Nonlinear Filtering Theory Research Articles

ARTIFICIAL NEURAL NETWORK MODELING FOR REFORESTATION DESIGN THROUGH THE DOMINANT TREES BOLE‐VOLUME ESTIMATION

Abstract In the management of restoration reforestations or recreational reforestations of trees, the density of the planted trees and the site conditions can influence the growth and bole volume of the dominant tree. The ability to influence growth of these trees in a reforestation contributes greatly to the formation of large dimension trees and thereby to the production of commercially valuable wood. The potential of two artificial neural network (ANN) architectures in modeling the dominant Pinus brutia tree bole volume in reforestation configuration at 12 years of age was investigated: (1) the multilayer perceptron architecture using a back‐propagation algorithm and (2) the cascade‐correlation architecture, utilizing (a) either the nonlinear Kalman's filter theory or (b) the adaptive gradient descent learning rule. The incentive for developing bole‐volume equations using ANN techniques was to demonstrate an alternative new methodology in the field of reforestation design, which would enable estimation and optimization of the bole volume of dominant trees in reforestations using easily measurable site and competition factors. The usage of the ANNs for the estimation of dominant tree bole volume through site and competition factors can be a very useful tool in forest management practice.

Determining a stable relationship between hedge fund index HFRI-Equity and S&P 500 behaviour, using filtering and maximum likelihood

In this article we test the ability of the stochastic differential model proposed by Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is, of the Hedge Fund Research performance Index (HFRI)-Equity index and of the S&P 500 (Standard & Poor 500 New York Stock Exchange) index, respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI-Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return of the market index can be satisfactorily modelled using the Heston stochastic volatility model. The model consists of a system of three stochastic differential equations, two of them are the Heston stochastic volatility model and the third one is the equation that models the behaviour of the hedge fund index and its relation with the market index. The model is calibrated on observed data using a method based on filtering and maximum likelihood proposed by Mariani et al. [Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory, Opt. Lett., 2 (2008), pp. 177–222] and further developed in Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362; The calibration of the Heston stochastic volatility model using filtering and maximum likelihood methodsin Proceedings of Dynamic Systems and Applications, Vol. 5, G.S. Ladde, N.G. Medhin, C. Peng, and M. Sambandham, eds., Dynamic Publishers, Atlanta, USA, 2008, pp. 170–181]. That is, an inverse problem for the stochastic dynamical system representing the model is solved using the calibration procedure. The data analysed is from January 1990 to June 2007, and are monthly data. For each observation time, they consist of the value at the observation time of the log-returns of the HFRI-Equity and of the S&P 500 indices. The calibration procedure uses appropriate subsets of data, that is the data observed in a 6 months time period. The 6 months data time period used in the calibration is rolled through the time series generating a sequence of calibration problems. The values of the HFRI-Equity and S&P 500 indices forecasted using the calibrated models are compared to the values of the indices observed. The result of the comparison is very satisfactory. The website http://www.econ.univpm.it/recchioni/finance/w8 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of some of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

Non-linear filtering for a dust-perturbed two-body model

In standard textbooks on classical mechanics, the two-body central forcing problem is formulated as a system of the coupled non-linear second-order deterministic differential equations. Uncertainties, introduced by the astronomical ‘dust’, are not assumed in the orbit dynamics. The dust population produces an additional random force on the orbiting particle. This work is a continuation of the paper (Sharma and Parthasarathy, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463:979–1003, [2007]) in which the authors developed and analyzed the dust-perturbed two-body model, which accounts for the dust perturbation felt by the orbiting particle. The theory of the dust-perturbed stochastic system was developed using the Fokker–Planck equation. This paper discusses the problem of realizing non-linear stochastic filters for estimating the states of the dust-perturbed planar two-body stochastic system, especially from noisy observations. This paper utilizes the Kushner’s theory of non-linear filtering, which involves stochastic observation term in the evolution of conditional probability density, for deriving the stochastic evolutions of the conditional mean and conditional covariance. The effectiveness of the non-linear filters of this paper is examined on the basis of their ability to preserve the perturbation effect, less random fluctuations in the mean trajectory and stability characteristics in the mean and variance trajectories. Most notably, this paper reveals the efficacy of the second-order approximate Kushner filter for the estimation procedure in contrast to the first-order approximate filter. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.

Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: an application of nonlinear filtering theory

Let us suppose that the dynamics of the stock prices and of their stochastic variance is described by the Heston model, that is by a system of two stochastic differential equations with a suitable initial condition. The aim of this paper is to estimate the parameters of the Heston model and one component of the initial condition, that is the initial stochastic variance, from the knowledge of the stock and option prices observed at discrete times. The option prices considered refer to an European call on the stock whose prices are described by the Heston model. The method proposed to solve this problem is based on a filtering technique to construct a likelihood function and on the maximization of the likelihood function obtained. The estimated parameters and initial value component are characterized as being a maximizer of the likelihood function subject to some constraints. The solution of the filtering problem, used to construct the likelihood function, is based on an integral representation of the fundamental solution of the Fokker–Planck equation associated to the Heston model, on the use of the wavelet expansions presented in (Fatone et al. in High performance algorithms based on a new wavelet expansion for time dependent acoustic obstacle scattering. Commun. Computat. Phys. (2007), Research Developments in Acoustics, vol. 2, pp. 39–69. Transworld Research Network, Kerala (2005), New wavelet bases made of piecewise polynomial functions: approximation theory, quadrature rules and applications to kernel sparsification and image compression. SIAM J. Sci. Comput. (submitted)) to approximate the integral kernel appearing in the representation formula of the fundamental solution, on a simple truncation procedure to exploit the sparsifying properties of the wavelet expansions and on the use of the fast Fourier transform (FFT). The use of these techniques generates a very efficient and fully parallelizable numerical procedure to solve the filtering problem, this last fact makes possible to evaluate very efficiently the likelihood function and its gradient. As a byproduct of the solution of the filtering problem we have developed a stochastic variance tracking technique that gives very good results in numerical experiments. The maximum likelihood problem used in the estimation procedure is a low dimensional constrained optimization problem, its solution with ad hoc techniques is justified by the computational cost of evaluating the likelihood function and its gradient. We use parallel computing and a variable metric steepest ascent method to solve the maximum likelihood problem. Some numerical examples of the estimation problem using synthetic and real data, that is data relative to an index of the Milano stock exchange (S&PMIB30), obtained with a parallel implementation of the previous numerical method are presented. Very impressive speed up factors are obtained in the numerical examples using the parallel implementation of the numerical method proposed. The website: http://www.econ.univpm.it/pacelli/mariani/finance/w1 contains animations and some auxiliary material that helps the understanding of this paper and makes available to the interested users the computer programs used to produce the numerical experience presented.

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.

Self-tuning and stochastic resonance in a simple threshold system—a filter theory approach

Fully nonlinear filter theory is applied to a simple threshold system, which plays the role of an observer of input signals from a Brownian dynamical system. Our filtering results are discussed within the frameworks of both stochastic resonance and self-tuning of the threshold values, which gathers a lot of attention in relation to information processing in auditory systems.

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported.

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhar’s work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.

Optimal quadratic quantization for numerics: the Gaussian case

Optimal quantization has been recently revisited in multi-dimensional numerical integration, multi-asset American option pricing, control theory and nonlinear filtering theory. In this paper, we enlighten some numerical procedures in order to get some accurate optimal quadratic quantization of the Gaussian distribution in one and higher dimensions. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ≥ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed from mathematical finance are used to test the accuracy of our Gaussian optimal quantizers.

Optimal quadratic quantization for numerics: the Gaussian case

Optimal quantization has been recently revisited in numerical integration in high dimension (see [18]), in Control Theory (see [19]) and in Nonlinear Filtering Theory (see [20]). In this paper, we enlight numerical procedures in order to get optimal quadratic quantization of the Gaussian law. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ‚ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed to mathematical finance are provided.

Non-Linear Kalman Filter in Simulation Modelling of Hydrologic Cycles in the Basin of Lake Prespa

The branch of systems theory in the state space aimed at the theory of process state estimation modelling appeared to play a keyrole, and has given incentives to be explored in applications to a variety of environmental problems. This paper presents a simulation model for medium and longer-term dynamics of water level of Prespanean Lake from precipitation to evaporation in a hydrologic cycle. Based on the observation data, a system-theoretic state estimation method using theory of non-linear Kalrnan filters is developed and applied within the model.

When is a MAP poisson?

The departure process of a queue is important in the analysis of networks of queues, as it may be the arrival process to another queue in the network. A simple description of the departure process could enable a tractable analysis of a network, saving costly simulation or avoiding the errors of approximation techniques. In a recent paper, Olivier and Walrand [1] conjectured that the departure process of a MAP/ PH/1 queue is not a MAP unless the queue is a stationary M/M/1 queue. This conjecture was prompted by their claim that the departure process of an MMPP/ M/1 queue is not a MAP unless the queue is a stationary M/M/1 queue. We note that their proof has an algebraic error, see [2], which leaves the above question of whether the departure process of an MMPP/ PH/1 queue is a MAP, still open. There is also a more fundamental problem with Olivier and Walrand's proof. In order to identify stationary M/M/1 queues, it is essential to be able determine from its generator when a stationary MAP is a Poisson process. This is not discussed in [1], nor does it appear to have been discussed elsewhere in the literature. This deficiency is remedied using ideas from nonlinear filtering theory, to give a characterisation as to when a stationary MAP is a Poisson process.

EXTENSION OF KALMAN FILTER THEORY TO NONLINEAR SYSTEMS WITH APPLICATION TO WING ROCK MOTION

ABSTRACTA formulation for nonlinear Kalman filter theory is developed in this paper. It is shown that the solution of the nonlinear Kalman filter problem is governed by a Hamilton‐Jacobi‐Bellman inequality (HJBI). Choosing the closed loop Lyapunov function in a symmetric matrix form of the state vector results in the reduction of the HJBI to an algebraic Riccati inequality along with several other algebraic inequalities. These inequalities are formulated into a series of closed loop Lyapunov inequalities which are turned into equalities by adding a positive state vector function. Closed loop Lyapunov functions are then obtained successively by solving these eqauations based on the powers of the state vector. Control of a nonlinear wing rock motion is employed as an example to illustrate the theory.

A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts

Abstract Knowledge of the probability distribution of initial conditions is central to almost all practical studies of predictability and to improvements in stochastic prediction of the atmosphere. Traditionally, data assimilation for atmospheric predictability or prediction experiments has attempted to find a single “best” estimate of the initial state. Additional information about the initial condition probability distribution is then obtained primarily through heuristic techniques that attempt to generate representative perturbations around the best estimate. However, a classical theory for generating an estimate of the complete probability distribution of an initial state given a set of observations exists. This nonlinear filtering theory can be applied to unify the data assimilation and ensemble generation problem and to produce superior estimates of the probability distribution of the initial state of the atmosphere (or ocean) on regional or global scales. A Monte Carlo implementation of the fully n...

Characterization of the optimal filter: the non markov case

We characterize the optimal filter in the nonlinear filtering theory as the unique solution to the Zakai equation. The results are very general as we allow the function h appearing in the filtering model to be discontinuous and unbounded. We consider the standard Markov model as well as the case when the signal X and the observation Y are solutions of a stochastic differential equation where the coefficients are allowed to depend on the past of Y. This is done via some results on existence of stationary solutions and solutions corresponding to certain measure valued evolution equations for controlled (and uncontrolled) martingale problems

Classification of four-dimensional estimation algebras

Of central importance in nonlinear filtering theory is the classification of finite-dimensional estimation algebras, through which we can construct recursive filters. We classify all t-dimensional estimation algebras, t/spl les/4, for arbitrary state space dimension.

Fractional Brownian motion: theory and applications

We present new theoretical results on the fractional Brownian motion, including different definitions (and their relationships) of the stochastic integral with respect to this process, Girsanov theorem, Clark representation formula, Ito formula and so on. Several applications (such as non-linear filtering theory, queuing networks, mathematical finance) are given as applied examples of these theoretical results.

Parameter estimation for the piecewice-polynomial mean value trend of a stochastic time series

We consider the problem of mean value trend estimation for a stochastic time series in a mixture with noise. The trend is simulated as a polynomial function of time with parameters changing discontinuously at a random instant. The proposed solution of this problem is based on the optimal nonlinear filtering theory for pulsed Markovian processes. An algorithm for approximate estimation of piesewise-linear trend parameters is devised as an example. The results of numerical simulation of a synthesized algorithm are presented.

Nonlinear filtering: Interacting particle resolution

In this Note, we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as in Propagation of Chaos Theory ( see [12] and [19]), and in Nonlinear Filtering Theory. The main contribution of this Note is to prove the convergences to the optimal filter of such approximations, yielding what seemed to be the first mathematically well-founded convergence results for such approximations of the nonlinear filtering equations. This new treatment was influenced primarily by the development of genetic algorithms ( see [16] and [3]), and secondarily by the papers of H. Kunita and L. Stettner, [17] and [18] respectively.

Registration of mobile sensors using the parallelized extended Kalman filter

The problem of registration for multiple mobile sensors in a fusion system is considered. We develop a registration model for mobile sensors and implement the registration based on the nonlinear Kalman filter theory. A parallelized extended Kalman filter (EKF) structure is also derived in which the algorithm is parallelized and processed at each sensor. The parallelized EKF is shown to have the performance of a centralized system. It does not rely on a single central facility to perform and thus is less sensitive to sensor defects than the centralized system. Finally, computer simulations are used to show the effectiveness of the proposed registration algorithm.