Local Linear Filtering Research Articles

Estimation from a multisensor environment for systems with multiple packet dropouts and correlated measurement noises

This paper addresses the problem of distributed fusion estimation from measurements with packet dropouts and cross-correlated noises acquired from different sensors. Assuming that the packet dropouts are modelled by independent Bernoulli random variables with different characteristics for each sensor and that measurement noises are cross-correlated at the same and at consecutive sampling times, filtering and smoothing algorithms are derived using the distributed fusion method. The distributed fusion filter and smoother are obtained as a matrix-weighted linear combination of corresponding local least-squares linear estimators, verifying that the mean squared error is minimum. The local linear filtering and fixed-point smoothing algorithms are derived using the first and second-order moments of the signal and the noises present in the observation model. Simulation results are provided to illustrate the feasibility of the proposed algorithms, using the error estimation covariance matrices as measure of the quality of the estimators.

Joint state and parameter estimation of the hemodynamic model by particle smoother expectation maximization method

Objective. In this paper, we aimed for the robust estimation of the parameters and states of the hemodynamic model by using blood oxygen level dependent signal. Approach. In the fMRI literature, there are only a few successful methods that are able to make a joint estimation of the states and parameters of the hemodynamic model. In this paper, we implemented a maximum likelihood based method called the particle smoother expectation maximization (PSEM) algorithm for the joint state and parameter estimation. Main results. Former sequential Monte Carlo methods were only reliable in the hemodynamic state estimates. They were claimed to outperform the local linearization (LL) filter and the extended Kalman filter (EKF). The PSEM algorithm is compared with the most successful method called square-root cubature Kalman smoother (SCKS) for both state and parameter estimation. SCKS was found to be better than the dynamic expectation maximization (DEM) algorithm, which was shown to be a better estimator than EKF, LL and particle filters. Significance. PSEM was more accurate than SCKS for both the state and the parameter estimation. Hence, PSEM seems to be the most accurate method for the system identification and state estimation for the hemodynamic model inversion literature. This paper do not compare its results with Tikhonov-regularized Newton—CKF (TNF-CKF), a recent robust method which works in filtering sense.

Joint parameter and state estimation of the hemodynamic model by iterative extended Kalman smoother

Abstract The joint estimation of the parameters and the states of the hemodynamic model from the blood oxygen level dependent (BOLD) signal is a challenging problem. In the functional magnetic resonance imaging (fMRI) literature, quite interestingly, many proposed algorithms work only as a filtering method. This makes the estimation of hidden states and parameters less reliable compared with the algorithms that use smoothing. In standard implementations, smoothing is performed only once. However, joint state and parameter estimation can be improved substantially by iterating smoothing schemes such as the extended Kalman smoother (IEKS). In the fMRI literature, extended Kalman filtering is thought to be less accurate than standard particle filtering (PF). We compared EKF with PF and observed that the contrary is true. We improved the EKF performance by adding smoother. By iterative scheme joint hemodynamic and parameter estimation is improved substantially. We compared IEKS performance with the square-root cubature Kalman smoother (SCKS) algorithm. We show that its accuracy for the state and the parameter estimation is better and much faster than iterative SCKS. SCKS was found to be a better estimator than the dynamic expectation maximization (DEM), EKF, local linearization filter (LLF) and PF methods. We show in this paper that IEKS is a better estimator than iterative SCKS under different process and measurement noise conditions. As a result, IEKS seems to be the best method we evaluated in all aspects.

The volatility structure of the fixed income market under the HJM framework: A nonlinear filtering approach

The dynamics for interest rate processes within the well-known multi-factor Heath, Jarrow and Morton (HJM) specification are considered. Despite the flexibility of and the notable advances in theoretical research about the HJM model, the number of empirical studies of it is still very sparse. This paucity is principally due to the difficulties in estimating models in this class, which are not only high-dimensional, but also nonlinear and involve latent state variables. The estimation of a fairly broad class of HJM models as a nonlinear filtering problem is undertaken by adopting the local linearization filter, which is known to have some desirable statistical and numerical features, so enabling the estimation of the model via the maximum likelihood method. The estimator is then applied to the US, the UK and the Australian markets. Different two- and three-factor models are found to be the best for each market, with the factors being the level, the slope and the “twist” effect. The contribution of each factor towards overall variability of the interest rates and the financial reward each factor claims are found to differ considerably from one market to another.

Nonlinear continuous time modeling approaches in panel research

Stochastic differential equations (SDE) are used as dynamical models for cross‐sectional discrete time measurements (panel data). Thus causal effects are formulated on a fundamental infinitesimal time scale. Cumulated causal effects over the measurement interval can be expressed in terms of fundamental effects which are independent of the chosen sampling intervals (e.g. weekly, monthly, annually). The nonlinear continuous–discrete filter is the key tool in deriving a recursive sequence of time and measurement updates. Several approximation methods including the extended Kalman filter (EKF), higher order nonlinear filters (HNF), the local linearization filter (LLF), the unscented Kalman filter (UKF), the Gauss–Hermite filter (GHF) and generalizations (GGHF), as well as simulated filters (functional integral filter FIF) are compared.

Nonlinear local electrovascular coupling. II: From data to neuronal masses

In the companion article a local electrovascular coupling (LEVC) model was proposed to explain the continuous dynamics of electrical and vascular states within a cortical unit. These states produce certain mesoscopic reflections whose discrete time series can be reconstructed from electroencephalography (EEG) and functional magnetic resonance imaging (fMRI). In this article we develop a recursive optimization algorithm based on the local linearization (LL) filter and an innovation method to make statistical inferences about the LEVC model from both EEG and fMRI data, i.e., to estimate the unobserved states and the unknown parameters of the model. For a better understanding, the LL filter is described from a Bayesian point of view, providing the particulars for the case of hybrid data (e.g., EEG and fMRI), which could be sampled at different rates. The dynamics of the exogenous synaptic inputs going into the cortical unit are also estimated by introducing a set of Gaussian radial basis functions. In order to study the dynamics of the electrical and vascular states in the striate cortex of humans as well as their local interrelationships, we applied this algorithm to EEG and fMRI recordings obtained concurrently from two subjects while passively observing a radial checkerboard with a white/black pattern reversal. The EEG and fMRI data from the first subject was used to estimate the electrical/vascular states and parameters of the LEVC model in V1 for a 4.0 Hz reversion frequency. We used the EEG data from the second subject to investigate the changes in the dynamics of the electrical states when the frequency of reversion is varied from 0.5-4.0 Hz. Then we made use of the estimated electrical states to predict the effects on the vasculature that such variations produce.

The Volatility Structure of the Fixed Income Market Under the Hjm Framework: A Nonlinear Filtering Approach

This paper considers the dynamics for interest rate processes within a multi-factor Heath, Jarrow and Morton (1992) specification. Despite the flexibility of and the notable advances in theoretical research about the HJM models, the number of empirical studies is still inadequate. This paucity is principally because of the difficulties in estimating models in this class, which are not only high-dimensional, but also nonlinear and involve latent state variables. This paper treats the estimation of a fairly broad class of HJM models as a nonlinear filtering problem, and adopts the local linearization filter of Jimenez and Ozaki (2003), which is known to have some desirable statistical and numerical features, to estimate the model via the maximum likelihood method. The estimator is then applied to the interbank offered-rates of the U.S, U.K, Australian and Japanese markets. The two-factor model, with the factors being the level and the slope effect, is found to be a reasonable choice for all of the markets. However, the contribution of each factor towards overall variability of the interest rates and the financial reward each factor claims differ considerably from one market to another. A revised version of the paper is now published in Computational Statistics and Data Analysis, Vol. 53, Issue 6, pp. 2075-2088.

A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals

In this paper, a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent (BOLD) responses. The proposed method constitutes an alternative to the recently proposed Friston [Neuroimage 16 (2002) 513] method and has some advantages over it. The procedure is based on recent groundbreaking time series analysis techniques that have been, in this case, adopted to characterize hemodynamic responses in functional magnetic resonance imaging (fMRI). This work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons. First, our model includes physiological noise. Previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation. Secondly, by using the innovation method and the local linearization filter, not only the parameters, but also the underlying states of the system generating responses can be estimated. These states can include things like a flow-inducing signal triggered by neuronal activation, de-oxyhemoglobine, cerebral blood flow and volume. Finally, radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach, which could be essential to understanding those brain areas indirectly related to the stimulus. Hence, thirdly, by inferring about the radial basis parameters, we are able to perform a blind deconvolution, which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also, to implicitly reconstruct the underlying synaptic dynamics, induced experimentally, which caused these states variations. From this study, we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps (SPM), nonlinear BOLD phenomena and unspecific input temporal sequences must be included in the fMRI analysis.