AsymTotalVarDist {distrEx}

Generic function for the computation of asymmetric total variation distance of two distributions

Description

Generic function for the computation of asymmetric total variation distance d_v(\rho) of two distributions P and Q where the distributions may be defined for an arbitrary sample space (\Omega,{\cal A}). For given ratio of inlier and outlier probability \rho, this distance is defined as

d_v(\rho)(P,Q)=\int (dQ-c\,dP)_+

for c defined by

\rho \int (dQ-c\,dP)_+ = \int (dQ-c\,dP)_-

It coincides with total variation distance for \rho=1.

Usage

AsymTotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
AsymTotalVarDist(e1,e2, rho = 1,
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
AsymTotalVarDist(e1, e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
AsymTotalVarDist(e1, e2, rho  = 1, ...)
## S4 method for signature 'numeric,AbscontDistribution'
AsymTotalVarDist(e1, e2, rho = 1, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
            up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,numeric'
AsymTotalVarDist(e1, e2,  rho = 1,
            asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
            up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
AsymTotalVarDist(e1, e2,
          rho = 1, rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)

Arguments

e1

object of class "Distribution" or "numeric"

e2

object of class "Distribution" or "numeric"

asis.smooth.discretize

possible methods are "asis", "smooth" and "discretize". Default is "discretize".

n.discr

if asis.smooth.discretize is equal to "discretize" one has to specify the number of lattice points used to discretize the abs. cont. distribution.

low.discr

if asis.smooth.discretize is equal to "discretize" one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.

up.discr

if asis.smooth.discretize is equal to "discretize" one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.

h.smooth

if asis.smooth.discretize is equal to "smooth" – i.e., the empirical distribution of the provided data should be smoothed – one has to specify this parameter.

rho

ratio of inlier/outlier radius

rel.tol

relative tolerance for distrExIntegrate and uniroot

maxiter

parameter for uniroot

Ngrid

How many grid points are to be evaluated to determine the range of the likelihood ratio?

,

TruncQuantile

Quantile the quantile based integration bounds (see details)

IQR.fac

Factor for the scale based integration bounds (see details)

...

further arguments to be used in particular methods – (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to distrExIntegrate).

diagnostic

logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

For distances between absolutely continuous distributions, we use numerical integration; to determine sensible bounds we proceed as follows: by means of min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile)), max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile)) we determine quantile based bounds c(low.0,up.0), and by means of s1 <- max(IQR(e1),IQR(e2)); m1<- median(e1); m2 <- median(e2) and low.1 <- min(m1,m2)-s1*IQR.fac, up.1 <- max(m1,m2)+s1*IQR.fac we determine scale based bounds; these are combined by low <- max(low.0,low.1), up <- max(up.0,up1).

Again in the absolutely continuous case, to determine the range of the likelihood ratio, we evaluate this ratio on a grid constructed as follows: x.range <- c(seq(low, up, length=Ngrid/3), q.l(e1)(seq(0,1,length=Ngrid/3)*.999), q.l(e2)(seq(0,1,length=Ngrid/3)*.999))

Finally, for both discrete and absolutely continuous case, we clip this ratio downwards by 1e-10 and upwards by 1e10

In case we want to compute the total variation distance between (empirical) data and an abs. cont. distribution, we can specify the parameter asis.smooth.discretize to avoid trivial distances (distance = 1).

Using asis.smooth.discretize = "discretize", which is the default, leads to a discretization of the provided abs. cont. distribution and the distance is computed between the provided data and the discretized distribution.

Using asis.smooth.discretize = "smooth" causes smoothing of the empirical distribution of the provided data. This is, the empirical data is convoluted with the normal distribution Norm(mean = 0, sd = h.smooth) which leads to an abs. cont. distribution. Afterwards the distance between the smoothed empirical distribution and the provided abs. cont. distribution is computed.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

Asymmetric Total variation distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

total variation distance of two absolutely continuous univariate distributions which is computed using distrExIntegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

total variation distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

total variation distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

total variation distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Total variation distance of mixed discrete and absolutely continuous univariate distributions.

Author(s)

Peter Ruckdeschel [e-mail]

References

to be filled; Agostinelli, C and Ruckdeschel, P. (2009): A simultaneous inlier and outlier model by asymmetric total variation distance.

See Also

TotalVarDist-methods, ContaminationSize, KolmogorovDist, HellingerDist, Distribution-class

Examples

AsymTotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)), rho=0.3)
AsymTotalVarDist(Norm(), Td(10), rho=0.3)
AsymTotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100), rho=0.3) # mutually singular
AsymTotalVarDist(Pois(10), Binom(size = 20), rho=0.3) 

x <- rnorm(100)
AsymTotalVarDist(Norm(), x, rho=0.3)
AsymTotalVarDist(x, Norm(), asis.smooth.discretize = "smooth", rho=0.3)

y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
AsymTotalVarDist(y, Norm(), rho=0.3)
AsymTotalVarDist(y, Norm(), asis.smooth.discretize = "smooth", rho=0.3)

AsymTotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5), rho=0.3)