1
Methods in climatology
II. Multivariate analysis
Multivariate analysis
• Large datasets
• Redundant information
• Stochastic character of processes
• Signal vs noise
Analysis examples
• Identification of climate modes (NAO)
• Climate zones definition (on different scales)
• Statistical downscalling (regional climate vs large-scale atmospheric
circulation.
Main aim:
• to separate climate signal from the background climate variability (noise) and
• to identify physical processes responsible for the generation of the signal
Multivariate analysis
• Abiltiy to represent spatio-temporal data in a compresed way
Four main goals of MA in climate research:
• to recognize the patterns of climate vartiability
• to identify physical processes and use them to construct CM
• to validate climate models with observations
• to use signals for predictions
Principal Component Analysis (PCA)
Multivariate analysis examples
PC1 75.6% PC2 22.8% PC3 1.1%
PC4 0.3% PC5 0.1% PC6 0.04%
Multivariate analysis examples
Číslo Vlastní Procenta Kumulov.
PC čísla rozptylu procenta TM1 TM2 TM3 TM4 TM5 TM7
1 2262,96 75,62 75,62 0,243 0,181 0,346 0,230 0,728 0,454
2 682,34 22,80 98,42 0,115 0,050 0,229 -0,936 -0,012 0,237
3 33,80 1,13 99,55 0,553 0,323 0,513 0,201 -0,531 -0,064
4 7,79 0,26 99,81 -0,264 -0,141 -0,037 0,168 -0,432 0,833
5 4,54 0,15 99,96 0,712 -0,102 -0,668 -0,034 0,000 0,186
6 1,21 0,04 100,00 -0,212 0,911 -0,343 -0,044 -0,022 0,069
Zátěže
7655443322111 TMaTMaTMaTMaTMaTMaPC +++++=
7655443322112 TMbTMbTMbTMbTMbTMbPC +++++=
…
eigenvalue = vlastní číslo
eigenvector = vlastní vektor
zátěž = loading
2
Multivariate analysis examples
SLP patterns of winter windsorms
• 98 cases of winter windstorms from
the 20th century
• 121 grided values describing MSLP
patterns during windstorms
• objective classification using principal
components analysis
Multivariate analysis examples
Component scores of the first
component of the SLP field on
days D-5 to D calculated by
the Principal Component
Analysis Method (PCA) for 37
floods (1881-2000) of the
winter synoptic type on the
river Vltava in Prague
(brackets - explained variance
in %)
Multivariate analysis examples
May, mean SLP 1961–1990 mean SLP in 31 dry Mays since 1850 SLP anomaly (map2 – map1)
SLP at extremely dry Mays defined in the 1850–2010 period and SLP differences in extremely dry
months compared to MSLP of the 1961–1990 reference period; black points mark statistically significant
(α=0.05) SLP decrease/incerese w.r.t the 1961–1990 reference period
Fluctuations of MAM drought indices (Z-index, SPEI-1) for the Czech Lands in the 1805–2012 period.
Multivariate analysis examples
Three the most similar monthly means acording to pc1 used for interpretation of
EOF_1 mode.
Multivariate analysis examples Patterns in Space: EOF analysis
(Empirical Orthogonal Functions)
EOF
F (x,y,t)
3
Patterns in Space: EOF analysis
(Empirical Orthogonal Functions)
• DATA: Instantaneous samples (maps) of geophysical fields (air
temperature) defined in a number of points (stations or grid-points)
recorded over period of time
• EOF (PCA – Principal Component Analysis) – technique for compressing
the variability in the data set
• Introduced by Edward Lorenz in 1956
• Widely applied in climatology and oceanography
• Goal: compact description of the spatial and temporal variability of data
series in terms of orthogonal functions – statistical „modes“
• Most of variability is in the first few orthogonal functions whose
patterns MAY BE be linked to possible dynamical mechanisms
EOF – data preparation
• A set of N maps at times t = 1 …N
• Each map contains measurements of the field ψ at locations m= 1…M
• We have M time series ψm(t), each of length N
• We assume that N > M (number of time steps is larger than the number
of locations
• Annual (seasonal) cycle is necessary to remove BEFORE EOF analysis –
subtract climatological cycle from the field ψm(t).
EOF – data preparation
• Data standardization:
EOF – data preparation
We construct M x N data matrix F with M rows (locations m) nad the N
columns (times t):
Two approaches for EOFs computing
• Covariance matrix decomposition to eigenvalues and eigenvectors (rozklad
kovarianční matice na vlastní čísla a vlastní vektory)
• Singular Value Decomposition of the data matrix (singulární rozklad
matice)
Vlastní čísla a vlastní vektory matice
• Existují pro čtvercové matice, které
neobsahují lineárně závislé proměnné
• Vlastní čísla informují o variabilitě
vyčerpané vytvářenými faktorovými osami
• Vlastní číslo představuje rozptyl „nové“
proměnné definované v souřadném systému
vlastních vektorů
• Vlastní vektory definují směr nových
faktorových os v prostoru původních
proměnných
• Vlastní vektory jsou navzájem ortogonální
– tj. nezávislé – tedy každý nese unikátní
informaci
• Vlastní vektory mohou být různým
způsobem standardizovány a jejich
interpretace se liší podle použité
standardizace
e1
e2
4
Geometric interpretation of eigenvalues
and eigenvectors
An n×n matrix A multiplied by n×1 vector x results in another n×1
vector y=Ax. Thus A can be considered as a transformation matrix.
In general, a matrix acts on a vector by changing both its magnitude
and its direction. However, a matrix may act on certain vectors by
changing only their magnitude, and leaving their direction unchanged (or
possibly reversing it). These vectors are the eigenvectors of the
matrix.
A matrix acts on an eigenvector by multiplying its magnitude by a
factor, which is positive if its direction is unchanged and negative if its
direction is reversed. This factor is the eigenvalue associated with
that eigenvector.
Geometric interpretation of eigenvalues
and eigenvectors
Possible configuration of the
data vectors fn (n = 1 … N
denote the time steps) and
the empirical orthogonal
vectors em, m = 1 … M. (from
Peixoto and Oort, 1992)
EOF – The Covariance Matrix Approach
Data matrix F is used to derive spatial covariance matrix RFF of the field Fm(t)
by multiplying F by its transpose Ft:
i,j = 1 .. M
RFF is square (M x M) and symetric
EOF – The Covariance Matrix Approach
We solve the eigenproblem:
• the eigenvalues are sorted in decreasing order
• all eigenvalues are greater or equal to zero
• typically only first K eigenvalues are non-zero, K ≤ min(N,M)
• thus only K EOF modes can be determined
EOF – The Covariance Matrix Approach
We solve the eigenproblem:
• each non-zero eigenvalue λk in matrix Λ is associated with a column
eigenvector Ek in matrix E.
• only K eigenvectors are used in decomposition
• K are modes of EOF decomposition
EOF – The Covariance Matrix Approach
The eigenvector matrix E has the property that E * Et = Et * E = I,
where I is Identity matrix.
This means that the eigenvectors are uncorrelated over space –
they are orthogonal to one another
Each eigenvector Ek represents the spatial EOF pattern of mode k
The spatial EOF patterns – Loadings
5
EOF – The Covariance Matrix Approach
where Et is K x M, F is M x N, A is K x N
Rows in A are time series of length N – Principal Components (Time
coefficients, Scores)
EOF – The Covariance Matrix Approach
Each eigenvalue λk is proportional to the percentage of the variance of the field
F that is accounted for by the mode k :
The original field F can be totally reconstructed by multiplying each EOF pattern
Ek by its corresponding principal component Ak and adding the products over all K
modes:
where F is M x N, E is M x K, A is K x N
EOF – The Covariance Matrix Approach
The goal of the EOF decomposition is reconstruction of compressed and less
noisy version of the original field F
This is done by truncating the decomposition ni 2.14 eq. using only first H modes
with H < K
The H first modes account for the largest fraction of the field variance:
This leads to a significant reduction of the amount of data while retaining most
of the variance of the field F.
The choice of H may be rather subjective
The first or the few first EOF modes sometimes represent meaningful physical
processes
The Singular Value Decomposition Approach
• one-step method to compute all components of eigenvalue problem
• Results are computationally more stable and robust
SVD is performed directly on the data matrix F with M rows (spatial points)
and N columns (samples in time)
SVD is based on the concept that any M x N matrix can be written as the
product of three matrices:
U is M x M matrix
Vt is transpose of the N x N matrix V
Г is M x N matrix with zero elements outside the diagonal and positive or
zero elements on the diagonal
Scalars γk on the diagonal are called singular values. They are placed in
decreasing order and they are proportional to eigenvalues λk λk = γk
2
There is a maximum of K ≤ min(N,M) non-zero singular values which defines
the maximum number of EOF modes that we can determine.
The Singular Value Decomposition Approach
The columns in U matrix are orthogonal and are called left singular vectors of F
They are identical to the eigenvectors E and they are the EOF patterns
associated with each singular value. There is only K useful left singular vectors
The rows in Vt matrix are orthogonal and are called right singular vectors of F
They are proportional to the principal components A obtained from equations
2.11 and 2.12 and the constant of proportionality are the singular values γk such
that:
Matrix A contains the principal coefficients of data matrix F and effective
size of A is K x N
The Singular Value Decomposition Approach
Using equation (2.19) we can reconstruct field F adding all K modes of the
decomposition:
Note the similarity between (2.22) and (2.14) where: U = E and A = γVt
6
Number of important modes
• Data compression is more important than physical interpretation
• We try to separate signal from noise
• Several methods:
• Scree plot
• Guttman criterion
• Eigenvalues > 1
• Modes with eigenvalue that is higher than mean of all eigenvlaues
• Modes that explain more that 70 – 90 % of total variability
Rotated EOFs
• physical interpretation is more important than data compression
• Due to orthogonality
• Orthogonal or obligue rotation
• Rules of „simple structure“
Notes on EOFs interpretation
• Some EOFs not necessarily correspond to real physical behavior of
dynamical modes
• A clue to the interpretation of EOF modes may be found in the principal
component.
• Their temporal variability may be similar to some known processes
• The physical interpretation is limited due to spatial orthogonality of the
EOF patterns
• Real world processes do not have orthogonal patterns or may not be
represented with uncorrelated indices
• Traditional EOFs can detect standing oscillations, however signal may be
propagating in space
Notes on EOFs interpretation
• The EOF patterns depend on the size of the study area
• Variable with uniform distribution of variance and with the spatial scale
comparable (or larger) to spatial domain produce monopole EOF 1 (the same
sign in all points)
• The need to be orthogonal to the first EOF creates a second EOF with
dipole pattern
• Thus the size of the domain should be greater than the typical spatial scale
of field analyzed
Units of presentation
• Units of field F are carried by the PCs while the EOFs are dimensionless
• It is common to re-normalize results (e.g. EOFs carry units of F and PCs
have variance of 1)
• Re-normalization is SW-specific – see e.g. Climate explorer application
• EOFs can be presented as a correlation maps – correlations between
principal component and the values of the field F at each location.
EOF analysis example
Main mode of SST vatiability in Central Pacific