Can we use PCA to detect small signals in noisy data?

Principal component analysis (PCA) is among the most commonly applied dimension reduction techniques suitable to denoise data. Focusing on its limitations to detect low variance signals in noisy data, we discuss how statistical and systematical errors occur in PCA reconstructed data as a function of the size of the data set, which extends the work of Lichtert and Verbeeck, (2013) [16]. Particular attention is directed towards the estimation of bias introduced by PCA and its influence on experiment design. Aiming at the denoising of large matrices, nullspace based denoising (NBD) is introduced.

[1]  R. Schneider,et al.  EELS of Niobium and Stoichiometric Niobium-Oxide Phases—Part I: Plasmon and Near-Edges Fine Structure , 2009, Microscopy and Microanalysis.

[2]  D. Alexander,et al.  Mapping chemical and bonding information using multivariate analysis of electron energy-loss spectrum images. , 2006, Ultramicroscopy.

[3]  L. Allen,et al.  Atomic-resolution electron energy loss spectroscopy imaging in aberration corrected scanning transmission electron microscopy. , 2003, Physical review letters.

[4]  S D Findlay,et al.  Two-dimensional mapping of chemical information at atomic resolution. , 2007, Physical review letters.

[5]  R. Klie,et al.  Position-sensitive change in the transition metal L-edge fine structures , 2015 .

[6]  J. Idrobo,et al.  Aberrated electron probes for magnetic spectroscopy with atomic resolution: Theory and practical aspects , 2016 .

[7]  Paul Cueva,et al.  Data Processing for Atomic Resolution Electron Energy Loss Spectroscopy , 2012, Microscopy and Microanalysis.

[8]  Antonino Staiano,et al.  Intrinsic dimension estimation: Advances and open problems , 2016, Inf. Sci..

[9]  G A Botton,et al.  Quantitative statistical analysis, optimization and noise reduction of atomic resolved electron energy loss spectrum images. , 2012, Micron.

[10]  N Bonnet,et al.  EELS elemental mapping with unconventional methods. I. Theoretical basis: image analysis with multivariate statistics and entropy concepts. , 1990, Ultramicroscopy.

[11]  Yuan Chen,et al.  Variance analysis of unbiased least ℓp-norm estimator in non-Gaussian noise , 2016, Signal Process..

[12]  R. Cattell.  The Scree Test For The Number Of Factors. , 1966, Multivariate behavioral research.

[13]  J Verbeeck,et al.  Model based quantification of EELS spectra. , 2004, Ultramicroscopy.

[14]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[15]  P. Schattschneider,et al.  Detection of magnetic circular dichroism using a transmission electron microscope , 2006, Nature.

[16]  Stijn Lichtert,et al.  Statistical consequences of applying a PCA noise filter on EELS spectrum images. , 2013, Ultramicroscopy.

[17]  Leslie J. Allen,et al.  Atomic-resolution chemical mapping using energy-dispersive x-ray spectroscopy , 2010 .

[18]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[19]  Paul Geladi,et al.  Random error bias in principal component analysis. Part I. derivation of theoretical predictions , 1995 .