EMutilities: Functions related to implementing the EM algorithm on...

In DOBAD: Analysis of Discretely Observed Linear Birth-and-Death(-and-Immigration) Markov Chains

Description

These are functions for the EM algorithm on a partially observed linear birth-death process where the immigration rate is a constant scalar times the birthrate. The ".SC" suffix refers to this constraint ("SC" stands for "Special Case").

E.step.SC performs the "Expectation step" and M.step.SC performs the maximization step.

BDloglikelihood.PO computes the log likelihood of a partially observed birth-death process.

Usage

M.step.SC(EMsuffStats, T,beta.immig)
E.step.SC(theData, oldParams, beta.immig,  dr=0.001, n.fft=1024,
                      r=4, prec.tol, prec.fail.stop)

Arguments

EMsuffStats	Vector with names "Nplus", "Nminus", and "Holdtime", which are the number of jumps up, number of jumps down, and the total holding time, respectively. These often come from the E.step.SC function.
T	total Time the chain was observed for (ie usually the last observation time).
beta.immig	Immigration rate is constrained to be a multiple of the birth rate. immigrationrate = beta.immig * lambda where lambda is birth rate.
oldParams	Parameters with which to compute the expectation
n.fft	Number of terms to use in the fast fourier transform or the riemann integration when using the generating functions to compute probabilities or joint expectations for the birth-death process. See the add.joint.mean.many, etc, functions.
theData	Partially observed chain. Has components $times and $states where dat$states[i] is the state observed at time dat$times[i]. (No $T component needed).
dr	Parameter for numerical differentiation
r	Parameter for differentiation; see numDeriv package documentation.
prec.tol	"Precision tolerance"; to compute conditional means, first the joint means are computed and then they are normalized by transition probabilities. The precision parameters govern the conditions under which the function will quit if these values are very small. If the joint-mean is smaller than prec.tol then the value of prec.fail.stop decides whether to stop or continue.
prec.fail.stop	If true, then when joint-mean values are smaller than prec.tol the program stops; if false then it continues, usually printing a warning.

Details

Assume we have a linear-birth-death process X_t with birth parameter lambda, death parameter mu, and immigration parameter beta*lambda (for some known, real beta). We observe the process at a finite set of times over a time interval [0,T].

E.step.SC computes the needed expectations for the EM algorithm. These are the expectations of the sufficient statistics, conditional on the data. These expectations are computed with respect to the measure given by oldParams, i.e. the chain governed by oldParams.

M.Step.SC maximizes the partial-data likelihood given the passed in expecatations of the sufficient statistics, to get the parameter iterates for the next step of the EM algorithm. (This is easy when we are in the "Special Case" where immigration is constrained.)

BDloglikelihood.PO computes the log likelihood of the passed in birth-death process.

Value

M.step.SC returns a length 2 vector with first element lambda-hat and second element mu-hat, the respective maximizers of the likelihood.

E.step.SC returns a vector with names "Nplus", "Nminus", and "Holdtime."

BDloglikelihood.PO returns a real number, the log-likelihood of the data.

Author(s)

Charles Doss

See Also

EM.BD.SC

DOBAD documentation built on May 2, 2019, 3:04 a.m.