Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where ...Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.展开更多
Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-à-vis a repre...Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-à-vis a representation featuring only a single structure. However, using multiple discrete and continuous structures often leads to ambiguous results that make it difficult to determine the most faithful representation of the proximity matrix in question. We propose to resolve this dilemma by using a nonmetric analogue of spectral matrix decomposition, namely, the decomposition of the proximity matrix into a sum of equally-sized matrices, restricted only to display an order-constrained patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique amount of the total variability of the original data. As our ultimate goal, we seek to extract a small number of matrices in AR form such that their sum allows for a parsimonious, but faithful reconstruction of the total variability among the original proximity entries. Subsequently, the AR matrices are treated as separate proximity matrices. Their specific patterning lends them immediately to the representation by a single (discrete non-spatial) ultrametric cluster dendrogram and a single (continuous spatial) unidimensional scale. Because both models refer to the same data base and involve the same number of parameters, estimated through least-squares, a direct comparison of their differential fit is legitimate. Thus, one can readily determine whether the amount of variability associated which each AR matrix is most faithfully represented by a discrete or a continuous structure, and which model provides in sum the most appropriate representation of the original proximity matrix. We propose an extension of the order-constrained anti-Robinson decomposition of square-symmetric proximity matrices to the analysis of individual differences of three-way data, with the third way representing individual data sources. An application to judgments of schematic face stimuli illustrates the method.展开更多
文摘Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.
文摘Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-à-vis a representation featuring only a single structure. However, using multiple discrete and continuous structures often leads to ambiguous results that make it difficult to determine the most faithful representation of the proximity matrix in question. We propose to resolve this dilemma by using a nonmetric analogue of spectral matrix decomposition, namely, the decomposition of the proximity matrix into a sum of equally-sized matrices, restricted only to display an order-constrained patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique amount of the total variability of the original data. As our ultimate goal, we seek to extract a small number of matrices in AR form such that their sum allows for a parsimonious, but faithful reconstruction of the total variability among the original proximity entries. Subsequently, the AR matrices are treated as separate proximity matrices. Their specific patterning lends them immediately to the representation by a single (discrete non-spatial) ultrametric cluster dendrogram and a single (continuous spatial) unidimensional scale. Because both models refer to the same data base and involve the same number of parameters, estimated through least-squares, a direct comparison of their differential fit is legitimate. Thus, one can readily determine whether the amount of variability associated which each AR matrix is most faithfully represented by a discrete or a continuous structure, and which model provides in sum the most appropriate representation of the original proximity matrix. We propose an extension of the order-constrained anti-Robinson decomposition of square-symmetric proximity matrices to the analysis of individual differences of three-way data, with the third way representing individual data sources. An application to judgments of schematic face stimuli illustrates the method.
