Metody pro periodické a nepravidelné časové řady

Title: Methods for periodic and irregular time series Author: Mgr. Tomáš Hanzák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Tomáš Cipra, DrSc. Abstract: The thesis primarily deals with modifications of exponential smoothing type methods for univariate time series with periodicity and/or certain types of irregularities. A modified Holt method for irregular times series robust to the problem of "time-close" observations is suggested. The general concept of seasonality modeling is introduced into Holt-Winters method including a linear interpolation of seasonal indices and usage of trigonometric functions as special cases (the both methods are applicable for irregular observations). The DLS estimation of linear trend with seasonal dummies is investigated and compared with the additive Holt-Winters method. An autocorrelated term is introduced as an additional component in the time series decomposition. The suggested methods are compared with the classical ones using real data examples and/or simulation studies. Keywords: Discounted Least Squares, Exponential smoothing, Holt-Winters method, Irregular observations, Time series periodicityNázev práce: Metody pro periodické a nepravidelné časové řady Autor: Mgr. Tomáš Hanzák Katedra: Katedra pravděpodobnosti a matematické statistiky Vedoucí disertační práce: Prof. RNDr. Tomáš Cipra, DrSc. Abstrakt: Disertační práce se primárně zabývá modifikacemi metod typu exponenciální vyrovnávání pro jednorozměrné časové řady s periodicitou a/nebo určitými typy nepravidelností. Je navržena modifikovaná Holtova metoda pro nepravidelné časové řady robustní vůči problému "časově blízkých" pozorování. Obecný koncept modelování sezónnosti je zaveden do Holtovy-Wintersovy metody včetně lineární interpolace sezónních indexů a použití goniometrických funkcí jako speciálních případů (obě metody jsou použitelné pro nepravidelná pozorování). Je zkoumán DLS odhad regrese s lineárním trendem a sezónními indexy a metoda je porovnána s aditivní Holtovou-Wintersovou metodou. Autokorelovaný člen je navržen jako další složka dekompozice časové řady. Navržené metody jsou porovnávány s klasickými na reálných datech a/nebo prostřednictvím simulačních studií. Klíčová slova: Diskontované nejmenší čtverce, exponenciální vyrovnávání, Holtova-Wintersova metoda, nepravidelná pozorování, periodicita časových řadKatedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Exponential smoothing for irregular time series

summary:The paper deals with extensions of exponential smoothing type methods for univariate time series with irregular observations. An alternative method to Wright’s modification of simple exponential smoothing based on the corresponding ARIMA process is suggested. Exponential smoothing of order m for irregular data is derived. A similar method using a DLS **discounted least squares** estimation of polynomial trend of order m is derived as well. Maximum likelihood parameters estimation for forecasting methods in irregular time series is suggested. The suggested methods are compared with the existing ones in a simulation numerical study

Methods for periodic and irregular time series

Title: Methods for periodic and irregular time series Author: Mgr. Tomáš Hanzák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Tomáš Cipra, DrSc. Abstract: The thesis primarily deals with modifications of exponential smoothing type methods for univariate time series with periodicity and/or certain types of irregularities. A modified Holt method for irregular times series robust to the problem of "time-close" observations is suggested. The general concept of seasonality modeling is introduced into Holt-Winters method including a linear interpolation of seasonal indices and usage of trigonometric functions as special cases (the both methods are applicable for irregular observations). The DLS estimation of linear trend with seasonal dummies is investigated and compared with the additive Holt-Winters method. An autocorrelated term is introduced as an additional component in the time series decomposition. The suggested methods are compared with the classical ones using real data examples and/or simulation studies. Keywords: Discounted Least Squares, Exponential smoothing, Holt-Winters method, Irregular observations, Time series periodicit

Methods for periodic and irregular time series

Title: Methods for periodic and irregular time series Author: Mgr. Tomáš Hanzák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Tomáš Cipra, DrSc. Abstract: The thesis primarily deals with modifications of exponential smoothing type methods for univariate time series with periodicity and/or certain types of irregularities. A modified Holt method for irregular times series robust to the problem of "time-close" observations is suggested. The general concept of seasonality modeling is introduced into Holt-Winters method including a linear interpolation of seasonal indices and usage of trigonometric functions as special cases (the both methods are applicable for irregular observations). The DLS estimation of linear trend with seasonal dummies is investigated and compared with the additive Holt-Winters method. An autocorrelated term is introduced as an additional component in the time series decomposition. The suggested methods are compared with the classical ones using real data examples and/or simulation studies. Keywords: Discounted Least Squares, Exponential smoothing, Holt-Winters method, Irregular observations, Time series periodicit

Decomposition methods for time series with irregular observations

This work deals with extensions of classical exponential smoothing type methods for univariate time series with irregular observations. Extensions of simple exponential smoothing, Holt method, Holt-Winters method and double exponential smoothing which have been developed in past are presented. An alternative method to Wright's modification of simple exponential smoothing for irregular data, based on the corresponding ARIMA process, is suggested. Exponential smoothing of order m for irregular data as a generalization of simple and double exponential smoothing is derived. A similar method using a DLS (discounted least squares) estimation of polynomial trend of order m is derived as well. In all cases the recursive character of these methods is preserved making them easy to implement and high computationally effective. A program in which most of the methods presented here are available is a part of the work. Some numerical examples of their application are also included

Exponential smoothing for time series with outliers

summary:Recursive time series methods are very popular due to their numerical simplicity. Their theoretical background is usually based on Kalman filtering in state space models (mostly in dynamic linear systems). However, in time series practice one must face frequently to outlying values (outliers), which require applying special methods of robust statistics. In the paper a simple robustification of Kalman filter is suggested using a simple truncation of the recursive residuals. Then this concept is applied mainly to various types of exponential smoothing (recursive estimation in Box-Jenkins models with outliers is also mentioned). The methods are demonstrated using simulated data

Seigniorage in continuous time

The government is able to acquire real goods through printing money. Because government does not create wealth through printing money, this revenue, the seigniorage, is at the expense of the public, as the purchasing power of monetary units decreases because of the issue of new money. The authors use the model of auctions to which the public comes with their money and the government with the newly issued money. The value of goods acquired by the government in such an auction equals the newly printed money divided by the sum of the newly printed money and the money spent by the public. Upon this auction model, the authors develop the formula for seigniorage in continuous time. The seigniorage calculated in this way is lower than the seigniorage calculated upon the assumption of discrete changes in economic variables.seigniorage, inflation tax, money issue