Package 'nspmix'

Title: Nonparametric and Semiparametric Mixture Estimation
Description: Mainly for maximum likelihood estimation of nonparametric and semiparametric mixture models, but can also be used for fitting finite mixtures. The algorithms are developed in Wang (2007) <doi:10.1111/j.1467-9868.2007.00583.x> and Wang (2010) <doi:10.1007/s11222-009-9117-z>.
Authors: Yong Wang
Maintainer: Yong Wang <[email protected]>
License: GPL (>= 2)
Version: 1.5-0
Built: 2025-03-14 05:29:12 UTC
Source: https://github.com/cran/nspmix

Help Index

Beta-blockers Data

Description

Contains the data of the 22-center clinical trial of beta-blockers for reducing mortality after myocardial infarction.

Format

A numeric matrix with four columns:

center: center identification code.

deaths: the number of deaths in the center.

total: the number of patients taking beta-blockers in the center.

treatment: 0 for control, and 1 for treatment.

Source

Aitkin, M. (1999). A general maximum likelihood analysis of variance components in generalized linear models. Biometrics, 55, 117-128.

References

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.

See Also

mlogit,cnmms.

Examples

data(betablockers)
x = mlogit(betablockers)
cnmms(x)

Z-values of BRCA Data

Description

Contains 3226 zz-values computed by Efron (2004) from the data obtained in a well-known microarray experiment concerning two types of genetic mutations causing increased breast cancer risk, BRCA1 and BRCA2.

Format

A numeric vector containing 3226 zz-values.

References

Efron, B. (2004). Large-scale simultaneous hypothesis testing: the choice of a null hypothesis. Journal of the American Statistical Association, 99, 96-104.

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. and C.-S. Chee (2012). Density estimation using nonparametric and semiparametric mixtures. Statistical Modelling: An International Journal, 12, 67-92.

See Also

npnorm,cnm.

Examples

data(brca)
x = npnorm(brca)
plot(cnm(x), x)

Maximum Likelihood Estimation of a Nonparametric Mixture Model

Description

Function cnm can be used to compute the maximum likelihood estimate of a nonparametric mixing distribution (NPMLE) that has a one-dimensional mixing parameter. or simply the mixing proportions with support points held fixed.

Usage

cnm(
  x,
  init = NULL,
  model = c("npmle", "proportions"),
  maxit = 100,
  tol = 1e-06,
  grid = 100,
  plot = c("null", "gradient", "probability"),
  verbose = 0
)

Arguments

x

a data object of some class that is fully defined by the user. The user needs to supply certain functions as described below.
init

list of user-provided initial values for the mixing distribution mix and the structural parameter beta.
model

the type of model that is to estimated: the non-parametric MLE (if npmle), or mixing proportions only (if proportions).
maxit

maximum number of iterations.
tol

a tolerance value needed to terminate an algorithm. Specifically, the algorithm is terminated, if the increase of the log-likelihood value after an iteration is less than tol.
grid

number of grid points that are used by the algorithm to locate all the local maxima of the gradient function. A larger number increases the chance of locating all local maxima, at the expense of an increased computational cost. The locations of the grid points are determined by the function gridpoints provided by each individual mixture family, and they do not have to be equally spaced. If needed, a gridpoints function may choose to return a different number of grid points than specified by grid.
plot

whether a plot is produced at each iteration. Useful for monitoring the convergence of the algorithm. If ="null", no plot is produced. If ="gradient", it plots the gradient curves and if ="probability", the plot function defined by the user for the class is used.
verbose

verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5).

Details

A finite mixture model has a density of the form

f(x;π,θ,β)=∑j=1kπjf(x;θj,β).f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j, \beta).

where πj≥0\pi_j \ge 0 and ∑j=1kπj\sum_{j=1}^k \pi_j=1=1.

A nonparametric mixture model has a density of the form

f(x;G)=∫f(x;θ)dG(θ),f(x; G) = \int f(x; \theta) d G(\theta),

where GG is a mixing distribution that is completely unspecified. The maximum likelihood estimate of the nonparametric GG, or the NPMLE of GG, is known to be a discrete distribution function.

Function cnm implements the CNM algorithm that is proposed in Wang (2007) and the hierarchical CNM algorithm of Wang and Taylor (2013). The implementation is generic using S3 object-oriented programming, in the sense that it works for an arbitrary family of mixture models defined by the user. The user, however, needs to supply the implementations of the following functions for their self-defined family of mixture models, as they are needed internally by function cnm:

initial(x, beta, mix, kmax)

valid(x, beta)

logd(x, beta, pt, which)

gridpoints(x, beta, grid)

suppspace(x, beta)

length(x)

print(x, ...)

weight(x, ...)

While not needed by the algorithm for finding the solution, one may also implement

plot(x, mix, beta, ...)

so that the fitted model can be shown graphically in a user-defined way. Inside cnm, it is used when plot="probability" so that the convergence of the algorithm can be graphically monitored.

For creating a new class, the user may consult the implementations of these functions for the families of mixture models included in the package, e.g., npnorm and nppois.

Value

family

the name of the mixture family that is used to fit to the data.
num.iterations

number of iterations required by the algorithm
max.gradient

maximum value of the gradient function, evaluated at the beginning of the final iteration
convergence

convergence code. =0 means a success, and =1 reaching the maximum number of iterations
ll

log-likelihood value at convergence
mix

MLE of the mixing distribution, being an object of the class disc for discrete distributions.
beta

value of the structural parameter, that is held fixed throughout the computation.

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86

Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.

See Also

nnls, npnorm, nppois, cnmms.

Examples

## Simulated data
x = rnppois(1000, disc(c(1,4), c(0.7,0.3))) # Poisson mixture
(r = cnm(x))
plot(r, x)

x = rnpnorm(1000, disc(c(0,4), c(0.3,0.7)), sd=1) # Normal mixture
plot(cnm(x), x)                        # sd = 1
plot(cnm(x, init=list(beta=0.5)), x)   # sd = 0.5
mix0 = disc(seq(min(x$v),max(x$v), len=100)) # over a finite grid
plot(cnm(x, init=list(beta=0.5, mix=mix0), model="p"),
    x, add=TRUE, col="blue")          # An approximate NPMLE

## Real-world data
data(thai)
plot(cnm(x <- nppois(thai)), x)     # Poisson mixture

data(brca)
plot(cnm(x <- npnorm(brca)), x)     # Normal mixture

Maximum Likelihood Estimation of a Semiparametric Mixture Model

Description

Functions cnmms, cnmpl and cnmap can be used to compute the maximum likelihood estimate of a semiparametric mixture model that has a one-dimensional mixing parameter. The types of mixture models that can be computed include finite, nonparametric and semiparametric ones.

Usage

cnmms(x, init=NULL, maxit=1000, model=c("spmle","npmle"), tol=1e-6,
     grid=100, kmax=Inf, plot=c("null", "gradient", "probability"),
     verbose=0)
cnmpl(x, init=NULL, tol=1e-6, tol.npmle=tol*1e-4, grid=100, maxit=1000,
     plot=c("null", "gradient", "probability"), verbose=0)
cnmap(x, init=NULL, maxit=1000, tol=1e-6, grid=100, plot=c("null",
     "gradient"), verbose=0)

Arguments

x

a data object of some class that can be defined fully by the user
init

list of user-provided initial values for the mixing distribution mix and the structural parameter beta
maxit

maximum number of iterations
model

the type of model that is to estimated: non-parametric MLE (npmle) or semi-parametric MLE (spmle).
tol

a tolerance value that is used to terminate an algorithm. Specifically, the algorithm is terminated, if the relative increase of the log-likelihood value after an iteration is less than tol. If an algorithm converges rapidly enough, then -log10(tol) is roughly the number of accurate digits in log-likelihood.
grid

number of grid points that are used by the algorithm to locate all the local maxima of the gradient function. A larger number increases the chance of locating all local maxima, at the expense of an increased computational cost. The locations of the grid points are determined by the function gridpoints provided by each individual mixture family, and they do not have to be equally spaced. If needed, an individual gridpoints function may return a different number of grid points than specified by grid.
kmax

upper bound on the number of support points. This is particularly useful for fitting a finite mixture model.
plot

whether a plot is produced at each iteration. Useful for monitoring the convergence of the algorithm. If null, no plot is produced. If gradient, it plots the gradient curves and if probability, the plot function defined by the user of the class is used.
verbose

verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5).
tol.npmle

a tolerance value that is used to terminate the computing of the NPMLE internally.

Details

Function cnmms can also be used to compute the maximum likelihood estimate of a finite or nonparametric mixture model.

A finite mixture model has a density of the form

f(x;π,θ,β)=∑j=1kπjf(x;θj,β).f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j, \beta).

where pij≥0pi_j \ge 0 and ∑j=1kpij\sum_{j=1}^k pi_j=1=1.

A nonparametric mixture model has a density of the form

f(x;G)=∫f(x;θ)dG(θ),f(x; G) = \int f(x; \theta) d G(\theta),

where GG is a mixing distribution that is completely unspecified. The maximum likelihood estimate of the nonparametric GG, or the NPMLE of $GG, is known to be a discrete distribution function.

A semiparametric mixture model has a density of the form

f(x;G,β)=∫f(x;θ,β)dG(θ),f(x; G, \beta) = \int f(x; \theta, \beta) d G(\theta),

where GG is a mixing distribution that is completely unspecified and β\beta is the structural parameter.

Of the three functions, cnmms is recommended for most problems; see Wang (2010).

Functions cnmms, cnmpl and cnmap implement the algorithms CNM-MS, CNM-PL and CNM-AP that are described in Wang (2010). Their implementations are generic using S3 object-oriented programming, in the sense that they can work for an arbitrary family of mixture models that is defined by the user. The user, however, needs to supply the implementations of the following functions for their self-defined family of mixture models, as they are needed internally by the functions above:

initial(x, beta, mix, kmax)

valid(x, beta)

logd(x, beta, pt, which)

gridpoints(x, beta, grid)

suppspace(x, beta)

length(x)

print(x, ...)

weight(x, ...)

While not needed by the algorithms, one may also implement

plot(x, mix, beta, ...)

so that the fitted model can be shown graphically in a way that the user desires.

For creating a new class, the user may consult the implementations of these functions for the families of mixture models included in the package, e.g., cvp and mlogit.

Value

family

the class of the mixture family that is used to fit to the data.
num.iterations

Number of iterations required by the algorithm
grad

For cnmms, it contains the values of the gradient function at the support points and the first derivatives of the log-likelihood with respect to theta and beta. For cnmpl, it contains only the first derivatives of the log-likelihood with respect to beta. For cnmap, it contains only the gradient of beta.
max.gradient

Maximum value of the gradient function, evaluated at the beginning of the final iteration. It is only given by function cnmap.
convergence

convergence code. =0 means a success, and =1 reaching the maximum number of iterations
ll

log-likelihood value at convergence
mix

MLE of the mixing distribution, being an object of the class disc for discrete distributions
beta

MLE of the structural parameter

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86

See Also

nnls, cnm, cvp, cvps, mlogit.

Examples

## Compute the MLE of a finite mixture
x = rnpnorm(100, disc(c(0,4), c(0.7,0.3)), sd=1)
for(k in 1:6) plot(cnmms(x, kmax=k), x, add=(k>1), comp="null", col=k+1,
                  main="Finite Normal Mixtures")
legend("topright", 0.3, leg=paste0("k = ",1:6), lty=1, lwd=2, col=2:7)

## Compute a semiparametric MLE
# Common variance problem 
x = rcvps(k=50, ni=5:10, mu=c(0,4), pr=c(0.7,0.3), sd=3)
cnmms(x)              # CNM-MS algorithm
cnmpl(x)              # CNM-PL algorithm
cnmap(x)              # CNM-AP algorithm

# Logistic regression with a random intercept
x = rmlogit(k=30, gi=3:5, ni=6:10, pt=c(0,4), pr=c(0.7,0.3),
           beta=c(0,3))
cnmms(x)

data(toxo)            # k = 136
cnmms(mlogit(toxo))

Class ‘cvps’

Description

These functions can be used to study a common variance problem (CVP), where univariate observations fall in known groups. Observations in each group are assumed to have the same mean, but different groups may have different means. All observations are assumed to have a common variance, despite their different means, hence giving the name of the problem. It is a random-effects problem.

Usage

cvps(x)
rcvp(k, ni=2, mu=0, pr=1, sd=1)
rcvps(k, ni=2, mu=0, pr=1, sd=1)
## S3 method for class 'cvps'
print(x, ...)

Arguments

x

CVP data in the raw form as an argument in cvps, or an object of class cvps in print.cvps.
k

the number of groups.
ni

a numeric vector that gives the sample size in each group.
mu

a numeric vector for all the theoretical means.
pr

a numeric vector for all the probabilities associated with the theoretical means.
sd

a scalar for the standard deviation that is common to all observations.
...

arguments passed on to function print.

Details

Class cvps is used to store the CVP data in a summarized form.

Function cvps creates an object of class cvps, given a matrix that stores the values (column 2) and their grouping information (column 1).

Function rcvp generates a random sample in the raw form for a common variance problem, where the means follow a discrete distribution.

Function rcvps generates a random sample in the summarized form for a common variance problem, where the means follow a discrete distribution.

Function print.cvps prints the CVP data given in the summarized form.

The raw form of the CVP data is a two-column matrix, where each row represents an observation. The two columns along each row give, respectively, the group membership (group) and the value (x) of an observation.

The summarized form of the CVP data is a four-column matrix, where each row represents the summarized data for all observations in a group. The four columns along each row give, respectively, the group number (group), the number of observations in the group (ni), the sample mean of the observations in the group (mi), and the residual sum of squares of the observations in the group (ri).

Author(s)

Yong Wang <[email protected]>

References

Neyman, J. and Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1-32.

Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat., 27, 886-906.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.

See Also

nnls, cnmms.

Examples

x = rcvps(k=50, ni=5:10, mu=c(0,4), pr=c(0.7,0.3), sd=3)
cnmms(x)              # CNM-MS algorithm
cnmpl(x)              # CNM-PL algorithm
cnmap(x)              # CNM-AP algorithm

Class ‘disc’

Description

Class disc is used to represent an arbitrary univariate discrete distribution with a finite number of support points.

Usage

disc(pt, pr=1)
## S3 method for class 'disc'
print(x, ...)

Arguments

pt

a numeric vector for support points.
pr

a numeric vector for probability values at the support points.
x

an object of class disc.
...

arguments passed on to function print.

Details

Function disc creates an object of class disc, given the support points and probability values at these points.

Function print.disc prints the discrete distribution.

Author(s)

Yong Wang <[email protected]>

See Also

cnm, cnmms.

Examples

(d = disc(pt=c(0,4), pr=c(0.3,0.7)))

Lung Cancer Data

Description

Contains the data of 14 studies of the effect of smoking on lung cancer.

Format

A numeric matrix with four columns:

study: study identification code.

lungcancer: the number of people diagnosed with lung cancer.

size: the number of people in the study.

smoker: 0 for smoker, and 1 for non-smoker.

Source

Booth, J. G. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society, Ser. B, 61, 265-285.

References

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.

See Also

mlogit,cnmms.

Examples

data(lungcancer)
x = mlogit(lungcancer)
cnmms(x)

Class ‘mlogit’

Description

These functions can be used to fit a binomial logistic regression model that has a random intercept to clustered observations. Observations in each cluster are assumed to have the same intercept, while different clusters may have different intercepts. This is a mixed-effects problem.

Usage

mlogit(x)
rmlogit(k, gi=2, ni=2, pt=0, pr=1, beta=1, X)

Arguments

x

a numeric matrix with four or more columns that stores clustered data.
k

the number of groups or clusters.
gi

a numeric vector that gives the sample size in each group.
ni

a numeric vector for the number of Bernoulli trials for each observation.
pt

a numeric vector for all the support points.
pr

a numeric vector for all the probabilities associated with the support points.
beta

a numeric vector for the fixed coefficients of the covariates of the observation.
X

the numeric matrix as the design matrix. If missing, a random matrix is created from a normal distribution.

Details

Class mlogit is used to store data for fitting the binomial logistic regression model with a random intercept.

Function mlogit creates an object of class mlogit, given a matrix with four or more columns that stores, respectively, the group/cluster membership (column 1), the number of ones or successes in the Bernoulli trials (column 2), the number of the Bernoulli trials (column 3), and the covariates (columns 4+).

Function rmlogit generates a random sample that is saved as an object of class mlogit.

An object of class mlogit contains a matrix with four or more columns, that stores, respectively, the group/cluster membership (column 1), the number of ones or successes in the Bernoulli trials (column 2), the number of the Bernoulli trials (column 3), and the covariates (columns 4+).

It also has two additional attributes that facilitate the computing by function cmmms. The first attribute is ui, which stores the unique values of group memberships, and the second is gi, the number of observations in each unique group.

It is convenient to use function mlogit to create an object of class mlogit.

Author(s)

Yong Wang <[email protected]>

References

Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat., 27, 886-906.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.

See Also

nnls, cnmms.

Examples

x = rmlogit(k=30, gi=3:5, ni=6:10, pt=c(0,4), pr=c(0.7,0.3),
           beta=c(0,3))    
cnmms(x)

### Real-world data
# Random intercept logistic model
data(toxo)
cnmms(mlogit(toxo))

data(betablockers)
cnmms(mlogit(betablockers))

data(lungcancer)
cnmms(mlogit(lungcancer))

Class ‘npnorm’

Description

Class npnorm can be used to store data that will be processed as those of a nonparametric normal mixture. There are several functions associated with the class.

Usage

npnorm(v, w=1)
rnpnorm(n, mix=disc(0), sd=1)
## S3 method for class 'npnorm'
plot(x, mix, beta, breaks=NULL, col="red", len=100,
    add=FALSE, border.col=NULL, border.lwd=1,
    fill="lightgrey", main, lwd=2, lty=1, xlab="Data",
    ylab="Density", components=c("proportions","curves","null"),
    lty.components=2, lwd.components=2, ...)

Arguments

v

a numeric vector that stores the values of a sample.
w

a numeric vector that stores the corresponding weights/frequencies of the observations.
n

the sample size.
mix

an object of class disc, for a discrete distribution.
beta

the structural parameter.
sd

a scalar for the component standard deviation that is common to all components.
x

an object of class npnorm.
breaks

the rough number bins used for plotting the histogram.
col

the color of the density curve to be plotted.
len

the number of points roughly used to plot the density curve over the interval of length 8 times the component standard deviation around each component mean.
add

if FALSE, creates a new plot; if TRUE, adds the plot to the existing one.
border.col

color for the border of histogram boxes.
border.lwd

line width for the border of histogram boxes.
fill

color to fill in the histogram boxes.
components

if proportions (default), also show the support points and mixing proportions; if curves, also show the component density curves; if null, components are not shown.
lty.components, lwd.components

line type and width for the component curves.
main, lwd, lty, xlab, ylab

arguments for graphical parameters (see par).
...

arguments passed on to function plot.

Details

Function npnorm creates an object of class npnorm, given values and weights/frequencies.

Function rnpnorm generates a random sample from a normal mixture and saves the data as an object of class npnorm.

Function plot.npnorm plots the normal mixture.

When components="proportions", the component means are shown on the horizontal line of density 0. The vertical lines going upwardly at the support points are proportional to the mixing proportions at these points.

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

See Also

nnls, cnm, cnmms, plot.nspmix.

Examples

mix = disc(pt=c(0,4), pr=c(0.3,0.7))  # a discrete distribution
x = rnpnorm(200, mix, sd=1)
plot(x, mix, beta=1)

Class ‘nppois’

Description

Class nppois is used to store data that will be processed as those of a nonparametric Poisson mixture.

Usage

nppois(v, w=1)
rnppois(n, mix=disc(1))
## S3 method for class 'nppois'
plot(x, mix, beta, col="red", add=FALSE,
    components=TRUE, main="nppois", lwd=1, lty=1, xlab="Data",
    ylab="Density", ...)

Arguments

v

a numeric vector that stores the values of a sample.
w

a numeric vector that stores the corresponding weights/frequencies of the observations.
n

the sample size.
x

an object of class nppois.
mix

an object of class disc.
beta

the structural parameter, which is not really needed for the Poisson mixture.
col

the color of the density curve to be plotted.
add

if FALSE, creates a new plot; if TRUE, adds the plot to the existing one.
components

if TRUE, also show the support points and mixing proportions.
main, lwd, lty, xlab, ylab

arguments for graphical parameters (see par).
...

arguments passed on to function plot.

Details

Function nppois creates an object of class nppois, given values and weights/frequencies.

Function rnppois generates a random sample from a Poisson mixture and saves the data as an object of class nppois.

Function plot.nppois plots the Poisson mixture.

When components=TRUE, the support points are shown on the horizontal line of density 0. The component density curves, weighted appropriately, are also shown.

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

See Also

nnls, cnm, cnmms, plot.nspmix.

Examples

mix = disc(pt=c(1,4), pr=c(0.3,0.7))
x = rnppois(200, mix)
plot(x, mix)

Class 'nspmix'

Description

Class nspmix is an object returned by function cnm, cnmms, cnmpl or cnmap.

Usage

## S3 method for class 'nspmix'
plot(x, data, type=c("probability","gradient"), ...)

Arguments

x

an object of a mixture model class
data

a data set from the mixture model
type

the type of function to be plotted: the probability model of the mixture family (probability), or the gradient function (gradient).
...

arguments passed on to the plot function called.

Details

Function plot.nspmix plots either the mixture model, if the family of the mixture provides an implementation of the generic plot function, or the gradient function.

data must belong to a mixture family, as specified by its class.

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86

See Also

nnls, cnm, cnmms, cnmpl, cnmap, npnorm, nppois.

Examples

## Poisson mixture
x = rnppois(200, disc(c(1,4), c(0.7,0.3)))
r = cnm(x)
plot(r, x, "p")
plot(r, x, "g")

## Normal mixture
x = rnpnorm(200, mix=disc(c(0,4), c(0.3,0.7)), sd=1)
r = cnm(x, init=list(beta=0.5))   # sd = 0.5
plot(r, x, "p")
plot(r, x, "g")

Plot the Gradient Function

Description

Function plotgrad plots the gradient function or its first derivative of a nonparametric mixture.

Usage

plotgrad(
  x,
  mix,
  beta,
  len = 500,
  order = 0,
  col = "blue",
  col2 = "red",
  add = FALSE,
  main = paste0("Class: ", class(x)),
  xlab = expression(theta),
  ylab = paste0("Gradient (order = ", order, ")"),
  cex = 1,
  pch = 1,
  lwd = 1,
  xlim,
  ylim,
  ...
)

Arguments

x

a data object of a mixture model class.
mix

an object of class 'disc', for a discrete mixing distribution.
beta

the structural parameter.
len

number of points used to plot the smooth curve.
order

the order of the derivative of the gradient function to be plotted. If 0, it is the gradient function itself.
col

color for the curve.
col2

color for the support points.
add

if FALSE, create a new plot; if TRUE, add the curve and points to the current one.
main, xlab, ylab, cex, pch, lwd, xlim, ylim

arguments for graphical parameters (see par).
...

arguments passed on to function plot.

Details

data must belong to a mixture family, as specified by its class.

The support points are shown on the horizontal line of gradient 0. The vertical lines going downwards at the support points are proportional to the mixing proportions at these points.

Author(s)

Yong Wang <[email protected]>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86

See Also

plot.nspmix, nnls, cnm, cnmms, npnorm, nppois.

Examples

## Poisson mixture
x = rnppois(200, disc(c(1,4), c(0.7,0.3)))
r = cnm(x)
plotgrad(x, r$mix)

## Normal mixture
x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1)
r = cnm(x, init=list(beta=0.5))   # sd = 0.5
plotgrad(x, r$mix, r$beta)

Illness Spells and Frequencies of Thai Preschool Children

Description

Contains the results of a cohort study in north-east Thailand in which 602 preschool children participated. For each child, the number of illness spells xx, such as fever, cough or running nose, is recorded for all 2-week periods from June 1982 to September 1985. The frequency for each value of xx is saved in the data set.

Format

A data frame with 24 rows and 2 variables:

x: values of xx.

freq: frequencies for each value of xx.

Source

Bohning, D. (2000). Computer-assisted Analysis of Mixtures and Applications: Meta-analysis, Disease Mapping, and Others. Boca Raton: Chapman and Hall-CRC.

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

See Also

nppois,cnm.

Examples

data(thai)
x = nppois(thai)
plot(cnm(x), x)

Toxoplasmosis Data

Description

Contains the number of subjects testing positively for toxoplasmosis in 34 cities of El Salvador, with various rainfalls.

Format

A numeric matrix with four columns:

city: city identification code.

y: the number of subjects testing positively for toxoplasmosis.

n: the number of subjects tested.

rainfall: the annual rainfall of the city, in meters.

References

Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81, 709-721.

Aitkin, M. (1996). A general maximum likelihood analysis of overdispersion in generalised linear models. Statistics and Computing, 6, 251-262.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86.

See Also

mlogit,cnmms.

Examples

data(toxo)
x = mlogit(toxo)
cnmms(x)

Weighted Histograms

Description

Plots or computes the histogram with observations with multiplicities/weights.

Usage

whist(
  x,
  w = 1,
  breaks = "Sturges",
  plot = TRUE,
  freq = NULL,
  xlim = NULL,
  ylim = NULL,
  xlab = "Data",
  ylab = NULL,
  main = NULL,
  add = FALSE,
  col = NULL,
  border = NULL,
  lwd = 1,
  ...
)

Arguments

x

a vector of values for which the histogram is desired.
w

a vector of multiplicities/weights for the values in x.
breaks, plot, freq, xlim, ylim, xlab, ylab, main, add, col, border, lwd

These arguments have similar functionalities to their namesakes in function hist.
...

arguments passed on to function plot.

Details

Just like hist, whist can either plot the histogram or compute the values that define the histogram, by setting plot to TRUE or FALSE.

The histogram can either be the one for frequencies or density, by setting freq to TRUE or FALSE.

Author(s)

Yong Wang <[email protected]>

See Also

hist.