17,996 research outputs found
More "normal" than normal: scaling distributions and complex systems
One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems
Magnitude Uncertainties Impact Seismic Rate Estimates, Forecasts and Predictability Experiments
The Collaboratory for the Study of Earthquake Predictability (CSEP) aims to
prospectively test time-dependent earthquake probability forecasts on their
consistency with observations. To compete, time-dependent seismicity models are
calibrated on earthquake catalog data. But catalogs contain much observational
uncertainty. We study the impact of magnitude uncertainties on rate estimates
in clustering models, on their forecasts and on their evaluation by CSEP's
consistency tests. First, we quantify magnitude uncertainties. We find that
magnitude uncertainty is more heavy-tailed than a Gaussian, such as a
double-sided exponential distribution, with scale parameter nu_c=0.1 - 0.3.
Second, we study the impact of such noise on the forecasts of a simple
clustering model which captures the main ingredients of popular short term
models. We prove that the deviations of noisy forecasts from an exact forecast
are power law distributed in the tail with exponent alpha=1/(a*nu_c), where a
is the exponent of the productivity law of aftershocks. We further prove that
the typical scale of the fluctuations remains sensitively dependent on the
specific catalog. Third, we study how noisy forecasts are evaluated in CSEP
consistency tests. Noisy forecasts are rejected more frequently than expected
for a given confidence limit. The Poisson assumption of the consistency tests
is inadequate for short-term forecast evaluations. To capture the
idiosyncrasies of each model together with any propagating uncertainties, the
forecasts need to specify the entire likelihood distribution of seismic rates.Comment: 35 pages, including 15 figures, agu styl
Break detection in the covariance structure of multivariate time series models
In this paper, we introduce an asymptotic test procedure to assess the
stability of volatilities and cross-volatilites of linear and nonlinear
multivariate time series models. The test is very flexible as it can be
applied, for example, to many of the multivariate GARCH models established in
the literature, and also works well in the case of high dimensionality of the
underlying data. Since it is nonparametric, the procedure avoids the
difficulties associated with parametric model selection, model fitting and
parameter estimation. We provide the theoretical foundation for the test and
demonstrate its applicability via a simulation study and an analysis of
financial data. Extensions to multiple changes and the case of infinite fourth
moments are also discussed.Comment: Published in at http://dx.doi.org/10.1214/09-AOS707 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Breaking the curse of dimensionality in regression
Models with many signals, high-dimensional models, often impose structures on
the signal strengths. The common assumption is that only a few signals are
strong and most of the signals are zero or close (collectively) to zero.
However, such a requirement might not be valid in many real-life applications.
In this article, we are interested in conducting large-scale inference in
models that might have signals of mixed strengths. The key challenge is that
the signals that are not under testing might be collectively non-negligible
(although individually small) and cannot be accurately learned. This article
develops a new class of tests that arise from a moment matching formulation. A
virtue of these moment-matching statistics is their ability to borrow strength
across features, adapt to the sparsity size and exert adjustment for testing
growing number of hypothesis. GRoup-level Inference of Parameter, GRIP, test
harvests effective sparsity structures with hypothesis formulation for an
efficient multiple testing procedure. Simulated data showcase that GRIPs error
control is far better than the alternative methods. We develop a minimax
theory, demonstrating optimality of GRIP for a broad range of models, including
those where the model is a mixture of a sparse and high-dimensional dense
signals.Comment: 51 page
Heavy-Tailed Features and Empirical Analysis of the Limit Order Book Volume Profiles in Futures Markets
This paper poses a few fundamental questions regarding the attributes of the
volume profile of a Limit Order Books stochastic structure by taking into
consideration aspects of intraday and interday statistical features, the impact
of different exchange features and the impact of market participants in
different asset sectors. This paper aims to address the following questions:
1. Is there statistical evidence that heavy-tailed sub-exponential volume
profiles occur at different levels of the Limit Order Book on the bid and ask
and if so does this happen on intra or interday time scales ?
2.In futures exchanges, are heavy tail features exchange (CBOT, CME, EUREX,
SGX and COMEX) or asset class (government bonds, equities and precious metals)
dependent and do they happen on ultra-high (<1sec) or mid-range (1sec -10min)
high frequency data?
3.Does the presence of stochastic heavy-tailed volume profile features evolve
in a manner that would inform or be indicative of market participant behaviors,
such as high frequency algorithmic trading, quote stuffing and price discovery
intra-daily?
4. Is there statistical evidence for a need to consider dynamic behavior of
the parameters of models for Limit Order Book volume profiles on an intra-daily
time scale ?
Progress on aspects of each question is obtained via statistically rigorous
results to verify the empirical findings for an unprecedentedly large set of
futures market LOB data. The data comprises several exchanges, several futures
asset classes and all trading days of 2010, using market depth (Type II) order
book data to 5 levels on the bid and ask
Diffusion-Based Adaptive Distributed Detection: Steady-State Performance in the Slow Adaptation Regime
This work examines the close interplay between cooperation and adaptation for
distributed detection schemes over fully decentralized networks. The combined
attributes of cooperation and adaptation are necessary to enable networks of
detectors to continually learn from streaming data and to continually track
drifts in the state of nature when deciding in favor of one hypothesis or
another. The results in the paper establish a fundamental scaling law for the
steady-state probabilities of miss-detection and false-alarm in the slow
adaptation regime, when the agents interact with each other according to
distributed strategies that employ small constant step-sizes. The latter are
critical to enable continuous adaptation and learning. The work establishes
three key results. First, it is shown that the output of the collaborative
process at each agent has a steady-state distribution. Second, it is shown that
this distribution is asymptotically Gaussian in the slow adaptation regime of
small step-sizes. And third, by carrying out a detailed large deviations
analysis, closed-form expressions are derived for the decaying rates of the
false-alarm and miss-detection probabilities. Interesting insights are gained.
In particular, it is verified that as the step-size decreases, the error
probabilities are driven to zero exponentially fast as functions of ,
and that the error exponents increase linearly in the number of agents. It is
also verified that the scaling laws governing errors of detection and errors of
estimation over networks behave very differently, with the former having an
exponential decay proportional to , while the latter scales linearly
with decay proportional to . It is shown that the cooperative strategy
allows each agent to reach the same detection performance, in terms of
detection error exponents, of a centralized stochastic-gradient solution.Comment: The paper will appear in IEEE Trans. Inf. Theor
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