# Triangular: Triangular Distribution Class In distr6: The Complete R6 Probability Distributions Interface

 Triangular R Documentation

## Triangular Distribution Class

### Description

Mathematical and statistical functions for the Triangular distribution, which is commonly used to model population data where only the minimum, mode and maximum are known (or can be reliably estimated), also to model the sum of standard uniform distributions.

### Details

The Triangular distribution parameterised with lower limit, a, upper limit, b, and mode, c, is defined by the pdf,

f(x) = 0, x < a
f(x) = 2(x-a)/((b-a)(c-a)), a ≤ x < c
f(x) = 2/(b-a), x = c
f(x) = 2(b-x)/((b-a)(b-c)), c < x ≤ b
f(x) = 0, x > b for a,b,c ε R, a ≤ c ≤ b.

### Value

Returns an R6 object inheriting from class SDistribution.

### Distribution support

The distribution is supported on [a, b].

### Default Parameterisation

Tri(lower = 0, upper = 1, mode = 0.5, symmetric = FALSE)

N/A

N/A

### Super classes

`distr6::Distribution` -> `distr6::SDistribution` -> `Triangular`

### Public fields

`name`

Full name of distribution.

`short_name`

Short name of distribution for printing.

`description`

Brief description of the distribution.

`packages`

Packages required to be installed in order to construct the distribution.

### Active bindings

`properties`

Returns distribution properties, including skewness type and symmetry.

### Methods

#### Public methods

Inherited methods

#### Method `new()`

Creates a new instance of this R6 class.

##### Usage
```Triangular\$new(
lower = NULL,
upper = NULL,
mode = NULL,
symmetric = NULL,
decorators = NULL
)```
##### Arguments
`lower`

`(numeric(1))`
Lower limit of the Distribution, defined on the Reals.

`upper`

`(numeric(1))`
Upper limit of the Distribution, defined on the Reals.

`mode`

`(numeric(1))`
Mode of the distribution, if `symmetric = TRUE` then determined automatically.

`symmetric`

`(logical(1))`
If `TRUE` then the symmetric Triangular distribution is constructed, where the `mode` is automatically calculated. Otherwise `mode` can be set manually. Cannot be changed after construction.

`decorators`

`(character())`
Decorators to add to the distribution during construction.

##### Examples
```Triangular\$new(lower = 2, upper = 5, symmetric = TRUE)
Triangular\$new(lower = 2, upper = 5, mode = 4, symmetric = FALSE)

# You can view the type of Triangular distribution with \$description
Triangular\$new(symmetric = TRUE)\$description
Triangular\$new(symmetric = FALSE)\$description
```

#### Method `mean()`

The arithmetic mean of a (discrete) probability distribution X is the expectation

E_X(X) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

##### Usage
`Triangular\$mean(...)`
`...`

Unused.

#### Method `mode()`

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

##### Usage
`Triangular\$mode(which = "all")`
##### Arguments
`which`

`(character(1) | numeric(1)`
Ignored if distribution is unimodal. Otherwise `"all"` returns all modes, otherwise specifies which mode to return.

#### Method `median()`

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns `self\$mean`, otherwise returns `self\$quantile(0.5)`.

##### Usage
`Triangular\$median()`

#### Method `variance()`

The variance of a distribution is defined by the formula

var_X = E[X^2] - E[X]^2

where E_X is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

##### Usage
`Triangular\$variance(...)`
`...`

Unused.

#### Method `skewness()`

The skewness of a distribution is defined by the third standardised moment,

sk_X = E_X[((x - μ)/σ)^3]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.

##### Usage
`Triangular\$skewness(...)`
`...`

Unused.

#### Method `kurtosis()`

The kurtosis of a distribution is defined by the fourth standardised moment,

k_X = E_X[((x - μ)/σ)^4]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

##### Usage
`Triangular\$kurtosis(excess = TRUE, ...)`
##### Arguments
`excess`

`(logical(1))`
If `TRUE` (default) excess kurtosis returned.

`...`

Unused.

#### Method `entropy()`

The entropy of a (discrete) distribution is defined by

- ∑ (f_X)log(f_X)

where f_X is the pdf of distribution X, with an integration analogue for continuous distributions.

##### Usage
`Triangular\$entropy(base = 2, ...)`
##### Arguments
`base`

`(integer(1))`
Base of the entropy logarithm, default = 2 (Shannon entropy)

`...`

Unused.

#### Method `mgf()`

The moment generating function is defined by

mgf_X(t) = E_X[exp(xt)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Triangular\$mgf(t, ...)`
##### Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

#### Method `cf()`

The characteristic function is defined by

cf_X(t) = E_X[exp(xti)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Triangular\$cf(t, ...)`
##### Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

#### Method `pgf()`

The probability generating function is defined by

pgf_X(z) = E_X[exp(z^x)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Triangular\$pgf(z, ...)`
##### Arguments
`z`

`(integer(1))`
z integer to evaluate probability generating function at.

`...`

Unused.

#### Method `clone()`

The objects of this class are cloneable with this method.

##### Usage
`Triangular\$clone(deep = FALSE)`
##### Arguments
`deep`

Whether to make a deep clone.

### References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

Other continuous distributions: `Arcsine`, `BetaNoncentral`, `Beta`, `Cauchy`, `ChiSquaredNoncentral`, `ChiSquared`, `Dirichlet`, `Erlang`, `Exponential`, `FDistributionNoncentral`, `FDistribution`, `Frechet`, `Gamma`, `Gompertz`, `Gumbel`, `InverseGamma`, `Laplace`, `Logistic`, `Loglogistic`, `Lognormal`, `MultivariateNormal`, `Normal`, `Pareto`, `Poisson`, `Rayleigh`, `ShiftedLoglogistic`, `StudentTNoncentral`, `StudentT`, `Uniform`, `Wald`, `Weibull`

Other univariate distributions: `Arcsine`, `Bernoulli`, `BetaNoncentral`, `Beta`, `Binomial`, `Categorical`, `Cauchy`, `ChiSquaredNoncentral`, `ChiSquared`, `Degenerate`, `DiscreteUniform`, `Empirical`, `Erlang`, `Exponential`, `FDistributionNoncentral`, `FDistribution`, `Frechet`, `Gamma`, `Geometric`, `Gompertz`, `Gumbel`, `Hypergeometric`, `InverseGamma`, `Laplace`, `Logarithmic`, `Logistic`, `Loglogistic`, `Lognormal`, `Matdist`, `NegativeBinomial`, `Normal`, `Pareto`, `Poisson`, `Rayleigh`, `ShiftedLoglogistic`, `StudentTNoncentral`, `StudentT`, `Uniform`, `Wald`, `Weibull`, `WeightedDiscrete`

### Examples

```
## ------------------------------------------------
## Method `Triangular\$new`
## ------------------------------------------------

Triangular\$new(lower = 2, upper = 5, symmetric = TRUE)
Triangular\$new(lower = 2, upper = 5, mode = 4, symmetric = FALSE)

# You can view the type of Triangular distribution with \$description
Triangular\$new(symmetric = TRUE)\$description
Triangular\$new(symmetric = FALSE)\$description
```

distr6 documentation built on March 28, 2022, 1:05 a.m.