Robust PCA for high-dimensional data based on characteristic transformation

In this paper, we propose a novel robust principal component analysis (PCA) for high-dimensional data in the presence of various heterogeneities, in particular strong tailing and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. The suggested method has the distinct advantage of dealing with heavy-tail-distributed data, whose covariances may be non-existent (positively infinite, for instance), in addition to the usual outliers. The proposed approach is also a case of kernel principal component analysis (KPCA) and employs the robust and non-linear properties via a bounded and non-linear kernel function. The merits of the new method are illustrated by some statistical properties, including the upper bound of the excess error and the behaviour of the large eigenvalues under a spiked covariance model. Additionally, using a variety of simulations, we demonstrate the benefits of our approach over the classical PCA. Finally, using data on protein expression in mice of various genotypes in a biological study, we apply the novel robust PCA to categorise the mice and find that our approach is more effective at identifying abnormal mice than the classical PCA.     

Robust optimal estimation of location from discretely sampled functional data

Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that under suitable assumptions the proposed family of estimators is minimax rate optimal both for commonly and independently observed trajectories and we illustrate its highly competitive performance and practical usefulness in a Monte-Carlo study and a real-data example involving recent Covid-19 data.     

估计灰色Verhulst模型参数的LS-SVM方法及应用

估计灰色Verhulst模型中的参数通常采用最小二乘法，这种基于大样本理论的经验风险最小化方法无法保证小样本预测下模型的推广性能.为提高灰色Verhulst模型的预测精度，本文提出了基于LS-SVM算法估计模型参数的方法.首先根据Verhulst灰色差分方程的特点，通过构造以背景值序列和原始序列为训练样本的LS-SVM模型，将一维样本空间里的Verhulst模型转化为一个二维特征空间里的LS-SVM模型，进而将Verhulst模型的灰参数的估计问题转化为一个LS-SVM模型的回归系数估计问题.然后通过核函数构造法，结合模型特点合理构造了LS-SVM模型的核函数，基于LS-SVM算法求解回归系数，进而得到Verhulst模型的参数估计.实验结果表明该方法是可行的有效的，可保证Verhulst模型具有良好推广性，相比于传统参数估计方法本文预测精度更高.     

Asymptotic Distribution Theory for Nonparametric Entropy Measures of Serial Dependence

Entropy is a classical statistical concept with appealing properties. Establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging. In this paper, we develop an asymptotic theory for a class of kernel‐based smoothed nonparametric entropy measures of serial dependence in a time‐series context. We use this theory to derive the limiting distribution of Granger and Lin's (1994) normalized entropy measure of serial dependence, which was previously not available in the literature. We also apply our theory to construct a new entropy‐based test for serial dependence, providing an alternative to Robinson's (1991) approach. To obtain accurate inferences, we propose and justify a consistent smoothed bootstrap procedure. The naive bootstrap is not consistent for our test. Our test is useful in, for example, testing the random walk hypothesis, evaluating density forecasts, and identifying important lags of a time series. It is asymptotically locally more powerful than Robinson's (1991) test, as is confirmed in our simulation. An application to the daily S&P 500 stock price index illustrates our approach.