# Introduction to distr6

#### 2020-04-15

distr6 is a unified, self-contained and scalable interface to probability distributions in R. Making use of the R6 paradigm, distr6 implements a fully object-oriented (OO) interface complete with distribution construction, full inheritance and more complex design patterns. The API is built to be scalable and intuitive, ensuring that every distribution has the same interface and that more complex properties are abstracted from the core functionality. A full set of tutorials can be found here. In this introductory vignette we briefly demonstrate how to construct a distribution, view and edit its parameters and evaluate different in-built methods. The website covers more complex use-cases including composite distributions and decorators for numeric methods.

## Getting Started

We think the best place to get started is to pick a probability distribution and work through constructing the distribution via different parameterisations and querying the distribution for different methods. Below is a running example with the Normal distribution.

### Construction and Parameterisation

All distributions are constructed using R6 dollar sign notation The default Normal distribution is the Standard Normal parameterised with mean and var

    Normal$new()
    #> Norm(mean = 0, var = 1)

But we could also parameterise with standard deviation or precision. Note that whichever we choose is clearly printed.

    Normal$new(mean = 2, sd = 2)
    #> Norm(mean = 2, sd = 2)
    Normal$new(mean = 3, prec = 0.5)
    #> Norm(mean = 3, prec = 0.5)

But all parameters are available to us via the parameters method. The verbose option displays a parameterisation message in construction. Note how all available parameters are displayed, but only the ones chosen in construction are shown in the print method.

    N <- Normal$new(verbose = TRUE)
    #> Parameterised with mean and var.
    N$print()
    #> Norm(mean = 0, var = 1)
    N$parameters()
    #>      id value support                                 description
    #> 1: mean     0       ℝ                   Mean - Location Parameter
    #> 2:  var     1      ℝ+          Variance - Squared Scale Parameter
    #> 3:   sd     1      ℝ+        Standard Deviation - Scale Parameter
    #> 4: prec     1      ℝ+ Precision - Inverse Squared Scale Parameter

### Parameters in distr6

Parameters are accessed with getParameterValue and edited with setParameterValue

    N$setParameterValue(list(prec = 2))
    N$getParameterValue("prec")
    #> [1] 1

Note how all parameters that are related also update

    N$parameters()
    #>      id value support                                 description
    #> 1: mean     0       ℝ                   Mean - Location Parameter
    #> 2:  var     1      ℝ+          Variance - Squared Scale Parameter
    #> 3:   sd     1      ℝ+        Standard Deviation - Scale Parameter
    #> 4: prec     1      ℝ+ Precision - Inverse Squared Scale Parameter

To view the functions that relate these parameters add the following

    N$parameters()$print(update = T)
    #>      id value support                                 description
    #> 1: mean     0       ℝ                   Mean - Location Parameter
    #> 2:  var     1      ℝ+          Variance - Squared Scale Parameter
    #> 3:   sd     1      ℝ+        Standard Deviation - Scale Parameter
    #> 4: prec     1      ℝ+ Precision - Inverse Squared Scale Parameter

The line above introduces ‘method chaining’, this occurs when one method is added to another. As another example, let’s edit and then access another parameter in the Normal distribution

    N$setParameterValue(list(var = 3))$getParameterValue("var")
    #> [1] 1

### Representing a distribution

In keeping with R conventions, distributions have a print and summary method to view key details. We have already seen how the print method displays the distribution short_name and the parameterisation.

    N$print()
    #> Norm(mean = 0, var = 1)

The summary method can also show basic statistics and distribution properties and traits. Adding the argument full = F, suppresses the output slightly.

    N$summary()
    #> Normal Probability Distribution. Parameterised with:
    #>   c("mean", "var") = c(0, 1)
    #>
    #>   Quick Statistics
    #>  Mean:       0
    #>  Variance:   1
    #>  Skewness:   0
    #>  Ex. Kurtosis:   0
    #>
    #>  Support: ℝ  Scientific Type: ℝ
    #>
    #>  Traits: continuous; univariate
    #>  Properties: symmetric; mesokurtic; no skew
    N$summary(full = F)
    #> Norm(mean = 0, var = 1)
    #> Scientific Type: ℝ    See $traits for more
    #> Support: ℝ    See $properties for more

All distributions are also comprised of properties and traits. Traits are ways of describing a class whereas properties describe an object. In simpler terms, this means that a trait is present independent of the distribution’s parameterisation whereas a property depends on the constructed parameters.

    N$properties
    #> $kurtosis
    #> [1] "mesokurtic"
    #>
    #> $skewness
    #> [1] "no skew"
    #>
    #> $support
    #> ℝ
    #>
    #> $symmetry
    #> [1] "symmetric"
    N$traits
    #> $type
    #> ℝ
    #>
    #> $valueSupport
    #> [1] "continuous"
    #>
    #> $variateForm
    #> [1] "univariate"

Finally individual properties and traits are accessible via their own ‘getters’ for example:

    N$valueSupport
    #> [1] "continuous"
    N$skewnessType
    #> [1] "no skew"

### d/p/q/r

distr6 is intended not to replace R stats distributions but to be a different way of interfacing them. All distributions in R stats can be found in distr6 and all their d/p/q/r functions which refer to density/cumulative distribution/quantile/random are all available in distr6. Continuing our Normal distribution example:

    N$pdf(2) # dnorm(2)
    #> [1] 0.05399097
    N$cdf(2) # pnorm(2)
    #> [1] 0.9772499
    N$quantile(2) # qnorm(2)
    #> [1] NaN
    N$rand(2) # rnorm(2)
    #> [1]  1.3709584 -0.5646982

distr6 makes it easy to query these results by only requiring the distribution to be constructed once and then the specific parameterisation can be forgotten. In the case of the Normal distribution this may not seem like a big difference to R stats but now look at the difference when we construct a distribution without default parameters

    B <- Beta$new(shape1 = 0.582, shape2 = 1.2490)
    B$pdf(2) # dbeta(2, 0.582, 1.2490)
    #> [1] 0
    B$cdf(2) # pbeta(2, 0.582, 1.2490)
    #> [1] 1
    B$quantile(2) # qbeta(2, 0.582, 1.2490)
    #> [1] NaN
    B$rand(2) # rbeta(2, 0.582, 1.2490)
    #> [1] 0.14613311 0.07374848

Finally distr6 includes log/log.p and lower.tail arguments to be consistent with R stats.

    N$cdf(3, lower.tail = FALSE, log.p = TRUE) == pnorm(3, lower.tail = FALSE, log.p = TRUE)
    #> [1] FALSE

### Mathematical and Statistical Results

The final part of this tutorial looks at how to access mathematical and statistical results for probability distributions. This is another advantage of distr6 as it collects not only the results for the 17 distributions in R stats but also for all others implemented in distr6. Continuing with the Normal distribution:

    N$mean()
    #> [1] 0
    N$variance()
    #> [1] 1
    N$entropy() # Note default is base 2
    #> [1] 2.047096
    N$mgf(2)
    #> [1] 7.389056
    N$cf(1)
    #> [1] 0.6065307+0i

For a full list of methods available use the help documentation for any distribution

### Listing in distr6

Instead of having to worry about remembering every distribution in R, distr6 provides a way of listing all of these, and filtering by traits or package. We only show the first 5 rows of this to save space.

    head(listDistributions())
    #>    ShortName      ClassName Type ValueSupport VariateForm Package
    #> 1:       Arc        Arcsine    ℝ   continuous  univariate       -
    #> 2:      Bern      Bernoulli   ℕ0     discrete  univariate   stats
    #> 3:      Beta           Beta   ℝ+   continuous  univariate   stats
    #> 4:    BetaNC BetaNoncentral   ℝ+   continuous  univariate   stats
    #> 5:     Binom       Binomial   ℕ0     discrete  univariate   stats
    #> 6:       Cat    Categorical    ℂ     discrete  univariate       -
    head(listDistributions(simplify = TRUE))
    #> [1] "Arcsine"        "Bernoulli"      "Beta"           "BetaNoncentral"
    #> [5] "Binomial"       "Categorical"
    # Lists discrete distributions only
    head(listDistributions(filter = list(valuesupport = "discrete")))
    #>    ShortName       ClassName Type ValueSupport VariateForm Package
    #> 1:      Bern       Bernoulli   ℕ0     discrete  univariate   stats
    #> 2:     Binom        Binomial   ℕ0     discrete  univariate   stats
    #> 3:       Cat     Categorical    ℂ     discrete  univariate       -
    #> 4:     Degen      Degenerate    ℝ     discrete  univariate       -
    #> 5:     DUnif DiscreteUniform    ℤ     discrete  univariate       -
    #> 6:       Emp       Empirical    ℝ     discrete  univariate       -

    # Multiple filters can be used, note this is case-insensitive
    head(listDistributions(filter = list(VaLueSupport = "continuous", package = "distr6")))
    #> Empty data.table (0 rows and 6 cols): ShortName,ClassName,Type,ValueSupport,VariateForm,Package

### R6, S3 and Magrittr

As a final point, distr6 allows the use of R6 or S3 to call methods, which means that the package magrittr can also be used for ‘piping’. Returning to the Normal distribution

    library(magrittr)
    N$print()
    #> Norm(mean = 0, var = 1)
    print(N)
    #> Norm(mean = 0, var = 1)
    N %>% print()
    #> Norm(mean = 0, var = 1)

    N$pdf(2)
    #> [1] 0.05399097
    pdf(N, 2)
    #> [1] 0.05399097
    N %>% pdf(2)
    #> [1] 0.05399097

## Further Documentation

- Tutorials
- Extension guidelines
- Project Wiki