Iterative Nonlinear Least-squares Research Articles

Efficient transient correlation of thermal lumped network models to reference data

In this paper, an efficient algorithm for correlating the transient behaviour of thermal lumped models to reference data is reported. The algorithm is based on an iterative non-linear least-squares minimization process that allows for fast convergence. The correlation process requires the calculation of the so-called Moore–Penrose pseudoinverse of the Jacobian matrix. However, obtaining the transient Jacobian matrix can be computationally expensive for large models. A new method for calculating the transient Jacobian matrix of lumped thermal models, called Jacobian Propagation, is also presented. For large models, the Jacobian Propagation method, when combined with an implicit solver, has a negligible computational cost compared to the solving process. The correlation algorithm and the Jacobian Propagation method have been successfully tested with two thermal models, obtaining better performance than previous approaches.

Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Nonresonant Nodes and Arbitrary Frequency-Variant Reactive Couplings

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this article. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary reactive frequency-variant coupling (AFVC) networks is proposed. The engineered formulation is more general than prior-art ones—with the only restriction that the coupling network is a reactive-type two-port circuit—and can be treated as an extension of previous synthesis models since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, which allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded nonresonating nodes (NRNs), frequency-dependent source–load coupling, multiple frequency-variant cross couplings, and/or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, the third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs).

Optimization of spin-lock times in T1ρ mapping of knee cartilage: Cramér-Rao bounds versus matched sampling-fitting.

To compare different optimization approaches for choosing the spin-lock times (TSLs), in spin-lattice relaxation time in the rotating frame (T1ρ ) mapping. Optimization criteria for TSLs based on Cramér-Rao lower bounds (CRLB) are compared with matched sampling-fitting (MSF) approaches for T1ρ mapping on synthetic data, model phantoms, and knee cartilage. The MSF approaches are optimized using robust methods for noisy cost functions. The MSF approaches assume that optimal TSLs depend on the chosen fitting method. An iterative non-linear least squares (NLS) and artificial neural networks (ANN) are tested as two possible T1ρ fitting methods for MSF approaches. All optimized criteria were better than non-optimized ones. However, we observe that a modified CRLB and an MSF based on the mean of the normalized absolute error (MNAE) were more robust optimization approaches, performing well in all tested cases. The optimized TSLs obtained the best performance with synthetic data (3.5-8.0% error), model phantoms (1.5-2.8% error), and healthy volunteers (7.7-21.1% error), showing stable and improved quality results, comparing to non-optimized approaches (4.2-13.3% error on synthetic data, 2.1-6.2% error on model phantoms, 9.8-27.8% error on healthy volunteers). A modified CRLB and the MSF based on MNAE are robust optimization approaches for choosing TSLs in T1ρ mapping. All optimized criteria allowed good results even using rapid scans with two TSLs when a complex-valued fitting is done with iterative NLS or ANN.

A New Iterative Algorithm for Estimating Parameters and Orders of Multiple-Input Single-Output Time Series Models

In this paper, we propose a new iterative algorithm for estimating the parameters and orders of a multiple-input single-output (MISO) time series model. This algorithm is based on a method suggested by Hannan and Rissanen (1982) for estimating an ARMA model. The key is the use of pseudo-linear regression techniques to derive the iterative nonlinear least-squares estimators by using the Gauss-Newton algorithm. Simulation results are presented to compare the new algorithm with the exact maximum likelihood method (EML) and the generalized least squares (GLS) method proposed by Sabiti (1997).

A spectral weighting function for abiotic photodegradation based on photochemical emission of CO2 from leaf litter in sunlight

Photodegradation can be a significant driver of leaf litter decomposition although the spectral effectiveness of sunlight in driving this process is not well characterized. We developed spectral weighting functions (WFs) for the photochemical emission of CO2 from three leaf litter types using 10 cutoff filters that provided contrasting polychromatic sunlight under clear skies in Tempe, AZ, USA. An iterative nonlinear least-squares regression fitting procedure was used to estimate how effective sunlight at a given wavelength was in eliciting CO2 emission. Although absolute CO2 emission rates varied appreciably among litter types, their WFs were very similar. Using the average WF of all litter types, the effectiveness of sunlight declined from 300 nm by one and two orders of magnitude at 399 and 498 nm, respectively. The slope of the WF was most similar to WFs for CO emission from terrestrial leaf litter and photobleaching of dissolved organic matter in lakes, and was much more gradual than WFs addressing UV damage to biotic processes. Peak effectiveness of clear-sky noon sunlight with our WF occurred at 330 nm, with UV-B (280–320 nm), UV-A (320–400 nm) and visible (400–550 nm) wavebands responsible for 9, 61 and 30% of CO2 emission, respectively. Results from past field studies suggest that solar UV is typically less effective in driving litter mass loss than our WF predicts; we discuss possible reasons for this discrepancy. The gradual slope of our WF implies that differences in UV-B irradiance associated with stratospheric ozone thickness or latitude are unlikely to significantly influence photochemical litter emission.

A Simplified Time-Domain Fitting Method Based on Fractional Operational Matrix for Cole Parameter Estimation

The Cole model is widely used in bioimpedance spectroscopy (BIS) applications for evaluating the contents and status of biological tissues. Traditional frequency-domain fitting methods for Cole parameter estimation (CPE) often need wideband direct or indirect BIS measurements. The newly emerged time-domain fitting methods do not require BIS measurement, but they usually have unsatisfactory estimation accuracy and heavily rely on some specific excitations. This paper proposes a novel time-domain fitting method based on the fractional operational matrix (FOM) for CPE. First, the Cole model is represented as a FOM-based algebraic equation. Then, the model-based voltage response can be calculated as the product of the sampled current excitation and the FOM-based Cole equation. Finally, the Cole parameter can be estimated by minimizing the sum of squares error between the sampled voltage response and the model-based voltage response using an iterative nonlinear least-squares fitting algorithm. Experimental measurements are verified on three 2R1C circuits and in vivo leg muscle of a healthy volunteer. The results show that the Cole parameter can be accurately estimated by the proposed FOM-based fitting method directly using the sampled current and voltage signals. This paper establishes an effective time-domain fitting method for CPE, which could greatly simplify the hardware and software designs without the need for the BIS measurement.

The Nonlinear Iterative Least Squares (NL-ILS) Estimator: An Application to Volatility Models

The paper proposes a new robust estimator for GARCH-type models: the nonlinear iterative least squares (NL-ILS). This estimator is especially useful on specications where errors have some degree of dependence over time or when the conditional variance is misspecied.

Two-dimensional Gaussian fitting for precise measurement of lattice constant deviation from a selected-area diffraction map.

Unlike X-ray diffraction or Raman techniques, which suffer from low spatial resolution, transmission electron microscopy can be used to obtain strain maps of nanoscaled materials and devices. Convergent-beam electron diffraction (CBED) and nanobeam electron diffraction (NBED) techniques detect the deviation of a lattice constant (i.e. an indicator of strain) within 0.01%; however, their use is restricted to beam-insensitive samples. Selected-area electron diffraction (SAED) does not have such limitations but has low spatial resolution and precision. The use of a spherical aberration corrector and a nanosized selected-area aperture improves the spatial resolution, but the precision is still low. In this study, a two-dimensional stage-scanning system is used to acquire arrays of diffraction patterns at different positions of the sample under fixed beam conditions. Data processing with iterative nonlinear least-squares fitting enabled the spot displacement for each point of the scan area to be measured with precision comparable to that of the CBED or NBED technique. The precise strain determination, in combination with the simplicity of the measurement process, makes the nanosized SAED technique competitive with other methods for strain mapping at nanoscale dimensions.

Improvement of depth resolution in depth-resolved wavenumber-scanning interferometry using wavenumber-domain least-squares algorithm: comparison and experiment.

It is important to improve the depth resolution in depth-resolved wavenumber-scanning interferometry (DRWSI) owing to the limited range of wavenumber scanning. In this work, a new nonlinear iterative least-squares algorithm called the wavenumber-domain least-squares algorithm (WLSA) is proposed for evaluating the phase of DRWSI. The simulated and experimental results of the Fourier transform (FT), complex-number least-squares algorithm (CNLSA), eigenvalue-decomposition and least-squares algorithm (EDLSA), and WLSA were compared and analyzed. According to the results, the WLSA is less dependent on the initial values, and the depth resolution δz is approximately changed from δz to δz/6. Thus, the WLSA exhibits a better performance than the FT, CNLSA, and EDLSA.

Image Processing of Illuminated Ellipsoid

This study introduces novel algorithms and the underlying mathematics to process photographs of planetary illuminated bodies and use them for navigation purposes. The goal is to accurately estimate the observer-to-body relative position in inertial coordinates. The main motivation is to provide autonomous navigation capabilities to spacecrafts by observing a planet or a moon. This is needed, for example, in human-rated vehicles in order to provide navigation capabilities in a loss-of-communications scenario. The algorithm is derived for the general case of a triaxial ellipsoid that is observed bounded by an elliptical cone. The orientation of the elliptical cone reference frame is obtained by eigenanalysis, and the offset between the elliptical cone axis and the body center direction as well as the equation of the terminator are quantified. The main contribution of this paper is in the image-processing approach adopted to derive centroid and distance to the body. This is done by selecting a set of pixels around the body limb and modeling the smooth limb transition using two-dimensional circular and elliptical sigmoid functions. More accurate estimates of the centroid and distance are then obtained by iterative nonlinear least-squares using these models. A sensitivity analysis is performed, and numerical examples using a real moon photograph taken from Earth are provided to clarify the image-processing steps and to validate the proposed theory.

Fundamental Initial Frequency and Frequency Rate Estimation of Random-Amplitude Harmonic Chirps

We consider the problem of estimating the fundamental initial frequency and frequency rate of a linear chirp with random amplitudes harmonic components. We develop an iterative nonlinear least squares estimator, which involves a large number of computations as it requires high resolution search in the initial frequency and frequency rate parameter space. As an alternative, we suggest two suboptimal low-complexity estimators. The first is based on the high-order ambiguity function, which reduces the problem to a one-dimensional search. The second method applies our recently published harmonic separate-estimate method, which was used for constant-amplitude harmonic chirps. We present modifications of both methods for harmonic chirps with random amplitudes. We also provide a framework for estimating the number of harmonic components. Numerical simulations show that the iterative nonlinear least-squares estimator achieves its asymptotic accuracy in medium to high signal-to-noise ratio, while the two sub-optimal low-complexity estimators perform well in high signal-to-noise ratio. Real data examples demonstrate the performance of the harmonic separate-estimate method on random amplitude real-life signals.

Fast and High-Accuracy Localization for Three-Dimensional Single-Particle Tracking

We report a non-iterative localization algorithm that utilizes the scaling of a three-dimensional (3D) image in the axial direction and focuses on evaluating the radial symmetry center of the scaled image to achieve the desired single-particle localization. Using this approach, we analyzed simulated 3D particle images by wide-field microscopy and confocal microscopy respectively, and the 3D trajectory of quantum dots (QDs)-labeled influenza virus in live cells. Both applications indicate that the method can achieve 3D single-particle localization with a sub-pixel precision and sub-millisecond computation time. The precision is almost the same as that of the iterative nonlinear least-squares 3D Gaussian fitting method, but with two orders of magnitude higher computation speed. This approach can reduce considerably the time and costs for processing the large volume data of 3D images for 3D single-particle tracking, which is especially suited for 3D high-precision single-particle tracking, 3D single-molecule imaging and even new microscopy techniques.

Laser pulse transient method for measuring the normal spectral emissivity of samples with arbitrary surface quality

A laser pulse transient method for measuring normal spectral emissivity is described. In this method, a laser pulse (λ=1064 nm) irradiates the top surface of a flat specimen. A two-dimensional temperature response of the bottom surface is measured with a calibrated thermographic camera. By solving an axisymmetric boundary value heat conduction problem, the normal spectral emissivity at 1064 nm is determined by using an iterative nonlinear least-squares estimation procedure. The method can be applied to arbitrary sample surface quality. The method is tested on a nickel specimen and used to determine the normal spectral emissivity of AISI 304 stainless steel. The expanded combined uncertainty of the method has been estimated to be 18%.

Spectral properties and supramolecular inclusion complex formation between 2-styrylbenzothiazolium dye and cyclodextrins

The effect of native and randomly methylated β-CDs on the absorption and steady-state fluorescence spectra of 2-(4-dimethylaminostyryl)-3-(2-hydroxyethyl)-benzothiazolium chloride (DHB) in aqueous buffer solutions with various pH values was studied. The inclusion with both CDs at pH 7.2 barely changed the UV spectra, whereas significant variations were produced in the emission spectra in all buffer solutions. In all cases the CDs increase guest fluorescence. The 1:1 stoichiometry of the inclusion complexes of the dye with both CDs was established according to the modified Benesi-Hildebrand method. Binding constant values were calculated using the iterative nonlinear least-squares regression approach. The pH of the solution and the type of the CD affected complex stability. The results indicate that native β-CD possesses better complexing ability towards DHB than randomly substituted β-CD and that the most stable inclusion complexes are formed in basic medium because of the structural changes in the guest molecule. In basic medium an attempt is made to interpret the proposed mechanism in terms of molecular rearrangement which take place as the dye penetrates the CD cavity.

Mechanistic Studies of Rhodobacter sphaeroides Me2SO Reductase

Studies of the molybdenum-containing dimethyl sulfoxide reductase from Rhodobacter sphaeroides have yielded new insight into its catalytic mechanism. A series of reductive titrations, performed over the pH range 6-10, reveal that the absorption spectrum of reduced enzyme is highly sensitive to pH. The reaction of reduced enzyme with dimethyl sulfoxide is found to be clearly biphasic throughout the pH range 6-8 with a fast, initial substrate-binding phase and substrate-concentration independent catalytic phase. The intermediate formed at the completion of the fast phase has the characteristic absorption spectrum of the established dimethyl sulfoxide-bound species. Quantitative reductive and oxidative titrations of the enzyme demonstrate that the molybdenum center takes up only two reducing equivalents, implying that the two pyranopterin equivalents of the molybdenum center are not formally redox active. Finally, the visible spectrum associated with the catalytically relevant "high-g split" Mo(V) species has been determined. Spectral deconvolution and EPR quantitation of enzyme-monitored turnover experiments with trimethylamine N-oxide as substrate reveal that no substrate-bound intermediate accumulates and that Mo(V) content remains near unity for the duration of the reaction. Similar experiments with dimethyl sulfoxide show that significant quantities of both the Mo(V) species and the dimethyl sulfoxide-bound complex accumulate during the course of reaction. Accumulation of the substrate-bound complex in the steady-state with dimethyl sulfoxide arises from partial reversal of the physiological reaction in which the accumulating product, dimethyl sulfide, reacts with oxidized enzyme to yield the substrate-bound intermediate, a process that significantly slows turnover.

THE DETERMINATION OF THE UNKNOWN THERMAL PROPERTIES OF HOMOGENEOUS HEAT CONDUCTORS

The aim of this paper is to determine the two constant parameters corresponding to the physical properties of a homogeneous heat conductor, namely, the heat capacity and the thermal conductivity, from heat flux and temperature measurements. An iterative nonlinear least-squares boundary element method is proposed. The inversion is performed for both exact and noisy measurements. Numerically, it is shown that the thermal properties can uniquely and stably be retrieved from two measurements containing at least one heat flux measurement for finite homogeneous heat conductors, whilst theoretically, for the semi-infinite conductor it is shown that one heat flux and one internal temperature measurement are necessary and sufficient.

Application of a genetic algorithm: near optimal estimation of the rate and equilibrium constants of complex reaction mechanisms

In iterative non-linear least-squares fitting, the reliable estimation of initial parameters that lead to convergence to the global optimum can be difficult. Irrespective of the algorithm used, poor parameter estimates can lead to abortive divergence if initial guesses are far from the true values or in rare cases convergence to a local optimum. For determination of the parameters of complex reaction mechanisms, where often little is known about what value these parameters should take, the task of determining good initial estimates can be time consuming and unreliable. In this contribution, the methodology of applying a genetic algorithm (GA) to the task of determining initial parameter estimates that lie near the global optimum is explained. A generalised genetic algorithm was implemented according to the methodology and the results of its application are also given. The parameter estimates obtained were then used as the starting parameters for a gradient search method, which quickly converged to the global optimum. The genetic algorithm was successfully applied to both simulated kinetic measurements where the reaction mechanism contained one equilibrium constant and two rate constants to be fitted, and to kinetic measurements of the complexation of Cu 2+ by 1,4,8,11-tetraazacyclotetradecane where two equilibrium and two rate constants were fitted. The implementation of the algorithm is such that it can be generally applied to any reaction mechanism that can be expressed by standard chemistry notation. The control parameters of the algorithm can be varied through a simple user interface to account for parameter range and the number of parameters involved.

Parameter estimation for non-linear environmental models using below-detection data

Problems requiring regression analysis of below-detection or censored data frequently arise in the field of environmental engineering. Detection limits of analyzing instruments are the main reason for censored observations of pollutant concentrations. An iterative least-squares method for regression analysis is developed to suit the doubly censored data for non-linear models commonly used in environmental engineering. The algorithm developed, non-linear iterative least squares (NILS), utilizes the values expected for censored observations estimated from probability density functions of doubly censored data in the regression process. It has been shown through extensive simulation that the NILS method proposed returns estimates of model parameters close to those estimated with complete information, even for probability of censoring (POC) equal to 0.7 (i.e. 70% of the data are censored). Simulation results of 500 runs (sample size 20, 40 and 80 data points with POC 0.2, 0.4 and 0.7, respectively) from the algorithm developed are also compared to results obtained using some obvious simplifications on censored data, such as: (i) ignoring censored data; (ii) taking censored data as the detection limit; and (iii) taking censored data as equal to half of the difference between the lower limit (e.g. background level) and the detection limit. These simplifications have no scientific basis. It is shown that these simplifications on censored data cause substantial bias in the parameters estimated, and thus model predictions become uncertain. In addition to simulation analysis, the methodology is also established for actual observations for environmental models taken from the literature.

The radial Hamiltonian for the ground electronic state of KrH +

All literature vibration-rotational and pure rotational transitions belonging to ten isotopomers of KrH + in the X 1 Σ + state (in total 351 lines) have been used in a iterative nonlinear least-squares fit of compact analytic Born-Oppenheimer potential in the form of generalized potential energy function (GPEF). Additional functions describing the adiabatic and nonadiabatic corrections have also been determined. The obtained radial Hamiltonian reproduces the spectra with sufficiently good accuracy. The GPEF of KrH + reproduces the experimental value of the dissociation energy, has a correct R −4 asymptotic behaviour, and reproduces long-range inverse-power potential coefficient C 4.

Determination of the size and location of a discontinuity in a waveguide by inversion of measured scattered wavefields

It is shown that the combined use of: 1) a rigorous, numerically efficient, model of the measurements, on a strip of the upper face of the waveguide, of the response to mono mode monochromatic probe radiation, and 2) an iterative non-linear least-squares scheme to minimize the discrepancy between measured and theoretical response, enables a fairly-accurate retrieval, in near real-time, of the two location and one size parameters of the discontinuity.