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Discrete Dynamics in Nature and Society, Vol. 1, pp. 255-264Reprints available directly from the publisherPhotocopying permitted by license only

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Predictability Problems of Global Change as Seenthrough Natural Systems Complexity Description

2. Approach

VLADIMIR V. KOZODEROV a,., VICTOR A. SADOVNICHII b, SERGEY A. USHAKOV b

and OLEG A. TIMOSHIN b

Institute of Computational Mathematics, Russian Academy of Sciences, Gubkine Street 8, 117333 Moscow, Russia,"b M.V. Lomonosov Moscow State University, 119899 Moscow, Vorobiovy Gory, MSU, Russia

(Received in finalform 2 June 1997)

Developing the general statements of the proposed global change theory, outlined inPart 1 of the publication, Kolmogorov’s probability space is used to study properties ofinformation measures (unconditional, joint and conditional entropies, information diver-gence, mutual information, etc.). Sets of elementary events, the specified algebra of theirsub-sets and probability measures for the algebra are composite parts of the space. Theinformation measures are analyzed using the mathematical expectance operator andthe adequacy between an additive function of sets and their equivalents in the form ofthe measures. As a result, explanations are given to multispectral satellite imageryvisualization procedures using Markov’s chains of random variables represented by pixelsof the imagery. The proposed formalism of the information measures application enablesto describe the natural targets complexity by syntactically governing probabilities.Asserted as that of signal]noise ratios finding for anomalies of natural processes, thepredictability problem is solved by analyses of temporal data sets of related measure-ments for key regions and their background within contextually coherent structures ofnatural targets and between particular boundaries of the structures.

Keywords: Probability space, Information measures, Complexity

General statements of research programmes, con-cerning global change issues, were outlined inPart of the publication. The statements wereconsidered as composite parts of possible discretedynamics application to study global change bythe induced representations about information

Corresponding author.

255

sub-spaces taking stringent terms of sets, mea-sures and metrics (SMM) into account. Orderand chaos categories in dynamic systems weredescribed to develop conceptual models of globalanalysis, interpretation and modelling using themajor framework concerning the SMM categories.

256 V.V. KOZODEROV et al.

Correct and incorrect problems were mathe-matically set up to find ways of their solutionsexistence, uniqueness and stability relative to dis-turbances of initial data, given by data of remotesensing measurements.Below we present our approach to employ in-

formation measures and entropy metrics for de-scribing the complexity of natural targets andstructures construction to find ordering procedureswhile processing multispectral satellite imagery ofthe targets/structures. This will be needed to cometo analysis of temporal data sets of the imagerythat should approach us to understanding thepredictability problems of global change.

KOLMOGOROV’S PROBABILITY SPACE

We shall operate in our considerations with thechaos and order characteristics of the conceptionof Kolmogorov’s probability space. This space isdefined when the following three categories areaccepted known (f,)c, #), f is a set of elemen-tary events with their composite elementswhich are only considered in the classical prob-ability theory; 9c is a special algebra, called as

a-algebra of the f sub-sets, that is associatedwith the a-algebra of the Az: events; the latter aredefined provided both their conjunction (sum)UAk and cross-section (product) NkA exist inthe infinite sequence of the events; # is a prob-ability measure on the 9c algebra.Random variables X with their particular mean-

ings x on a finite set 32 can be then defined as a

result of the following transformation X: f 32,so that X-(x)E " for all x E 32. The probabilityof such an event in terms of these random vari-ables is the #-measure of a corresponding sub-setA of the $2 set, i.e.

any random variable X with its meanings on the32 and any distribution P on 32 x y (this notationmeans the Cartesian product of the infinite sets),the striction of which on the 32 is coincident withthe given Px distribution, the Y random variableexists such as Pxr--P. This supposition is sure tobe valid if f is the unit interval (0, 1), U is a fam-ily of its Borel’s sub-sets in the Euclidean spaceand # is the Lebesgue measure (Kolmogorov andFomin, 1976).

Let us identify sets of all probability distribu-tions on the finite set 32 as sub-sets of then= 1321-measure Euclidean space that consists inall vectors with components Pk>0 such as

kPz:= 1. Linear combinations and convexityare then understood in accordance with the sup-position. For instance, the convexity of a realfunction f(P) from the probability distributionon 32 means that

f(cP, + (1 c)P2) _< cf(P1) + (1 c)f(P2)

for any distributions P1, P2 and c (0, 1).It is possible to use topological terms for prob-

ability distributions on 32 assuming that they arerelated to a metrical topology characterized byEuclidean distance. In particular, the convergencePn - P implies that Pn(x) --+ P(x) for any x 32.We can introduce after these notations and

definitions information measures having in mindtheir relevance to the well-known probabilitytheory, from one side, and remote sensing dataapplications, from the other side. Of particularinterest here for us are the forthcomingmeasures: unconditional, joint and conditionalentropies, information divergence, mutual infor-

mation, etc.

P{X A} =/({cv: X(cv) A}).

We shall suppose hereafter that the major prob-ability space (f,)c, #) is "sufficiently rich" in thesense that for any pair of infinite sets 32 and y,

INFORMATION MEASURES

The scalar quantity of the amount of informationin any set of measurements is defined throughthe mathematical expectance operator E in the

PREDICTABILITY PROBLEMS: PART 2 257

following way:

) E(-logP(X)) H(P)H(X) E logp(x)P(x) log P(x).

xEX"

This is the alternative representation of the entropyof a random variable X or a probability distribu-tion Px P. The entropy is the measure of ana priori uncertainty that is contained in the vari-able X before its measuring or observing; themain property of the measure of the amount ofinformation is

0 <_ H(P) < log

Understanding of the entropy as the measure

of uncertainty about a process under study is

meant that a "more homogeneous" distributionwould possess of larger entropy values. If twodistributions P and Q are given on X, then sayingabout the above "homogeneity", we imply thatP> Q provided that for any two non-decreasingorderings Pl >_ P2 >_ >--Pn, ql >_ q2 > >_ q,(n--]A’]) of probabilities from these distributions,the following inequality is valid for any k,<k<n:

k k

i=1 i=1

so that from the condition P>Q, the otherinequality entails:

H(P) > H(Q).

The information divergence, that is connectedwith statistical hypotheses testing, is the measure

of differing between distributions P and Q and is

also given by the E operator:

D(P, Q) E logQ(x)J P(x) logQ(x)

The conditional entropy is the measure ofadditional amount of information that containsin the random variable Y, if X has already beenknown, and is expressed through the joint entropyof pair of the variables and the unconditionalentropy:

H(YI X) H(X, Y)- H(X).

Due to the definition

PxI(x Y)Pxy(x,y)

for any Px(x)> 0, the conditional entropy can bealso written as:

H( Y X) Px(x)H( Y X x),x ,.V

where

H(YIX- x) Pylx(ylx)logPylx(ylx),y3

i.e. properties of the information categories en-

able to express the conditional entropy H(YIXas the mathematical expectance of the entropy ofthe conditional distribution Y under the condi-tion X x.

Mutual information of the X and Y variables

I(X A Y) H( Y) H( Y X) H(X) H(X Y)

serves as the measure of a stochastic dependencebetween these variables. We use the other letterfor the measure notation (I instead of H in allother cases) just to follow traditional principlesto do that. In particular, the formula

I(X A X) H(X)

expresses the amount of information that is

contained in X relative to its own.


Other information measures of the type

/-/(x z z)

ZP{Y-TIZ-z}H(XIY-y’Z-z)’yEY

I(X A YIZ)P{Z z)I(X/ YIZ z),...,

zEZ

are often studied in various theoretical ap-proaches (Csiszar and Korner, 1981).

It can be found that all the listed and anyother information measures have the followinggeneral properties:

(1) are non-negative;(2) are additive, i.e.

H(X, Y) H(X) + H( Y X),H(X, YIX) H(X[Z) + H(YIX, Z),...;

(3) satisfy the "chain rules" for sequences of ran-dom variables:

k

H(X,, X) H(X X, X_, ),i=1

k

I(X1, ,Xk A Y) Z I(Xi A Y IXI,...,Xi_I),...i=1

(4) H(P) is the concave function of P andD(P, Q) is the convex function of the pair(P, (2), i.e. if

P(x) aP1 (x) + (1 a)P2(x)

and

Q(x) aQ1 (x) + (1 a)Qz(x)for anyxE,’and0<a< 1,

then

oH(P) + (1 -o)H(P2) <_ H(P)

and

aD(P, Q1) + (1 c)D(P2, Q2) _> D(P, Q).’

Owing to their additive properties, these infor-mation measures can be considered as formalidentities for the random variables. The adequacyhas been proven (Csiszar and Korner, 1981) toexist between these identities to be valid for anoptional additive function f and their equivalentsin the form of the information measures. Denot-ing the sign of such adequacy by e=, we canrepresent the proven facts as

H(X) = f(A), H(X, Y) =f(A t2 B),H(XI Y e: f(A\B), I(X A Y)e,f(A V B),

where U and N mean the conjunction and pro-duct of the A and B sets, respectively. In general,the theorem was proven in the cited referencethat any pair of the information measures wouldbe adequate to an expression of the followingtypef((A Cq B)\C) with the sign "backslash" denot-ing the sets disjunction, where A, B, C are infi-nite conjunctions of sets (supposed that A and Bare not empty, C may be empty). And vice versa:any expression of the same type is adequate tothe related information measure.As a result of the facts, the following quantity

(A\B) (B\A) A/B, called as the symmetricaldifference between sets A and B (Kolmogorovand Fomin, 1976), can be used in the studies.This quantity is a metrics of sub-sets A and B onthe initial f set of Kolmogorov’s space. Thefunctionf((A\B) LJ (B\A)) does not have any directanalog in the theory of information. However, itis the metrics for the quantity

d(X, Y) H(xI Y) %- H( Y X)

on the random variables space. The said can beconvinced by realizing that the metrics propertiesare correspondent to those given by initial ax-ioms of metrical spaces:

d(X, Y) >_ O, d(X,X)=O,d(X, Y) d( Y, X),d(X, Y)+ d(Y,Z) > d(X, Z).


It is not difficult to find that the informationmeasures are continuous relative to the entropymetrics:

]H(X1) H(X2) M(X1, X2),]H(X IV1) H(X2 Y2)] _< d(X, X2) + d( Y, Y2),

II(X1 A Y1) I(X2 A Y2)I -< a(x1, Y1) + a(x2, Y2).

However, the information divergence D(P, Q) can-not serve as the measure that satisfies the Euclideandistance requirements, on the probability distribu-tion space since it is not symmetrical. Even the"symmetrized" divergence J(P, Q) D(P, Q) +D(Q, P) is not the distance once such probabilitydistributions P1, P2, P3 can be found, for whichsimultaneously the following inequalities are valid:

D(P1, P2) + D(P2, P3) < D(P, P3),D(P,P) + D(P2, P) < D(P3, P1).

Following the listed results, we can study prop-erties of the information sub-spaces as imbeddedinto the main probability space that comprisesrandom variables. This would require additionalexplanations to consider the sub-spaces relativeto the distribution space because of the aboveconcavity of the entropy and the convexity of theinformation divergence in the space. In fact, an

opportunity is emerged in the first case to inventa unified description of different data sets repre-sentation for selected classes of natural targetsusing their transformed images, given by remotesensing measurements. The description is basedon the SMM categories giving rise to imageryvisualization procedures, which are usually impliedwhile saying about the thematical interpretationof the images in a particular subject area. Theseprocedures enable to find an analog to the sub-jective analysis of single satellite pictures by eyesof an experienced interpretor when an analyzedpicture is displayed on the computer screen or asa hard copy. The rigorous definition of such visual-ization that also includes multispectral analysis,practically not accessible for the subjective inter-pretation, would originate from the SMM consid-erations in the metrical information sub-spaces.

IMAGERY VISUALIZATION

Of particular importance for an objective inter-pretation of natural targets variability on theirspace imagery are Markov’s chains as an effec-tive tool to identify "a recipe" of pixels orderingwithin a spatial structure. The above mutual in-formation is the quantity that is the most profit-able for an analysis of alterations on sets of pixelsto be considered as sequences of these randomvariables. In accordance with its definition, a finiteor infinite sequence of variables X1, X2,... withfinal sets of their values is called Markov’s chain(Pougachev, 1979) if for any the variable Xi+lis conditionally independent on (X1,...,X;_I)relative to Xi. The latter notation is common-used in the information theory: if informationmeasures are dependent on a set of random vari-ables and these variables can be represented bythe only symbol, the set is written as an argumentwithout any parentheses. The parentheses areused to emphasize mutual information betweenthe variables. Random variables X, X2,... wouldgenerate a conditional Markov’s chain relative tothe variable Y if for any the variable Xi+l isconditionally independent on (Xi,...Xi_) pro-vided (X;, Y). Both types of the chains serve tofind elements of ordering on the images.

Since according to the definition of mutualinformation I(XA Y) 0 if variables X and Y areindependent, and I(XA Y Z) 0 if X and Y are

conditionally independent variables relative to Z,then it can be stated that elements (pixels) of a

multispectral image represented by X,X2,...would make up Markov’s chain there and onlythere where

I(X1, Xi_ / Xi+ll Xi) 0

for any i. The similar form of ascertaining theconditional Markov’s chain relative to Y looks as

I(X1, Xi_ Y) o

Assuming random variables X_l, X;, X+ asthree levels of delineating pixels of one spectral


band for an image while identifying rules of deci-sion making as to the separability of the pixelsand variable(s) Y in the above sense as related tothe second band of the image, the search of pixelsof its structure, for which the last "chain rule" isvalid, would represent the essence of the visuali-zation procedure for the two-band image. Findingof contextually coherent structures (Kozoderov,1997) of natural targets and accounting for therelated measures of the targets complexity de-scription would be the result of these rules appli-cation. Considering random variables X1,...,Xi,

Xi+ 1, Y as characteristics of a particular spectralband, the number of which is / 2, this search ofthe ordering measures using the chain ruleswould serve to elucidate the optimal selection ofthe number of bands and the efficiency of therelevant instruments called imaging spectrometers(Mission to Planet Earth, 1996). Both these aspectsof the information measures applicability areneeded to be realized in constructing new versionsof special computer languages.

NATURAL TARGETS COMPLEXITY

The most pattern recognition and scene analysistechniques that would present the scientific basisfor multispectral imagery processing are dividedinto two groups: one of them is tackled from thedecision making position (Tou and Gonzalez,1978) and the second is considered within thesyntax approach (Fu, 1977). Natural objects(specific targets) are characterized by sets of num-bers in the first case. These numbers are digitalequivalents of results of remote sensing measure-ments. Pattern recognition as a procedure of attrib-uting of each pixel on an image to some classesis carried out in this case by sub-dividing theentire space of characteristic features on selectedareas to be delineated by sets of such rather stan-dard procedures. Classes are to be defined inaccordance with the probability distributionfunctions for sets of pixels on the scene underprocessing. It is required in the second case (the

structural description of each pattern) that therecognition procedure would enable not only totake an object to a particular class, but to de-scribe those peculiarities of the object, whichwould exclude its taking to any other class.

Developing the known metrical pattern recogni-tion theory (Grenander, 1976; 1978), we can extendthe definition of pixels in the techniques of thefirst case to elements x E A" in the second case.The recognition of the images in the second caseis based on an analogy between "the structuralpatterns" (hierarchical or in a tree form) and thesyntax of a computer language. The recognitionin this case is in a syntax analysis of "a grammati-cal sentence" that describes a concrete scene underanalysis. This scene is reflected by sets of variousobjects to be quantitatively described by the infor-mation measures. Such elements that can be calledas generators are natural to be used for con-structing configurations. It means that inducing a

group of transformations on the set A’, a set ofobjects to be recognized is divided on classes oftheir equivalency. The configurations are deter-mined by the composition and structure of theirgenerators and by the combinatorial theory ofthe configurations construction on particularimagery of natural objects to be analyzed by theproposed treatments.

If it is possible to assume a structural combi-nation of the generators into configurations, thenthese combined objects being characterized bythe composition of bonds between the elementsand by their own structures are initial to studynew classes of the metrical images. The directproblem of studying processes of the images for-mation through mathematical operations of com-bination, identification and deformation isusually called the imagery synthesis whilst theinverse problem of selecting particular configura-tions on the images is called the imagery analysis(Grenander, 1978).Denoting by a system of rules or restrictions

that are to define what configurations are regular,we can write the following symbolic expressionfor a computer language representation on a set


of such regular configuration Q(7-.):

(a, s,

where G is the generators set, S is the transfor-mation set of the generators, N is the type of thebonds for the taken sets of generators, p is theratio of consistency between the possible bondsin their structural connections. These bonds andconnections may serve in the first approximationas a measure of the .structure complexity. Morecomprehensive definition of the complexity in thematrix form will be given below. The regularityof these configurations on particular imagery issupposed in the studies as their consisting in spe-cific structural connections not purely randomfor the elements; otherwise, no opportunity couldbe found in traditional supervising procedures ofpattern recognition techniques and all attemptsto create "an artificial intelligence" by findingregular rules of the element connections wouldbe ambiguous.

There is the theorem (Grenander, 1976) thatthe tree type bonds and the equality for theratio of consistency p induce the above Markov’sproperties of the probability measures on sets oftheir regular configurations. These sets are under-stood in the sense of the existence of a topology7- on Kolmogorov’s initial space (on the f set)when any system of sub-sets F should satisfy thefollowing requirements (Kolmogorov and Fomin,1976):

(i) the set f and the empty set belong to 7-;

(ii) the sum UF of any finite or infinite setand the cross-section nk_lFk of any finitenumbers of such sets from 7- belong to thetopology.

Three known axioms of separability are valid forsuch topological spaces T=(f,7-) (Sadovnichii,1979):

(1) neighbourhood O(x) of a point x, not contain-ing another point y, and neighbourhoodO(y) of the point y, not containing the pointx, exist for any two points of the T space;

(2) any two points x and y of the T space havedisjoint neighbourhoods O(x) and O(y) (theknown Hausdorff’s axiom);

(3) any point and any closed set, not containingthe point, have disjoint neighbourhoods.

The above regularity of the configurative proba-bility space is accepted by us as satisfying theaxioms (1) and (3). Having these rules in mind,the syntactically governing probabilities can bedefined as

> Pr > O, Z Pr for any E N,rET

where r is altered from to p, N is the numberof elements of the syntactical variables.Now it is possible to introduce the complexity

matrix for the grammatical rules of the descrip-tion of the bonds for the regular configurations:

M-- {rnij: i, j 1,2,..., v},

where mij rEiPrn.i(r); 7i is the set of the per-mutation rules for the variable c with the/-index;n(r), n2(r),..., nv(r) are numbers of the appearanceofthe first, second,..., vth syntactical variables, thetotal number ofwhich is v for a testing grammatics.

Recalling that the entropy is a measure of anyordering, we can use it to write the following ex-

pression for the syntactically governing probabili-ties Pr in the attempts to order the grammaticrules:

hi h(i) Pr logpr,Ti

i= 1,2,...,v.

The entropy of a style of imagery descriptioncan then be introduced as:

tI- H()- Z p/(B) log P(B),B

where probabilities Pj. are to be known for anypossible chain J E Q(74.) for sets of outputs whilefitting the style of the imagery description in thetree form. B denotes that using the proposed


conception of generators, bonds and sets of regularconfigurations for the information measures, therules of forming the description style of the takencomputer language are to be selected only from theinduced syntactically governing probabilities.

Returning to the above information measuresrelation

H(X, Y) H(X) + H( Y IX)

and considering X as a random variable for a

resulting grammatics and Y as that for the styleof its description, one can obtain

Hi=hi+

or in the matrix form

H-[I- M]-h,

where the matrix H of ordering of the styledescription is expressed through the matrix h ofordering of the syntax of imagery by the inversematrix that is equal to the difference between theunit matrix I and the complexity matrix M. Thisgives a rule for testing a special language of thestructural imagery description (Kozoderov, 1997).To sum up the results, we can say about the

applications of the related languages for testingthem while describing the contextually coherentstructures mentioned above. It is worthwhile toevolve the techniques for computer work stationsto proceed from these improvements to the finalstage of the multispectral satellite imagery analysis.This stage is concerned the signal/noise ratios ex-traction from temporal sets of the consequentimages with the structures, predescribed in accor-dance with the given complexity procedures.Analyzing temporal sets of satellite imagery, givenby multispectral radiometers of different spatialresolutions, and describing the imagery structuresby the proposed techniques, we are able to pro-ceed from the regional structures description forselected natural targets to their global changes.

Let us add a few words about the complexitycategory.

Discussing the incorrect problems of data setsinterpretation in Part of the paper, we used thecomplexity functionals in the regularization tech-niques. These functionals were applied for findingthe models, which satisfying to the general forma-lism of solutions of the inverse problems wouldbe of "the minimal complexity". The last term wasutilized there to select those specific models fromsets of similar other models, which would becomparable with accuracies of data of the obser-vations that were to be fitted to theoretical resultsof modelling. More complicated models in thissense could be less consistent with the relevant setof observations than these models of the minimalcomplexity.

Saying about rules and restrictions in thestructural conjunction of the proposed "standardblocks" (generators) here, we have introduced thestructural complexity of configurations, whichare regular in the sense of how one set of these"details" could be imbedded into the other of thehigher level construction. We can use, if necessary,the definition of "the quantitative complexity" ofa configuration, just simply counting the numberof generators in the configuration. By the generalexpression of the complexity in the matrix formabove, we determined grammatic rules of thecomplexity description by using the syntax andstyle of the language in the analysis of the struc-tures on multispectral satellite images for selectedclasses of natural targets. The term "complexityof terrestrial ecosystems" in the InternationalGlobal Change and Terrestrial Ecosystems(GCTE) project (Kozoderov, 1995) is in factidentical to the biological diversity. Our inten-tions are to describe changes in terrestrial ecosys-tems by the overall natural systems complexitymatrix while using regular observations of thesystems. Biodiversity changes would inevitablyresult in the observable changes on remote sens-ing images. Thus, we do have an opportunity tofilter out all seasonal harmonics of vegetationgrowth on the images and deal with "signals" of


their possible change by the proposed belowapplication of discrete dynamics techniques.

PREDICTABILITY OF GLOBAL CHANGE

The scientific basis for solving the predictabilityproblems is given by the cross-correlation tech-niques to find asynchronous correlations ofanomalies of the fields under study (outgoinglong-wave radiation, the biomass amount of vege-tation, etc.). These correlations are represented inthe following form of the signal/noise ratio fortwo autoregression Markov’s processes of thefirst order (Marchuk et al., 1990):

a(7-) SD(Rxr(7-))

Here

Rxg(7-)n- 7-k=

are cross-correlations of the anomalies (devia-tions from the quadric standard) of the studiedquantities for n observations and different shiftswith time 7-, SD(Cxy) is the standard deviationof observation covariances for two intervals on

time,

Cxr Rxr(7-)Sxy

is the sampling estimate of the cross-covariancefor the two discrete processes X and Y,

(Xk (Yk r)Sxg-k=l

’T 1Xk, r---Hk= Hk=

The index T characterizes the averaging procedurefor the two data sets.The analytical solution for the problem with

two Markov’s random processes is known as (see

the cited reference)

2 ) 1/2SD(Cxy) CrV n(1 -exp[-(Ax + Ay)])

where

is the covariance of the processes under twoshifts 7-1 and 7-2, Nxy is the effective number ofpairs of the processes under the shifts,

7xx(7-) oS(X)exp(-xT-),,),gg(7-) @(Y) exp(-)yT-);

@(X) and aF2(Y) are dispersions of the analyzedfields for different periods of time.The statistical confidence of the mutual corre-

lation coefficients from the formulae on the 95%of the confidence level is given by

Rxg(O) > 2SD(Cxy)/OF.

The predictability of the process X(t) via Y(t),both are represented in the discrete form withtime t, is then determined by the expression

Pxg(7-)axy(7-)

/2Rxy(7-) (Nxy(1 exp[-(Ax +

The statistical significance of the anomalies ofthe fields under study is defined here by the num-ber of independent samples that is equal to

NxyRxx(k)Rgg(k)

In the final run, the predictability problem isreduced to finding the cross-correlations between"particular points" of multispectral remote


sensing imagery and their background. Selectionof these points and analysis of the structures,that comprise the points, is the subject of theabove visualization techniques. The next stage ofimagery processing is to retrieve state parametersof natural objects, classified in accordance withroutine pattern recognition and scene analysistechniques using relevant imagery transformationprocedures (Curran et al., 1990). The final stageis in the temporal data sets analysis that wouldenable to understand the predictability problemin the way presented here. All these stages ofinformation and mathematical applications fordata sets of satellite imagery interpretation arean example of advances in the multidisciplinarydescription of natural processes.

CONCLUSION

The information and mathematical aspects ofglobal change, presented in the publication, aredemonstrated by the unified approach how to

compare sets of data in information sub-spacesand to understand predictive capabilities in solv-ing the problem. In spite of the formal informa-tion theory does not enable to solve all problemsof satellite imagery interpretation, we have elabo-rated techniques to describe the complexity problemof natural structures using the mathematicalformalism of tackling with sets of data, informa-tion measures and entropy metrics. We employthe known axiomatics of Kolmogorov’s probabil-ity space to emphasize the discrepancy of our

approach with tendencies of pure numerical appli-cations in current international scientific pro-grammes of global change. Our studies are designedto remove deficiencies given by the common-usedinformation theory, which are due to the assump-tion that a structure under study is finite. Ourimprovements of the classical theory gets possibleowing to the proven possibilities to unify differ-ent data of observations in terms of sets, mea-sures and metrics. The opportunity to accountfor scientifically the comparability, ordering and

calculation measures enables us to extend existingknowledge about the natural structures descrip-tion. Adherent to updated views on order, chaosand similar other categories in natural dynamicsystems, we have shown how these categories arerepresented in metrical spaces for our purpose tofind a regularity on structures of natural targetsrepresented in information sub-spaces by thesetargets imagery. Our main intention in future isto elucidate the problem of "genesis" of informa-tion and situations using the unified descriptionof natural processes by the proposed way. Thus,we are approaching the understanding of globalchange based on major achievements in informa-tion and mathematical sciences.

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