Functional Canonical Correlation Analysis Research Articles

Bootstrap Methods for Canonical Correlation Analysis of Functional Data

The bootstrap method is a very general resampling procedure for investigating the distributional property of statistics. In this paper, we present two bootstrap methods with the aim of studying the functional canonical components for functional data. The bootstrap I method constructs the bootstrap replications by resampling from the raw data, while the bootstrap II algorithm samples with replacement from the principal component scores. Simulation studies are conducted to examine the performance of the proposed bootstrap methods. The method is also applied to the motion analysis dataset, which consists of the angles formed by the hip and knee of each of 39 children over each child’s gait cycle. Numerical simulations and real data analysis show the good performance of both bootstrap methods for functional canonical correlation analysis. Moreover, as measured by the mean error and mean squared error, the bootstrap II algorithm performs better in approximating sample canonical components than the bootstrap I method.

Functional data analysis: An application to COVID‐19 data in the United States in 2020

BackgroundIn this paper, we conduct an analysis of the COVID‐19 data in the United States in 2020 via functional data analysis methods. Through this research, we investigate the effectiveness of the practice of public health measures, and assess the correlation between infections and deaths caused by the COVID‐19. Additionally, we look into the relationship between COVID‐19 spread and geographical locations, and propose a forecasting method to predict the total number of confirmed cases nationwide.MethodsThe functional data analysis methods include functional principal analysis methods, functional canonical correlation analysis methods, an expectation‐maximization (EM) based clustering algorithm and a functional time series model used for forecasting.ResultsIt is evident that the practice of public health measures helps to reduce the growth rate of the epidemic outbreak over the nation. We have observed a high canonical correlation between confirmed and death cases. States that are geographically close to the hot spots are likely to be clustered together, and population density appears to be a critical factor affecting the cluster structure. The proposed functional time series model gives more reliable and accurate predictions of the total number of confirmed cases than standard time series methods.ConclusionsThe results obtained by applying the functional data analysis methods provide new insights into the COVID‐19 data in the United States. With our results and recommendations, the health professionals can make better decisions to reduce the spread of the epidemic, and mitigate its negative effects to the national public health.

Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data

It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates. Leveraging the fact that the corresponding manifolds are submanifolds of unit Hilbert spheres, we describe how finite-dimensional representations of the functional data objects can be easily computed, which then facilitates use of standard multivariate canonical correlation analysis methods. We further construct and visualize canonical variate directions directly on the space of densities or shapes. Utility of the method is demonstrated through numerical simulations and performance on a magnetic resonance imaging dataset of glioblastoma multiforme brain tumors.

Functional cluster and canonical correlation analysis of EU countries by number of daily deaths and stringency index during Covid-19 pandemic

The danger of a global pandemic, such as the new Coronavirus (Covid-19),is obvious. This study aims to investigate the behavior and relationship of thenumber of daily new conrmed deaths per million and the stringency indexof twenty-seven European Union (EU) countries by utilizing functional clusteranalysis and functional canonical correlation analysis. Functional clusteranalysis was used to observe how countries cluster together according to dailydeaths during the time interval between March and July 2020. Functionalcanonical correlation analysis was also utilized to measure the correlationbetween the frequency index and daily deaths, and also to determine therelative positions of countries concerning their respective variability structure.The data is obtained from OWID. Here, it is seen that Italy, Spain,Belgium, and France are particularly aected by the pandemic during thetime interval within the EU countries, and the course of daily deaths is in adierent position compared to other EU countries. At the same time, a veryhigh relationship emerged between the stringency index and daily deaths asexpected.

Nonlinear functional canonical correlation analysis via distance covariance

Functional canonical correlation analysis (FCCA) is a tool for exploring the associations between a pair of functional data. However, when the association is nonlinear or even nonmonotone, FCCA can fail to discover any meaningful relationship between the pair. In this paper, nonlinear FCCA estimators are constructed based on some popular measures of dependence — distance covariance and distance correlation. Consistency of the estimators is shown. Numerical studies are presented that demonstrate nonlinear FCCA can uncover new association patterns between functional covariates.

The optimal rate of canonical correlation analysis for stochastic processes

Functional canonical correlation analysis (FCCA) has been applied in many contexts, but the asymptotic properties have not yet been studied enough. In this paper we consider a general setup resembling that of Eubank and Hsing (2008). We focus on the convergence of estimates of the associated canonical functions to their population counterparts. A regularized estimator is proposed based on the Tikhinov regularized approach. Under some conditions, an upper bound is derived for this estimator and furthermore, a sharp lower bound is established. Consequently, the minimax optimal rate is obtained and depends on the level of dependence between the two stochastic processes. A simple simulation is performed to illustrate our methods.

Robust estimation in canonical correlation analysis for multivariate functional data

In this paper, the canonical correlation analysis for multivariate functional data is considered. The analysis is based on the basis functions representation of the data. The use of non-orthogonal bases is available in contrast to the approach given in the literature. The robust estimation methods of the covariance matrix are also studied in the multivariate functional canonical correlation analysis. Simulation studies and breakdown analysis suggest that the proposed methods may perform better than the classical estimator under non-normal models and in the presence of outlying observations.

A quadratically regularized functional canonical correlation analysis for identifying the global structure of pleiotropy with NGS data.

Investigating the pleiotropic effects of genetic variants can increase statistical power, provide important information to achieve deep understanding of the complex genetic structures of disease, and offer powerful tools for designing effective treatments with fewer side effects. However, the current multiple phenotype association analysis paradigm lacks breadth (number of phenotypes and genetic variants jointly analyzed at the same time) and depth (hierarchical structure of phenotype and genotypes). A key issue for high dimensional pleiotropic analysis is to effectively extract informative internal representation and features from high dimensional genotype and phenotype data. To explore correlation information of genetic variants, effectively reduce data dimensions, and overcome critical barriers in advancing the development of novel statistical methods and computational algorithms for genetic pleiotropic analysis, we proposed a new statistic method referred to as a quadratically regularized functional CCA (QRFCCA) for association analysis which combines three approaches: (1) quadratically regularized matrix factorization, (2) functional data analysis and (3) canonical correlation analysis (CCA). Large-scale simulations show that the QRFCCA has a much higher power than that of the ten competing statistics while retaining the appropriate type 1 errors. To further evaluate performance, the QRFCCA and ten other statistics are applied to the whole genome sequencing dataset from the TwinsUK study. We identify a total of 79 genes with rare variants and 67 genes with common variants significantly associated with the 46 traits using QRFCCA. The results show that the QRFCCA substantially outperforms the ten other statistics.

FCCAC: functional canonical correlation analysis to evaluate covariance between nucleic acid sequencing datasets

SummaryComputational evaluation of variability across DNA or RNA sequencing datasets is a crucial step in genomic science, as it allows both to evaluate reproducibility of biological or technical replicates, and to compare different datasets to identify their potential correlations. Here we present fCCAC, an application of functional canonical correlation analysis to assess covariance of nucleic acid sequencing datasets such as chromatin immunoprecipitation followed by deep sequencing (ChIP-seq). We show how this method differs from other measures of correlation, and exemplify how it can reveal shared covariance between histone modifications and DNA binding proteins, such as the relationship between the H3K4me3 chromatin mark and its epigenetic writers and readers.Availability and ImplementationAn R/Bioconductor package is available at http://bioconductor.org/packages/fCCAC/.Supplementary information Supplementary data are available at Bioinformatics online.

A Unified Approach to Functional Principal Component Analysis and Functional Multiple-Set Canonical Correlation.

Functional principal component analysis (FPCA) and functional multiple-set canonical correlation analysis (FMCCA) are data reduction techniques for functional data that are collected in the form of smooth curves or functions over a continuum such as time or space. In FPCA, low-dimensional components are extracted from a single functional dataset such that they explain the most variance of the dataset, whereas in FMCCA, low-dimensional components are obtained from each of multiple functional datasets in such a way that the associations among the components are maximized across the different sets. In this paper, we propose a unified approach to FPCA and FMCCA. The proposed approach subsumes both techniques as special cases. Furthermore, it permits a compromise between the techniques, such that components are obtained from each set of functional data to maximize their associations across different datasets, while accounting for the variance of the data well. We propose a single optimization criterion for the proposed approach, and develop an alternating regularized least squares algorithm to minimize the criterion in combination with basis function approximations to functions. We conduct a simulation study to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of multiple-subject functional magnetic resonance imaging data to obtain low-dimensional components of blood-oxygen level-dependent signal changes of the brain over time, which are highly correlated across the subjects as well as representative of the data. The extracted components are used to identify networks of neural activity that are commonly activated across the subjects while carrying out a working memory task.

Smoothed functional canonical correlation analysis of humidity and temperature data

This paper focuses on smoothed functional canonical correlation analysis (SFCCA) to investigate the relationships and changes in large, seasonal and long-term data sets. The aim of this study is to introduce a guideline for SFCCA for functional data and to give some insights on the fine tuning of the methodology for long-term periodical data. The guidelines are applied on temperature and humidity data for 11 years between 2000 and 2010 and the results are interpreted. Seasonal changes or periodical shifts are visually studied by yearly comparisons. The effects of the ‘number of basis functions’ and the ‘selection of smoothing parameter’ on the general variability structure and on correlations between the curves are examined. It is concluded that the number of time points (knots), number of basis functions and the time span of evaluation (monthly, daily, etc.) should all be chosen harmoniously. It is found that changing the smoothing parameter does not have a significant effect on the structure of curves and correlations. The number of basis functions is found to be the main effector on both individual and correlation weight functions.

Canonical correlation analysis for irregularly and sparsely observed functional data

Several approaches for functional canonical correlation analysis have been developed to measure the association between paired functional data. However, the existing methods in the literature have been developed for dense and balanced functional data, and they cannot be directly applicable to the situations where the observed curves are recorded in the irregular and sparse fashion. In this paper, we model the associations between paired functional data into a linear mixed-effects model framework by relating two sets of curves using canonical correlation analysis. The proposed approach automatically deals with irregularly or sparsely observed functional data, and brings a new insight into the interpretation of canonical correlation analysis. Numerical studies are carried out to demonstrate finite sample behavior. Two real data applications are provided to illustrate the methodology.

Some asymptotic properties for functional canonical correlation analysis

We consider convergence rates of functional canonical correlation analysis (FCCA). There are already several studies on FCCA in the literature, which focused on its population properties as well as consistency. Our setup most closely resembles that of He et al. (2003). Under an assumption that controls the level of dependence (roughly that the dependence between the two functional objects is not too high), we derive convergence rates of the weight functions to their population counterpart. Both upper bound and lower bound are derived for the L2-norm and the prediction risk (also called Σ-norm) of the weight functions.

Borsa İstanbul’da İşlem Gören Şirketlerin Kapanış Fiyatları İle İşlem Miktarları Arasındaki İlişkinin Fonksiyonel Kanonik Korelasyon İle Analizi

Having found a large scope of application in international literature in recent years, functional data analysis methods are used for the purpose of studying the functional data structures. In this study, closing price of the companies trading on Istanbul Stock Exchange and their trading volumes are turned into functional data and then the relationship between the aforementioned data sets is researched through functional canonical correlation analysis. Regarded as important indicators related to activities performed in the capital markets reveal the relationship between the closing prices and trading volumes, the market is very important for giving information about the structure. As a result of the analysis, closing price and trading volume, average functions with regard to data sets have been obtained and average change on trading volume has been seen the highest level in comparison to closing price. In the smoothing level of 0,01, functional canonical correlation coefficient has been measured as 0,88. This value indicates that there is a strong correlation among these two data sets.

Functional data analysis view of functional near infrared spectroscopy data

Functional near infrared spectroscopy (fNIRS) is a powerful tool for the study of oxygenation and hemodynamics of living tissues. Despite the continuous nature of the processes generating the data, analysis of fNIRS data has been limited to discrete-time methods. We propose a technique, namely functional data analysis (fDA), that converts discrete samples to continuous curves. We used fNIRS data collected on forehead during a cold pressor test (CPT) from 20 healthy subjects. Using functional principal component analysis, oxyhemoglobin (HbO2) and deoxyhemoglobin (Hb) curves were decomposed into several components based on variability across the subjects. Each component corresponded to an experimental condition and provided qualitative and quantitative information of the shape and weight of that component. Furthermore, we applied functional canonical correlation analysis to investigate the interaction between Hb and HbO2 curves. We showed that the variation of Hb and HbO2 was positively correlated during the CPT, with a "far" channel on right forehead showing a smaller and faster HbO2 variation than Hb. This research suggests the fDA platform for the analysis of fNIRS data, which solves problem of high dimensionality, enables study of response dynamics, enhances characterization of the evoked response, and may improve design of future fNIRS experiments.

An examination of the articulatory characteristics of prominence in function and content words using real-time magnetic resonance imaging

We examine the functional coupling between articulatory characteristics of prominence, such as articulator speed, and its acoustic characteristics, such as F0 and acoustic energy, for content words (nouns) and function words (e.g., articles, prepositions and conjunctions) using real-time magnetic resonance imaging data. We use Granger causality ideas to test the degree and direction of causal influence between the chosen articulatory and acoustic measures for function and content words. We further apply functional canonical correlation analysis to these measures to understand the covariant behavioral modes of the articulatory and acoustic measures. After controlling for word duration, we observe that articulatory speed generally has a significant causal influence on F0, especially for longer content words, but observe no such effect in the opposite direction. Notably, we do not observe this effect for function words in most cases. We further observe a tighter coupling of canonical weight functions of articulatory speed and acoustic-prominence characteristics (F0 and energy) for the content words as compared to the function words considered. These observations provide support for the hypothesis that prominence realized during content words may result from a close coupling between articulatory and acoustic characteristics such as articulatory speed and F0, with suggestions of a directional causal relationship. [Work supported by NIH.]

Functional Multiple-Set Canonical Correlation Analysis

We propose functional multiple-set canonical correlation analysis for exploring associations among multiple sets of functions. The proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered. As in classical multiple-set canonical correlation analysis, computationally, the method solves a matrix eigen-analysis problem through the adoption of a basis expansion approach to approximating data and weight functions. We apply the proposed method to functional magnetic resonance imaging (fMRI) data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task.