On-line inference for hidden Markov models via particle filters

Summary.  We consider the on-line Bayesian analysis of data by using a hidden Markov model, where inference is tractable conditional on the history of the state of the hidden component. A new particle filter algorithm is introduced and shown to produce promising results when analysing data of this type. The algorithm is similar to the mixture Kalman filter but uses a different resampling algorithm. We prove that this resampling algorithm is computationally efficient and optimal, among unbiased resampling algorithms, in terms of minimizing a squared error loss function. In a practical example, that of estimating break points from well-log data, our new particle filter outperforms two other particle filters, one of which is the mixture Kalman filter, by between one and two orders of magnitude.     

A multifactor transformed diffusion model with applications to VIX and VIX futures

Abstract

Transformed diffusions (TDs) have become increasingly popular in financial modeling for their model flexibility and tractability. While existing TD models are predominately one-factor models, empirical evidence often prefers models with multiple factors. We propose a novel distribution-driven nonlinear multifactor TD model with latent components. Our model is a transformation of a underlying multivariate Ornstein–Uhlenbeck (MVOU) process, where the transformation function is endogenously specified by a flexible parametric stationary distribution of the observed variable. Computationally efficient exact likelihood inference can be implemented for our model using a modified Kalman filter algorithm and the transformed affine structure also allows us to price derivatives in semi-closed form. We compare the proposed multifactor model with existing TD models for modeling VIX and pricing VIX futures. Our results show that the proposed model outperforms all existing TD models both in the sample and out of the sample consistently across all categories and scenarios of our comparison.     

Monte Carlo Kalman filter and smoothing for multivariate discrete state space models

The author studies state space models for multivariate binomial time series, focussing on the development of the Kalman filter and smoothing for state variables. He proposes a Monte Carlo approach employing the latent variable representation which transplants the classical Kalman filter and smoothing developed for Gaussian state space models to discrete models and leads to a conceptually simple and computationally convenient approach. The method is illustrated through simulations and concrete examples.     

基于辅助粒子滤波的DSGE模型识别方法及应用

动态随机一般均衡模型中涵盖无法直接观测的变量,同时跨方程约束涉及复杂的非线性关系使方程的解析估计难以实现。在贝叶斯框架下识别动态随机一般均衡模型,基于状态空间方法建立度量方程和状态转移方程,采用辅助粒子滤波预测条件后验分布,建立贝叶斯误差带描述宏观经济变量脉冲响应函数的动态特征。实际数据分析验证了贝叶斯识别方法的有效性。     

Robust estimation of linear state space models

The model parameters of linear state space models are typically estimated with maximum likelihood estimation, where the likelihood is computed analytically with the Kalman filter. Outliers can deteriorate the estimation. Therefore we propose an alternative estimation method. The Kalman filter is replaced by a robust version and the maximum likelihood estimator is robustified as well. The performance of the robust estimator is investigated in a simulation study. Robust estimation of time varying parameter regression models is considered as a special case. Finally, the methodology is applied to real data.     

An alternative derivation of the Kalman filter using the quasi-likelihood method

The Kalman filter gives a recursive procedure for estimating state vectors. The recursive procedure is determined by a matrix, so-called gain matrix, where the gain matrix is varied based on the system to which the Kalman filter is applied. Traditionally the gain matrix is derived through the maximum likelihood approach when the probability structure of underlying system is known. As an alternative approach, the quasi-likelihood method is considered in this paper. This method is used to derive the gain matrix without the full knowledge of the probability structure of the underlying system. Two models are considered in this paper, the simple state space model and the model with correlated between measurement and transition equation disturbances. The purposes of this paper are (i) to show a simple way to derive the gain matrix; (ii) to give an alternative approach for obtaining optimal estimation of state vector when underlying system is relatively complex.     

A new robust Kalman filter for filtering the microstructure noise

We propose a robust Kalman filter (RKF) to estimate the true but hidden return when microstructure noise is present. Following Zhou's definition, we assume the observed return Y_t is the result of adding microstructure noise to the true but hidden return X_t. Microstructure noise is assumed to be independent and identically distributed (i.i.d.); it is also independent of X_t. When X_t is sampled from a geometric Brownian motion process to yield Y_t, the Kalman filter can produce optimal estimates of X_t from Y_t. However, the covariance matrix of microstructure noise and that of X_t must be known for this claim to hold. In practice, neither covariance matrix is known so they must be estimated. Our RKF, in contrast, does not need the covariance matrices as input. Simulation results show that the RKF gives essentially identical estimates to the Kalman filter, which has access to the covariance matrices. As applications, estimated X_t can be used to estimate the volatility of X_t.     

Analytical uses of Kalman filtering in econometrics — A survey

This paper surveys the different uses of Kalman filtering in the estimation of statistical (econometric) models. The Kalman filter will be portrayed as (i) a natural generalization of exponential smoothing with a time-dependent smoothing factor, (ii) a recursive estimation technique for a variety of econometric models amenable to a state space formulation in particular for econometric models with time varying coefficients (iii) an instrument for the recursive calculation of the likelihood of the (constant) state space coefficients (iv) a means of helping to implement the scoring⁻ and EM-method for iteratively maximizing this likelihood (v) an analytical tool in asymptotic estimation theory. The concluding section points to the importance of Kalman filtering for alternatives to maximum⁻ likelihood estimation of state space parameters.     

Modeling Unequally Spaced Bivariate Growth Curve Data Using a Kalman Filter Approach

ABSTRACT

In many clinical studies, patients are followed over time with their responses measured longitudinally. Using mixed model theory, one can characterize these data using a wide array of across subject models. A state-space representation of the mixed effects model and use of the Kalman filter allows one to have great flexibility in choosing the within error correlation structure even in the presence of missing or unequally spaced observations. Furthermore, using the state-space approach, one can avoid inverting large matrices resulting in efficient computation. The approach also allows one to make detailed inference about the error correlation structure. We consider a bivariate situation where the longitudinal responses are unequally spaced and assume that the within subject errors follows a continuous first-order autoregressive (CAR(1)) structure. Since a large number of nonlinear parameters need to be estimated, the modeling strategy and numerical techniques are critical in the process. We developed both a Visual Fortran^® and a SAS^® program for modeling such data. A simulation study was conducted to investigate the robustness of the model assumptions. We also use data from a psychiatric study to demonstrate our model fitting procedure.     

Nonlinear filters based on taylor series expansions ∗

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable α_t and α_t-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., I_t-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of α_t and α_t-1 given I_t-1 and those of the error terms θ and η_t. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.     

Reml and best linear unbiased prediction in state space models

This article takes a hierarchical model approach to the estimation of state space models with diffuse initial conditions. An initial state is said to be diffuse when it cannot be assigned a proper prior distribution. In state space models this occurs either when fixed effects are present or when modelling nonstationarity in the state transition equation. Whereas much of the literature views diffuse states as an initialization problem, we follow the approach of Sallas and Harville (1981,1988) and incorporate diffuse initial conditions via noninformative prior distributions into hierarchical linear models. We apply existing results to derive the restricted loglike-lihood and appropriate modifications to the standard Kalman filter and smoother. Our approach results in a better understanding of De Jong's (1991) contributions. This article also shows how to adjust the standard Kalman filter, the fixed inter- val smoother and the state space model forecasting recursions, together with their mean square errors, for he presence of diffuse components. Using a hierarchical model approach it is shown that the estimates obtained are Best Linear Unbiased Predictors (BLUP).     

Adaptive Kalman Filtering with Multivariate Generalized Laplace System Noise

An adaptive Kalman filter is proposed to estimate the states of a system where the system noise is assumed to be a multivariate generalized Laplace random vector. In the presence of outliers in the system noise, it is shown that improved state estimates can be obtained by using an adaptive factor to estimate the dispersion matrix of the system noise term. For the implementation of the filter, an algorithm which includes both single and multiple adaptive factors is proposed. A Monte-Carlo investigation is also carried out to access the performance of the proposed filters in comparison with other robust filters. The results show that, in the sense of minimum mean squared state error, the proposed filter is superior to other filters when the magnitude of a system change is moderate or large.     

Generalized Gauss–Hermite filtering

We consider a generalization of the Gauss–Hermite filter (GHF), where the filter density is represented by a Hermite expansion with leading Gaussian term (GGHF). Thus, the usual GHF is included as a special case. The moment equations for the time update are solved stepwise by Gauss–Hermite integration, and the measurement update is computed by the Bayes formula, again using numerical integration. The performance of the filter is compared numerically with the GHF, the UKF (unscented Kalman filter) and the EKF (extended Kalman filter) and leads to a lower mean squared filter error.     

Estimation and tests of hypotheses for the initial mean and covariance in the kalman filter model

Kalman filtering techniques are widely used by engineers to recursively estimate random signal parameters which are essentially coefficients in a large-scale time series regression model. These Bayesian estimators depend on the values assumed for the mean and covariance parameters associated with the initial state of the random signal. This paper considers a likelihood approach to estimation and tests of hypotheses involving the critical initial means and covariances. A computationally simple convergent iterative algorithm is used to generate estimators which depend only on standard Kalman filter outputs at each successive stage. Conditions are given under which the maximum likelihood estimators are consistent and asymptotically normal. The procedure is illustrated using a typical large-scale data set involving 10-dimensional signal vectors.     

Asymptotic distribution theory for the kalman filter state estimator

An asymptotic distribution theory for the state estimate from a Kalman filter in the absence of the usual Gaussian assumption is presented. It is found that the stability properties of the state transition matrix playa key role in the distribution theory. Specifically, when the state equation is neutrally stable (i.e., borderline stable-unstable) the state estimate is asymptotically normal when the random terms in the model have arbitrary distributions. This case includes the popular random walk state equation. However, when the state equation is either stable or unstable, at least some of the random terms in the model must be normally distributed beyond some finite time before the state estimate is asymptotically normal.     

The ensemble Kalman filter is an ABC algorithm

The ensemble Kalman filter is the method of choice for many difficult high-dimensional filtering problems in meteorology, oceanography, hydrology and other fields. In this note we show that a common variant of the ensemble Kalman filter is an approximate Bayesian computation (ABC) algorithm. This is of interest for a number of reasons. First, the ensemble Kalman filter is an example of an ABC algorithm that predates the development of ABC algorithms. Second, the ensemble Kalman filter is used for very high-dimensional problems, whereas ABC methods are normally applied only in very low-dimensional problems. Third, recent state of the art extensions of the ensemble Kalman filter can also be understood within the ABC framework.     

New sequential Monte Carlo methods for nonlinear dynamic systems

In this paper we present several new sequential Monte Carlo (SMC) algorithms for online estimation (filtering) of nonlinear dynamic systems. SMC has been shown to be a powerful tool for dealing with complex dynamic systems. It sequentially generates Monte Carlo samples from a proposal distribution, adjusted by a set of importance weight with respect to a target distribution, to facilitate statistical inferences on the characteristic (state) of the system. The key to a successful implementation of SMC in complex problems is the design of an efficient proposal distribution from which the Monte Carlo samples are generated. We propose several such proposal distributions that are efficient yet easy to generate samples from. They are efficient because they tend to utilize both the information in the state process and the observations. They are all Gaussian distributions hence are easy to sample from. The central ideas of the conventional nonlinear filters, such as extended Kalman filter, unscented Kalman filter and the Gaussian quadrature filter, are used to construct these proposal distributions. The effectiveness of the proposed algorithms are demonstrated through two applications—real time target tracking and the multiuser parameter tracking in CDMA communication systems.This work was supported in part by the U.S. National Science Foundation (NSF) under grants CCR-9875314, CCR-9980599, DMS-9982846, DMS-0073651 and DMS-0073601.     

Inference for Adaptive Time Series Models: Stochastic Volatility and Conditionally Gaussian State Space Form

In this paper we model the Gaussian errors in the standard Gaussian linear state space model as stochastic volatility processes. We show that conventional MCMC algorithms for this class of models are ineffective, but that the problem can be alleviated by reparameterizing the model. Instead of sampling the unobserved variance series directly, we sample in the space of the disturbances, which proves to lower correlation in the sampler and thus increases the quality of the Markov chain.

Using our reparameterized MCMC sampler, it is possible to estimate an unobserved factor model for exchange rates between a group of n countries. The underlying n + 1 country-specific currency strength factors and the n + 1 currency volatility factors can be extracted using the new methodology. With the factors, a more detailed image of the events around the 1992 EMS crisis is obtained.

We assess the fit of competitive models on the panels of exchange rates with an effective particle filter and find that indeed the factor model is strongly preferred by the data.     

The reverse kalman filter

Equations for the reverse Kalman filter are provided. This enables calculation of the distribution of the state given the future data in the state space representation of a time series model. This complements the conventional Kalman filter recursions for the distribution of the state given the past data. The usefulness of these two recursions is demonstrated by showing how they can be combined to efficiently calculate leave-k-out diagnostics or the likelihood under a model incorporating a patch of anomalies in a time series.     

Autoregressive Model with Partial Forgetting within Rao-Blackwellized Particle Filter

The authors are concerned with Bayesian identification and prediction of a nonlinear discrete stochastic process. The fact that a nonlinear process can be approximated by a piecewise linear function advocates the use of adaptive linear models. They propose a linear regression model within Rao-Blackwellized particle filter. The parameters of the linear model are adaptively estimated using a finite mixture, where the weights of components are tuned with a particle filter. The mixture reflects a priori given hypotheses on different scenarios of (expected) parameters' evolution. The resulting hybrid filter locally optimizes the weights to achieve the best fit of a nonlinear signal with a single linear model.