Usage
spate.mcmc(y,coord=NULL,lengthx=NULL,lengthy=NULL,Sind=NULL,n=NULL, IncidenceMat=FALSE,x=NULL,SV=c(rho0=0.2,sigma2=0.1, zeta=0.25,rho1=0.2,gamma=1,alpha=0.3,muX=0,muY=0,tau2=0.005), betaSV=rep(0,dim(x)[1]),RWCov=NULL,parh=NULL,tPred=NULL, sPred=NULL,P.rho0=Prho0,P.sigma2=Psigma2,P.zeta=Pzeta,P.rho1=Prho1, P.gamma=Pgamma,P.alpha=Palpha,P.mux=Pmux,P.muy=Pmuy,P.tau2=Ptau2, lambdaSV=1,sdlambda=0.01,P.lambda=Plambda,DataModel="Normal", DimRed=FALSE,NFour=NULL,indEst=1:9,Nmc=10000,BurnIn =1000, path=NULL,file=NULL,SaveToFile=FALSE,PlotToFile=FALSE, FixEffMetrop=TRUE,saveProcess=FALSE,Nsave=200,seed=NULL, Padding=FALSE,adaptive=TRUE,NCovEst=500,BurnInCovEst=500, MultCov=0.5,printRWCov=FALSE,MultStdDevLambda=0.75, Separable=FALSE,Drift=!Separable,Diffusion=!Separable, logInd=c(1,2,3,4,5,9),nu=1,plotTrace=TRUE, plotHist=FALSE,plotPairs=FALSE,trueVal=NULL, plotObsLocations=FALSE,trace=TRUE,monitorProcess=FALSE, tProcess=NULL,sProcess=NULL)
Arguments
y
Observed data in an T x N matrix with columns and rows corresponding to
time and space (observations on a grid stacked into a vector),
respectively. By default, at each time point, the observations are
assumed to lie on a square grid with each axis scaled so that it has unit length.
coord
If specified, this needs to be a matrix of dimension N x 2 with coordinates of the N observation points. Observations in 'y' can either be on a square grid or not. If not, the coordinates of each observation point need to be specified in 'coord'. According to these coordinates, each observation location is then mapped to a grid cell. If 'coord' is not specified, the observations in 'y' are assumed to lie on a square grid with each axis scaled so that it has unit length.
lengthx
Use together with 'coord' to specify the length of the x-axis. This is
usefull if the observations lie in a rectangular area instead of a
square. The length needs to be at least as large as the largest
x-distance in 'coord.
lengthy
Use together with 'coord' to specify the length of the y-axis. This is
usefull if the observations lie in a rectangular area instead of a
square. The length needs to be at least as large as the largest
y-distance in 'coord.
Sind
Vector of indices of grid cells where observations are made, in case,
the observation are not made at every grid cell. Alternatively, the
coordinates of the observation locations can be specfied in 'coord'.
n
Number of point per axis of the square into which the points are
mapped. In total, the process is modeled on a grid of size n*n.
IncidenceMat
Logical; if 'TRUE' an incidence matrix relating the latent process to
observation locations is used. This is only recommended to use when the
observations are relatively low-dimensional and when the latent process
is modeled in a reduced dimensional space as well.
x
Covariates in an array of dimensions p x T X N, where p denotes the
number of covariates, T the number of time points, and N the number of
spatial points.
SV
Starting values for parameters. Parameters for the SPDE in the following
order: rho_0, sigma^2, zeta, rho_1, gamma, alpha, mu_x, mu_y, tau^2. rho_0 and sigma^2 are the range
and marginal variance of the Matern covariance funtion for the
innovation term epsilon. zeta is the damping parameter. rho_1, gamma,
and alpha parametrize the diffusion matrix with rho_1 being a range
parameter, gamma and alpha determining the amount and the direction,
respectively, of anisotropy. mu_x and mu_y are the two components of
the drift vector. tau^2 denotes the nugget effect or measurment error.
betaSV
Starting values for regression coefficients.
RWCov
Covariance matrix of the proposal distribution used in the random walk
Metropolis-Hastings step for the hyper-parameters.
parh
Only used in prediction mode. If 'parh'
is not 'NULL', this indicates that 'spate.mcmc' is used for making
predictions at locations (tPred,sPred) instead of applying the traditional MCMC
algorithm. In case 'parh' is not 'NULL', it is a Npar x Nsim
matrix containing Nsim samples from the posterior of the Npar
parameters. This argument is used by the wrapper function 'spate.predict'.
tPred
Time points where predictions are made.This needs to be a vector if
predictions are made at multiple times. For instance, if T is the number
of time points in the data 'y', then tPred=c(T+1, T+2) means that
predictions are made at time 'T+1' and 'T+2'. This argument is used by
the wrapper function 'spate.predict'.
sPred
Vector of indices of grid cells (positions of locations in the stacked
spatial vector) where predictions are made. This argument is used by
the wrapper function 'spate.predict'.
P.rho0
Function specifying the prior for rho0.
P.sigma2
Function specifying the prior for sigma2.
P.zeta
Function specifying the prior for zeta.
P.rho1
Function specifying the prior for rho1.
P.gamma
Function specifying the prior for gamma.
P.alpha
Function specifying the prior for alpha.
P.mux
Function specifying the prior for mux.
P.muy
Function specifying the prior for muy.
P.tau2
Function specifying the prior for tau2.
lambdaSV
Starting value for transformation parameter lambda in the Tobit model.
sdlambda
Standard deviation of the proposal distribution used in the random walk
Metropolis-Hastings step for lambda.
P.lambda
Function specifying the prior for lambda.
DataModel
Specifies the data model. "Normal" or "SkewTobit" are available options.
DimRed
Logical; if 'TRUE' dimension reduction is applied. This means that not
the full number (n*n) of Fourier functions is used but rather only a
reduced dimensional basis of dimension 'NFour'.
NFour
If 'DimRed' is 'TRUE', this specifies the number of Fourier functions.
indEst
A vector of numbers specifying which for which parameters the posterior
should be computed and which should be held fix (at their starting
value). If the corresponding to the index of rho_0, sigma^2, zeta,
rho_1, gamma, alpha, mu_x, mu_y, tau^2 is present in the vector, the
parameter will be estimated otherwise not. Default is indEst=1:9 which
means that one samples from the posterior for all parameters.
Nmc
Number of MCMC samples.
BurnIn
Length of the burn-in period.
path
Path, in case plots and / or the spateMCMC object should be save in a file.
file
File name, in case plots and / or the spateMCMC object should be save in a file.
SaveToFile
Indicates whether the spateMCMC object should be save in a file.
PlotToFile
Indicates whether the MCMC output analysis plots should be save in a file.
FixEffMetrop
The fixed effects, i.e., the regression coefficients, can either be
sampled in a Gibbs step or updated together with the hyperparameters in
the Metropolis-Hastings step. The latter is the default and recommended
option since correlations between fixed effects and the random process
can result in slow mixing.
saveProcess
Logical; if 'TRUE' samples from the posterior of the latent
spatio-temporal process xi are saved.
Nsave
Number of samples from the posterior of the latent
spatio-temporal process xi that should be save.
seed
Seed for random generator.
Padding
Indicates whether padding is applied or not. If the range parameters are
large relative to the domain, this is recommended since otherwise
spurious periodicity can occur.
adaptive
Indicates whether an adaptive Metropolis-Hastings algorithm is used or
not. If yes, the proposal covariance matrix 'RWCov' is adaptively
estimated during the algorithm and tuning does not need to be done by hand.
NCovEst
Minimal number of samples to be used for estimating the proposal matrix.
BurnInCovEst
Burn-in period for estimating the proposal matrix.
MultCov
Numeric used as multiplier for the adaptively estimated
proposal cocariance matrix 'RWCov' of the hyper-parameters. I.e., the estimated covariance
matrix is multiplied by 'MultCov'.
printRWCov
Logical, if 'TRUE' the estimated
proposal cocariance matrix is printed each time.
MultStdDevLambda
Numeric used as multiplier for the adaptively
estimated proposal
standard deviation of the Tobit transformation parameter lambda. I.e.,
the estimated standard deviation is multiplied by 'MultStdDevLambda'.
Separable
Indicates whether a separable model, i.e., no transport / drift and no
diffusion, should be estimated.
Drift
Indicates whether a drift term should be included.
Diffusion
Indicates whether a diffusion term should be included.
logInd
Indicates which parameters are sampled on the log-scale. Default is
logInd=c(1, 2, 3, 4, 5, 9) corresponding to rho_0, sigma2, zeta,
rho_1, gamma, and tau^2.
nu
Smoothness parameter of the Matern covariance function for the innovations. By
default this equals 1 corresponding to the Whittle covariance function.
plotTrace
Indicates whether trace plots are made.
plotHist
Indicates whether histograms of the posterior distributions are made.
plotPairs
Indicates whether scatter plots of the hyper-parameters and the
regression coefficients are made.
trueVal
In simulations, true values can be supplied for comparison with the MCMC output.
plotObsLocations
Logical; if 'TRUE' the observations locations are ploted together with
the grid cells.
trace
Logical; if 'TRUE' tracing information on the progress of the MCMC algorithm is produced.
monitorProcess
Logical; if 'TRUE' in addition to the trace plots of the hyper-parameters, the mixing
properties of the latent process xi=Phi*alpha is monitored. This is
done by plotting the current sample of the process. More specifically,
the time series at locations 'sProcess' and the spatial fieldd at time points 'tProcess'.
tProcess
To be secified if 'monitorProcess=TRUE'. Time points at which
spatial fields of the sampled process should be plotted.
sProcess
To be secified if 'monitorProcess=TRUE'. Locations at which time
series of the sampled process should be plotted.