Bayesian Dynamic Linear Modelling for. Complex Computer Models

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1 Bayesian Dynamic Linear Modelling for Complex Computer Models Fei Liu, Liang Zhang, Mike West Abstract Computer models may have functional outputs. With no loss of generality, we assume that a single computer run is generating a function of time. For complex computer models, Bayarri et al considers the time as a computer model associated input parameter, and uses the Gaussian Response Surface Approximation method GaSP with the Kronecker product correlation matrix in the augumented space. However, this approach is only applicable when there are only a few time points. In this paper, we consider the Bayesian Dynamic Linear Model West and Harrison 1997 as an alternative approach when there are many time points. Our method also allows the forecasting for the future. Keywords: Computer model; Bayesian Dynamic Linear Model; Gaussian stochastic process; Bayesian analysis; Forwarding filtering and backward sampling; MCMC. 1 Introduction The computer models can be represented as deterministic functions of the associated parameters. There are generally two types of parameters: a calibration parameters u are 1

2 only associated with the computer codes. They may be uncertain physical properties. b unknown parameters x are associated with both the computer models and the field experiments. They are characteristics associated with the real experiments. For simplicity, we use x to represent x, u. As a result, we can represent the computer model as a function of x, y x. On the other hand, exsercising the code is very time consuming for complex computer models. Consequently, the function y x is only evaluated at selected locations x i, i = 1,..., n. In this paper, we focus on the computer models with the functional outputs. We assume that the computer model outputs are functions of time t, t = 1,..., T. We represent such computer model output as y x, t. This type of computer models has been studied both in Bayarri et al and Bayarri et al The SAVE model in Bayarri et al uses the Gaussian Response Surface Approximation method GaSP on the augmented space of x, t by assuming seperable correlation in the space of x and t. They assume that the computer model outputs are realizations from a Gaussian Stochastic Process defined on the x, t space, i.e., 1 y, GP µ, Corr,,, λm where Corr y x, t, y x, t = exp βi x i x i α i exp βt t t α t. We use y x to represent the functional output of a single computer run whose inputs is x, y x t = y x, t j, j = 1,..., T. The likelihood in SAVE is represented as, y x 1. N y x n µ 1, 1 λ Σ M 1 Σ 2 1 where Σ 1 k,l = exp β i x ki x li α i and Σ2 k,l = exp β t t k t l α t. 2

3 To implement the SAVE model, one needs to invert the matrices Σ 1 and Σ 2, where Σ 1 is a n by n matrix, and Σ 2 is T by T. In the context of complex computer models, inverting Σ 1 is feasible because n is generally small. But however, the dimension of Σ 2 may be too huge to invert. Bayarri et al uses basis expansion method SAVE2, i.e., y x, = I w i x φ i i=1 where {φ i } is a basis library, and they use a wavelet for their application. Then, they 1 model the coefficients as independent spatial processes, w i GP µ i, Corr λ M i,. i SAVE2 can give predictions with confidence bounds for the computer model output at any values of x by spatial interpolation. However, it can only handle the computer models with fixed time grids t = 1,..., T. Some applications of the computer model may require forecasting for the future, weather forecasting models for instance. In this paper, we will discuss modelling the computer model code by Dynamic Linear Models DLM, as to capture the temporal structures in the data. The paper is organized as follows. We will first introduce our DLM model and make connections with the SAVE model in section 2. In section 3, we will give the likelihood and specify the prior distributions for the unknown parameters associated with the DLM model. Section 4 discusses the MCMC method to get draws from the posterior distributions of the unknown quantities, and also gives spatial interpolation for the computer model at arbitray locations in the x space. The method will be applied on an example data set in section 5. 2 the DLM for the computer model outputs For a single computer model run at x, we use the time varying autoregressive model TVAR West and Harrison, 1997 to model its temporal structure. 3

4 y x, t = p φ t,j yx, t j + ɛ t x 2 j The computer model runs are correlated by assuming a Gaussian stochastic processes for the evolutions ɛ t x in equation 2, i.e., ɛ t GP0, v t Corr t, 3 where, we are assuming that Corr t, = Corr, is the same for all t. And we use seperable power exponential function for the evolution correlation, i.e., Corrx, x = exp i β i x i x i α i The model in equation 2 can be connected with the SAVE model given in equation 1, in an approximation sense. Consider the likelihood for the SAVE model in equation 1. Let y t = y x 1, t,..., y x n, t. We represent the likelihood in equation 1 by the product of conditional likelihoods, L y T, y T 1,..., y 1 Θ = p+1 L y i y i 1,..., y 1, Θ L y p, y p 1,..., y 1 Θ 4 i=t Next, at any time t, we approximate the conditional likelihood as, L y t y t 1,..., y 1, Θ L y t y t 1,..., y t p, Θ 5 Let ρk, l = exp β t k l α t, ρt,t 1:t p = ρt, t 1,..., ρt, t p, Σ2 ρk, l, k, l = t 1,..., t p. The conditional likelihoods in equation 5 are multivariate normals with mean vectors, k,l = 4

5 E y t y t 1,,..., y t p, Θ = = ρt,t 1:t p Σ2 1 Σ 1 Σ 1 Σ2 ρt,t 1:t p 1 I n n y t 1. y t p y t 1. y t p This implies the auto-regressive term in equation 2, y M x, t = Σ2 y M x, t 1 1 ρ t,t 1:t p. y M x, t p We assume that Corr t, = Corr, in equation 3 because the covariance matrices of the conditional likelihoods L y t y t 1,..., y t p, Θ is time-independent. To see this, we representat Cov y t y t 1,..., y t p, Θ as, Cov y t y t 1,..., y t p, Θ = 1 λ M = 1 λ M Σ 1 ρ t,t 1:t p 1 Σ 1 Σ2 Σ 1 ρ t,t 1:t p Σ 1 Σ2 1 ρ t,t 1:t p 1 ρt,t 1:t p Σ 1 Finally, realizing that the functinal outputs of the computer models are usually temporally inhomogenous, we adapt our model to such inhomogenienty by allowing time-varying autoregressive coefficients and time-varying variances of the innovations in equation 2. 5

6 3 Likelihood and the Prior Distributions 3.1 The Multivariate DLM representation We can represent the likelihood in the matrix form, i.e., yx 1, t yx 1, t 1 yx 1, t 2... yx 1, t p yx 2, t yx 2, t 1 yx 2, t 2... yx 2, t p = yx n, t yx n, t 1 yx n, t 2... yx n, t p φ t,1 φ t,2. φ t,p ɛ t x 1 ɛ t x 2 +. ɛ t x n 6 And we model the TVAR coefficients Φ t = φ t,1 φ t,2. φ t,p as, Φ t = Φ t 1 + w t where w t N0, W t. Let G t be the identity matrix of size p, V t = v t Σ 1, and F t = yx 1, t 1 yx 1, t 2... yx 1, t p yx 2, t 1 yx 2, t 2... yx 2, t p yx n, t 1 yx n, t 2... yx n, t p We can represent the likelihood in the way of Multivariate DLM West and Harrison, 1997, {F t, G t, V t, W t } T t=1 6

7 3.2 The Prior distributions Let D t be the data up to time t. We sequentially specify the prior distribtions for W t and V t by two discounting factors δ 1, δ 2. For W t, we assume, v 1 t D t 1 Gδ 1 n t 1 /2, δ 1 d t 1 /2 W t D t 1 = 1 δ 2 C t 1 /δ 2, C t 1 = CovΦ t 1 D t 1 where C t 1 = Cov Φ t 1 D t 1 and will be specified recursively in section A. The values for n 0, d 0, C 0 will be prespecified. Finally, for the spatial parameters α = {α i } and β = {β i }, we use the Jeffereys rule prior π α, β discussed in Berger et al and Paulo π α, β I α, β 1/2 trσ 1 1 Σ 1 2 where I α, β is the Fisher information matrix, and Σ 1 = Σ 1 α,β. 4 MCMC method for the Multivariate DLM We use the Monte Carlo Markov Chain method MCMC to draw samples from the posterior distributions, π {v 1,... v T }; {Φ 1,..., Φ T }; {α, β} D T. We first give the algorithm as follows. At the i th iteration, 1. Sample algorithm. {α i, β i } D T, {v i 1 1,..., v i 1 T }, {Φ i 1 1,..., Φ i 1 T } by the Metroplis-Hastings 7

8 2. Sample {v i 1,..., v i 2.1 Sample T }, {Φi 1,..., Φ i T } D T, {α i, β i } as, {v i 1,... v i T } D T, {α i, β i }. This will be discussed in section Sample {Φ i 1,..., Φ i T } D T, {v i 1,... v i T }, {αi, β i } as in section Sampling the variances We give the algorithm to update the variances {v i 1,... v i T } D T, {α i, β i }. 1. Do the Forward filtering assuming {v 1,..., v T } unknown, as discussed in the appendix B. 2. Sample v 1 i T DT, {α i, β } i G n T /2, d T /2. 3. Sample v t, t = T 1,..., 1 recursively as, v 1 t = δ 1 v 1 t+1 + G 1 δ 1 n t /2, d t /2 4.2 Sampling the TVAR coefficients Below is the algorithm to make draws from π {Φ 1,..., Φ T } D T, {v 1,... v T }, {α, β}. 1. Do the Forward filtering conditional on {v 1,..., v T }. This will be discussed in the appendix A. 2. Sample Φ T D T, {v 1,..., v T } MVN m T, C T. 8

9 3. Sample Φ t, t = T 1,..., 1 recursively from, Φ t D T, Φ t+1, {v 1,..., v T } MVN 1 δ 2 m t + δ 2 Φ t+1, 1 δ 2 C t 4.3 Spatial interpolation We predict the output of a computer model at a new input value by spatial interpolation. Suppose x is the new unexsercised input value. Let e t x i = y t x i j y t jx i φ t,j and ρ x x, x 1:n = Corrx, x 1,..., Corrx, x n, we have, y t x {y t 1 x,... y t p x}, Data, {v 1,..., v T }, {α, β} N µ t x, σ 2 t x where, µ t x = j y t j xφ t,j + v 1 t e t x 1 ρ x x, x 1:n Σ 1 e t x 2 1. e t x n and, σ 2 t x = v t 1 ρ x x, x 1:n Σ 1 1 ρ x x, x 1:n As all the computer model emulators do, the DLM modelling approach gives back the computer model output, when we are trying to make predictions for the exsercised computer input values. In other words, if x {x 1,..., x n }, we have µ t x = y t x and σ 2 t x = 0. 9

10 5 An example 5.1 The data Figure 1 gives an example of the functional outputs of computer models. Each time series is associated with an x value located to the left of the series. The x values are considered as the computer model inputs. The data with x = 0.5 in red is obtained from some real physical experiment. This data is observed at T = 3000 time points. We use y t 0.5 = y t 0.5, t = 1,..., T to represent it. Given y t 0.5 and its TVAR 20 fit {φ t,j, v t }, we simulate the data for x = 0.25,..., 0.75 by fixing α = 2, β = 1.6. The details are discussed in Appendix C. Figure 1: The simulated computer model data at various input values 10

11 5.2 MCMC Results In section 4, we can perfectly sample {v i 1,..., v i T }, {Φi This implies that, we do not need to update {v i 1,..., v i In particular, we update {v i 1,..., v i T }, {Φi 1,..., Φ i T } conditional on {αi, β i }. T }, {Φi 1,..., Φ i } in every iteration. 1,..., Φ i } after every 200 iterations of sampling {α i, β i } by the Metroplis-Hastings algorithm. We fix {α i } at 2 for the example data set. For the other unknowns, starting the MCMC from true parameter values, we obtained N = 2000 samples, among which the first 1000 are treated as burnin samples and will be discarded in all the posterior inferences. Figure 2 gives the trace plot, prior distribution up to a normalizing constant, posterior distribution, autocorrelation function for β. For the purpose of making comparison between the prior and the posterior distribution for β, we highlight with red line the prior distribution in the interval 1, 2, within which the posterior draws are concentrated. { } Suppose is the i th MCMC draw for the TVAR coefficients {φ t,j }, where i = 1,..., N, φ i t,j t = 1,..., T, and j = 1,..., 20. We calculate the posterior mean for φ t,j, ˆφ t,j by, T T ˆφ t,j = 1 N And the point-wise posterior means of the TVAR coefficients are shown in the left panel of figure 3. The right panel shows {ˆv t, t = 1,..., T }, the point-wise posterior means of {v t }. i φ i t,j 5.3 Spatial interpolation One direct application of the multivariate DLM, as we discussed in section 4.3, is to get the prediction for the computer model at input other than the design points. In figure 4, we give our prediction for the dynamic computer model outputs at input value x = 0.5. We also make comparison between the true outputs and our prediction at the time intervals 11

12 Figure 2: Upper-left: trace plot of the MCMC samples for β; Upper-Right: autocorrelation functions of the MCMC samples for β; Lower-Left: posterior distribution of β; Lower-Right: prior density of β. 1100, 1300 and 2700, 2900, where the data is exhibiting interesting features. 5.4 Wave and modular decomposition We can decompose the process {yt} as 12

13 Figure 3: Left: posterior means for the TVAR coefficients {φ t,j }; Right: posterior means for the time varying variances {v t } Figure 4: Posterior predictive curve green, true computer model output red, and 90% piece-wise predictive intervals for spatial interpolation with input value x = 0.5. y t = c z t,l + l=1 r l=1 x t,l 13

14 where the latent processes {z t,l } are TVAR s with lag 1 and x t,l are stochastically timevarying damped harmonic components, each of which is associated with the modulars damping parameters {a t,l } and the wavelengths periods {λ t,l } West and Harrison, Such decomposition can help to understand the physics meanings of the computer model outputs. In Figure 5, we show the decompositions for the posterior mean of the process {y t 0.5}. In Figure 6, we show the modulars and the wavelengths of the first 5 components, as a function of t. Figure 5: The true computer model output data {y t 0.5}bottom, posterior mean for {y t 0.5}second to the bottom, and decomposition of the posterior mean the rest curves are the first to the third components from bottom to the top. A Forward filtering with known variances We briefly review the forward filtering algorithm with known variances for multivariate DLM. For more details, refer to the Chapter 16 in West and Harrison With m 0, C 0, 14

15 Figure 6: Left: wave decompositions; Right: modular decompositions a. Posterior at t 1: Φ t 1 D t 1 Nm t 1, C t 1 b. Prior at t: Φ t D t 1 Na t, R t, with, a t = m t 1, R t = C t 1 /δ 2 c. One-step forecast: y t D t 1 Nf t, Q t, with, f t = F t a t = F t m t 1 ; Q t = F t C t 1 F t /δ 2 + v t d. Poserior at t: Φ t D t Nm t, C t with, m t = a t + A t e t and C t = R t A t Q t A t where, 15

16 A t = R t F t Q 1 t and e t = Y t f t B Forward filtering with unknown variances We first describe the forward filtering algorithm with unknown variances for multivariate DLM. With m 0, C 0, s 0, n 0, a. Posterior at t 1: Φ t 1 D t 1 Nm t 1, C t 1 b. Prior at t: Φ t D t 1 Na t, R t, with, a t = m t 1, R t = C t 1 /δ 2 c. One-step forecast: y t D t 1 Nf t, Q t, with, f t = F t a t = F t m t 1 ; Q t = F t C t 1 F t /δ 2 + s t 1 Σ 1 d. Posterior at t: Φ t D t T nt m t, C t, and, V 1 t D t Gn t /2, d t /2, with, A t = R t F t Q 1 t = C t 1 F t Q 1 t /δ 2 where m t = m t 1 + A t e t, e t = y t F t m t 1, and C t = st Ct 1 s t 1 δ 2 A t Q t A t, and, n t = δ 1 n t 1 + n; d t = δ 1 d t 1 + s t 1 e t tq 1 t e t 7 16

17 Now, we derive the relationship in equation 7. At time t, the prior for vt 1 is, v 1 t D t 1 Gδ1 n t 1 /2, δ 1 d t 1 /2 The likelihood, et D t 1, vt 1 N0, Qt Therefore, the posterior distribution for v 1 t is, πv 1 t D t 1 v t Q t exp s t 1 1/2 v t ɛ t tq 1 t ɛ t vt 1 δ 1n t 1 /2 1 exp δ 1 d t 1 vt 1 /2 This implies that, v 1 t D t G n + δ 1 n t 1 /2, δ 1 d t 1 + s t 1 ɛ t tq 1 t ɛ t /2 In other words, n t = δ 1 n t 1 + n; d t = δ 1 d t 1 + s t 1 e t tq 1 t e t ; s t = d t /n t 17

18 C Data simulation Suppose we have a functional data y t x, t = 1,..., T. Given α, β, {Φ t }, {v t }, we simulate the data as, y t x 1 µ t x 1 y t x 2 µ t x 2 {y t 1 x i }, {Φ t }, y t x MVN µ... t =, v t Σ... y t x n µ t x n with, µ t x i = j φ t,j Y t j x i + v t 1 ρ x x, x 1:n y t x j φ t,j y t j x and, Σ = Σ 1 ρ x x; x 1:n ρ x x; x 1:n Σ 1 is n n matrix with the i, j element Σ 1 i,j = Corrx i, x j. And ρ x x; x 1:n is an n by 1 vector with the i th element ρ x x; x 1:n i = Corrx, x i. 18

19 References Bayarri, M., Berger, J., Garcia-Donato, G., Liu, F., Palomo, J., Paulo, R., Sacks, J., Walsh, D., Cafeo, J., and Parthasarathy, R Computer model validation with functional outputs. Niss tech. report. Bayarri, M., Berger, J., Higdon, D., Kennedy, M., Kottas, A., Paulo, R., Sacks, J., Cafeo, J., Cavendish, J., Lin, C., and Tu, J A framework for validation of computer models. In D. Pace and S. S. Eds., eds., Proceedings of the Workshop on Foundations for V&V in the 21st Century. Society for Modeling and Simulation International. Berger, J., Oliveira, V. D., and Sanso, B Objective bayesian analysis of spatially correlated data. JASA Paulo, R Default priors for gaussian processes. Annals West, M. and Harrison, P Bayesian Forecasting and Dynamic Models. Springer, New York, USA. 19