Rob J Hyndman (1992)  PhD thesis, The University of Melbourne.

This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes.

The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stochastic differential equations. These processes are well-understood and there is a large body of literature devoted to their study. I summarise some of the relevant material and develop some further results. In particular, I propose a new and very fast method of estimation using an approach analogous to the Yule–Walker estimates for discrete time autoregressive processes. The models so estimated may be used for preliminary analysis of the appropriate model structure and as a starting point for maximum likelihood estimation.

A natural extension of CAR processes is the class of continuous timethreshold autoregressive (CTAR) processes defined by piecewise linear stochastic differential equations. Very little attention has been given to these processes with a few isolated papers in the engineering and probability literature over the past 30 years and some recent publications by Tong and Yeung (summarised in Tong, 1990). I consider the order one case in detail and derive conditions for stationarity, equations for the stationary density, equations for the first two moments and discuss various approximating Markov chains. Higher order processes are also discussed and several approaches to estimation and forecasting are developed.

Both CAR and CTAR models are fitted to several real data sets and the results are compared. A rule-based procedure is suggested for fitting these models to a given time series.

One difficulty with using non-linear models (such as CTAR models) is that the forecast densities are not Gaussian, not symmetric and often multi-modal. Therefore, it is inappropriate to just consider the mean and standard deviation of forecasts. It is argued that for non-Gaussian forecast densities, highest density regions should be used when describing forecasts. An algorithm which enables the rapid construction of highest density regions given a probability density function is developed. The methods described are applicable to univariate data with a continuous density containing any number of local modes. The density may be known analytically or may be estimated from observations or simulations. Code for the algorithm is provided in the C language.

PDF of thesis (without figures)


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February 28th, 2017

Forecasting with temporal hierarchies

February 14th, 2017

The Australian Macro Database: An online resource for macroeconomic research in Australia

February 14th, 2017

Macroeconomic forecasting for Australia using a large number of predictors

January 13th, 2017

Visualising forecasting algorithm performance using time series instance spaces

January 1st, 2017

Associations between outdoor fungal spores and childhood and adolescent asthma hospitalisations

January 1st, 2017

Grouped functional time series forecasting: an application to age-specific mortality rates

December 7th, 2016

Exploring the influence of short-term temperature patterns on temperature-related mortality: a case-study of Melbourne, Australia

October 13th, 2016

Reconciling forecasts: the hts and thief packages

September 20th, 2016

smoothAPC package for R

September 20th, 2016

stR package for R

September 15th, 2016

Forecasting large collections of related time series

August 22nd, 2016

thief package for R

June 21st, 2016

Exploring time series collections used for forecast evaluation

May 25th, 2016

ISCRR time series workshop

May 6th, 2016

Automatic foRecasting using R

February 29th, 2016

On sampling methods for costly multi-objective black-box optimization

February 19th, 2016

Dynamic Algorithm Selection for Pareto Optimal Set Approximation

February 4th, 2016

Forecasting uncertainty in electricity smart meter data by boosting additive quantile regression

January 30th, 2016

Bayesian rank selection in multivariate regressions

January 25th, 2016

Probabilistic Energy Forecasting: Global Energy Forecasting Competition 2014 and Beyond

January 24th, 2016

Long-term forecasts of age-specific participation rates with functional data models

January 1st, 2016

Bagging exponential smoothing methods using STL decomposition and Box-Cox transformation

January 1st, 2016

Fast computation of reconciled forecasts for hierarchical and grouped time series

December 31st, 2015

Measuring forecast accuracy

November 26th, 2015

Forecasting hierarchical and grouped time series through trace minimization

November 2nd, 2015

Forecasting big time series data using R

October 7th, 2015

Optimal forecast reconciliation for big time series data

October 5th, 2015

Google workshop: Forecasting and visualizing big time series data

September 16th, 2015


August 25th, 2015

New IJF editors

August 17th, 2015

Machine learning bootcamp

August 7th, 2015

Statistical issues with using herbarium data for the estimation of invasion lag-phases

June 30th, 2015

Exploring the feature space of large collections of time series

June 26th, 2015

Exploring the boundaries of predictability: what can we forecast, and when should we give up?

June 25th, 2015

Automatic algorithms for time series forecasting

June 23rd, 2015

MEFM: An R package for long-term probabilistic forecasting of electricity demand

June 19th, 2015

Probabilistic forecasting of peak electricity demand

June 10th, 2015

Do human rhinovirus infections and food allergy modify grass pollen–induced asthma hospital admissions in children?

June 8th, 2015

STR: A Seasonal-Trend Decomposition Procedure Based on Regression

June 4th, 2015

Probabilistic time series forecasting with boosted additive models: an application to smart meter data

June 1st, 2015

Large-scale unusual time series detection

May 26th, 2015

Visualization of big time series data

May 22nd, 2015

Probabilistic forecasting of long-term peak electricity demand

April 20th, 2015

A note on the validity of cross-validation for evaluating time series prediction

April 4th, 2015

Discussion of “High-dimensional autocovariance matrices and optimal linear prediction”

April 1st, 2015

Change to the IJF editors

February 23rd, 2015

Visualization and forecasting of big time series data

January 12th, 2015

Visualizing and forecasting big time series data

December 24th, 2014

Bivariate data with ridges: two-dimensional smoothing of mortality rates

December 17th, 2014

MEFM package for R