132,795 research outputs found
On Measure Transformed Canonical Correlation Analysis
In this paper linear canonical correlation analysis (LCCA) is generalized by
applying a structured transform to the joint probability distribution of the
considered pair of random vectors, i.e., a transformation of the joint
probability measure defined on their joint observation space. This framework,
called measure transformed canonical correlation analysis (MTCCA), applies LCCA
to the data after transformation of the joint probability measure. We show that
judicious choice of the transform leads to a modified canonical correlation
analysis, which, in contrast to LCCA, is capable of detecting non-linear
relationships between the considered pair of random vectors. Unlike kernel
canonical correlation analysis, where the transformation is applied to the
random vectors, in MTCCA the transformation is applied to their joint
probability distribution. This results in performance advantages and reduced
implementation complexity. The proposed approach is illustrated for graphical
model selection in simulated data having non-linear dependencies, and for
measuring long-term associations between companies traded in the NASDAQ and
NYSE stock markets
Asymptotic theory of multiple-set linear canonical analysis
This paper deals with asymptotics for multiple-set linear canonical analysis
(MSLCA). A definition of this analysis, that adapts the classical one to the
context of Euclidean random variables, is given and properties of the related
canonical coefficients are derived. Then, estimators of the MSLCA's elements,
based on empirical covariance operators, are proposed and asymptotics for these
estimators are obtained. More precisely, we prove their consistency and we
obtain asymptotic normality for the estimator of the operator that gives MSLCA,
and also for the estimator of the vector of canonical coefficients. These
results are then used to obtain a test for mutual non-correlation between the
involved Euclidean random variables
Testing Dependence Among Serially Correlated Multi-category Variables
The contingency table literature on tests for dependence among discrete multi-category variables assume that draws are independent, and there are no tests that account for serial dependencies − a problem that is particularly important in economics and finance. This paper proposes a new test of independence based on the maximum canonical correlation between pairs of discrete variables. We also propose a trace canonical correlation test using dynamically augmented reduced rank regressions or an iterated weighting method in order to account for serial dependence. Such tests are useful, for example, when testing for predictability of one sequence of discrete random variables by means of another sequence of discrete random variables as in tests of market timing skills or business cycle analysis. The proposed tests allow for an arbitrary number of categories, are robust in the presence of serial dependencies and are simple to implement using multivariate regression methods
Asymptotic study of canonical correlation analysis: from matrix and analytic approach to operator and tensor approach
Asymptotic study of canonical correlation analysis gives the opportunity to present the different steps of an asymptotic study and to show the interest of an operator and tensor approach of multidimensional asymptotic statistics rather than the classical, matrix and analytic approach. Using the last approach, Anderson (1999) assumes the random vectors to have a normal distribution and the non zero canonical correlation coefficients to be distinct. The new approach we use, Fine (2000), is coordinate-free, distribution-free and permits to have no restriction on the canonical correlation coefficients multiplicity order. Of course, when vectors have a normal distribution and when the non zero canonical correlation coefficients are distinct, it is possible to find again Anderson's results but we diverge on two of them. In this methodological presentation, we insist on the analysis frame (Dauxois and Pousse, 1976), the sampling model (Dauxois, Fine and Pousse, 1979) and the different mathematical tools (Fine, 1987, Dauxois, Romain and Viguier, 1994) which permit to solve problems encountered in this type of study, and even to obtain asymptotic behavior of the analyses random elements such as principal components and canonical variables.
Testing Dependence among Serially Correlated Multi-category Variables
The contingency table literature on tests for dependence among discrete multi-category variables is extensive. Existing tests assume, however, that draws are independent, and there are no tests that account for serial dependencies−a problem that is particularly important in economics and finance. This paper proposes a new test of independence based on the maximum canonical correlation between pairs of discrete variables. We also propose a trace canonical correlation test using dynamically augmented reduced rank regressions or an iterated weighting method in order to account for serial dependence. Such tests are useful, for example, when testing for predictability of one sequence of discrete random variables by means of another sequence of discrete random variables as in tests of market timing skills or business cycle analysis. The proposed tests allow for an arbitrary number of categories, are robust in the presence of serial dependencies and are simple to implement using multivariate regression methods. Monte Carlo experiments show that the proposed tests have good finite sample properties. An empirical application to survey data on forecasts of GDP growth demonstrates the importance of correcting for serial dependencies in predictability tests.contingency tables, canonical correlations, serial dependence, tests of predictability
Sparse CCA: Adaptive Estimation and Computational Barriers
Canonical correlation analysis is a classical technique for exploring the
relationship between two sets of variables. It has important applications in
analyzing high dimensional datasets originated from genomics, imaging and other
fields. This paper considers adaptive minimax and computationally tractable
estimation of leading sparse canonical coefficient vectors in high dimensions.
First, we establish separate minimax estimation rates for canonical coefficient
vectors of each set of random variables under no structural assumption on
marginal covariance matrices. Second, we propose a computationally feasible
estimator to attain the optimal rates adaptively under an additional sample
size condition. Finally, we show that a sample size condition of this kind is
needed for any randomized polynomial-time estimator to be consistent, assuming
hardness of certain instances of the Planted Clique detection problem. The
result is faithful to the Gaussian models used in the paper. As a byproduct, we
obtain the first computational lower bounds for sparse PCA under the Gaussian
single spiked covariance model
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