In the thesis, we study the data collected from twelve-note music of Webern and Schoenberg, including opus 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 and opus 31 of Webern and opus 25, 33a and opus 37 of Schoenberg. The data consists of the following two kinds. The data of the first kind consists of the four basic forms of the twelve-tone music. And the data of the second kind consists of the twelve-tone derived from the matrix of the twelve-note music. We will introduce the twelve-note music first and then study two main topics about twelve-note music in this thesis. In the first part, we consider the Markov properties of the first kind data. We compare the sample autocorrelation function and autocorrelation function of the fitted model to determine the fitness of the Markovian model. In the second part, we build the time series model for the second kind data. Sample autocorrelation function¡Bpartial autocorrelation function and extended autocorrelation function are used to determine the orders of the models. The best model is selected based on the AICC. Finally, we check the fitness of the models using sample autocorrelation function and partial autocorrelation function of the residuals.

In this thesis, identification of periodically varying orders of univariate Periodic Autoregressive Moving-Average (PARMA) processes is mainly studied. The identification of the varying orders of PARMA process is carried out by generalizing the well-known Box-Jenkins techniques to a seasonwise manner. The identification of pure periodic moving-average (PMA) and pure periodic autoregressive (PAR) models are considered only. For PARMA model identification, the Periodic Autocorrelation Function (PeACF) and Periodic Partial Autocorrelation Function (PePACF), which play the same role as their ARMA counterparts, are employed. For parameter estimation, which is considered only to refine model identification, the conditional least squares estimation (LSE) method is used which is applicable to PAR models. Estimation becomes very complicated, difficult and may give unsatisfactory results when a moving-average (MA) component exists in the model. On account of overcoming this difficulty, seasons following PMA processes are tried to be modeled as PAR processes with reasonable orders in order to employ LSE. Diagnostic checking, through residuals of the fitted model, is also performed stating its reasons and methods. The last part of the study demonstrates application of identification techniques through analysis of two seasonal hydrologic time series, which consist of average monthly streamflows. For this purpose, computer programs were developed specially for PARMA model identification.

Στο πρώτο κεφάλαιο εισάγουμε τις βασικές έννοιες της διπλωματικής εργασίας. Αναφέρουμε τους ορισμούς και τον σκοπό της ανάλυσης χρονολογικών σειρών. Επίσης εισάγονται ορισμένα βασικά χαρακτηριστικά των χρονολογικών σειρών όπως η έννοια της στασιμότητας και της συνάρτησης αυτοσυσχέτισης και αναφέρουμε τις τρεις βασικές κατηγορίες στοχαστικών υποδειγμάτων χρονολογικών σειρών που αφορούν στις στάσιμες στοχαστικές διαδικασίες, οι οποίες θα αναλυθούν στα επόμενα κεφάλαια. Στο δεύτερο κεφάλαιο αναλύουμε τα αυτοπαλίνδρομα υποδείγματα, πρώτης, δεύτερης και γενικά p τάξης. Αναφέρονται παραδείγματα. Στο τρίτο κεφάλαιο αναλύουμε τα υποδείγματα κινητού μέσου πρώτης και γενικά q τάξης καθώς και μεικτά υποδείγματα πρώτης και γενικά (p,q) τάξης. Αναφέρονται παραδείγματα. Στο τέταρτο κεφάλαιο αναλύουμε χρονολογικές σειρές που δεν έχουν τα χαρακτηριστικά στάσιμων στοχαστικών διαδικασιών. Επίσης αναλύουμε την μεθοδολογία Box-Jenkins, η οποία είναι μία μέθοδος εξεύρεσης ενός στατιστικού υποδείγματος (ARIMA). Τέλος εφαρμόζεται η παραπάνω μέθοδος σε ένα παράδειγμα με τη χρήση του SPSS. / At the first chapter we introduce the basic concepts. We present the main definitions and the objectives of the time series analysis. Furthermore, we introduce some basic characteristics of the time series such as the concepts of “stationary process” and “autocorrelation”. Finally we mention three basic categories of time series models that concern stationary stochastic processes. Following in the second chapter we analyze the autoregressive models of first, second and generally “p” order. We present various relative examples. At the third chapter we analyze the moving average models of first and generally “q” order. Additionally, we analyze the mixed models of first and generally (p,q) order. Various relative examples are presented. Finally, at the forth chapter we analyze time series that don’t have the characteristics of stationary stochastic proceedings. Also we analyze the method Box-Jenkins. Furthermore, the later method is studied using the statistic software package SPSS.