Computation of canonical correlation and best predictable aspect of future for time series

The canonical correlation between the (infinite) past and future of a stationary time series is shown to be the limit of the canonical correlation between the (infinite) past and (finite) future, and computation of the latter is reduced to a (generalized) eigenvalue problem involving (finite) matrices. This provides a convenient and essentially, finite-dimensional algorithm for computing canonical correlations and components of a time series. An upper bound is conjectured for the largest canonical correlation

Causal Patterns: Extraction of multiple causal relationships by Mixture of Probabilistic Partial Canonical Correlation Analysis

In this paper, we propose a mixture of probabilistic partial canonical correlation analysis (MPPCCA) that extracts the Causal Patterns from two multivariate time series. Causal patterns refer to the signal patterns within interactions of two elements having multiple types of mutually causal relationships, rather than a mixture of simultaneous correlations or the absence of presence of a causal relationship between the elements. In multivariate statistics, partial canonical correlation analysis (PCCA) evaluates the correlation between two multivariates after subtracting the effect of the third multivariate. PCCA can calculate the Granger Causal- ity Index (which tests whether a time-series can be predicted from an- other time-series), but is not applicable to data containing multiple partial canonical correlations. After introducing the MPPCCA, we propose an expectation-maxmization (EM) algorithm that estimates the parameters and latent variables of the MPPCCA. The MPPCCA is expected to ex- tract multiple partial canonical correlations from data series without any supervised signals to split the data as clusters. The method was then eval- uated in synthetic data experiments. In the synthetic dataset, our method estimated the multiple partial canonical correlations more accurately than the existing method. To determine the types of patterns detectable by the method, experiments were also conducted on real datasets. The method estimated the communication patterns In motion-capture data. The MP- PCCA is applicable to various type of signals such as brain signals, human communication and nonlinear complex multibody systems.Comment: DSAA2017 - The 4th IEEE International Conference on Data Science and Advanced Analytic

Estimation and Hypothesis Testing of Cointegration

abstract: Estimating cointegrating relationships requires specific techniques. Canonical correlations are used to determine the rank and space of the cointegrating matrix. The vectors used to transform the data into canonical variables have an eigenvector representation, and the associated canonical correlations have an eigenvalue representation. The number of cointegrating relations is chosen based upon a theoretical difference in the convergence rates of the eignevalues. The number of cointegrating relations is consistently estimated using a threshold function which places a lower bound on the eigenvalues associated with cointegrating relations and an upper bound on the eigenvalues on the eigenvalues not associated with cointegrating relations. The proposed estimator performs better with a large number of cross-sectional observations and moderate time series length.Dissertation/ThesisPh.D. Economics 201

Continuous Time Quantum Monte Carlo Method for Fermions: Beyond Auxiliary Field Framework

Numerically exact continuous-time Quantum Monte Carlo algorithm for finite fermionic systems with non-local interactions is proposed. The scheme is particularly applicable for general multi-band time-dependent correlations since it does not invoke Hubbard-Stratonovich transformation. The present determinantal grand-canonical method is based on a stochastic series expansion for the partition function in the interaction representation. The results for the Green function and for the time-dependent susceptibility of multi-orbital super-symmetric impurity model with a spin-flip interaction are presented

Continuous Time Quantum Monte Carlo method for fermions

We present numerically exact continuous-time Quantum Monte Carlo algorithm for fermions with a general non-local in space-time interaction. The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation. The method is particularly applicable for multi-band time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation. The test calculations for exactly solvable models as well results for the Green function and for the time-dependent susceptibility of the multi-band super-symmetric model with a spin-flip interaction are discussed.Comment: 10 pages, 7 Figure

CANONICAL CORRELATIONS OF PAST AND FUTURE FOR TIME-SERIES - BOUNDS AND COMPUTATION

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Discovering Potential Correlations via Hypercontractivity

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.Comment: 30 pages, 19 figures, accepted for publication in the 31st Conference on Neural Information Processing Systems (NIPS 2017

Decadal rainfall variability modes in observed rainfall records over East Africa and their relations to historical sea surface temperature changes.

Detailed knowledge about the long-term interface of climate and rainfall variability is essential for managing agricultural activities in Eastern African countries. To this end, the space-time patterns of decadal rainfall variability modes over East Africa and their predictability potentials using Sea Surface Temperature (SST) are investigated. The analysis includes observed rainfall data from 1920-2004 and global SSTs for the period 1950-2004. Simple correlation, trend and cyclical analyses, Principal Component Analysis (PCA) with VARIMAX rotation and Canonical Correlation Analysis (CCA) are employed. The results show decadal signals in filtered observed rainfall record with 10 years period during March - May (MAM) and October – December (OND) seasons. During June - August (JJA), however, cycles with 20 years period are common. Too much / little rainfall received in one or two years determines the general trend of the decadal mean rainfall. CCA results for MAM showed significant positive correlations between the VARIMAX-PCA of SST and the canonical component time series over the central equatorial Indian Ocean. Positive loadings were spread over the coastal and Lake Victoria regions while negative loading over the rest of the region with significant canonical correlation skills. For the JJA seasons, Atlantic SSTs had negative loadings centred on the tropical western Atlantic Ocean associated with the wet / dry regimes over western / eastern sectors. The highest canonical correlation skill between OND rainfall and the Pacific SSTs showed that El Niño-Southern Oscillation (ENSO)/La Niña phases are associated with wet/dry decades over the region

Functional factor analysis for periodic remote sensing data

We present a new approach to factor rotation for functional data. This is achieved by rotating the functional principal components toward a predefined space of periodic functions designed to decompose the total variation into components that are nearly-periodic and nearly-aperiodic with a predefined period. We show that the factor rotation can be obtained by calculation of canonical correlations between appropriate spaces which make the methodology computationally efficient. Moreover, we demonstrate that our proposed rotations provide stable and interpretable results in the presence of highly complex covariance. This work is motivated by the goal of finding interpretable sources of variability in gridded time series of vegetation index measurements obtained from remote sensing, and we demonstrate our methodology through an application of factor rotation of this data.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS518 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org