# staircase.EM: Estimate gauged sites hyperparameters In EnviroStat: Statistical Analysis of Environmental Space-Time Processes

## Description

Estimate \cal{H}_g hyperparameters of the gauged sites using the EM algorithm, using the staircase of the missing data to determine the default block structure.

## Usage

 1 2 3 staircase.EM(data, p = 1, block = NULL, covariate = NULL, B0 = NULL, init = NULL, a = 2, r = 0.5, verbose = FALSE, maxit = 20, tol = 1e-06) 

## Arguments

 data data matrix, grouped by blocks each with stations having the same number of missing observations. The blocks are organized in order of decreasing number of missing observations, ie. block 1 has more missing observations than block2. Default structure: Each column represent data from a station; rows are for time Blocks are decided based on the number of missing observations p number of pollutants measured at each stations. (first p columns of y are for p pollutants from station 1, block 1). block a vector indicating the number of stations in each block - from 1 to K covariate design matrix for covariates created with model.matrix with as.factor B0 Provided if the hyperparameter β_0 (B0) is known and not estimated init Initial values for the hyperparameters; output of this function can be used for that a When p=1, the type-II MLE's for delta's are not available. Delta's are assumed to follow a gamma distribution with parameters (a,r) r When p=1, the type-II MLE's for delta's are not available. Delta's are assumed to follow a gamma distribution with parameters (a,r) verbose flag for writing out the results at each iteration maxit the default maximum number of iterations tol the convergence level.

## Details

The estimated model is as follows:

• data \sim MVN ( z \times β , {\rm kronecker}(I, Σ) )

• β \sim MVN (β_0 , {\rm kronecker}(F^{-1} , Σ ) )

• Σ \sim GIW (Θ , δ )

Θ is a collection of hyperparameters including ξ_0, Ω, Λ, H^{-1}.

## Value

A list with following elements:

 Delta The estimated degrees freedom for each of the blocks (list) Omega The estimated covariance matrix between pollutants Lambda The estimated conditional covariance matrix between stations in each block given data at stations in higher blocks (less missing data) - (list) Xi0 The estimated slopes of regression between stations in each blocks and those in higher blocks (list). Note that τ_{0i} = {\rm kronecker}(ξ_0, diag(p)) - same across stations for each pollutants. Beta0 Coefficients - assumed to be the same across stations for each pollutant Finv Scale associated with β_0 Hinv The estimated hyperparameters (list) - inverse of H_j Psi The estimated (marginal) covariance matrix between stations block From input data From input covariate From input Lambda.1K The inverse Bartlett decomposition (eqn 23?)

## See Also

staircase.hyper.est

EnviroStat documentation built on May 30, 2017, 5:38 a.m.