AsymTotalVarDist: Generic function for the computation of asymmetric total...
In distrEx: Extensions of Package 'distr'
AsymTotalVarDist R Documentation
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 peter.ruckdeschel@uni-oldenburg.de
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)
distrEx documentation built on Jan. 30, 2024, 3:01 a.m.
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| AsymTotalVarDist | R Documentation |
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 |
e2 |
object of class |
asis.smooth.discretize |
possible methods are |
n.discr |
if |
low.discr |
if |
up.discr |
if |
h.smooth |
if |
rho |
ratio of inlier/outlier radius |
rel.tol |
relative tolerance for |
maxiter |
parameter for |
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 |
diagnostic |
logical; if |
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
supportandsum. - 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 peter.ruckdeschel@uni-oldenburg.de
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)
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