Component Analysis Of Functional Data Research Articles

Quantile-based Clustering for Functional Data via Modelling Functional Principal Components Scores

Clustering tasks of functional data arise naturally in many applications, and efficient classification approaches are needed to find groups. The current paper combines the quantile-based model with the principal component analysis of functional data (FPCA). In our proposed procedures, the projection of functional data is first approximated based on (rotated) FPCA. The quantile-based model is then implemented on the space of rotated scores to identify the potential features of underlying clusters. The proposed method overcomes the limitation of using direct basis function expansion such as Fourier, B-spline, or linear fitting, besides representing a nonparametric clustering alternative based on a quantile approach. The proposed method’s performance has been evaluated in a comprehensive simulation study and afterward compared with existing functional and non-functional clustering methods. The simulation study results showed that the proposed method performs well in terms of correct classification rate and computing time average. Finally, a real-world application concerning temporal wind speed data has been analyzed to demonstrate the proposed method’s advantages and usefulness.

Spatial-Temporal Characteristics Analysis of Air Pollution in Hubei Province Based on Functional Data Clustering

In recent years, China has attached great importance to ecological and environmental protection, upholding the important concept that clear waters and lush mountains are valuable assets. In order to explore the characteristics of air pollution in this area, the air quality index (AQI) is studied by clustering analysis of functional data. Firstly, the change curve of the air quality index is reduced by principal component analysis of functional data. Secondly, the basis coefficients of principal components are clustered by K -means. Finally, the characteristics of air pollution in Hubei Province are analyzed from the aspects of the time dimension and space dimension. The results show that 13 prefectural administrative regions in Hubei Province can be divided into four types of areas according to air quality changes, showing a trend of increasing pollution from north to south and showing a significant similar pattern in the time dimension, with significant seasonal characteristics.

Nonlinear and additive principal component analysis for functional data

We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature of the data, and the second space captures the nonlinear dependence. In the meantime, additivity is imposed so that we can avoid high-dimensional kernels in the functional space, which causes the curse of dimensionality. Along with the NAFPCA, we also develop a method of selection of the number of principal components and the tuning parameters that determines the degree of nonlinearity, as well as the asymptotic results for both the fully observed and the incompletely observed functional data. Simulation results show that the new method performs better than functional principal component analysis when the relations among random functions are nonlinear. We apply the new method to online handwritten digits and electroencephalogram (EEG) data sets.

A Method of L1-Norm Principal Component Analysis for Functional Data

Recently, with the popularization of intelligent terminals, research on intelligent big data has been paid more attention. Among these data, a kind of intelligent big data with functional characteristics, which is called functional data, has attracted attention. Functional data principal component analysis (FPCA), as an unsupervised machine learning method, plays a vital role in the analysis of functional data. FPCA is the primary step for functional data exploration, and the reliability of FPCA plays an important role in subsequent analysis. However, classical L2-norm functional data principal component analysis (L2-norm FPCA) is sensitive to outliers. Inspired by the multivariate data L1-norm principal component analysis methods, we propose an L1-norm functional data principal component analysis method (L1-norm FPCA). Because the proposed method utilizes L1-norm, the L1-norm FPCs are less sensitive to the outliers than L2-norm FPCs which are the characteristic functions of symmetric covariance operator. A corresponding algorithm for solving the L1-norm maximized optimization model is extended to functional data based on the idea of the multivariate data L1-norm principal component analysis method. Numerical experiments show that L1-norm FPCA proposed in this paper has a better robustness than L2-norm FPCA, and the reconstruction ability of the L1-norm principal component analysis to the original uncontaminated functional data is as good as that of the L2-norm principal component analysis.

Principal component analysis for functional data on Riemannian manifolds and spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered, for example, as movement trajectories on the surface of the earth are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the Frechet mean function, and then performing a classical functional principal component analysis (FPCA) on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation yields upper bounds to residual variances if the Riemannian manifold has nonnegative curvature. We derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to outperform an unrestricted FPCA in terms of trajectory recovery and prediction in applications and simulations.

Principal component analysis for functional data on grain yield of winter wheat cultivars

Summary The aim of this paper is to present a statistical methodology to assess patterns of cultivars' adaptive response to agricultural environments (agroecosystems) on the basis of complete Genotype x Crop Management x Location x Year (GxMxLxY) data obtained from 3-year multi-location twofactor trials conducted within the framework of the Polish post-registration trials (PDOiR), with an illustration of the application and usefulness of this methodology in analyzing winter wheat grain yield. Producing specific varieties for each subregion of a target region, from widely adapted varieties, may exploit positive genotype x location (GL) interactions to increase crop yields. Experiments designed to examine combinations of environment (E), management practices (M) and cultivars (G) also provide evidence of the relative importance of each of these factors for yield improvement. The evidence shows that variation due to E far outweighs the variation of grain yield that can be attributed to M or G, or the interactions between these factors, and between these factors and E (Anderson, 2010). This statistical method involves the use of functional PCA and cluster analysis. A total of 24 cultivars were evaluated over 3 years in 20 environments using randomized incomplete split-block designs with two replications per trial. The methodology proved an efficient tool for the reliable classification of 24 winter wheat cultivars, distinguishing cultivar groups that exhibited homogeneous adaptive response to environments. It enables the identification of cultivars displaying wide or specific adaptation. The remaining cultivars were locally adapted to some testing environments, or some of them were not relatively adapted to the environments because they always yielded substantially below the environmental means. Performing earlier specific selection, or adopting distinct genetic bases for each agro-ecosystem, may further increase the advantage of specific breeding.

Robust SIMCA-bounding influence of outliers

In this article a robust version of SIMCA, based on spherical principal component analysis [N. Locantore, J.S. Marron, D.G. Simpson, N. Tripoli, J.T. Zhang, K.L. Cohen, Robust principal component analysis for functional data (with comments), Test 8 (1999) 1–74], is introduced for chemometrics community. The efficiency of the new approach is compared to the classical SIMCA and to its robust version proposed by Vanden Branden et al. [K. Vanden Branden, M. Hubert, Robust classification in high dimensions based on the SIMCA method, Chemometrics and Intelligent Laboratory Systems 79 (2005) 10–21]. The performances of the presented approaches are evaluated on simulated and real data sets. The results obtained from a simulation study give evidence that the proposed robust SIMCA approach offers a satisfactory efficiency when the model set does not contain outliers and is also robust, what ensures a proper classification of new objects even, when the model set used to derive classification rules is contaminated to a large extent by outlying objects.