Probabilistic multidimensional scaling: An anisotropic model for distance judgments

Abstract A probabilistic multidimensional scaling model is proposed in which the coordinates of each stimulus may be correlated and have distinct variances on each dimension. Maximum likelihood estimates of the parameters are obtained. The model is compared to deterministic and isotropic models. Properties of the model are described.

[1]  Distribution of multivariate quadratic forms under certain covariance structures , 1987 .

[2]  Herbert Solomon,et al.  Distribution of Quadratic Forms and Some Applications , 1955 .

[3]  H. Ruben,et al.  Probability Content of Regions Under Spherical Normal Distributions, IV: The Distribution of Homogeneous and Non-Homogeneous Quadratic Functions of Normal Variables , 1961 .

[4]  Donald R. Lehmann,et al.  Television Show Preference: Application of a Choice Model , 1971 .

[5]  R. Davies.  The distribution of a linear combination of 2 random variables , 1980 .

[6]  John W. Keon.  Product Positioning: TRINODAL Mapping of Brand Images, Ad Images, and Consumer Preference , 1983 .

[7]  David B. MacKay,et al.  Probabilistic Multidimensional Analysis of Preference Ratio Judgments , 1989 .

[8]  Carl W. Helstrom.  Comment: Distribution of Quadratic Forms in Normal Random Variables—Evaluation by Numerical Integration , 1983 .

[9]  S. Rice.  Distribution of Quadratic Forms in Normal Random Variables—Evaluation by Numerical Integration , 1980 .

[10]  Yoshio Takane,et al.  Analysis of Covariance Structures and Probabilistic Binary Choice Data , 1989 .

[11]  J. Sargan,et al.  On the theory and application of the general linear model , 1970 .

[12]  David B. MacKay,et al.  Probabilistic Scaling of Spatial Distance Judgments , 1981 .

[13]  Allan D. Shocker,et al.  Multiattribute Approaches for Product Concept Evaluation and Generation: A Critical Review , 1979 .

[14]  Morris L. Eaton,et al.  On the Projections of Isotropic Distributions , 1981 .

[15]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[16]  Joseph L. Zinnes,et al.  Probabilistic multidimensional scaling: Complete and incomplete data , 1983 .

[17]  J. Imhof.  Computing the distribution of quadratic forms in normal variables , 1961 .

[18]  R. Farebrother.  The Distribution of a Positive Linear Combination of X2 Random Variables , 1984 .

[19]  Kenneth Mullen,et al.  A multivariate model for discrimination methods , 1986 .

[20]  D. R. Jensen,et al.  A Gaussian Approximation to the Distribution of a Definite Quadratic Form , 1972 .

[21]  Ulf Grenander,et al.  The Distribution of Quadratic Forms in Normal Variates: A Small Sample Theory with Applications to Spectral Analysis , 1959 .

[22]  J. Sheil,et al.  The Distribution of Non‐Negative Quadratic Forms in Normal Variables , 1977 .

[23]  C. G. Khatri,et al.  14 Quadratic forms in normal variables , 1980 .

[24]  J. Eliashberg,et al.  A New Stochastic Multidimensional Unfolding Model for the Investigation of Paired Comparison Consumer Preference/Choice Data , 1987 .