Mathematical and statistical functions for the Arrdist distribution, which is commonly used in matrixed Bayesian estimators such as Kaplan-Meier with confidence bounds over arbitrary dimensions.
Returns an R6 object inheriting from class SDistribution.
The Arrdist distribution is defined by the pmf, $$f(x_{ijk}) = p_{ijk}$$ for \(p_{ijk}, i = 1,\ldots,a, j = 1,\ldots,b; \sum_i p_{ijk} = 1\).
This is a generalised case of Matdist with a third dimension over an arbitrary length.
By default all results are returned for the median curve as determined by
(dim(a)[3L] + 1)/2 where a is the array and assuming third dimension is odd,
this can be changed by setting the which.curve parameter.
Given the complexity in construction, this distribution is not mutable (cannot be updated after construction).
The distribution is supported on \(x_{111},...,x_{abc}\).
Arrdist(array(0.5, c(2, 2, 2), list(NULL, 1:2, NULL)))
N/A
N/A
McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.
Other discrete distributions:
Bernoulli,
Binomial,
Categorical,
Degenerate,
DiscreteUniform,
EmpiricalMV,
Empirical,
Geometric,
Hypergeometric,
Logarithmic,
Matdist,
Multinomial,
NegativeBinomial,
WeightedDiscrete
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,
Triangular,
Uniform,
Wald,
Weibull,
WeightedDiscrete
distr6::Distribution -> distr6::SDistribution -> Arrdist
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
aliasAlias of the distribution.
propertiesReturns distribution properties, including skewness type and symmetry.
Inherited methods
distr6::Distribution$cdf()distr6::Distribution$confidence()distr6::Distribution$correlation()distr6::Distribution$getParameterValue()distr6::Distribution$iqr()distr6::Distribution$liesInSupport()distr6::Distribution$liesInType()distr6::Distribution$parameters()distr6::Distribution$pdf()distr6::Distribution$prec()distr6::Distribution$print()distr6::Distribution$quantile()distr6::Distribution$rand()distr6::Distribution$setParameterValue()distr6::Distribution$stdev()distr6::Distribution$summary()distr6::Distribution$workingSupport()
new()Creates a new instance of this R6 class.
Arrdist$new(pdf = NULL, cdf = NULL, which.curve = 0.5, decorators = NULL)pdfnumeric()
Probability mass function for corresponding samples, should be same length x.
If cdf is not given then calculated as cumsum(pdf).
cdfnumeric()
Cumulative distribution function for corresponding samples, should be same length x. If
given then pdf calculated as difference of cdfs.
which.curvenumeric(1) | character(1)
Which curve (third dimension) should results be displayed for? If
between (0,1) taken as the quantile of the curves otherwise if greater than 1 taken as the curve index, can also be 'mean'. See examples.
decorators(character())
Decorators to add to the distribution during construction.
mean()The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions. If distribution is improper (F(Inf) != 1, then E_X(x) = Inf).
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).
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).
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. If distribution is improper (F(Inf) != 1, then var_X(x) = Inf).
skewness()The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. If distribution is improper (F(Inf) != 1, then sk_X(x) = Inf).
kurtosis()The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3. If distribution is improper (F(Inf) != 1, then k_X(x) = Inf).
entropy()The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions. If distribution is improper then entropy is Inf.
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. If distribution is improper (F(Inf) != 1, then mgf_X(x) = Inf).
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. If distribution is improper (F(Inf) != 1, then cf_X(x) = Inf).
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. If distribution is improper (F(Inf) != 1, then pgf_X(x) = Inf).
x <- Arrdist$new(pdf = array(0.5, c(3, 2, 4),
dimnames = list(NULL, 1:2, NULL)))
Arrdist$new(cdf = array(c(0.5, 0.5, 0.5, 1, 1, 1), c(3, 2, 4),
dimnames = list(NULL, 1:2, NULL))) # equivalently
#> Arrdist(3x2x4)
# d/p/q/r
x$pdf(1)
#> [,1] [,2] [,3]
#> 1 0.5 0.5 0.5
x$cdf(1:2) # Assumes ordered in construction
#> [,1] [,2] [,3]
#> 1 0.5 0.5 0.5
#> 2 1.0 1.0 1.0
x$quantile(0.42) # Assumes ordered in construction
#> [,1] [,2] [,3]
#> [1,] 1 1 1
x$rand(10)
#> [,1] [,2] [,3]
#> [1,] 2 1 2
#> [2,] 1 2 1
#> [3,] 1 2 1
#> [4,] 2 2 2
#> [5,] 2 2 1
#> [6,] 2 2 1
#> [7,] 2 2 2
#> [8,] 2 2 1
#> [9,] 1 2 1
#> [10,] 1 1 1
# Statistics
x$mean()
#> [1] 1.5 1.5 1.5
x$variance()
#> [1] 0.25 0.25 0.25
summary(x)
#> Array Probability Distribution.
#> Parameterised with:
#>
#> Id Support
#> <char> <char>
#> 1: cdf [0,1]^n
#> 2: pdf [0,1]^n
#> 3: which.curve {mean} ∪ ℤ ∪ (0,1)
#> 4: x ℤ
#> Value
#> <list>
#> 1:
#> 2: 0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,...
#> 3: 0.5
#> 4:
#> Tags
#> <list>
#> 1: required,linked
#> 2: required,linked
#> 3: required
#> 4: immutable
#>
#>
#> Quick Statistics
#> Mean: 1.5, 1.5, 1.5
#> Variance: 0.25, 0.25, 0.25
#> Skewness: 000
#> Ex. Kurtosis: -2-2-2
#>
#> Support: {1, 2} Scientific Type: ℝ^n
#>
#> Traits: discrete; univariate
#> Properties: asymmetric; platykurtic platykurtic platykurtic; no skew no skew no skew
# Changing which.curve
arr <- array(runif(90), c(3, 2, 5), list(NULL, 1:2, NULL))
arr <- aperm(apply(arr, c(1, 3), function(x) x / sum(x)), c(2, 1, 3))
arr[, , 1:3]
#> , , 1
#>
#> 1 2
#> [1,] 0.04677327 0.9532267
#> [2,] 0.32013436 0.6798656
#> [3,] 0.41040922 0.5895908
#>
#> , , 2
#>
#> 1 2
#> [1,] 0.7650905 0.2349095
#> [2,] 0.5823992 0.4176008
#> [3,] 0.2655225 0.7344775
#>
#> , , 3
#>
#> 1 2
#> [1,] 0.45597150 0.5440285
#> [2,] 0.46175050 0.5382495
#> [3,] 0.08836011 0.9116399
#>
x <- Arrdist$new(arr)
x$mean() # default 0.5 quantile (in this case index 3)
#> [1] 1.544029 1.417601 1.634341
x$setParameterValue(which.curve = 3) # equivalently
x$mean()
#> [1] 1.544029 1.538250 1.911640
# 1% quantile
x$setParameterValue(which.curve = 0.01)
x$mean()
#> [1] 0.5319016 0.4988487 1.1706783
# 5th index
x$setParameterValue(which.curve = 5)
x$mean()
#> [1] 1.591033 1.077222 1.634341
# mean
x$setParameterValue(which.curve = "mean")
x$mean()
#> [1] 1.513675 1.404545 1.681100