In this talk, we propose a novel clustering method of functional data that significantly extends the distribution scope of functional data. Most existing methods have been developed for a specific distribution structure of functional data and, therefore, only provide good results if the distribution assumptions are met. In addition, most methods based on the basis expansion approach use only one curve representing the centrality of the data as a clustering input, which prevents these methods from performing well when clustering functional data with more complex distribution structures. To solve this problem, we consider several types of curves, including mean and quantile curves, as input variables for clustering, which might better understand the distribution of the data. Moreover, we apply the concept of sparse clustering to several curve types, resulting in good clustering performance if at least one curve type can divide the data well into several subgroups. Results from numerical experiments and real data analysis are also presented well.

Keywords: Functional principal component analysis, Functional data clustering, Non-Gaussianity, Quantile curve

The PLS regression of multivariate functional data X with real-valued response Y is considered. Unlike principal component analysis (PCA), the partial least square (PLS) approach obtained PLS components using the relationship between predictors and the response. For computational aspects, I used a relationship between the PLS regression with univariate functional data (FPLS) and the PLS regression with multivariate functional data (MFPLS). Simulation studies compare the performances of our approaches with previous methods. 

Keywords: Multivariate functional data analysis, Supervised learning, Regression, Partial least squares regression

Recently, due to the climate change and global warming, interests in marine heatwave (MHW) are increasing, which denotes the phenomenon that sea surface temperature (SST) becomes extraordinarily high. However, there are only few studies on MHW occurred in Korea. This talk presents spatial and temporal characteristics of SST of the East Sea. We group MHW events into several clusters via K-means clustering using the strength, frequency, and duration of the events as features. In addition, vertical distribution of the temperature is also presented based on ESROB data observed in the southeastern area. We demonstrate that MHW events become more frequent and their durations are also increasing. Furthermore, strength of MHW occurred in summer or winter is also growing. Finally, we obtain return period and return level by implementing the extreme value theory. This work was exhibited at the 3rd Ocean Science Big Data Contest.

Keywords: Marine Heatwaves, K-means clustering, Extreme Value Analysis

We propose a quantile based fitting method for analyzing graph signals. Unlike traditional approaches for data fitting such as smoothing splines and quantile smoothing splines working on Euclidean space, the proposed method is designed for graph domain, considering the inherent graph structure. In contrast to prevalent graph signal denoising methods that rely on optimization problem with L2-norm fidelity, our approach provides denoised signals that are robust to the existence of outliers, and identifies varying structural relationships within graph signals. We validate the efficacy of our method through comprehensive simulation studies and real data analysis.

Keywords: Denoising, Graph Signal, Quantiles, Regularization

This study proposes a new decomposition method for graph signals (or graph-valued data), termed `statistical graph empirical mode decomposition (SGEMD),' which adopts graph denoising and boundary treatment. The main contribution of SGEMD is that it extends the scope of decomposition to noisy graph signals that cannot be efficiently handled by existing graph decomposition methods, such as graph Fourier-based decomposition and graph empirical mode decomposition. The proposed SGEMD can efficiently decompose various graph signals into several components without distortions and achieve stable decomposition results near the boundaries. Finally, the effectiveness of SGEMD is demonstrated through simulation experiments and real data analysis.

Keywords: Graph signal, Graph Fourier transform, Graph empirical mode decomposition, Graph denoising, Boundary treatment

다중척도연구실의 역사소개 및 졸업생 진로 현황, 그리고 통계학 대학원 졸업 후 진로에 대한 세미나를 진행합니다. 

A new measure to assess the centrality of vertices in an undirected and connected graph is proposed. The proposed measure, $L_1$ centrality, can adequately handle graphs with weights assigned to vertices and edges. The study provides tools for graphical and multiscale analysis based on the $L_1$ centrality. Specifically, the suggested analysis tools include the target plot, $L_1$ centrality-based neighborhood, local $L_1$ centrality, multiscale edge representation, and heterogeneity plot and index. Most importantly, our work is closely associated with the concept of data depth for multivariate data, which allows for a wide range of practical applications of the proposed measure. Throughout the paper, we demonstrate our tools with two interesting examples: the Marvel Cinematic Universe movie network and the bill cosponsorship network of the 21st National Assembly of South Korea. An R package L1centrality, available from the Comprehensive R Archive Network (CRAN), provides all methods and data sets used in this paper.

Keywords: Graph centrality, Data depth, Multiscale analysis, Visualization, Network data

This presentation is an outline of my doctoral dissertation. When doing statistics with metric space-valued data, we sometimes need a notion of direction. This exists in Euclidean space, but not necessarily on general metric spaces. On Hadamard spaces (also known as spaces of global non-positive curvature or complete CAT(0) spaces), the so-called boundary at infinity gives a canonical sense of direction at each point, which can be used to do statistical inference on these spaces. I will detail how this property can be used to define quantiles on Hadamard spaces and introduce some large-sample properties of sample quantiles. The presentation concludes with a discussion on further statistical applications of this boundary at infinity, including expectiles on Hadamard spaces and treatment effects on Hadamard spaces.

Keywords: Geometric statistics

Many examples of tensor data have smooth aspects in one or more of the modes, and often contain outliers beyond the level of noise. Therefore, considering the smoothness and robustness simultaneously for tensor decomposition helps to identify continuously varying factors that are less sensitive to the contamination of the data. We present a novel rank-one approximation approach incorporating these two viewpoints in the objective of CP decomposition model. Our formulation adopts a robust loss and penalization in each mode of the tensor to ensure robustness and smoothness. We develop an iterative reweighted least square algorithm as an extension of Zhang et al. (2013). Furthermore, we increase the flexibility of our models by applying spline constraints to each factor of the tensor. We demonstrate the advantages of our methods over non-robust or non-smooth decomposition methods via simulation studies and an analysis of Korean sea weather data.

Keywords: Low-rank approximation, CP decomposition, Smoothness, Robustness

The boxplot visualizes the distribution of continuous unimodal data, providing information about the location, spread, skewness, and tails of the data. However, when the data are skewed, conventional observations often exceed the fence and are erroneously classified as outliers. 

 In this study, a method for adjusting the boxplot is proposed, incorporating a robust measure of skewness in the determination of the fence. This allows for a more accurate representation of outliers in the data. As a result, this adjusted boxplot can be used as a fast and automatic outlier detection tool without making parametric assumptions about the distribution of the bulk of the data. Several examples and simulation results show the advantages of this new procedure.

 Furthermore, we propose an outlier detection algorithm suitable for time series data. Seasonal and trend patterns are decomposed using LOESS (local regression). The remaining residuals are then utilized for outlier detection using an adjusted boxplot that can handle asymmetrical distributions. We introduce a new outlier detection algorithm for time series data, expected to be effective in identifying outliers in nonlinear, non-stationary, and asymmetric time series data with trends and seasonal variations.

Keywords: Outlier detection, Medcouple, LOESS, Skewness

Predicting electricity consumption is an important issue with regard to stable and efficient electricity supply. Therefore, in this study, one time series data for electricity was divided into days with similar characteristics, and more accurate prediction was achieved by applying LSTM to major trends in each divided data set. I would like to discuss the overall introduction to this method and the expandable parts.

Keywords: Extreme gradient boosting, Similar day, Empirical mode decomposition, Long short-term memory neural networks