ABSTRACT
OBSERVA:TIO::-J SIGI'\AL + ::-JOISE (10.1) The corresponding equation that is more familiar to statisticians is \Vrittcn as
(10.3) where the (rn x rn) matrix G 1 is <-1Ssumed known, and w 1 denotes a (rn X 1) vector of deviations such that wT = ( Wr ,t, W:z,t, ... , Wm,t)·
1 0.1.1 The Ta.ndorn walk plus noise model
Xt = f.lt + Ht (10.4) where the unobservable local level l'·t. is assumed to follow a random walk given by
1 0.1.2 The linear· growth model
(10.7)
(10.10)
10.1. 7 Model building
(10.11)
1 ' ' et = X t - h t Bt lt-1
flt = fl t -l + CH:'t (10.16) where the smoothing constant o is a (complicated) fundion of the signal-tonoise ratio IJ.;j a~ (see Exercise 10.1). Equation (10.16) is. of course, simple
where
fttlt-1 ii.·t-1 + /Jt-1
P< = il111-1 + "'u el = r"/-1 + Sl-1 + ku e1 and
An intuitively reasonable \vay to initialize the bvo state variables from the first hvo observations is to take /f2 = X2 and J2 = X 2 - X 1 (see Exercise 10.2).
