dma {dma}

Dynamic model averaging for continuous outcomes

Description

Implemtent dynamic model averaging for continuous outcomes as described in Raftery, A.E., Karny, M., and Ettler, P. (2010). Online Prediction Under Model Uncertainty Via Dynamic Model Averaging: Application to a Cold Rolling Mill. Technometrics 52:52-66. Along with the values described below, plot() creates a plot of the posterior model probabilities over time and model-averaged fitted values and print() returns model matrix and posterior model probabilities. There are TT time points, K models, and d total covariates.

Usage

dma(x, y, models.which, lambda=0.99, gamma=0.99, 
 eps=.001/nrow(models.which), delay=0, initialperiod=200)

Arguments

x

TTxd matrix of system inputs

y

TT-vector of system outputs

models.which

Kxd matrix, with 1 row per model and 1 col per variable indicating whether that variable is in the model (the state theta is of dim (model.dim+1); the extra 1 for the intercept)

lambda

parameter forgetting factor

gamma

flatterning parameter for model updating

eps

regularization parameter for regularizing posterior model model probabilities away from zero

delay

When y_t is controlled, only y_t-delay-1 and before are available. This is determined by the machine. Note that delay as defined here corresponds to (k-1) in the Ettler et al (2007, MixSim) paper. Thus k=25 in the paper corresponds to delay=24.

initialperiod

length of initial period. Performance is summarized with and without the first initialperiod samples.

Value

yhat.bymodel

TTxK matrix whose tk element gives yhat for yt for model k

yhat.ma

TT vector whose t element gives the model-averaged yhat for yt

pmp

TTxK matrix whose tk element is the post prob of model k at t

thetahat

KxTTx(nvar+1) array whose ktj element is the estimate of theta_j-1 for model k at t

Vtheta

KxTTx(nvar+1) array whose ktj element is the variance of theta_j-1 for model k at t

thetahat.ma

TTx(nvar+1) matrix whose tj element is the model-averaged estimate of theta_j-1 at t

Vtheta.ma

TTx(nvar+1) matrix whose tj element is the model-averaged variance of thetahat_j-1 at t

mse.bymodel

MSE for each model

mse.ma

MSE of model-averaged prediction

mseinitialperiod.bymodel

MSE for each model excluding the first initialperiod samples

mseinitialperiod.ma

MSE of model averaging excluding the first initialperiod samples

model.forget

forgetting factor for the model switching matrix

Author(s)

Adrian Raftery, Tyler H. McCormick

References

Raftery, A.E., Karny, M., and Ettler, P. (2010). Online Prediction Under Model Uncertainty Via Dynamic Model Averaging: Application to a Cold Rolling Mill. Technometrics 52:52-66.

Examples

#simulate some data to test
#first, static coefficients
coef<-c(1.8,3.4,-2,3,-2.8,3)
coefmat<-cbind(rep(coef[1],200),rep(coef[2],200),
            rep(coef[3],200),rep(coef[4],200),
            rep(coef[5],200),rep(coef[6],200))
#then, dynamic ones
coefmat<-cbind(coefmat,seq(1,2.45,length.out=nrow(coefmat)),
            seq(-.75,-2.75,length.out=nrow(coefmat)),
            c(rep(-1.5,nrow(coefmat)/2),rep(-.5,nrow(coefmat)/2)))
npar<-ncol(coefmat)-1
dat<-matrix(rnorm(200*(npar),0,1),200,(npar))
ydat<-rowSums((cbind(rep(1,nrow(dat)),dat))[1:100,]*coefmat[1:100,])
ydat<-c(ydat,rowSums((cbind(rep(1,nrow(dat)),dat)*coefmat)[-c(1:100),c(6:9)]))
mmat<-matrix(c(c(1,0,1,0,0,rep(1,(npar-7)),0,0),
            c(rep(0,(npar-4)),rep(1,4)),rep(1,npar)),3,npar,byrow=TRUE)
dma.test<-dma(dat,ydat,mmat,lambda=.99,gamma=.99,initialperiod=20)
plot(dma.test)