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2021 | OriginalPaper | Chapter

15. Methods for Coupled Patterns

Author : Abdelwaheb Hannachi

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

Publisher: Springer International Publishing

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Abstract

Previous chapters focussed mostly on single fields, such as EOFs of seal level pressure. This chapter is an extension of previous methods. It describes different methods that mostly deal with two fields to identify coupled patterns that covary coherently. The chapter discusses both the conventional and regularised problems. It also explores the predictive power of coupled pattern analysis.

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previous chapter Functional and Regularised EOFs
next chapter Further Topics
Footnotes
1
It is possible to apply these techniques to two combined fields, e.g. SLP and SST, by combining them into a single space–time field. In this way the method does not explicitly take into account the co-variability of both fields.
 
2
If Σ xx = U ΛU T, where U is orthogonal, then one can define this square root by \(\boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}} = \mathbf {U} \boldsymbol {\Lambda }^{\frac {1}{2}}\). In this case we have \(\boldsymbol {\Sigma }_{\mathbf {xx}}= \boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}} \left [ \boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}}\right ]^T\). Note that this square root is not symmetric. A symmetric square root can be defined by \(\boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}} = \mathbf {U} \boldsymbol {\Lambda }^{\frac {1}{2}} {\mathbf {U}}^T\), in which case \(\boldsymbol {\Sigma }_{\mathbf {xx}} = \boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}} \boldsymbol {\Sigma }_{\mathbf {xx}}^{\frac {1}{2}}\), and hence the square root of Σ xx is not unique.
 
3
Generalised inverse can be used as an alternative, see e.g. Khatri (1976), but as pointed out by Bretherton et al. (1992) the results in this case will be difficult to interpret.
 
4
Ridge regression is closely related to Tikhonov regularisation in Hilbert spaces. Tikhonov regularisation consists of finding an approximate solution to an “ill-posed” problem, Af = u, by solving instead a “regularised” problem, (A + λ I) = u. This yields the approximate solution \(\hat {\mathbf {f}} = ( {\mathbf {A}}^* \mathbf {A} + \lambda \mathbf {I} )^{-1} {\mathbf {A}}^* \mathbf {u}\), which is obtained using “penalised least squares” by minimising ∥Af −u∥2 + λ∥f∥2. The matrix A ∗ is the adjoint of A.
 
5
This is like removing the ensemble mean of each field from each curve. Note that ′t′ here plays the role of variables in the discrete case and the index k refers to observations or realisation.
 
6
In the standard notation of stochastic processes x(t) may be better noted as x(t, ω), where ω refers to the random part. That is, for fixed ω, i.e. ω = ω 0 (a realisation), we get a usual function x(t, ω 0) of t, and for fixed t, i.e. t = t 0, we get a random variable x(t 0, ω).
 
7
This is a formal differentiation noted as δa and operates as in the usual case. Note that the differential δa is also a function of the same type.
 
8
If μ is fixed to 1, (15.31) becomes of first kind.
 
9
The residuals here do not have the same meaning as those used to construct EOFs via minimisation. Here instead, these residuals are used as an approximation to compute the “mis-fit”.
 
10
In fact, the absolute value of the elements of the matrix.
 
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Metadata
Title
Methods for Coupled Patterns
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_15

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