Nonlinear Least-squares Approach Research Articles

Head Capsule Width and Instar Determination for Larvae of <I>Streblote panda</I> (Lepidoptera: Lasiocampidae)

Abstract The moth Streblote panda Hubner (1820) (Lepidoptera: Lasiocampidae) is an ornamental and fruit plant pest in southwestern Andalusia. Analysis of head capsule width of 387 larvae reared on different host plants (Pistacea lentisco L., Arbutus unedo L., Citrus reticulata Blanco, Retama sphaerocarpa (L.) Boiss, Spartium junceum L., Tamarix gallica L., and Vaccinium spp.) indicates that growth rate between instars does not fit well to Dyar’s rule. The number of instars varied from 5 to 8. Head capsule width could be used directly to determine developmental stage of individual in almost all host plants used. Only for larvae fed on S. junceum, R. sphaerocarpa, Vaccinium corymbosum L. ‘Sharpblue’, and Vaccinium ashei Reade ‘Bonita’, beyond third instar, did the range of measurements overlap. These data were analyzed using a combination of statistical methods. A nonlinear least-squares approach was used to determine width means and ranges for each instar, and misclassification probabilities were calculate...

Power system frequency estimation method

A nonlinear least-squares approach for measuring the power system frequency is presented, wherein the voltage at the point of common coupling is modelled using the Fourier series. The fundamental frequency is estimated by performing a one-dimensional search over the range of allowed frequency variation. The robustness of this algorithm is studied and validated by hardware implementation using an FPGA. The performance of this method is also compared with that of the phase-locked loop.

On Nonparametric Identification of Wiener Systems

In this paper, a new method for the identification of the Wiener nonlinear system is proposed. The system, being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element, is identified by a two-step semiparametric approach. The impulse response function of the linear part is identified via the nonlinear least-squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate. The obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the Wiener system. The proposed method permits recovery of a wide class of nonlinearities which need not be invertible. As a result, the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate. Furthermore, our approach allows non-Gaussian input signals and the presence of additive measurement noise. However, only linear systems with a finite memory are admissible. The conditions for the convergence of the proposed estimates are given. Computer simulations are included to verify the basic theory

Non-linear least-squares solution to the moiré hole method problem in orthotropic materials. Part 1: Residual stresses

The hole method problem relates to two inverse problems of interest: the first, most commonly addressed by practitioners, is to obtain residual stresses; the other, generally neglected, inverse problem can be posed as either a stress separation problem or a material elastic properties identification problem. In both this Paper I and Paper II, we pose and solve this dual hole method problem in an orthotropic plate, using computer generated moire isothetics, by means of a non-linear least-squares approach. In Paper I we address the residual stress problem. In Paper II we pose the use of moire isothetics as a means to achieve separation of stresses, but we deal with the determination of the five orthotropic elastic constants, four of which are independent.

Sequential non-linear least-square estimation for damage identification of structures

An early detection of structural damage is an important goal of any structural health monitoring system. In particular, the ability to detect damages on-line, based on vibration data measured from sensors, will ensure the reliability and safety of the structures. In this connection, innovative data analysis techniques for the on-line damage detection of structures have received considerable attentions recently, although the problem is quite challenging. In this paper, we proposed a new data analysis method, referred to as the sequential non-linear least-square (SNLSE) approach, for the on-line identification of structural parameters. This new approach has significant advantages over the extended Kalman filter (EKF) approach in terms of the stability and convergence of the solution as well as the computational efforts involved. Further, an adaptive tracking technique recently proposed has been implemented in the proposed SNLSE to identify the time-varying system parameters of the structure. The accuracy and effectiveness of the proposed approach have been demonstrated using the Phase I ASCE structural health monitoring benchmark building, a non-linear elastic structure and non-linear hysteretic structures. Simulation results indicate that the proposed approach is capable of tracking on-line the changes of structural parameters leading to the identification of structural damages.

Geometric algorithms for least squares estimation of 3-D information from monocular image

Estimating the three-dimensional (3-D) information of an object from a sequence of projections is of paramount importance in diverse domains such as autonomous navigation, robot vision, object recognition, world modeling, and human-computer interaction. We present geometric optimization-based algorithms for simultaneous 3-D shape (depth) and motion (linear and angular velocities) recovery that minimizes the squared distance between the observed and predicted optical flow measurements. A principal advantage of our approach is that, using a separable nonlinear least-squares approach in various ways, the search space dimension in the numerical optimization is reduced. For improvement of performance in the numerical optimization, we also develop various methods to take advantage of the separable nonlinear least-squares approach effectively. A second advantage is that, by explicitly accounting for the existence of an one-parameter family of solutions, a geometric optimization algorithms on the Cartesian product manifold can be applied. The Hessian is straightforwardly evaluated as a by-product of the algorithm, making sensitivity analysis straightforward.

Estimating soil hydraulic properties and their uncertainty: the use of stochastic simulation in the inverse modelling of the evaporation method

The evaporation method is an easy and reliable procedure for measuring soil hydraulic properties. Inverse modelling has been used to analyse the evaporation data facilitating the prediction of soil hydraulic properties. The conventional approach to solve the inverse solution is the use of the nonlinear least-squares (NLLS) technique. A major problem with such inverse modelling approaches is nonidentifiability, which is a condition where different parameter sets can lead to a similar flow response. In addition, NLLS only attempts to find optimum parameters that give the best fit to the observed data while not recognising the uncertainties in the models and observations made. The discrepancies of the NLLS approach may be alleviated by adopting the concept of generalized likelihood uncertainty estimation (GLUE), which acknowledges that many parameter sets within a model will give similar output to satisfy a given objective function. This is illustrated in this paper by using the GLUE method for estimating soil water retention and hydraulic conductivity from an evaporation experiment encompassing a range of soil types ranging from sand to heavy clay. Results show that the GLUE methodology can provide the estimates of the hydraulic properties and along with its uncertainty. The water retention curve can be predicted well from the range of potential 0 to −1000 cm, but the uncertainty associated with the unsaturated hydraulic conductivity prediction is quite high.

Modelling of cyclic plasticity and viscoplasticity of a nickel-based alloy using Chaboche constitutive equations

Cyclic plasticity and viscoplasticity of an advanced nickel-based superalloy have been described using the Chaboche unified constitutive model. Experiments have been developed to obtain the full cyclic stress–strain and stress relaxation data at 650 °C. A step-by-step method has been used to obtain the initial material parameters, while a non-linear least-square approach has been used to obtain the optimised material parameters. Compared with our early work [Int J Fatigue 25 (2003) 413; Mech Mater 33 (2001) 593], much improved results have been obtained for the simulation of the transient cyclic stress–strain and stress relaxation as well as the steady-state behaviour with no additional terms or parameters introduced. The depiction of creep and combined cyclic stress–strain and stress relaxation curves seems also to be very encouraging.

Accounting for heteroscedasticity in rating curve estimates

The non-linear least squares (NLS) method has long been the standard technique used by hydrologists for constructing rating curves. The reasons for its adaptation are vague, and its appropriateness as a method of describing discharge measurement uncertainty has not been well investigated. It is shown in this paper that the classical method of NLS can model only a very limited class of variance heterogeneity. Furthermore, this lack of flexibility often leads to unaccounted heteroscedasticity, resulting in dubious values for the rating curve parameters and estimated discharge. By introducing a heteroscedastic maximum likelihood model, the variance heterogeneity is treated more generally. The maximum likelihood model stabilises the variance better than the NLS approach, and thus is a more robust and appropriate way to fit a rating curve to a set of discharge measurements. Finally, it is shown that an extreme value analysis of the discharge can be affected by the model used to construct the rating curve.

A study of creep-fatigue interaction in a new nickel-based superalloy

The behaviour of creep-fatigue interaction of a new nickel-based superalloy has been described using the Chaboche unified model. Simple cyclic and dwell experiments have been carried out at three strain rates at 650°C. The optimised material parameters have been determined by a simultaneous numerical (non-linear least-square) approach as well as a step-by-step procedure using basic material data, and applied to dwell tests. The predictive capability of the Chaboche model for this alloy will be evaluated.

Transition strengths and potential curves for the valence transitions in Br2 from a reanalysis of the ultraviolet-visible absorption at low resolution

The absorption profile of gaseous Br2 in the 320–675 nm region is reanalyzed using a quantum mechanical nonlinear least-squares approach. For the first time the A(1u 3Π)←X(1Σg+) transition is included along with the two stronger transitions, B(0u+ 3Π)←X and C(1u 1Π)←X. The analyzed data include absorption spectra at temperatures between 23 °C and 440 °C along with specific estimates of B←X and (A+C)←X absorption. A new computational device facilitates inclusion of the discrete region of the spectrum (λ>510 nm) in the analysis: The A←X and B←X transitions are treated as entirely bound-free, which is accomplished computationally by just removing the attractive branches of the A and B potentials. The new analysis determines the molar absorptivities of all three transitions within ∼1 L mol−1 cm−1. Although the high precision is partly due to the inclusion of the specific B←X and (A+C)←X data, test computations show that the resolution can be achieved reliably by fitting just the T-dependent spectra. At 23 °C the component peaks are: 10.6 L mol−1 cm−1 at 525 nm (A←X), 82.5 L mol−1 cm−1 at 480 nm (B←X), and 158 L mol−1 cm−1 at 412 nm (C←X).

The Sensitivity of Saturation Transfer Electron Paramagnetic Resonance Spectra to Restricted Amplitude Uniaxial Rotational Diffusion

Computational methods have been developed to model the effects of constrained or restricted amplitude uniaxial rotational diffusion (URD) on saturation transfer electron paramagnetic resonance (ST-EPR) signals observed from nitroxide spin labels. These methods, which have been developed to model the global rotational motion of intrinsic membrane proteins that can interact with the cytoskeleton or other peripheral proteins, are an extension of previous work that described computationally efficient algorithms for calculating ST-EPR spectra for unconstrained URD (Hustedt and Beth, 1995, Biophys. J. 69:1409–1423). Calculations are presented that demonstrate the dependence of the ST-EPR signal ( V′ 2) on the width (Δ) of a square-well potential as a function of the microwave frequency, the correlation time for URD, and the orientation of the spin-label with respect to the URD axis. At a correlation time of 10 μs, the V′ 2 signal is very sensitive to Δ in the range from 0 to 60°, marginally sensitive from 60° to 90°, and insensitive beyond 90°. Sensitivity to Δ depends on the correlation time for URD with higher sensitivity to large values of Δ at the shorter correlation times, on the microwave frequency, and on the orientation of the spin-label relative to the URD axis. The computational algorithm has been incorporated into a global nonlinear least-squares analysis approach, based upon the Marquardt–Levenberg method (Blackman et al., 2001, Biophys. J. 81:3363–3376). This has permitted determination of the correlation time for URD and the width of the square-well potential by automated fitting of experimental ST-EPR data sets obtained from a spin-labeled membrane protein and provided a new automated method for analysis of data obtained from any system that exhibits restricted amplitude URD.

On frequency offset estimation for flat-fading channels

We consider the problem of estimating the frequency offset between the transmitter and receiver in flat-fading communication channels using a training sequence. Kuo and Fitz (1997) and Morelli, Mengali and Vitetta (see IEEE Commun. Lett., vol.2, p.327-30, 1998), proposed two frequency estimators to solve the aforementioned problem. They rely on weighted linear regression for the phases of the correlation sequence or their differences. The method of Kuo et al. needs phase unwrapping and both methods require knowledge of the correlation of the fading process. In this paper, two simple methods which do not require any of these assumptions are proposed. The first is the unweighted version of the method of Morelli et al. The second is drawn from the array processing framework. It is based on a nonlinear least-squares approach that assumes an unstructured correlation of the fading process. The performance of the proposed methods is as good as that of the methods of Kuo et al. and Morelli et al., but the robustness of our methods to modeling errors is much enhanced.

Complexation of metal ions in brines: application of electronic spectroscopy in the study of the Cu(II)-LiCl-H2O system between 25 and 90°C

The concentration and transport of metals in hydrothermal solutions depend on how metals ions combine with ligands to form complexes, and experimental methods are necessary to identify the important complexes. UV-Vis-NIR spectrophotometry was used to study the formation of Cu(II)-chloride complexes in LiCl brines up to very high chlorinities (18 m LiCl), at temperatures between 25°C and 90°C. The number of Cu(II)-chloride complexes necessary to account for the variation in spectra with varying chloride molality at each temperature was estimated using principal component analysis. The molar absorptivity coefficients and concentrations of each complex were then determined using a “model-free” analysis, which does not require any assumption about the chemistry of the system, other than the number of absorbing species present. Subsequently, the results from the “model-free” analysis were integrated with independent experimental evidence to develop a thermodynamic speciation model, where the logarithms of the equilibrium constants for Cu(II)-chloride formation reactions were fitted to the data using a non-linear least-squares approach. Maps of the residual function were used to estimate uncertainties in the fitted equilibrium constants.The results of this study are similar to published properties of distorted octahedral [CuCl(OH2)5]+ and [CuCl2(OH2)4]0 at all temperatures, but diverge for [CuCl3(OH2)3]− and distorted tetrahedral [CuCl4]2−. Moreover, the data suggest the presence of [CuCl5]3−, probably with D3h point group, at very high salt concentration. This study demonstrates that it is possible to determine apparent thermodynamic equilibrium constants for the formation of complexes of trace amount of metals in highly concentrated brines, such as those associated with many ore deposits. The results are dependent on the choice of activity coefficients for charged and neutral aqueous complexes, but this influence is relatively small compared with the experimental uncertainty. This study shows that Cu2+ chloro-complexes, predominantly [CuCl2(OH2)4]0 and [CuCl4]2−, will play a dominant role in nature where free oxygen is available (near-surface), and where chloride activities are very high (evaporitic basins; hypersaline soils).

Solving the generalized indirect Fourier transformation (GIFT) by Boltzmann simplex simulated annealing (BSSA)

The structure of colloidal particles can be studied with small-angle X-ray and neutron scattering (SAXS and SANS). In the case of randomly oriented systems, the indirect Fourier transformation (IFT) is a well established technique for the calculation of model-free real-space information. Interaction leads to an overlap of inter- and intraparticle scattering effects, preventing most detailed interpretations. The recently developed generalized indirect Fourier transformation (GIFT) technique allows these effects to be separated by assuming various models for the interaction,i.e.the so-called structure factors. The different analytical behaviour of these structure factors from that of the form factors, describing the intraparticle scattering, allows this separation. The mean-deviation surface is defined by the quality of the fit for different parameter sets of the structure factor. Its global minimum represents the solution. The former non-linear least-squares approach has proved to be inefficient and not very reliable. In this paper, the incorporation of the completely different Boltzmann simplex simulated annealing (BSSA) algorithm for finding the global minimum of the hypersurface is presented. This new method increases not only the calculation speed but also the reliability of the evaluation.

Estimating signal parameters using the nonlinear instantaneous least squares approach

A novel method for signal parameter estimation is presented, termed the nonlinear instantaneous least squares (NILS) estimator. The basic idea is to use the observations in a sliding window to compute an instantaneous (short-term) estimate of the amplitude used in the separated nonlinear least squares (NLLS) criterion. The effect is a significant improvement of the numerical properties in the criterion function, which becomes well-suited for a signal parameter search. For small-sized sliding windows, the global minimum in the NLIS criterion function is wide and becomes easy to find. For maximum size windows, the NILS is equivalent to the NLLS estimator, which implies statistical efficiency for Gaussian noise. A blind signal parameter search algorithm that does not use any a priori information is proposed. The NILS estimator can be interpreted as a signal-subspace projection-based algorithm. Moreover, the NILS estimator can be interpreted as an estimator based on the prediction error of a (structured) linear predictor. Hereby, a link is established between NLLS, signal-subspace fitting, and linear prediction-based estimation approaches. The NILS approach is primarily applicable to deterministic signal models. Specifically, polynomial-phase signals are studied, and the NILS approach is evaluated and compared with other approaches. Simulations show that the signal-to-noise ratio (SNR) threshold is significantly lower than that of the other methods, and it is confirmed that the estimates are statistically efficient. Just as the NLLS approach, the NILS estimator can be applied to nonuniformly sampled data.

Parameter estimation for random amplitude chirp signals

We consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise, i.e., whose amplitude is randomly time-varying. Two methods for solving this problem are presented. First, an unstructured nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional (2-D) maximization problem. A theoretical analysis of the NLS estimator is presented, and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance that is very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HBF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of the HAF-based estimator is also carried out, and closed-form expressions are derived for the asymptotic variance of the HAF estimators based on the data and on the squared data. Numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods.

Transfer function estimation using elemental sets

Nonlinear least-squares (NLS) fitting of rational transfer functions to frequency response data yields the maximum likelihood estimator (MLE) of the transfer function coefficient vector under mild conditions on the observation noise. Furthermore, the NLS approach is robust to errors in the modeling of data. However the NLS criterion is in general difficult to minimize. Here we show that an asymptotic realization of the NLS estimator can be obtained from the elemental set parameter estimates by simple linear operations. The latter estimates are derived by matching the frequency response data on many "elemental sets" comprising a number of frequencies equal to half the number of unknown parameters.

Real-time estimation of turning movement proportions from partial counts on urban networks

Modern traffic signal control systems require reliable estimates of turning flows in real time to formulate effective control actions, and accommodate disturbances in traffic demand without deteriorating the system performance. The more accurate the estimation is, the more effective the control plan is. Most of the previous research works assumed that a full set of detector counts is available and employed the least-squares methods to produce unbiased estimates of the turning movement proportions. However, in practice, such a dense detector configuration is expensive to install and maintain. Also, the least-squares estimates are not feasible when the travel time between inflows and outflows is significant, or when intervening traffic conditions change the travel time. This study proposes a nonlinear least-square (NLS) approach and a quasi maximum likelihood (QML) approach to recursively estimate turning movement proportions in a network of intersections where only a partial set of detector counts are available. Using large population approximation technique, a class of nonlinear, discrete-time traffic flow models are transformed into a linear state–space model tractable for on-line applications. The quality of estimates is demonstrated by implementing the proposed algorithms with simulation and real data. As a comparison, the NLS estimator shows less bias but with higher variance than the QML estimator. The QML estimator outperforms the NLS estimator in terms of total mean square error, due to an increase in bias being traded for a decrease in variance.

Preliminary estimators for robust non-linear regression estimation

In this paper, the robustness of weighted non-linear least-squares estimation based on some preliminary estimators is examined. The preliminary estimators are the Lnorm estimates proposed by Schlossmacher, by El-Attar et al., by Koenker and Park, and by Lawrence and Arthur. A numerical example is presented to compare the robustness of the weighted non-linear least-squares approach when based on the preliminary estimators of Schlossmacher (HS), El-Attar et al. (HEA), Koenker and Park (HKP), and Lawrence and Arthur (HLA). The study shows that the HEA estimator is as robust as the HKP estimator. However, the HEA estimator posed certain computational problems and required more storage and computing time.