M.psi: Psi / Chi / Wgt / Rho Functions for *M-Estimation
In robustbase: Basic Robust Statistics
Mpsi R Documentation
Psi / Chi / Wgt / Rho Functions for *M-Estimation
Description
Compute Psi / Chi / Wgt / Rho functions for M-estimation,
i.e., including MM, etc. For definitions and details, please use the
vignette
“\psi-Functions Available in Robustbase”.
MrhoInf(x) computes \rho(\infty), i.e., the
normalizing or scaling constant for the transformation
from \rho(\cdot) to
\tilde\rho(\cdot), where the latter, aka as
\chi() fulfills \tilde\rho(\infty) = 1
which makes only sense for “redescending” psi functions, i.e.,
not for "huber".
Mwgt(x, *) computes \psi(x)/x (fast and numerically accurately).
Usage
Mpsi(x, cc, psi, deriv = 0)
Mchi(x, cc, psi, deriv = 0)
Mwgt(x, cc, psi)
MrhoInf(cc, psi)
.Mwgt.psi1(psi, cc = .Mpsi.tuning.default(psi))
.regularize.Mpsi(psi, redescending = TRUE)
Arguments
x
numeric (“abscissa” values) vector, possibly with
attributes such as dim or
names, etc. These are preserved for the
M*() functions (but not the .M() ones).
cc
numeric tuning constant, for some psi of length
> 1.
psi
a string specifying the psi / chi / rho / wgt function;
either "huber", or one of the same possible specifiers as for
psi in lmrob.control, i.e. currently,
"bisquare", "lqq", "welsh", "optimal",
"hampel", or "ggw".
deriv
an integer, specifying the order of derivative to
consider; particularly, Mpsi(x, *, deriv = -1) is the
principal function of \psi(), typically denoted
\rho() in the literature. For some psi functions,
currently "huber", "bisquare", "hampel", and "lqq",
deriv = 2 is implemented, for the other psi's only
d \in \{-1,0,1\}
redescending
logical indicating in .regularize.Mpsi(psi,.)
if the psi function is redescending.
Details
Theoretically, Mchi() would not be needed explicitly as it can be computed
from Mpsi() and MrhoInf(), namely, by
Mchi(x, *, deriv = d) == Mpsi(x, *, deriv = d-1) / MrhoInf(*)
for d = 0, 1, 2 (and ‘*’ containing par, psi, and
equality is in the sense of all.equal(x,y, tol) with a
small tol.
Similarly, Mwgt would not be needed strictly, as it could be
defined via Mpsi), but the explicit definition takes care of
0/0 and typically is of a more simple form.
For experts, there are slightly even faster versions,
.Mpsi(), .Mwgt(), etc.
.Mwgt.psi1() mainly a utility for nlrob(),
returns a function with similar semantics as
psi.hampel, psi.huber, or
psi.bisquare from package MASS. Namely,
a function with arguments (x, deriv=0), which for
deriv=0 computes Mwgt(x, cc, psi) and otherwise computes
Mpsi(x, cc, psi, deriv=deriv).
.Mpsi(), .Mchi(), .Mwgt(), and .MrhoInf() are
low-level versions of
Mpsi(), Mchi(), Mwgt(), and MrhoInf(), respectively,
and .psi2ipsi() provides the psi-function integer codes needed
for ipsi argument of the .M*() functions.
For psi = "ggw", the \rho() function has no closed
form and must be computed via numerical integration, apart from 6
special cases including the defaults, see the ‘Details’ in
help(.psi.ggw.findc).
.Mpsi.regularize() may (rarely) be used to regularize a psi function.
Value
a numeric vector of the same length as x, with corresponding
function (or derivative) values.
Author(s)
Manuel Koller, notably for the original C implementation;
tweaks and speedup via .Call and .M*() etc by
Martin Maechler.
References
See the vignette about
“\psi-Functions Available in Robustbase”.
See Also
psiFunc and the psi_func class, both
of which provide considerably more on the R side, but are less
optimized for speed.
.Mpsi.tuning.defaults, etc, for tuning constants'
defaults forlmrob(), and .psi.ggw.findc()
utilities to construct such constants' vectors.
Examples
x <- seq(-5,7, by=1/8)
matplot(x, cbind(Mpsi(x, 4, "biweight"),
Mchi(x, 4, "biweight"),
Mwgt(x, 4, "biweight")), type = "l")
abline(h=0, v=0, lty=2, col=adjustcolor("gray", 0.6))
hampelPsi
(ccHa <- hampelPsi @ xtras $ tuningP $ k)
psHa <- hampelPsi@psi(x)
## using Mpsi():
Mp.Ha <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(all.equal(Mp.Ha, psHa, tolerance = 1e-15))
psi.huber <- .Mwgt.psi1("huber")
if(getRversion() >= "3.0.0")
stopifnot(identical(psi.huber, .Mwgt.psi1("huber", 1.345),
ignore.env=TRUE))
curve(psi.huber(x), -3, 5, col=2, ylim = 0:1)
curve(psi.huber(x, deriv=1), add=TRUE, col=3)
## and show that this is indeed the same as MASS::psi.huber() :
x <- runif(256, -2,3)
stopifnot(all.equal(psi.huber(x), MASS::psi.huber(x)),
all.equal( psi.huber(x, deriv=1),
as.numeric(MASS::psi.huber(x, deriv=1))))
## and how to get MASS::psi.hampel():
psi.hampel <- .Mwgt.psi1("Hampel", c(2,4,8))
x <- runif(256, -4, 10)
stopifnot(all.equal(psi.hampel(x), MASS::psi.hampel(x)),
all.equal( psi.hampel(x, deriv=1),
as.numeric(MASS::psi.hampel(x, deriv=1))))
## "lqq" / "LQQ" and its tuning constants:
ctl0 <- lmrob.control(psi = "lqq", tuning.psi=c(-0.5, 1.5, 0.95, NA))
ctl <- lmrob.control(psi = "lqq", tuning.psi=c(-0.5, 1.5, 0.90, NA))
ctl0$tuning.psi ## keeps the vector _and_ has "constants" attribute:
## [1] -0.50 1.50 0.95 NA
## attr(,"constants")
## [1] 1.4734061 0.9822707 1.5000000
ctl$tuning.psi ## ditto:
## [1] -0.5 1.5 0.9 NA \ .."constants" 1.213726 0.809151 1.500000
stopifnot(all.equal(Mpsi(0:2, cc = ctl$tuning.psi, psi = ctl$psi),
c(0, 0.977493, 1.1237), tol = 6e-6))
x <- seq(-4,8, by = 1/16)
## Show how you can use .Mpsi() equivalently to Mpsi()
stopifnot(all.equal( Mpsi(x, cc = ctl$tuning.psi, psi = ctl$psi),
.Mpsi(x, ccc = attr(ctl$tuning.psi, "constants"),
ipsi = .psi2ipsi("lqq"))))
stopifnot(all.equal( Mpsi(x, cc = ctl0$tuning.psi, psi = ctl0$psi, deriv=1),
.Mpsi(x, ccc = attr(ctl0$tuning.psi, "constants"),
ipsi = .psi2ipsi("lqq"), deriv=1)))
## M*() preserving attributes :
x <- matrix(x, 32, 8, dimnames=list(paste0("r",1:32), col=letters[1:8]))
comment(x) <- "a vector which is a matrix"
px <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(identical(attributes(x), attributes(px)))
## The "optimal" psi exists in two versions "in the litterature": ---
## Maronna et al. 2006, 5.9.1, p.144f:
psi.M2006 <- function(x, c = 0.013)
sign(x) * pmax(0, abs(x) - c/dnorm(abs(x)))
## and the other is the one in robustbase from 'robust': via Mpsi(.., "optimal")
## Here are both for 95% efficiency:
(c106 <- .Mpsi.tuning.default("optimal"))
c1 <- curve(Mpsi(x, cc = c106, psi="optimal"), -5, 7, n=1001)
c2 <- curve(psi.M2006(x), add=TRUE, n=1001, col=adjustcolor(2,0.4), lwd=2)
abline(0,1, v=0, h=0, lty=3)
## the two psi's are similar, but really quite different
## a zoom into Maronna et al's:
c3 <- curve(psi.M2006(x), -.5, 1, n=1001); abline(h=0,v=0, lty=3);abline(0,1, lty=2)
robustbase documentation built on Jan. 27, 2024, 3 p.m.
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| Mpsi | R Documentation |
Psi / Chi / Wgt / Rho Functions for *M-Estimation
Description
Compute Psi / Chi / Wgt / Rho functions for M-estimation,
i.e., including MM, etc. For definitions and details, please use the
vignette
“\psi-Functions Available in Robustbase”.
MrhoInf(x) computes \rho(\infty), i.e., the
normalizing or scaling constant for the transformation
from \rho(\cdot) to
\tilde\rho(\cdot), where the latter, aka as
\chi() fulfills \tilde\rho(\infty) = 1
which makes only sense for “redescending” psi functions, i.e.,
not for "huber".
Mwgt(x, *) computes \psi(x)/x (fast and numerically accurately).
Usage
Mpsi(x, cc, psi, deriv = 0)
Mchi(x, cc, psi, deriv = 0)
Mwgt(x, cc, psi)
MrhoInf(cc, psi)
.Mwgt.psi1(psi, cc = .Mpsi.tuning.default(psi))
.regularize.Mpsi(psi, redescending = TRUE)
Arguments
x |
numeric (“abscissa” values) vector, possibly with
|
cc |
numeric tuning constant, for some |
psi |
a string specifying the psi / chi / rho / wgt function;
either |
deriv |
an integer, specifying the order of derivative to
consider; particularly, |
redescending |
logical indicating in |
Details
Theoretically, Mchi() would not be needed explicitly as it can be computed
from Mpsi() and MrhoInf(), namely, by
Mchi(x, *, deriv = d) == Mpsi(x, *, deriv = d-1) / MrhoInf(*)
for d = 0, 1, 2 (and ‘*’ containing par, psi, and
equality is in the sense of all.equal(x,y, tol) with a
small tol.
Similarly, Mwgt would not be needed strictly, as it could be
defined via Mpsi), but the explicit definition takes care of
0/0 and typically is of a more simple form.
For experts, there are slightly even faster versions,
.Mpsi(), .Mwgt(), etc.
.Mwgt.psi1() mainly a utility for nlrob(),
returns a function with similar semantics as
psi.hampel, psi.huber, or
psi.bisquare from package MASS. Namely,
a function with arguments (x, deriv=0), which for
deriv=0 computes Mwgt(x, cc, psi) and otherwise computes
Mpsi(x, cc, psi, deriv=deriv).
.Mpsi(), .Mchi(), .Mwgt(), and .MrhoInf() are
low-level versions of
Mpsi(), Mchi(), Mwgt(), and MrhoInf(), respectively,
and .psi2ipsi() provides the psi-function integer codes needed
for ipsi argument of the .M*() functions.
For psi = "ggw", the \rho() function has no closed
form and must be computed via numerical integration, apart from 6
special cases including the defaults, see the ‘Details’ in
help(.psi.ggw.findc).
.Mpsi.regularize() may (rarely) be used to regularize a psi function.
Value
a numeric vector of the same length as x, with corresponding
function (or derivative) values.
Author(s)
Manuel Koller, notably for the original C implementation;
tweaks and speedup via .Call and .M*() etc by
Martin Maechler.
References
See the vignette about
“\psi-Functions Available in Robustbase”.
See Also
psiFunc and the psi_func class, both
of which provide considerably more on the R side, but are less
optimized for speed.
.Mpsi.tuning.defaults, etc, for tuning constants'
defaults forlmrob(), and .psi.ggw.findc()
utilities to construct such constants' vectors.
Examples
x <- seq(-5,7, by=1/8)
matplot(x, cbind(Mpsi(x, 4, "biweight"),
Mchi(x, 4, "biweight"),
Mwgt(x, 4, "biweight")), type = "l")
abline(h=0, v=0, lty=2, col=adjustcolor("gray", 0.6))
hampelPsi
(ccHa <- hampelPsi @ xtras $ tuningP $ k)
psHa <- hampelPsi@psi(x)
## using Mpsi():
Mp.Ha <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(all.equal(Mp.Ha, psHa, tolerance = 1e-15))
psi.huber <- .Mwgt.psi1("huber")
if(getRversion() >= "3.0.0")
stopifnot(identical(psi.huber, .Mwgt.psi1("huber", 1.345),
ignore.env=TRUE))
curve(psi.huber(x), -3, 5, col=2, ylim = 0:1)
curve(psi.huber(x, deriv=1), add=TRUE, col=3)
## and show that this is indeed the same as MASS::psi.huber() :
x <- runif(256, -2,3)
stopifnot(all.equal(psi.huber(x), MASS::psi.huber(x)),
all.equal( psi.huber(x, deriv=1),
as.numeric(MASS::psi.huber(x, deriv=1))))
## and how to get MASS::psi.hampel():
psi.hampel <- .Mwgt.psi1("Hampel", c(2,4,8))
x <- runif(256, -4, 10)
stopifnot(all.equal(psi.hampel(x), MASS::psi.hampel(x)),
all.equal( psi.hampel(x, deriv=1),
as.numeric(MASS::psi.hampel(x, deriv=1))))
## "lqq" / "LQQ" and its tuning constants:
ctl0 <- lmrob.control(psi = "lqq", tuning.psi=c(-0.5, 1.5, 0.95, NA))
ctl <- lmrob.control(psi = "lqq", tuning.psi=c(-0.5, 1.5, 0.90, NA))
ctl0$tuning.psi ## keeps the vector _and_ has "constants" attribute:
## [1] -0.50 1.50 0.95 NA
## attr(,"constants")
## [1] 1.4734061 0.9822707 1.5000000
ctl$tuning.psi ## ditto:
## [1] -0.5 1.5 0.9 NA \ .."constants" 1.213726 0.809151 1.500000
stopifnot(all.equal(Mpsi(0:2, cc = ctl$tuning.psi, psi = ctl$psi),
c(0, 0.977493, 1.1237), tol = 6e-6))
x <- seq(-4,8, by = 1/16)
## Show how you can use .Mpsi() equivalently to Mpsi()
stopifnot(all.equal( Mpsi(x, cc = ctl$tuning.psi, psi = ctl$psi),
.Mpsi(x, ccc = attr(ctl$tuning.psi, "constants"),
ipsi = .psi2ipsi("lqq"))))
stopifnot(all.equal( Mpsi(x, cc = ctl0$tuning.psi, psi = ctl0$psi, deriv=1),
.Mpsi(x, ccc = attr(ctl0$tuning.psi, "constants"),
ipsi = .psi2ipsi("lqq"), deriv=1)))
## M*() preserving attributes :
x <- matrix(x, 32, 8, dimnames=list(paste0("r",1:32), col=letters[1:8]))
comment(x) <- "a vector which is a matrix"
px <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(identical(attributes(x), attributes(px)))
## The "optimal" psi exists in two versions "in the litterature": ---
## Maronna et al. 2006, 5.9.1, p.144f:
psi.M2006 <- function(x, c = 0.013)
sign(x) * pmax(0, abs(x) - c/dnorm(abs(x)))
## and the other is the one in robustbase from 'robust': via Mpsi(.., "optimal")
## Here are both for 95% efficiency:
(c106 <- .Mpsi.tuning.default("optimal"))
c1 <- curve(Mpsi(x, cc = c106, psi="optimal"), -5, 7, n=1001)
c2 <- curve(psi.M2006(x), add=TRUE, n=1001, col=adjustcolor(2,0.4), lwd=2)
abline(0,1, v=0, h=0, lty=3)
## the two psi's are similar, but really quite different
## a zoom into Maronna et al's:
c3 <- curve(psi.M2006(x), -.5, 1, n=1001); abline(h=0,v=0, lty=3);abline(0,1, lty=2)
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