7 research outputs found
Robust Principal Component Analysis using Density Power Divergence
Principal component analysis (PCA) is a widely employed statistical tool used
primarily for dimensionality reduction. However, it is known to be adversely
affected by the presence of outlying observations in the sample, which is quite
common. Robust PCA methods using M-estimators have theoretical benefits, but
their robustness drop substantially for high dimensional data. On the other end
of the spectrum, robust PCA algorithms solving principal component pursuit or
similar optimization problems have high breakdown, but lack theoretical
richness and demand high computational power compared to the M-estimators. We
introduce a novel robust PCA estimator based on the minimum density power
divergence estimator. This combines the theoretical strength of the
M-estimators and the minimum divergence estimators with a high breakdown
guarantee regardless of data dimension. We present a computationally efficient
algorithm for this estimate. Our theoretical findings are supported by
extensive simulations and comparisons with existing robust PCA methods. We also
showcase the proposed algorithm's applicability on two benchmark datasets and a
credit card transactions dataset for fraud detection
rSVDdpd: A Robust Scalable Video Surveillance Background Modelling Algorithm
A basic algorithmic task in automated video surveillance is to separate
background and foreground objects. Camera tampering, noisy videos, low frame
rate, etc., pose difficulties in solving the problem. A general approach that
classifies the tampered frames, and performs subsequent analysis on the
remaining frames after discarding the tampered ones, results in loss of
information. Several robust methods based on robust principal component
analysis (PCA) have been introduced to solve this problem. To date,
considerable effort has been expended to develop robust PCA via Principal
Component Pursuit (PCP) methods with reduced computational cost and visually
appealing foreground detection. However, the convex optimizations used in these
algorithms do not scale well to real-world large datasets due to large matrix
inversion steps. Also, an integral component of these foreground detection
algorithms is singular value decomposition which is nonrobust. In this paper,
we present a new video surveillance background modelling algorithm based on a
new robust singular value decomposition technique rSVDdpd which takes care of
both these issues. We also demonstrate the superiority of our proposed
algorithm on a benchmark dataset and a new real-life video surveillance dataset
in the presence of camera tampering. Software codes and additional
illustrations are made available at the accompanying website rSVDdpd Homepage
(https://subroy13.github.io/rsvddpd-home/
Analysis of the rSVDdpd Algorithm: A Robust Singular Value Decomposition Method using Density Power Divergence
The traditional method of computing singular value decomposition (SVD) of a
data matrix is based on a least squares principle, thus, is very sensitive to
the presence of outliers. Hence the resulting inferences across different
applications using the classical SVD are extremely degraded in the presence of
data contamination (e.g., video surveillance background modelling tasks, etc.).
A robust singular value decomposition method using the minimum density power
divergence estimator (rSVDdpd) has been found to provide a satisfactory
solution to this problem and works well in applications. For example, it
provides a neat solution to the background modelling problem of video
surveillance data in the presence of camera tampering. In this paper, we
investigate the theoretical properties of the rSVDdpd estimator such as
convergence, equivariance and consistency under reasonable assumptions. Since
the dimension of the parameters, i.e., the number of singular values and the
dimension of singular vectors can grow linearly with the size of the data, the
usual M-estimation theory has to be suitably modified with concentration bounds
to establish the asymptotic properties. We believe that we have been able to
accomplish this satisfactorily in the present work. We also demonstrate the
efficiency of rSVDdpd through extensive simulations.Comment: arXiv admin note: substantial text overlap with arXiv:2109.1068
Robust and Efficient Estimation in Ordinal Response Models using the Density Power Divergence
In real life, we frequently come across data sets that involve some
independent explanatory variable(s) generating a set of ordinal responses.
These ordinal responses may correspond to an underlying continuous latent
variable, which is linearly related to the covariate(s), and takes a particular
(ordinal) label depending on whether this latent variable takes value in some
suitable interval specified by a pair of (unknown) cut-offs. The most efficient
way of estimating the unknown parameters (i.e., the regression coefficients and
the cut-offs) is the method of maximum likelihood (ML). However, contamination
in the data set either in the form of misspecification of ordinal responses, or
the unboundedness of the covariate(s), might destabilize the likelihood
function to a great extent where the ML based methodology might lead to
completely unreliable inferences. In this paper, we explore a minimum distance
estimation procedure based on the popular density power divergence (DPD) to
yield robust parameter estimates for the ordinal response model. This paper
highlights how the resulting estimator, namely the minimum DPD estimator
(MDPDE), can be used as a practical robust alternative to the classical
procedures based on the ML. We rigorously develop several theoretical
properties of this estimator, and provide extensive simulations to substantiate
the theory developed
Asymptotic Breakdown Point Analysis for a General Class of Minimum Divergence Estimators
Robust inference based on the minimization of statistical divergences has
proved to be a useful alternative to classical techniques based on maximum
likelihood and related methods. Basu et al. (1998) introduced the density power
divergence (DPD) family as a measure of discrepancy between two probability
density functions and used this family for robust estimation of the parameter
for independent and identically distributed data. Ghosh et al. (2017) proposed
a more general class of divergence measures, namely the S-divergence family and
discussed its usefulness in robust parametric estimation through several
asymptotic properties and some numerical illustrations. In this paper, we
develop the results concerning the asymptotic breakdown point for the minimum
S-divergence estimators (in particular the minimum DPD estimator) under general
model setups. The primary result of this paper provides lower bounds to the
asymptotic breakdown point of these estimators which are independent of the
dimension of the data, in turn corroborating their usefulness in robust
inference under high dimensional data.Comment: Keywords: Breakdown Point, Minimum S-divergence Estimator, Density
Power Divergence, Power Divergence Details: 28 pages, 7 figure