widals (version 0.6.2)

H.als.b: Adaptive Least Squares

Description

Adaptive Least Squares expecially for large spacio-temporal data

Usage

H.als.b(Z, Hs, Ht, Hst.ls, rho, reg, b.lag = -1, Hs0 = NULL, Ht0 = NULL, Hst0.ls = NULL)

Value

A named list.

Z.hat

A \(\tau\) x n matrix, the ith row of which is the ALS prediction of the supporting sites at time i.

B

A \(\tau\) x \((p_s+p_t+p_st)\) matrix, the ith row of which is the ALS state (partial slopes) prediction at time i.

Z0.hat

A \(\tau\) x n* matrix, the ith row of which is the ALS prediction of the interpolation sites at time i.

inv.LHH

A \((p_s+p_t+p_st)\) x \((p_s+p_t+p_st)\) matrix. This is the (ALS predicted) covariate precision matrix at time \(\tau\).

ALS.g

The ALS gain at time \(\tau\).

Arguments

Z

Space-time data. A \(\tau\) x \(n\) numeric matrix.

Hs

Spacial covariates (of supporting sites). An \(n\) x \(p_s\) numeric matrix.

Ht

Temporal covariates (of supporting sites). A \(\tau\) x \(p_t\) numeric matrix.

Hst.ls

Space-time covariates (of supporting sites). A list of length \(\tau\), each element should be a \(n\) x \(p_st\) numeric matrix.

rho

ALS signal-to-noise ratio (SNR). A non-negative scalar.

reg

ALS regularizer. A non-negative scalar.

b.lag

ALS lag. A scalar integer, typically -1 (a-prior), or 0 (a-posteriori).

Hs0

Spacial covariates (of interpolation sites). An \(n\)* x \(p_s\) matrix, or NULL.

Ht0

Temporal covariates (of interpolation sites). A \(\tau\) x \(p_t\) matrix, or NULL. If not NULL, I cannot imagine a scenario where this shouldn't be Ht.

Hst0.ls

Space-time covariates (of interpolation sites). A list of length \(\tau\), each element should be a numeric \(n\) x \(p_st\) matrix.

Examples

set.seed(99999)

library(SSsimple)

tau <- 70
n.all <- 14

Hs.all <- matrix(rnorm(n.all), nrow=n.all)
Ht <- matrix(rnorm(tau*2), nrow=tau)
Hst.ls.all <- list()
for(i in 1:tau) { Hst.ls.all[[i]] <- matrix(rnorm(n.all*2), nrow=n.all) }

Hst.combined <- list()
for(i in 1:tau) { 
    Hst.combined[[i]] <- cbind( Hs.all, matrix(Ht[i, ], nrow=n.all, ncol=ncol(Ht), 
    byrow=TRUE), Hst.ls.all[[i]] ) 
}

######## use SSsimple to simulate
sssim.obj <- SS.sim.tv( 0.999, Hst.combined, 0.01, diag(1, n.all), tau )


ndx.support <- 1:10
ndx.interp <- 11:14

Z.all <- sssim.obj$Z
Z <- Z.all[ , ndx.support]
Z0 <- Z.all[ , ndx.interp]

Hst.ls <- subsetsites.Hst.ls(Hst.ls.all, ndx.support)
Hst0.ls <- subsetsites.Hst.ls(Hst.ls.all, ndx.interp)

Hs <- Hs.all[ ndx.support, , drop=FALSE]
Hs0 <- Hs.all[ ndx.interp, , drop=FALSE]

xrho <- 1/10
xreg <- 1/10
xALS <- H.als.b(Z=Z, Hs=Hs, Ht=Ht, Hst.ls=Hst.ls, rho=xrho, reg=xreg, b.lag=-1, 
Hs0=Hs0, Ht0=Ht, Hst0.ls=Hst0.ls) 



test.rng <- 20:tau

errs.sq <- (Z0 - xALS$Z0.hat)^2
sqrt( mean(errs.sq[test.rng, ]) )


################ calculate the 'effective standard errors' (actually 'effective prediction
################ errors') of the ALS partial slopes
rmse <- sqrt(mean((Z[test.rng, ] - xALS$Z.hat[test.rng, ])^2))
rmse
als.se <- rmse * sqrt(xALS$ALS.g) * sqrt(diag(xALS$inv.LHH))
cbind(xALS$B[tau, ], als.se, xALS$B[tau, ]/als.se)