{r}
class.learner.list <- list(
  if(requireNamespace("rpart"))mlr3::LearnerClassifRpart$new(),
  mlr3::LearnerClassifFeatureless$new())
size_cv <- mlr3resampling::ResamplingVariableSizeTrainCV$new()
(class.bench.grid <- mlr3::benchmark_grid(
  class.task.list,
  class.learner.list,
  size_cv))
```

Below we run the learning algorithm for each of the train/test splits
defined by our benchmark grid:

```{r}
if(FALSE){
  if(require(future))plan("multisession")
}
if(require(lgr))get_logger("mlr3")$set_threshold("warn")
(class.bench.result <- mlr3::benchmark(
  class.bench.grid, store_models = TRUE))
```

Below we compute scores (test error) for each resampling iteration,
and show the first row of the result.

```{r}
class.bench.score <- mlr3resampling::score(class.bench.result)
class.bench.score[1]
```

The output above has columns which are very similar to the regression
example in the previous section. The main difference is the
`classif.ce` column, which is the classification error on the test
set.

Finally we plot the test error values below.

```{r}
if(require(animint2)){
  ggplot()+
    geom_line(aes(
      train_size, classif.ce,
      group=paste(algorithm, seed),
      color=algorithm),
      shape=1,
      data=class.bench.score)+
    geom_point(aes(
      train_size, classif.ce, color=algorithm),
      shape=1,
      data=class.bench.score)+
    facet_grid(
      task_id ~ test.fold,
      labeller=label_both,
      scales="free")+
    scale_x_log10()
}
```

It is clear from the plot above that 

* in constant task, rpart does not have significantly lower error
  rates than featureless, which is expected, because the best
  prediction function is constant (predict the most frequent class, no
  relationship between inputs and output).
* in xor task, more than 30 samples are required for rpart to be more
  accurate than featureless, which indicates it has learned a
  non-trivial relationship between inputs and output.
  
Exercise for the reader: compute and plot mean and SD for these
classification tasks, similar to the plot for the regression tasks in
the previous section.
  
### Interactive visualization of data, test error, and splits

The code below can be used to create an interactive data visualization
which allows exploring how different functions are learned during
different splits.

```{r ResamplingVariableSizeTrainCVAnimintClassification}
class.grid.vec <- seq(-class.abs.x, class.abs.x, l=21)
class.grid.dt <- CJ(x1=class.grid.vec, x2=class.grid.vec)
class.pred.dt.list <- list()
class.point.dt.list <- list()
for(score.i in 1:nrow(class.bench.score)){
  class.bench.row <- class.bench.score[score.i]
  task.dt <- data.table(
    class.bench.row$task[[1]]$data(),
    class.bench.row$resampling[[1]]$instance$id.dt)
  set.ids <- data.table(
    set.name=c("test","train")
  )[
  , data.table(row_id=class.bench.row[[set.name]][[1]])
  , by=set.name]
  i.points <- set.ids[
    task.dt, on="row_id"
  ][
    is.na(set.name), set.name := "unused"
  ][]
  class.point.dt.list[[score.i]] <- data.table(
    class.bench.row[, .(task_id, iteration)],
    i.points)
  if(class.bench.row$algorithm!="featureless"){
    i.learner <- class.bench.row$learner[[1]]
    i.learner$predict_type <- "prob"
    i.task <- class.bench.row$task[[1]]
    grid.class.task <- mlr3::TaskClassif$new(
      "grid", class.grid.dt[, label:=factor(NA,levels(task.dt$y))], target="label")
    pred.grid <- as.data.table(
      i.learner$predict(grid.class.task)
    )[, data.table(class.grid.dt, prob.spam)]
    pred.wide <- dcast(pred.grid, x1 ~ x2, value.var="prob.spam")
    prob.mat <- as.matrix(pred.wide[,-1])
    if(length(table(prob.mat))>1){
      contour.list <- contourLines(
        class.grid.vec, class.grid.vec, prob.mat, levels=0.5)
      class.pred.dt.list[[score.i]] <- data.table(
        class.bench.row[, .(
          task_id, iteration, algorithm
        )],
        data.table(contour.i=seq_along(contour.list))[, {
          do.call(data.table, contour.list[[contour.i]])[, .(level, x1=x, x2=y)]
        }, by=contour.i]
      )
    }
  }
}
(class.pred.dt <- rbindlist(class.pred.dt.list))
(class.point.dt <- rbindlist(class.point.dt.list))

set.colors <- c(
  train="#1B9E77",
  test="#D95F02",
  unused="white")
algo.colors <- c(
  featureless="blue",
  rpart="red")
if(require(animint2)){
  viz <- animint(
    title="Variable size train sets, classification",
    pred=ggplot()+
      ggtitle("Predictions for selected train/test split")+
      theme(panel.margin=grid::unit(1, "lines"))+
      theme_animint(width=600)+
      coord_equal()+
      scale_fill_manual(values=set.colors)+
      scale_color_manual(values=c(spam="black","not spam"="white"))+
      geom_point(aes(
        x1, x2, color=y, fill=set.name),
        showSelected="iteration",
        size=3,
        stroke=2,
        shape=21,
        data=class.point.dt)+
      geom_path(aes(
        x1, x2, 
        group=paste(algorithm, iteration, contour.i)),
        showSelected=c("iteration","algorithm"),
        color=algo.colors[["rpart"]],
        data=class.pred.dt)+
      facet_grid(
        . ~ task_id,
        labeller=label_both,
        space="free",
        scales="free"),
    err=ggplot()+
      ggtitle("Test error for each split")+
      theme_animint(height=400)+
      theme(panel.margin=grid::unit(1, "lines"))+
      scale_y_continuous(
        "Classification error on test set")+
      scale_color_manual(values=algo.colors)+
      scale_x_log10(
        "Train set size")+
      geom_line(aes(
        train_size, classif.ce,
        group=paste(algorithm, seed),
        color=algorithm),
        clickSelects="seed",
        alpha_off=0.2,
        showSelected="algorithm",
        size=4,
        data=class.bench.score)+
      facet_grid(
        test.fold~task_id,
        labeller=label_both,
        scales="free")+
      geom_point(aes(
        train_size, classif.ce,
        color=algorithm),
        size=5,
        stroke=3,
        fill="black",
        fill_off=NA,
        showSelected=c("algorithm","seed"),
        clickSelects="iteration",
        data=class.bench.score),
    source="https://github.com/tdhock/mlr3resampling/blob/main/vignettes/ResamplingVariableSizeTrainCV.Rmd")
  viz
}
if(FALSE){
  animint2pages(viz, "2023-12-27-train-sizes-classification")
}
```

If you are viewing this in an installed package or on CRAN, 
then there will be no data viz on this page, 
but you can view it on:
<https://tdhock.github.io/2023-12-27-train-sizes-classification/>

The interactive data viz consists of two plots

* The first plot shows the data, with each point colored according to
  its label/y value (black outline for spam, white outline for not),
  and the set it was assigned (fill color) in the currently selected
  split/iteration. The red lines additionally show the learned
  decision boundary for rpart, given the currently selected
  split/iteration.  For constant, the ideal decision boundary is none
  (always predict the most frequent class), and for xor, the ideal
  decision boundary looks like a plus sign.
* The second plot shows the test error rates, as a function of train
  set size. Clicking a line selects the corresponding random seed,
  which makes the corresponding points on that line appear. Clicking a
  point selects the corresponding iteration (seed, test fold, and train
  set size).

## Conclusion

In this section we have shown how to use mlr3resampling for comparing
test error of models trained on different sized train sets.

# Session info 

```{r}
sessionInfo()
```
