Classical Least-squares Method Research Articles

Accuracy Comparison of LS and Squared-Range LS for Source Localization

In this correspondence, we compute a closed-form expression for the asymptotic (large-sample) accuracy of the recently proposed squared-range least-squares (SR-LS) method for source localization [A. Beck, P. Stoica, and J. Li, ¿Exact and approximate solutions of source localization problems,¿ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Trans. Signal Process.</i> , vol. 56, no. 5, pp. 1770-1778, May 2008]. We compare its accuracy to that of the classical least-squares (LS) method and show that LS and SR-LS perform differently in general. We identify geometries where the performances of the methods are identical but also geometries when the difference in performance is unbounded.

HPLC and chemometrics-assisted UV-spectroscopy methods for the simultaneous determination of ambroxol and doxycycline in capsule

High-performance liquid chromatography (HPLC) and multivariate spectrophotometric methods are described for the simultaneous determination of ambroxol hydrochloride (AM) and doxycycline (DX) in combined pharmaceutical capsules. The chromatographic separation was achieved on reversed-phase C 18 analytical column with a mobile phase consisting of a mixture of 20 mM potassium dihydrogen phosphate, pH 6–acetonitrile in ratio of (1:1, v/v) and UV detection at 245 nm. Also, the resolution has been accomplished by using numerical spectrophotometric methods as classical least squares (CLS), principal component regression (PCR) and partial least squares (PLS-1) applied to the UV spectra of the mixture and graphical spectrophotometric method as first derivative of the ratio spectra ( 1DD) method. Analytical figures of merit (FOM), such as sensitivity, selectivity, analytical sensitivity, limit of quantitation and limit of detection were determined for CLS, PLS-1 and PCR methods. The proposed methods were validated and successfully applied for the analysis of pharmaceutical formulation and laboratory-prepared mixtures containing the two component combination.

Content uniformity of pharmaceutical solid dosage forms by near infrared hyperspectral imaging: A feasibility study

Near-infrared imaging systems simultaneously record spectral and spatial information. Each measurement generates a data cube containing several thousand spectra. Chemometric methods are therefore required to extract qualitative and quantitative information. The aim of this study was to determine the feasibility of quantifying active pharmaceutical ingredient (API) and excipient content in pharmaceutical formulations using hyperspectral imaging. Two kinds of tablets with a range of API content were analysed: a binary mixture of API and cellulose, and a pharmaceutical formulation with seven different compounds. Two pixel sizes, 10 μm/pixel and 40 μm/pixel, were compared, together with two types of spectral pretreatment: standard normal variate (SNV) normalization and Savitzky–Golay smoothing. Two methods of extracting concentrations were compared: the partial least squares 2 (PLS2) algorithm, which predicts the content of several compounds simultaneously, and the multivariate classical least squares (CLS) algorithm based on pure compound reference spectra without calibration. Best content predictions were achieved using 40 μm/pixel resolution and the PLS2 method with SNV normalized spectra. However, the CLS method extracted distribution maps with higher contrast and was less sensitive to noisy spectra and outliers; its API predictions were also highly correlated to real content, indicating the feasibility of predicting API content using hyperspectral imaging without calibration.

Simultaneous determination of Nifuroxazide and Drotaverine hydrochloride in pharmaceutical preparations by bivariate and multivariate spectral analysis

The quantitative predictive abilities of the new and simple bivariate spectrophotometric method are compared with the results obtained by the use of multivariate calibration methods [the classical least squares (CLS), principle component regression (PCR) and partial least squares (PLS)], using the information contained in the absorption spectra of the appropriate solutions. Mixtures of the two drugs Nifuroxazide (NIF) and Drotaverine hydrochloride (DRO) were resolved by application of the bivariate method. The different chemometric approaches were applied also with previous optimization of the calibration matrix, as they are useful in simultaneous inclusion of many spectral wavelengths. The results found by application of the bivariate, CLS, PCR and PLS methods for the simultaneous determinations of mixtures of both components containing 2–12 μg ml −1 of NIF and 2–8 μg ml −1 of DRO are reported. Both approaches were satisfactorily applied to the simultaneous determination of NIF and DRO in pure form and in pharmaceutical formulation. The results were in accordance with those given by the EVA Pharma reference spectrophotometric method.

Overview of total least-squares methods

We review the development and extensions of the classical total least-squares method and describe algorithms for its generalization to weighted and structured approximation problems. In the generic case, the classical total least-squares problem has a unique solution, which is given in analytic form in terms of the singular value decomposition of the data matrix. The weighted and structured total least-squares problems have no such analytic solution and are currently solved numerically by local optimization methods. We explain how special structure of the weight matrix and the data matrix can be exploited for efficient cost function and first derivative computation. This allows to obtain computationally efficient solution methods. The total least-squares family of methods has a wide range of applications in system theory, signal processing, and computer algebra. We describe the applications for deconvolution, linear prediction, and errors-in-variables system identification.

Remote sensing by open-path FTIR spectroscopy. Comparison of different analysis techniques applied to ozone and carbon monoxide detection

The analysis of FTIR spectra is usually carried out by commercial programs that use methods based on classical least-squares (CLS) procedures These procedures are normally appropriate to analyse O-P FTIR spectra, but some occasional discrepancies with standard extractive methods have been observed. In this work, a line-by-line method (SFIT) is additionally used. Our purpose is to explain the sporadic discrepancies observed between CLS and standard extractive method results and to study the capability of both methods (CLS and SFIT) to analyse open-path measurements. This capability has been studied for ozone and carbon monoxide. It has been demonstrated that in CLS-based methods discrepancies are induced by the experimental background reference spectrum. In these cases, SFIT results are in very good agreement with the standard extractive methods.

Determination of Carbon Monoxide Concentration and Total Pressure in Gas Cavities in the Silica Glass Body of Light Bulbs by FT-IR Spectrometry

Fourier transform infrared (FT-IR) spectroscopy has been adapted to control the quality of light bulbs made from silica glass. Such light bulbs contain a molybdenum accessory which, if contaminated with carbon, during the melting procedure of bulb fabrication, can cause the production of carbon monoxide. This CO can be trapped in small gas cavities in the silica glass body of the bulb. A method has been developed for the detection of CO and the total pressure within these gas cavities by traditional FT-IR spectrometry using a spectral resolution of 0.5 cm(-1). The concentration of CO was determined by using a classical least-squares (CLS) method, and the accuracy of concentration determination is reported for the case with sample and reference spectra recorded at different pressures. The total pressure in the cavities was established by two different methods: either by CLS fitting of reference spectra to sample spectra or fitting a Voigt line shape function to the spectral lines within the CO fundamental stretching band. In the latter method, the width of the lines was determined and pressure-broadening coefficients are given and compared with high-resolution data from the literature. According to the measurements, 0.55-0.80 atm total pressure and 0.8-4.0% (v/v) CO was determined in the gas cavities. This method can also be applied to determine the total pressure in similar enclosed spaces in which an appropriate indicator gas component exists.

Simultaneous ultraviolet–spectrophotometric determination of sulfonamides by multivariate calibration approaches

Anti-bacterial drugs such as sulfonamides and trimethoprim, are commonly used in medical and veterinary applications. The tablets often contain mixtures of drugs such as sulfadiazine, sulfadimidine, sulfamethoxazole, sulfanilamide and trimethoprim. Thus, a simple, rapid and inexpensive method for the simultaneous determination of these drugs was researched and developed with the aid of chemometrics methods comparing the classical least squares (CLS), principal component regression (PCR) and partial least squares (PLS) models. UV-spectra were collected in the 200–350 nm range from a set of samples containing aqueous ethanolic solutions (pH = 9.9) of drug mixtures prepared from the above compounds with the aid of an orthogonal array design. Initial studies of the UV-spectra of the individual compounds showed that satisfactory linear regression calibration models could be constructed in the concentration range of approximately 1.0–24.0 mg l − 1 for all compounds with LOD values in the range of ca. 0.2–1.0 mg l − 1 for each of the five drugs. These studies also indicated the significant spectral overlap of the UV- spectra of the drugs showing the need for chemometrics modeling for simultaneous analysis of the mixtures. Calibration models for the three multivariate methods were constructed with the use of the raw and derivative spectral data sets. These were verified with the aid of UV-spectral data obtained from a separately prepared set of drug mixture samples. The %RPE values for analysis of the individual drugs in mixtures were in the range of 2–10 with %Recoveries being close to 100 for the PCR and PLS models while the CLS method performed relatively poorly. The four PCR and the PLS models constructed could not be distinguished on prediction performance for practical purposes, and the PCR calibration based on the raw data matrix was utilised to illustrate successfully the application of the proposed method for analysis of the drugs in commonly available pharmaceutical tablets.

In-Phase Error Estimation of Experimental Data and Optimal First Derivatives

The development of in-phase, a posteriori error estimates and the best functional representation for the solution and its derivative are examined. Several new and fundamental findings pertinent to both the experimentalist and analyst are presented. Applications requiring a high degree of precision such as associated with the defense, aerospace, and heat treatment arenas may benefit from these new findings. The basic framework involves the classical least-squares method, which is reexamined with the intent of 1) approximating local measurement errors necessary for estimating uncertainty and 2) predicting optimal solutions in both the primitive function and its derivative. With accurate local-error estimates, the initial data can be corrected by adding these estimates to the initial data to substantially reduce the root-mean-square error, that is, uncertainty. Additionally, the clarification between best curve fit and best prediction or solution is elucidated to provide insight into the meaning of a desirable resolution. The framework offered presents an elegant but practical method for approximating local measurement errors and for developing a functional representation of the desired function that best represents the solution in the presence of errorless data

A simple but efficient algorithm for multiple-image deblurring

We consider the simultaneous deblurring of a set of noisy images whose point spread functions are different but known and spatially invariant, and with Gaussian noise. Currently available iterative algorithms that are typically used for this type of problem are computationally expensive, which makes their application for very large images impractical. We present a simple extension of a classical least-squares (LS) method where the multi-image deblurring is efficiently reduced to a computationally efficient single-image deblurring. In particular, we show that it is possible to remarkably improve the ill-conditioning of the LS problem by means of stable operations on the corresponding normal equations, which in turn speed up the convergence rate of the iterative algorithms. The performance and limitations of the method are analyzed through numerical simulations. Its connection with a column weighted least-squares approach is also considered in an appendix.

Load Updating for Finite Element Models

A method for reducing the differences between experimental testing or real-world events and their finite element counterparts is presented. In this method system identification or finite element model updating are not performed; instead, the loads on the finite element model that would create equivaient displacements or strains are identified by various means, thus saving a large amount of computational effort. This method of finding an applied load is also extremely useful in the analysis of testing data for vehicles and other structures. Using methods based on classical least-squares methods, we present the basis for finite element load updating and sample application through the use of a computer program for a beam under static loads. We examine the conditioning number of the resulting system of equations and provide a least-squares solution that is more robust through the use of singular value decomposition. This allows an easier analysis of structural models, for example, of a prototype vehicle, by determining the loads that the prototype is subjected to in testing. These loads can then be used on other models of that vehicle during the design process to decrease the time spent in testing while increasing design's reliability by reducing uncertainties in the applied load.

The Long‐Term Evolution of AR 7978: The Scalings of the Coronal Plasma Parameters with the Mean Photospheric Magnetic Field

We analyze the evolution of the fluxes observed in X-rays and correlate them with the magnetic flux density in active region (AR) NOAA 7978 from its birth throughout its decay, for five solar rotations. We use Solar and Heliospheric Observatory Michelson Doppler Imager (MDI) data, together with Yohkoh Soft X-Ray Telescope (SXT) and Yohkoh Bragg Crystal Spectrometer (BCS) data, to determine the global evolution of the temperature and the emission measure of the coronal plasma at times when no significant brightenings were observed. We show that the mean X-ray flux and derived parameters, temperature and emission measure ( together with other quantities deduced from them, such as the density and the pressure), of the plasma in the AR follow power-law relationships with the mean magnetic flux density ((B) over bar). The exponents (b) of these power-law functions (a (B) over bar (b)) are derived using two different statistical methods, a classical least-squares method in log-log plots and a nonparametric method, which takes into account the fact that errors in the data may not be normally distributed. Both methods give similar exponents, within error bars, for the mean temperature and for both instruments (SXT and BCS); in particular, b stays in the range [0.27, 0.31] and [0.24, 0.57] for full-resolution SXT images and BCS data, respectively. For the emission measure, the exponent b lies in the range [0.85, 1.35] and [0.45, 1.96] for SXT and BCS, respectively. The determination of such power-law relations, when combined with the results from coronal heating models, can provide us with powerful tools for determining the mechanism responsible for the existence of the high-temperature corona.

Identification of Physical Parameters of Unstable Systems: Theoretical Background

Identification of physical parameters of unstable system is an important problem, for example in the design of its diagnostic system. In the diagnostics we usually observe the trends in physical parameters changes. Two identification methods of the state space model resulting from the physics laws are presented in the paper. First one is a modification of the Observer/Kalman Filter Identification (OKID) method (J.-N. Juang (1994). Applied System Identification, Prentice-Hall, Englewood Cliffs, NJ) and the observer/controller Markov parameters are computed. They are used to obtain the physical state-space model of the open-loop system. In the second method an AutoRegressive model with eXogenous inputs (ARX) model of the observer/controller system is identified first. To identify an ARX model on-line the classical Recursive Least-Squares (RLS) method is used.

Application of a classical least-squares regression method to the assay of 1,4-dihydropyridine antihypertensives and their photoproducts

A multicomponent UV spectrophotometric method using a classical least-squares (CLS) algorithm has been developed for the quantitative determination of 1,4-dihydropyridine (DHP) calcium antagonists and respective photoproducts. The procedure was optimized by defining a fractionation scheme for selecting the more useful wavelength ranges to be used in the calibration model. The method is potentially able to be extended to the other drugs of the same family. The recovery values from synthetic mixtures and commercial formulations were verified to be 99.09 and 97.85% for drug parents and degradation products, respectively. The sensitivity for the photoproducts determination was found to be below 0.8%. The results obtained from laboratory mixtures and commercial formulations were compared with those provided by UV derivative spectrophotometry.

Estimation of symmetric positive-definite matrices from imperfect measurements

In a number of contexts relevant to control problems, including estimation of robot dynamics, covariance, and smart structure mass and stiffness matrices, we need to solve an overdetermined set of linear equations AX /spl ap/ 8 with the constraint that the matrix X be symmetric and positive definite. In the classical least-squares method, the measurements of A are assumed to be free of error, hence, all errors are confined to B. Thus, the optimal solution is given by minimizing the optimization criterion /spl par/AX /spl par//sub F//sup 2/. However, this assumption is often impractical. Sampling errors, modeling errors, and, sometimes, human errors bring inaccuracies to A as well. In this note, we introduce a different optimization criterion, based on area, which takes the errors in both A and B into consideration. Under the condition that the data matrices A and B are full rank, which in practice is easy to satisfy, the analytic expression of the global optimizer is derived. A method to handle the case that A is full rank and B loses rank is also discussed. Experimental results indicate that the new approach is practical, and improves performance.

Concentration Residual Augmented Classical Least Squares (CRACLS): A Multivariate Calibration Method with Advantages over Partial Least Squares

A significant extension to the classical least-squares (CLS) algorithm called concentration residual augmented CLS (CRACLS) has been developed. Previously, unmodeled sources of spectral variation have rendered CLS models ineffective for most types of problems, but with the new CRACLS algorithm, CLS-type models can be applied to a significantly wider range of applications. This new quantitative multivariate spectral analysis algorithm iteratively augments the calibration matrix of reference concentrations with concentration residuals estimated during CLS prediction. Because these residuals represent linear combinations of the unmodeled spectrally active component concentrations, the effects of these components are removed from the calibration of the analytes of interest. This iterative process allows the development of a CLS-type calibration model comparable in prediction ability to implicit multivariate calibration methods such as partial least squares (PLS) even when unmodeled spectrally active components are present in the calibration sample spectra. In addition, CRACLS retains the improved qualitative spectral information of the CLS algorithm relative to PLS. More importantly, CRACLS provides a model compatible with the recently presented prediction-augmented CLS (PACLS) method. The CRACLS/PACLS combination generates an adaptable model that can achieve excellent prediction ability for samples of unknown composition that contain unmodeled sources of spectral variation. The CRACLS algorithm is demonstrated with both simulated and real data derived from a system of dilute aqueous solutions containing glucose, ethanol, and urea. The simulated data demonstrate the effectiveness of the new algorithm and help elucidate the principles behind the method. Using experimental data, we compare the prediction abilities of CRACLS and PLS during cross-validated calibration. In combination with PACLS, the CRACLS predictions are comparable to PLS for the prediction of the glucose, ethanol, and urea components for validation samples collected when significant instrument drift was present. However, the PLS predictions required recalibration using nonstandard cross-validated rotations while CRACLS/PACLS was rapidly updated during prediction without the need for time-consuming cross-validated recalibration. The CRACLS/PACLS algorithm provides a more general approach to removing the detrimental effects of unmodeled components.

Simultaneous determination of domperidone maleate and cinnarizine in a binary mixture using derivative ratio spectrophotometry and classical least squares calibration

This work is concerned with the simultaneous determination of domperidone maleate (DOM) and cinnarizine (CINN) in a binary mixture form without previous separation by two different methods. The first method is the application of derivative ratio spectrophotometry where the linearity range was 2.5–30 μg/ml, 2.5–25 μg/ml for DOM and CINN, respectively, and percentage recoveries were 100.26±1.308 and 99.86±0.939 for DOM and CINN, respectively, in their laboratory prepared mixtures. The second method depends on the application of classical least squares (CLS) calibration model. Two training sets were constructed and the best model was used for the prediction of the concentrations of both drugs. The proposed procedures were successfully applied for the simultaneous determination of both drugs in laboratory prepared mixtures and in commercial tablet preparations. The validity of the proposed methods was assessed by applying the standard addition technique where the percentage recovery of the added standard was found to be 99.83±1.861 and 98.38±0.871 for DOM and CINN, respectively, using the derivative ratio method and 99.53±0.916 and 99.39±0.599 for DOM and CINN, respectively, using the CLS method. The proposed procedures are rapid, simple, require no preliminary separation steps and can, therefore, be used routine analysis of both drugs in quality control laboratories.

Comparison of an Innovative Nonlinear Algorithm to Classical Least-Squares for Analyzing Open-Path Fourier Transform Infrared Spectra Collected at a Concentrated Swine Production Facility

Open-path Fourier transform infrared (OP/FT-IR) spectrometry was used to measure the concentrations of ammonia, methane, and other atmospheric gases at an integrated swine production facility. The concentration-pathlength products of the target gases at this site often exceeded the linear dynamic range of the OP/FT-IR spectrometer. To minimize the effect of this nonlinearity on the accuracy of the reported concentrations, a piecewise, linear classical least-squares (CLS) analysis method, which used a calibration set containing multiple reference spectra for each target gas at concentration-pathlength products that encompassed those found in the field spectra, was used to predict the path-averaged concentrations of the target gases. The predicted concentrations reported by this piecewise, linear CLS method were compared to those predicted by a conventional linear CLS method, which used a calibration set consisting of only one reference spectrum at a single concentration-pathlength product for each target gas, and an innovative nonlinear algorithm (NLA), which performs an iterative fit of the convolved spectral line data from the high-resolution transmission molecular absorption (HITRAN) database to the single-beam field spectra. The conventional, linear CLS method generally under-reported the target gas concentrations relative to those predicted by the piecewise, linear CLS method when the field spectra exhibited concentration-pathlength products larger than those of the reference spectra in the single-level calibration set. In extreme cases, for example, during measurements of methane along the waste lagoon, the concentrations predicted by the conventional CLS methods were more than 30% lower than those predicted by the piecewise, linear CLS method. In contrast, the concentrations of methane predicted by the NLA were, on average, within 4% of those predicted by the piecewise, linear CLS method. For ammonia, however, the concentrations predicted by the NLA were slightly higher than those predicted by the piecewise, linear CLS method at the lower range of observed concentration-pathlength products and slightly lower than those predicted by the piecewise, linear CLS method at the upper range of concentration-pathlength products. The NLA also consistently predicted higher concentrations of nitrous oxide and carbon dioxide relative to that predicted by the piecewise, linear CLS method. Differences in the background spectra and spectral ranges over which the analyses were conducted apparently did not contribute to the differences in the path-averaged concentrations predicted by these two analysis methods.

New augmented classical least squares methods for improved quantitative spectral analyses

A series of new augmented classical least squares (ACLS) methods have been developed that show considerable promise for improving multivariate spectral calibrations. The normal limitations of classical least squares (CLS) methods are largely overcome by using information derived from the spectral residuals during the CLS calibration in the development of the new ACLS models. The resulting models are able to provide prediction ability comparable to that of partial least squares (PLS) even when only concentration information about a single analyte is known in the calibration data set. By combining ACLS calibration with our recently described prediction augmented CLS (PACLS) method, we are able to rapidly update spectral calibration models during prediction for the presence of unmodeled chemical components in the unknown samples, system drift, or changes in spectrometers. With PACLS updating, prediction ability is often better than possible with PLS. One of the new calibration augmented CLS methods is combined with PACLS to demonstrate the ability of the method to transfer multivariate calibrations between near-infrared (NIR) spectrometers on a system of multi-component organic solvents.

Simultaneous determination of nitrobenzene and nitro-substituted phenols by differential pulse voltammetry and chemometrics

The voltammetric behaviour of five nitro-substituted aromatic compounds, such as nitrobenzene, 2-, 3-, 4-nitrophenol and 2,4-dinitrophenol, was investigated, and a method was developed for the simultaneous determination of these compounds, based on their reduction at a hanging mercury drop electrode (HMDE). It was found that by applying the differential pulse voltammetric (DPV) approach in a Britton–Robinson buffer of pH 5.4, nitrobenzene, 2-, 3- and 4-nitrophenol had well defined voltammetric reduction waves with peak potentials at −344, −284, −292 and −376 mV, respectively. However, 2,4-dinitrophenol had two peaks with potentials at −240 and −364 mV under the same conditions. For each compound linear calibration graphs were obtained in the general concentration range of 0.05–3.0 mg l −1. In mixtures of the five compounds, serious overlapping of voltammetric peaks was observed, and with conventional analytical methodology pre-separation steps would be required. In this study, chemometrics methods of data analysis, such as partial least squares (PLS), principal component regression (PCR) and classical least squares (CLS), were applied to resolve the overlapped voltammograms. Orthogonal experimental design was used for construction of the training sets containing the five aromatic compounds in the concentration range of 0.1–2.6 mg l −1. Five significant factors or principal components were modelled for prediction by the PLS and PCR methods, respectively. The percent of relative prediction error (RPE) was similar and acceptable for both methods being approximately ±10%. The percent of recoveries were within ±10% of the target value. The CLS method performed poorly (RPE T=38%). The developed method was then applied to the analysis of these nitro-substituted aromatic compounds in field samples with similar satisfactory results.