[Econometria (Econometrics)] Piet M.T. Broersen   Automatic Autocorrelation and Spectral Analysis
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[Econometria (Econometrics)] Piet M.T. Broersen Automatic Autocorrelation and Spectral Analysis


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Automatic Autocorrelation and Spectral Analysis
Piet M.T. Broersen
Automatic
Autocorrelation and
Spectral Analysis
With 104 Figures
123
Piet M.T. Broersen, PhD
Department of Multi Scale Physics
Delft University of Technology
Kramers Laboratory
Prins Bernhardlaan 6
2628 BW, Delft
The Netherlands
British Library Cataloguing in Publication Data
Broersen, Piet M. T.
Automatic autocorrelation and spectral analysis
1.Spectrum analysis - Statistical methods 2.Signal
processing - Statistical methods 3.Autocorrelation
(Statistics) 4.Time-series analysis
I.Title
543.5\u20190727
ISBN-13: 9781846283284
ISBN-10: 1846283280
Library of Congress Control Number: 2006922620
ISBN-10: 1-84628-328-0 e-ISBN 1-84628-329-9 Printed on acid-free paper
ISBN-13: 978-1-84628-328-4
© Springer-Verlag London Limited 2006
MATLAB® is a registered trademarkofTheMathWorks, Inc., 3AppleHillDrive,Natick,MA01760-2098,
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The publisher makes no representation, express or implied, with regard to the accuracy of the infor-
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To Rina
Preface
If different people estimate spectra from the same finite number of stationary 
stochastic observations, their results will generally not be the same. The reason is 
that several subjective decisions or choices have to be made during the current 
practice of spectral analysis, which influence the final spectral estimate. This 
applies also to the analysis of unique historical data about the atmosphere and the 
climate. That might be one of the reasons that the debate about possible climate 
changes becomes confused. The contribution of statistical signal processing can be 
that the same stationary statistical data will give the same spectral estimates for 
everybody who analyses those data. That unique solution will be acceptable only if 
it is close to the best attainable accuracy for most types of stationary data. The 
purpose of this book is to describe an automatic spectral analysis method that 
fulfills that requirement. It goes without saying that the best spectral description 
and the best autocorrelation description are strongly related because the Fourier 
transform connects them. 
Three different target groups can be distinguished for this book. 
Students in signal processing who learn how the power spectral density and the 
autocorrelation function of stochastic data can be estimated and interpreted 
with time series models. Several applications are shown. The level of mathe-
matics is appropriate for students who want to apply methods of spectral 
analysis and not to develop them. They may be confident that more thorough 
mathematical derivations can be found in the referenced literature. 
Researchers in applied fields and all practical time series analysts who can 
learn that the combination of increased computer power, robust algorithms, and 
the improved quality of order selection have created a new and automatic time 
series solution for autocorrelation and spectral estimation. The increased 
computer power gives the possibility of computing enough candidate models 
such that there will always be a suitable candidate for given data. The improved 
order-selection quality always guarantees that one of the best candidates will be 
selected automatically and often the very best. The data themselves decide 
which is their best representation, and if desired, they suggest possible alter-
natives. The automatic computer program ARMAsel provides their language. 
viii Preface 
Time series scientists who will observe that the methods and algorithms that are 
used to find a good spectral estimate are not always the methods that are 
preferred in asymptotic theory. The maximum likelihood theory especially has 
very good asymptotic theoretical properties, but the theory fails to indicate 
what sample sizes are required to benefit from those properties in practice. 
Maximum likelihood estimation often fails for moving average parameters. 
Furthermore, the most popular order-selection criterion of Akaike and the 
consistent criteria perform rather poorly in extensive Monte Carlo simulations. 
Asymptotic theory is concerned primarily with the optimal estimation of a 
single time series model with the true type and the true order, which are 
considered known. It should be a challenge to develop a sound mathematical 
background for finite-sample estimation and order selection. In finite-sample 
practice, models of different types and orders have to be computed because the 
truth is not yet known. This will always include models of too low orders, of 
too high orders, and of the wrong type. A good selection criterion has to pick 
the best model from all candidates. Good practical performance of simplified 
algorithms as a robust replacement for truly nonlinear estimation problems is 
not yet always understood. 
The time series theory in this book is limited to that part of the theory that I 
consider relevant for the user of an automatic spectral analysis method. Those 
subjects are treated that have been especially important in developing the program 
required to perform estimation automatically. The theory of time series models 
presents estimated models as a description of the autocorrelation function and the 
power spectral density of stationary stochastic data. A selection is made from the 
numerous estimation algorithms for time series models. A motivation of the choice 
of the preferred algorithms is given, often supported by simulations. For the 
description of many other methods and algorithms, references to the literature are 
given. 
The theory of windowed and tapered periodograms for spectra and lagged 
products for autocorrelation is considered critically. It is shown that those methods 
are not particularly suitable for stochastic processes and certainly not for automatic 
estimation. Their merit is primarily historical. They have been the only general, 
feasible, and practical solutions for spectral analysis for a long period until about 
2002. In the last century, computers were not fast enough to compute many time 
series models, to select only one of them, and to forget the rest of the models. 
ARMAsel has become a useful time series solution for autocorrelation and spectral 
estimation by increased computer power, together with robust algorithms and 
improved order selection. 
Piet M.T. Broersen 
November 2005
Contents 
1 Introduction ...................................................................................................... 1
 1.1 Time Series Problems................................................................................ 1 
2 Basic Concepts ................................................................................................ 11 
 2.1 Random Variables ................................................................................... 11 
 2.2 Normal Distribution ................................................................................