# Density Estimation in mlr3proba

#### Nurul Ain Toha

This vignette is an introduction to performing density estimation in mlr3proba.

## A very quick introduction to density estimation

Density estimation is the learning task to find the unknown distribution from which an i.i.d. data set is generated. We interpret this broadly, with this distribution not necessarily being continuous (so may possess a mass not density). The conditional case, where a distribution is predicted conditional on covariates, is known as ‘probabilistic supervised regression’, and will be implemented in mlr3proba in the near-future. In mlr3proba, (unconditional) density estimation is viewed as an unsupervised task, whereas probabilistic supervised regression (or conditional density estimation) is a supervised task

## Density Task

Unconditional density estimation is an unsupervised method. Hence, `TaskDens` is an unsupervised task which inherits directly from `Task` unlike `TaskClassif` and `TaskRegr`. However, `TaskDens` still has a `target` and a `$truth` field defined by:

- `target` - the variable for which to estimate density
- `truth` - the `target`. This is not the true density which is always unknown.

    library(mlr3proba); library(mlr3)
    #> Registered S3 methods overwritten by 'mlr3proba':
    #>   method                       from
    #>   as.data.table.PredictionRegr mlr3
    #>   c.PredictionRegr             mlr3
    #>
    #> Attaching package: 'mlr3'
    #> The following object is masked from 'package:mlr3proba':
    #>
    #>     PredictionRegr

    task = TaskDens$new(id = "mpg", backend = datasets::mtcars, target = "mpg")

    task
    #> <TaskDens:mpg> (32 x 11)
    #> * Target: mpg
    #> * Properties: -
    #> * Features (10):
    #>   - dbl (10): am, carb, cyl, disp, drat, gear, hp, qsec, vs, wt

    task$truth()[1:10]
    #>  [1] 21.0 21.0 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2

## Train and Predict

Density learners have `train` and `predict` methods, though being unsupervised, ‘prediction’ is actually ‘estimation’. In training, a distr6 object is created, see here for full tutorials on how to access the `pdf`, `cdf`, and other important fields and methods. The predict method is simply a wrapper around `self$model$pdf` and if available `self$model$cdf`, i.e. evaluates the pdf/cdf at given points. Note that in prediction the points to evaluate the pdf and cdf are determined by the `target` column in the `TaskDens` object used for testing.

    # create task and learner

    task_faithful = TaskDens$new(id = "eruptions", backend = datasets::faithful,
                                 target = "eruptions")
    learner = lrn("dens.kde")

    # train/test split

    train_set = sample(task_faithful$nrow, 0.8 * task_faithful$nrow)
    test_set = setdiff(seq_len(task_faithful$nrow), train_set)

    # fitting KDE and model inspection

    learner$train(task_faithful, row_ids = train_set)
    learner$model
    #> Norm_KDE
    class(learner$model)
    #> [1] "Distribution" "R6"

    # make predictions for new data

    prediction = learner$predict(task_faithful, row_ids = test_set)

Every `PredictionDens` object can estimate:

- `pdf` - probability density function

Some learners can estimate:

- `cdf` - cumulative distribution function

    prediction
    #> <PredictionDens> for 55 observations:
    #>     row_id truth       pdf
    #>          3 3.333 0.1094527
    #>          8 3.600 0.2057676
    #>         11 1.833 0.3015347
    #> ---
    #>        241 4.150 0.4651316
    #>        242 2.350 0.2310265
    #>        272 4.467 0.4800183

    # `pdf` is evaluated using the `log-loss`

    prediction$score()
    #> dens.logloss
    #>     1.145351