Group Comparisons on Cognitive Attributes Using the Least Squares Distance Model of Cognitive Diagnosis

2010 Mathematics Subject Classification: 62P15.

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

A brief look at the least-squares approach as a classifier applied to restricted-vocabulary speech recognition

In this letter, a speech recognition algorithm based on the least-squares method is presented. Particularly, the intention is to exemplify how such a traditional numerical technique can be applied to solve a signal processing problem that is usually treated by using more elaborated formulations.

A simplified implementation of the least squares solution for pairwise comparisons matrices

This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by Bozóki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method.

Validation of the Item-Attribute Matrix in TIMSS-Mathematics Using Multiple Regression and the LSDM

The purposes of this study were (1) to validate of the item-attribute matrix using two levels of attributes (Level 1 attributes and Level 2 sub-attributes), and (2) through retrofitting the diagnostic models to the mathematics test of the Trends in International Mathematics and Science Study (TIMSS), to evaluate the construct validity of TIMSS mathematics assessment by comparing the results of two assessment booklets. Item data were extracted from Booklets 2 and 3 for the 8th grade in TIMSS 2007, which included a total of 49 mathematics items and every student's response to every item. The study developed three categories of attributes at two levels: content, cognitive process (TIMSS or new), and comprehensive cognitive process (or IT) based on the TIMSS assessment framework, cognitive procedures, and item type. At level one, there were 4 content attributes (number, algebra, geometry, and data and chance), 3 TIMSS process attributes (knowing, applying, and reasoning), and 4 new process attributes (identifying, computing, judging, and reasoning). At level two, the level 1 attributes were further divided into 32 sub-attributes. There was only one level of IT attributes (multiple steps/responses, complexity, and constructed-response). Twelve Q-matrices (4 originally specified, 4 random, and 4 revised) were investigated with eleven Q-matrix models (QM1 ~ QM11) using multiple regression and the least squares distance method (LSDM). Comprehensive analyses indicated that the proposed Q-matrices explained most of the variance in item difficulty (i.e., 64% to 81%). The cognitive process attributes contributed to the item difficulties more than the content attributes, and the IT attributes contributed much more than both the content and process attributes. The new retrofitted process attributes explained the items better than the TIMSS process attributes. Results generated from the level 1 attributes and the level 2 attributes were consistent. Most attributes could be used to recover students' performance, but some attributes' probabilities showed unreasonable patterns. The analysis approaches could not demonstrate if the same construct validity was supported across booklets. The proposed attributes and Q-matrices explained the items of Booklet 2 better than the items of Booklet 3. The specified Q-matrices explained the items better than the random Q-matrices.

A modified PNN algorithm with optimal PD modeling using the orthogonal least squares method

In this paper a modified algorithm is suggested for developing polynomial neural network (PNN) models. Optimal partial description (PD) modeling is introduced at each layer of the PNN expansion, a task accomplished using the orthogonal least squares (OLS) method. Based on the initial PD models determined by the polynomial order and the number of PD inputs, OLS selects the most significant regressor terms reducing the output error variance. The method produces PNN models exhibiting a high level of accuracy and superior generalization capabilities. Additionally, parsimonious models are obtained comprising a considerably smaller number of parameters compared to the ones generated by means of the conventional PNN algorithm. Three benchmark examples are elaborated, including modeling of the gas furnace process as well as the iris and wine classification problems. Extensive simulation results and comparison with other methods in the literature, demonstrate the effectiveness of the suggested modeling approach.

Design of Digital Fractional-Order Integrators and Differentiators by Least Squares

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Least squares regression with errors in both variables: case studies

Analytical curves are normally obtained from discrete data by least squares regression. The least squares regression of data involving significant error in both x and y values should not be implemented by ordinary least squares (OLS). In this work, the use of orthogonal distance regression (ODR) is discussed as an alternative approach in order to take into account the error in the x variable. Four examples are presented to illustrate deviation between the results from both regression methods. The examples studied show that, in some situations, ODR coefficients must substitute for those of OLS, and, in other situations, the difference is not significant.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Fast sequential multi-element determination of Ca, Mg, K, Cu, Fe, Mn and Zn for foliar diagnosis using high-resolution continuum source flame atomic absorption spectrometry: Feasibility of secondary lines, side pixel registration and least-squares background correction

The fast sequential multi-element determination of Ca, Mg, K, Cu, Fe, Mn and Zn in plant tissues by high-resolution continuum source flame atomic absorption spectrometry is proposed. For this, the main lines for Cu (324.754 nm), Fe (248.327 nm), Mn (279.482 nm) and Zn (213.857 nm) were selected, and the secondary lines for Ca (239.856 nm), Mg (202.582 nm) and K (404.414 nm) were evaluated. The side pixel registration approach was studied to reduce sensitivity and extend the linear working range for Mg by measuring at wings (202.576 nm; 202.577 nm; 202.578 nm; 202.580 nm: 202.585 nm; 202.586 nm: 202.587 nm; 202.588 nm) of the secondary line. The interference caused by NO bands on Zn at 213.857 nm was removed using the least-squares background correction. Using the main lines for Cu, Fe, Mn and Zn, secondary lines for Ca and K, and line wing at 202.588 nm for Mg, and 5 mL min(-1) sample flow-rate, calibration curves in the 0.1-0.5 mg L-1 Cu, 0.5-4.0 mg L-1 Fe, 0.5-4.0 mg L-1 Mn, 0.2-1.0 mg L-1 Zn, 10.0-100.0 mg L-1 Ca, 5.0-40.0 mg L-1 Mg and 50.0-250.0 mg L-1 K ranges were consistently obtained. Accuracy and precision were evaluated after analysis of five plant standard reference materials. Results were in agreement at a 95% confidence level (paired t-test) with certified values. The proposed method was applied to digests of sugar-cane leaves and results were close to those obtained by line-source flame atomic absorption spectrometry. Recoveries of Ca, Mg, K, Cu, Fe, Mn and Zn in the 89-103%, 84-107%, 87-103%, 85-105%, 92-106%, 91-114%, 96-114% intervals, respectively, were obtained. The limits of detection were 0.6 mg L-1 Ca, 0.4 mg L-1 Mg, 0.4 mg L-1 K, 7.7 mu g L-1 Cu, 7.7 mu g L-1 Fe, 1.5 mu g L-1 Mn and 5.9 mu g L-1 Zn. (C) 2009 Elsevier B.V. All rights reserved.

Internal standardization and least-squares background correction in high-resolution continuum source flame atomic absorption spectrometry to eliminate interferences on determination of Pb in phosphoric acid

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

PRACTICAL LAND GRADING BASED ON LEAST SQUARES.

Practical methods for land grading design of a plane surface for rectangular and irregularly shaped fields based on a least squares analysis are presented. The least squares procedure leads to a system of three linear equations with three unknowns for determination of the best-fit plane. The equations can be solved by determinants (Cramer's rule) using a procedure capable of solution by many programmable calculators. The detailed computational process for determining the equation of the plane and a simple method to find the centroid location of an irregular field are also given. An illustrative example and design instructions are included to demonstrate the application of the design procedure.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Integrated marine gravity field in the Brazilian coast from altimeter-derived sea surface gradient and shipborne gravity

Sea surface gradients derived from the Geosat and ERS-1 satellite altimetry geodetic missions were integrated with marine gravity data from the National Geophysical Data Center and Brazilian national surveys. Using the least squares collocation method, models of free-air gravity anomaly and geoid height were calculated for the coast of Brazil with a resolution of 2` x 2`. The integration of satellite and shipborne data showed better statistical results in regions near the coast than using satellite data only, suggesting an improvement when compared to the state-of-the-art global gravity models. Furthermore, these results were obtained with considerably less input information than was used by those reference models. The least squares collocation presented a very low content of high-frequency noise in the predicted gravity anomalies. This may be considered essential to improve the high resolution representation of the gravity field in regions of ocean-continent transition. (C) 2010 Elsevier Ltd. All rights reserved.