M. West and J. Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, 2nd edition, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Unfortunately, in many cases such explicit error models do not exist since it is impossible to predict all errors a user might make. Another common approach in dynamic systems is to monitor the residuals of observations, thereby testing the appropriateness of the underlying model assumptions [11, 166]. We propose an alternative, more capable approach to overcome the limitations of these methods. Our technique is based on online model selection, which aims at identifying the model that is best suited to explain the observed data [166, 125] The quality of a model is given by its predictive ....

....the appropriateness of the underlying model assumptions [11, 166] We propose an alternative, more capable approach to overcome the limitations of these methods. Our technique is based on online model selection, which aims at identifying the model that is best suited to explain the observed data [166, 125]. The quality of a model is given by its predictive performance, i.e. the likelihood of the observed data given the model. Multiple models can be compared using Bayes factors [55, 166] which then yield the ratio of posterior model probabilities. To apply model selection in our context, we will ....

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M. West and P.J. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York, 2nd edition, 1997.

....from the instantiations of a class are considered (virtual) cases of that class. 39] give both theoretical as well as empirical evidence that this learning method is superior to Note that this approach can be seen as a generalization of the method for parameter learning in DBNs, see e.g. [57]. conventional parameter learning in object oriented domains. 5.1 Structural OO learning The goal of our learning algorithm is to nd a good estimate of the unknown underlying statistical distribution function, i.e. the task of density estimation [52] Note that if focus had been on e.g. ....

Mike West and Je Harrison. Bayesian Forecasting and Dynamic Models. Springer Verlag, New York, 2nd edition, 1997.

....m1 . Merging the two components of this submixture is equivalent to finding the parameters of the closest Gaussian density. If closeness is taken in the KL sense, then # C ) arg min #C D (yj m1 # Cm1 ) ff (yj m1 # Cm1 )# (yj# C) which has a simple solution (see [34], Chapp. 12) are the global mean and covariance of the given two component mixture, i.e. m1 m1 ff m2 m2 (18) ff ) 19) This means that when merging components m 1 and m 2 of the mixture, the resulting component must retain the combined probability, mean, and ....

M. West and J Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, New York, 1989.

....the theory of the general dynamic linear model is presented. Mainly, we will be engaged with definitions and the Kalman filter. At the end of this chapter the dynamic linear model is applied to the pigs data. The theory of this chapter is based on [Gammelgaard et al. 1995, Chapter 2] and [West and Harrison, 1989, Chapter 2, 3, and 4] 7.1 Bayesian learning and dynamic linear models A statistical analysis of a set of data principally comprises of making inference about the underlying structure described by the distribution from which the data were generated. If this distribution is assumed known in ....

....a weighted expression of the one step forecast variance matrix Q t , i.e. the expected variance matrix of the observation Y t . For further elaborations of the adaption coefficient A t in the special case of the Kalman filter applied to a simple time series DLM see [Gammelgaard et al. 1995] and [West and Harrison, 1989] . In this section, a DLM will be applied to model the pigs data. We will confine ourselves to model the weight of a single pig, as this immediately can be extended to model the weights of a class of pigs, e.g. the complete set of data or a single breed. First, the DLM used in the following will ....

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M. West and J. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, 1989.

....and models of reasoning. In some domains, the detection of relevant events depends on analyzing past as well as present data; methodologies that reason explicitly about time include trend detection [1, 15] knowledgebased interval abstraction [47] and time series analysis and forecasting [6, 53]. These temporal reasoning methodologies may be used to warn of impending events, as well as transpiring events. Alarms that utilize the above inference methodologies are often described as intelligent sometimes because of the complexity of inferences, sometimes because the inference mechanism ....

Mike West and Jeff Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York, 1989.

....a time series of observations, and then updating that model as new observations arrive. Probability forecasting methods, in particular, can reason about uncertain system states and uncertain future effects due to unmodeled exogenous influences. Despite recent advances in probability forecasting [12, 14, 66], however, they are largely limited to predicting future states assuming no interventions. Thus, a PCM must integrate desirable features that are found in mathematical models of dynamical systems and forecasting models. If we represent actions with a time dependent control variable # t and system ....

....prediction task: 1) transfer function models, and 2) delay coordinate embedding (DCE) models. Transfer function models, which are rooted in engineering mathematics, are stochastic linear difference equations that can be used to predict intervention effects for a broad class of dynamic processes [29, 66]. DCE models, which are grounded in theoretical results from mathematical physics [61] can be used to make similar predictions for complex nonlinear dynamical systems with nonnormal uncertainties [15] We will explore and optimize automatic methods for constructing these models from time series ....

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Mike West and Jeff Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, New York, 1989.

....this submixture is equivalent to finding the parameters and C of the closest Gaussian density. If closeness is taken in the KL sense, then ( C ) arg min ;C D Theta ff 0 m1 N (yj m1 ; Cm1 ) ff 0 m2 N (yj m1 ; Cm1 ) N (yj; C) which has a simple solution (see [34], Chapp. 12) and C are the global mean and covariance of the given two component mixture, i.e. ff 0 m1 m1 ff 0 m2 m2 (18) C = ff 0 m1 (Cm1 m1 T m1 ) ff 0 m2 (Cm2 m2 T m2 ) Gamma T : 19) This means that when merging components m 1 and m ....

M. West and J Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, New York, 1989.

....we are interested in developing algorithms to process data arriving on line. This is a complex optimal nonlinear ltering problem. Many approximation algorithms, such as the extended Kalman lter and Gaussian sum approximations, have been proposed to surmount this problem (Anderson and Moore 1979, West and Harrison 1997). However, in many realistic problems, these approximating methods are notoriously unreliable and faults are dicult to diagnose on line. Recently there has been a surge of interest in sequential Monte Carlo (SMC) methods (also known as particle ltering when the objective of the analysis is to ....

West, M. and Harrison, J. (1997). Bayesian forecasting and dynamic models, Springer Series in Statistics, second edn, Springer-Verlag, New York.

....time series analysis which can be found in the literature. In an early attempt to apply statistical time series analysis to online monitoring data, Smith and West [13] used a multiprocess dynamic linear model to monitor patients after renal transplantation. In dynamic linear models (DLMs) [58] the observation X t at time t is regarded as a linear transform of an unobservable state parameter. This state is assumed to change dynamically in time according to a simple regression model. Particularly, the linear growth model X t = t # t t = t 1 # t 1 # t,1 # t = # t 1 # t,2 is ....

West, M., and Harrison, J. (1989), Bayesian Forecasting and Dynamic Models, Springer, New York.

.... Subset selection; Volterra expansion; Polynomial autoregressive models; Cascade modelling 1 Introduction Autoregressive (AR) processes are widely used to model a variety of signals, including audio (see e.g. 17, 34, 48] speech (see e.g. 9] and statistical and econometric time series (see e.g. [5, 49]) Their source filter structure is analogous to the physical processes involved in speech and many musical instruments [9] We consider the problem of reconstructing such a signal from a Preprint submitted to Elsevier Preprint 31 October 1999 PSfrag replacements e t AR k, a (k) x t NAR b, ....

M. West, J. Harrison, Bayesian Forecasting and Dynamic Models, SpringerVerlag, 1997, 2nd edition.

....[14, 37] Unfortunately, in the internal delay tracking problem, it is extremely difficult to sample from the optimal importance distribution. We can, however, achieve a slight improvement over the use of the prior distribution by considering a local linearization of the optimal distribution [36]. We consider the function l( m ) log p( m j m 1 ; y(m) and reparameterise using r m = log m.We have: l(r m ) ln p(y(m) jr m ) ln p(r m jr m 1 ) ln p(y(m) ln p(y(m) jr m ) 1 2 2 (r m r m 1 ) T (r m r m 1 ) ln p(y m ) 15 0 0.1 0.2 0 0.1 0.2 0 5 10 15 0 0.15 ....

M. West and J. F. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, 1997.

....using examples from three di erent areas of currently active research: econonometrics, sheries, and physics. 1 Introduction The state space approach is one of the most powerful tools for dynamic modeling and forecasting of time series and longitudinal data. Excellent overviews are given in West and Harrison (1997) and Fahrmeir and Tutz (1994) A state space model consists of observation and state equations. The observation equations specify the conditional distributions of the observations y t at time t as a function of unknown states t . But unlike a static model, the state of nature, t , changes over ....

WEST, M. and HARRISON, P.J. (1997): Bayesian Forecasting and Dynamic Models. Springer, New York.

.... # = 0;# 2 # =0;and # 2 # = Nominal Level Change : # 2 # # 0;# 2 # =0;and # 2 # = Nominal Slope Change : # 2 # = 0;# 2 # #0;and # 2 # = Nominal Transient : # 2 # = 0;# 2 # =0;and # 2 # = Large Additional details and discussion can be found in Gordon and Smith (1988) and West and Harrison (1989). As waste level observations are obtained, the state definitions given above are used in a modified Kalman filter to update the description of the waste level. The description consists of (a) the probability of being in each of the four states and (b) current best estimates of the parameters # ....

....to update the description of the waste level. The description consists of (a) the probability of being in each of the four states and (b) current best estimates of the parameters # t and # t for each of the four models. Updating equations are derived and presented in Gordon and Smith (1988) and West and Harrison (1989). t steady t level change t slope change t transient Figure 2. States for a linear growth model. These states are idealized analogues of corresponding states shown in Figure 1. The sets of probabilities for the four states of the model are used to evaluate the state of the system after ....

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West, M. and J. Harrison. 1989. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York.

....the fact that political attitudes in the contemporary United States do not change much over the course of a single week. For known W , this is a special case of a model for which one can use the Kalman filter to obtain the posterior moments of the state vectors, # t for t = 0, T (see e.g. West and Harrison (1997)) 2.3 Analytic Expressions for Posterior Inference In order to obtain samples from the posterior distribution of the weights for our poststratification estimate, we first obtain samples from the posterior distribution of the 6 state process in our dynamic model given all of the data up to time ....

....problems. The advantage of this technique for averaging over our uncertainty in the model parameters compared to simply using the Gibbs sampler to simulate the state process given the model parameters and then simulate the model parameters given the state process (as is frequently done, see e.g. West and Harrison (1997)) is that in our method, no iterative simulation is required for the state vectors. This is a great simplification since adjacent state vectors are highly correlated in their joint posterior distribution, hence obtaining convergence of the chain can be di#cult if we must use an iterative ....

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West, M., and Harrison, J. (1997), Bayesian Forecasting and Dynamic Models, New York: Springer-Verlag.

....of the meteorological variables, temperature, humidity and wind velocity. Our goal is to propose a statistical model that forecasts temporally, interpolates spatially and show its performance at both levels. We elaborate our models within the Bayesian paradigm using Dynamic Linear Models as in West and Harrison (1997). We strongly believe our modeling approach could assist in the implementation of an environmental contingency strategy. Previous analyses of ground level ozone data for multiple sites, modeled jointly, appears in the paper by Carroll et al. 1997) which uses a spatially homogeneous and ....

....the matrix of euclidean distances between monitoring stations. W 1;t is speci ed with a discount factor approach and W 2;t as a block diagonal matrix with blocks of the form j exp( D= j ) j = 1; q. This spatio temporal model can be easily written in the state space form notation of West and Harrison (1997). Thus, conditional on the hyperparameters that de ne the covariance structure, the ltering and recurrence equations of the DLM produces predictive values and restropective inferences for observed values. Formal Bayesian inference on the hyperparameters leads to the Forward Filtering Backward ....

West, M. and Harrison, P.J. (1997) Bayesian Forecasting and Dynamic Models, 2nd edn. New York: Springer.

....spatio temporal data are often constructed by combining traditional time series techniques with methods from spatial statistics. In the time series context, popular approaches include ARIMA models (see Box, Jenkins, and Reinsel, 1994) for stationary 1 data, and dynamic linear models (DLMs; see West and Harrison, 1997), which allow for nonstationary components such as temporal trends and seasonality. In the spatial setting, much of the literature revolves around isotropic models (see for example, Cressie, 1993) These models grew out of applications in geostatistics, where the objective is spatial prediction or ....

....(3) 4) so that t1 = fi t and F 0 t1 = X. Let the remaining blocks represent the additional components, and denote their state vector and design matrix by tk and F tk ; k = 2; K. A brief discussion of multi component models is given in Section 4. For a more detailed discussion, see West and Harrison (1997, chapters 7 9) In the case of a single component, K = 1, the simplest structure for G t is diagonal, implying autoregressions for each element of t . A random walk prior, with G t = I, is a natural choice when no prior information is available or if no temporal trends are expected. With ....

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West, M. and Harrison, P. (1997), Bayesian Forecasting and Dynamic Models, New York: Springer, 2nd ed.

....by Engle and Smith (1998) which seems relevant for the parameter shift modeling chapter. Third, the book does not contain any discussion of Bayesian forecasting methods, despite their apparent success both in univariate forecast competitions and in vector autoregression environments (see e.g. West and Harrison, 1989). Recognizing the importance of density forecasts, rather than simple point forecasts, the Bayesian approach has clear advantages in terms of delivering posterior odds distributions even across unit root regions of the parameter space. Finally, Stock and Watson (1998) find that forecast ....

West, M., and Harrison, P.J. (1989), Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.

.... in a posterior distribution arises frequently in Bayesian analyses of time series, where flat priors over the stationary region for autoregressive parameters give rise to posteriors known only up to a constant factor of proportionality (e.g. Chib and Greenberg 1994; Marriot et al. 1996; West and Harrison 1997). The extension to the discrete time series case introduces no new complications since all the calculations take place with respect to the estimates of the continuous latent quantities (conditional on the observed discrete responses) Previous implementations include work by and has been addressed ....

West, Mike and Jeff Harrison. 1997. Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.

....model (20) is said to be time invariant when the system matrices are constant over time t, that is Z t = Z, T t = T , G t = G and H t = H, for t = 1; T . An introduction to statistical analysis based on state space models is given by, for example, Harvey (1989) Kitagawa and Gersch (1996) West and Harrison (1997) and Durbin and Koopman (2000) The contemporaneous form of the state space model is y t = Z t t G t t ; t WN(0; I) t = 1; T ; t = T t t 1 H t t ; 0 WN(a;P ) 21) By setting t = t 1 , the future state model is obtained with y t = Z ....

West, M. and P.J. Harrison (1997) Bayesian Forecasting and Dynamic Models, 2nd ed. New York: Springer-Verlag.

....components. Models for spatio temporal data are often constructed by combining time series models with variogram based models from spatial statistics. In the time series context, popular approaches include ARIMA models (Box et al. 1994) for stationary data, and dynamic linear models (DLMs; West and Harrison, 1997), which allow for nonstationary components such as temporal trends and seasonality. In the spatial setting, much of the literature revolves around isotropic models (Cressie, 1993) These models grew out of applications in geostatistics, where the main objective is prediction or kriging; thus we ....

....t1 = X. The rst block corresponds to the spatial model (3) 4) The remaining blocks represent the additional components, with state vector and design matrix tk and F tk ; k = 2; K. A brief discussion of multi component models is given in Section 4. For a more detailed discussion, see West and Harrison (1997, chapters 7 9) In the case of a single component, K = 1, the simplest structure for G t is diagonal, implying autoregressions for each element of t . A random walk prior, with G t = I, is a natural choice when no prior information is available or if no temporal trends are expected. With ....

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West, M. and Harrison, P. (1997) Bayesian Forecasting and Dynamic Models. New York: Springer, 2nd edn.

....First we determine an auxiliary discrete mixture model to approximate the given model. The weights in the mixture model are allowed to be adaptive and depend on the state vector. Conditional on the mixture component we assume that the auxiliary model reduces to a linear Gaussian state space model (West and Harrison, 1997) and so implementation is based on using a block sampling Metropolis algorithm. We implement two SDV models: a stochastic volatility model with jumps and an ane term structure model. SVOL models are used to describe the evolution of asset returns and a common approach is to assume that volatility ....

West, M. and Harrison, P. (1997) Bayesian Forecasting and Dynamic Models. New York: Springer, 2nd edn.

....3. Adaptive Bayesian Designs for Dose Ranging Drug Trials 11 where the study is carried out has been developed. Details of the interface are given in Section 8. 2. 3 Estimating dose response We use a dose response model based on Normal Dynamic Linear Models (NDLM) as described, for example, in West and Harrison (1997). This is essentially a piecewise linear model. It provides the necessary exibility to encompass both monotonic and non monotonic dose response relationships. An additional advantage of the NDLM is the existence of analytical results for the determination of the posterior distribution of the ....

....for non monotonicity and other irregular features. This is particularly important since death is a possible consequence of stroke. The drug may have a di erent dose e ect as regards rehabilitating patients and mortality. Based on these considerations we chose a normal dynamic linear model (NDLM) West and Harrison (1997) give a formal de nition and discussion of NDLMs. Before we describe details of the model, we outline some important features. Denote by Z j ; j = 0; J , the range of allowable doses, and by j = f(Z j ; j = 0; J , the vector of mean responses at the allowable doses. The ....

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West, M. and Harrison, P.J. (1997). Bayesian Forecasting and Dynamic Models. Second Edition. New York: Springer-Verlag.

....sequentially update posterior distributions in various mixture modeling frameworks. This literature has involved methods for both time evolving states and xed parameters, and is exempli ed by the important class of adaptive multi process models used in Bayesian forecasting since the early 1970s [42, 85, 94]. During the 1980s, this naturally led to larger scale analyses using discrete grids of parameter values, though the combinatorial explosion of grid sizes with increasing parameter dimension limited this line of development. Novel methods using ecient quadrature based, adaptive numerical ....

....Series Modeling Bayesian dynamic linear models, a class of dynamic linear regression models, have experienced growth in real life application in recent years. Among those dedicated to apply Bayesian models to nancial time series, 73] proposed using multivariate Bayesian DLMs in the framework of [94], 72] and [76] to forecast currency returns. In contrast to the widely used static regression models, Bayesian DLMs allows the regression coecients to change over time, thus are better at capturing market trends. In addition, Bayesian DLMs provide the mechanism to naturally incorporate external ....

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M West and J Harrison. Bayesian Forecasting and Dynamic Models (2nd edition) . Springer-Verlag, New York, 1997.

....models ARIMA(p; d; q) can be used to model Y t , i.e. OE(B) 1 Gamma B) d Y t = B)ffl t . Non stationary processes exhibiting polynomial trends, seasonal or cyclical patterns and dynamic regression components can be handled via dynamic linear models or DLMs. The DLM theory is developed in West and Harrison (1997). One important class of models among DLMs is that of time varying autoregressions or TVAR models. This class of models is considered here to describe non stationarities in long time series arising in a particular application setting. In order to motivate the use of TVAR models, consider the ....

....over time and to assume a stochastic structure on the 3 innovations. Typically, it is assumed that the innovations are normally distributed; other distributions considered in the literature include heavy tailed error distributions and mixture error distributions (see Carter and Kohn, 1994 and West 1997a) Sophisticated prior structures have been developed for standard AR models. Barnett et al. 1996a) and Barnett et al. 1996b) develop priors for AR and ARMA models on partial regression coefficients rather than on ARMA coefficients; Huerta and West (1997) propose a novel class of priors for ....

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West, M. and Harrison, J. (1997) Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.

....in such experiments are needed. In previous work we have demonstrated the usefulness of various classes of dynamic models for the analysis of univariate EEG series, and in exploratory studies of the relationships between multiple EEG traces on a single subject (West et al. 1999, Prado and West 1997 and Krystal et al. 1996) This article describes further development and application of more formal multivariate dynamic models. Some data description sets the stage. The EEG data studied here is part of a full data set, code named Ictal19, that corresponds to records of 19 EEG channels recorded ....

.... is usually dominated by alpha and beta rhythms (Dyro 1989) anaesthesia effects typically induce mixtures of slow and fast activity (Weiner et al. 1991) and effective ECT therapies induce brain seizure activity dominated by so called seizure slow waves in the delta range (Dyro 1989; Prado and West 1997). The left frame in Figure 1 shows a schematic of the approximate locations of the 19 electrodes over the scalp. By convention, electrodes located on the left side are odd numbered while the ones located on the right side are even numbered. The capital letters F, Fp, P, T, C and O refer to the ....

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West, M. and Harrison, J. (1997) Bayesian Forecasting and Dynamic Models, 2nd edn. New York: Springer-Verlag.

....pp. 000 000 ISBA and Eurostat, 2001 Intrinsic Bayes Factors for Dynamic Linear Models. Abel Rodriguez and Luis R. Pericchi 1 CESMa, Universidad Simon Bolivar, Venezuela SUMMARY In Rodriguez and Pericchi (2000) Local Bayes Factors are defined and developed for Dynamic Linear Models (DLM) West and Harrison (1997). Local Intrinsic Bayes Factors are based on simulated replicas of the initial set of observations. This generates well calibrated Intrinsic Priors, leaving the whole set of observations for model determination. When the observational variance is known, we give exact results, and when they are ....

....) is a default improper prior, that we will denote by N i ( 0i ) It is well known that Bayes Factors are typically undetermined when improper priors are employed, at least when the dimensions of the parameters are 1 different. The standard procedure for model specification, as describred in West and Harrison (1997), is to leave aside a (minimal) real training sample of the data points closest to the initial point t = 0 that makes all the entertained models proper, and then use these trained priors to compute the marginal densities for the rest of the observations. The ratio of these marginals are the ....

West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.

....activity in these bands. Comparing such characteristics across EEG channels and between seizures under di erent ECT protocols provides insights into both basic scienti c and clinical issues. Time series decomposition methodology is based on extensions of existing time series decomposition theory (West 1997a) to time varying parameter models. The new theoretical results underlying the methodology arise in general classes of dynamic linear models (DLMs) of which TVAR models form a very important subclass. In Section 2 we introduce TVAR models and describe the decomposition structure and theory. ....

....of model tting and posterior computation. We rst discuss the theoretical basis of our decomposition methodology and its implications in the class of TVAR models de ned by equation (1) irrespective of the speci c forms of evolution of t and 2 t : The theory of time series decompositions in West (1997a) has important generalizations that are relevant to TVAR models and other dynamic models with timevarying parameters (including various classes of nonlinear models) The theory is accessed by casting the TVAR model (1) in dynamic linear model (DLM) form, as x t = F 0 x t ; x t = G t x t 1 ....

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West, M., and Harrison, P.J. (1997), Bayesian Forecasting and Dynamic Models (2nd Edition), New York: Springer-Verlag.

....to the overall mixture density. In the context of raw density estimation, West (1990) discusses the reduction of kernel estimates to mixtures of much smaller numbers of components, often lower than 10 of the original sample size number, using a particular form of clustering. The discussion in West and Harrison (1989, Section 12.3) on issues and techniques involved in approximating mixtures generally, is also relevant. A very basic method of clustering mixture components, combining ideas from each of these two references, is used in West (1992a) At the simplest, it involves reducing the number of ....

....With discrete approximations, this involves mapping a prior set of points and weights to a posterior set, possibly with some form of smoothing involved at both prior and posterior stages. These issues, and others, are sharply evident in sequential modelling of time series using dynamic models (West and Harrison, 1989), where the progressive revision of posterior and predictive distributions requires calculations that are typically impossible to perform exactly. Modellers have developed a variety of approaches to analytic and numerical approximation of such distributions, including direct quadrature methods ....

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West M., and Harrison, P.J. (1989) Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York.

....developments and current research frontiers. 2. UNIVARIATE TIME SERIES DECOMPOSITIONS AND LATENT STRUCTURE 2.1. Introduction Much of the recent development in latent structure analysis is based on novel extensions and exploitation of the fundamental component structure of dynamic models (West and Harrison 1997, chapter 6) Begin with a general dynamic linear model (DLM) in which the scalar time series y t ; observed at equally spaced time points t = 1; 2; is modelled as y t = x t t ; x t = F 0 t t ; t = G t t Gamma1 t (2:1) for each t: Here x t is the latent signal process, t ....

....result is evident through their definitions and interpretations in specific special cases. Some key special cases exemplify this in the following sections. Full background and mathematical details of this construction are given in West, Prado and Krystal (1997) and in special cases in West (1997c) and West and Harrison (1997, sections 9.5, 9.5 and 15.3) 2.2. Latent Structure, Prior Specifications and Model Uncertainty in Autoregressions The simplest, and important, special case is that of an autoregressive signal in noise, in which x t = P d j=1 OE j x t Gammaj t : This is a ....

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West, M. and Harrison, P.J. (1997). Bayesian Forecasting and Dynamic Models (2nd Edn). New York: SpringerVerlag. West, M., Prado, R. and Krystal, A. (1997). Latent structure in non-stationary time series with application in studies of EEG traces. ISDS Discussion Paper 97-14, Duke University.

....for improvements in short term predictive ability. 3 Dynamic hierarchical regression models Improvements in day specific, short term forecasts can be expected to arise from refined models that adapt to the observed within day variability in flows using dynamic modelling methods from time series (West and Harrison 1997). As it stands the basic regression model of equations (2) and (3) provides day specific effects that are constant over the course of the day. This constancy is certainly most appropriate for the basic spline parameters representing the in out flows from ramps, but perhaps a little rigid in ....

....; whereas their effects are, in part, made physically evident through the transfer responses to downstream flows and hence in a modelling framework through changes in regression parameters. This is a traditional concept that underlies the entire field of dynamic modelling and Bayesian forecasting (West and Harrison 1997). A typical immediate benefit of recognising and modelling time varying regression parameters is increased shortterm forecasting accuracy and reduced forecast uncertainty, as the examples in West and Harrison (1997, chapters 2 and 3, particularly) vividly demonstrate. Here we detail and explore ....

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West, M. and P.J. Harrison, Bayesian Forecasting and Dynamic Models, (2nd Edn.), Springer Verlag, New York, 1997.

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M. West and J. Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, 2nd edition, 1997.

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M. West and J. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York, 1997.

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West M. and Harrison J.F. (1997) Bayesian Forecasting and Dynamic Models, Springer Verlag Series in Statistics, 2 nd edition.

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West, M. and Harrison, J. (1997). Bayesian forecasting and dynamic models, Springer Series in Statistics, second edn, Springer-Verlag, New York.

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West, M. and Harrison, P. J. (1997). Bayesian Forecasting and Dynamic Models, 2 nd ed., Springer-Verlag, New York.

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M. West and J. Harrison. Bayesian forecasting and dynamic models. Springer-Verlag, 1997. 42

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Mike West and Jeff Harrison. Bayesian forecasting and dynamic models. Springer, 1997. 15

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West, M. and Harrison, P.J. (1997) Bayesian Forecasting and Dynamic Models, 2nd edn. Springer-Verlag, New York.

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West, M. and Harrison, P.J. (1997) Bayesian Forecasting and Dynamic Models. 2nd edition. Springer Verlag, New York.

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M. West and J. Harrison, Bayesian forecasting and dynamic models, Springer, 1999.

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M. West and P. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, 1989.

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M. West and P.J. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York, 2nd edition, 1997.

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M. West and P.J. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, New York, 2nd edition, 1997.

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M. West and P. Harrison. Bayesian Forecasting and Dynamic Models. Springer-Verlag, 1989.

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West M. and Harrison P.J. (1997) Bayesian Forecasting and Dynamic Models. 2nd edition, New York: Springer-Verlag.

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West, Mike, and Je# Harrison. 1997. Bayesian Forecasting and Dynamic Models . New York: Springer.

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West, M. & Harrison, P. J. (1989). Bayesian Forecasting and Dynamic Models. New York: Springer--Verlag.

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West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, Second Edition. Berlin: Springer Verlag. 30

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West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models (Second Edition). New York: Springer-Verlag.

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West, M. and Harrison, J. (1997), Bayesian Forecasting and Dynamic Models (2nd Edn), New York: Springer-Verlag.

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