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.
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)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.
(input) data
underlying type density function
Likelihood of the final (best) iteration
Likelihood of the final (best) iteration
number of components obtained
probability of each component
parameter of distribution (normal distr. -> mean, poisson distr. -> lambda, binomial distr. -> prob)
variance of each component (ONLY if family == "gaussian")
probabilies, belonging to each component
classification labels for each observation (which.max of @prob).
number of steps performed (EM, VEM).
result of VEM algorithm.
the matched call.
parameter specifying, whether a meta analysis was performed.
Outcome of the VEM-algorithm, which was run before the EM.
deltaLL of the final iteration (for VEM and EM)
observed / dependent variable. Vector or colname of data. Must be specified!
weights of the data. Vector or colname of data. Default is NULL.
the underlying type density function as a character ("gaussian", "poisson" or "binomial")!
an optional data frame. obs, weights, pop.at.risk and var.lnOR can be specified as column name of the data frame.
population at risk: These data could be used to determine a mixture model for Poisson data. Vector or colname of data. Default isNULL.
variances of the data: These variances might be given when working with meta analyses! Vector or colname of data. Default is NULL.
parameter to control the limit of union several components. Default is 0.01.
convergence criterion. VEM and EM loops stop when deltaLL<acc. Default is 10^(-7).
parameter to control the maximal number of iterations in the VEM and EM loops. Default is 5000.
starting/maximal number of components. This number will be used to compute the grid in the VEM. Default is 50.
Peter Schlattmann and Johannes Hoehne
The documentation of leukDat contains a disease mapping example using mixalg and the documentation of golubMerge contains a microarray analysis example.
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.
mixalg.EM, mixalg.VEM, anova.CAMAN.object, mixcov, mixalg.boot
### 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
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