# 0distr-package: distr - Object Orientated Implementation of Distributions

Description Details Classes Methods Functions Extension Packages in distrXXX family Package versions Acknowledgement Start-up-Banner Demos Note Author(s) References Examples

### Description

distr provides a conceptual treatment of distributions by means of S4 classes. A mother class `Distribution` is introduced with slots for a parameter and —most important— for the four constitutive methods `r`, `d`, `p`, and `q` for simulation respectively for evaluation of density / c.d.f.\ and quantile function of the corresponding distribution.

Most distributions of package stats (like normal, Poisson, etc.) are implemented as subclasses of either `AbscontDistribution` or `DiscreteDistribution`, which themselves are again subclasses of `Distribution`.

Up to arguments referring to a parameter of the distribution (like `mean` for the normal distribution), these function slots have the same arguments as those of package stats, i.e.; for a distribution object `X` we may call these functions as

• `r(X)(n)`

• `d(X)(x, log = FALSE)`

• `p(X)(q, lower.tail = TRUE, log.p = FALSE)`

• `q(X)(p, lower.tail = TRUE, log.p = FALSE)`

For the arguments of these function slots see e.g. `rnorm`. Note that, as usual, slots `d`, `p`, and `q` are vectorized in their first argument, but are not on the subsequent ones.

Arithmetics and unary mathematical transformations for distributions are available: For `Distribution` objects `X` and `Y` expressions like `3*X+sin(exp(-Y/4+3))` have their natural interpretation as corresponding image distributions.

### Details

 Package: distr Version: 2.6 Date: 2016-04-23 Depends: R(>= 2.14.0), methods, graphics, startupmsg, sfsmisc, SweaveListingUtils Suggests: distrEx, svUnit (>= 0.7-11) Imports: stats, grDevices, utils, MASS LazyLoad: yes License: LGPL-3 URL: http://distr.r-forge.r-project.org/ SVNRevision: 1088

### Classes

Distribution classes have a slot `param` the class of which is is specialized for the particualar distributions. The parameter classes for the particular distributions have slots with names according to the corresponding [rdpq]<name> functions of package base. From version 1.9 on, `AbscontDistribution` and descendants have a slot `gaps` for gaps in the support. `DiscreteDistribution` and descendants have an additional slot `support`, which is again specialized to be a lattice for class `LatticeDistribution`.

For saved objects from earlier versions, we provide the methods `isOldVersion`, and `conv2NewVersion` to check whether the object was generated by an older version of this package and to convert such an object to the new format, respectively. This applies to objects of subclasses of `AbscontDistribution` lacking a `gap`-slot as well as to to objects of subclasses of `LatticeDistribution` lacking a `lattice`-slot.

To enhance accuracy, from version 1.9 on, we also provide subclasses `AffLinAbscontDistribution`, `AffLinDiscreteDistribution`, and `AffLinLatticeDistribution`, as well as the class union `AffLinDistribution`, so that in particular functionals like `E` from package distrEx can recur to exact formula more frequently: These classes have additional slots `a`, `b`, and `X0` to reflect the fact, that a distribution object of theses classes has the same distribution as `a*X0+b`.

For all particular distributions, as well as for classes `AbscontDistribution`, `DiscreteDistribution`, `LatticeDistribution`, `UnivarDistrList` and `DistrList` generating functions are provided, e.g. `X <- Norm(mean = 3, sd = 2)`. The same goes for the space classes. All slots should be inspected / modified by means of corresponding accessor- /replacement functions; e.g. `mean(X) <- 3`

Again to enhance accuracy, from version 2.0 on, we also provide subclasses `UnivarMixingDistribution` to support mixing distributions, `UnivarLebDecDistribution`, to support Lebesgue decomposed distributions (with a discrete and an a.c. part) as well as `AffLinUnivarLebDecDistribution`, for corresponding affine linear transformations. Class `UnivarLebDecDistribution` is closed under arithmetical operations + `/`, `*`, `^` for pairs of independent variables + `+`, `-` for pairs of independent variables + affine linear transformations + truncation, huberization, min/max which are all now available analytically.

(see Parameter classes).

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126``` ```[*]: there is a generating function with the same name ########################## Distribution classes ########################## slots: [()] img(rSpace), param(OptionalParameter), r(function), d(OptionalFunction), p(OptionalFunction), q(OptionalFunction), .withSim(logical), .withArith(logical), .logExact(logical), .lowerExact(logical), Symmetry(DistributionSymmetry) "Distribution" |>"UnivariateDistribution" |>|>"UnivarMixingDistribution" [*] |>|>|>"UnivarLebDecDistribution" [*] |>|>|>|>"AffLinUnivarLebDecDistribution" |>|>|>"CompoundDistribution" [*] |>|>"AbscontDistribution" [*] |>|>|>"AffLinAbscontDistribution" |>|>|>"Arcsine" [*] |>|>|>"Beta" [*] |>|>|>"Cauchy" [*] |>|>|>"ExpOrGammaOrChisq" (VIRTUAL) |>|>|>|>"Exp" [*] |>|>|>|>"Gammad" [*] |>|>|>|>"Chisq" [*] |>|>|>"Fd" [*] |>|>|>"Lnorm" [*] |>|>|>"Logis" [*] |>|>|>"Norm" [*] |>|>|>"Td" [*] |>|>|>"Unif" [*] |>|>|>"Weibull" [*] |>|>|"DiscreteDistribution" [*] |>|>|>"AffLinDiscreteDistribution" |>|>|>"LatticeDistribution" [*] |>|>|>|>"AffLinLatticeDistribution" |>|>|>|>"Binom" [*] |>|>|>|>"Dirac" [*] |>|>|>|>"Hyper" [*] |>|>|>|>"NBinom" [*] |>|>|>|>|>"Geom" [*] |>|>|>|>"Pois" [*] "AffLinDistribution" = union ( "AffLinAbscontDistribution", "AffLinDiscreteDistribution", "AffLinUnivarLebDecDistribution" ) "DistrList" |>"UnivarDistrList" [*] "AcDcLc" = union ( "AbscontDistribution", "DiscreteDistribution", "UnivarLebDecDistribution" ) ########################## Parameter classes ########################## "OptionalParameter" |>"Parameter" |>|>"BetaParameter" |>|>"BinomParameter" |>|>"CauchyParameter" |>|>"ChisqParameter" |>|>"DiracParameter" |>|>"ExpParameter" |>|>"FParameter" |>|>"GammaParameter" |>|>"GeomParameter" |>|>"HyperParameter" |>|>"LnormParameter" |>|>"LogisParameter" |>|>"NbinomParameter" |>|>"NormParameter" |>|>"UniNormParameter" |>|>|>"PoisParameter" |>|>"TParameter" |>|>"UnifParameter" |>|>"WeibullParameter" ########################## Space classes ########################## "rSpace" |>"EuclideanSpace" |>|>"Reals" |>"Lattice" |>"Naturals" ########################## Symmetry classes ########################## slots: type(character), SymmCenter(ANY) "Symmetry" |>"NoSymmetry" [*] |>"EllipticalSymmetry" [*] |>|>"SphericalSymmetry" [*] |>"DistributionSymmetry" |>"FunctionSymmetry" |>|>"NonSymmetric" [*] |>|>"EvenSymmetric" [*] |>|>"OddSymmetric" [*] list thereof "DistrSymmList" [*] "FunSymmList" [*] ########################## Matrix classes ########################## slots: none "PosSemDefSymmMatrix" [*] is subclass of class "matrix" of package "base". |>"PosDefSymmMatrix" [*] ########################## Class unions ########################## "OptionalNumeric" = union("numeric", "NULL") "OptionalMatrix" = union("matrix","NULL") ```

### Methods

The group `Math` of unary (see Math) as well as convolution are made available for distributions, see operators-methods; in particular for convolution powers, we have method convpow. Besides, there are `plot` and `print`-methods for distributions. For the space classes, we have `liesIn`, for the `DicreteDistribution` class, we have `liesInSupport`, as well as a generating function. The "history" of distributions obtained by chaining operations may be shortened using `simplifyr`.

### Functions

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```RtoDPQ Default procedure to fill slots d,p,q given r for a.c. distributions RtoDPQ.d Default procedure to fill slots d,p,q given r for discrete distributions RtoDPQ.LC Default procedure to fill slots d,p,q given r for Lebesgue decomposed distributions decomposePM decomposes a distribution into positive and negative part and, if discrete, into part '0' simplifyD tries to reduce/simplify mixing distribution using that certain weights are 0 flat.LCD makes a single UnivarLebDecDistribution out of a list of UnivarLebDecDistribution with corresp. weights flat.mix makes a single UnivarLebDecDistribution out of a list of a UnivarMixingDistribution distroptions Functions to change the global variables of the package 'distr' standardMethods Utility to automatically generate accessor and replacement functions ```

### Extension Packages in distrXXX family

Please note that there are extension packages of this packages available on CRAN,

distrDoc

a documentation package providing joint documentation for all packages of the distrXXX family of packages in the form of vignette 'distr'; try `require(distrDoc); vignette("distr")`.

distrEx

provides functionals (like `E`, `sd`, `mad`) operating on distributions, as well as distances between distributions and basic support for multivariate and conditional distributions.

distrSim

for the standardized treatment of simulations, also under contaminations.

distrTEst

with classes and methods for evaluations of statistical procedures on simulations generated by distrSim.

distrTeach

embodies illustrations for basic stats courses using our distribution classes.

distrMod

provides classes for parametric models and hence covers, in an object orientated way, estimation in statistical models.

distrEllipse

provides classes for elliptically symmetric distributions.

### Package versions

Note: The first two numbers of package versions do not necessarily reflect package-individual development, but rather are chosen for the distrXXX family as a whole in order to ease updating "depends" information.

### Acknowledgement

We thank Martin Maechler, Josef Leydold, John Chambers, Duncan Murdoch, Gregory Warnes, Paul Gilbert, Kurt Hornik, Uwe Ligges, Torsten Hothorn, and Seth Falcon for their help in preparing this package.

### Start-up-Banner

You may suppress the start-up banner/message completely by setting `options("StartupBanner"="off")` somewhere before loading this package by `library` or `require` in your R-code / R-session.

If option `"StartupBanner"` is not defined (default) or setting `options("StartupBanner"=NULL)` or `options("StartupBanner"="complete")` the complete start-up banner is displayed.

For any other value of option `"StartupBanner"` (i.e., not in `c(NULL,"off","complete")`) only the version information is displayed.

The same can be achieved by wrapping the `library` or `require` call into either `suppressStartupMessages()` or `onlytypeStartupMessages(.,atypes="version")`.

As for general `packageStartupMessage`'s, you may also suppress all the start-up banner by wrapping the `library` or `require` call into `suppressPackageStartupMessages()` from startupmsg-version 0.5 on.

### Demos

Demos are available — see `demo(package="distr")`

### Note

Arithmetics on distribution objects are understood as operations on corresponding (independent) r.v.'s and not on distribution functions or densities.

See also `distrARITH()`.

Some functions of package stats have intentionally been masked, but completely retain their functionality — see `distrMASK()`.

Accuracy of these arithmetics is controlled by global options which may be inspected / set by `distroptions()` and `getdistrOption()`, confer distroptions .

### Author(s)

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,
Thomas Stabla statho3@web.de,
Florian Camphausen fcampi@gmx.de,
Matthias Kohl Matthias.Kohl@stamats.de

Maintainer: Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de

### References

P. Ruckdeschel, M. Kohl, T. Stabla, F. Camphausen (2006): S4 Classes for Distributions, R News, 6(2), 2-6. http://CRAN.R-project.org/doc/Rnews/Rnews_2006-2.pdf

P. Ruckdeschel and M. Kohl (2014): General purpose convolution algorithm for distributions in S4-Classes by means of FFT. J. Statist. Softw. 59(4): 1-25.

a vignette for packages distr, distrSim, distrTEst, and distrEx is included into the mere documentation package distrDoc and may be called by `require("distrDoc");vignette("distr")`

a homepage to this package is available under
http://distr.r-forge.r-project.org/

### Examples

 ```1 2 3 4 5 6``` ```X <- Unif(2,3) Y <- Pois(lambda = 3) Z <- X+Y # generates Law of corresponding independent variables p(Z)(0.2) r(Z)(1000) plot(Z+sin(Norm())) ```

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