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)
Omitted Methods
N/A
Also known as
N/A
Super classes
distr6::Distribution -> distr6::SDistribution -> Triangular
Public fields
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
packagesPackages required to be installed in order to construct the distribution.
Active bindings
propertiesReturns distribution properties, including skewness type and symmetry.
Methods
Public methods
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Inherited methods
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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(...)
Arguments
...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$variance(...)
Arguments
...Unused.
Method skewness()
The skewness of a distribution is defined by the third standardised moment,
sk_X = E_X[((x - μ)/σ)^3]
where E_X is the expectation of distribution X, μ is the mean of the
distribution and σ is the standard deviation of the distribution.
Usage
Triangular$skewness(...)
Arguments
...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
deepWhether to make a deep clone.
References
McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01).
Michael P. McLaughlin.
See Also
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)$description
distr6 documentation built on March 28, 2022, 1:05 a.m.
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| Triangular | R Documentation |
Triangular Distribution Class
Description
Mathematical and statistical functions for the Triangular distribution, which is commonly used to model population data where only the minimum, mode and maximum are known (or can be reliably estimated), also to model the sum of standard uniform distributions.
Details
The Triangular distribution parameterised with lower limit, a, upper limit, b, and mode, c, is defined by the pdf,
f(x) = 0, x < a
f(x) = 2(x-a)/((b-a)(c-a)), a ≤ x < c
f(x) = 2/(b-a), x = c
f(x) = 2(b-x)/((b-a)(b-c)), c < x ≤ b
f(x) = 0, x > b for a,b,c ε R, a ≤ c ≤ b.
Value
Returns an R6 object inheriting from class SDistribution.
Distribution support
The distribution is supported on [a, b].
Default Parameterisation
Tri(lower = 0, upper = 1, mode = 0.5, symmetric = FALSE)
Omitted Methods
N/A
Also known as
N/A
Super classes
distr6::Distribution -> distr6::SDistribution -> Triangular
Public fields
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
packagesPackages required to be installed in order to construct the distribution.
Active bindings
propertiesReturns distribution properties, including skewness type and symmetry.
Methods
Public methods
Inherited methods
Method new()
Creates a new instance of this R6 class.
Usage
Triangular$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, ifsymmetric = TRUEthen determined automatically.symmetric(logical(1))
IfTRUEthen the symmetric Triangular distribution is constructed, where themodeis automatically calculated. Otherwisemodecan 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(...)
Arguments
...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$variance(...)
Arguments
...Unused.
Method skewness()
The skewness of a distribution is defined by the third standardised moment,
sk_X = E_X[((x - μ)/σ)^3]
where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.
Usage
Triangular$skewness(...)
Arguments
...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$kurtosis(excess = TRUE, ...)
Arguments
excess(logical(1))
IfTRUE(default) excess kurtosis returned....Unused.
Method entropy()
The entropy of a (discrete) distribution is defined by
- ∑ (f_X)log(f_X)
where f_X is the pdf of distribution X, with an integration analogue for continuous distributions.
Usage
Triangular$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
deepWhether to make a deep clone.
References
McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.
See Also
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
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