Contribution to the theory of random environment integer-valued autoregressive processes: doctoral dissertation

Abstract

This dissertation has 2 basic goals. The first goal is to construct the random environment INAR time series that can take both positive and negative values. The realization of such goal would create new possibilities in integer-valued data modeling. In addition, since the environment state estimation of each individual realization is a crucial step in real-life data modeling using models in random environment, the goal is to adapt existing clustering techniques in order to make the environment state estimates more accurate. Both goals, if realized, would represent an original authorial contribution to the integervalued time series analysis. The dissertation contains 4 chapters. Chapter 1 is the introductory one and provides a historical overview of the INAR models development. Also, this chapter offers important theorems and distributions known from before, necessary to adduce proofs in subsequent chapters. Relying on results given in [15], Chapter 2 discusses possibilities of extracting and predicting latent components of the true INAR time series with skewed Skellam marginal distribution. In Chapter 3, a construction of the new non-stationary random environment INAR model with values over entire Z is given. Unknown model parameters are estimated using adapted estimation techniques. The efficiency of estimates is tested on simulated data. A quality of the introduced model is examined on appropriate real-life data. In Chapter 4, the K-means clustering technique adaptation is provided, in order to make it suitable for estimating environment states of realizations corresponding to the generalized random environment INAR time series. The adaptation efficiency is tested on simulated and real-life data and compared to clustering results obtained using standard K-means.

Ova disertacija ima 2 cilja. Najpre, cilj disertacije je konstrukcija novih INAR vremenskih serija u sluˇcajnoj okolini koji mogu uzeti kako pozitivne, tako i negativne vrednosti. Uspeˇsna realizacija ovog cilja donela bi nove mogu´cnosti u modeliranju celobrojnih nizova podataka. Dodatno, kako je ocena stanja okoline svake realizacije kljuˇcni korak u modeliranju stvarnih procesa pomo´cu novouvedenih modela u sluˇcajnoj okolini, cilj disertacije je prilagod¯avanje postoje´cih metoda klasterovanja sa namerom da ocene stanja budu ˇsto preciznije. Oba navedena cilja bi, u sluˇcaju realizacije, predstavljala originalan doprinos autora analizi celobrojnih vremenskih serija. Disertacija sadrˇzi 4 glave. Glava 1 je uvodnog karaktera i daje istorijski pregled razvoja INAR modela. Takod¯e, ova glava nudi neke bitne teoreme i raspodele poznate od ranije, neophodne za izvod¯enje dokaza u narednim glavama. Oslanjaju´ci se na rezultate date u [15], u Glavi 2 su razmotrene mogu´cnosti identifikovanja i predvid¯anja latentnih komponenti INAR vremenske serije sa asimetriˇcnom Skelamovom marginalnom raspodelom. U Glavi 3 pristupa se konstrukciji novog nestacionarnog INAR modela u sluˇcajnoj okolini koji moˇze uzeti vrednosti na ˇcitavom skupu Z. Nepoznati parametri modela ocenjeni su pomo´cu prilagod¯enih tehnika ocenjivanja. Efikasnost ocena je testirana na simuliranim podacima. Kvalitet modela ispitan je na odgovaraju´cim realnim nizovima podataka. U Glavi 4 pristupa se adaptaciji K-means tehnike klasterovanja, sa ciljem da se ona prilagodi ocenjivanju stanja okoline realizacija koje odgovaraju uopˇstenoj INAR vremenskoj seriji u sluˇcajnoj okolini. Efikasnost adaptacije testirana je na simuliranim podacima i upored¯ena sa rezultatima klasterovanja dobijenim pomo´cu standardne K-means tehnike.