T.W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covariance matrices. Annals of Mathematical Statistics, pages 420--431, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the same computational pitfalls. The second goal of this paper is to have the proposed class of algorithms give explicit estimates on the probability of misclassification on future data, without resorting to unrealistic distribution assumptions or computationally expensive density estimation [5]. The Minimax Probability Machine for Classification (MPMC) due to Lanckriet et al. 2; 3] is a recent algorithm that has this charac teristic. Given the means and covariance matrices of two classes, MPMC calculates a hyperplane that separates the data by minimizing the maximum probability of ....

T.W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covariance matrices. Annals of Mathematical Statistics, 33(2):420-431, 1962.

....are normally dis tributed and that one wants a reduction to one dimension, Kazakos [10] reduces the LDR problem to a one dimensional search problem. Finding the optimal solution for this search problem, is equivalent to finding the optimal linear feature. The work of Kazakos is closely related to [1]. Three other HLDR approaches for two class problems, that generalize upon Fisher, were proposed in [13] 4] and [5] of which the latter is also applicable in the multi class case. 13] uses scatter measures different to the one used in LDA. In [4] and [5] the criterions to be optimized ....

T.W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covariance matrices. Annals of Mathematical Statistics, 33:420431, 1962.

....are normally distributed and that one wants a reduction to one dimension, Kazakos [10] reduces the LDR problem to a one dimensional search problem. Finding the optimal solution for this search problem, is equivalent to finding the optimal linear feature. The work of Kazakos is closely related to [1]. Other HLDR approaches for two class problems were proposed by Malina [13] and Decell et al. 4, 5] of which the latter is also applicable in the multi class case. These three approaches are also heteroscedastic generalizations of the Fisher criterion. 13] uses scatter measures different to ....

T.W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covari- ance matrices. Annals of Mathematical Statistics, 33:420431, 1962.

....is constructed: I(x) sign X i ff i z i (x) 4) by adjusting the weights ff i from the i th hidden unit to the output unit so as to minimize some error measure over the training data. As a result of Rosenblatt s approach, 3 The optimal coefficient for was found in the sixties [2]. 2 construction of decision rules was again associated with the construction of linear hyperplanes in some space. An algorithm that allows for all weights of the neural network to adapt in order locally to minimize the error on a set of vectors belonging to a pattern recognition problem was ....

.... classification function f in (31) for an unknown vector x only depends on the dot products: f(x) OE(x) Delta w b = X i=1 y i ff i OE(x) Delta OE(x i ) b : 33) The idea of constructing support vectors networks comes from considering general forms of the dot product in a Hilbert space [2]: OE(u) Delta OE(v) j K(u; v) 34) According to the Hilbert Schmidt Theory [6] any symmetric function K(u; v) with K(u; v)2L 2 , can be expanded in the form K(u; v) 1 X i=1 i OE i (u) Delta OE i (v) 35) 13 where i 2 and OE i are eigenvalues and eigenfunctions Z K(u; v)OE i ....

T. W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covariance matrices. Ann. Math. Stat., 33:420--431, 1966.

....31.2 73.8 33.0 74.6 76.1 Heart 42.8 81.4 50.5 83.6 unknown 5 Conclusions The problem of linear discrimination has a long and distinguished history. Many results on misclassification rates have been obtained by making distributional assumptions (e.g. Anderson and Bahadur [1]) Our results, on the other hand, make use of recent work on moment problems and semidefinite optimization to obtain distribution free results for linear discriminants. We have also shown how to exploit Mercer kernels to generalize our algorithm to nonlinear classification. The computational ....

Anderson, T. W. and Bahadur, R. R. (1962) Classification into two multivariate Normal distributions with different covariance matrices. Annals of Mathematical Statistics 33(2): 420-431.

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T.W. Anderson and R. R. Bahadur. Classification into two multivariate normal distributions with different covariance matrices. Annals of Mathematical Statistics, pages 420--431, 1962.

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