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In this paper, the Polynomial NonLinear State Space (PNLSS) approach is applied to model a nonlinear system with a Wiener-Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behavior. The identification procedure is applied to measurement data

This paper describes the identification of Wiener-Hammerstein models and two recently suggested algorithms are applied to the SYSID'09 benchmark data. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks so that local minima are avoided. Both of the considered algorithms obtain good initial estimates by using the best linear approximation (BLA) which can easily be estimated from data. Given the BLA, the two algorithms differ in the way the dynamics are separated into two linear parts. The first algorithm simply considers all possible splits of the dynamics. Each of the splits is used to initialize one Wiener-Hammerstein model using linear least-squares and the best performing model is selected. In the second algorithm, both linear blocks are initialized with the entire BLA model using basis function expansions of the poles and zeros of the BLA. This gives over-parameterized linear blocks and their order is decreased in a model reduction step. Both algorithms are explained and their properties are discussed. They both give good, comparable models on the benchmark data.

The general approach for modeling systems assumes that the measured signals are (weakly) stationary, i.e., the power spectrum is time invariant. However, the stationarity assumption is violated when: 1) transient effects due to experimental conditions are dominant; 2) data are missing due to, for instance, sensor failure; or 3) the amplitude of the excitation signals smoothly varies over time due to, for instance, actuator problems. Although different methods exist to deal with each of these nonstationary effects specifically, no unified approach is available. In this paper, a new and general technique is presented to handle nonstationary effects, based on processing overlapping subrecords of the measured data. The proposed method is a simple preprocessing step where the user does not need to specify which nonstationary effect is present, nor the time interval where the nonstationary effect appears. The merits of the proposed approach are demonstrated on an operational wireless system suffering from interrupted link effects.

In this paper, we propose an identification procedure for nonlinear feedback systems by introducing a new type of model structure. The idea is to combine the advantages of two different model structures: the Polynomial NonLinear State Space (PNLSS) model and the block-structured feedback model. By imposing structural information to the existing PNLSS model, we obtain a flexible model with a lower number of parameters compared with the unconstrained structure. The structured-PNLSS model consists of a linear block in the feedforward path and a PNLSS block in the feedback path. The presented identification procedure for the structured-PNLSS model is applied to input/output measurements of a nonlinear device.

In this paper, we focus on the nonlinear modelling of Wiener-Hammerstein structures. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks. Here, good initial estimates are obtained by using the Best Linear Approximation of the system under test, which can easily be extracted from the data. The idea is to write the linear dynamic blocks as a linear combination of basis functions containing the poles and the zeros of the Best Linear Approximation. Next, these initial values are further tuned via a nonlinear optimization procedure. The proposed identification method is applied to real measurement data.

Based on simple Frequency Response Function (FRF) measurements, we give the user some guidance in the selection of an appropriate nonlinear block structure for the system to be modeled. The method consists in measuring the FRF using a Gaussian-like input signal and varying in a first experiment the root-mean-square (rms) value of this signal while maintaining the coloring of the power spectrum. Next, in a second experiment, the coloring of the power spectrum is varied while keeping the rms value constant. Based on the resulting behavior of the FRF, an appropriate nonlinear block structure can be selected to approximate the real system. The identification of the selected block-oriented model itself is not addressed in this paper. A theoretical analysis and two practical applications of this structure identification method are presented for some nonlinear block structures.

Dynamical systems describing a physical process with a dominant diffusion phenomenon require a large dimensional model due to their long memory. Without prior knowledge, it is however not straightforward to know if/whether one deals with a fractional order system or long memory effects. Since the parametric modeling of a fractional system is very involved, we tackle the question whether fractional insight can be gathered in a non-parametric way. In this paper we show that the classical Fourier basis leading to the Frequency Response Function (FRF) lacks fractional insight. Therefore, we introduce a TaylorFourier basis to obtain non-parametric insight in the fractional system. This analysis proposes a novel type of spectrum to visualize the spectral content of a fractional system: the Taylor-Fourier spectrum.

When analyzing fMRI data to study the brain process, one faces two challenges: (i) the correct noise distribution and (ii) the brain dynamics. In general, the brain dynamics are modeled under the simplifying, but wrong assumption that the noise follows a Gaussian distribution. In this paper, we model the brain dynamics under the correct Rice distribution. We implement the hemodynamic response function into a Rice framework and apply the standard General Linear Model (GLM) which is linear-in-the-parameters and can easily be solved. Next, the statistical properties of the least squares estimator are investigated via a simulation experiment.

Engineers find interpreting plots of a measured physical variable more straightforward than doing a formal statistical analysis. The default choice to display the data behavior is the histogram. The histogram’s performance has proved to be sufficient. However, histograms have a number of limitations including sensitivity to the binwidth and a non-physical roughness. Over the past years, statisticians have developed different techniques to address these problems. These techniques provide a much clearer visualization of the probability density and a more accurate estimation of the statistical properties of the measured data. Despite their increasing use in other fields, these techniques are rarely used in the measurement community. For instance, most measurement instruments provide histograms only. This review article revisits these techniques from an engineer viewpoint to encourage its use. Different examples that include known and unknown densities result in practical guidelines that help the measurement engineer to visualize the probability content.

The problem of detecting significant brain activity upon stimulus in functional Magnetic Resonance Imaging (fMRI) data is tackled by a statistical data analysis for which the signal's amplitude is required. From literature, it is known that fMRI data follow a Rice distribution. Hence, for fMRI signal detection first the parameters of the Rice distribution need to be estimated. Different methods exist and each has their own pros and cons. In this paper, a novel estimation approach is presented in order to overcome the drawbacks of the existing methods. The proposed estimator is based on the harmonic median. Its performance is verified via a simulation experiment and compared with the state-of-the-art approaches.

The use of the histogram to characterize the random component in raw measurements is widely known, applied and applauded. However, its correct use to show the features hidden in the data may require some caution and insight. The most important degree of freedom specified by the user is the binwidth. Although standard rules for binwidth selection exist, they offer no guarantees that the histogram reveals all the desired features. Furthermore, the histogram is a discontinuous representation of the underlying probability density function (pdf) of the data but measured data are usually continuous. Smooth alternatives to the histogram have been developed since the 1970s but still require significant user interaction and insight into the true data probability density. In this paper, we investigate a novel technique that offers a smooth estimate of the pdf without any necessary interaction of the user. The method is fully blind and adaptive such that the best graphical representation of the probability density is ensured.

The world is often considered to behave approximately linearly. However, many real-life phenomena are inherently nonlinear! Hence, in order to accurately model the true behavior of a radio-frequency device or system, its nonlinear characteristics can no longer be ignored and should be taken into account. To do so, one should first be able to measure these nonlinear effects. After some years of hesitation, the high-frequency measurement world finally acknowledged the necessity to accurately measure the in- and out-of-band nonlinear behavior of a radio-frequency (RF) device or system. Since then, different measurement approaches have been developed to achieve this goal. The two major measurement principles being pursued are the sampler-based and the mixer-based methodology. The calibration and de-embedding process of nonlinear measurements is quite involved and requires special calibration standards. From the acquired “nonlinear” measurement data, one can then build a model that accurately describes the in-band and out-of-band nonlinear behavior of an RF system. This chapter will show the reader how to do accurate nonlinear RF measurements and how to obtain a simple and robust characterization of the nonlinear behavior of an RF system.

Functional Magnetic Resonance Imaging (fMRI) data consist of relatively weak signals with a complicated noise structure. To reduce the effects of noise arising from both instrumental and physiological sources, a series of standard preprocessing steps is performed. Nevertheless, fMRI signals will show an undesired offset due to the measurement setup. Prior to fMRI data analysis, this offset component needs to be removed in an additional preprocessing step. Classically, one assumes the data to be Gaussian distributed which eases this preprocessing step. However, this assumption is only valid for high signal-to-noise ratios (SNRs). For low SNRs, it is known that fMRI data follow a Rice distribution. Hence, to perform a proper data preprocessing, we need to take into account the correct characteristics of the Rice distributed data.

When modeling a linear system in a parametric way, one needs to deal with (i) model structure selection, (ii) model order selection as well as (iii) an accurate fit of the model. The most popular model structure for linear systems has a rational form which reveals crucial physical information and insight due to the accessibility of poles and zeros. In the model order selection step, one needs to specify the number of poles and zeros in the model. Automated model order selectors like Akaike@?s Information Criterion (AIC) and the Minimum Description Length (MDL) are popular choices. A large model order in combination with poles and zeros lying closer to each other in frequency than the frequency resolution indicates that the modeled system exhibits some fractional behavior. Classical integer order techniques cannot handle this fractional behavior due to the fact that the poles and zeros are lying to close to each other to be resolvable and not enough data is available for the classical integer order identification procedure. In this paper, we study the use of fractional order poles and zeros and introduce a fully automated algorithm which (i) estimates a large integer order model, (ii) detects the fractional behavior, and (iii) identifies a fractional order system.

In measurement science mathematical models are often used as an indirect measurement of physical properties which are mapped to measurands through the mathematical model. Dynamical systems describing a physical process with a dominant diffusion or dispersion phenomenon requires a large dimensional model due to its long memory. Ignoring a dominant difussion or dispersion component acts as a confounder which may introduce a bias in the estimated quantities of interest. For linear systems it has been observed that fractional order models outperform classical rational forms in terms of the number of parameters for the same fitting error. However it is not straightforward to deal with a fractional order system or long memory effects without prior knowledge. Since the parametric modeling of a fractional system is very involved, we put forward the question whether fractional insight can be gathered in a non-parametric way. In this paper we show that classical Fourier basis leading to the frequency response function lacks fractional insight. To circumvent this problem, we introduce a fractional Taylor–Fourier basis to obtain non-parametric insight in the fractional system. This analysis proposes a novel type of spectrum to visualize the spectral content of a fractional system: Taylor–Fourier spectrum. This spectrum is fully measurement driven which can be used as a first to explore the fractional dynamics of a measured diffusion or dispersion system.

Modeling systems based on measurements is a well established field of research and engineering practice. The techniques available to build and identify these models operate under the assumption that “sufficiently many” measurements are available. In most cases, the model quality improves when the number of measurements increases. Unfortunately, measurement time is expensive and in some applications it is even infeasible to increase the number of measurements. For these kinds of applications, classical modeling tools become untrustworthy and no alternatives are available. In this paper, we introduce fractional order differential equations instead of ordinary differential equations to model linear systems. The major advantage of the presented technique is that only a small number of parameters is needed to obtain a very flexible model. We propose an identification technique which replaces the ordinary differential equations by fractional order differential equations with a smaller number of parameters.

In this paper, we tackle the problem of signal detection in functional Magnetic Resonance Imaging (fMRI) data by means of a statistical analysis. The main problem of the commonly used statistical tests is that they are based on the assumption that the data are Gaussian distributed, which is only valid for high signal-to-noise ratios (SNRs). Hence, for low SNRs the classical statistical tests are inadequate due to the wrong normality assumption, since it is known from literature that fMRI data follow a Rice distribution. In order to handle both high and low SNRs, we present in this paper a correction for the simplest and most widely used t-test by incorporating the correct Rice conditions. The performance of the Rice-corrected statistical test is shown through simulations and compared with its uncorrected counterpart.

In this paper, the oscillometric waveform measured by automatic non-invasive blood pressure meters (NIBP) is analyzed by transforming the data from the time domain to the frequency domain. The signal's spectrum of the oscillometric waveform is in current literature badly understood or explored. The only known link between the oscillometric waveform and the blood pressure is the maximum of the oscillometry's envelope equalling the mean arterial pressure (MAP). This link is established under the assumption that the oscillometry is an AM-signal. Unfortunately, computing the MAP is difficult in practice due to the non-sinusoidal nature of the actual measured signals. In this paper, we construct the best AM-signal approximation of the oscillometry and explore its use to compute the MAP.

Nonlinear block-oriented models have successfully been used in many engineering applications to identify chemical, mechanical, and biological systems or processes. Due to their simple structure, these models are very attractive from a user’s point of view. However, the block-oriented approach also has some disadvantages over black-box modelling approaches.

In this paper, the Polynomial NonLinear State Space (PNLSS) approach is applied to model a nonlinear system with a Wiener–Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behavior. The identification procedure is applied to measurement data.

The problem of detecting a periodic signal buried in zero-mean Gaussian noise is present in various fields of engineering. It is well-known that the amplitude of the disturbed signal follows a Rice distribution which is characterized by two parameters. In this paper, an alternative Bayesian approach is proposed to tackle this two-parameter estimation problem. By incorporating prior knowledge into a mathematical framework, the drawbacks of the existing methods (i.e., the maximum likelihood approach and the method of moments) can be overcome. The performance of the proposed Bayesian estimator is shown through simulations.

In this paper, a method is proposed to initialize the linear dynamic blocks of a Wiener-Hammerstein model. The idea is to build these blocks from the poles and the zeros of the best linear approximation of the system under test, which can easily be extracted from the data. This approach results in an easy to solve problem (linear-in-the-parameters) from which initial estimates for the linear dynamics can be obtained. The proposed method is applied to simulation data from a Wiener-Hammerstein system.

Analyzing functional MRI data is often a hard task due to the fact that these periodic signals are strongly disturbed with noise. In many cases, the signals are buried under the noise and not visible, such that detection is quite impossible. However, it is well known that the amplitude measurements of such disturbed signals follow a Rice distribution which is characterized by two parameters. In this paper, an alternative Bayesian approach is proposed to tackle this two-parameter estimation problem. By incorporating prior knowledge into a mathematical framework, the drawbacks of the existing methods (i.e. the maximum likelihood approach and the method of moments) can be overcome. The performance of the proposed Bayesian estimator is analyzed theoretically and illustrated through simulations. Finally, the developed approach is successfully applied to measurement data for the analysis of functional MRI.

In this paper, we propose a method to model nonlinear systems using polynomial nonlinear state space equations. Obtaining good initial estimates is a major problem in nonlinear modelling. It is solved here by identifying first the best linear approximation of the system under test. The proposed identification procedure is successfully applied to measurements of two physical systems.

In this paper, a structure identification method based upon the best linear approximation which was presented in earlier work is applied to three different physical systems: the silverbox, a crystal detector, and an RF amplifier. The approach of this method consists in applying a Gaussian-like input signal, and to vary in a first experiment the root mean square (rms) value of this signal while maintaining the colouring of the power spectrum. Next, in a second experiment the colouring of the power spectrum is varied while keeping the rms value constant. Based on the resulting behaviour of the best linear approximation, a distinction can be made between candidate nonlinear block structures for approximating the real system.

A method is proposed to initialize the linear dynamic blocks of a Wiener-Hammerstein model. The idea is to build these blocks from the poles and the zeros of the Best Linear Approximation of the system under test, which can easily be extracted from the data. This approach results in an easy to solve problem (linear-in-the-parameters) from which initial estimates for the linear dynamics can be obtained. The proposed method is applied to measurement data from a Wiener-Hammerstein system.