Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: In this thesis we study a wavelet construction procedure based on a multiresolutional method, before specializing to the case of spline wavelets. First, we introduce and analyze the concepts of scaling functions and their duals, after which we analyze the multiresolutional analysis (MM) which they generate. The advantages of orthonormality in scaling functions are pointed out and discussed. Following the methods which were introduced in two standard texts of Chui, we next show how a minimally supported wavelet and its dual can be explicitly constructed from a given MM, thereby yielding an orthogonal decomposition of the space of square-(Lebesgue)integrable functions on the real line. We show that our method applied to orthonormal scaling functions also yields orthonormal wavelets, including as a special case the Daubechies wavelet. General decomposition and reconstruction algorithms are explicitly formulated, and the importance of the vanishing moments of a wavelet in practical applications is shown. We next introduce and analyze cardinal B-splines, in particular showing that these functions are refinable, and that they satisfy the criteria of Riesz stability. Thus the cardinal B-spline is an admissible choice for a scaling function, so that the previously developed wavelet construction procedure based on a MM yields an explicit formula for the minimally supported B-spline wavelet. The corresponding vanishing moment order is calculated, and the resulting ability of the B-spline wavelet to detect singularities in a given function is demonstrated by means of a numerical example. Finally, we develop an explicit procedure for the construction of minimally supported B-spline wavelets on a bounded interval. This method, as developed in work by de Villiers and Chui, is then compared with a previous boundary wavelet construction method introduced in work by Chui and Quak. / AFRIKAANSE OPSOMMING: In hierdie tesis bestudeer ons 'n golfie konstruksieprosedure wat gebaseer is op 'n multiresolusiemetode, voordat ons spesialiseer na die geval van latfunksie-golfies. Eerstens word die konsepte van skaalfunksies en hulle duale bekendgestel en geanaliseer, waarna ons die multiresolusie analise (MM) wat sodoende gegenereer word, analiseer. Die voordeel van ortonormaliteit by skaalfunksies word uitgewys en bespreek. Deur die metodes te volg wat bekendgestel is in twee standaardtekste van Chui, wys ons vervolgens hoe 'n minimaal-gesteunde golfie en die duaal daarvan eksplisiet gekonstrueer kan word vanuit 'n gegewe MM, en daarmee 'n ortogonale dekomposisie van die ruimte van kwadraties-(Lebesgue)integreerbare funksies op die reële lyn lewer. Ons wys dat ons metode toegepas op ortonormale skaalfunksies ook ortonormale golfies oplewer, insluitende as 'n spesiale geval die Daubechies golfie. Algemene dekomposisie en rekonstruksie algoritmes word eksplisiet geformuleer, en die belangrikheid in praktiese toepassings van 'n golfie met die nulmomenteienskap word aangetoon. Vervolgens word kardinale B-Iatfunksies bekendgestel, en word daar in die besonder aangetoon dat hierdie funksies verfynbaar is, en dat hulle aan die Rieszstabiliteit vereiste voldoen. Dus is die kardinale B-Iatfunksie 'n toelaatbare keuse vir 'n skaalfunksie, sodat die golfie konstruksieprosedure gebaseer op 'n MM, soos vantevore ontwikkel, 'n eksplisiete formule vir die minimaal-gesteunde Blatfunksiegolfie oplewer. Die ooreenkomstige nulmomentorde word bereken, en die gevolglike vermoë van 'n B-Iatfunksiegolfie om singulariteite in 'n gegewe funksie raak te sien en uit te wys word gedemonstreer deur middel van 'n numeriese voorbeeld. Laastens ontwikkelons 'n eksplis.iete prosedure vir die konstruksie van minimaal-gesteunde B-Iatfunksiegolfies op 'n begrensde interval. Hierdie metode, soos ontwikkel in werk deur de Villiers en Chui, word dan vergelyk met 'n vorige randgolfie konstruksie wat bekendgestel is in werk deur Chui en Quak.

Thesis (MSc)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: In this thesis we study the underlying mathematical principles of stationary subdivision, which can be regarded as an iterative recursion scheme for the generation of smooth curves and surfaces in computer graphics. An important tool for our work is Fourier analysis, from which we state some standard results, and give the proof of one non-standard result. Next, since cardinal spline functions have strong links with subdivision, we devote a chapter to this subject, proving also that the cardinal B-splines are refinable, and that the corresponding Euler-Frobenius polynomial has a certain zero structure which has important implications in our eventual applications. The concepts of a stationary subdivision scheme and its convergence are then introduced, with as motivating example the de Rahm-Chaikin algorithm. Standard results on convergence and regularity for the case of positive masks are quoted and graphically illustrated. Next, we introduce the concept of interpolatory stationary subdivision, in which case the limit curve contains all the original control points. We prove a certain set of sufficient conditions on the mask for convergence, at the same time also proving the existence and other salient properties of the associated refinable function. Next, we show how the analysis of a certain Bezout identity leads to the characterisation of a class of symmetric masks which satisfy the abovementioned sufficient conditions. Finally, we show that specific special cases of the Bezout identity yield convergent interpolatory symmetric subdivision schemes which are identical to choosing the corresponding mask coefficients equal to certain point evaluations of, respectively, a fundamental Lagrange interpolation polynomial and a fundamental cardinal spline interpolant. The latter procedure, which is known as the Deslauriers-Dubuc subdivision scheme in the case of a polynomial interpolant, has received attention in recent work, and our approach provides a convergence result for such schemes in a more general framework. Throughout the thesis, numerical illustrations of our results are provided by means of graphs. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons die onderliggende wiskundige beginsels van stasionêre onderverdeling, wat beskou kan word as 'n iteratiewe rekursiewe skema vir die generering van gladde krommes en oppervlakke in rekenaargrafika. 'n Belangrike stuk gereedskap vir ons werk is Fourieranalise, waaruit ons sekere standaardresuJtate formuleer, en die bewys gee van een nie-standaard resultaat. Daarna, aangesien kardinale latfunksies sterk bande het met onderverdeling, wy ons 'n hoofstuk aan hierdie onderwerp, waarin ons ook bewys dat die kardinale B-Iatfunksies verfynbaar is, en dat die ooreenkomstige Euler-Frobenius polinoom 'n sekere nulpuntstruktuur het wat belangrike implikasies het in ons uiteindelike toepassings. Die konsepte van 'n stasionêre onderverdelingskema en die konvergensie daarvan word dan bekendgestel, met as motiverende voorbeeld die de Rahm-Chaikin algoritme. Standaardresultate oor konvergensie en regulariteit vir die geval van positiewe maskers word aangehaal en grafies geïllustreer. Vervolgens stelons die konsep van interpolerende stasionêre onderverdeling bekend, in welke geval die limietkromme al die oorspronklike kontrolepunte bevat. Ons bewys 'n sekere versameling van voldoende voorwaardes op die masker vir konvergensie, en bewys terselfdertyd die bestaan en ander toepaslike eienskappe van die ge-assosieerde verfynbare funksie. Daarna wys ons hoedat die analise van 'n sekere Bezout identiteit lei tot die karakterisering van 'n klas simmetriese maskers wat die bovermelde voldoende voorwaardes bevredig. Laastens wys ons dat spesifieke spesiale gevalle van die Bezout identiteit konvergente interpolerende simmetriese onderverdelingskemas lewer wat identies is daaraan om die ooreenkomstige maskerkoëffisientegelyk aan sekere puntevaluasies van, onderskeidelik, 'n fundamentele Lagrange interpolasiepolinoom en 'n kardinale latfunksie-interpolant te kies. Laasgenoemde prosedure, wat bekend staan as die Deslauriers-Dubuc onderverdelingskema in die geval van 'n polinoominterpolant, het aandag ontvang in onlangse werk, en ons benadering verskaf 'n konvergensieresultaat vir sulke skemas in 'n meer algemene raamwerk. Deurgaans in die tesis word numeriese illustrasies van ons resultate met behulp van grafieke verskaf.

Thesis (PhD)--Stellenbosch University, 1975. / ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of aerodynamics or electron optics, involve certain singular integrals which do not exist classically. The problems can, however, be solved pLovided that such integrals are interpreted as finite-part integrals. Although the concept of a finite-part integral has existed for about fifty years, it was possible to define it rigorously only by means of distribution theory, developed about twenty-five years ago. But, to the best of our knowledge, no quadrature formula for the numerical eva= luation of finite-part integrals ha~ been given in the literature. The main concern of this thesis is the study and discussion of.two kinds of quadrature formulae for evaluating finite-part integrals in= volving an algebraic singularity. Apart from a historical introduction, the first chapter contains some physical examples of finite-part integrals and their definition based on distribution theory. The second chapter treats the most im= portant properties of finite-part integrals; in particular we study their behaviour under the most common rules for ordinary integrals. In chapters three and four we derive a quadrature formula for equispaced stations and one which is optimal in the sense of the Gauss-type quadra= ture. In connection with the latter formula, we also study a new class of orthogonal polynomials. In the fifth and.last chapter we give a derivative-free error bound for the equispaced quadrature formula. The error quantities which are independent of the integrand were computed for the equispaced quadrature formula and are also given. In the case of some examples, we compare the computed error bounds with the actual errors. ~esides this theoretical investigation df finite-part integrals, we also computed - for several orders of the algebraic singularity the coefficients for both of the aforesaid quadrature formulae, in which the number of stations ranges from three up to twenty. In the case of the equispaced quadrature fortnu1a,we give the weights and - for int~ger order of the singularity - the coefficients for a numerical derivative of the integrand function. For the Gauss-type quadrature, we give the stations, the corresponding weights and the coefficients of the orthogonal polynomials. These data are being published in a separate report [18] which also contains detailed instructions on the use of the tables.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / See full text for abstract.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2007. / An exposition is given of [12], a paper by N. Krupnik, which is a discussion of the minimum number of idempotent generators of a complete matrix algebra Mn(F) over a field F, as well as direct sums of complete matrix algebras over F. It will, for example, be proved that, if n ≥ 2, then the minimum number of idempotent generators of a n × n matrix algebra is equal to 2 or 3. Krupnik made an incorrect statement in ([12], Theorem 5), namely that the minimum number of idempotent generators of m copies of an infinite field F, as an algebra over F, is m−1. This error was identified and corrected by A.V. Kelarev, A.B. van der Merwe and L. van Wyk in [11]. The thesis also includes an exposition of this correction. Furthermore an exposition will be given of the main result of [5], where E. Formanek showed that, if n ≥ 2, then there is a non-vanishing central polynomial for Mn(F), with F any field. The last mentioned result will be used in the exposition of [12].

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The high sickle cell gene frequency has been hypothesised to be related to the protective advantage against malaria disease among heterozygous individuals. In this thesis, we study the interaction between the dynamics of malaria and sickle cell gene. The main aim is to investigate the impact of malaria treatment on the frequency of sickle cell gene. For this, we develop a mathematical model that describes the interactions between malaria and sickle cell gene under malaria treatment. The model includes both homozygous for the normal gene (AA) and heterozygous for sickle cell gene (AS) and assumes that AS individuals are not treated since they do not show clinical symptoms. We first analyse the model without malaria treatment, using singular perturbation techniques, basing on the fact that epidemiological and demographical dynamics occur on two different time scales (fast and slow dynamics). Our analysis on the fast time scale shows that high sickle cell gene frequency leads to high endemic levels for longer duration of parasitemia among heterozygous individuals. However, if the duration of parasitemia is reduced then high sickle cell gene frequency is associated with low endemic levels. We also note that on the slow time scale, the invasion ability of sickle cell gene is dependent on the malaria epidemiological parameters. The invasion coefficient given as the difference in the weighted death rates of AA and AS individuals is used as a measure to determine the establishment of sickle cell gene in the population. Results show that, the gene may establish itself if the weighted death rate of AA individuals is greater than that of AS individuals otherwise it fails. We note that, high mortality of AA individuals leads to establishment of sickle cell gene in the population. Then we analysed the model with treatment, our results indicate that the frequency of sickle cell gene decreases with an increase in the recovery rate of AA individuals. We thus conclude that eradication of malaria disease will lead to a reduction in sickle cell gene frequency. / AFRIKAANSE OPSOMMING: Daar word veronderstel dat die hoë sekelsel geenfrekwensie onder heterosigotiese individue verwant is aan die beskermende voordeel teen malaria siekte. In hierdie verhandeling ondersoek ons die wisselwerking tussen die dinamika van malaria en die sekelsel geen. Die hoofdoel is om die invloed van malaria behandeling op die frekwensie van die sekelsel geen te ondersoek. Hiervoor het ons ‘n wiskundige model ontwikkel, wat die wisselwerking tussen die dinamika van malaria en die sekelsel geen met malaria behandeling, beskryf. Die model sluit beide homosigotiese vir die normale geen (AA) en heterosigotiese vir die sekelsel geen (AS) in, en neem aan dat AS individue nie behandel is nie omdat hulle nie die eerste kliniese simptome getoon het nie. Ons ontleed eers die model sonder malaria behandeling, deur gebruik te maak van enkelvoudige pertubasie tegnieke, wat gegrond is op die feit dat epidemiologiese en demografiese dinamika plaasvind op twee verskillende tydskale (vinnige en stadige dinamika). Ons ontleding op die vinnige tydskaal dui dat hoë sekelsel geenfrekwensie onder heterosigotiese individue lei tot hoë endemiese vlakke vir ‘n langer duur van parasitemie. Nietemin, as die duur van parasitemie afneem, dan word hoë sekelsel geenfrekwensie verbind met lae endemiese vlakke. Ons neem ook waar dat op die stadige skaal die indringingsvermoë van die sekelsel afhanklik is van malaria se epidemiologiese parameters. Die indringingskoëffisiënt wat bereken word as die verskil van die geweegde sterftekoerse van AA en AS individue, word gebruik as ‘n maatstaf om die vestiging van die sekelsel geen in die bevolking te bepaal. Resultate toon dat die geen homself kan vestig as die geweegde sterftekoers van AA individue groter is as di e van die AS individue, andersins misluk dit. Ons let op dat hoë mortaliteit van AA individue lei tot die vestiging van die sekelsel geen in die bevolking. Daarna het ons die model wat behandeling insluit ge-analiseer en ons resultate toon dat die frekwensie van die sekelsel geen afneem met ‘n toename in die herstelkoers van AA individue. Ons kom dus tot die gevolgtrekking dat die uitwissing van malaria siekte sal lei tot die afname in sekelsel geenfrekwensie.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis. / AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

Thesis (MSc)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: Buying and selling in financial markets are driven by demand. The demand can be quantified by the imbalance in the number of shares QB and QS transacted by buyers and sellers respectively over a given time interval t. The demand in an interval t is given by (t) = QB − QS. The local noise intensity is given by = h|aiqi − haiqii|i where i = 1, . . . ,N labels the transactions in t, qi is the number of shares traded in transaction i, ai = ±1 denotes buyer- initiated and seller- initiated trades respectively and h· · · i is the local expectation value computed from all the transactions during the interval t. In a paper [1] based on data from the New York Stock Exchange Trade and Quote database during the period 1995-1996, Plerou, Gopikrishnan and Stanley [1] reported that the analysis of the probability distribution P( | ) of demand conditioned on the local noise intensity revealed the surprising existence of a critical threshold c. For < c, the most probable value of demand is roughly zero; they interpreted this as an equilibrium phase in which neither buying nor selling predominates. For > c two most probable values emerge that are symmetrical around zero demand, corresponding to excess demand and excess supply; they interpreted this as an out-of-equilibrium phase in which the market behaviour is buying for half of the time, and selling for the other half. It was suggested [1] that the two-phase behaviour indicates a link between the dynamics of a financial market with many interacting participants and the phenomenon of phase transitions that occurs in physical systems with many interacting units. This thesis reproduces the two-phase behaviour by means of experiments using sequences of random variables. We reproduce the two-phase behaviour based on correlated and uncorrelatd data. We use a Markov modulated Bernoulli process to model the transactions and investigate a simple interpretation of the two-phase behaviour. We sample data from heavy-tailed distributions and reproduce the two-phase behaviour. Our experiments show that the results presented in [1] do not provide evidence for the presence of complex phenomena in a trading market; the results are a consequence of the sampling method employed. / AFRIKAANSE OPSOMMING: Aankope en verkope in finansi¨ele markte word deur aanvraag gedryf. Aanvraag kan gekwantifiseer word in terme van die ongebalanseerdheid in die getal aandele QB en QB soos onderskeidelik verhandel deur kopers en verkopers in ’n gegewe tyd-interval t. Die aanvraag in ’n interval t word gegee deur (t) = QB −QS. Die lokale geraasintensiteit word gegee deur = h|aiqi − haiqii|i waar i = 1, . . . ,N die transaksies in t benoem, qi die getal aandele verhandel in transaksies verwys, en h· · · i op die lokale verwagte waarde dui, bereken van al die tansaksies tydens die interval t. In ’n referaat [1] wat op data van die New York Effektebeurs se Trade and Quote databasis in die periode tussen 1995 en 1996 geskoei was, het Plerou, Gopikrishnan en Stanley [1] gerapporteer dat ’n analise van die waarskynlikheidsverspreiding P( | ) van aanvraag gekondisioneer op die lokale geraasintensiteit , die verrassende bestaan van ’n kritieke drempelwaarde c na vore bring. Vir < c is die mees waarskynlike aanvraagwaarde nagenoeg nul; hulle het dit ge¨ınterpreteer as ’n ekwilibriumfase waartydens n`og aankope n`og verkope die oormag het. Vir > c is die twee mees waarskynlike aanvraagwaardes wat te voorskyn kom simmetries rondom nul aanvraag, wat oorenstem met ’n oormaat aanvraag en ’n oormaat aanbod; hulle het dit geinterpreteer as ’n buite-ewewigfase waartydens die markgedrag die helfte van die tyd koop en die anderhelfte verkoop. Daar is voorgestel [1] dat die tweefase gedrag op ’n verband tussen die dinamiek van ’n finansiele mark met baie deelnemende partye, en die verskynsel van fase-oorgange wat in fisieke sisteme met baie wisselwerkende eenhede voorkom, dui. Hierdie tesis reproduseer die tweefase gedrag deur middel van eksperimente wat gebruik maak van reekse van lukrake veranderlikes. Ons reproduseer die tweefase gedrag gebaseer op gekorreleerde en ongekorreleerde data. Ons gebruik ’n Markov-gemoduleerde Bernoulli proses om die transaksies te moduleer en ondersoek ’n eenvoudige interpretasie van die tweefase gedrag. Ons seem steekproefdata van “heavy-tailed” verspreidings en reproduseer die tweefase gedrag. Ons ekperimente wys dat die resultate in [1] voorgested is nie bewys lewer vir die teenwoordigheid van komplekse verskynsel in’n handelsmark nie; die resultate is as gevolg van die metode wat gebruik is vir die generering van die steekproefdata.

Thesis (MSc (Mathematics))--University of Stellenboswch, 2010. / ENGLISH ABSTRACT: In the work of Green and Matignon it was shown that the Oort-Sekiguchi conjecture is equivalent to a local question of lifting automorphisms of power series rings. The Oort-Sekiguchi conjecture asks when an algebraic curve in characteristic p can be lifted to a relative curve in characteristic 0, while keeping the same automorphism group. The local formulation asks when an automorphism of a power series ring over a field k of characteristic p can be lifted to an automorphism of a power series ring over a discrete valuation ring with residue field k of the same order as the original automorphism. This thesis looks at the local formulation and surveys many of the results for this case. At the end it presents a new theorem giving a Hensel's Lemma type sufficient condition under which lifting is possible. / AFRIKAANSE OPSOMMING: Green en Matignon het bewys dat die Oort-Sekiguchi vermoede ekwivalent is aan `n lokale vraag oor of outomorfismes van magsreeksringe gelig kan word. Die Oort-Sekiguchi vermoede vra of `n algebra ese kromme in karakteristiek p gelig kan word na `n relatiewe kromme in karakteristiek 0, terwyl dit dieselfde outomorfisme groep behou. Die lokale vraag vra wanneer `n outomorfisme van `n magsreeksring oor `n liggaam k van karakteristiek p gelig kan word na `n outomorfisme van `n magsreeksring oor `n diskrete waarderingsring met residuliggaam k, terwyl dit dieselfde orde behou as die aanvanklike outomorfisme. Hierdie tesis fokus op die lokale vraag en bied `n opsomming van baie bekende resultate vir hierdie geval. Aan die einde word `n nuwe stelling aangebied wat voorwaardes stel waaronder hierdie vraag positief beantwoord kan word.

Thesis (MSc)--Stellenbosch University, 2000. / ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, paying specific attention to the Method of Surrogate Data and its possibilities. Within the field of nonlinear modelling, we examined the following areas of study: attractor reconstruction, general model building techniques, cost functions, description length, and a specific modelling methodology. The Method of Surrogate Data was initially applied in a more conventional application, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in a less conventional application; i.e. testing the residual vectors of a nonlinear model for membership of identically and independently distributed (i.i.d) noise. The importance of the initial surrogate analysis of a time series (determining whether the apparent structure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. This study confrrmed that omitting this crucial step could lead to a flawed conclusion. If evidence of nonlinear structure in the time series was identified, a radial basis model was constructed, using sophisticated software based on a specific modelling methodology. The model is an iterative algorithm using minimum description length as the stop criterion. The residual vectors of the models generated by the algorithm, were tested for membership of the dynamic class described as i.i.d noise. The results of this surrogate analysis illustrated that, as the model captures more of the underlying dynamics of the system (description length decreases), the residual vector resembles Li.d noise. It also verified that the minimum description length criterion leads to models that capture the underlying dynamics of the time series, with the residual vector resembling Li.d noise. In the case of the "worst" model (largest description length), the residual vector could be distinguished from Li.d noise, confirming that it is not the "best" model. The residual vector of the "best" model (smallest description length), resembled Li.d noise, confirming that the minimum description length criterion selects a model that captures the underlying dynamics of the time series. These applications were illustrated through analysis and modelling of three time series: a time series generated by the Lorenz equations, a time series generated by electroencephalograhpic signal (EEG), and a series representing the percentage change in the daily closing price of the S&P500 index. / AFRIKAANSE OPSOMMING: In hierdie studie ondersoek ons nie-lineere modelleringstegnieke soos toegepas op dinamiese sisteme. Spesifieke aandag word geskenk aan die Metode van Surrogaat Data en die moontlikhede van hierdie metode. Binne die veld van nie-lineere modellering het ons die volgende terreine ondersoek: attraktor rekonstruksie, algemene modelleringstegnieke, kostefunksies, beskrywingslengte, en 'n spesifieke modelleringsalgoritme. Die Metode and Surrogaat Data is eerstens vir 'n meer algemene toepassing gebruik wat die gekose tydsreeks vir aanduidings van nie-lineere, dimanise struktuur toets. Tweedens, is dit vir 'n minder algemene toepassing gebruik wat die residuvektore van 'n nie-lineere model toets vir lidmaatskap van identiese en onafhanlike verspreide geraas. Die studie illustreer die noodsaaklikheid van die aanvanklike surrogaat analise van 'n tydsreeks, wat bepaal of die struktuur van die tydsreeks toegeskryf kan word aan nie-lineere, dalk chaotiese gedrag. Ons bevesting dat die weglating van hierdie analise tot foutiewelike resultate kan lei. Indien bewyse van nie-lineere gedrag in die tydsreeks gevind is, is 'n model van radiale basisfunksies gebou, deur gebruik te maak van gesofistikeerde programmatuur gebaseer op 'n spesifieke modelleringsmetodologie. Dit is 'n iteratiewe algoritme wat minimum beskrywingslengte as die termineringsmaatstaf gebruik. Die model se residuvektore is getoets vir lidmaatskap van die dinamiese klas wat as identiese en onafhanlike verspreide geraas bekend staan. Die studie verifieer dat die minimum beskrywingslengte as termineringsmaatstaf weI aanleiding tot modelle wat die onderliggende dinamika van die tydsreeks vasvang, met die ooreenstemmende residuvektor wat nie onderskei kan word van indentiese en onafhanklike verspreide geraas nie. In die geval van die "swakste" model (grootse beskrywingslengte), het die surrogaat analise gefaal omrede die residuvektor van indentiese en onafhanklike verspreide geraas onderskei kon word. Die residuvektor van die "beste" model (kleinste beskrywingslengte), kon nie van indentiese en onafhanklike verspreide geraas onderskei word nie en bevestig ons aanname. Hierdie toepassings is aan die hand van drie tydsreekse geillustreer: 'n tydsreeks wat deur die Lorenz vergelykings gegenereer is, 'n tydsreeks wat 'n elektroenkefalogram voorstel en derdens, 'n tydsreeks wat die persentasie verandering van die S&P500 indeks se daaglikse sluitingsprys voorstel.