Estimation of the factor model by unweighted least squares (ULS) is distribution free, yields consistent estimates, and is computationally fast if the Minimum Residuals (MinRes) algorithm is employed. MinRes algorithms produce a converging sequence of monotonically decreasing ULS function values. Various suggestions for algorithms of the MinRes type are made for confirmatory as well as for exploratory factor analysis. These suggestions include the implementation of inequality constraints and the prevention of Heywood cases. A simulation study, comparing the bootstrap standard deviations for the parameters with the standard errors from maximum likelihood, indicates that these are virtually equal when the score vectors are sampled from the normal distribution. Two empirical examples demonstrate the usefulness of constrained exploratory and confirmatory factor analysis by ULS used in conjunction with the bootstrap method.
Several models in data analysis are estimated by minimizing the objective function defined as the residual sum of squares between the model and the data. A necessary and sufficient condition for the existence of a least squares estimator is that the objective function attains its infimum at a unique point. It is shown that the objective function for Parafac-2 need not attain its infimum, and that of DEDICOM, constrained Parafac-2, and, under a weak assumption, SCA and Dynamals do attain their infimum. Furthermore, the sequence of parameter vectors, generated by an alternating least squares algorithm, converges if it decreases the objective function to its infimum which is attained at one or finitely many points.
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Author supplied: "This paper gives a linearised adjustment model for the affine, similarity and congruence transformations in 3D that is easily extendable with other parameters to describe deformations. The model considers all coordinates stochastic. Full positive semi-definite covariance matrices and correlation between epochs can be handled. The determination of transformation parameters between two or more coordinate sets, determined by geodetic monitoring measurements, can be handled as a least squares adjustment problem. It can be solved without linearisation of the functional model, if it concerns an affine, similarity or congruence transformation in one-, two- or three-dimensional space. If the functional model describes more than such a transformation, it is hardly ever possible to find a direct solution for the transformation parameters. Linearisation of the functional model and applying least squares formulas is then an appropriate mode of working. The adjustment model is given as a model of observation equations with constraints on the parameters. The starting point is the affine transformation, whose parameters are constrained to get the parameters of the similarity or congruence transformation. In this way the use of Euler angles is avoided. Because the model is linearised, iteration is necessary to get the final solution. In each iteration step approximate coordinates are necessary that fulfil the constraints. For the affine transformation it is easy to get approximate coordinates. For the similarity and congruence transformation the approximate coordinates have to comply to constraints. To achieve this, use is made of the singular value decomposition of the rotation matrix. To show the effectiveness of the proposed adjustment model total station measurements in two epochs of monitored buildings are analysed. Coordinate sets with full, rank deficient covariance matrices are determined from the measurements and adjusted with the proposed model. Testing the adjustment for deformations results in detection of the simulated deformations."
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