The notion of scale freeness does not seem to have been well understood in the factor analytic literature. It has been believed that if the loss function that is minimized to obtain estimates of the parameters in the factor model is scale invariant, then the estimates are scale free. It is shown that scale invariance of the loss function is neither a necessary nor a sufficient condition for scale freeness. A theorem that ensures scale freeness in the orthogonal factor model is given in this paper.

Given two matrices, A and B, each with n rows, and with p and q columns, respectively, a very common goal is the comparison of these matrices in a subspace of dimension s, 1 ≤ s ≤ min p, q. That is, assuming linear mappings of A and B into the subspace, the comparison involves the choice of two transformation matrices L and M of dimensions p by s and q by s, respectively, and the subsequent comparisons of the images AL and BM in some fashion.

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.

Bayesian least squares techniques are adapted to estimation of stimulus-response curves, rather broadly conceived. Illustrative examples deal with estimation of person characteristic curves and item characteristic curves in the context of mental testing, and estimation of a stimulus-response curve using data from a psychophysical experiment.

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.

“Improper linear models” (see Dawes, Am. Psychol. 34:571–582, 1979), such as equal weighting, have garnered interest as alternatives to standard regression models. We analyze the general circumstances under which these models perform well by recasting a class of “improper” linear models as “proper” statistical models with a single predictor. We derive the upper bound on the mean squared error of this estimator and demonstrate that it has less variance than ordinary least squares estimates. We examine common choices of the weighting vector used in the literature, e.g., single variable heuristics and equal weighting, and illustrate their performance in various test cases.

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

The paper derives sufficient conditions for the consistency and asymptotic normality of the least squares estimator of a trilinear decomposition model for multiway data analysis.

NEWMOD®, developed by R.C. Reynolds, Jr., has been an important tool for evaluating quantitatively X-ray diffraction (XRD) patterns from interstratified clay minerals for more than 20 years. However, a significant drawback to the NEWMOD® approach is that analyses are done by forward simulation, making results sensitive to user input and starting-model assumptions. In the present study, a reverse-fitting procedure was implemented in a new program, FITMOD, which automatically minimizes the differences between experimental and simulated XRD patterns. The differences are minimized by varying model parameters (such as Reichweite, crystal-size distribution, cation content, type of disorder, etc.) using the downhill simplex method. The downhill simplex method is a non-linear optimization technique for determining minima of functions. This method does not require calculation of the derivatives of the functions being minimized, an important consideration with many of the parameters in NEWMOD- type simulations. Instead, the downhill simplex method calculates pseudo-derivatives by evaluating sufficient points to define a derivative for each independent variable. The performance of FITMOD was evaluated by fitting a series of synthetic XRD patterns generated by NEWMOD+, yielding agreement factors, Rwp, of <0.3%. As long as the correct interstratified system was specified (e.g. illite-smectite), excellent fits were obtained irrespective of the starting parameters for the simulations. FITMOD was also tested using experimental XRD patterns, which gave very good fits, in agreement with previously published results. The optimization routine yields good fits for both synthetic and experimental XRD profiles in a reasonable time, with the possibility of varying all important structural parameters. FITMOD automatically provides optimum fits to experimental XRD data without operator bias, and fitting efficiency and accuracy were, therefore, significantly improved.

In the era of Unmanned Aerial Systems (UAS), an onboard autopilot occupies a prominent place and is inevitable for many of their modern applications. The efficacy of autopilot heavily relies upon the accuracy of the sensors employed and the capability of the onboard flight controller. In general, aerodynamic behaviour and flight dynamic capabilities of Unmanned Aerial Vehicles (UAVs) govern the selection and the design of flight controllers. Precise modeling of linear aerodynamic characteristics from flight data can be achieved using many of the existing classical parameter estimation techniques such as Output Error Method (OEM), Equation Error Method (EEM), and Filter Error Method (FEM). However, all the classical methods may not be readily applicable for aerodynamic modeling in nonlinear flight envelopes. The current manuscript is an attempt to exploit the capabilities of the Artificial Intelligence (AI) technique, named Particle Swarm Optimisation (PSO), in combination with Least Squares (LS) cost function to perform linear as well as nonlinear aerodynamic parameter estimation. The aforementioned task is accomplished by considering flight data from manoeuvers pertaining to linear angles of attack, moderate and near stall flight envelopes of two different UAVs with cropped delta planform geometry. Parameters estimated using the proposed LS-PSO method are consistent with minimum standard deviation and are on a par with OEM estimates. The proposed LS-PSO method enhances the capabilities of LS-based EEM while estimating stall characteristic parameters, which was not possible with LS alone. The longitudinal and lateral-directional static parameters estimated from the full-scale wind tunnel testing of the two UAVs were also used to corroborate the results obtained from the flight data using the LS-PSO method.

This paper contributes by presenting a parameter identification procedure for n-degrees-of-freedom flexible joint robot manipulators. An advantage of the given procedure is the obtaining of robot parameters in a single experiment. Guidelines are provided for the computing of the joint position filtering and velocity estimation. The method relies in the filtered robot model, for which no acceleration measurements are required. The filtered model is expressed in regressor form, which allows applying a parameter identification procedure based on the least squares algorithm. In order to assess the performance of the proposed parameter identification scheme, an implementation of a least squares with forgetting factor (LSFF) parameter identification method is carried out. In order to assess the reliability of the tested identification schemes, a model-based trajectory tracking controller has been implemented twice in different conditions: one control experiment using the estimated parameters provided by the proposed scheme, and another experiment using the parameters given by the LSFF method. These real-time control experiments are compared with respect to numerical simulations using the estimated parameters for each identification method. For the proposed scheme, the comparison between experiments and numerical simulations indicates better accuracy in the torque and position prediction.

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under H–1-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.