The overarching goal of this thesis is to use distance covariance based methods to extend asymptotic results from the i.i.d. case to general time series settings. Accounting for dependence may make already difficult statistical inference all the more challenging. The distance covariance is an increasingly popular measure of dependence between random vectors that goes beyond linear dependence as described by correlation. It is defined by a squared integral norm of the difference between the joint and marginal characteristic functions with respect to a specific weight function. Distance covariance has the advantage of being able to detect dependence even for uncorrelated data. The energy distance is a closely related quantity that measures distance between distributions of random vectors. These statistics can be used to establish asymptotic limit theory for stationary ergodic time series. The asymptotic results are driven by the limit theory for the empirical characteristic functions. In this thesis we apply the distance covariance to three problems in time series modeling: (i) Independent Component Analysis (ICA), (ii) multivariate time series clustering, and (iii) goodness-of-fit using residuals from a fitted model. The underlying statistical procedures for each topic uses the distance covariance function as a measure of dependence. The distance covariance arises in various ways in each of these topics; one as a measure of independence among the components of a vector, second as a measure of similarity of joint distributions and, third for assessing serial dependence among the fitted residuals. In each of these cases, limit theory is established for the corresponding empirical distance covariance statistics when the data comes from a stationary ergodic time series. For Topic (i) we consider an ICA framework, which is a popular tool used for blind source separation and has found application in fields such as financial time series, signal processing, feature extraction, and brain imaging. The Structural Vector Autogregression (SVAR) model is often the basic model used for modeling macro time series. The residuals in such a model are given by e_t = A S_t, the classical ICA model. In certain applications, one of the components of S_t has infinite variance. This differs from the standard ICA model. Furthermore the e_t's are not observed directly but are only estimated from the SVAR modeling. Many of the ICA procedures require the existence of a finite second or even fourth moment. We derive consistency when using the distance covariance for measuring independence of residuals under the infinite variance case.Extensions to the ICA model with noise, which has a direct application to SVAR models when testing independence of residuals based on their estimated counterparts is also considered. In Topic (ii) we propose a novel methodology for clustering multivariate time series data using energy distance. Specifically, a dissimilarity matrix is formed using the energy distance statistic to measure separation between the finite dimensional distributions for the component time series. Once the pairwise dissimilarity matrix is calculated, a hierarchical clustering method is then applied to obtain the dendrogram. This procedure is completely nonparametric as the dissimilarities between stationary distributions are directly calculated without making any model assumptions. In order to justify this procedure, asymptotic properties of the energy distance estimates are derived for general stationary and ergodic time series. Topic (iii) considers the fundamental and often final step in time series modeling, assessing the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are inherently dependent since they are based on the same parameter estimates and thus standard tests of serial independence, such as those based on the autocorrelation function (ACF) or distance correlation function (ADCF) of the fitted residuals need to be adjusted. We apply sample splitting in the time series setting to perform tests of serial dependence of fitted residuals using the sample ACF and ADCF. Here the first f_n of the n data points in the time series are used to estimate the parameters of the model. Tests for serial independence are then based on all the n residuals. With f_n = n/2 the ACF and ADCF tests of serial independence tests often have the same limit distributions as though the underlying residuals are indeed i.i.d. That is, if the first half of the data is used to estimate the parameters and the estimated residuals are computed for the entire data set based on these parameter estimates, then the ACF and ADCF can have the same limit distributions as though the residuals were i.i.d. This procedure ameliorates the need for adjustment in the construction of confidence bounds for both the ACF and ADCF, based on the fitted residuals, in goodness-of-fit testing. We also show that if f_n < n/2 then the asymptotic distribution of the tests stochastically dominate the corresponding asymptotic distributions for the true i.i.d. noise; the stochastic order gets reversed under f_n > n/2.

Thesis (M.Statistics) North-West University, Mafikeng Campus, 2010

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.

For a company to stay profitable and be competitive, the customer satisfaction must be very high. This means that the company must provide the right item at the right place at the right time, or the customer may bring its business to the competitor. But these factors bring uncertainty for the company in the supply chain of when, what and how much of the item to produce and distribute. For reducing this uncertainty and for making better plans for future demand, some sort of forecasting method must be provided. A forecast can however be statistically based and also completed with a judgmental knowledge if the statistics are not sufficient. This thesis has been done in cooperation with the Sales and Operations (S&amp;OP) department at Sandvik Mining Rock Tools in Sandviken, where a statistical forecast is currently used in combination with manual changes from sales. The forecasts are used as base for planning inventory levels and making production plans and are created by looking at the history of sales. This is done in order to meet market expectations and continuously be in sync with market fluctuations. The purpose with this thesis has been to study the item- customer combination demand and the statistical forecasting process that is currently used at the S&amp;OP department. One problem when creating forecast is how to forecast irregular demand accurately. This thesis has therefore been examining the history of sales too see in what extent irregular demand exists and how it can be treated. The result is a basic tool for mapping customers' demand behavior, where the behavior is decomposed into average monthly demand and volatility. Another result is that history of sales can get decomposed into Volatility, Volume, Value, Number of sales and Sales interval for better analysis. These variables can also be considered whenever analyzing and forecasting irregular demand. A third result is a classification of time series working as a guideline if demand should be statistically or judgmentally forecasted or being event based. The study analyzed 36 months history of sales for 56 850 time series of item- customer specific demand. The findings were that customers should have at least one year of continuous sales before the demand can be entirely statistically forecasted. The limits for demand to even be forecasted, the history of sales should at least occur every third month in average and contain at least six sales. Then the demand is defined as irregular and the forecast method is set to judgmental forecasting, which can be forecasted using statistical methods with manual adjustments. The results showed that the class of irregular demand represents approximately 70 percent in the aspect of revenue and therefore requires attention. / För att ett företag ska kunna vara lönsamt och konkurrenskraftigt måste kundnöjdheten vara mycket hög. Detta betyder att ett företag måste kunna förse rätt produkt i rätt tid på rätt plats, annars kommer kunden troligtvis att vända sig till konkurrenten. Men dessa faktorer kommer med osäkerhet för företaget i försörjningskedjan i när, vad och hur mycket av produkten de ska producera och distribuera. För att minska osäkerheten och för att planera bättre för framtida efterfrågan, måste någon typ av prognos upprättas. En prognos kan vara baserad på statistiska metoder men också kompletterad med subjektiv marknadsinformation om statistiken inte är tillräcklig. Studien som denna rapport beskriver är gjord i samarbete med Sales och Operations- avdelning (S&amp;OP) på Sandvik Mining Rock Tools i Sandviken. Där används statistiska prognoser i kombination med manuella förändringar av säljare samt regionala planerare som bas för planering av lagernivåer och produktion. Detta gör man för att möta marknadens efterfråga och för att kontinuerligt vara uppdaterad med marknadens variationer. Syftet med detta arbete har varit att studera kunders efterfrågan av produkt- kund kombination och den metod som används vid statistiska prognoser hos S&amp;OP- avdelningen. Ett problem som finns när man vill skapa prognoser är hur man ska prognostisera oregelbunden försäljning korrekt. Detta arbete har därför analyserat historisk försäljning för att se i vilken utsträckning oregelbunden efterfrågan finns och hur den kan hanteras. Resultatet är ett enkelt verktyg för att kunna kartlägga kunders köpbeteende. Ett till resultat är att historisk försäljning kan bli uppdelat i Volatilitet, Volym, Värde, Antalet köptillfällen och Tidsintervallet mellan köptillfällena. Dessa variabler kan även tas till hänsyn när man analyserar och prognostiserar oregelbunden försäljning. Ett tredje resultat är en klassificering av tidsserier som kan fungera som riktmärken om efterfrågan ska vara statistisk eller manuellt prognostiserade eller inte bör ha en prognos över huvud taget. Denna studie analyserade 36 månaders historik för 56 850 tidsserier av försäljning per produkt- kund kombination. Resultaten var att en kund bör ha åtminstone ett år av kontinuerlig efterfrågan innan man kan ha en prognos med statistiska modeller. Gränsen för att ens ha en prognos är att efterfrågan bör återkomma var tredje månad i genomsnitt och ha en historik av åtminstone sex försäljningstillfällen. Då klassificeras efterfrågan som oregelbunden och prognosen kan vara baserad på statistiska metoder men med manuella ändringar. I resultatet framkom det att oregelbunden efterfrågan representerar cirka 70 procent i avseende på intäkter och kräver således mycket uppmärksamhet.

Dissertation (PhD)--University of Stellenbosch, 2002 / ENGLISH ABSTRACT: Singular spectrum analysis (SSA) originated in the field of Physics. The technique is non-parametric by nature and inter alia finds application in atmospheric sciences, signal processing and recently in financial markets. The technique can handle a very broad class of time series that can contain combinations of complex periodicities, polynomial or exponential trend. Forecasting techniques are reviewed in this study, and a new coordinate free joint-horizon k-period-ahead forecasting formulation is derived. The study also considers model selection in SSA, from which it become apparent that forward validation results in more stable model selection. The roots of SSA are outlined and distributional assumptions of signal senes are considered ab initio. Pitfalls that arise in the multivariate statistical theory are identified. Different approaches of recurrent one-period-ahead forecasting are then reviewed. The forecasting approaches are all supplied in algorithmic form to ensure effortless adaptation to computer programs. Theoretical considerations, underlying the forecasting algorithms, are also considered. A new coordinate free joint-horizon kperiod- ahead forecasting formulation is derived and also adapted for the multichannel SSA case. Different model selection techniques are then considered. The use of scree-diagrams, phase space portraits, percentage variation explained by eigenvectors, cross and forward validation are considered in detail. The non-parametric nature of SSA essentially results in the use of non-parametric model selection techniques. Finally, the study also considers a commercial software package that is available and compares it with Fortran code, which was developed as part of the study. / AFRIKAANSE OPSOMMING: Singulier spektraalanalise (SSA) het sy oorsprong in die Fisika. Die tegniek is nieparametries van aard en vind toepassing in velde soos atmosferiese wetenskappe, seinprossesering en onlangs in finansiële markte. Die tegniek kan 'n wye verskeidenheid tydreekse hanteer wat kombinasies van komplekse periodisiteite, polinomiese- en eksponensiële tendense insluit. Vooruitskattingstegnieke word ook in hierdie studie beskou, en 'n nuwe koërdinaatvrye gesamentlike horison k-periodevooruitskattingformulering word afgelei. Die studie beskou ook model seleksie in SSA, waaruit duidelik blyk dat voorwaartse validasie meer stabiele model seleksie tot gevolg het. Die agtergrond van SSA word ab initio geskets en verdelingsaannames van seinreekse beskou. Probleemgevalle wat voorkom in die meervoudige statistiese teorie word duidelik geïdentifiseer. Verskeie tegnieke van herhalende toepassing van een-periode-vooruitskatting word daarna beskou. Die benaderings tot vooruitskatting word in algororitmiese formaat verskaf wat die aanpassing na rekenaarprogrammering vergemaklik. Teoretiese vraagstukke, onderliggend aan die vooruitskattings-algortimes, word ook beskou. 'n Nuwe koërdinaatvrye gesamentlike horison k-periode-vooruitskattingsformulering word afgelei en aangepas vir die multikanaal SSA geval. Verskillende model seleksie tegnieke is ook beskou. Die gebruik van "scree"- diagramme, fase ruimte diagramme, persentasie variasie verklaar deur eievektore, kruis- en voorwaartse validasie word ook aangespreek. Die nie-parametriese aard van SSA noop die gebruik van nie-parametriese model seleksie tegnieke. Die studie vergelyk laastens 'n kommersiële sagtewarepakket met die Fortran bronkode wat as deel van hierdie studie ontwikkel is.

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / Environmental Management / Master / Master of Science in Environmental Management

published_or_final_version / abstract / Statistics and Actuarial Science / Master / Master of Philosophy