STARMAX Revisted. Spatio-Temporal Modeling for Biosurveillance

Transcription

1 Spatio-Temporal Modeling for Biosurveillance D.S. Stoffer University of Pittsburgh STARMAX Revisted Stoffer (1986). Estimation and Identification of Space-Time ARMAX Models with Missing Data JASA, (An approach to space-time modeling using a spatially constrained state space model.) 1

2 The Data: The Pennsylvania portion of CDCs National Influenza Surveillance Effort data set. These are weekly mortality reports for pneumonia and influenza in various locations, 1962 present. Scranton Allentown Pittsburgh Reading Philadelphia 2

3 3 Weeks Mortality Allentown Philadelphia Pittsburgh 7 Three Series:

4 Poisson models: The basic model for time series of counts is yt y t 1, yt 2,... Poisson(µt), where log µt = Γut + p φiyt i. i=1 Here, ut are covariates (or inputs). Some difficulties: not stationary except under restrictive conditions no obvious way to analyze multiple series interpretation difficult: E{yt past} = exp(γut) exp( p i=1 φ iyt i) including correlated errors is difficult (glarma). Davis, Dunsmuir & Wang (1999). Modelling Time Series of Count Data. Asymptotics, Nonparametrics, and Time Series, Marcel Dekker, Fahrmeir & Tutz (1994) Multivariate Statistical Modeling Based on Generalised Linear Models, Springer-Verlag. Zeger (1988). A regression model for time series of counts. Biometrika,

5 It would be difficult to model the original data (even under normality) without some transformation, which isn t allowed in Poisson models, because it would destroy the Poisonness. The ACF for Pittsburgh (local trends, long memory, persistent seasonality): 5

6 An approach is to transform the data. For example, for Pittsburgh, let z t be the original observations. Let zt = z t + 1 (variance stabilizing transformation), and finally, consider the weekly changes: yt = zt zt 1 Pittsburgh Transformed Pittsburgh Transformed and Standardized

7 The ACF and PACF of yt suggest a simple MA(1) model: yt = wt θwt 1 where wt is white noise (or perhaps ARCH). Similary, this is an IMA(1,1) for the transformed series; that is, zt = zt 1 + wt θwt 1. (recall yt = zt zt 1 where z = data + 1 ) 7

8 8 Recall zt = zt 1 + wt θwt 1 where z = data + 1. Weeks Mortality Pittsburgh Predicted Pred + 3 SEs 6 A simple estimate yields θ =.6 and the fitted model (to the actual data) is:

9 How about Allentown? 4 Allentown Transformed Weeks 9

10 1 Weeks Mortality Allentown Predicted Pred + 3 SEs 16 Allentown? Same model as Pittsburgh with θ =.8. Not bad!

11 Toward a more general (spatial) model: A model that fits the data better is an ARMA(1, 1) for yt, that is, yt = φyt 1 + wt θwt 1 where yt = zt zt 1 and z = data + 1. This model can be written in state-space form : xt+1 = φxt + (φ θ)wt yt = xt + wt xt: state equation (unobserved - factor) yt: observation equation (observed) wt: white noise (unobserved - error). y t φyt 1 = xt + wt φ(xt 1 + wt 1) = [(φ θ)wt 1] + wt φwt 1 11

12 The General State-Space Model Notation: xt+1 = Φxt + Υut + Gwt t =, 1,..., n y t = Atxt + Γut + vt t = 1,..., n xt: p-dimensional state vector y t : q-dimensional observation vector ut: r-dimensional fixed input vector var(wt) = Q, var(vt) = R, cov(wt, vt) = S Model uniquely parameterized by Θ (k-dimensional): Φ = Φ(Θ), Υ = Υ(Θ), G = G(Θ), Q = Q(Θ), At = At(Θ), Γ = Γ(Θ), R = R(Θ), S = S(Θ). Shumway & Stoffer (2, Ch 4). Time Series Analysis and Its Applications. New York: Springer. 12

13 The Kalman filter yields Prediction: y t 1 t = BLP{y t yt 1,..., y 1 ; Θ} Innovations: ɛt(θ) = y t y t 1 t ; var{ɛt(θ)} = Σt(Θ) Estimation of Θ: The innovations form of the Gaussian likelihood (ignoring a constant) is ln LY (Θ) = 1 2 n { ln Σ t(θ) + ɛt(θ) Σt(Θ) 1 ɛt(θ) } t=1 where LY (Θ) denotes the likelihood of Θ given the data y 1,..., y n assuming normality. Quasi-GML via Newton-Raphson: Θ = argmax Θ LY (Θ) Notes: Can use a mixture of normal if the data are markedly non-normal. Can include stochastic volatility. cusum possible on standardized innovations Σ 1/2 t ɛt. 13

14 A Model for an Individual Location (e.g. Pittsburgh): xt+1 = φxt + (φ θ)wt, t =, 1,..., n yt = xt + Γut + wt t = 1,..., n. yt = zt zt 1 is a univariate process ( z = data could also try yt = zt zt 52), Γ is a 1 r vector of (constrained/unconstrained) regression parameters, ut is an r 1 vector of inputs, including mortality rates from nearby locations at various time lags (contemporaneous values included). Here, Θ = (φ, θ, γ1,..., γr, σ 2 w). 14

15 15 Weeks Mortality Pittsburgh Predicted Pred + 3 SEs 6 And φ =.1 (.5), θ =.7 (.2), σw =.7.3 alt 2,.1 pht 2,.6 rdt 2,.3 sct 2) [largest SE =.3] Γut = (.4 alt,.4 pht,.9 rdt,.11 sct,.7 alt 1,.2 pht 1,.1 rdt 1,.2 sct 1, The previous model was fit to the Pittsburgh data.

16 The Innovations (residuals): Histogram of the Residuals Empirical Distribution of the Standardized Residuals Normal cdf Empirical df

17 17 Weeks Mortality Philadelphia Predicted Pred + 3 SEs 9 Similar model for Philadelphia:

18 STARMAX: A Spatially Constrained Multivariate Approach: xt+1 = DΦxt + D(Φ Θ)wt t =, 1,..., n y t = xt + Γut + wt t = 1,..., n xt is the p-dimensional state vector y t is the p-dimensional observation vector ut is the r-dimensional vector of exogenous variables wt is the p-dimensional noise vector Φ and Θ are diagonal p p parameter matrices Γ is a p r matrix of regression parameters D is a p p matrix of specified spatial constraints This state-space model implies y t = DΦy t 1 + Γut + wt DΘwt 1 18

19 The model is easily generalized to arbitrary orders and spatial constraints. For example: [ ] [ ] D 1Φ1 D2Φ2 D 1(Φ1 Θ1) xt+1 = I xt + D2Φ2 wt y t = [ I, ] xt + Γut + wt yields the STARMAX(2, 1) model: y t = D1Φ1y t 1 + D2Φ2y t 2 + Γut + wt D1Θ1wt 1 where D1 and D2 are first order and second order spatial constraint matrices and Φ1, Φ2, Θ1 are diagonal matrices, as before. The exogenous variables [inputs] are ut. Note: xt is 2p 1 and y t is p 1 19

20 Pittsburgh 6 5 Pittsburgh Predicted Pred + 3 SEs

21 Philadelphia 9 8 Philadelphia Predicted Pred + 3 SEs

22 Allentown Allentown Predicted Pred + 3 SEs

23 D Matrix [Dij = ĉorr(y t, y t 1 )]: Alln(t-1) Phil(t-1) Pitt(t-1) Read(t-1) Scrn(t-1) Alln(t) Phil(t) Pitt(t) Read(t) Scrn(t) Estimates & Errors: original data units phi se theta se rmspe stdev improvement Allentown % Philadelphia % Pittsburgh % Reading % Scranton % 23