9,337 research outputs found
Big Data and Reliability Applications: The Complexity Dimension
Big data features not only large volumes of data but also data with
complicated structures. Complexity imposes unique challenges in big data
analytics. Meeker and Hong (2014, Quality Engineering, pp. 102-116) provided an
extensive discussion of the opportunities and challenges in big data and
reliability, and described engineering systems that can generate big data that
can be used in reliability analysis. Meeker and Hong (2014) focused on large
scale system operating and environment data (i.e., high-frequency multivariate
time series data), and provided examples on how to link such data as covariates
to traditional reliability responses such as time to failure, time to
recurrence of events, and degradation measurements. This paper intends to
extend that discussion by focusing on how to use data with complicated
structures to do reliability analysis. Such data types include high-dimensional
sensor data, functional curve data, and image streams. We first provide a
review of recent development in those directions, and then we provide a
discussion on how analytical methods can be developed to tackle the challenging
aspects that arise from the complexity feature of big data in reliability
applications. The use of modern statistical methods such as variable selection,
functional data analysis, scalar-on-image regression, spatio-temporal data
models, and machine learning techniques will also be discussed.Comment: 28 pages, 7 figure
Applications of a finite-dimensional duality principle to some prediction problems
Some of the most important results in prediction theory and time series
analysis when finitely many values are removed from or added to its infinite
past have been obtained using difficult and diverse techniques ranging from
duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear
regression in statistics (Box and Tiao, 1975). We unify these results via a
finite-dimensional duality lemma and elementary ideas from the linear algebra.
The approach reveals the inherent finite-dimensional character of many
difficult prediction problems, the role of duality and biorthogonality for a
finite set of random variables. The lemma is particularly useful when the
number of missing values is small, like one or two, as in the case of
Kolmogorov and Nakazi prediction problems. The stationarity of the underlying
process is not a requirement. It opens up the possibility of extending such
results to nonstationary processes.Comment: 15 page
Selfdecomposability and selfsimilarity: a concise primer
We summarize the relations among three classes of laws: infinitely divisible,
selfdecomposable and stable. First we look at them as the solutions of the
Central Limit Problem; then their role is scrutinized in relation to the Levy
and the additive processes with an emphasis on stationarity and selfsimilarity.
Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and
their selfdecomposable stationary distributions, and we end with a few
particular examples.Comment: 24 pages, 3 figures; corrected misprint in the title; redactional
modifications required by the referee; added references from [16] to [28];.
Accepted and in press on Physica
Range descriptions for the spherical mean Radon transform
The transform considered in the paper averages a function supported in a ball
in \RR^n over all spheres centered at the boundary of the ball. This Radon
type transform arises in several contemporary applications, e.g. in
thermoacoustic tomography and sonar and radar imaging. Range descriptions for
such transforms are important in all these areas, for instance when dealing
with incomplete data, error correction, and other issues. Four different types
of complete range descriptions are provided, some of which also suggest
inversion procedures. Necessity of three of these (appropriately formulated)
conditions holds also in general domains, while the complete discussion of the
case of general domains would require another publication.Comment: LATEX file, 55 pages, two EPS figure
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