H. Hotelling. Relations between two sets of variates. Biometrika, 28, 1936. 4  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... function k(x# x ) The principal angles 0 1 : k ( 2) between the two subspaces are uniquely defined as: cos( k ) max u2UA v2UB v (1) subject to: u = v v =1# u u i =0# v v i =0# i =1#: #k; 1 The concept of principal angles is due to Jordan in 1875, where [13] is the first to introduce the recursive definition above. The quantities cos( i ) are sometimes referred to as canonical correlations of the matrix pair (A# B) There are various ways of formulating this problem, which are all equivalent, but some are more suitable for numerical stability than ....

....the most intensive part consisting of a single application of SVD on a k Theta k matrix. For the sake of completeness, in the following two sections we will describe two other approaches for kernalizing the computation of principal angles. The first approach is based on the Lagrange formulation [13] where the principal angles are the generalized eigenvalues of an expanded 2k Theta 2k matrix. The second approach is based on an eigendecomposition formulation for generating Q A and QB instead of the QR step. Both approaches are less efficient than the QR SVD approach where the Lagrange ....

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--372, 1936.

....the representation of the semantics. The main di#erence between CCA and the other three methods is that CCA is closely related to mutual information (Borga 1998 [3] Hence CCA can be easily motivated in information based tasks and is our natural selection. Proposed by H. Hotelling in 1936 [12], CCA can be seen as the problem of finding basis vectors for two sets of variables such that the correlation between the projections of the variables onto these basis vectors are mutually maximised. In an attempt to increase the flexibility of the feature selection, kernelisation of CCA (KCCA) ....

....the computational problems that arose in Section 3 are presented in Section 4. Our experimental results are presented In Section 5. In Section 6 we present the generalisation framework for CCA while in Section 7 draws final conclusions. 2 Theoretical Foundations Proposed by H. Hotelling in 1936 [12], Canonical correlation analysis can be seen as the problem of finding basis vectors for two sets of variables such that the correlation between the projections of the variables onto these basis vectors are mutually maximised. Correlation analysis is dependent on the co ordinate system in which ....

H. Hotelling. Relations between two sets of variates. Biometrika, 28:312-- 377, 1936.

.... function k(x# x ) The principal angles 0 1 : k ( 2) between the two subspaces are uniquely defined as: cos( k ) max u2UA v2UB v (1) subject to: u = v v =1# u u i =0# v v i =0# i =1#: #k; 1 The concept of principal angles is due to Jordan in 1875, where [13] is the first to introduce the recursive definition above. The quantities cos( i ) are sometimes referred to as canonical correlations of the matrix pair (A# B) There are various ways of formulating this problem, which are all equivalent, but some are more suitable for numerical stability than ....

....(DA ) jj =0and obtain R A whose number of columns are equal to the rank of A. Likewise for B. 2.1 An Alternative Formulation The QR approach for formulating the principal angles problem is known as the most numerically stable approach, yet there other approaches. The original formulation by [13] was based on Lagrange multipliers generating the principal angles as the set of generalized eigenvalues of a block diagonal matrix. The approach suffers from numerical stability issues, however, the extension to computation in feature space is immediate as shown next. Problem (1) can written ....

H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--372, 1936.

....between the di#erent sets of canonical correlations can be easily deduced. 1 Introduction Canonical correlation analysis (CCA) is a well developed tool in statistical analysis that is used for measuring the linear relationship between two sets of random variables. It was developed by H. Hotelling [10]. Although a wide variety of applications exists in econometrics, biometrics, chemometrics, statistics, meteorology, etc. the technique has only got introduced quite recently in the communities of signal processing, system theory and identification and neural networks [4, 14, 20] In a classic ....

H. Hotelling, "Relations between two sets of variates", Biometrika 28, 321--372, 1936.

....way that (5) be computationally solvable. If f 1 and f 2 were restricted to be linear functionals obtained by projecting two di erent vector representations of the genes on particular directions, then the maximization of (4) would be the exactly the rst canonical correlation between f 1 and f 2 [Hot36] as obtained by classical canonical correlation analysis (CCA) Linear algebra algorithms involving eigenvector decomposition exist to perform CCA. However f 1 is not restricted to be a linear feature, and (4) is consequently ill posed. Formulated as (5) however, we recover a slight ....

H. Hotelling. Relation between two sets of variates. Biometrika, 28:322-377, 1936.

....centering. Finally, in Section 4 the nonlinear version is given which leads to kernel PCA. 2 Classical principal component analysis formulation Consider a given set of data N Xk k 1 with xk and N given data points for which one aims at finding projected variables wTxk with maximal variance [7, 8, 10]. This means maxVar(wTx) Cov(wx,wx) wx) 2 w wTCw N by definition. One optimizes this objective function under the where C k 1 XkXk constraint that wTw 1. This gives the constrained optimization (w; A) lwTCw A(w w 1) 2) with Lagrange multiplier A where the solution ....

Hotelling H., "Relations between two sets of variates," Biometrica, 28,321 377, 1936.

.... where the cross covariance matrix is the diagonal matrix of singular values determined from the SVD: diag (10) The matrix is called the canonical correlation matrix of canonical correlations , and the matrix is called the squared canonical correlation matrix of squared canonical correlations [4] [5]. These squared canonical correlations are eigenvalues of the squared coherence matrix Fig. 3. Geometry of canonical coordinates. or, equivalently, of the matrix , as the following calculation shows: 11) These eigenvalues are invariant to the choice of a square root for . The eigenvalues are ....

H. Hotelling, "Relations between two sets of variates," Bimetrika, vol. 28, pp. 321--377, 1936. SCHARF AND MULLIS: CANONICAL COORDINATES AND THE GEOMETRY OF INFERENCE, RATE, AND CAPACITY 831

....models. 2 Canonical Correlation Analysis (CCA) We used Canonical Correlation Analysis (CCA) to identify the dynamics involved in a signal, viewed as a set of random variables partitioned as a set of past random variables and a set of future random variables. CCA was developed by H. Hotelling[10]. It is a standard tool for statistical analysis of two sets of random variables, of which the inter dependence should be assessed. In classical CCA, we nd the linear combination of the variables in 3 each set which gives the maximum correlation between the linear combinations. There may exist ....

H. Hotelling. Relations between two sets of variates. Biometrika,28, pages 321 377, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28, 1936. 4

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--372, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--377, 1936.

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Hotelling, H. (1936). Relation between two sets of variates. Biometrika, 28:322--377.

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H. Hotelling, "Relations between two sets of variates," Biometrika, vol. 28, pp. 321--377, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--377, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--377, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321-- 377, 1936.

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H. Hotelling, "Relations between two sets of variates," Biometrika, vol. 28, pp. 321--377, 1936.

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H. Hotelling, \Relations between two sets of variates", Biometrika 28, 321-372, 1936.

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H. Hotelling, Relations between two sets of variates, Biometrika 28 (1936) 321{ 372.

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H. Hotelling, Relations between two sets of variates, Biometrika, Vol. 28 (1936), pp. 321--372.

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H. Hotelling "Relations Between Two Sets of Variates" Biometrica, 1936, vol 28, p 321-377.

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H. Hotelling. Relations between two sets of variates. Biometrika, 8:321--377, 1936.

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Hotelling H. Relations between two sets of variates. Biometrika, Vol.28, pp. 321-377, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:321--372, 1936.

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H. Hotelling. Relations between two sets of variates. Biometrika, 28:312--377, 1936.

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