Mohamed N. Nounou

Multiscale fuzzy Kalman filtering

Abstract Measured data are usually contaminated with errors which sometimes mask their important features. Therefore, data filtering is needed for effective utilization of such measurements. For nonlinear systems which can be described by a Takagi–Sugeno (TS) fuzzy model, several fuzzy Kalman (FK) filtering algorithms have been developed to extend Kalman filtering to such systems. Also, multiscale representation of data is a powerful data analysis tool, which has been successfully used to solve several data filtering problems. In this paper, a multiscale fuzzy Kalman (MSFK) filtering algorithm, in which multiscale representation is utilized to improve the performance of fuzzy Kalman filtering, is developed. The idea is to apply FK filtering at multiple scales to combine the advantages of the FK filter with those of the low pass filters used in multiscale data representation. Starting with a fuzzy model in the time domain, a similar fuzzy model is derived at each scale using the scaled signal approximation of the data obtained by stationary wavelet transform (SWT). These multiscale fuzzy models are then used in FK filtering, and the FK filter with the least cross validation mean square error among all scales is selected as the optimum filter. Also, theoretically, it has been shown that applying FK filtering at a coarser scale than the time domain is equivalent to using a time-averaged FK filter. Finally, the performance of the developed MSFK filtering algorithm is illustrated through a simulated example.

Multiscale finite impulse response modeling

Multiscale representation of data has shown great shrinkage abilities when used in data filtering. This paper shows that multiscale representation has similar advantages when used in empirical process modeling. One advantage is that it helps separate noise from important features, which helps improve the accuracy of estimated models. Another advantage is the fact that the number of significant cross-correlation function (CCF) coefficients relating the scaled signal approximations of the input and output data shrinks in half (i.e., decreases dyadically) at every subsequent coarser scale. This advantage is very important in FIR modeling because it means that smaller FIR models are needed at coarse scales. This advantage is exploited to develop a Multiscale Finite Impulse Response (MSFIR) modeling algorithm that helps deal with the collinearity problem often encountered in FIR models. The idea is to decompose the input-output data at multiple scales, and using the scaled signals at each scale, construct smaller FIR models with less collinearity, and then select among all scales the optimum estimated model. The developed MSFIR modeling algorithm is finally shown to outperform some of the existing FIR model estimation methods, such as Ordinary Least Squares (OLS) regression, Ridge Regression (RR), and Principal Component Regression (PCR). A key reason for the advantage of MSFIR over RR is that MSFIR shrinks the statistically insignificant CCF coefficients and noise wavelet coefficients (which are statistically zero) towards zero, while RR shrinks the FIR coefficients towards zero, while they are not. are not.

Multiscale Kernel PLS-Based Exponentially Weighted-GLRT and Its Application to Fault Detection

In this paper, we develop an improved statistical technique in order to enhance monitoring of biological processes. To improve the performance of monitoring, a detection statistic that exploits the advantages of the generalized likelihood ratio test (GLRT) statistic with those of the exponentially weighted moving average filter, kernel partial least square (KPLS) model, and multiscale representation is developed. The advantages of multiscale (MS) KPLS-based exponentially weighted GLRT (EW-GLRT) are threefold: First, the developed EW-GLRT statistic takes into account the information given by the current and previous data by giving high importance to the more recent data; second, the dynamical multiscale representation is proposed to extract accurate deterministic features and decorrelate autocorrelated measurements; third, the MSKPLS model evaluates the KPLS of the wavelet coefficients at each scale. Due to its multiscale nature, MSKPLS is appropriate for modeling of data that contain contributions from events whose behavior changes over time and frequency. The detection performance is studied using Cad System in E. coli model and genomic copy number data for detecting small and moderate shifts. MSKPLS-based EW-GLRT is used to enhance fault detection of the Cad System in E. coli model through monitoring some of the key variables involved in this model, such as enzymes, lysine, and cadaverine. The proposed technique is also applied to detect diseases using genomic copy number data through better detection of aberrations in the genetic information of patients, which can help medical doctors make more accurate diagnosis of diseases.

Process Monitoring Using Data-Based Fault Detection Techniques: Comparative Studies

Data based monitoring methods are often utilized to carry out fault detection (FD) when process models may not necessarily be available. The partial least square (PLS) and principle component analysis (PCA) are two basic types of multivariate FD methods, however, both of them can only be used to monitor linear processes. Among these extended data based methods, the kernel PCA (KPCA) and kernel PLS (KPLS) are the most well-known and widely adopted. KPCA and KPLS models have several advantages, since, they do not require nonlinear optimization, and only the solution of an eigenvalue problem is required. Also, they provide a better understanding of what kind of nonlinear features are extracted: the number of the principal components (PCs) in a feature space is fixed a priori by selecting the appropriate kernel function. Therefore, the objective of this work is to use KPCA and KPLS techniques to monitor nonlinear data. The improved FD performance of KPCA and KPLS is illustrated through two simulated examples, one using synthetic data and the other using simulated continuously stirred tank reactor (CSTR) data. The results demonstrate that both KPCA and KPLS methods are able to provide better detection compared to the linear versions.

Nonlinear State and Parameter Estimation Using Iterated Sigma Point Kalman Filter: Comparative Studies

In this chapter, iterated sigma‐point Kalman filter (ISPKF) methods are used for nonlinear state variable and model parameter estimation. Different conventional state estimation methods, namely the unscented Kalman filter (UKF), the central difference Kalman filter (CDKF), the square‐root unscented Kalman filter (SRUKF), the square‐ root central difference Kalman filter (SRCDKF), the iterated unscented Kalman filter (IUKF), the iterated central difference Kalman filter (ICDKF), the iterated square‐root unscented Kalman filter (ISRUKF) and the iterated square‐root central difference Kalman filter (ISRCDKF) are evaluated through a simulation example with two comparative studies in terms of state accuracies, estimation errors and convergence. The state variables are estimated in the first comparative study, from noisy measure‐ ments with the several estimation methods. Then, in the next comparative study, both of states and parameters are estimated, and are compared by calculating the estimation root mean square error (RMSE) with the noise‐free data. The impacts of the practical challenges (measurement noise and number of estimated states/ parameters) on the performances of the estimation techniques are investigated. The results of both comparative studies reveal that the ISRCDKF method provides better estimation accuracy than the IUKF, ICDKF and ISRUKF. Also the previous methods provide better accuracy than the UKF, CDKF, SRUKF and SRCDKF techniques. The ISRCDKF method provides accuracy over the other different estimation techniques; by iterating maximum a posteriori estimate around the updated state, it re‐linearizes the measurement equation instead of depending on the predicted state. The results also represent that estimating more parameters impacts the estimation accuracy as well as the convergence of the estimated parameters and states. The ISRCDKF provides improved state accuracies than the other techniques even with abrupt changes in estimated states.

Dealing with Collinearity in FIR models using Multiscale Estimation

In this paper, multiscale representation of data is utilized to reduce the collinearity problem often encountered in Finite Impulse Response (FIR) modeling. The idea is to decompose the input-output data at multiple scales, use the scaled signal approximations of the data to construct a FIR model at each scale, and then select among all scales the optimum estimated FIR model. The rationale behind this approach is that the number of significant cross correlation function (CCF) coefficients estimated using the scaled signal approximations of the input-output data decreases at coarser scales. This means that more parsimonious FIR models, with less collinearity and improved estimation accuracy, can be constructed at coarser scales. Of course, the estimation accuracy will deteriorate at very coarse scales. Therefore, it is very important to select the most appropriate scale for modeling purposes, which can be done by selecting the scale which results in the maximum prediction signal to noise ratio. The developed multiscale FIR modeling approach is shown to outperform existing methods, such as ordinary least squares (OLS) regression and ridge regression (RR).

Enhanced prediction accuracy of fuzzy models using multiscale estimation

The presence of measurement noise in the data used in empirical modeling can have a drastic effect on the accuracy of estimated models, and thus need to be removed for improved model accuracy. Multiscale representation of data has shown great noise-removal ability when used in data filtering. In this paper, this ability is exploited to improve the prediction accuracy of the Takagi-Sugeno (TS) fuzzy model by developing a multiscale fuzzy (MSF) system identification algorithm. The algorithm relies on constructing multiple fuzzy models at multiple scales using the scaled signal approximations of the input-output data, and then selecting the optimum multiscale model which maximizes the prediction signal-to-noise ratio. The developed algorithm is shown to outperform its time domain counterpart through a simulated example.

Empirical Bayesian finite impulse response modeling

This paper presents a Bayesian modeling technique, called empirical Bayesian finite response (EBFIR) modeling, that helps deal with the collinearity problem usually encountered in FIR models, and helps improve the estimation accuracy of their coefficients. The developed technique iteratively solves for the prior density used in estimation and the FIR coefficients. The advantages of the developed EBFIR modeling technique are also illustrated though a simulated example.