90,212 research outputs found
Predictability, complexity and learning
We define {\em predictive information} as the mutual
information between the past and the future of a time series. Three
qualitatively different behaviors are found in the limit of large observation
times : can remain finite, grow logarithmically, or grow
as a fractional power law. If the time series allows us to learn a model with a
finite number of parameters, then grows logarithmically with
a coefficient that counts the dimensionality of the model space. In contrast,
power--law growth is associated, for example, with the learning of infinite
parameter (or nonparametric) models such as continuous functions with
smoothness constraints. There are connections between the predictive
information and measures of complexity that have been defined both in learning
theory and in the analysis of physical systems through statistical mechanics
and dynamical systems theory. Further, in the same way that entropy provides
the unique measure of available information consistent with some simple and
plausible conditions, we argue that the divergent part of
provides the unique measure for the complexity of dynamics underlying a time
series. Finally, we discuss how these ideas may be useful in different problems
in physics, statistics, and biology.Comment: 53 pages, 3 figures, 98 references, LaTeX2
Informational and Causal Architecture of Discrete-Time Renewal Processes
Renewal processes are broadly used to model stochastic behavior consisting of
isolated events separated by periods of quiescence, whose durations are
specified by a given probability law. Here, we identify the minimal sufficient
statistic for their prediction (the set of causal states), calculate the
historical memory capacity required to store those states (statistical
complexity), delineate what information is predictable (excess entropy), and
decompose the entropy of a single measurement into that shared with the past,
future, or both. The causal state equivalence relation defines a new subclass
of renewal processes with a finite number of causal states despite having an
unbounded interevent count distribution. We use these formulae to analyze the
output of the parametrized Simple Nonunifilar Source, generated by a simple
two-state hidden Markov model, but with an infinite-state epsilon-machine
presentation. All in all, the results lay the groundwork for analyzing
processes with infinite statistical complexity and infinite excess entropy.Comment: 18 pages, 9 figures, 1 table;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dtrp.ht
Stambaugh correlations, monkey econometricians and redundant predictors
We consider inference in a widely used predictive model in empirical finance. "Stambaugh Bias" arises when innovations to the predictor variable are correlated with those in the predictive regression. We show that high values of the "Stambaugh Correlation" will arise naturally if the predictor is actually predictively redundant, but emerged from a randomised search by data mining econometricians. For such predictors even bias-corrected conventional tests will be severely distorted. We propose tests that distinguish well between redundant predictors and the true (or "perfect") predictor. An application of our tests does not reject the null that a range of predictors of stock returns are redundant
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