Mathematical and statistical functions for the Matdist distribution, which is commonly used in vectorised empirical estimators such as Kaplan-Meier.

Value

Returns an R6 object inheriting from class SDistribution.

Details

The Matdist distribution is defined by the pmf, $$f(x_{ij}) = p_{ij}$$ for \(p_{ij}, i = 1,\ldots,k, j = 1,\ldots,n; \sum_i p_{ij} = 1\).

This is a special case distribution in distr6 which is technically a vectorised distribution but is treated as if it is not. Therefore we only allow evaluation of all functions at the same value, e.g. $pdf(1:2) evaluates all samples at '1' and '2'.

Sampling from this distribution is performed with the sample function with the elements given as the x values and the pdf as the probabilities. The cdf and quantile assume that the elements are supplied in an indexed order (otherwise the results are meaningless).

Distribution support

The distribution is supported on \(x_{11},...,x_{kn}\).

Default Parameterisation

Matdist(matrix(0.5, 2, 2, dimnames = list(NULL, 1:2)))

Omitted Methods

N/A

Also known as

N/A

References

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

Super classes

distr6::Distribution -> distr6::SDistribution -> Matdist

Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

Methods

Inherited methods


Method new()

Creates a new instance of this R6 class.

Usage

Matdist$new(pdf = NULL, cdf = NULL, decorators = NULL)

Arguments

pdf

numeric()
Probability mass function for corresponding samples, should be same length x. If cdf is not given then calculated as cumsum(pdf).

cdf

numeric()
Cumulative distribution function for corresponding samples, should be same length x. If given then pdf calculated as difference of cdfs.

decorators

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

x

numeric()
Data samples, must be ordered in ascending order.


Method strprint()

Printable string representation of the Distribution. Primarily used internally.

Usage

Matdist$strprint(n = 2)

Arguments

n

(integer(1))
Ignored.


Method 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).

Usage

Matdist$mean(...)

Arguments

...

Unused.


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

Matdist$median()


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

Matdist$mode(which = 1)

Arguments

which

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


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. If distribution is improper (F(Inf) != 1, then var_X(x) = Inf).

Usage

Matdist$variance(...)

Arguments

...

Unused.


Method 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).

Usage

Matdist$skewness(...)

Arguments

...

Unused.


Method 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).

Usage

Matdist$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 $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions. If distribution is improper then entropy is Inf.

Usage

Matdist$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. If distribution is improper (F(Inf) != 1, then mgf_X(x) = Inf).

Usage

Matdist$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. If distribution is improper (F(Inf) != 1, then cf_X(x) = Inf).

Usage

Matdist$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. If distribution is improper (F(Inf) != 1, then pgf_X(x) = Inf).

Usage

Matdist$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

Matdist$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples

x <- Matdist$new(pdf = matrix(0.5, 2, 2, dimnames = list(NULL, 1:2)))
Matdist$new(cdf = matrix(c(0.5, 1), 2, 2, TRUE, dimnames = list(NULL, c(1, 2)))) # equivalently
#> Matdist() 

# d/p/q/r
x$pdf(1:5)
#>   [,1] [,2]
#> 1  0.5  0.5
#> 2  0.5  0.5
#> 3  0.0  0.0
#> 4  0.0  0.0
#> 5  0.0  0.0
x$cdf(1:5) # Assumes ordered in construction
#>   [,1] [,2]
#> 1  0.5  0.5
#> 2  1.0  1.0
#> 3  1.0  1.0
#> 4  1.0  1.0
#> 5  1.0  1.0
x$quantile(0.42) # Assumes ordered in construction
#>      [,1] [,2]
#> [1,]    1    1
x$rand(10)
#>       [,1] [,2]
#>  [1,]    2    2
#>  [2,]    2    2
#>  [3,]    1    1
#>  [4,]    2    2
#>  [5,]    2    2
#>  [6,]    1    2
#>  [7,]    2    2
#>  [8,]    2    2
#>  [9,]    1    2
#> [10,]    2    2

# Statistics
x$mean()
#> [1] 1.5 1.5
x$variance()
#> [1] 0.25 0.25

summary(x)
#> Matrix Probability Distribution. 
#> Parameterised with:
#> 
#>     Id Support           Value            Tags
#> 1: cdf [0,1]^n                 required,linked
#> 2: pdf [0,1]^n 0.5,0.5,0.5,0.5 required,linked
#> 3:   x       ℤ                       immutable
#> 
#> 
#> Quick Statistics
#> 	Mean:		1.5, 1.5
#> 	Variance:	0.25, 0.25
#> 	Skewness:	00
#> 	Ex. Kurtosis:	-2-2
#> 
#> Support: {1, 2} 	Scientific Type: ℝ^n 
#> 
#> Traits:		discrete; univariate
#> Properties:	asymmetric; platykurtic platykurtic; no skew no skew