756 research outputs found
Information Bottlenecks, Causal States, and Statistical Relevance Bases: How to Represent Relevant Information in Memoryless Transduction
Discovering relevant, but possibly hidden, variables is a key step in
constructing useful and predictive theories about the natural world. This brief
note explains the connections between three approaches to this problem: the
recently introduced information-bottleneck method, the computational mechanics
approach to inferring optimal models, and Salmon's statistical relevance basis.Comment: 3 pages, no figures, submitted to PRE as a "brief report". Revision:
added an acknowledgements section originally omitted by a LaTeX bu
Statistical Complexity of Simple 1D Spin Systems
We present exact results for two complementary measures of spatial structure
generated by 1D spin systems with finite-range interactions. The first, excess
entropy, measures the apparent spatial memory stored in configurations. The
second, statistical complexity, measures the amount of memory needed to
optimally predict the chain of spin values. These statistics capture distinct
properties and are different from existing thermodynamic quantities.Comment: 4 pages with 2 eps Figures. Uses RevTeX macros. Also available at
http://www.santafe.edu/projects/CompMech/papers/CompMechCommun.htm
Structure and Randomness of Continuous-Time Discrete-Event Processes
Loosely speaking, the Shannon entropy rate is used to gauge a stochastic
process' intrinsic randomness; the statistical complexity gives the cost of
predicting the process. We calculate, for the first time, the entropy rate and
statistical complexity of stochastic processes generated by finite unifilar
hidden semi-Markov models---memoryful, state-dependent versions of renewal
processes. Calculating these quantities requires introducing novel mathematical
objects ({\epsilon}-machines of hidden semi-Markov processes) and new
information-theoretic methods to stochastic processes.Comment: 10 pages, 2 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ctdep.ht
Reductions of Hidden Information Sources
In all but special circumstances, measurements of time-dependent processes
reflect internal structures and correlations only indirectly. Building
predictive models of such hidden information sources requires discovering, in
some way, the internal states and mechanisms. Unfortunately, there are often
many possible models that are observationally equivalent. Here we show that the
situation is not as arbitrary as one would think. We show that generators of
hidden stochastic processes can be reduced to a minimal form and compare this
reduced representation to that provided by computational mechanics--the
epsilon-machine. On the way to developing deeper, measure-theoretic foundations
for the latter, we introduce a new two-step reduction process. The first step
(internal-event reduction) produces the smallest observationally equivalent
sigma-algebra and the second (internal-state reduction) removes sigma-algebra
components that are redundant for optimal prediction. For several classes of
stochastic dynamical systems these reductions produce representations that are
equivalent to epsilon-machines.Comment: 12 pages, 4 figures; 30 citations; Updates at
http://www.santafe.edu/~cm
Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models
We recently derived analytical expressions for the pairwise (auto)correlation
functions (CFs) between modular layers (MLs) in close-packed structures (CPSs)
for the wide class of stacking processes describable as hidden Markov models
(HMMs) [Riechers \etal, (2014), Acta Crystallogr.~A, XX 000-000]. We now use
these results to calculate diffraction patterns (DPs) directly from HMMs,
discovering that the relationship between the HMMs and DPs is both simple and
fundamental in nature. We show that in the limit of large crystals, the DP is a
function of parameters that specify the HMM. We give three elementary but
important examples that demonstrate this result, deriving expressions for the
DP of CPSs stacked (i) independently, (ii) as infinite-Markov-order randomly
faulted 2H and 3C stacking structures over the entire range of growth and
deformation faulting probabilities, and (iii) as a HMM that models
Shockley-Frank stacking faults in 6H-SiC. While applied here to planar faulting
in CPSs, extending the methods and results to planar disorder in other layered
materials is straightforward. In this way, we effectively solve the broad
problem of calculating a DP---either analytically or numerically---for any
stacking structure---ordered or disordered---where the stacking process can be
expressed as a HMM.Comment: 18 pages, 6 figures, 3 tables;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dplcps.ht
Occam's Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel
A stochastic process's statistical complexity stands out as a fundamental
property: the minimum information required to synchronize one process generator
to another. How much information is required, though, when synchronizing over a
quantum channel? Recent work demonstrated that representing causal similarity
as quantum state-indistinguishability provides a quantum advantage. We
generalize this to synchronization and offer a sequence of constructions that
exploit extended causal structures, finding substantial increase of the quantum
advantage. We demonstrate that maximum compression is determined by the
process's cryptic order---a classical, topological property closely allied to
Markov order, itself a measure of historical dependence. We introduce an
efficient algorithm that computes the quantum advantage and close noting that
the advantage comes at a cost---one trades off prediction for generation
complexity.Comment: 10 pages, 6 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/oqs.ht
Islands in the Gap: Intertwined Transport and Localization in Structurally Complex Materials
Localized waves in disordered one-dimensional materials have been studied for
decades, including white-noise and correlated disorder, as well as
quasi-periodic disorder. How these wave phenomena relate to those in
crystalline (periodic ordered) materials---arguably the better understood
setting---has been a mystery ever since Anderson discovered disorder-induced
localization. Nonetheless, together these revolutionized materials science and
technology and led to new physics far beyond the solid state. We introduce a
broad family of structurally complex materials---chaotic crystals---that
interpolate between these organizational extremes---systematically spanning
periodic structures and random disorder. Within the family one can tune the
degree of disorder to sweep through an intermediate structurally disordered
region between two periodic lattices. This reveals new transport and
localization phenomena reflected in a rich array of energy-dependent
localization degree and density of states. In particular, strong localization
is observed even with a very low degree of disorder. Moreover, markedly
enhanced localization and delocalization coexist in a very narrow range of
energies. Most notably, beyond the simply smoothed bands found in previous
disorder studies, islands of transport emerge in band gaps and sharp band
boundaries persist in the presence of substantial disorder. Finally, the family
of materials comes with rather direct specifications of how to assemble the
requisite material organizations.Comment: 7 pages, 3 figures, supplementary material;
http://csc.ucdavis.edu/~cmg/compmech/pubs/talisdm.ht
Inferring Pattern and Disorder in Close-Packed Structures from X-ray Diffraction Studies, Part II: Structure and Intrinsic Computation in Zinc Sulphide
In the previous paper of this series [D. P. Varn, G. S. Canright, and J. P.
Crutchfield, Physical Review B, submitted] we detailed a
procedure--epsilon-machine spectral reconstruction--to discover and analyze
patterns and disorder in close-packed structures as revealed in x-ray
diffraction spectra. We argued that this computational mechanics approach is
more general than the current alternative theory, the fault model, and that it
provides a unique characterization of the disorder present. We demonstrated the
efficacy of computational mechanics on four prototype spectra, finding that it
was able to recover a statistical description of the underlying modular-layer
stacking using epsilon-machine representations. Here we use this procedure to
analyze structure and disorder in four previously published zinc sulphide
diffraction spectra. We selected zinc sulphide not only for the theoretical
interest this material has attracted in an effort to develop an understanding
of polytypism, but also because it displays solid-state phase transitions and
experimental data is available.Comment: 15 pages, 14 figures, 4 tables, 57 citations;
http://www.santafe.edu/projects/CompMech/papers/ipdcpsii.htm
Inferring Pattern and Disorder in Close-Packed Structures from X-ray Diffraction Studies, Part I: epsilon-Machine Spectral Reconstruction Theory
In a recent publication [D. P. Varn, G. S. Canright, and J. P. Crutchfield,
Phys. Rev. B {\bf 66}:17, 156 (2002)] we introduced a new technique for
discovering and describing planar disorder in close-packed structures (CPSs)
directly from their diffraction spectra. Here we provide the theoretical
development behind those results, adapting computational mechanics to describe
one-dimensional structure in materials. By way of contrast, we give a detailed
analysis of the current alternative approach, the fault model (FM), and offer
several criticisms. We then demonstrate that the computational mechanics
description of the stacking sequence--in the form of an
epsilon-machine--provides the minimal and unique description of the crystal,
whether ordered, disordered, or some combination. We find that we can detect
and describe any amount of disorder, as well as materials that are mixtures of
various kinds of crystalline structure. Underlying this approach is a novel
method for epsilon-machine reconstruction that uses correlation functions
estimated from diffraction spectra, rather than sequences of microscopic
configurations, as is typically used in other domains. The result is that the
methods developed here can be adapted to a wide range of experimental systems
in which spectroscopic data is available.Comment: 26 pages, 23 figures, 8 tables, 110 citations;
http://www.santafe.edu/projects/CompMech/papers/ipdcpsi.htm
- …