School of Mathematics and Statistics - Theses
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ItemTransforms and truncations of time series( 2015)A time series can be defined as a collection of random variables indexed according to the order they are obtained in time. Examples of time series are monthly Australian retail sales, or quarterly GDP data. Forecasting of time series is generally considered much more important than fitting. Models that use exponential smoothing methods have been found to perform well on time series. Chapter 2 describes the estimation and forecasting procedure of additive forms of time series models; these include the local level model, local trend model, damped trend model, and seasonal equivalents. This chapter also briefly discusses some other time series methods, and introduces the M3-competition data that is extensively used in this thesis. Models that include multiplicative components for time series are considered in Chapter 3, increasing the total number of possible models from 6 to 30. While multiplicative models are often better than purely additive models, model selection methods using all combinations of multiplicative and additive models are found to be no better statistically than just selecting using the purely additive models; model selection methods are confused by the large number of possible models. In this thesis, transforms and truncations are used with exponential smoothing, in the quest for better forecasts of time series. Two types of transforms are explored: those applied directly to a time series; and those applied indirectly, to the prediction errors. The various transforms are tested on a large number of time series from the M3-competition data, and analysis of variance (ANOVA) is applied to the results. We find that the non-transformed time series is significantly worse than some transforms on the monthly data, and on a distribution-based performance measure for both annual and quarterly data. To try to understand why the transforms perform as they do, a simulation study was carried out, using simulations from a paper on outliers. Three types of simulations were used: a Level Shift permanently shifts the series to a new level; an Additive Outlier increases the series for only one time period; and a Transitory Change gradually reverts the series to the old level after the jump point. The non-transformed time series were significantly worse than some transforms on some simulation types. Truncations are applied so that there is no possibility of obtaining an observation below zero on a positive-definite time series. There are two types of truncations: those applied only to the forecasts, and those applied to the fits and forecasts. By using the same methods as for the transforms, we found that the truncations worked better when applied only to the forecasts, but the non-truncated model was never significantly worse than any truncation. Chapter 7 combines transforms with truncations. We find that applying the heteroscedastic state space transform with a truncated normal significantly improved forecasts over the non-transformed results. The final chapter of this thesis investigates how various properties of time series affect the forecasting performance. Of particular interest is the finding that a measure commonly used to assess prediction performance is flawed.