correlation: Correlation structures

In mcmcsae: Markov Chain Monte Carlo Small Area Estimation

Correlation structures

Description

Element 'factor' of a model component created using function gen is a formula composed of several possible terms described below. It is used to derive a (sparse) precision matrix for a set of coefficients, and possibly a matrix representing a set of linear constraints to be imposed on the coefficient vector.

iid(f)

Independent effects corresponding to the levels of factor f.

RW1(f, circular=FALSE, w=NULL)

First-order random walk over the levels of factor f. The random walk can be made circular and different (fixed) weights can be attached to the innovations. If specified, w must be a positive numeric vector of length one less than the number of factor levels. For example, if the levels correspond to different times, it would often be reasonable to choose w proportional to the reciprocal time differences. For equidistant times there is generally no need to specify w.

RW2(f)

Second-order random walk.

AR1(f, phi, w=NULL)

First-order autoregressive correlation structure among the levels of f. Required argument is the (fixed) autoregressive parameter phi. For irregularly spaced AR(1) processes weights can be specified, in the same way as for RW1.

season(f, period)

Dummy seasonal with period period.

spatial(f, poly.df, snap, queen, derive.constraints=FALSE)

CAR spatial correlation. Argument poly.df can either be an object of (S4) class SpatialPolygonsDataFrame or an object of (S3) class sf. The latter can be obtained, e.g., from reading in a shape file using function st_read. Arguments snap and queen are passed to poly2nb. If derive.constraints=TRUE the constraint matrix for an IGMRF model component is formed by computing the singular vectors of the precision matrix.

spline(f, knots, degree)

P-splines, i.e. penalized B-splines structure over the domain of a quantitative variable f. Arguments knots and degree are passed to splineDesign. If knots is a single value it is interpreted as the number of knots, otherwise as a vector of knot positions. By default 40 equally spaced knots are used, and a degree of 3.

custom(f, D=NULL, Q=NULL, R=NULL, derive.constraints=NULL)

Either a custom precision or incidence matrix associated with factor f can be passed to argument Q or D. Optionally a constraint matrix can be supplied as R, or constraints can be derived from the null space of the precision matrix by setting derive.constraints=TRUE.

References

B. Allevius (2018). On the precision matrix of an irregularly sampled AR(1) process. arXiv:1801.03791.

H. Rue and L. Held (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC.

Examples

# example of CAR spatial random effects
if (requireNamespace("sf")) {
  # 1. load a shape file of counties in North Carolina
  nc <- sf::st_read(system.file("shape/nc.shp", package="sf"))
  # 2. generate some data according to a model with a few regression
  # effects, as well as spatial random effects
  gd <- generate_data(
    ~ reg(~ AREA + BIR74, prior=pr_normal(precision=1), name="beta") +
      gen(factor = ~ spatial(NAME, poly.df=nc), name="vs"),
    sigma.mod = pr_invchisq(df=10, scale=1),
    data = nc
  )
  # add the generated target variable and the spatial random effects to the
  # spatial dataframe object
  nc$y <- gd$y
  nc$vs_true <- gd$pars$vs
  # 3. fit a model to the generated data, and see to what extent the
  #    parameters used to generate the data, gd$pars, are reproduced
  sampler <- create_sampler(
    y ~ reg(~ AREA + BIR74, prior=pr_normal(precision=1), name="beta") +
    gen(factor = ~ spatial(NAME, poly.df=nc), name="vs"),
    block=TRUE, data=nc
  )
  sim <- MCMCsim(sampler, store.all=TRUE, n.iter=600, n.chain=2, verbose=FALSE)
  (summ <- summary(sim))
  nc$vs <- summ$vs[, "Mean"]
  plot(nc[c("vs_true", "vs")])
  plot(gd$pars$vs, summ$vs[, "Mean"]); abline(0, 1, col="red")
}

mcmcsae documentation built on Oct. 11, 2023, 1:06 a.m.