pomp {pomp}R Documentation

Partially-observed Markov process object.

Description

Create a new pomp object.

Usage

  pomp(data, times, t0, rprocess, dprocess, rmeasure, dmeasure,
       initializer, ...)

Arguments

data An array holding the data. This array is of dimensions nobs x ntimes, where nobs is the number of observed variables and ntimes is the number of times at which observations were made. It is also possible to specify data as a data-frame, in which case the argument times must be the name of the column of observation times. Note that if data is a data-frame, it will be coerced to an array with storage-mode `double'.
times The times corresponding to the observations. times must be a strictly increasing numeric vector. If data is a data-frame, times should be the name of the column of observation times.
t0 The zero-time. This must be no later than the time of the first observation, times[1].
rprocess Function of prototype rprocess(xstart,times,params,...) which simulates from the unobserved process.
dprocess Function of prototype dprocess(x,times,params,log=FALSE,...) which evaluates the likelihood of a sequence of consecutive state transitions.
rmeasure Function of prototype rmeasure(x,times,params,...) which simulates from the observation process.
dmeasure Function of prototype dmeasure(y,x,times,params,log=FALSE,...) which gives the likelihood of y given x.
initializer Function of prototype initializer(params,t0,...) which yields initial conditions for the state process when given a vector, params, of parameters. By default, i.e., if it is unspecified when pomp is called, the initializer assumes any names in params that end in ".0" correspond to initial value parameters. These are simply copied over as initial conditions when init.state is called (see init.state-pomp). The names of the state variables are the same as the corresponding initial value parameters, but with the ".0" dropped.
... Any additional arguments are stored in a slot `userdata' and are passed as arguments to each of the functions rprocess, dprocess, rmeasure, dmeasure, and initializer whenever they are evaluated. Using this mechanism, the user can store additional information necessary for the definition of the model.

Details

It is the user's responsibility to ensure that the rprocess, dprocess, rmeasure, dmeasure, and initializer functions satisfy the following conditions:

rprocess
rprocess must have at least the following arguments: xstart, times, params, and .... It can also take additional arguments. It is guaranteed that these will be filled with the corresponding elements the user has included as additional arguments in the construction of the pomp object.

In calls to rprocess, xstart will be a rank-2 array (matrix) with rows corresponding to state variables and columns corresponding to independent realizations of the process. params will similarly be a rank-2 array with rows corresponding to parameters and columns corresponding to independent realizations. The columns of params are to be matched up with those of xstart; in particular, they will agree in number. Both xstart and params must have rownames, which are available for use by the user.

rprocess must return a rank-3 array with rownames. Suppose x is the array returned. Then dim(x)=c(nvars,nreps,ntimes), where nvars is the number of state variables (=nrow(xstart)), nreps is the number of independent realizations simulated (=ncol(xstart)), and ntimes is the length of the vector times. x[,j,k] is the value of the state process in the j-th realization at time times[k]. In particular, x[,,1] must be identical to xstart. The rownames of x must correspond to those of xstart.

dprocess
dprocess must have at least the following arguments: x, times, params, log, and .... It may take additional arguments. It is guaranteed that these will be filled with the corresponding elements the user has included as additional arguments in the construction of the pomp object.

In calls to dprocess, x will be an nvars x nreps x ntimes array, where these terms have the same meanings as above. params will be a rank-2 array with rows corresponding to individual parameters and columns corresponding to independent realizations. The columns of params are to be matched up with those of x; in particular, they will agree in number. Both x and params must have rownames, available for use by the user.

dprocess must return a rank-2 array (matrix). Suppose d is the array returned. Then dim(d)=c(nreps,ntimes-1). d[j,k] is the probability density of the transition from state x[,j,k-1] at time times[k-1] to state x[,j,k] at time times[k]. If log=TRUE, then the log of the p.d.f. is returned. In writing this function, you may assume that the transitions are consecutive.

rmeasure
rmeasure must have at least the arguments x, times, params, and .... It may take additional arguments, which will be filled with user-specified data as above. x must be a rank-3 array dimension c(nvars,nreps,ntimes), where these variables have the same meanings as above. times is the corresponding set of times. params is a rank-2 array of dimension c(npars,nreps) as above.

rmeasure must return a rank-3 array with rownames. If y is the returned array, then dim(y)=c(nobs,nreps,ntimes), where nobs is the number of observable variables and nreps, ntimes agree with the corresponding dimensions of x. y[,j,k] must be the vector of observables in the j-th realization at time times[k].

dmeasure
dmeasure must have at least the arguments y, x, times, params, log, and .... y is a rank-2 array of observations (nobs x ntimes); x, a rank-3 array of states; params, a rank-2 array containing parameters; and times, the corresponding observation times. It may take additional arguments which will be filled with user-specified data as above. dmeasure must return a rank-2 array of dimension nreps x ntimes. If d is the returned array, then d[j,k] is the p.d.f. of y[,k] given x[,j,k] at time times[k]. If log=TRUE, then the log of the p.d.f. is returned.
initializer
initializer must have at least the arguments params, t0, and .... params is a named vector of parameters. t0 will be the time at which initial conditions are desired. initializer must return a named vector of initial conditions.

Value

An object of class pomp.

Warning

Some error checking is done, but complete error checking is impossible. If the user-specified functions do not conform to the above specifications (see Details), then the results may be invalid.

Author(s)

Aaron A. King (kingaa at umich dot edu)

References

See Also

pomp-class, euler, vignette(intro_to_pomp)

Examples

## Simple example: a 2-D Brownian motion, observed with normal error
## first, set up the pomp object:

rw2 <- pomp(
            rprocess = function (xstart, times, params, ...) { 
              ## this function simulates two independent random walks with intensities s1, s2
              nsims <- ncol(params)
              ntimes <- length(times)
              dt <- diff(times)
              x <- array(0,dim=c(2,nsims,ntimes))
              rownames(x) <- rownames(xstart)
              noise.sds <- params[c('s1','s2'),]
              x[,,1] <- xstart
              for (j in 2:ntimes) {
                ## we are mimicking a continuous-time process, so the increments have SD ~ sqrt(dt)
                ## note that we do not have to assume that 'times' are equally spaced
                x[,,j] <- rnorm(n=2*nsims,mean=x[,,j-1],sd=noise.sds*dt[j-1])
              }
              x
            },
            dprocess = function (x, times, params, log = FALSE, ...) { 
              ## given a sequence of consecutive states in 'x', this function computes the p.d.f.
              nsims <- ncol(params)
              ntimes <- length(times)
              dt <- diff(times)
              d <- array(0,dim=c(2,nsims,ntimes-1))
              noise.sds <- params[c('s1','s2'),]
              for (j in 2:ntimes)
                d[,,j-1] <- dnorm(x[,,j]-x[,,j-1],mean=0,sd=noise.sds*dt[j-1],log=TRUE)
              if (log) {
                apply(d,c(2,3),sum)
              } else {
                exp(apply(d,c(2,3),sum))
              }
            },
            rmeasure = function (x, times, params, ...) {
              ## noisy observations of the two walks with common noise SD 'tau'
              nsims <- dim(x)[2]
              ntimes <- dim(x)[3]
              y <- array(0,dim=c(2,nsims,ntimes))
              rownames(y) <- c('y1','y2')
              for (j in 1:nsims) {
                for (k in 1:ntimes) {
                  y[,j,k] <- rnorm(2,mean=x[,j,k],sd=params['tau',j])
                }
              }
              y
            },
            dmeasure = function (y, x, times, params, log = FALSE, ...) {
              ## noisy observations of the two walks with common noise SD 'tau'
              d1 <- dnorm(
                          x=y['y1',],
                          mean=x['x1',,],
                          sd=params['tau',],
                          log=TRUE
                          )
              d2 <- dnorm(
                          x=y['y2',],
                          mean=x['x2',,],
                          sd=params['tau',],
                          log=TRUE
                          )
              if (log) {
                d1+d2
              } else {
                exp(d1+d2)
              }
            },
            times=1:100,
            data=rbind(
              y1=rep(0,100),
              y2=rep(0,100)
              ),
            t0=0
            )

## specify some parameters
p <- rbind(s1=c(2,2,3),s2=c(0.1,1,2),tau=c(1,5,0),x1.0=c(0,0,5),x2.0=c(0,0,0))

## simulate
examples <- simulate(rw2,params=p)
rw2 <- examples[[1]]  ## by default, simulate produces a list of pomp objects
plot(rw2)

t <- seq(0,20)
X <- simulate(rw2,params=p[,1],nsim=10,states=TRUE,obs=TRUE,times=t)

## compute the process model likelihoods
f <- dprocess(
              rw2,
              x=X$states,
              times=t,
              params=matrix(
                p[,1],
                nrow=nrow(p),
                ncol=10,
                dimnames=list(rownames(p),NULL)
                ),
              log=TRUE
              )
apply(f,1,sum)

## compute the measurement likelihoods
g <- dmeasure(
              rw2,
              y=X$obs[,7,],
              x=X$states,
              times=t,
              params=matrix(
                p[,1],
                nrow=nrow(p),
                ncol=10,
                dimnames=list(rownames(p),NULL)
                ),
              log=TRUE
              )
apply(g,1,sum)

## For more examples, see the vignettes.

[Package pomp version 0.19-1 Index]