Bayesian Inference in Cyclical Component Dynamic Linear Models

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1302 Journal of the American Statistical Association, December 1995 45 - sov 40- .50 100 .150 35 - 30 - 25- 20- -200 0 20 40 60 80 100 time Figure 1. 1 10 Observations on an EEG Trace. Figure 2. 5 10 15 wavelength Smoothed Periodogram for the Detrended EEG Time Series. In addition to the specific focus on cyclical component models, this article develops, for the first time, simulation analyses for state evolution matrix parameters and variance components in DLM's. This builds on and extends recent work on Gibbs sampling and related methods for posterior distributions for state vectors themselves (Carter and Kohn 1993; Fruhwirth-Schnatter 1994) and provides the basis for wider application of state-space models with uncertain defining evolution and variance parameters, in some generality. 2. DYNAMIC LINEAR MODEL AUTOREGRESSIONS 2.1 Basic Model Structure Using Cyclical Autoregressive Component Dynamic Linear Models Of the several possible representations of time-varying cycles, the simple cyclical autoregressive form (West and Harrison 1989, sec. 9.4) is the focus here, for reasons that will become apparent. Consider each of the latent model components xt,j in (1). The basic representation of interest is = a,jcos(27rt/Aj + b,ij), where Aj = 27r/acos(C3/2), or Ij = 2cos(27r/Aj) = 2cos(aj), and the amplitude al,j and phase bl,j are constants determined by the starting value zij, and /3j. Thus we have a model component defining a cyclical process of constant period Aj. Over time, the innovations w,j impact the process, inducing stochastic time variation in the amplitude and phase characteristics, with such variations related directly to the innovations variance wj (Priestley 1981, sec. 3.5.3; West and Harrison 1989, p. 253). Thus (1) and (2) combine to give the overall model 10 - Yt =Xt,0 + E xt,j + vt j=1 k Xtj = jXt1 -,j - Xt-2,j + Wtj (2) where /j is a model parameter to be discussed later and w,j is an innovation term at time t. As usual in dynamic linear modeling, the w,j, (t = 1, 2, .. .), are assumed to be independent and mutually independent sequences, with Wt,j -N(tj w IO, jw). This model for xt,j is a standard AR(2) form with a statespace representation (West and Harrison 1989, sec. 9.4), Xt,j = (1, O)zt,j and Zt, = Gjztl,3 + (t,j, 0)' where Zt, = (xt,j, Xti,j)' for all t. Conditional on (an arbitrary starting value) z1,j, the implied forecast function for the future of the process is E(x+,;Iz,i) = (l,O)Gt-1z, , and this has a form, as a function of t, determined by the eigenstructure of Gj using standard theory (West and Harrison 1989, chap. 5). It easily follows that E(xt,j|Zi,;) 8 s 0) 0J 0 5 10 15 20 wavelength Figure 3. Bayesian "Periodogram" for the EEG Series, Based on the Simple Harmonic Regression Model with a Single Cycle, Following Bretthorst (1988). The plotted curve is the log-likelihood function, equivalently the reference posterior density under a uniform prior, for the single wavelength in such a model.
West: Inference on Cycles in Time Series 1303 and Xt,; =3jXt-1,j -Xt-2,j + Wtj (j 1, , k), (3) 100 - so - ++ + with /3 = 2 cos(27r/Aj) for j = ,... , k. In addition to the earlier assumptions about the innovations process, it is further assumed that the observational errors form an independent sequence and are independent of the Wt,j, with Vt N(vtIO,v). It remains to model the acyclic trend process xt,0. To take a simple and specific model, though one of much utility and used in an illustration that follows, we suppose here a first-order polynomial. Other DLM forms are possible, of course, but the first-order model provides for simple but flexible local smoothing of underlying, acyclical, and nonstationary trends in the absence of more informed trend models such as regression DLM's. Then (following West and Harrison 1989, chap. 2), Xt,O = Xt-1,0 + W'tO, where wt,O N(w)t,o 0, wo) independently over time and independently of the innovations wtj in (3). This trend model combines with (3) to give the general DLM representation and Yt = F'zt + Vt with the following ingredients: Zt = Gzt_l + 6t, (4) Xt,O 1 Xt,1 Xt,21 Zt= St12 F= 0 Xt,k 1 Xt-l,k 0 /1 :1 -1 1 0 /32 1 G= 1 0 50 + + + -200 + + ++ l 40 o 2 6 >0 ~~~~~~ + Lmt + +~~~ 4k Here - is+ b so t + -200 ~ ~ ~ ~ + + + 0 20 40 60 80 100 -100 vaiac mari W igw;w ;W,0 N . *** ) time Figure 4. Estimated Acyctlc Trend, With One Standard Deviation Limits, for the EEG Series. Here G is block-diagonal, so the "empty" entries in (5) are all zero. Note further that 6t -N (6t 0O, W) with diagonal variance matrix W = diag(wo; wl, 0; W2, o; ... ; Wk, 0). 2.2 Posterior Structure and Framework of Simulation Analysis Assume the model (4) for the series of n observations {Yt; t = 1, ... , n}. Inference is required on the wavelengths A, equivalently the state parameters 3 = {i3,- ... , !k }, the variance components v and W, the individual "latent" stochastic subcycles Xj d {xt,j; t = O.... ,n} for each j = 1,...,k, and the trend component Xo df {Xt,o; t = O,... ,}. In standard notation, define Dt = {yt, Dt-i} for the information set available at time t, having observed the time series value at that time; at t = 0, Do represents all initial information including the model specification and assumptions. Assume the analysis to be closed to external inputs and interventions, so that Dt is a sufficient summary of the history at time t. Hence analysis involves specifying 100 50 - 0 4 Wt ,O Wt, 1 0 wt,2 nt ,k /3k -1 1 0 t= 0' 5 0 -50 -150 -' -200 - ., , , , I 0 20 40 60 80 100 time Figure 5. Fitted Values for the EEG Series.
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Bayesian Inference in Cyclical Component Dynamic Linear Models

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