staircase.EM: Estimate gauged sites hyperparameters

Description Usage Arguments Details Value See Also

View source: R/LZ-EM.est.R

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

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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)

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:

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.