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2021 | OriginalPaper | Chapter

6. Principal Oscillation Patterns and Their Extension

Author : Abdelwaheb Hannachi

Published in: Patterns Identification and Data Mining in Weather and Climate

Publisher: Springer International Publishing

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Abstract

EOF method is essentially an exploratory method to analyse the modes of variability of multivariate weather and climate data, with no model is involved. This chapter describes a different method, Principal Oscillation Pattern (POP) analysis, that seeks the simplest dynamical system that can explain the main features of the space–time data. The chapter also provides further extension of POPs by including nonlinearity. Examples from climate data are also given.

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previous chapter Complex/Hilbert EOFs
next chapter Extended EOFs and SSA
Footnotes
1
There are exceptions, and these happen when, for example, there is a pair of equal eigenvalues separated from the rest of the spectrum.
 
2
This decomposition can be sometimes problematic due to the possible existence of small eigenvalues of Σ 0. In general it is common practice to filter the data first using, for example, EOFs and keeping the leading PCs (Barnett and Preisendorfer 1987).
 
3
Note that there is no natural order for the POPs so far.
 
4
The Dirac function is defined by the property \(\int \delta _a(u) f(u) du = f(a)\), and in general, δ 0(u) is simply noted as δ(u) .
 
5
If the noise term ε t is Gaussian, then (6.15) implies that if x t is given the state vector x t+τ is also normal with mean G(τ)x t and covariance matrix σ(τ), i.e.
$$\displaystyle \begin{aligned}Pr \left( {\mathbf{x}}_{t+\tau} = \mathbf{x } | {\mathbf{x}}_t \right) = ( 2 \pi)^{-\frac{p}{2}} |\boldsymbol{\sigma}(\tau)|{}^{- \frac{1}{2}} \mbox{ exp} \left[ -\frac{1}{2} \left( \mathbf{x} - \mathbf{G} (\tau) {\mathbf{x}}_t \right)^T [\boldsymbol{\sigma}(\tau) ]^{-1} \left( \mathbf{x} - \mathbf{G} (\tau) {\mathbf{x}}_t \right) \right], \end{aligned}$$
which, under stationarity, tends to the multinormal distribution \(N \left ( \mathbf {0}, \boldsymbol {\Sigma }_0 \right )\) when τ tends to infinity.
 
6
In practice POP analysis has been applied to many time series not necessarily strictly stationary.
 
7
see von Storch et al. (1995) for other unpublished works on CPOPs.
 
8
Often taken to be the mean state or climatology, although conceptually it should be a stationary state of the system. This choice is adopted because of the difficulties in finding stationary states.
 
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Metadata
Title
Principal Oscillation Patterns and Their Extension
Author
Abdelwaheb Hannachi
Copyright Year
2021
Book
Patterns Identification and Data Mining in Weather and Climate

Print ISBN: 978-3-030-67072-6

Electronic ISBN: 978-3-030-67073-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-67073-3_6

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