11,693 research outputs found
Precursors of extreme increments
We investigate precursors and predictability of extreme increments in a time
series. The events we are focusing on consist in large increments within
successive time steps. We are especially interested in understanding how the
quality of the predictions depends on the strategy to choose precursors, on the
size of the event and on the correlation strength. We study the prediction of
extreme increments analytically in an AR(1) process, and numerically in wind
speed recordings and long-range correlated ARMA data. We evaluate the success
of predictions via receiver operator characteristics (ROC-curves). Furthermore,
we observe an increase of the quality of predictions with increasing event size
and with decreasing correlation in all examples. Both effects can be understood
by using the likelihood ratio as a summary index for smooth ROC-curves
Importance sampling the union of rare events with an application to power systems analysis
We consider importance sampling to estimate the probability of a union
of rare events defined by a random variable . The
sampler we study has been used in spatial statistics, genomics and
combinatorics going back at least to Karp and Luby (1983). It works by sampling
one event at random, then sampling conditionally on that event
happening and it constructs an unbiased estimate of by multiplying an
inverse moment of the number of occuring events by the union bound. We prove
some variance bounds for this sampler. For a sample size of , it has a
variance no larger than where is the union
bound. It also has a coefficient of variation no larger than
regardless of the overlap pattern among the
events. Our motivating problem comes from power system reliability, where the
phase differences between connected nodes have a joint Gaussian distribution
and the rare events arise from unacceptably large phase differences. In the
grid reliability problems even some events defined by constraints in
dimensions, with probability below , are estimated with a
coefficient of variation of about with only sample
values
Zero-Crossing Statistics for Non-Markovian Time Series
In applications spaning from image analysis and speech recognition, to energy
dissipation in turbulence and time-to failure of fatigued materials,
researchers and engineers want to calculate how often a stochastic observable
crosses a specific level, such as zero. At first glance this problem looks
simple, but it is in fact theoretically very challenging. And therefore, few
exact results exist. One exception is the celebrated Rice formula that gives
the mean number of zero-crossings in a fixed time interval of a zero-mean
Gaussian stationary processes. In this study we use the so-called Independent
Interval Approximation to go beyond Rice's result and derive analytic
expressions for all higher-order zero-crossing cumulants and moments. Our
results agrees well with simulations for the non-Markovian autoregressive
model
Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification
Stochastic spectral methods are efficient techniques for uncertainty
quantification. Recently they have shown excellent performance in the
statistical analysis of integrated circuits. In stochastic spectral methods,
one needs to determine a set of orthonormal polynomials and a proper numerical
quadrature rule. The former are used as the basis functions in a generalized
polynomial chaos expansion. The latter is used to compute the integrals
involved in stochastic spectral methods. Obtaining such information requires
knowing the density function of the random input {\it a-priori}. However,
individual system components are often described by surrogate models rather
than density functions. In order to apply stochastic spectral methods in
hierarchical uncertainty quantification, we first propose to construct
physically consistent closed-form density functions by two monotone
interpolation schemes. Then, by exploiting the special forms of the obtained
density functions, we determine the generalized polynomial-chaos basis
functions and the Gauss quadrature rules that are required by a stochastic
spectral simulator. The effectiveness of our proposed algorithm is verified by
both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201
Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal noise
To understand the sample-to-sample fluctuations in disorder-generated
multifractal patterns we investigate analytically as well as numerically the
statistics of high values of the simplest model - the ideal periodic
Gaussian noise. By employing the thermodynamic formalism we predict the
characteristic scale and the precise scaling form of the distribution of number
of points above a given level. We demonstrate that the powerlaw forward tail of
the probability density, with exponent controlled by the level, results in an
important difference between the mean and the typical values of the counting
function. This can be further used to determine the typical threshold of
extreme values in the pattern which turns out to be given by
with . Such observation provides a
rather compelling explanation of the mechanism behind universality of .
Revealed mechanisms are conjectured to retain their qualitative validity for a
broad class of disorder-generated multifractal fields. In particular, we
predict that the typical value of the maximum of intensity is to be
given by , where is the
corresponding singularity spectrum vanishing at . For the
noise we also derive exact as well as well-controlled approximate
formulas for the mean and the variance of the counting function without
recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints
corrected, editing done and references adde
Failure Probability Estimation and Detection of Failure Surfaces via Adaptive Sequential Decomposition of the Design Domain
We propose an algorithm for an optimal adaptive selection of points from the
design domain of input random variables that are needed for an accurate
estimation of failure probability and the determination of the boundary between
safe and failure domains. The method is particularly useful when each
evaluation of the performance function g(x) is very expensive and the function
can be characterized as either highly nonlinear, noisy, or even discrete-state
(e.g., binary). In such cases, only a limited number of calls is feasible, and
gradients of g(x) cannot be used. The input design domain is progressively
segmented by expanding and adaptively refining mesh-like lock-free geometrical
structure. The proposed triangulation-based approach effectively combines the
features of simulation and approximation methods. The algorithm performs two
independent tasks: (i) the estimation of probabilities through an ingenious
combination of deterministic cubature rules and the application of the
divergence theorem and (ii) the sequential extension of the experimental design
with new points. The sequential selection of points from the design domain for
future evaluation of g(x) is carried out through a new learning function, which
maximizes instantaneous information gain in terms of the probability
classification that corresponds to the local region. The extension may be
halted at any time, e.g., when sufficiently accurate estimations are obtained.
Due to the use of the exact geometric representation in the input domain, the
algorithm is most effective for problems of a low dimension, not exceeding
eight. The method can handle random vectors with correlated non-Gaussian
marginals. The estimation accuracy can be improved by employing a smooth
surrogate model. Finally, we define new factors of global sensitivity to
failure based on the entire failure surface weighted by the density of the
input random vector.Comment: 42 pages, 24 figure
Lost in translation: Toward a formal model of multilevel, multiscale medicine
For a broad spectrum of low level cognitive regulatory and other biological phenomena, isolation from signal crosstalk between them requires more metabolic free energy than permitting correlation. This allows an evolutionary exaptation leading to dynamic global broadcasts of interacting physiological processes at multiple scales. The argument is similar to the well-studied exaptation of noise to trigger stochastic resonance amplification in physiological subsystems. Not only is the living state characterized by cognition at every scale and level of organization, but by multiple, shifting, tunable, cooperative larger scale broadcasts that link selected subsets of functional modules to address problems. This multilevel dynamical viewpoint has implications for initiatives in translational medicine that have followed the implosive collapse of pharmaceutical industry 'magic bullet' research. In short, failure to respond to the inherently multilevel, multiscale nature of human pathophysiology will doom translational medicine to a similar implosion
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