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Estimating good discrete partitions from observed data symbolic false nearest neighbors

Estimating good discrete partitions from observed data symbolic false nearest neighbors
Estimating good discrete partitions from observed data symbolic false nearest neighbors

a r X i v :n l i n /0304054v 1 [n l i n .C D ] 26 A p r 2003

Estimating good discrete partitions from observed data:

symbolic false nearest neighbors

Matthew B.Kennel ?and Michael Buhl ?

Institute For Nonlinear Science University of California,San Diego

La Jolla,CA 92093-0402(Dated:April 2,2002)

A symbolic analysis of observed time series data requires making a discrete partition of a con-tinuous state space containing observations of the dynamics.A particular kind of partition,called “generating”,preserves all dynamical information of a deterministic map in the symbolic represen-tation,but such partitions are not obvious beyond one dimension,and existing methods to ?nd them require signi?cant knowledge of the dynamical evolution operator or the spectrum of unstable periodic orbits.We introduce a statistic and algorithm to re?ne empirical partitions for symbolic state reconstruction.This method optimizes an essential property of a generating partition:avoid-ing topological degeneracies.It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.Because of its resemblance to a geometrical statistic frequently used for reconstructing valid time-delay embeddings,we call the algorithm “symbolic false nearest neighbors”.

PACS numbers:05.45b

Why might one want to represent observed time series of dynamical systems as sequences of low-precision dis-crete symbols?In this representation,there are often in-teresting techniques—often (but not exclusively)derived from information theory and its associated technology—which may illuminate data in novel ways [1].The ini-tial step for all these methods requires making a par-tition :a coloring of the state space [2],x ∈R d ,into non-overlapping regions and associated symbols so that any x is assigned a unique symbol s in a discrete alpha-bet.The symbol may be represented as an integer in the set 0,1,...A ?1.A partition P de?nes a discretization of the observed sequence x i ,i =1...N into a symbolic sequence,s i ,i =1...N .

What partitions are “good”?Which discretizations re-tain the full structure of the original dynamics in the x space in the sequence of symbols?Unfortunately,the situation is unlike the remarkable time-delay embedding method for continuous dynamics:simple partitions are not generically satisfactory.The mathematics of sym-bolic dynamics speci?es what we want:a “generating partition”(GP),where symbolic orbits uniquely iden-tify one continuous space orbit,and thus the symbolic dynamics is fully equivalent to the continuous space dy-namics.

Unfortunately there is no satisfactory mathematical theory about how to ?nd a GP as a general procedure (ex-cept for one dimensional dynamics (d =1),where parti-tioning at the critical points works).Are the ad-hoc par-titions often used still satisfactory?Unfortunately they are often not so.Bollt et al [3]examined the degrada-tion in the symbolic dynamics which results from the fre-quently used “histogram partition”,as opposed to a GP.A less optimal partition will induce improper projections or degeneracies,where a given symbolic segment may

correspond to more than one topologically distinct state space orbit.This resulted in ?nding the wrong topolog-ical entropy.Chaotic communication with symbolic tar-geting works most satisfactorily knowing a GP,because then the transmitted symbolic message may be directly mapped into a desired orbit in the attractor.(see,e.g.[7])

Since a partition is a critical ?rst step for any sym-bolic data analysis,a poor partition yield poor results,a method to approximate good partitions from observed data alone is urgently needed.With apparently no ex-isting satisfactory solutions,this is the problem we at-tack.Davidchack et al [4]recently presented a parti-tioning method which works by successively coloring un-stable periodic orbits (UPOs)to ensure unique codings (all UPOs have unique codes under a GP).The neces-sary high-order UPOs are very di?cult to obtain from observed data alone,unfortunately.

The Kolmogorov-Sinai entropy rate h KS of the dynam-ics can be found from the supremum,over all increas-ingly ?ne partitions,of Shannon’s entropy rate evaluated on the information source implied by the discretization.More strikingly,a GP also achieves this supremum with a ?nite,and,one hopes,small alphabet.This suggests a naive strategy whereby one maximizes a statistical es-timator of the entropy rate evaluated on the sequence induced by candidate partitions [5].This apparently at-tractive idea is ?awed as demonstrated by the follow-ing counterexample.Consider a partition of the state space with a ?ne box-size ?where each region is ran-domly assigned a symbol from the alphabet.For su?-ciently small ?,the symbol sequence of any ?nite time series will appear indistinguishable from a memoryless and structureless information source with the maximum possible entropy rate h =log 2A ,since each observed da-

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tum could have encountered a di?erent partition element with a new random symbol..With the highest entropy possible this partition would be selected over competitors but is clearly useless for data analysis as the resulting symbolic stream says nothing about the original time se-ries or its particular dynamics.Even if one believes this pathology to be irrelevant to coarser encodings,there are practical problems with the maximum-estimated-entropy idea.First,estimation of h is not trivial to do well;sec-ond,when there is observational noise(inevitable with data acquisition equipment)larger alphabets will appear to give signi?cantly higher entropies even if they are not actually much better at encoding the dynamics.As the true entropy rate of the system is not already known(and often a key quantity one wants to estimate given a good partition),there is no absolute statistical target which con?rms whether the proposed partition is at all close or far from the ideal.In practice,selecting partitions with entropy does not seem to work well in general.

We assert our practical criterion for a good partition: short sequences of consecutive symbols ought to localize the corresponding continuous state space point as well as possible.A good coding ought to maintain the bene?ts of a low-precision symbolic representation with minimum distortion of the original state space dynamics.Our cen-tral idea is to form a particular geometrical embedding of the symbolic sequence under the candidate partition and evaluate,and minimize,a statistic which quanti?es the apparent errors in localizing state space points.

We embed the symbol sequence into the unit square[6]: y i= k max k=1s i?(k?1)/A k,k max k=1s i+k/A k .(1)

(k max is chosen such that A?k max is as small as the com-putational precision.)For a binary alphabet(A=2),the ?rst coordinate of y i is the binary fraction whose digits start at s i and go backwards in time,the second is with the sequence going forward from s i+1.Intuitively,the distribution on y is like a P-dependent symbolic version of the invariant measure.

Given x i and a partition P,the symbolic embed-ding(1)yields a parallel series y i,de?ning points on some map y=φP(x).We wantφP to be injective,i.e.φP(x)=φP(x′)implies x=x′.With?nite data,we de-sire that if||φP(x)?φP(x′)||is small,so is||x?x′||.By construction,su?ciently near points in x have close sym-bolic sequences in their most signi?cant digits.In a good partition,additionally,nearby points in y remain close when mapped back into the x-space.By contrast,bad partitions induce topological degeneracies where similar symbolic words map back to globally distant regions of state space.As shown in[3],this phenomenon confounds proper analysis of the observed symbolic dynamics.

We need to quantify how well any candidate par-tition achieves our ideal.We?nd the nearest neigh-bor,in Euclidean distance,to each point y i.Conven-tional k-d tree algorithms[8]e?ciently provide the in-dex of the nearest neighbor to any point in a data set: N[i]=arg min k=i||y k?y i||.Knowing symbolic neigh-bors,we?nd distances of those same points back in x-space,D i=||x N[i]?x i||.We normalize the set of D i by a monotonic transformation:given any D,?nd its rank R∈[0,1]in the cumulative distribution of random two-point distances||xα?xβ||.Large R means that localizing well in symbol space did not localize well in the original state space.

Better partitions give a smaller proportion of sym-bolic false nearest neighbors,that fraction of R i which are greater than some thresholdη,denoted J sfnn.This resembles the false neighbors statistic for time-delay embeddings[9]:both count large-deviation“mistakes”in a related space which result from topological mis-embedding in the tested space.Appropriate values for ηwhich de?ning a“large”deviation areη≈0.01?0.3, depending on the noise in x i.An alternative to J sfnn is K sfnn,de?ned as the arithmetic average of the largestγpercentile of the set of R https://www.wendangku.net/doc/a61445496.html,ing J sfnn,ηmay need tun-ing depending on the noise scale and dynamical system, but the e?ect of changingγis lower.On the downside, K sfnn does not necessarily converge to near zero for the optimal partition.We typically?nd good results with γ≈0.01?0.05.

For concrete numerical calculations,we need to param-eterize partitions with a relatively small number of free parameters.Inspired by[4],we de?ne partitions with re-spect to a set of radial-basis“in?uence”functions of the form f k(x)=αk/||x?z k||2,the set ofαand z being the free variables.For any particular x,one f l(x)will generi-cally result in the largest value versus other f k(x),k=l, and then x is assigned to that symbol which was pre-assigned to in?uence function f l.The z parameters are initialized to random examples from the x i andαto inde-pendent random variates[0,1),and n f functions assigned to each of the A symbols.In[4]the z k were?xed on the UPOs and their symbols varied;here,the centers and co-e?cients vary but their symbols are?xed.We minimize J sfnn or K sfnn over the An f(d+1)free parameters using “di?erential evolution”[10],a genetic algorithm suitable for continuous parameter spaces.

Figure1shows the?nal P on2000data points from the Ikeda map[11].It shows the best result(lowest J sfnn)out of six restarts changing only the random seed governing the initial conditions;the results were not much worse on the other runs,however.The result is very close to the partition knowing the dynamics.

In a stationary information source,the number of dis-tinct length-p codewords will scale,for asymptotic p,as N p∝e h T p where h T is the topological entropy,a dy-namical invariant.We validate P with an estimate of the de?ciency between h T implied by P and the correct

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FIG.1:Left:partition estimated by SFNN optimization on

2000data points from the Ikeda map.Right:partition cal-

culated with foreknowledge of UPOs,numerically extracted

from the equation of motion.The partition we estimate from

observed data alone is quite close to a presumably correct

one,calculated from the method of[4].The measure on the

two?gures is not the same:the left?gure is a sample of the

natural measure,whereas the right shows UPOs up to period

16.They avoid regions of homoclinic tangencies,contributing

to the blank spaces.

FIG.2:For each new best partition:Minimization target

J sfnn(circles,and right scale),estimated de?ciency in topo-

logical entropyδh T(asterisks and left scale)Minimizing J sfnn

generally minimizesδh T and thus maximizes topological en-

tropy of the symbolic language.

h T:δh T=p max?1 p max p=1p?1log N p/?N p .N p is the

number of distinct period-p UPOs(which were computed

knowing the equations of motion),?N p the number of such

UPOs with unique p-symbol codes in some P.A GP gives

δh T=0,andδh T→0for better(less UPO-degenerate)

partitions.Figure2showsδh T on each new best partition

found during the optimization.The optimization target,

J sfnn,decreases strictly monotonically by construction;

FIG.3:Minimizing K sfnn withγ=0.01:estimated partition

for a time series of5000data points from the Lozi map with

10%additive by amplitude Gaussian noise.Either the x1or

x2axes are GPs for the noiseless map.Here despite the noise

the algorithm?nds a partition close to what would be a GP.

FIG.4:Top:estimated binary partition for time-delay em-

bedding of inter-bubble time intervals(arbitrary units),min-

imizing K sfnn.Bottom:J sfnn(η)vsηfor the optimized par-

tition(circles)and for a naive equiprobable histogram parti-

tion with the same alphabet(asterisks).For the optimized

partition there are very few large distance errors,e.g.J sfnn

observed aboveη=0.1.

thoughδh T does not decrease quite monotonically,the

trend toward very small values is clear.This gives evi-

dence that minimizing J sfnn also re?nes approximations

to GPs.

Figures3–5demonstrate applications of the algorithm.

Fig.3shows the e?ect of noise on a system where the GP

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FIG.5:Same as Fig.4but with combustion engine heat release time series(energy,arbitrary units),and A=3.The noise level is higher thus there remain more moderately sized distances,even with a larger alphabet which usually results in better localization.

is analytically known.The Lozi map(see analysis in[6]) is similar to the Henon map but replaces the quadratic nonlinearity with a piecewise linear one.We?nd a par-tition which is close to the noise-free GP even when the data have been contaminated by signi?cant amounts of additive noise.Though complete localization to a single point is not possible here,minimizing large divergences is still a desirable criterion.Figures4to5show estimated partitions on experimental data sets where no analytical form of the equations(much less partitions)are known. On account of noise,dynamical or observational,a cer-tain amount of divergence D for symbolic nearest neigh-bors is inevitable.Still,minimizing large deviations is a reasonable goal even for noisy data.There are very few rank distances with R≥0.2or0.3compared to a basic histogram partition.

It must be kept in mind that GPs are not necessarily unique for any attractor.Distinctly di?erent partitions may be found,all of which are reasonably satisfactory. (At a minimum any iterate of a GP is also a GP).There are many coexisting solutions(roughly like?nding low-energy states of a spin glass),which is why the optimiza-tion problem is hard,requiring a global search method. We conjecture this is one reason why understanding the structure of generating partitions has been so di?cult for mathematicians.

We also point out that it is possible to partition one space x,but quantify distances D i=||z N[i]?z i||in some di?erent space as long as there is some relation between each x i and z i.For example,one may be in-terested in a simple symbolic control scheme,say where z i=x i+T(?nd the best partition of observables now that best predicts some future),or perhaps when two di?erent variables are measured simultaneously and one wants to cross-predict.In the?rst case,in principle,a GP should be optimal for predicting the future as well as the present but the inevitable issues of?nite data and noise may make the best empirical partition di?erent for the two cases.In the second case,generating partitions are irrelevant entirely.

?Electronic address:mkennel@https://www.wendangku.net/doc/a61445496.html,

?Electronic address:mbuhl@https://www.wendangku.net/doc/a61445496.html,

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(2000),R.Steuer,W.Ebeling,D.F.Russell,S.Bahar,

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E.R.Tracy,“Symbolic analysis of experimental data”,

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