Continuous canonical correlation analysis

Given a bivariate distribution, the set of canonical correlations and functions is in general finite or countable. By using an inner product between two functions via an extension of the covariance, we find all the canonical correlations and functions for the so-called Cuadras-Aug´e copula and prove the continuous dimensionality of this distribution

Robust Sparse Canonical Correlation Analysis

Canonical correlation analysis (CCA) is a multivariate statistical method which describes the associations between two sets of variables. The objective is to find linear combinations of the variables in each data set having maximal correlation. This paper discusses a method for Robust Sparse CCA. Sparse estimation produces canonical vectors with some of their elements estimated as exactly zero. As such, their interpretability is improved. We also robustify the method such that it can cope with outliers in the data. To estimate the canonical vectors, we convert the CCA problem into an alternating regression framework, and use the sparse Least Trimmed Squares estimator. We illustrate the good performance of the Robust Sparse CCA method in several simulation studies and two real data examples

Correlation-Compressed Direct Coupling Analysis

Learning Ising or Potts models from data has become an important topic in statistical physics and computational biology, with applications to predictions of structural contacts in proteins and other areas of biological data analysis. The corresponding inference problems are challenging since the normalization constant (partition function) of the Ising/Potts distributions cannot be computed efficiently on large instances. Different ways to address this issue have hence given size to a substantial methodological literature. In this paper we investigate how these methods could be used on much larger datasets than studied previously. We focus on a central aspect, that in practice these inference problems are almost always severely under-sampled, and the operational result is almost always a small set of leading (largest) predictions. We therefore explore an approach where the data is pre-filtered based on empirical correlations, which can be computed directly even for very large problems. Inference is only used on the much smaller instance in a subsequent step of the analysis. We show that in several relevant model classes such a combined approach gives results of almost the same quality as the computationally much more demanding inference on the whole dataset. We also show that results on whole-genome epistatic couplings that were obtained in a recent computation-intensive study can be retrieved by the new approach. The method of this paper hence opens up the possibility to learn parameters describing pair-wise dependencies in whole genomes in a computationally feasible and expedient manner.Comment: 15 pages, including 11 figure

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

Permutation Inference for Canonical Correlation Analysis

Canonical correlation analysis (CCA) has become a key tool for population neuroimaging, allowing investigation of associations between many imaging and non-imaging measurements. As other variables are often a source of variability not of direct interest, previous work has used CCA on residuals from a model that removes these effects, then proceeded directly to permutation inference. We show that such a simple permutation test leads to inflated error rates. The reason is that residualisation introduces dependencies among the observations that violate the exchangeability assumption. Even in the absence of nuisance variables, however, a simple permutation test for CCA also leads to excess error rates for all canonical correlations other than the first. The reason is that a simple permutation scheme does not ignore the variability already explained by previous canonical variables. Here we propose solutions for both problems: in the case of nuisance variables, we show that transforming the residuals to a lower dimensional basis where exchangeability holds results in a valid permutation test; for more general cases, with or without nuisance variables, we propose estimating the canonical correlations in a stepwise manner, removing at each iteration the variance already explained, while dealing with different number of variables in both sides. We also discuss how to address the multiplicity of tests, proposing an admissible test that is not conservative, and provide a complete algorithm for permutation inference for CCA.Comment: 49 pages, 2 figures, 10 tables, 3 algorithms, 119 reference

The Correlation Analysis Between Brand Equity and the Customer Decision Buying an Automotive Product

Some reasons have made the customer buying an automotive product, and one of it is a brand. Brand image is the power increasing the total sales of automotive products in Indonesia, so that more aspects affecting the power of the brand image need to be analyzed. This research is different with the previous one. It aims to recognize the effect of brand equity (brand awareness, brand association, quality perception, and brand loyalty) upon the consumer decision buying Toyota Kijang Innova which is one of the family premium high involvement products at Auto 2000 – Yasmin Branch in Bogor. This research has applied a descriptive quantitative approach refers to a survey upon 85 respondents as the customers of Auto 2000 Yasmin Branch in Bogor. Data analysis has implemented a multiple linear analysis using SPSS 16.0. The result of this research has identified partially that only the variable of brand loyalty has significantly affected the customer decision, though simultaneously all the dimensions of brand equity have affected the customer decision buying the product. Furthermore, the result can be used as a basic of marketing strategy regarding a high involvement product

Multi-set canonical correlation analysis for 3D abnormal gait behaviour recognition based on virtual sample generation

Small sample dataset and two-dimensional (2D) approach are challenges to vision-based abnormal gait behaviour recognition (AGBR). The lack of three-dimensional (3D) structure of the human body causes 2D based methods to be limited in abnormal gait virtual sample generation (VSG). In this paper, 3D AGBR based on VSG and multi-set canonical correlation analysis (3D-AGRBMCCA) is proposed. First, the unstructured point cloud data of gait are obtained by using a structured light sensor. A 3D parametric body model is then deformed to fit the point cloud data, both in shape and posture. The features of point cloud data are then converted to a high-level structured representation of the body. The parametric body model is used for VSG based on the estimated body pose and shape data. Symmetry virtual samples, pose-perturbation virtual samples and various body-shape virtual samples with multi-views are generated to extend the training samples. The spatial-temporal features of the abnormal gait behaviour from different views, body pose and shape parameters are then extracted by convolutional neural network based Long Short-Term Memory model network. These are projected onto a uniform pattern space using deep learning based multi-set canonical correlation analysis. Experiments on four publicly available datasets show the proposed system performs well under various conditions