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1987 Applied Stochastic Models and Data Analysis
A large collection of factorial data analysis methods can be characterized by the following matrices: X, the k x n matrix of data, and A, B the symmetric positive definite matrices of size n, k which represent the chosen norms of ℝn, ℝk, respectively. All methods amount to computing the largest eigenvalues of U = XAXTB or the largest singular values of E = B1/2XA1/2. In Part I of this paper we begin by a geometrical and probabilistic interpretation of the various methods, showing how U and E are defined in each case. We then define the computational kernel for factorial data analysis. We conclude by devising the numerical aspects of software implementation for this kernel on microcomputers and presenting the package INDA.
1987 Applied Stochastic Models and Data Analysis
The analysis of variance (ANOVA) is widely used in biological studies, yet there remains considerable confusion among researchers about the interpretation of hypotheses being tested. Ambiguities arise when statistical designs are unbalanced, and in particular when not all combinations of design factors are represented in the data. This paper clarifies the relationship among hypothesis testing, statistical modelling and computing procedures in ANOVA for unbalanced data. A simple two‐factor fixed effects design is used to illustrate three common parametrizations for ANOVA models, and some associations among these parametrizations are developed. Biologically meaningful hypotheses for main effects and interactions are given in terms of each parametrization, and procedures for testing the hypotheses are described. The standard statistical computing procedures in ANOVA are given along with their corresponding hypotheses. Throughout the development unbalanced designs are assumed and attention is given to problems that arise with missing cells.
O'quigley, John; Moreau, Thierry
1987 Applied Stochastic Models and Data Analysis
In medical applications of survival analysis we are often interested in determining prognosis, and the adoption of statistical models in the establishment of a prognostic index has become common practice. The aim of this presentation is to examine the problem of re‐evaluation or the modification of any such index in the light of information subsequent to the initial assessment. Examples are given to highlight the approach.
1987 Applied Stochastic Models and Data Analysis
The second part of this paper deals with the systolic implementation of the computational kernel for factorial data analysis, defined in Part I, on special‐purpose hardware. The framework of the study is that a sequence of different algorithms has to be performed on a unique hardware array. This fact has led us to the design of the programmable systolic array SARDA: this is a triangular array which consists of programmable nodes with local memory and programmable orthogonal connections.
Hillion, Hervé; Proth, Jean‐Marie
1987 Applied Stochastic Models and Data Analysis
In this paper we present an original top‐down hierarchical classification algorithm. In our approach we associate to each decomposition a ‘degree of separability’ which is used to evaluate the efficiency of the decomposition. In particular, it shows whether or not the classes are well separated and also whether or not they are homogeneous. This algorithm can be applied to items defined by real and bounded numbers.
1987 Applied Stochastic Models and Data Analysis
The problem of choosing strategies which minimize the cost of decision tree testing is considered in the case where the cost of each test depends on its outcome. The classical entropy bound on the cost is derived and we propose an algorithm for decision tree design based on an extension of Huffman's construction. By comparing the mean costs for a large sample of randomly chosen situations, we show that this algorithm performs well compared with a number of previously published algorithms. The algorithms are also compared with optimal performance for a range of simple cases.
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