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  • Electronic Resource  (4)
  • 1980-1984  (4)
  • 1
    Electronic Resource
    Electronic Resource
    Optimally scaled matrices, necessary and sufficient conditions (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 239-245 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65F35 CR ; 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We shall in this paper consider the problem of determination a row or column scaling of a matrixA, which minimizes the condition number ofA. This problem was studied by several authors. For the cases of the maximum norm and of the sum norm the scale problem was completely solved by Bauer [1] and Sluis [5]. The condition ofA subordinate to the pair of euclidean norms is the ratio Λ/λ, where Λ and λ are the maximal and minimal eigenvalue of (A H A)1/2 respectively. The euclidean case was considered by Forsythe and Strauss [3]. Shapiro [6] proposed some approaches to a numerical solution in this case. The main result of this paper is the presentation of necessary and sufficient conditions for optimal scaling in terms of maximizing and minimizing vectors. A uniqueness proof for the solution is offered provided some normality assumption is satisfied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408697
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis (1982)
    Springer
    Psychometrika 47 (1982), S. 187-199 
    Details
    ISSN: 1860-0980
    Keywords: reduced rank ; reliability ; sample estimates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Psychology
    Notes: Abstract One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02296274
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Weighted minimum trace factor analysis (1982)
    Springer
    Psychometrika 47 (1982), S. 243-264 
    Details
    ISSN: 1860-0980
    Keywords: reliability ; reduced rank ; sample estimates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Psychology
    Notes: Abstract In the last decade several authors discussed the so-called minimum trace factor analysis (MTFA), which provides the greatest lower bound (g.l.b.) to reliability. However, the MTFA fails to be scale free. In this paper we propose to solve the scale problem by maximization of the g.l.b. as the function of weights. Closely related to the primal problem of the g.l.b. maximization is the dual problem. We investigate the primal and dual problems utilizing convex analysis techniques. The asymptotic distribution of the maximal g.l.b. is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions. Finally we outline computational algorithms and consider numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02294158
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Minimum rank and minimum trace of covariance matrices (1982)
    Springer
    Psychometrika 47 (1982), S. 443-448 
    Details
    ISSN: 1860-0980
    Keywords: factor analysis ; covariance matrices ; minimum trace ; constrained minimum trace ; minimum rank
    Source: Springer Online Journal Archives 1860-2000
    Topics: Psychology
    Notes: Abstract This paper considers some mathematical aspects of minimum trace factor analysis (MTFA). The uniqueness of an optimal point of MTFA is proved and necessary and sufficient conditions for a point x to be optimal are established. Finally, some results about the connection between MTFA and the classical minimum rank factor analysis will be presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293708
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    Paper (German National Licenses)
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