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Indirect measurements of physical parameters of interest require a mathematical model in which these parameters are estimated from the gathered measurements. Within the least squares (LS) estimation, the parameters are estimated through a regression problem. The presence of dynamics, multiple sensors, and high sampling rates leads to high-dimensional regression matrices. This paper deals with solving such large-scale regression problems time efficiently. We revisit Renaut’s least squares multisplitting (LSMS) technique aimed at solving the ordinary LS problem in parallel. The LSMS decomposes the design matrix column-wise into several blocks. The global LS solution is subsequently replaced by an equivalent set of local LS problems that are to be solved in parallel. We study how the user should configure the partition of the multisplitting. We propose a partition design based on a clustering analysis and prove the consistency of this approach. The method is illustrated with dedicated numerical simulations for a highly scalable LS-based problem within engineering: frequency response function (FRF) estimation in the presence of missing output samples. Finally, its practical utility is shown with a laboratory measurement application.

In this paper, we discuss nonparametric estimation of the probability density function (PDF) of a univariate random variable. This problem has been the subject of a vast amount of scientific literature in many domains, while statisticians are mainly interested in the analysis of the properties of proposed estimators, and engineers treat the histogram as a ready-to-use tool for a data set analysis. By considering histogram data as a numerical sequence, a simple approach for PDF estimation is presented in this paper. It is based on basic notions related to the reconstruction of a continuous-time signal from a sequence of samples. When estimating continuous PDFs, it is shown that the proposed approach is as accurate as kernel-based estimators, widely adopted in the statistical literature. Conversely, it can provide better accuracy when the PDF to be estimated exhibits a discontinuous behavior. The main statistical properties of the proposed estimators are derived and then verified by simulations related to the common cases of normal and uniform density functions. The obtained results are also used to derive optimal, i.e., minimum integral of the mean square error, estimators.

It has been argued that due to the bias at low SNR, the Gaussian approach is unsuitable for modelling Rician fMRI data. As a result several estimators incorporating the Rician nature of the data have been proposed to measure the signal as accurately as possible. However, within fMRI the main objective is not to measure the signal, but rather to measure changes within the signal. As an increasing function of the signal, the mean can be used for this purpose as well. In this paper it is argued that, due to its lower variance, the sample average is a more suitable tool to detect changes in the amplitude than several conventional Rician parameter estimators at those SNR values common within fMRI measurements. While the interpretation is slightly different, this Rician mean-based approach is essentially equivalent to the Gaussian approach. Despite its bias, the Gaussian approach is therefore preferable within fMRI analysis.

The article presents a methodology to discriminate glucose levels using electrical impedance spectroscopy technology. The method is based on an adequate estimation and assessment of the impedance data followed by identification of a general rational fractional model. The methodology is illustrated on a group of saline–protein–glucose solutions at physiological concentrations, and shows the ability of the fractional models to discriminate glucose levels. The method exhibit significant differences in the zero position of the fractional model for different glucose concentrations allowing discriminate the glucose effect on the impedance data along different matrix solutions. The results based on fractional model method are compared with classical circuit models and rational integer models. Even when the different methodologies perceive variations in the impedance data given glucose alterations, the fractional models present low uncertainty allowing discriminating small glucose alterations in the physiological range.

The power spectrum has served as a quick and fairly robust method to obtain access to the underlying dynamics of measured weakly stationary signals. As periodograms are noisy estimates of the underlying power spectrum, periodogram smoothing is performed in the frequency domain to obtain a more accurate description of the power spectrum. Frequency smoothing operates under the assumption that the power spectrum is continuously differentiable. Fractional order signals are becoming more important as they serve to describe important diffusion and dispersion effects encountered in mechanical, electrochemical and biomedical applications. Fractional time series of the Riemann-Liouville type have the property that the underlying power spectrum exists but it is not continuously differentiable. In this paper, we mathematically study these signals and propose a new frequency smoother to overcome the problem such that these signals can still be studied in a non-parametric way.

Indirect measurements of physical parameters of interest often require a mathematical model in which these parameters are estimated accordingly to the gathered measurements. Within the Least Squares estimation, the parameters are estimated through a regression problem. The presence of dynamics, multiple sensors and high sampling rates lead to high dimensional regression matrices. This paper deals with solving such massive regression problems efficiently. We revisit Renaut's Least Squares Multisplitting (LSMS) technique aimed at solving the ordinary least squares problem in parallel. The LSMS decomposes the design matrix column-wise into several (possibly overlapping) blocks. The global least squares solution is subsequently replaced by an equivalent set of local least squares problems which are to be solved in parallel. At every iteration step the local solutions are recombined using an appropriate weighting scheme. This allows for a scalable and highly parallel implementation aimed at distributed systems. We study how to decompose the global estimation problem in smaller subproblems and quantify the effects of various possible partitions. The method is illustrated with dedicated numerical simulations and a toy application in the area of sensor fusion.

In this paper, we present a new spectrum sensing technique for cognitive radios based on discriminant analysis called spectrum discriminator. The presented technique uses the knowledge of the noise uncertainty and a probabilistic validation to overcome the limitations of the discriminant analysis. A comparative study between the proposed technique and the maximum-minimum eigenvalue detection has been performed based on two performance metrics: the probability of false alarm and the probability of detection. The spectrum discriminator has been further developed to a peel-off technique where all primary users can be detected. The performance of the spectrum discriminator and the peel-off technique has been tested on simulations and experimentally verified. The comparative study is based on simulations as well as measurements.

A time series approach is often applied to obtain insight in the system dynamics or the signal filtering. Dynamical systems may exhibit a diffusion or dispersion components which reveals a long memory or long time dependency. This class of systems is called fractional order systems. It is typically difficult to detect that one is dealing with a fractional order system. In this paper, we show that one can start with a classical time series analysis based on Autoregressive-Moving average modeling. In the presence of a diffusion or dispersion component the fractional behavior forces the estimated poles and zeros of the ARMA model to form an alternating chain. This chain can be detected and consecutively be compressed to finally result in a fractional order version of the ARMA model which is called the Autoregressive Fractionally Integrated Moving Average or ARFIMA model. Note that the presented approach is fully measurement driven and requires no interaction from the user.

The steady state response of a system to a periodic input is still subject to noise transients. For lightly damped systems these noise transients can significantly increase the variance of the estimated frequency response function (FRF). This paper presents a method that suppresses the influence of the noise transients (leakage errors) in nonparametric FRF and noise (co-)variance estimates of dynamic systems excited by periodic signals. The method is based on a local polynomial approximation of the noise leakage errors on the FRF. Compared with the classical approaches, the proposed procedure is more robust and needs less measurement time (two signal periods are sufficient). The theory is supported by simulation and real measurement examples.

This is Part II of a series of two papers on the estimation of nonparametric noise and frequency response function models for multiple input, multiple output systems excited by arbitrary inputs. Part I (Pintelon et al. (2009)) [1] develops the methodology for linear dynamic multivariable systems with exactly known input and where the output is disturbed by stationary noise ( = output error problem ) . The contributions of Part II are the following. First, it is shown that the proposed method can be used for nonlinear systems and for parametric identification of the system dynamics in a generalized output error framework. Next, the methodology is generalized to handle noisy input–output data ( = errors -in-variables problem), and identification in feedback. Finally, the approach is illustrated on simulations and real measurements examples.

This paper analyses the variance of linear models that are identified in the presence of nonlinear distortions. The best linear approximation G BLA to a nonlinear system, driven by a Gaussian distributed random input, is identified in least squares sense. Even in the absence of disturbing noise, the estimated model Ĝ BLA will vary from one realization of the excitation signal to the other. In this paper it will be shown that the variance of the nonparametric frequency response function (FRF) Ĝ BLA ( k ) can still be obtained from the experimental data using the classical variance formulas. However, for parametric estimates Ĝ BLA ( q , θ), an increased variance is observed. The contributing terms to this additional variance are studied in this paper.

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.

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.

Despite the extensive literature, describing the probability content of measurements remains an important topic for engineering problems. The histogram remains the golden standard, even though kernel density estimation is a strong competitor when smooth estimates are desired. Critical user interaction is required for the use of both the histograms and kernel densities. A good choice for the bandwidth is essential in both cases. On top of that the kernel density method requires a proper choice of the kernel. Incorrect choices may lead to incorrect results generated by either masking important details or introducing false details. In this paper, we propose a new approach which requires no user-defined choices. The method is therefore fully automatic and provides the user a smooth density estimate of the probability content.

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.

In this paper we present a new spectrum sensing technique for cognitive radios based on discriminant analysis called spectrum discriminator and compare it with the maximum minimum eigenvalue detector. The common feature between those two techniques is that neither prior knowledge about the system noise level nor the primary user signal, that might occupy the band under sensing, is required. Instead the system noise level will be derived from the received signal. The main difference between both techniques is that the spectrum discriminator is a non-parametric technique while the maximum minimum eigenvalue detector is a parametric technique. The comparative study between both has been done based on two performance metrics: the probability of false alarm and the probability of detection. For the spectrum discriminator an accuracy factor called noise uncertainty is defined as the level over which the noise energy may vary. Simulations are performed for different values of noise uncertainty for the spectrum discriminator and different values for the number of received samples and smoothing factor for the maximum minimum eigenvalue detector.

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.

Spectrum sensing is an essential pre-processing step for cognitive radio technology. This paper presents a novel method to detect the significant spectral components in measured nonflat spectra, and to estimate the magnitude of the spectral components. Furthermore, the probability that the spectral component was incorrectly classified is available. The algorithm is able to detect the presence or absence of signals in any kind of spectrum since no prior knowledge about the measured signal is needed. Hence, this method becomes a strong basis for a high-quality operation mode of cognitive radios. Simulation results prove the advantages of the presented technique.

Measuring the blood pressure as accurately as possible can save a lot of human lives. Hence, it is very important to find an optimal method to determine the systolic and diastolic pressures out of the measured oscillometric blood pressure waveform. Recently, studies have been showing that, by working in the frequency domain, outperforming results could be obtained. Using the digital Taylor-Fourier transform (DTFT) even allows separating the breathing and cardiac activity that is present in the oscillometric waveform. Furthermore, an estimate of the frequency fluctuation can easily be obtained. In this paper, we will investigate whether or not a Kalman filtering implementation can provide better results than the DTFT analysis. In theory both approaches should be equally performing. Both techniques will be compared on measured oscillometric waveforms. Even if the alternating Kalman filter does not excel the DTFT algorithm in interharmonic rejection, it offers interesting signal decomposition alternatives.

In this paper, we study the asymptotic properties (consistency and asymptotic efficiency) of a frequency-domain errors-in-variables estimator using data extracted from overlapping subrecords. While the classical approach without overlap needs six consecutive periods, we show in this paper that, using overlapping subrecords, consistent estimators can be found with only two periods of the steady state response to a periodic excitation. This is done using overlapping subrecords. Moreover, the system identification procedure developed for data extracted from independent periods is shown to be valid for data extracted from overlapping subrecords. This allows the user to reduce the measurement time considerably without changing the identification procedure.

The detection of periodic components buried in noise is a general problem in various engineering fields. The amplitudes in the frequency domain of a disturbed signal follow Rice distribution, which is fully described by two parameters. Most methods are restricted to automatically detecting the harmonic components. In this paper, we extend the methodology to detect significant harmonics in measured spectra such that, aside from detection, the magnitude of the harmonic component is also estimated, together with the probability that the harmonic component was incorrectly detected.

This series of two papers presents a method for estimating nonparametric noise and frequency response function models of multivariable linear dynamic systems excited by arbitrary inputs. It extends the results of Schoukens et al. (2006) [1] and Schoukens and Pintelon (2009) [2] from single input, single output systems with known input and noisy output observations ( = output error problem ), to multiple input, multiple output systems where both the input and output are disturbed by noise ( = errors‐in‐variables problem ). In Part I, the theory is developed for linear dynamic multivariable output error problems. The results are supported by simulations. A detailed comparison with the classical spectral analysis based on correlation techniques shows that the proposed procedures are more robust. In Part II (Pintelon et al., 2009) [3] , the method first is applied to nonlinear systems, and parametric identification within a generalized output error framework. Next, it is extended to handle errors-in-variables problems, and identification in feedback. Finally, it is illustrated on four real measurement examples.

The objective of this paper is twofold. The first part provides further insight in the statistical properties of the Welch power spectrum estimator. A major drawback of the Welch method reported in the literature is that the variance is not a monotonic decreasing function of the fraction of overlap. Selecting the optimal fraction of overlap, which minimizes the variance, is in general difficult since it depends on the window used. We show that the explanation for the nonmonotonic behavior of the variance, as reported in the literature, does not hold. In the second part, this extra insight allows one to eliminate the nonmonotonic behavior of the variance for the Welch power spectrum estimator (PSE) by introducing a small modification to the Welch method. The main contributions of this paper are providing extra insight in the statistical properties of the Welch PSE; modifying the Welch PSE to circular overlap-the variance is a monotonically decreasing function of the fraction of overlap, making the method more user friendly; and an extra reduction of variance with respect to the Welch PSE without introducing systematic errors-this reduction in variance is significant for a small number of data records only.

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.

This paper studies non-parametric power spectrum estimation with circular overlap. Although circular overlap imposes a discontinuity on the stochastic process under investigation; it is shown that for a (Gaussian) ergodic weakly stationary process the power spectrum estimator with circular overlap is asymptotically unbiased. Also the variance decreases due to the overlap. Furthermore, a closed form expression is derived for the lowest reachable variance, with respect to all possible fractions of overlap.

In this paper, we study the consistency of a Frequency domain errors-ln-variables estimator using data extracted from overlapping sub-records. We show that consistent models can be found using periodic excitations. Only two periods of the steady state response are needed to extract all required information from the data. This is done using overlapping sub-records. Moreover, the system identification procedure used for data extracted from independent experiments is shown to be valid for data extracted from overlapping sub-records. This allows the user to reduce the measurement time considerably without changing the identification procedure.

In this paper we propose a method to extract a non-parametric noise model starting from periodic data with a small number of measured periods. Overlapping windows are used to avoid singular results. A full theoretic analysis is made.

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.

This paper analyzes the variance of the estimated frequency response function (FRF) GBLA of the best linear approximation GBLA to a nonlinear system that is driven by random excitations. GBLA varies not only due to the disturbing measurement and process noise but also over different realizations of the random excitation because the nonlinear distortions depend on the input realization. It will be shown that the variance expression σGBLA2 that is obtained in the linear framework can also be used to calculate the variance that is induced by the nonlinear distortions. This validates the common engineering practice, where the linear FRF methodology is often used under nonlinear conditions.

Electrical impedance spectroscopy (EIS) has been used to characterize different biological materials. This article exposes a methodology oriented to estimate glucose levels of a solution based on a rational fractional parametric model of the impedance data. The methodology is applied over saline–glucose solutions at five physiological glucose levels, using three sensors and five repetitions for each glucose concentration and employed sensor. The results suggest that changes in the glucose concentration produce significant changes in the impedance that should be reflected in the parametric model. The modeling procedure shows that the poles and zeros of an integer model presents a degree of correlation. However, the correlation is clearly explicit employing fractional models where the mean location of the complex zeros is highly related to the glucose content in the sample.

It is well known that under very general assumptions the discrete Fourier coefficients of filtered noise is asymptotically independent circular complex Gaussian distributed, based on a generalized central limit theorem (CLT). The standard results on the consistency and the asymptotic uncertainty of the frequency domain Errors-in-Variables (EIV) estimator are derived under the assumption that the Fourier coefficients are circular complex Gaussian distributed and independent over the different frequency bins. In this paper, we shall study the influence of this assumption on the consistency and the efficiency of the frequency domain EIV-estimator. We show that a slightly stronger form of the CLT is needed to preserve the classically obtained uncertainty bounds if independent complex Gaussian Fourier coefficients are not assumed. Our analysis reveals that the classical derived asymptotic uncertainty bounds are valid for a very wide class of distributions.