21,986 research outputs found
Pseudo-nonstationarity in the scaling exponents of finite-interval time series
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series
Chance, long tails, and inference: a non-Gaussian, Bayesian theory of vocal learning in songbirds
Traditional theories of sensorimotor learning posit that animals use sensory
error signals to find the optimal motor command in the face of Gaussian sensory
and motor noise. However, most such theories cannot explain common behavioral
observations, for example that smaller sensory errors are more readily
corrected than larger errors and that large abrupt (but not gradually
introduced) errors lead to weak learning. Here we propose a new theory of
sensorimotor learning that explains these observations. The theory posits that
the animal learns an entire probability distribution of motor commands rather
than trying to arrive at a single optimal command, and that learning arises via
Bayesian inference when new sensory information becomes available. We test this
theory using data from a songbird, the Bengalese finch, that is adapting the
pitch (fundamental frequency) of its song following perturbations of auditory
feedback using miniature headphones. We observe the distribution of the sung
pitches to have long, non-Gaussian tails, which, within our theory, explains
the observed dynamics of learning. Further, the theory makes surprising
predictions about the dynamics of the shape of the pitch distribution, which we
confirm experimentally
Gaussian Tests of "Extremal White Noise" for Dependent, Heterogeneous, Heavy Tailed Strochastic Processes with an Application
We develop a non-parametric test of tail-specific extremal serial dependence for possibly heavy-tailed time series. The test statistic is asymptotically chi-squared under a null of "extremal white noise", as long as extremes of the time series are Near-Epoch-Dependent on the extremes of some mixing process. The theory covers ARFIMA, FIGARCH, bilinear, and Extremal Threshold processes, and a wide range of nonlinear distributed lags. In this setting the test statistic obtains an asymptotic power of one under the alternative. Of separate interest, we deliver a joint distribution limit for an arbitrary vector of tail index estimators under extraordinarily gene ral conditions, complete with a consistent kernel estimator of the covariance matrix. We apply tail specific tests to equity market and exchange rate returns data.extremal dependence; white-noise; near-epoch-dependence; regular variation; infinite variance; portmanteau test
Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models
Long-Range Dependence (LRD) and heavy-tailed distributions are ubiquitous in natural and socio-economic data. Such data can be self-similar whereby both LRD and heavy-tailed distributions contribute to the self-similarity as measured by the Hurst exponent. Some methods widely used in the physical sciences separately estimate these two parameters, which can lead to estimation bias. Those which do simultaneous estimation are based on frequentist methods such as Whittle’s approximate maximum likelihood estimator. Here we present a new and systematic Bayesian framework for the simultaneous inference of the LRD and heavy-tailed distribution parameters of a parametric ARFIMA model with non-Gaussian innovations. As innovations we use the α-stable and t-distributions which have power law tails. Our algorithm also provides parameter uncertainty estimates. We test our algorithm using synthetic data, and also data from the Geostationary Operational Environmental Satellite system (GOES) solar X-ray time series. These tests show that our algorithm is able to accurately and robustly estimate the LRD and heavy-tailed distribution parameters
Understanding the source of multifractality in financial markets
In this paper, we use the generalized Hurst exponent approach to study the
multi- scaling behavior of different financial time series. We show that this
approach is robust and powerful in detecting different types of multiscaling.
We observe a puzzling phenomenon where an apparent increase in multifractality
is measured in time series generated from shuffled returns, where all
time-correlations are destroyed, while the return distributions are conserved.
This effect is robust and it is reproduced in several real financial data
including stock market indices, exchange rates and interest rates. In order to
understand the origin of this effect we investigate different simulated time
series by means of the Markov switching multifractal (MSM) model,
autoregressive fractionally integrated moving average (ARFIMA) processes with
stable innovations, fractional Brownian motion and Levy flights. Overall we
conclude that the multifractality observed in financial time series is mainly a
consequence of the characteristic fat-tailed distribution of the returns and
time-correlations have the effect to decrease the measured multifractality
On the return period of the 2003 heat wave
Extremal events are difficult to model since it is difficult to characterize formally those events. The 2003 heat wave in Europe was not characterized by very high temperatures, but mainly the fact that night temperature were no cool enough for a long period of time. Hence, simulation of several models (either with heavy tailed noise or long range dependence) yield different estimations for the return period of that extremal event.Heat wave, long range dependence, return period, heavy tails, GARMA processes, SARIMA processes
A study of memory effects in a chess database
A series of recent works studying a database of chronologically sorted chess
games --containing 1.4 million games played by humans between 1998 and 2007--
have shown that the popularity distribution of chess game-lines follows a
Zipf's law, and that time series inferred from the sequences of those
game-lines exhibit long-range memory effects. The presence of Zipf's law
together with long-range memory effects was observed in several systems,
however, the simultaneous emergence of these two phenomena were always studied
separately up to now. In this work, by making use of a variant of the
Yule-Simon preferential growth model, introduced by Cattuto et al., we provide
an explanation for the simultaneous emergence of Zipf's law and long-range
correlations memory effects in a chess database. We find that Cattuto's Model
(CM) is able to reproduce both, Zipf's law and the long-range correlations,
including size-dependent scaling of the Hurst exponent for the corresponding
time series. CM allows an explanation for the simultaneous emergence of these
two phenomena via a preferential growth dynamics, including a memory kernel, in
the popularity distribution of chess game-lines. This mechanism results in an
aging process in the chess game-line choice as the database grows. Moreover, we
find burstiness in the activity of subsets of the most active players, although
the aggregated activity of the pool of players displays inter-event times
without burstiness. We show that CM is not able to produce time series with
bursty behavior providing evidence that burstiness is not required for the
explanation of the long-range correlation effects in the chess database.Comment: 18 pages, 7 figure
Strong Orthogonal Decompositions and Nonlinear Impulse Response Functions for Infinite-Variance Processes
In this paper we prove Wold-type decompositions with strongorthogonal prediction innovations exist in smooth, re‡exive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. Our theory includes as special cases all previous Wold-type decompositions of discrete time processes; completely characterizes when nonlinear heavy-tailed processes obtain a strong-orthogonal moving average representation; and easily promotes a theory of nonlinear impulse response functions for infinite variance processes. We exemplify our theory by developing a nonlinear impulse response func tion for smooth transition threshold processes, we discuss how to test de composition innovations for strong orthogonality and whether the proposed model represents the best predictor, and we apply the methodology to currency exchange rates.Orthogonal decompositions, Banach spaces, projection iterations, infinite variance, moving average, nonlinear impulse response function, smooth transition autoregression, Lp-metric projection, Lp-GMM.
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