This dissertation addresses the delicate problem of establishing the statistical mechanical foundation of complex processes. These processes are characterized by a delicate balance of randomness and order, and a correct paradigm for them seems to be the concept of sporadic randomness. First of all, we have studied if it is possible to establish a foundation of these processes on the basis of a generalized version of thermodynamics, of non-extensive nature. A detailed account of this attempt is reported in Ignaccolo and Grigolini (2001), which shows that this approach leads to inconsistencies. It is shown that there is no need to generalize the Kolmogorov-Sinai entropy by means of a non-extensive indicator, and that the anomaly of these processes does not rest on their non-extensive nature, but rather in the fact that the process of transition from dynamics to thermodynamics, this being still extensive, occurs in an exceptionally extended time scale. Even, when the invariant distribution exists, the time necessary to reach the thermodynamic scaling regime is infinite. In the case where no invariant distribution exists, the complex system lives forever in a condition intermediate between dynamics and thermodynamics. This discovery has made it possible to create a new method of analysis of non-stationary time series which is currently applied to problems of sociological and physiological interest.

Due to the rapid improvements in the information technology, high dimensional time series datasets are frequently encountered in a variety of fields such as macroeconomics, finance, neuroscience, and meteorology. Some examples in economics and finance include forecasting low frequency macroeconomic indicators, such as GDP or inflation rate, or financial asset returns using a large number of macroeconomic and financial time series and their lags as possible covariates. In these settings, the number of candidate predictors (pT) can be much larger than the number of samples (T), and accurate estimation and prediction is made possible by relying on some form of dimension reduction. Given this ubiquity of time series data, it is surprising that few works on high dimensional statistics discuss the time series setting, and even fewer works have developed methods which utilize the unique features of time series data. This chapter consists of three chapters, and each one is self contained. The first chapter deals with high dimensional predictive regressions which are widely used in economics and finance. However, the theory and methodology is mainly developed assuming that the model is stationary with time invariant parameters. This is at odds with the prevalent evidence for parameter instability in economic time series. To remedy this, we present two L2 boosting algorithms for estimating high dimensional models in which the coefficients are modeled as functions evolving smoothly over time and the predictors are locally stationary. The first method uses componentwise local constant estimators as base learner, while the second relies on componentwise local linear estimators. We establish consistency of both methods, and address the practical issues of choosing the bandwidth for the base learners and the number of boosting iterations. In an extensive application to macroeconomic forecasting with many potential predictors, we find that the benefits to modeling time variation are substantial and are present across a wide range of economic series. Furthermore, these benefits increase with the forecast horizon and with the length of the time series available for estimation. This chapter is jointly written with Serena Ng. The second chapter deals with high dimensional non-linear time series models, and deals with the topic of variable screening/targeting predictors. Rather than assume a specific parametric model a priori, this chapter introduces several model free screening methods based on the partial distance correlation and developed specifically to deal with time dependent data. Methods are developed both for univariate models, such as nonlinear autoregressive models with exogenous predictors (NARX), and multivariate models such as linear or nonlinear VAR models. Sure screening properties are proved for our methods, which depend on the moment conditions, and the strength of dependence in the response and covariate processes, amongst other factors. Finite sample performance of our methods is shown through extensive simulation studies, and we show the effectiveness of our algorithms at forecasting US market returns. This chapter is jointly written with Yang Feng. The third chapter deals with variable selection for high dimensional linear stationary time series models. This chapter analyzes the theoretical properties of Sure Independence Screening (SIS), and its two stage combination with the adaptive Lasso, for high dimensional linear models with dependent and/or heavy tailed covariates and errors. We also introduce a generalized least squares screening (GLSS) procedure which utilizes the serial correlation present in the data. By utilizing this serial correlation when estimating our marginal effects, GLSS is shown to outperform SIS in many cases. For both procedures we prove two stage variable selection consistency when combined with the adaptive Lasso.

published_or_final_version / Statistics / Master / Master of Social Sciences

The research is based on financial time series modelling with special application / Thesis (M.Sc.) - University of KwaZulu-Natal, Pietermaritzburg, 2009.

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&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&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&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&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.

published_or_final_version / Civil and Structural 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