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This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties. [ABSTRACT FROM AUTHOR]

This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties.

This paper proposes a nonlinear friction identification method for friction compensation in servo-driven mechanisms. The proposed identification method captures the nonlinear Stribeck effect at low velocity which exacerbates the degradation on accuracy performance. Without the requirement of closed loop scheme, the proposed method approximates the friction along a specific stroll with Stribeck model in least squares sense, utilizes a separable least squares (SLS) approach for nonlinear optimization, uses the kinematic orthogonal property (KOP) to avoid inertial force estimation, and takes the features: (1) Computational cost is considerably reduced compared with conventional least squares approaches. (2) It identifies the friction model parameters without any inertial information, and noise introduced by acceleration estimation is avoided. (3) It can be implemented under open loop scheme which does not require controller tuning like most of other identifications do. Component-wise sensitivity analysis is also applied to evaluate the derivation of the model parameters. Additionally this identification method of combining SLS and KOP can also be generalized to other nonlinear mechanical systems while keeping these favorable features. The proposed method is implemented on the friction compensation in the feed drive control of a machine tool. The accuracy of the identified Stribeck model is evaluated and the compensation performance with implementation feasibility considered is validated by experimental comparisons. [ABSTRACT FROM AUTHOR]

Ordinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a "Big System" with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector-based two-stage method that are currently available. We applied this new method to two real datasets—a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters—to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Abstract: We propose a method for the estimation of the overall heat transfer coefficient to/from a fluid in a cylindrical pipe at high Peclet numbers from/to a medium with unknown temperature profile in the axial direction. The method uses an analytical solution of the heat equation in cylindrical coordinates subject to a Robin boundary condition and a parametrised, piecewise linear approximation of the external temperature profile. The velocity profiles of both viscous and turbulent flows are considered and compact solutions of the temperature profile of the fluid are evaluated in both cases as functions of the heat transfer coefficient and of the unknown external temperature profile. It is shown that these solutions provide an efficient method for the estimation of the heat transfer coefficient from fluid temperature data based on separable least squares. The overall procedure is illustrated by a numerical example using simulated data. [Copyright &y& Elsevier]

The force and position data used to construct models of limb impedance are often obtained from closed-loop experiments. If the system is tested in a stiff environment, it is possible to treat the data as if they were obtained in open loop. However, when limb impedance is studied in a compliant environment, the presence of feedback cannot be ignored. While unbiased estimates of a system can be obtained directly using the prediction error method, the same cannot be said when linear regression or correlation analysis is used to fit nonparametric time- or frequency-domain models. We develop a prediction error minimization-based identification method for a nonparametric time-domain model augmented with a parametric noise model. The identification algorithm is tested on a dynamic mass–spring–damper system and returns consistent estimates of the system's properties under both stiff and compliant feedback control. The algorithm is then used to estimate the impedance of a human elbow joint in both stiff and compliant environments. [ABSTRACT FROM PUBLISHER]

Abstract: Repetitive peripheral magnetic stimulation (RPMS) is an innovative approach in treatment of central paresis, e.g. after stroke. In this article we present a neuromuscular model for the RPMS-induced isometric muscle contraction. This model is the basis for our two recent goals in research: improvement and assessment of the RPMS therapy by means of position controlled induction of functional movements and with automated system identification based therapy evaluation. In order to adapt the model parameters to the individual a nonlinear on-line system identification method is proposed. Physiological systems may be complex and detailed so that in many cases macroscopic models of the dominant characteristics are built. Therefore, mathematical descriptions of these models and respective parameter identification methods have to cope with uncertainties. This paper presents a separable nonlinear regression model of Hammerstein structures that maximizes the possibility of incorporation of a priori knowledge and is still flexible to structural uncertainties. A robust on-line identification method based on the Levenberg–Marquardt algorithm is presented that works in a reduced parameter space, due to the separability of the model equation. [Copyright &y& Elsevier]

By using signal processing techniques, an estimate of activity in the brain from the electro- or magneto-encephalogram (EEG or MEG) can be obtained. For a proper analysis, a test is required to indicate whether the model for brain activity fits. A problem in using such tests is that often, not all assumptions are satisfied, like the assumption of the number of shells in an EEG. In such a case, a test on the number of sources (model order) might still be of interest. A detailed analysis is presented of the Wald test for these cases. One of the advantages of the Wald test is that it can be used when not all assumptions are satisfied. Two different, previously suggested, Wald tests in electromagnetic source analysis (EMSA) are examined: a test on source amplitudes and a test on the closeness of source pairs. The Wald test is analytically studied in terms of alternative hypotheses that are close to the null hypothesis (local alternatives). It is shown that the Wald test is asymptotically unbiased, that it has the correct level and power, which makes it appropriate to use in EMSA. An accurate estimate of the Cramér-Rao bound (CRB) is required for the use of the Wald test when not all assumptions are satisfied. The sandwich CRB is used for this purpose. It is defined for nonseparable least squares with constraints required for the Wald test on amplitudes. Simulations with EEG show that when the sensor positions are incorrect, or the number of shells is incorrect, or the conductivity parameter is incorrect, then the CRB and Wald test are still good, with a moderate number of trials. Additionally, the CRB and Wald test appear robust against an incorrect assumption on the noise covariance. A combination of incorrect sensor positions and noise covariance affects the possibility of detecting a source with small amplitude. [ABSTRACT FROM AUTHOR]

A new method for analyzing three-state protein unfolding equilibria is described that overcomes the difficulties created by direct effects of denaturants on circular dichroism (CD) and fluorescence spectra of the intermediate state. The procedure begins with a singular value analysis of the data matrix to determine the number of contributing species and perturbations. This result is used to choose a fitting model and remove all spectra from the fitting equation. Because the fitting model is a product of a matrix function which is nonlinear in the thermodynamic parameters and a matrix that is linear in the parameters that specify component spectra, the problem is solved with a variable projection algorithm. Advantages of this procedure are perturbation spectra do not have to be estimated before fitting, arbitrary assumptions about magnitudes of parameters that describe the intermediate state are not required, and multiple experiments involving different spectroscopic techniques can be simultaneously analyzed. Two tests of this method were performed: First, simulated three-state data were analyzed, and the original and recovered thermodynamic parameters agreed within one standard error, whereas recovered and original component spectra agreed within 0.5%. Second, guanidine-induced unfolding titrations of the human retinoid-X-receptor ligand-binding domain were analyzed according to a three-state model. The standard unfolding free energy changes in the absence of guanidine and the guanidine concentrations at zero free-energy change for both transitions were determined from a joint analysis of fluorescence and CD spectra. Realistic spectra of the three protein states were also obtained. [ABSTRACT FROM AUTHOR]

This paper considers the use of cubic splines, instead of polynomials, to represent the static nonlinearities in block structured models. It introduces a system identification algorithm for the Hammerstein structure, a static nonlinearity followed by a linear filter, where cubic splines represent the static nonlinearity and the linear dynamics are modeled using a finite impulse response filter. The algorithm uses a separable least squares Levenberg-Marquardt optimization to identify Hammerstein cascades whose nonlinearities are modeled by either cubic splines or polynomials. These algorithms are compared in simulation, where the effects of variations in the input spectrum and distribution, and those of the measurement noise are examined. The two algorithms are used to fit Hammerstein models to stretch reflex electromyogram (EMG) data recorded from a spinal cord injured patient. The model with the cubic spline nonlinearity provides more accurate predictions of the reflex EMG than the polynomial based model, even in novel data. [ABSTRACT FROM AUTHOR]

The Hammerstein cascade, consisting of a zero-memory nonlinearity followed by a linear filter, is often used to model nonlinear biological systems. This structure can represent some high-order nonlinear systems accurately with relatively few parameters. However, it is not possible, in general, to estimate the parameters of a Hammerstein cascade in closed form. The most effective method available to date uses an iterative approach, which alternates between estimating the linear element from a crosscorrelation, and then fitting a polynomial to the nonlinearity via linear regression. This paper proposes the use of separable least squares optimization methods to estimate the linear and nonlinear elements simultaneously in a least squares framework. A separable least squares algorithm for the identification of Hammerstein cascades is developed and used to analyze stretch reflex electromyogram data from two experimental subjects. The results show that in each case the proposed algorithm produced a better model, in that it predicted the system’s response to novel inputs more accurately, than did models estimated using the traditional iterative algorithm. Monte-Carlo simulations demonstrated that when the input is a non-Gaussian, nonwhite signal, as is often the case experimentally, the traditional iterative identification approach produces biased models, whereas the separable least squares approach proposed in this paper does not. © 2001 Biomedical Engineering Society. PAC01: 8719Ff, 8719Nn, 8710+e [ABSTRACT FROM AUTHOR]