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

Abstract:
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)

  Posts

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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”

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

October 21st, 2014

Optimally reconciling forecasts in a hierarchy

September 23rd, 2014

Forecasting: principles and practice (UWA course)

September 1st, 2014

Outdoor fungal spores are associated with child asthma hospitalisations – a case-crossover study

August 1st, 2014

Efficient identification of the Pareto optimal set

July 1st, 2014

Fast computation of reconciled forecasts in hierarchical and grouped time series

June 24th, 2014

Functional time series with applications in demography

June 17th, 2014

Challenges in forecasting peak electricity demand

June 5th, 2014

Low-dimensional decomposition, smoothing and forecasting of sparse functional data

June 5th, 2014

Fast computation of reconciled forecasts for hierarchical and grouped time series

May 30th, 2014

State space models

May 24th, 2014

Common functional principal component models for mortality forecasting

May 22nd, 2014

Monash Electricity Forecasting Model

May 8th, 2014

forecast package for R

May 2nd, 2014

“Facts” may still be artefacts, since models can make unrealistic assumptions: statistical methods for the estimation of invasion lag-phases from herbarium data

April 9th, 2014

hts package for R

April 1st, 2014

A gradient boosting approach to the Kaggle load forecasting competition

March 31st, 2014

Measuring forecast accuracy

March 21st, 2014

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

February 13th, 2014

Automatic time series forecasting

February 1st, 2014

demography: Forecasting mortality, fertility, migration and population data

January 10th, 2014

Boosting multi-step autoregressive forecasts

January 1st, 2014

Forecasting: principles and practice

January 1st, 2014

Prospective life tables

October 31st, 2013

Nonparametric and semiparametric response surface methodology: a review of designs, models and optimization techniques

October 19th, 2013

hdrcde package for R

October 11th, 2013

Coherent mortality forecasting using functional time series

October 10th, 2013

Forecasting hierarchical time series

August 29th, 2013

ftsa package for R

August 28th, 2013

rainbow package for R

August 22nd, 2013

fds package for R

July 4th, 2013

R tools for hierarchical time series

June 25th, 2013

Forecasting without forecasters

June 19th, 2013

Mcomp package for R