Title:	Normal aka Gaussian 1-d Mixture Models
Version:	1.3-3
Date:	2024-04-05
Description:	Onedimensional Normal (i.e. Gaussian) Mixture Models (S3) Classes, for, e.g., density estimation or clustering algorithms research and teaching; providing the widely used Marron-Wand densities. Efficient random number generation and graphics. Fitting to data by efficient ML (Maximum Likelihood) or traditional EM estimation.
Imports:	stats, graphics
Suggests:	cluster, copula
URL:	https://curves-etc.r-forge.r-project.org/, https://r-forge.r-project.org/R/?group_id=846, https://r-forge.r-project.org/scm/viewvc.php/pkg/nor1mix/?root=curves-etc, svn://svn.r-forge.r-project.org/svnroot/curves-etc/pkg/nor1mix
BugReports:	https://r-forge.r-project.org/R/?group_id=846
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
NeedsCompilation:	no
Packaged:	2024-04-05 18:14:12 UTC; maechler
Author:	Martin Maechler [aut, cre], Friedrich Leisch [ctb] (norMixEM()), Erik Jørgensen [ctb] (pnorMix(), qnorMix())
Maintainer:	Martin Maechler <maechler@stat.math.ethz.ch>
Repository:	CRAN
Date/Publication:	2024-04-06 09:53:04 UTC

Normal aka Gaussian 1-d Mixture Models

Description

Onedimensional Normal (i.e. Gaussian) Mixture Models (S3) Classes, for, e.g., density estimation or clustering algorithms research and teaching; providing the widely used Marron-Wand densities. Efficient random number generation and graphics. Fitting to data by efficient ML (Maximum Likelihood) or traditional EM estimation.

Details

The DESCRIPTION file:

Index of help topics:

MarronWand              Marron-Wand Densities as 'norMix' Objects
clus2norMix             Transform Clustering / Grouping to Normal
                        Mixture
dnorMix                 Normal Mixture Density
llnorMix                Likelihood, Parametrization and EM-Steps For 1D
                        Normal Mixtures
nor1mix-package         Normal aka Gaussian 1-d Mixture Models
norMix                  Mixtures of Univariate Normal Distributions
norMix2call             Transform "norMix" object into Call, Expression
                        or Function
norMixEM                EM and MLE Estimation of Univariate Normal
                        Mixtures
plot.norMix             Plotting Methods for 'norMix' Objects
pnorMix                 Normal Mixture Cumulative Distribution and
                        Quantiles
r.norMix                Ratio of Normal Mixture to Corresponding Normal
rnorMix                 Generate 'Normal Mixture' Distributed Random
                        Numbers
sort.norMix             Sort Method for "norMix" Objects

Note that direct Maximum Likelihood ML (via optim()) is typically much faster converging (and more reliably detecting convergence correctly), notably thanks to a smart re-parametrization: use norMixMLE().

Author(s)

Martin Maechler [aut, cre] (<https://orcid.org/0000-0002-8685-9910>), Friedrich Leisch [ctb] (norMixEM(), <https://orcid.org/0000-0001-7278-1983>), Erik Jørgensen [ctb] (pnorMix(), qnorMix())

Maintainer: Martin Maechler <maechler@stat.math.ethz.ch>

See Also

The Marron-Wand examples of normal (gaussian) mixtures MarronWand.

Multivariate distributions from copulas Mvdc from the copula package can use norMix marginals.

Examples

   example(dnorMix)
   

Marron-Wand Densities as 'norMix' Objects

Description

The fifteen density examples used in Marron and Wand (1992)'s simulation study have been used in quite a few subsequent studies, can all be written as normal mixtures and are provided here for convenience and didactical examples of normal mixtures. Number 16 has been added by Jansen et al.

Usage

 MW.nm1	 # Gaussian
 MW.nm2	 # Skewed
 MW.nm2.old # Skewed(old)
 MW.nm3	 # Str Skew
 MW.nm4	 # Kurtotic
 MW.nm5	 # Outlier
 MW.nm6	 # Bimodal
 MW.nm7	 # Separated (bimodal)
 MW.nm8	 # Asymmetric Bimodal
 MW.nm9	 # Trimodal
 MW.nm10 # Claw
 MW.nm11 # Double Claw
 MW.nm12 # Asymmetric Claw
 MW.nm13 # Asymm. Double Claw
 MW.nm14 # Smooth   Comb
 MW.nm15 # Discrete Comb
 MW.nm16 # Distant Bimodal

Author(s)

Martin Maechler

Source

They have been translated from Steve Marron's Matlab code, now at https://marronwebfiles.sites.oasis.unc.edu/OldResearch/parameters/nmpar.m, however for number 2, the Matlab code had MW.nm2.old; and I've defined MW.nm2 as from the Annals paper; see also the last example below.

References

Marron, S. and Wand, M. (1992) Exact Mean Integrated Squared Error; Annals of Statistcs 20, 712–736.

For number 16,
P. Janssen, J. S. Marron, N. Veraverbeke and W. Sarle (1995) Scale measures for bandwidth selection; Journal of Nonparametric Statistics 5, 359–380. doi:10.1080/10485259508832654

Examples

MW.nm10
plot(MW.nm14)

## These are defined as norMix() calls in  ../R/zMarrWand-dens.R
nms <- ls(pattern = "^MW.nm", "package:nor1mix")
nms <- nms[order(as.numeric(substring(nms,6)))] # w/ warning for "2.old"
for(n in nms) {
   cat("\n",n,":\n"); print(get(n, "package:nor1mix"))
}

## Plot all of them:
op <- par(mfrow=c(4,4), mgp = c(1.2, 0.5, 0), tcl = -0.2,
          mar = .1 + c(2,2,2,1), oma = c(0,0,3,0))
for(n in nms[-17]) plot(get(n, "package:nor1mix"))
mtext("The Marron-Wand Densities", outer= TRUE, font= 2, cex= 1.6)

## and their Q-Q-plots (not really fast):
prob <- ppoints(N <- 100)
for(n in nms[-17]) # qnorMix() using monotone spline inversion ==> warning
   qqnorm(qnorMix(prob, get(n, "package:nor1mix")), main = n)
mtext("QQ-plots of Marron-Wand Densities", outer = TRUE,
      font = 2, cex = 1.6)
par(op)

## "object" overview:
cbind(sapply(nms, function(n) { o <- get(n)
      sprintf("%-18s: K =%2d; rng = [%3.1f, %2.1f]",
              attr(o, "name"), nrow(o),
              min(o[,"mu"] - 3*o[,"sigma"]),
              max(o[,"mu"] + 3*o[,"sigma"]) )
             }))


## Note that Marron-Wand (1992), p.720  give #2 as
MW.nm2
## the parameters of which at first look quite different from
MW.nm2.old
## which has been the definition in the above "Source" Matlab code.
## It's easy to see that mu_{nm2} = -.3 + 1.2   * mu_{paper},
## and correspondigly,   s2_{nm2} =       1.2^2 * s2_{paper}
## such that they are "identical" apart from scale and location:
op. <- par(mfrow=2:1, mgp= c(1.2,0.5,0), tcl= -0.2, mar=.1+c(2,2,2,1))
plot(MW.nm2)
plot(MW.nm2.old)
par(op.)

Transform Clustering / Grouping to Normal Mixture

Description

Simple transformation of a clustering or grouping to a normal mixture object (class "norMix", see, norMix.

Usage

clus2norMix(gr, x, name = deparse(sys.call()))

Arguments

Value

A call to norMix() with (mu, sig2, w) set to the empirical values of the groups (as defined by split(x,gr).

Note

Via this function, any simple clustering algorithm (such pam) can be used as simple mixture model fitting procedure.

Author(s)

Martin Maechler, Dec. 2007

See Also

norMix; further pam() (or clara()) from package cluster for sensible clusterings.

Examples

x9 <- rnorMix(500, MW.nm9)
require("cluster")
pxc <- pam(x9, k=3)
plot(pxc, which = 2)# silhouette

(nm.p9 <- clus2norMix(pxc$clustering, x9))
plot(nm.p9, p.norm=FALSE)
lines(MW.nm9, col="thistle")

Normal Mixture Density

Description

Evaluate the density function of the normal mixture specified as norMix object.

Usage

dnorMix (x, obj,        log = FALSE)

dnorMixL(obj, x = NULL, log = FALSE, xlim = NULL, n = 511)
dpnorMix(x, obj, lower.tail = TRUE)

Arguments

Value

dnorMix(x) returns the numeric vector of density values f(x), logged if log is TRUE.

dnorMixL() returns a list with components

dpnorMix() returns a list with components

See Also

rnorMix for random number generation, and norMix for the construction and further methods, particularly plot.norMix which makes use dnorMix.

Examples

 ff <- dnorMixL(MW.nm7)
 str(ff)
 plot(ff, type = "h", ylim = c(0,1)) # rather use plot(ff, ...)

 x <- seq.int(-4,5, length.out = 501)
 dp <- dpnorMix(x, MW.nm7)
 lines(x, dp$d, col = "tomato", lwd=3)
 lines(x, dp$p, col = 3, lwd=2)# does not fit y-wise
 stopifnot(all.equal(dp$d, dnorMix(x, MW.nm7), tolerance=1e-12),
           all.equal(dp$p, pnorMix(x, MW.nm7), tolerance=1e-12))

Likelihood, Parametrization and EM-Steps For 1D Normal Mixtures

Description

These functions work with an almost unconstrained parametrization of univariate normal mixtures.

llnorMix(p, *)

computes the log likelihood,

obj <- par2norMix(p)

maps parameter vector p to a norMix object obj,

p <- nM2par(obj)

maps from a norMix object obj to parameter vector p,

where p is always a parameter vector in our parametrization.

Partly for didactical reasons, the following functions provide the basic ingredients for the EM algorithm (see also norMixEM) to parameter estimation:

estep.nm(x, obj, p)

computes 1 E-step for the data x, given either a "norMix" object obj or parameter vector p.

mstep.nm(x, z)

computes 1 M-step for the data x and the probability matrix z.

emstep.nm(x, obj)

computes 1 E- and 1 M-step for the data x and the "norMix" object obj.

where again, p is a parameter vector in our parametrization, x is the (univariate) data, and z is a n \times m matrix of (posterior) conditional probabilities, and \theta is the full parameter vector of the mixture model.

Usage

llnorMix(p, x, m = (length(p) + 1)/3, trafo = c("clr1", "logit"))

par2norMix(p, trafo = c("clr1", "logit"), name = )

nM2par(obj, trafo = c("clr1", "logit"))

 estep.nm(x, obj, par)
 mstep.nm(x, z)
emstep.nm(x, obj)

Arguments

Details

We use a parametrization of a (finite) m-component univariate normal mixture which is particularly apt for likelihood maximization, namely, one whose parameter space is almost a full \mathbf{I\hskip-0.22em R}^M, M = 3m-1.

For an m-component mixture, we map to and from a parameter vector \theta (== p as R-vector) of length 3m-1. For mixture density

\sum_{j=1}^m \pi_j \phi((t - \mu_j)/\sigma_j)/\sigma_j,

we transform the \pi_j (for j \in 1,\dots,m) via the transform specified by trafo (see below), and log-transform the \sigma_j. Consequently, \theta is partitioned into

p[ 1:(m-1)]:

For

trafo = "logit":

p[j]= \mbox{logit}(\pi_{j+1}) and \pi_1 is given implicitly as \pi_1 = 1-\sum_{j=2}^m \pi_j.

trafo = "clr1":

(centered log ratio, omitting 1st element): Set \ell_j := \ln(\pi_j) for j = 1, \dots, m, and p[j]= \ell_{j+1} - 1/m\sum_{j'=1}^m \ell_{j'} for j = 1, \dots, m-1.

p[ m:(2m-1)]:

p[m-1+ j]= \mu_j, for j=1:m.

p[2m:(3m-1)]:

p[2*m-1+ j] = \log(\sigma_j), i.e., \sigma_j^2 = exp(2*p[.+j]).

Value

llnorMix() returns a number, namely the log-likelihood.

par2norMix() returns "norMix" object, see norMix.

nM2par() returns the parameter vector \theta of length 3m-1.

estep.nm() returns z, the matrix of (conditional) probabilities.

mstep.nm() returns the model parameters as a list with components w, mu, and sigma, corresponding to the arguments of norMix(). (and see the 'Examples' on using do.call(norMix, *) with it.)

emstep.nm() returns an updated "norMix" object.

Author(s)

Martin Maechler

See Also

norMix, logLik. Note that the log likelihood of a "norMix" object is directly given by sum(dnorMix(x, obj, log=TRUE)).

To fit, using the EM algorithm, rather use norMixEM() than the e.step, m.step, or em.step functions.

Note that direct likelihood maximization, i.e., MLE, is typically considerably more efficient than the EM, and typically converges well with our parametrization, see norMixMLE.

Examples

(obj <- MW.nm10) # "the Claw" -- m = 6 components
length(pp <- nM2par(obj)) # 17 == (3*6) - 1
par2norMix(pp)
## really the same as the initial 'obj' above

## Log likelihood (of very artificial data):
llnorMix(pp, x = seq.int(-2, 2, length.out = 1000))
set.seed(47)## of more realistic data:
x <- rnorMix(1000, obj)
llnorMix(pp, x)

## Consistency check :  nM2par()  and  par2norMix()  are inverses
all.EQ <- function(x,y, tol = 1e-15, ...) all.equal(x,y, tolerance=tol, ...)
stopifnot(all.EQ(pp, nM2par(par2norMix(pp))),
          all.EQ(obj, par2norMix(nM2par(obj)),
                    check.attributes=FALSE),
          ## Direct computation of log-likelihood:
          all.EQ(sum(dnorMix(x, obj, log=TRUE)),
                    llnorMix(pp, x))  )

## E- and M- steps : ------------------------------
rE1 <- estep.nm(x, obj)
rE2 <- estep.nm(x, par=pp) # the same as rE1
z <- rE1
str( rM <-  mstep.nm(x, z))
   (rEM <- emstep.nm(x, obj))

stopifnot(all.EQ(rE1, rE2),
          all.EQ(rEM, do.call(norMix, c(rM, name=""))))

Mixtures of Univariate Normal Distributions

Description

Objects of class norMix represent finite mixtures of (univariate) normal (aka Gaussian) distributions. Methods for construction, printing, plotting, and basic computations are provided.

Usage

norMix(mu, sig2 = rep(1,m), sigma = rep(1,m),
       w = NULL, name = NULL, long.name = FALSE)

is.norMix(obj)
 m.norMix(obj)
var.norMix(x, ...)
## S3 method for class 'norMix'
mean(x, ...)
## S3 method for class 'norMix'
print(x, ...)
## S3 method for class 'norMix'
x[i,j, drop=TRUE]

Arguments

Details

The (one dimensional) normal mixtures, R objects of class "norMix", are constructed by norMix and tested for by is.norMix. m.norMix() returns the number of mixture components; the mean() method for class "norMix" returns the (theoretical / true) mean E[X] and var.norMix() the true variance E[(X- E[X])^2] where X \sim \langle\mathit{norm.mixt}\rangle.

The subsetting aka “extract” method (x[i,j]; for generic [)—when called as x[i,]—will typically return a "norMix" object unless matrix indexing selects only one row in which case x[i, , drop=FALSE] will return the normal mixture (of one component only).

For further methods (density, random number generation, fitting, ...), see below.

Value

norMix returns objects of class "norMix" which are currently implemented as 3-column matrix with column names mu, sigma, and w, and further attributes. The user should rarely need to access the underlying structure directly.

Note

For estimation of the parameters of such a normal mixture, we provide a smart parametrization and an efficient implementation of the direct MLE or also the EM algorithm, see norMixMLE() which includes norMixEM().

Author(s)

Martin Maechler

See Also

dnorMix for the density, pnorMix for the cumulative distribution and the quantile function (qnorMix), and rnorMix for random numbers and plot.norMix, the plot method.

MarronWand has the Marron-Wand densities as normal mixtures.

norMixMLE() and norMixEM() provide fitting of univariate normal mixtures to data.

Examples

ex <- norMix(mu = c(1,2,5))# defaults: sigma = 1, equal proportions ('w')
ex
plot(ex, p.comp = TRUE)# looks like a mixture of only 2; 'p.comp' plots components

## The 2nd Marron-Wand example, see also  ?MW.nm2
ex2 <- norMix(name = "#2 Skewed",
                mu = c(0, .5, 13/12),
	     sigma = c(1, 2/3, 5/9),
		 w = c(.2, .2, .6))

m.norMix  (ex2)
mean      (ex2)
var.norMix(ex2)
(e23 <- ex2[2:3,]) # (with re-normalized weights)
stopifnot(is.norMix(e23),
          all.equal(var.norMix(ex2),     719/1080, tol=1e-14),
          all.equal(var.norMix(ex ),      35/9,    tol=1e-14),
          all.equal(var.norMix(ex[2:3,]), 13/4,    tol=1e-14),
          all.equal(var.norMix(e23), 53^2/(12^3*4),tol=1e-14)
)

plot(ex2, log = "y")# maybe "revealing"

Transform "norMix" object into Call, Expression or Function

Description

E.g., for taking symbolic derivatives, it may be useful to get an R call, expression, or function in / of x from a specific "norMix" object.

Usage

norMix2call(obj, oneArg = TRUE)
## S3 method for class 'norMix'
as.expression(x, oneArg = TRUE, ...)
## S3 method for class 'norMix'
  as.function(x, oneArg = TRUE, envir = parent.frame(), ...)

Arguments

Value

according to the function used, an R ‘language’ object, i.e., a call, expression, or function, respectively.

Author(s)

Martin Maechler

See Also

norMix. Note that deriv() currently only works correctly in case of the default oneArg = TRUE.

Examples

(cMW2 <- norMix2call(MW.nm2))
deriv(cMW2, "x")

(fMW1 <- as.function  (MW.nm1))
(eMW3 <- as.expression(MW.nm3))
stopifnot(is.call      (cMW2),  is.call(norMix2call(MW.nm2, FALSE)),
          is.function  (fMW1),  is.function  (as.function  (MW.nm4)),
          is.expression(eMW3),  is.expression(as.expression(MW.nm5))
)

EM and MLE Estimation of Univariate Normal Mixtures

Description

These functions estimate the parameters of a univariate (finite) normal mixture using the EM algorithm or Likelihood Maximimization via optim(.., method = "BFGS").

Usage

norMixEM(x, m, name = NULL, sd.min = 1e-07* diff(range(x))/m,
         trafo = c("clr1", "logit"),
         maxiter = 100, tol = sqrt(.Machine$double.eps), trace = 1)

norMixMLE(x, m, name = NULL, 
          trafo = c("clr1", "logit"),
          maxiter = 100, tol = sqrt(.Machine$double.eps), trace = 2)

Arguments

Details

Estimation of univariate mixtures can be very sensitive to initialization. By default, norMixEM and norMixLME cut the data into m groups of approximately equal size. See examples below for other initialization possibilities.

The EM algorithm consists in repeated application of E- and M- steps until convergence. Mainly for didactical reasons, we also provide the functions estep.nm, mstep.nm, and emstep.nm.

The MLE, Maximum Likelihood Estimator, maximizes the likelihood using optim, using the same advantageous parametrization as llnorMix.

Value

An object of class norMix.

Author(s)

EM: Friedrich Leisch, originally; Martin Maechler vectorized it in m, added trace etc.

MLE: M.Maechler

Examples

## use (mu, sigma)
ex  <- norMix(mu = c(-1,2,5), sigma = c(1, 1/sqrt(2), sqrt(3)))
tools::assertWarning(verbose=TRUE,
           ## *deprecated* (using 'sig2' will *NOT* work in future!)
           ex. <- norMix(mu = c(-1,2,5), sig2 = c(1, 0.5, 3))
       )
stopifnot(all.equal(ex, ex.))
plot(ex, col="gray", p.norm=FALSE)

x <- rnorMix(100, ex)
lines(density(x))
rug(x)

## EM estimation may fail depending on random sample
ex1 <- norMixEM(x, 3, trace=2) #-> warning (sometimes)
ex1
plot(ex1)

## initialization by cut() into intervals of equal length:
ex2 <- norMixEM(x, cut(x, 3))
ex2

## initialization by kmeans():
k3 <- kmeans(x, 3)$cluster
ex3 <- norMixEM(x, k3)
ex3

## Now, MLE instead of EM:
exM <- norMixMLE(x, k3, tol = 1e-12, trace=4)
exM

## real data
data(faithful)
plot(density(faithful$waiting, bw = "SJ"), ylim=c(0,0.044))
rug(faithful$waiting)

(nmF <- norMixEM(faithful$waiting, 2))
lines(nmF, col=2)
## are three components better?
nmF3 <- norMixEM(faithful$waiting, 3, maxiter = 200)
lines(nmF3, col="forestgreen")

Plotting Methods for 'norMix' Objects

Description

The plot and lines methods for norMix objects draw the normal mixture density, optionally additonally with a fitted normal density.

Usage

## S3 method for class 'norMix'
plot(x, type = "l", n = 511, xout = NULL, xlim = NULL, ylim,
      xlab = "x", ylab = "f(x)", main = attr(x, "name"), lwd = 1.4,
      p.norm = !p.comp, p.h0 = TRUE, p.comp = FALSE,
      parNorm = list(col = 2, lty = 2, lwd = 0.4),
      parH0 = list(col = 3, lty = 3, lwd = 0.4),
      parComp = list(col= "blue3", lty = 3, lwd = 0.4), ...)

## S3 method for class 'norMix'
lines(x, type = "l", n = 511, xout = NULL,
      lwd = 1.4, p.norm = FALSE, parNorm = list(col = 2, lty = 2, lwd = 0.4),
      ...)

Arguments

Author(s)

Martin Maechler

See Also

norMix for the construction and further methods, particularly dnorMix which is used here.

Examples

plot(norMix(m=c(0,3), sigma = c(2,1))) # -> var = c(2^2, 1) = c(4, 1)

plot(MW.nm4, p.norm=FALSE, p.comp = TRUE)
plot(MW.nm4, p.norm=FALSE, p.comp = TRUE, ylim = c(0, 2))# now works
stopifnot(all.equal(c(0,2), par("yaxp")[1:2], tol= 1e-15))

## Further examples in  ?norMix and  ?rnorMix

Normal Mixture Cumulative Distribution and Quantiles

Description

Compute cumulative probabilities or quantiles (the inverse) for a normal mixture specified as norMix object.

Usage

pnorMix(q, obj, lower.tail = TRUE, log.p = FALSE)

qnorMix(p, obj, lower.tail = TRUE, log.p = FALSE,
        tol = .Machine$double.eps^0.25, maxiter = 1000, traceRootsearch = 0,
        method = c("interpQspline", "interpspline", "eachRoot", "root2"),
        l.interp = pmax(1, pmin(20, 1000 / m)), n.mu.interp = 100)

Arguments

.

Details

Whereas the distribution function pnorMix is the trivial sum of weighted normal probabilities (pnorm), its inverse, qnorMix is computed numerically: For each p we search for q such that pnorMix(obj, q) == p, i.e., f(q) = 0 for f(q) := pnorMix(obj, q) - p. This is a root finding problem which can be solved by uniroot(f, lower,upper,*). If length(p) <= 2 or method = "eachRoot", this happens one for one for the sorted p's. Otherwise, we start by doing this for the outermost non-trivial (0 < p < 1) values of p.

For method = "interQpspline" or "interpspline", we now compute p. <- pnorMix(q., obj) for values q. which are a grid of length l.interp in each interval [q_j,q_{j+1}], where q_j are the “X-extremes” plus (a sub sequence of length n.mu.interp of) the ordered mu[j]'s. Then, we use montone inverse interpolation (splinefun(q., p., method="monoH.FC")) plus a few (maximally maxiter, typically one!) Newton steps. The default, "interQpspline", additionally logit-transforms the p. values to make the interpolation more linear. This method is faster, particularly for large length(p).

Value

a numeric vector of the same length as p or q, respectively.

Author(s)

Very first version (for length-1 p,q) by Erik Jørgensen Erik.Jorgensen@agrsci.dk.

See Also

dnorMix for the density function.

Examples

MW.nm3 # the "strange skew" one
plot(MW.nm3)
## now the cumlative :
x <- seq(-4,4, length.out = 1001)
plot(x, pnorMix(x, MW.nm3), type="l", col=2)
## and some of its inverse :
pp <- seq(.1, .9, by=.1)
plot(qnorMix(pp, MW.nm3), pp)

## The "true" median of a normal mixture:
median.norMix <- function(x) qnorMix(1/2, x)
median.norMix(MW.nm3) ## -2.32

Ratio of Normal Mixture to Corresponding Normal

Description

Compute r(x) = f(x)/ f0(x) where f() is a normal mixture density and f0 the normal density with the same mean and variance as f.

Usage

r.norMix(obj, x = NULL, xlim = NULL, n = 511, xy.return = TRUE)

Arguments

Value

It depends on xy.return. If it's false, a numeric vector of the same length as x, if true (as per default), a list that can be plotted, with components

Note

The ratio function is used in certain semi-parametric density estimation methods (and theory).

Examples

  d3 <- norMix(m = 5*(0:2), w = c(0.6, 0.3, 0.1))
  plot(d3)
  rd3 <- r.norMix(d3)
  str(rd3)
  stopifnot(rd3 $ y  == r.norMix(d3, xy.ret = FALSE))
  par(new = TRUE)
  plot(rd3, type = "l", col = 3, axes = FALSE, xlab = "", ylab="")
  axis(4, col.axis=3)

Generate 'Normal Mixture' Distributed Random Numbers

Description

Generate n random numbers, distributed according to a normal mixture.

Usage

rnorMix(n, obj)

Arguments

Details

For a mixture of m, i.e., m.norMix(obj), components, generate the number in each component as multinomial, and then use rnorm for each.

Note that the these integer (multinomial) numbers are generated via sample(), which is by .Random.seed, notably from RNGkind(sample.kind = ..) which changed with R version 3.6.0.

Value

numeric vector of length n.

See Also

dnorMix for the density, and norMix for the construction and further methods.

Examples

x <- rnorMix(5000, MW.nm10)
hist(x)# you don't see the claw
plot(density(x), ylim = c(0,0.6),
     main = "Estim. and true 'MW.nm10' density")
lines(MW.nm10, col = "orange")

Sort Method for "norMix" Objects

Description

Sorting a "norMix" object (see norMix), sorts along the mu values; i.e., for the default decreasing = FALSE the resulting x[,"mu"] are sorted from left to right.

Usage

## S3 method for class 'norMix'
sort(x, decreasing = FALSE, ...)

Arguments

Value

a "norMix" object like x.

Examples

sort(MW.nm9)
stopifnot(identical(MW.nm2, sort(MW.nm2)))