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

Description Usage Arguments Details Value See Also

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.

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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)

`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.

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}.

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?)