We propose a novel factor model in the graph frequency domain for multivariate data residing on the vertices of a graph, referred to as a multivariate graph signal. By utilizing graph filters, our model extends the frequency-domain approach of the dynamic factor model from time series to graphs, enabling a graph-aware and multiscale interpretation of factors across graph frequencies. This latent modeling approach reduces the dimensionality of graph signals, thereby improving the understanding of their structure. It also allows the use of the extracted factors for subsequent analyses, such as clustering. We describe the estimation of factors and their loadings and investigate the consistency of the factor estimator. In addition, we propose two consistent estimators for determining the number of factors. The finite sample performance of the proposed method is demonstrated through simulation studies across various graph structures. We also compare it with classical factor analysis and examine how the choice of graph structure affects the results. The findings show that our model achieves lower reconstruction errors and successfully incorporates the graph structure. Furthermore, we illustrate the effectiveness of the proposed method by applying it to G20 economic data, water quality data from the Geum River, and passenger data from the Seoul Metropolitan subway.

With the advancements in technology and monitoring tools, we often encounter multivariate graph signals, which can be seen as the realizations of multivariate graph processes, and revealing the relationship between their constituent quantities is one of the important problems. To address this issue, we propose a cross-spectral analysis tool for bivariate graph signals. The main goal of this study is to extend the scope of spectral analysis of graph signals to bivariate graph signals. In this study, we define joint weak stationarity graph processes and introduce graph cross-spectral density and coherence for bivariate graph processes. We propose several estimators for the cross-spectral density and investigate the theoretical properties of the proposed estimators. Furthermore, we demonstrate the effectiveness of the proposed estimators through numerical experiments, including simulation studies and a real data application. Finally, as an interesting extension, we discuss robust spectral analysis of graph signals in the presence of outliers.

This paper studies a new curve-fitting approach to data on Riemannian manifolds. We define a principal curve based on a mixture model for observations and unobserved latent variables and propose a new algorithm to estimate the principal curve for given data points on Riemannian manifolds.

This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach.

This paper considers expanding the scope of geodesic regression by proposing a robust estimation method. Most existing geodesic regressions have been developed based on the least-squares approach. However, these methods are sensitive to outliers, as in the classic Euclidean case. In this paper, we propose a robust regression approach on Riemannian manifolds through a novel combination of Euclidean robust approaches and geodesic regression when errors follow heavy-tailed distributions or include outliers. We also discuss how to sample from various distributions on Riemannian manifolds, and further conduct numerical experiments for empirical performance evaluation of the proposed method. Note that MATLAB codes used to implement the methods and to carry out some experiments are available at https://github.com/sangyulism/robust-multivariate-regression-on-Riemannian-manifolds in order that one can reproduce the same results.

In this study, a new spectrum measurement modelling method is proposed for several important frequency bands by using the Daubechies wavelets. On the basis of this method, spectrum measurement prediction is also proposed by using regression. Unlike most previous works that model or predict the occupancy rate of the frequency band of interest, this study models and predicts the power measurements directly to remove the dependence of the model on the challenging detection threshold and also to provide more comprehensive descriptions of the licenced user signals for performance improvement in cognitive radios (CRs). Numerical results show that the new spectrum models have acceptable accuracies. They also show that the proposed spectrum measurement prediction method tracks the trend of the true values well. Therefore, these results are very useful in CR designs.

This letter proposes a new algorithm, termed bidimensional statistical empirical mode decomposition (BSEMD) that adopts a smoothing procedure instead of an interpolation when constructing 2-D upper and lower envelopes. For this purpose, we investigate the sifting process effect of conventional bidimensional empirical mode decomposition (BEMD) on the decomposition results, and propose a modified BEMD via the smoothing sifting process coupling with a new identification method of 2-D local extrema. Furthermore, theoretical rationale for smoothing sifting is investigated.

This letter proposes a wavelet denoising method in the presence of missing data. This approach is based on a coupling of wavelet shrinkage and hierarchical (or h)-likelihood method. The h-likelihood provides an effective imputation methodology of missing data to give wavelet estimators for signals and motivates a fast and simple algorithm. The method can be easily extended to other settings, such as image denoising. Simulation studies demonstrate empirical properties of the proposed method.

In this paper, we focus on denoising images for which observations are equally spaced except around the boundaries which are irregular. Such images are very common in many fields, for example in geophysics. The advantages of adding a low-order polynomial term when implementing a wavelet regression for such images are presented. Besides removing the classical restriction of having a dyadic of number of observations, this strategy reduces the bias at the edges without significantly increasing the risk. In addition, this method is simple to implement, fast and efficient. Its utility is illustrated with simulation studies and a real example.

The wavelet spectrum of a random process comprises the variances of the wavelet coefficients of the process computed within each scale. This paper investigates the possibility of using the wavelet spectrum, obtained from a continuous wavelet transform (CWT), to uniquely represent the second-order statistical properties of random processes-particularly, stationary processes and long-memory nonstationary processes. As is well known, the Fourier spectrum of a stationary process is mathematically equivalent to the autocovariance function (ACF) and thus uniquely determines the second-order statistics of the process. This characterization property is shown to be possessed also by the wavelet spectrum under very mild regularity conditions that are easily satisfied by many widely used wavelets. It is also shown that under suitable regularity conditions, the characterization property remains valid for processes with stationary increments including 1/f noise.