Stochastic State Space Model Research Articles

Runoff modelling using radar data and flow measurements in a stochastic state space approach

In urban drainage the estimation of runoff with the help of models is a complex task. This is in part due to the fact that rainfall, the most important input to urban drainage modelling, is highly uncertain. Added to the uncertainty of rainfall is the complexity of performing accurate flow measurements. In terms of deterministic modelling techniques these are needed for calibration and evaluation of the applied model. Therefore, the uncertainties of rainfall and flow measurements have a severe impact on the model parameters and results. To overcome these problems a new methodology has been developed which is based on simple rain plane and runoff models that are incorporated into a stochastic state space model approach. The state estimation is done by using the extended Kalman filter in combination with a maximum likelihood criterion and an off-line optimization routine. This paper presents the results of this new methodology with respect to the combined consideration of uncertainties in distributed rainfall derived from radar data and uncertainties in measured flows in an urban catchment within the Emscher river basin, Germany.

An analysis of hippocampal spatio-temporal representations using a Bayesian algorithm for neural spike train decoding

Neural spike train decoding algorithms are important tools for characterizing how ensembles of neurons represent biological signals. We present a Bayesian neural spike train decoding algorithm based on a point process model of individual neurons, a linear stochastic state-space model of the biological signal, and a temporal latency parameter. The latency parameter represents the temporal lead or lag between the biological signal and the ensemble spiking activity. We use the algorithm to study whether the representation of position by the ensemble spiking activity of pyramidal neurons in the CA1 region of the rat hippocampus is more consistent with prospective coding, i.e., future position, or retrospective coding, past position. Using 44 simultaneously recorded neurons and an ensemble delay latency of 400 ms, the median decoding error was 5.1 cm during 10 min of foraging in an open circular environment. The true coverage probability for the algorithm's 0.95 confidence regions was 0.71. These results illustrate how the Bayesian neural spike train decoding paradigm may be used to investigate spatio-temporal representations of position by an ensemble of hippocampal neurons.

Combined use of point rain gauges, radar, microwave link and level measurements in urban hydrological modelling

A methodology has been developed for the combined exploitation of the information content in several different types of rainfall and hydrological measurements. The methodology is based on simple rain plane and runoff models that are incorporated into a stochastic state–space model approach. State estimation is done using the extended Kalman filter in combination with a maximum likelihood estimation criterion and an off-line optimisation routine. This paper presents the preliminary results of the application of the methodology for the combined use of multiple rainfall and runoff data on a 4800 ha urban catchment within the Emscher river basin in Germany. The time series used come from 12 rain gauges, a C-band and an X-band radar covering the same area and a microwave link. The preliminary results support the methodology's potential as a technique for the combined use of different types of rainfall and runoff data. The results are discussed in terms of applicability, the methodology's limitations and future improvements.

LOCAL LINEAR FORECASTS USING CUBIC SMOOTHING SPLINES

SummaryThis paper shows how cubic smoothing splines fitted to univariate time series data can be used to obtain local linear forecasts. The approach is based on a stochastic state‐space model which allows the use of likelihoods for estimating the smoothing parameter, and which enables easy construction of prediction intervals. The paper shows that the model is a special case of an ARIMA(0, 2, 2) model; it provides a simple upper bound for the smoothing parameter to ensure an invertible model; and it demonstrates that the spline model is not a special case of Holt's local linear trend method. The paper compares the spline forecasts with Holt's forecasts and those obtained from the full ARIMA(0, 2, 2) model, showing that the restricted parameter space does not impair forecast performance. The advantage of this approach over a full ARIMA(0, 2, 2) model is that it gives a smooth trend estimate as well as a linear forecast function.

NEW RECURSIVE LEAST SQUARE ALGORITHMS WITHOUT USING THE INITIAL INFORMATION

In this paper, new types of recursive least square (RLS) algorithms, without using the initial information of a parameter or a state to be estimated, are proposed. The proposed RLS algorithm is first obtained for a generic linear model and is then extended to a state estimator for a stochastic state-space model. Compared with the existing algorithms, the proposed RLS algorithms are simpler and more numerically stable. It is shown, by simulation studies, that the proposed RLS algorithms have better numerical stability for digital computation than existing algorithms.

Estimation of parameters in a linear state space model using a Rao-Blackwellised particle filter

A Rao-Blackwellised particle filter is used in the estimation of the parameters of a linear stochastic state space model. The proposed method combines the particle filtering technique with a Kalman filter using marginalisation so as to make full use of the analytically tractable structure of the model. Simulation studies are performed on a simple illustrative example and the results demonstrate the effectiveness of the proposed method in comparison with the conventional extended-Kalman-filler-based method. The proposed method is then applied in the estimation of the parameters in a railway vehicle dynamic model for condition monitoring and the desired results have been obtained.

Computer-Intensive Time-Varying Model Approach to the Systematic Risk of Australian Industrial Stock Returns

This paper aims to investigate the form of systematic risk of Australian industrial stock returns. We suggest using four stochastic state-space models for the analysis. The stochastic properties of systematic risk are studied by examining four classes of state-space models: random walk model, random coefficient model, ARMA(1, 1) model and mean reverting model (or moving mean model). We have found that the industrial portfolio betas are unstable. The variation of industrial portfolio beta is either random or mean-reverting. Among the nineteen industrial groups, ten of them have the mean-reverting process betas but six of them seem to have a moving long-term mean. Five of the industrial groups have the random process betas, more specifically; the betas of three of them are the random walk processes while the betas of the other two are just the random coefficients. We have also identified that the betas of five industrial groups seem to follow an ARMR(1,1) process.

Process monitoring for continuous process with periodic characteristics

AbstractApplication of conventional statistical monitoring methods to periodic processes can result in frequent false alarms and/or missed faults due to the non‐stationary behavior seen over a period. To address this problem, we propose to identify and use a stochastic state space model that describes the statistical behavior of changes occurring from period to period. This model, when retooled as a periodically time‐varying model, can be used for on‐line monitoring and estimation with the aid of a Kalman filter. The same model can also be used for inferential estimation of the variables that are difficult or slow to measure on‐line. The proposed approach is applied to a simulation benchmark of a waste water treatment process, which exhibits strong diurnal changes in the feed stream, and is compared against the principal component analysis (PCA) and partial least squares (PLS) methods. Copyright © 2004 John Wiley & Sons, Ltd.

Identification of continuous time models using discrete time data

Continuous-discrete stochastic state space models in the form of nonlinear partially observed Itô stochastic differential equations (SDE's) with measurement noise are advocated for modelling dynamic systems in continuous time using discrete time data. Such models provide a decomposition of the noise affecting the system into a process noise term and a measurement noise term, and this prediction error decomposition (PED) allows unknown parameters to be estimated from experimental data in a prediction error (PE) setting, which gives less biased and more reproductible results in the presence of significant process noise than the more commonly used output error (OE) setting. To facilitate the use of continuous-discrete stochastic state space models, a PE estimation scheme that features maximum likelihood (ML) and maximum a posteriori (MAP) estimation is presented along with a software implementation. To illustrate the superiority of PE estimation over OE estimation a case study is given, which demonstrates the higher sensitivity of OE estimates to process noise.

Model Validation in Non-Linear Continuous-Discrete Grey-Box Models

The validation part of the modeling procedure is used in the iterative process of model building and as an acceptance test of the obtained model. This paper presents a number of tools for validating non-linear continuous-discrete time stochastic state space models. For equidistantly sampled data a minor transformation of the one-step prediction errors is appropriate and the tools applied can also be used to suggest extensions of the model. Alternatively, we may regard the diffusion term as unmodelled systematic variation and in this case use the estimated diffusion as a tool to suggest possible sources of such variation. Finally, in the general case we may apply a transformation, based on numerical solution of the Fokker-Planck equation, in order to obtain standardized residuals which under the model are i.i.d. Gaussian. Standard tests can then be applied to these residuals.

A Nonlinear State Space Model for the Blood Glucose Metabolism of a Diabetic (Ein nichtlineares Zustandsraummodell für den Blutglukosemetabolismus eines Diabetikers)

The blood glucose metabolism of a diabetic is a complex nonlinear process closely linked to a number of internal factors which are not easily accessible to measurement. Based on accessible information – such as occasional blood glucose measurements and information about food intake and physical exercise – the system appears highly stochastic and the quantity of main interest, the blood glucose concentration, is very difficult to model and to predict. In this paper we describe a stochastic nonlinear state space model for modeling the blood glucose concentration of a diabetic patient. The model structure is based on physiological prior knowledge and the main nonlinearities are modeled using artificial neural networks. Offline training of the model is performed using a newly developed Monte-Carlo generalized EM (expectation maximization) algorithm. Online prediction is performed using particle filters. Our experimental results show that our approach provides better prediction results than a number of competing approaches.

Subspace identification for continuous-time stochastic systems via distribution-based approach

This paper presents a novel approach for system identification of continuous-time stochastic state space models from random input–output continuous data. The approach is based on the introduction of random distribution theory in describing (higher) time derivatives of stochastic processes, and the input–output algebraic relationship is derived which is treated in the time-domain. The efficacy of the approach is examined by comparing with other approaches employing the filters.

A STOCHASTIC REALISATION ALGORITHM WITH APPLICATION TO MODAL PARAMETER ESTIMATION

This paper presents procedures that allow us to estimate the frequencies and damping coefficients of a vibrating system in time domain, from multioutput data only. The system is excited by a random force and a stochastic state space model is considered. Two algorithms are used to determine the transition matrix of the system which contains all modal information. The first algorithm uses a forward-innovation representation, and the transition matrix is connected to the controllability matrix of the system. The second algorithm uses a backward-innovation representation and the transition matrix is used to the observability matrix of the system. A numerical example is treated and compare these algorithms.

Estimating and Testing an Exponential-Affine Term Structure Model by Nonlinear Filtering

In general, the interest rate is not directly observable in the financial market. Short term interest rates are quoted in the money market for maturities up to approximately one year, but longer term interest rates are traded only indirectly through the bond market. In this paper the spot interest rate is described by a bivariate stochastic differential equation state space model that gives rise to an exponential-affine term structure model. We propose a new maximum likelihood method for estimating parameters and interest rates in stochastic differential equations from observed coupon-bearing bond prices. The method utilizes continuous discrete second order nonlinear filtering techniques. As a preliminary analysis the method is applied to a cross-section of time series of Danish government bond prices.

Continuous time state space modeling of panel data by means of sem

Maximum likelihood parameter estimation of the continuous time linear stochastic state space model is considered on the basis of largeN discrete time data using a structural equation modeling (SEM) program. Random subject effects are allowed to be part of the model. The exact discrete model (EDM) is employed which links the discrete time model parameters to the underlying continuous time model parameters by means of nonlinear restrictions. The EDM is generalized to cover not only time-invariant parameters but also the cases of stepwise time-varying (piecewise time-invariant) parameters and parameters varying continuously over time according to a general polynomial scheme. The identification of the continuous time parameters is discussed and an educational example is presented.

Subspace-based Identification for Continuous-Time Stochastic Systems via Distribution-Theoretic Approach

This paper presents a novel approach for system identification of continuous-time stochastic state space models from random input-output continuous data. The approach is based on the introduction of the random distribution theory in describing the (higher) time derivatives of stochastic processes, and the input-output algebraic relationship is derived which is treated in the time-domain. The efficacy of the approach proposed is examined by comparing with other approaches employing the filters.

Dynamic harmonic regression

This paper describes in detail a flexible approach to nonstationary time series analysis based on a Dynamic Harmonic Regression (DHR) model of the Unobserved Components (UC) type, formulated within a stochastic state space setting. The model is particularly useful for adaptive seasonal adjustment, signal extraction and interpolation over gaps, as well as forecasting or backcasting. The Kalman Filter and Fixed Interval Smoothing algorithms are exploited for estimating the various components, with the Noise Variance Ratio and other hyperparameters in the stochastic state space model estimated by a novel optimization method in the frequency domain. Unlike other approaches of this general type, which normally exploit Maximum Likelihood methods, this optimization procedure is based on a cost function defined in terms of the difference between the logarithmic pseudo-spectrum of the DHR model and the logarithmic autoregressive spectrum of the time series. The cost function not only seems to yield improved convergence characteristics when compared with the alternative ML cost function, but it also has much reduced numerical requirements. Copyright © 1999 John Wiley & Sons, Ltd.

A modern approach to catch-age analysis for Hecate Strait rock sole ( Pleuronectes bilineatus)

We analyse catch-age data for Hecate Strait rock sole ( Pleuronectes bilineatus) over the 1945–1995 period. We employ a stochastic state-space model to assess uncertainty in stock abundance and recruitment trends. Rock sole biomass appears to have increased substantially since 1945. However, the apparent increase in biomass may be an artifact of inconsistencies in the age proportion data. Broad confidence intervals for model estimates of spawning stock biomass and corresponding age-4 recruits illustrate the uncertainty in the stock-recruitment relationship. This uncertainty must be acknowledged when results from catch-age analysis are translated into fishery yields. Thus, the tools we demonstrate here provide one step in implementing the precautionary approach to fish stock assessment.

Predictive Control of Air Temperature in Greenhouses

This paper describes a new method for improving the control of air temperature in greenhouses. The proposed controller is an extended version of the generalized predictive controller (GPC). The prediction of air temperature is facilitated by a continuous time stochastic state space model identified for the heat dynamics of the greenhouse. A simulation study illustrates that the proposed GPC gives a more active controller with less control error than the implemented PID controller.

State space model for river temperature prediction

A stochastic state space model for the estimation of river temperature is presented, setting the stage for its implementation in optimal control algorithms. Physical processes modeled include advection, diffusion, and environmental heat exchange; the mathematical formulation is one dimensional. The deterministic formulation uses a hybrid characteristics‐finite differences numerical scheme for the solution of the governing equations. The stochastic formulation accounts for uncertainty due to model assumptions, errors in the model inputs and parameters, and river temperature measurements. Model use is demonstrated by the estimation of temperatures in a 5‐mile (8 km) reach of the Des Plaines River, Illinois, below the Joliet power station. River temperature measurements and hydrological and meteorological data are used to estimate the model parameters; model formulation and limitations are assessed based on the model application results. The stochastic formulation improves river temperature estimation, as measured by the mean, variance‐covariance, correlation, and range of the residuals.