Version: 0.5-5
Date: 2018-06-04
Title: Finite Mixture Distribution Models
Packaged: 2018-06-04 18:18:27 UTC; pdmmac1
Author: Peter Macdonald <pdmmac@mcmaster.ca>, with contributions from Juan Du <duduyy@hotmail.com>
Maintainer: Peter Macdonald <pdmmac@mcmaster.ca>
Depends: R (≥ 1.4.0)
Imports: graphics, stats
Description: Fit finite mixture distribution models to grouped data and conditional data by maximum likelihood using a combination of a Newton-type algorithm and the EM algorithm.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL: https://www.r-project.org/, https://ms.mcmaster.ca/peter/mix/mix.html
NeedsCompilation: no
Repository: CRAN
Date/Publication: 2018-06-04 18:30:51 UTC

ANOVA Tables for Mixture Model Objects

Description

Compute analysis of variance tables for one or two mixture model objects.

Usage

## S3 method for class 'mix'
anova(object, mixobj2, ...)

Arguments

object

an object of class "mix", usually, a result of a call to the mixture model fitting function mix.

mixobj2

an object of the same type to be compared with object, which contains the results of fitting another model with more or fewer parameters fitted.

...

additional objects of the same type.

Value

An object of class "anova" inheriting from class "data.frame". When given a single argument this function produces a table which tests whether the model is significant. The table contains the residual degrees of freedom, Chi-square statistic and P value. If the class of the argument is not "mix", this function returns NULL. When given two objects, it tests the models against one another and lists them in the order of number of parameters fitted. For the model with fewer parameters fitted, the change in degrees of freedom is given. This only make statistical sense if the models are nested. If one of arguments does not belong to the class "mix", the function will give the anova table for the other argument; if both of them do not, it returns NULL.

Warning

The comparison between two models will only be valid if they are fitted to the same dataset. And the two models should be nested.

See Also

The model fitting function mix, the generic function anova.

Examples

data(pike65) # load the grouped data `pike65'
data(pikepar) # load the initial values of parameters for the data `pike65'
fitpike3 <- mix(pike65, pikepar, "lnorm", mixconstr(conmu = "MFX", 
                fixmu = c(FALSE, FALSE, FALSE, FALSE, TRUE), consigma = "CCV"), emstep = 3)
anova(fitpike3)
fitpike4 <- mix(pike65, pikepar, "lnorm", mixconstr(consigma = "CCV"), emsteps = 3)
anova(fitpike4)
anova(fitpike3, fitpike4)
anova(fitpike4, fitpike3)

Grouped Binomial Data

Description

We randomly generate four groups of binomial distribution data with means 4, 8, 12, 16, and corresponding variances 3.2, 4.8, 4.8 and 3.2. Then we mix the four data groups with 100 observations for each group, i.e., with equal proportions. After grouping the mixture data, we obtain the grouped data bindat.

The bindat data frame has 21 rows and 2 columns.

Usage

data(bindat)

Format

This data frame contains the following columns:

x

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Examples

data(bindat)
data(binpar)
plot.mixdata(bindat)
fit <- mix(bindat, binpar, "binom", mixconstr(conpi = "PFX",
           fixpi = c(TRUE, TRUE, TRUE, TRUE), consigma = "BINOM", size = c(20, 20, 20, 20)))
fit
plot(fit)

Starting Values of Parameters for the Binomial Data Set

Description

Starting values of parameters for fitting a mixture distribution to the data set bindat.

The binpar data frame has 4 rows and 3 columns.

Usage

data(binpar)

Format

This data frame contains the following columns:

pi

the starting values for proportions.

mu

the starting values for means.

sigma

the starting values for standard deviations.

Examples

data(binpar)

Cassie's Length-Frequency Example

Description

Data for Cassie's (1954) analysis of size frequency distributions.

The cassie data frame has 40 rows and 2 columns.

Usage

data(cassie)

Format

This data frame contains the following columns:

length

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Source

Cassie, R.M. (1954). Some uses of probability paper in the analysis of size frequency distributions. Aust. J. Mar. Freshwater Res. 5 , 513-522.

The data, lengths (in) of 256 snapper (Chrysophrys auratus Forster) taken by a trawl with a mesh of about 1.5 in, are given in Table 5 of that paper. Cassie's results are given in his Table 1.

References

http://www.math.mcmaster.ca/peter/mix/demex/excass.html

Examples

data(cassie)
plot.mixdata(cassie)

Extract Mixture Model Coefficients

Description

coef.mix is a function which extracts mixture model coefficients from objects returned by the model fitting function mix. It is called via the generic function coef.

Usage

## S3 method for class 'mix'
coef(object, natpar = FALSE, ...)

Arguments

object

an object of class "mix", usually, the results returned by the model fitting function mix.

natpar

a logical scalar specifying whether the natural parameters should be given.

...

other arguments.

Value

A data frame containing three variables, which are, in order, the proportions, means, and standard deviations, respectively. If natpar is TRUE, then the natural parameters of component distributions are also displayed.

See Also

mix for model fitting.

Examples

data(pike65) # load the grouped data `pike65'
data(pikepar) # load the initial values of parameters for the data `pike65'
fit <- mix(pike65, pikepar, "lnorm", mixconstr(consigma = "CCV"), emsteps = 3)
coef(fit)
coef(fit, natpar = TRUE)

Add Conditional Data to Grouped Data

Description

It combines automatically grouped data with conditional data when enter the conditional samples.

Usage

conditdat(mixdat, k, conditsamples)

Arguments

mixdat

a data frame containing grouped data, whose first column should be the right boundaries of grouping intervals, and the second one should be the numbers of observations falling into each interval.

k

the number of components.

conditsamples

a vector containing conditional data, which consists of the conditional samples, the first element of each sample is a number indicating which interval this sample comes from.

Value

A data frame containing the grouped data with conditional data.

See Also

mixgroup for constructing grouped and conditional data.

Examples

data(pike65) # load the data set `pike65'
pike65 # display the data set `pike65'
conditdat(pike65, k = 5, conditsamples =
          c(c(4, 9, 2, 0, 0, 0), c(5, 8, 6, 0, 0,0),
          c(12, 0, 2, 34, 0, 0), c(13, 0, 0, 21, 0, 0),
          c(15, 0, 0, 5, 5, 0), c(16, 0, 0, 6, 5, 1),
          c(17, 0, 0, 5, 7, 0), c(18, 0, 0, 4, 4, 3),
          c(19, 0, 0, 0, 8, 0), c(20, 0, 0, 0, 2, 1),
          c(21, 0, 0, 0, 1, 5), c(22, 0, 0, 0, 2, 4)))
# add conditional data to the grouped data `pike65'

A Mixture Data of Three Exponential Distributions

Description

A total of 1000 observations was generated by computer to follow the mixture distribution 1/3 E(1) + 1/3 E(4) + 1/3 E(16) where E(m) denotes an exponential distribution with mean m.

The expdat data frame has 25 rows and 2 columns.

Usage

data(expdat)

Format

This data frame contains the following columns:

x

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Source

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

http://www.math.mcmaster.ca/peter/mix/demex/exexp.html

Examples

data(expdat)
plot.mixdata(expdat)

A Mixed Data with Fifteen Normal Components

Description

Fifteen normal components grouped over eighty intervals.

The fiftn80 data frame has 80 rows and 2 columns.

Usage

data(fiftn80)

Format

This data frame contains the following columns:

x

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Details

A total of 820 observations were generated by computer to follow the distribution 1/15 N(5, 1) + 1/15 N(10, 1) + ... + 1/15 N(75, 1) where N(m, s) denotes a normal distribution with mean m and standard deviation s.

Source

http://www.math.mcmaster.ca/peter/mix/demex/ex1580.html

Examples

data(fiftn80)
plot.mixdata(fiftn80)

Compute Mixture Model Fitted Values

Description

fitted.mix is a function which computes fitted values from objects returned by the modeling function mix. It is called via the generic function fitted.

Usage

## S3 method for class 'mix'
fitted(object, digits = NULL, ...)

Arguments

object

an object of class "mix", usually, the results returned by the model fitting function mix.

digits

a specified number of decimal places to be reserved.

...

other arguments.

Value

List with the following components:

mixed

the estimated mixed data, that is, the fitted numbers of observations falling into each interval.

joint

the estimated joint data, that is, the fitted numbers of observations from each component falling into every interval.

conditional

the estimated conditional data to be returned if usecondit of object is TRUE, which are the fitted numbers of observations from given intervals belonging to each component.

conditprob

the estimated conditional probabilities of observations from given interval belonging to each component.

See Also

mix for fitting mixture distributions.

Examples

data(pike65)
data(pikepar)
fit1 <- mix(pike65, pikepar, "lnorm", mixconstr(consigma = "CCV"), emsteps = 3)
fitted(fit1)
data(pike65sg)
fit2 <- mix(pike65sg, pikepar, "gamma", mixconstr(consigma = "CCV"), usecondit = TRUE)
fitted(fit2, digits = 2)

Estimate Parameters of One-Component Mixture Distribution

Description

groupstats is a function which estimates the proportion, mean and standard deviation for a mixture distribution with one component.

Usage

groupstats(mixdat)

Arguments

mixdat

A data frame containing grouped data, whose first column should be right boundaries of grouping intervals where the first and last intervals are open-ended; whose second column should consist of the frequencies indicating numbers of observations falling into each interval.

Value

A list containing the following components:

pi

the value is 1 because of only one component.

mu

the estimated mean of mixdat.

sigma

the estimated standard deviation of mixdat.

See Also

mixgroup for grouping data, mixparam for constructing starting values of parameters.

Examples

data(pike65)
groupstats(pike65)

Compute Probabilities of an Observation Falling into a Grouping Interval

Description

Compute probabilities of an observation falling into a grouping interval when given component distribution which the observation comes from.

Usage

grpintprob(mixdat, mixpar, dist, constr)

Arguments

mixdat

a data frame containing grouped data, whose first column should be right boundaries of grouping intervals where the first and last intervals are open-ended; whose second column should consist of the frequencies indicating numbers of observations falling into each interval.

mixpar

a data frame containing the parameter values of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be one of "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

constr

a list of constraints on parameters of component distributions.

Value

It produces a matrix, whose each column contains the probabilities of the observations from one component falling into each grouping interval.

Examples

data(bindat)
data(binpar)
grpintprob(bindat, binpar, "binom", mixconstr(consigma = "BINOM", size = c(20, 20, 20, 20)))

Estimate Parameters of Mixture Distributions

Description

Find a set of overlapping component distributions that gives the best fit to grouped data and conditional data, using a combination of a Newton-type method and EM algorithm.

Usage

mix(mixdat, mixpar, dist = "norm", constr = list(conpi = "NONE", 
    conmu = "NONE", consigma = "NONE", fixpi = NULL, fixmu = NULL, 
    fixsigma = NULL, cov = NULL, size = NULL), emsteps = 1, 
    usecondit = FALSE, exptol = 5e-06, print.level = 0, ...)

Arguments

mixdat

A data frame containing grouped data, whose first column should be right boundaries of grouping intervals where the first and last intervals are open-ended; whose second column should consist of the frequencies indicating numbers of observations falling into each interval. If conditional data are available, this data frame should have k + 2 columns, where k is the number of components, whose element in row j and column i + 2 is the number of observations from the jth interval belonging to the ith component.

mixpar

A data frame containing starting values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be one of "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

constr

a list of constraints on parameters of component distributions. See function mixconstr.

emsteps

a non-negative integer specifying the number of EM steps to be performed.

usecondit

logical. If usecondit is TRUE and mixdat includes conditional data, then conditional data will be used with grouped data to estimate parameters of mixtures.

exptol

a positive scalar giving the tolerance at which the scaled fitted value is considered large enough to be a degree of freedom.

print.level

this argument determines the level of printing which is done during the optimization process. The default value of 0 means that no printing occurs, a value of 1 means that initial and final details are printed and a value of 2 means that full tracing information is printed.

...

additional arguments to the optimization function nlm

.

Value

A list containing the following items:

parameters

A data frame containing estimated values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

se

A data frame containing estimated values for standard errors of parameters of component distributions.

distribution

the distribution used to fit the data.

constraint

the constraints on parameters.

chisq

the goodness-of-fit chi-square statistic.

df

degrees of freedom of the fitted mixture model.

P

a significance level (P-value) for the goodness-of-fit test.

vmat

covariance matrix for the estimated parameters.

mixdata

the original data, i.e. the argument mixdat.

usecondit

the value of the argument usecondit.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mixgroup for grouping data, mixparam for organizing the parameter values, mixconstr for constructing constraints. nlm for additional arguments.

Examples

data(pike65)
data(pikepar)
fitpike1 <- mix(pike65, pikepar, "lnorm", constr = mixconstr(consigma = "CCV"), emsteps = 3)
fitpike1
plot(fitpike1)
data(pike65sg)
fitpike2 <- mix(pike65sg, pikepar, "lnorm", emsteps = 3, usecondit = TRUE)
fitpike2
plot(fitpike2)
data(bindat)
data(binpar)
fitbin1 <- mix(bindat, binpar, "binom",
               constr = mixconstr(consigma = "BINOM", size = c(20, 20, 20, 20)))
plot(fitbin1)
fitbin2 <- mix(bindat, binpar, "binom", constr = mixconstr(conpi = "PFX",
               fixpi = c(TRUE, TRUE, TRUE, TRUE),
               consigma = "BINOM", size = c(20, 20, 20, 20)))
plot(fitbin2)

Construct Constraints on Parameters

Description

Construct constraints on parameters and check if the constraints are invalid. See the reference for details.

Usage

mixconstr(conpi = "NONE", conmu = "NONE", consigma = "NONE", 
          fixpi = NULL, fixmu = NULL, fixsigma = NULL, cov = NULL, 
          size = NULL)

Arguments

conpi

a constraint on proportions, it can be either "NONE" denoting no constraint on proportions, or "PFX" indicating some proportions being fixed.

conmu

a constraint on means, it can be "NONE", "MFX", "MEQ", "MES" and "MGC", which denote no constraint on means, specified means fixed, means equal, means with equal spaces and means lying along a growth curve, respectively.

consigma

a constraint on standard deviations, it can be "NONE", "SFX", "SEQ", "FCV", "CCV", "BINOM", "NBINOM" and "POIS", which denote no constraint on standard deviations, specified standard deviations fixed, standard deviations equal, fixed coefficient of variation, constant coefficient of variation, the means and standard deviations have the same relation as that of Binomial distribution, as that of Negative Binomial distribution and as that of Possion distribution.

fixpi

NULL or a vector with TRUE and FALSE as its elements, indicating which proportions are fixed when conpi is "PFX". If an element is TRUE, the corresponding proportion is fixed at the starting value.

fixmu

similar to fixpi. NULL or a vector indicating which means are fixed when conmu is "MFX".

fixsigma

similar to fixpi. NULL or a vector indicating which standard deviations are fixed when consigma is "SFX".

cov

NULL or a scalar if consigma is "FCV", then the coefficients of variation are fixed at this scalar.

size

NULL or a vector of numbers of trials for each component when consigma is "BINOM" or "NBINOM".

Value

A list containing the following components, which are, in order, conpi, conmu, consigma, fixpi, fixmu, fixsigma, cov, size.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mixgroup for grouping data, mixparam for constructing starting values of parameters.

Examples

mixconstr()
mixconstr(conmu = "MEQ", consigma = "SFX", fixsigma = c(TRUE, FALSE, TRUE, TRUE, FALSE))
mixconstr(consigma = "BINOM", size = c(25, 25, 25))

Mixed Data

Description

as.mixdata checks if its argument is mixed data, if true, it returns the data with class "mixdata", if false, it returns NULL.

is.mixdata returns TRUE if its argument is of class "mixdata" and FALSE otherwise.

Usage

as.mixdata(x)
is.mixdata(x)

Arguments

x

object to be tested.

Details

Mixed data consist of grouped data and conditional data (if available). Grouped data is either a data frame or a matrix, whose first column should be right boundaries of grouping intervals where the first and last intervals are open-ended; whose second column should consist of the frequencies indicating numbers of observations falling into each interval. If conditional data are available, mixed data should have k + 2 columns, where k is the number of components, whose element in row j and column i + 2 is the number of observations from the jth interval belonging to the ith component.

See Also

mixgroup to construct mixed data.

Examples

data(pike65) # load data set `pike65'
pike65 # display the mixed data `pike65'
data(pike65sg) # load data set `pike65sg'
pike65sg # display the mixed data `pike65sg'
data(pikepar)
as.mixdata(pikepar)
as.mixdata(pike65)
is.mixdata(pike65)
is.mixdata(as.mixdata(pike65))

Construct Grouped Data from Raw Data

Description

Group raw data in the form of numbers of observations over successive intervals.

Usage

mixgroup(x, breaks = NULL, xname = NULL, k = NULL, usecondit = FALSE)

Arguments

x

a data frame or matrix containing raw data, whose first column should be the measurements to be grouped, and second column, if available, includes the numbers indicating which component each individual belongs to.

breaks

one of: * a vector giving the boundaries of intervals which raw data are grouped into, * a single number giving the number of intervals, * a character string naming an algorithm to compute the number of intervals, * a function to compute the number of intervals. In the last three cases the number is a suggestion only.

xname

the name of measurement.

k

the number of components.

usecondit

if usecondit is TRUE and x has two columns, then conditional data will be displayed with grouped data.

Value

A data frame containing grouped data derived from raw data, whose first column includes the right boundaries of grouping intervals, where the first and last intervals are open-ended; whose second column consists of the frequencies which are the numbers of observations falling into each interval. If usecondit is TRUE and the numbers indicating which component the individual comes from are available, conditional data which can be regarded as a table, whose element in row j and column i is the number of observations from the jth interval belonging to the ith component, will be displayed with grouped data.

See Also

hist for more information about the argument breaks, is.mixdata for checking the class of data sets, mixparam for organizing the parameter values, mixconstr for constructing constraints.

Examples

data(pikeraw) # load raw data `pikeraw'
pikeraw # display the data set `pikeraw'
mixgroup(pikeraw) # group raw data
pikemd <- mixgroup(pikeraw, breaks = c(0, seq(19.75, 65.75, 2), 80))
plot(pikemd)
mixgroup(pikeraw, breaks = c(0, seq(19.75, 65.75, 2), 80), usecondit = TRUE, k = 5)
# construct grouped data associated with conditional data
mixgroup(pikeraw, usecondit = TRUE)
mixgroup(pikeraw, usecondit = TRUE, k = 3) # grouping data with a warning message
mixgroup(pikeraw, usecondit = TRUE, k = 8)

Find the Parameters to be Estimated

Description

When there are constraints on parameters, we only estimate some parameters in terms of the constraints. This function is to find the parameters to be estimated. See the reference for details.

Usage

mixpar2theta(mixpar, constr, mixprop = TRUE)

Arguments

mixpar

A data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

constr

a list of constraints on parameters of component distributions.

mixprop

if TRUE, the proportions will be estimated.

Value

A vector containing the values for the parameters to be estimated.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mix for fitting mixture model, mixtheta2par for computing all the parameters from the estimated parameters.

Construct Starting Values for Parameters

Description

Construct starting values for parameters of a mixture model.

Usage

mixparam(mu, sigma, pi = NULL)

Arguments

mu

a vector of means of component distributions, which should be in ascending order.

sigma

a vector of standard deviations of component distributions, which are corresponding to the means. sigmas must be in ascending order when means are equal.

pi

the corresponding mixing proportions of components. If NULL, the proportions will be taken as 1/k, where k is the number of elements of mu.

Value

A data frame containing three variables, which are, in order, the proportions, means, and standard deviations.

See Also

mixgroup for grouping data, mixconstr for constructing constraints.

Examples

mixparam(mu = c(20, 30, 40), sigma = c(2, 3, 4))
mixparam(c(20, 30, 40), c(3), c(0.15, 0.78, 0.07))

Compute All of Parameters from the Estimated Parameters

Description

When there are constraints on parameters, we only estimate some parameters in terms of the constraints. This function is to compute all of parameters from the estimated ones.

Usage

 mixtheta2par(mixtheta, mixpar, constr, mixprop = TRUE)

Arguments

mixtheta

a vector containing the values for the estimated parameters, usually, a result of the function mixpar2theta.

mixpar

A data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

constr

a list of constraints on parameters of component distributions. See function mixconstr.

mixprop

if TRUE, the proportions will be estimated.

Value

A data frame containing three variables, which are, in order, the proportions, means, and standard deviations, respectively.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mixpar2theta for finding the parameters to be estimated.

Scale Mixture Data with Three Normal Components

Description

Scale mixture of three normal distributions.

The normals data frame has 25 rows and 2 columns.

Usage

data(normals)

Format

This data frame contains the following columns:

x

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Details

A total of 249 observations were generated by computer to follow the mixture distribution 1/3 N(12.5, 1) + 1/3 N(12.5, 3) + 1/3 N(12.5, 5) where N(m, s) denotes a normal distribution with mean m and standard deviation s.

Source

http://www.math.mcmaster.ca/peter/mix/demex/exscle.html

Examples

data(normals)
plot.mixdata(normals)

Karl Pearson's Crab Data

Description

The data give the ratio of "forehead" breadth to body length for 1000 crabs sampled at Naples by Professor W.F.R. Weldon.

The pearson data frame has 29 rows and 2 columns.

Usage

data(pearson)

Format

This data frame contains the following columns:

ratio

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Source

Pearson, K. (1894). Contributions to the mathematical theory of evolution. Phil. Trans. Roy. Soc. London A 185, 71-110.

References

http://www.math.mcmaster.ca/peter/mix/demex/excrabs.html

Examples

data(pearson)
plot.mixdata(pearson)

Starting Values of Parameters for the Pearson's Data

Description

Starting values of parameters for fitting a mixture distribution to the data set pearson.

The pearsonpar data frame has 2 rows and 3 columns.

Usage

data(pearsonpar)

Format

This data frame contains the following columns:

pi

the starting values for proportions.

mu

the starting values for means.

sigma

the starting values for standard deviations.

Source

Pearson, K. (1894). Contributions to the mathematical theory of evolution. Phil. Trans. Roy. Soc. London A 185, 71-110.

References

http://www.math.mcmaster.ca/peter/mix/demex/excrabs.html

Examples

data(pearsonpar)

Heming Lake Pike Data

Description

The raw data pikeraw give the lengths of 523 pike (Esox lucius), and there are known to be five age-groups in the sample. We grouped the lengths over 25 intervals to obtain the grouped data given as separate samples for each age group determined by scale reading.

The pikdat5 data frame has 25 rows and 6 columns.

Usage

data(pikdat5)

Format

This data frame contains the following columns:

length

the boundaries of grouping intervals.

age1

the numbers of observation from each interval belonging to the first age group.

age2

the numbers of observation from each interval belonging to the second age group.

age3

the numbers of observation from each interval belonging to the third age group.

age4

the numbers of observation from each interval belonging to the fourth age group.

age5

the numbers of observation from each interval belonging to the fifth age group.

Source

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

References

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

http://www.math.mcmaster.ca/peter/mix/demex/expike.html

Examples

data(pikdat5)

Length-Frequency Data for Heming Lake Pike

Description

The raw data pikeraw give the lengths of 523 pike (Esox lucius). We grouped the lengths over 25 intervals to obtain this length-frequency data.

The pike65 data frame has 25 rows and 2 columns.

Usage

data(pike65)

Format

This data frame contains the following columns:

length

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

Source

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

References

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

http://www.math.mcmaster.ca/peter/mix/demex/expike.html

Examples

data(pike65)
data(pikepar)
plot.mixdata(pike65)
fit <- mix(pike65, pikepar, "lnorm", constr = mixconstr(consigma = "CCV"), emsteps = 3)
plot(fit)

Length-Frequency Data with Subsamples for Heming Lake Pike

Description

The raw data pikeraw give the lengths of 523 pike (Esox lucius), and there are known to be five age-groups in the sample. After grouping the data, we take subsamples from some intervals to determine the age group, and then obtain this data set.

The pike65sg data frame has 25 rows and 7 columns.

Usage

data(pike65sg)

Format

This data frame contains the following columns:

length

the boundaries of grouping intervals.

freq

the frequencies of observation falling into each interval.

age1

the numbers of observation in the subsamples belonging to the first age group.

age2

the numbers of observation in the subsamples belonging to the second age group.

age3

the numbers of observation in the subsamples belonging to the third age group.

age4

the numbers of observation in the subsamples belonging to the fourth age group.

age5

the numbers of observation in the subsamples belonging to the fifth age group.

Source

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

References

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

http://www.math.mcmaster.ca/peter/mix/demex/expike.html

Examples

data(pike65sg)
data(pikepar)
fit1 <- mix(pike65sg, pikepar, "gamma", mixconstr(consigma = "CCV"), usecondit = TRUE)
plot(fit1)
fit2 <- mix(pike65sg, pikepar, "gamma", usecondit = TRUE)
plot(fit2)

Starting Values of Parameters for the Pike Data

Description

Starting values of parameters for fitting a mixture distribution to the data set pike65.

The pikepar data frame has 5 rows and 3 columns.

Usage

data(pikepar)

Format

This data frame contains the following columns:

pi

the starting values for proportions.

mu

the starting values for means.

sigma

the starting values for standard deviations.

Source

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

References

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

http://www.math.mcmaster.ca/peter/mix/demex/expike.html

Examples

data(pikepar)

A Sample of Pike Lengths

Description

The data give the lengths of 523 pike (Esox lucius), sampled in 1965 from Heming Lake, Manitoba, Canada. There are known to be five age-groups in the sample. For each fish, the age group is determined by scale reading.

The pikeraw data frame has 523 rows and 2 columns.

Usage

data(pikeraw)

Format

This data frame contains the following columns:

length

the lengths of 523 pike

age

the age groups of 523 pike

Source

Macdonald, P.D.M. and T.J. Pitcher (1979). Age-groups from size-frequency data: a versatile and efficient method of analysing distribution mixtures. Journal of the Fisheries Research Board of Canada 36, 987-1001.

References

Macdonald, P.D.M. (1987). Analysis of length-frequency distributions. In R.C. Summerfelt and G.E. Hall [editors], Age and Growth of Fish, Iowa State University Press, Ames, Iowa. pp 371-384.

http://www.math.mcmaster.ca/peter/mix/demex/expike.html

Examples

data(pikeraw)

Mix Object Plotting

Description

A function for plotting of Mix objects. It is called via the generic function plot.

Usage

## S3 method for class 'mix'
plot(x, mixpar = NULL, dist = "norm", root = FALSE, ytop = NULL, 
     clwd = 1, main, sub, xlab, ylab, bty, BW = FALSE, ...)

Arguments

x

an object of class "mix", usually, the results returned by the model fitting function mix.

mixpar

NULL or a data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

root

if TRUE, a hanging rootogram will be displayed.

ytop

a scalar which determines the top of the y-axis.

clwd

a positive number denoting line width, defaulting to 1.

main

an overall title for the plot.

sub

a subtitle for the plot.

xlab

a title for the x-axis.

ylab

a title for the y-axis.

bty

A character string which determined the type of box which is drawn about plots. If bty is one of "o", "l", "7", "c", "u", or "]" the resulting box resembles the corresponding upper case letter. A value of "n" suppresses the box.

BW

logical; if TRUE the plot will be drawn in black and white.

...

additional arguments to the function plot.default.

Details

If the argument x gives an object of class "mix", the plot will be a histogram for the grouped data which come from the element mixdata of x. Although the leftmost (first) and rightmost (mth) intervals are always open-ended, on the histogram the first interval is shown as being twice the width of the second interval and the mth is shown as being twice the width of the m - 1st interval. When the fitted distribution is one of "lnorm", "gamma" and "weibull", the left boundary of the first interval will be taken zero since negative values and zeroes are not allowed for these distribution. For the distributions "binom", "nbinom" and "pois" negative data are not permitted, so the left boundary of the first interval is taken -0.5. The component distributions weighted by their respect proportions and the mixture distribution are computed by the estimated parameter values from the element parameters of x, and superimposed on the histogram. The distribution of components will be taken the value of the element distribution. If sub, xlab, ylab and bty are not specified, the default values will be used. The positions of the means are indicated with triangles. When the argument root is TRUE, a hanging rootogram will be displayed, that is, if only grouped data are given, this option plots the histogram with the square root of relative frequency on the y-axis. If there is a model as well as data, not only is the y-axis the square root of relative frequency, also the bars of the histogram, instead of rising from 0, are shifted up or down so that the mid-point of the top of the bar is exactly on the curve indicating the mixture distribution and the bottom of the bar may therefore be above or below the x-axis. If the bar goes below the x-axis, the portion below is shown as a blue rectangle. If the bar does not reach the x-axis, the space between the bottom of the bar and the x-axis is shown as a blue rectangle. If the blue rectangles are almost above or below in an area of the x-axis, we may say that the mixture curve around that area is not fitting well.

See Also

mixparam for organizing the parameter values, mix for fitting mixture model, plot.mixdata for plotting Mixdata objects, plot.default for additional arguments.

Examples

data(pike65)
data(pikepar)
fit1 <- mix(pike65, pikepar, "lnorm",
            constr = mixconstr(consigma = "CCV"), emsteps = 3)
plot(fit1)
plot(fit1, root = TRUE)
data(bindat)
data(binpar)
fit2 <- mix(bindat, binpar, "binom",
            constr = mixconstr(consigma = "BINOM", size = c(20, 20, 20, 20)))
plot(fit2)
plot(fit2, root = TRUE)

Mixdata Object Plotting

Description

A function for plotting of Mixdata objects. It is called via the generic function plot.

Usage

## S3 method for class 'mixdata'
plot(x, mixpar = NULL, dist = "norm", root = FALSE, ytop = NULL, 
     clwd = 1, main, sub, xlab, ylab, bty, ...)

Arguments

x

an object of class "mixdata", usually, the results returned by the function mixgroup.

mixpar

NULL or a data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

root

if TRUE, a hanging rootogram will be displayed.

ytop

a scalar which determines the top of the y-axis.

clwd

a positive number denoting line width, defaulting to 1.

main

an overall title for the plot.

sub

a subtitle for the plot.

xlab

a title for the x-axis.

ylab

a title for the y-axis.

bty

A character string which determined the type of box which is drawn about plots. If bty is one of "o", "l", "7", "c", "u", or "]" the resulting box resembles the corresponding upper case letter. A value of "n" suppresses the box.

...

additional arguments to the function plot.default.

Details

If the argument mixpar is NULL, then only the histogram of the data will be displayed; if mixpar gives the values of parameters, the component distributions and the mixture distribution are computed from the parameter values and superimposed on the histogram.

See Also

plot.mix for plotting Mix objects, plot.default for additional arguments.

Examples

data(cassie)
as.mixdata(cassie) # if the result isn't `NULL', then cassie is mixed data
plot.mixdata(cassie)
data(pikeraw)
data(pikepar)
pikemd <- mixgroup(pikeraw, breaks = c(0, seq(19.75, 65.75, 2), 80))
plot(pikemd)
plot(pikemd, pikepar, "lnorm")
fit <- mix(pikemd, pikepar, "lnorm", constr = mixconstr(consigma = "CCV"), emsteps = 3)
plot(fit)
plot(pikemd, pikepar, "lnorm", root = TRUE)
plot(fit, root = TRUE)

Grouped Poisson Data

Description

The poisdat data frame has 15 rows and 2 columns.

Usage

data(poisdat)

Format

This data frame contains the following columns:

X

the boundaries of grouping intervals.

samppois

the frequencies of observation falling into each interval.

Examples

data(poisdat)
plot.mixdata(poisdat)

Starting Values of Parameters for the Poisson Data Set

Description

Starting values of parameters for fitting a mixture distribution to the data set poisdat.

The poispar data frame has 4 rows and 3 columns.

Usage

data(poispar)

Format

This data frame contains the following columns:

pi

the starting values for proportions.

mu

the starting values for means.

sigma

the starting values for standard deviations.

Examples

data(poispar)

Print Mix Object

Description

print.mix is a function which prints objects of class "mix" and returns it invisibly. It is called via the generic function print.

Usage

## S3 method for class 'mix'
print(x, digits = 4, ...)

Arguments

x

an object of class "mix", usually, the results returned by the model fitting function mix.

digits

how many significant digits are to be used.

...

further arguments passed to or from other methods.

Details

This function only prints information about the mixture model, which are the estimated parameters of the mixture, the distribution of components and the constraints on the parameters. Also, the values for the parameters are rounded to the specified number of decimal places (default 4). The whole object can be printed out using the function print.default.

See Also

mix for model fitting. print.default for printing the whole object.

Examples

data(pike65)
data(pikepar)
fit <- mix(pike65, pikepar, "gamma", mixconstr(consigma = "CCV"), emsteps = 3)
fit
print(fit)
print.mix(fit)
print.default(fit)

Summarizing Mixture Model Fits

Description

summary method for class "mix". It is called via the generic function summary.

Usage

## S3 method for class 'mix'
summary(object, digits = 4, ...)

Arguments

object

an object of class "mix", usually, the results returned by the model fitting function mix.

digits

how many significant digits are to be used.

...

additional arguments affecting the summary produced.

Value

A list containing the following items:

parameters

a data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

standard errors

a data frame giving the standard errors of estimated parameters.

anova table

analysis of variance table for the mixobj, that is, the results from the function anova.mix.

See Also

mix for model fitting, summary for summarizing other kinds of object. anova.mix for information about anova table.

Examples

data(pike65)
data(pikepar)
fit <- mix(pike65, pikepar, "lnorm", mixconstr(consigma = "CCV"), emsteps = 3)
fit
summary(fit)

Check Constraints

Description

Check if constraints on parameters are valid. See the reference for details.

Usage

testconstr(mixdat, mixpar, dist, constr)

Arguments

mixdat

a data frame containing grouped data, whose first column should be right boundaries of grouping intervals, whose second column should consist of the frequencies indicating numbers of observations falling into each interval. If conditional data are available, this data frame should have k + 2 columns, where k is the number of components, whose element in row j and column i + 2 is the number of observations from the jth interval belonging to the ith component.

mixpar

a data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be one of "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

constr

a list of constraints on parameters of component distributions. See function mixconstr.

Value

If the constraints are valid, this function will give a logical value TRUE. If not, it will give an error message to illustrate the reason.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mixgroup for grouping data, mixparam for organizing the parameter values, mixconstr for constructing constraints.

Examples

## Not run: 
testconstr(pike65, pikepar, "lnorm", constr = mixconstr(consigma = "CCV"))
testconstr(bindat, binpar, "binom", constr = mixconstr())
testconstr(bindat, binpar, "binom", constr = mixconstr(consigma = "BINOM"))
testconstr(bindat, binpar, "pois", constr = mixconstr(conmu = "MEQ", consigma = "POIS"))

## End(Not run)

Check Parameters

Description

Check if the values of parameters are valid. See the reference for details.

Usage

testpar(mixpar, dist, constr)

Arguments

mixpar

a data frame containing the values for parameters of component distributions, which are, in order, the proportions, means, and standard deviations.

dist

the distribution of components, it can be one of "norm", "lnorm", "gamma", "weibull", "binom", "nbinom" and "pois".

constr

a list of constraints on parameters of component distributions. See function mixconstr.

Details

Any of the parameter values can not be missing value (NA or NaN) or infinity (Inf), and the proportions can only take the values between 0 and 1. Besides, the standard deviations can not be negative. The components must be indexed so that the means are in non-decreasing order. If any two consecutive means are equal, then the corresponding standard deviations must be in strictly ascending order. Furthermore, the parameter values should be consistent with the constraints and the distribution of components. For example, if one wants to constrain the means to lie along a growth curve, then (\mu_3 - \mu_2) < (\mu_2 - \mu_1) is required. Also, negative means are not permitted by the constraints "FCV", "CCV", "BINOM", "NBINOM", "POIS" and all the distributions but Normal. If the Binomial distribution components with the constraint "BINOM" are fitted, then the relation \mu_i > (\sigma_i)^2 need to be satisfied. And the Negative Binomial components with the constraint "NBINOM" require \mu_i < (\sigma_i)^2.

Value

logical. If TRUE, the parameters are valid. If FALSE, some of the parameters are invalid. Since this function is for internal use, it doesn't give error messages.

References

Macdonald, P.D.M. and Green, P.E.J. (1988) User's Guide to Program MIX: An Interactive Program for Fitting Mixtures of Distributions. ICHTHUS DATA SYSTEMS.

See Also

mixparam for organizing the parameter values, mixconstr for constructing constraints, testconstr for checking constraints.

Compute Shape and Scale Parameters for Weibull Distribution

Description

Compute the parameters shape and scale for Weibull distribution given the mean, standard deviation and location.

Usage

weibullpar(mu, sigma, loc = 0)

Arguments

mu

the mean of weibull distribution.

sigma

the standard deviation of weibull distribution.

loc

the location parameter of weibull distribution defaulting to 0.

Value

A data frame containing three parameters, which are, in order, shape, scale, and location.

See Also

weibullparinv for computing mean and standard deviation from the parameters shape, scale and location.

Examples

weibullpar(2, 1.2)
weibullpar(2, 1.2, 1)

Compute the Mean and Standard Deviation of Weibull Distribution

Description

Compute mean and standard deviation of weibull distribution given the values of shape, scale and location.

Usage

weibullparinv(shape, scale, loc = 0)

Arguments

shape

the shape parameter of weibull distribution.

scale

the scale parameter of weibull distribution.

loc

the location parameter of weibull distribution defaulting to 0.

Value

A data frame containing three parameters, which are, in order, mean, standard deviation and location.

See Also

weibullpar for computing the parameters shape and scale from mean and standard deviation.

Examples

weibullparinv(weibullpar(2, 1.2)$shape, weibullpar(2, 1.2)$scale)