PCA Consistency for Non-Gaussian Data in High Dimension, Low Sample Size Context

Abstract

In this paper, we investigate both sample eigenvalues and Principal Component (PC) directions along with PC scores when the dimension d and the sample size n both grow to infinity in such a way that n is much lower than d. We consider general settings that include the case when the eigenvalues are all in the range of sphericity. We do not assume either the normality or a ρ-mixing condition. We attempt finding a difference among the eigenvalues by choosing n with a suitable order of d. We give the consistency properties for both the sample eigenvalues and the PC directions along with the PC scores. We also show that the sample eigenvalue has a Gaussian limiting distribution when the population counterpart is of multiplicity one.

Motion process monitoring using optical flow–based principal component analysis-independent component analysis method

“…Then, KDE is used for fitting these discrete points for DOF-PCA-ICA in Figure 8. The same is done to interval [0,7]. The frequency ratio of I 2 within each subinterval and the KDE for fitting these discrete points for SOF-PCA-ICA is shown in Figure 9.…”

“…Therefore, the design of motion process monitoring method for solving these problems is of great significance. Data-driven multivariate statistical process monitoring (MSPM) methods have been widely used to capture the characteristics of the process for further establishing accurate and reliable monitoring model, [4][5][6][7][8][9][10] such as principal component analysis (PCA) and partial least squares (PLS). However, there is a pre-assumption that the data follow Gaussian distribution which can be hardly satisfied in practical application.…”

Effective Methodologies for Statistical Inference on Microarray Studies

“…Hence, from Theorem 4.1, the discriminant rule given by (10) with γ = 0 was expected to hold (11). In Table 1, we investigated the performance of the discriminant rule by using test data sets consisting of 24 − n 1 = 14 remaining samples from Π 1 and 28 − n 2 = 18 remaining samples from Π 2 .…”

“…For the details including the limiting distribution ofλ j , see Yata and Aoshima (2009). If the population distribution is N p (µ, Σ), one may consider that z jk , j = 1, ..., p (k = 1, ..., n) are independent.…”

“…In this chapter, we present modern statistical methodologies that are very effective to draw statistical inference from HDLSS data. We focus on a series of effective HDLSS methodologies developed by Aoshima and Yata (2011) and Yata and Aoshima (2009, 2010a,b, 2011a. We demonstrate how those methodologies perform well and bring a new insight into researches on prostate cancer.…”