CONTINUOUS-TIME FILTERS FOR STATE ESTIMATION FROM POINT PROCESS MODELS OF NEURAL DATA

Neural spike trains, the primary communication signals in the brain, can be accurately modeled as point processes. For many years, significant theoretical work has been done on the construction of exact and approximate filters for state estimation from point process observations in continuous-time. We have previously developed approximate filters for state estimation from point process observations in discrete-time and applied them in the study of neural systems. Here, we present a coherent framework for deriving continuous-time filters from their discrete-counterparts. We present an accessible derivation of the well-known unnormalized conditional density equation for state evolution, construct a new continuous-time filter based on a Gaussian approximation, and propose a method for assessing the validity of the approximation following an approach by Brockett and Clark. We apply these methods to the problem of reconstructing arm reaching movements from simulated neural spiking activity from the primary motor cortex. This work makes explicit the connections between adaptive point process filters for analyzing neural spiking activity in continuous-time, and standard continuous-time filters for state estimation from continuous and point process observations.     

On the Discretization of Continuous-Time Filters for Nonstationary Stock and Flow Time Series

This article discusses the discretization of continuous-time filters for application to discrete time series sampled at any fixed frequency. In this approach, the filter is first set up directly in continuous-time; since the filter is expressed over a continuous range of lags, we also refer to them as continuous-lag filters. The second step is to discretize the filter itself. This approach applies to different problems in signal extraction, including trend or business cycle analysis, and the method allows for coherent design of discrete filters for observed data sampled as a stock or a flow, for nonstationary data with stochastic trend, and for different sampling frequencies. We derive explicit formulas for the mean squared error (MSE) optimal discretization filters. We also discuss the problem of optimal interpolation for nonstationary processes – namely, how to estimate the values of a process and its components at arbitrary times in-between the sampling times. A number of illustrations of discrete filter coefficient calculations are provided, including the local level model (LLM) trend filter, the smooth trend model (STM) trend filter, and the Band Pass (BP) filter. The essential methodology can be applied to other kinds of trend extraction problems. Finally, we provide an extended demonstration of the method on CPI flow data measured at monthly and annual sampling frequencies.     

A Bootstrap Test for Circular Data

In this article, variance stabilizing filters are discussed. A new filter with nice properties is proposed which makes use of moving averages and moving standard deviations, the latter smoothed with the Hodrick-Prescott filter. This filter is compared to a GARCH-type filter. An ARIMA model is estimated for the filtered GDP series, and the parameter estimates are used in forecasting the unfiltered series. These forecasts compare well with those of ARIMA, ARFIMA, and GARCH models based on the unfiltered data. The filter does not color white noise.     

Estimation of unknown parameters in nonlinear and non-Gaussian state-space models

For the last decade, various simulation-based nonlinear and non-Gaussian filters and smoothers have been proposed. In the case where the unknown parameters are included in the nonlinear and non-Gaussian system, however, it is very difficult to estimate the parameters together with the state variables, because the state-space model includes a lot of parameters in general and the simulation-based procedures are subject to the simulation errors or the sampling errors. Therefore, clearly, precise estimates of the parameters cannot be obtained (i.e., the obtained estimates may not be the global optima). In this paper, an attempt is made to estimate the state variables and the unknown parameters simultaneously, where the Monte Carlo optimization procedure is adopted for maximization of the likelihood function.     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

时变三因子模型风险系数的动态研究

由Fama和French提出的三因子模型能够较好地解释股票的收益率风险溢价。文章以状态空间模型为框架,将风险因子系数作为状态变量,市场风险溢价作为观测变量,构建时变三因子模型来应对股票市场价格的时变特征。研究结果显示,利用卡尔曼滤波来估计时变风险因子系数,增强了估计结果的准确性与连贯性;风险因子系数变化规律与中国A股市场政策和环境影响相吻合,消除非理性噪声后的时变三因子模型更具有解释力度。     

Seasonal Extraction in the Presence of Feedback

Time series seasonal extraction techniques are quite often applied in the context of a policy aimed at controlling the nonseasonal components of a time series. Monetary policies targeting the nonseasonal components of monetary aggregates are an example. Such policies can be studied as a quadratic optimal control model in which observations are contaminated by seasonal noise. Optimal extraction filters in such models do not correspond to univariate time series seasonal extraction filters. The linear quadratic control model components are nonorthogonal due to the presence of control feedback. This article presents the Kalman filter as a conceptual and computational device used to extract seasonal noise in the presence of feedback.     

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.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

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.     

Threshold Estimation for Stochastic Processes with Small Noise

Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.     

A Unified View of Nonparametric Trend-Cycle Predictors Via Reproducing Kernel Hilbert Spaces

We provide a common approach for studying several nonparametric estimators used for smoothing functional time series data. Linear filters based on different building assumptions are transformed into kernel functions via reproducing kernel Hilbert spaces. For each estimator, we identify a density function or second order kernel, from which a hierarchy of higher order estimators is derived. These are shown to give excellent representations for the currently applied symmetric filters. In particular, we derive equivalent kernels of smoothing splines in Sobolev and polynomial spaces. The asymmetric weights are obtained by adapting the kernel functions to the length of the various filters, and a theoretical and empirical comparison is made with the classical estimators used in real time analysis. The former are shown to be superior in terms of signal passing, noise suppression and speed of convergence to the symmetric filter.     

DELAY ESTIMATION BY A HILBERT TRANSFORM METHOD

This paper describes an estimation of the time delay between two stationary time series signals, in which an input signal is measured with little noise and an output signal is the sum of a noise and the response from a linear system. We use the Hilbert transform relation for minimum delay systems to estimate the time delay. Some computer simulation results are given to evaluate the performance of the proposed method.     

Multivariate Real-Time Signal Extraction by a Robust Adaptive Regression Filter

We propose a new regression-based filter for extracting signals online from multivariate high frequency time series. It separates relevant signals of several variables from noise and (multivariate) outliers.

Unlike parallel univariate filters, the new procedure takes into account the local covariance structure between the single time series components. It is based on high-breakdown estimates, which makes it robust against (patches of) outliers in one or several of the components as well as against outliers with respect to the multivariate covariance structure. Moreover, the trade-off problem between bias and variance for the optimal choice of the window width is approached by choosing the size of the window adaptively, depending on the current data situation.

Furthermore, we present an advanced algorithm of our filtering procedure that includes the replacement of missing observations in real time. Thus, the new procedure can be applied in online-monitoring practice. Applications to physiological time series from intensive care show the practical effect of the proposed filtering technique.     

Dynamic generalized extreme value modeling via particle filters

This article addresses how particle filters compare to MCMC methods for posterior density approximations of a model that allows for a dynamic state with fixed parameters and where the observation equation is nonlinear. This is a problem that was not been well studied in the specialized literature. We prove that these state and parameter estimations can be achieved via particle filter methods without the need of more expensive Forward Filtering Backward Sampling (FFBS) simulation. Estimation of a time-varying extreme value model via the generalized extreme value distribution is considered using these particle filter methods and compared to a MCMC algorithm that involves a variety of Metropolis-Hastings steps. We illustrate and compare the different methodologies with simulated data and some minimum daily stock returns occurring monthly from January 4, 1990 to December 28, 2007 using the Tokyo Stock Price Index (TOPIX).     

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.     

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.     

On the Estimation of Integrated Volatility With Jumps and Microstructure Noise

In this article, we propose a nonparametric procedure to estimate the integrated volatility of an Itô semimartingale in the presence of jumps and microstructure noise. The estimator is based on a combination of the preaveraging method and threshold technique, which serves to remove microstructure noise and jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore, asymptotic properties of the proposed estimator, such as consistency and a central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison with other alternative methods.     

Adaptive methods for sequential importance sampling with application to state space models

In this paper we discuss new adaptive proposal strategies for sequential Monte Carlo algorithms—also known as particle filters—relying on criteria evaluating the quality of the proposed particles. The choice of the proposal distribution is a major concern and can dramatically influence the quality of the estimates. Thus, we show how the long-used coefficient of variation (suggested by Kong et al. in J. Am. Stat. Assoc. 89(278–288):590–599, 1994) of the weights can be used for estimating the chi-square distance between the target and instrumental distributions of the auxiliary particle filter. As a by-product of this analysis we obtain an auxiliary adjustment multiplier weight type for which this chi-square distance is minimal. Moreover, we establish an empirical estimate of linear complexity of the Kullback-Leibler divergence between the involved distributions. Guided by these results, we discuss adaptive designing of the particle filter proposal distribution and illustrate the methods on a numerical example. This work was partly supported by the National Research Agency (ANR) under the program “ANR-05-BLAN-0299”.     

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.