mixalg {CAMAN}

Fitting Finite Mixture Models

Description

This hybrid mixture algorithm combines the VEM algorithm for flexible support size and the EM algorithm for a fixed number of components. The solution of the VEM algorithm provides starting values for the EM algorithm. By the NPMLE theorem the EM algorithm thus starts very close to the global maximum and proper convergence of the EM algorithm to a global maximum is ensured.

The algorithm proceeds as follows

Step 1: Define an approximating grid lambda[1], ..., lambda[L]

Step 2: Use the VEM algorithm to maximize L(P) in the simplex \Omega and identify grid points with positive support. Here positive support is defined as p[j] >= epsilon (often epsilon = 10^-2).
This gives an initial estimate of k.

Step 3: Use these k points and corresponding mixing weights p[j] as starting values for the EM algorithm

Step 4: Collapse identical components if | lambda[j]- lambda[i] | < delta (often delta=0.05) for i != j

Step 5: Obtain the final number of components k

This sequential algorithm leads to an initial estimate of the NPMLE and a proper solution for the subsequent EM algorithm. Crucial points are the definitions of \delta and \epsilon. Depending on these settings different solutions could result from this algorithm.

Usage

mixalg(obs, weights=NULL, family="gaussian", data=NULL, pop.at.risk=NULL, 
       var.lnOR=NULL, limit=0.01, acc=10^(-7), numiter=5000, startk=50)

Arguments

obs

observed / dependent variable. Vector or colname of data. Must be specified!

weights

weights of the data. Vector or colname of data. Default is NULL.

family

the underlying type density function as a character ("gaussian", "poisson" or "binomial")!

data

an optional data frame. obs, weights, pop.at.risk and var.lnOR can be specified as column name of the data frame.

pop.at.risk

population at risk: These data could be used to determine a mixture model for Poisson data. Vector or colname of data. Default isNULL.

var.lnOR

variances of the data: These variances might be given when working with meta analyses! Vector or colname of data. Default is NULL.

limit

parameter to control the limit of union several components. Default is 0.01.

acc

convergence criterion. VEM and EM loops stop when deltaLL<acc. Default is 10^(-7).

numiter

parameter to control the maximal number of iterations in the VEM and EM loops. Default is 5000.

startk

starting/maximal number of components. This number will be used to compute the grid in the VEM. Default is 50.

Details

The documentation of leukDat contains a disease mapping example using mixalg and the documentation of golubMerge contains a microarray analysis example.

Value

The function returns a CAMAN.object, describing a finite mixture model. The main information about the mixture model is printed by just typing the <object>. Additional information is given in summary(object) (summary.CAMAN.object). Single attributes can be accessed using the @, e.g. mix@LL.

dat

(input) data

family

underlying type density function

LL

Likelihood of the final (best) iteration

BIC

Likelihood of the final (best) iteration

num.k

number of components obtained

p

probability of each component

t

parameter of distribution (normal distr. -> mean, poisson distr. -> lambda, binomial distr. -> prob)

component.var

variance of each component (ONLY if family == "gaussian")

prob

probabilies, belonging to each component

classification

classification labels for each observation (which.max of @prob).

steps

number of steps performed (EM, VEM).

VEM_result

result of VEM algorithm.

cl

the matched call.

is_metaAnalysis

parameter specifying, whether a meta analysis was performed.

VEM_result

Outcome of the VEM-algorithm, which was run before the EM.

finalacc

deltaLL of the final iteration (for VEM and EM)

Author(s)

Peter Schlattmann and Johannes Hoehne

References

D. B\"ohning, P. Schlattmann, B.G. Lindsay: C.A.MAN - Computer Assisted Analysis of Mixtures: Statistical Algorithms.Biometrics, 1992, 48, 283-303

P. Schlattmann: On bootstrapping the unknown number of components in finite mixtures of Poisson distributions. Statistics and Computing, 2005, 15, 179-188

Schlattmann, P. (2009). Medical Applications of Finite Mixture Models. Berlin: Springer.

See Also

mixalg.EM, mixalg.VEM, anova.CAMAN.object, mixcov, mixalg.boot

Examples

### POISSON data with weights: thai_cohort
data(thai_cohort)
mix <- mixalg(obs="counts", weights="frequency", family="poisson", 
              data=thai_cohort, numiter=18000, acc=0.00001, startk=25)


# meta analysis
data(aspirin)
mix <- mixalg(obs="logrr", var.lnOR="var", data=aspirin)


## See the documentation of golub.Merge for a
## microarray analysis example using mixalg

## See the documentation of leukDat for a disease
## mapping example using mixalg