EXPLORING SPATIOTEMPORAL VARIABILITY BY EIGENDECOMPOSITION TECHNIQUES

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EXPLORING SPATIO-TEMPORAL VARIABILITY BY
EIGEN-DECOMPOSITION TECHNIQUES
Luigi Ippoliti - Lara Fontanella
Department of Quantitative Methods and Economic
Theory University G. dAnnunzio of
Chieti-Pescara
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Outline of the talk

• 1. Identification of oscillation spatial patterns
  by means of multivariate techniques
• principal component analysis (PCA)
• canonical correlations analysis (CCA)
• partial least square (PLS)
• redundancy analysis (RA)
• 2. Unified approach in the framework of the
• Generalized Eigenvalue Decomposition
• 3. Dynamic models for predictions aims
• 4. Conclusions

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Expansion of a spatial process

• Spatio-temporal process X(st) s?D??2 ,t ?? .
• For a given time tt0, assume that X(st0) is a
  zero mean second order spatial stochastic process
  with covariance function Q(s,s').
• X(st0) can be expanded in a set of deterministic
  functions ?k, k ??, which form a complete
  orthonormal system in the space L2(D) of square
  integrable functions on the domain D

where
are the time-dependent expansion coefficients.
This result is generally related to the
probabilistic corollary of Mercer's theorem known
as Karhunen-Loéve expansion.
In this talk, we consider an extension of the
preceding expansion and discuss the case in which
two kernels, Q1(s,s ') and Q2(s,s ') are defined
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Estimating spatial patterns using Simultaneous
Diagonalization

• Let us assume that
• Q1(s,s') and Q2(s,s') are real, symmetric and
  square integrable functions
• Q1 and Q2 are the integral operators with kernels
  Q1(s,s') and Q2(s,s')
• Q1 is positive definite
• Q2 is nonnegative definite
• is densely
  defined, bounded, and its extension to the whole
  of L2(D) has eigenfunctions which span L2(D)
• If lk and vk, k ??, are the eigenvalues and the
  orthonormalized eigenfunctions of Q, we have the
  following simultaneous diagonalization of the two
  kernels (Kadota 1976)

where
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Finite approximation

• Given the observed space-time field x(si,t),
  i1,,n, t1,T, a finite approximation for
• equations (1), (2) and (3) is required.
• At each time t, the observed spatial series x(t)
  is expanded in terms of a set of n column vectors
  called patterns

Where W is the matrix of the patterns, and a(t)
the vector of the sample expansion coefficients,
obtained as a weighted linear combination of the
data
The simultaneous diagonalization of the two
kernels can be approximated as
Where Q1 and Q2 are the matrix of the value of
the kernel functions onto the spatial sample
points. Equation (6) constitutes a Generalized
Eigenvalue Equation (GED)
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Principal Component Analysis

• Criterion for the selection of the spatial
  patterns to preserve as much variance as
  possible given a certain dimensionality of the
  model.
• PCA projects the data on the subspace of maximum
  data variation
• The spectral decomposition of the symmetric
  covariance matrix C is a special case of the
  generalized eigenvalue decomposition where

In this case, the k-th column of matrix W
represents the k-th EOF which is associated to
the corresponding expansion coefficient ak,
known as principal component. Sorting the EOFs
according to the magnitude of their eigenvalues
the process can be reconstructed following
equation in a reduced space if a truncation level
K in the expansion equation is chosen. PCA gives
a data dependent set of basis vectors that is
optimal in statistical mean square sense.
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Partial Least Square

• The partial least square allows for the
  identification of pairs of spatial patterns and
  time coefficients which account for a fraction of
  the covariance between two processes analized
  jointly.
• PLS projects the data on the subspace of maximum
  data covariation
• Given two data matrices, X and Y, of zero-mean
  spatio-temporal series x(si,t) and y(si,t), with
  between-sets spatial covariance matrix Cxy, the
  goal is to find the two directions of maximal
  data covariation i.e. the directions wxk and wyk
  such that the linear combination axkXwxk and
  aykYwyk have maximum covariance.
• The spatial patterns Wx and Wy can be obtained by
  generalized eigenvalue decomposition (6) where

The eigenvalue lk represents the covariance
between axk and ayk.
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Canonical Correlation Analysis

• The canonical correlation analysis allows for the
  identification of pairs of spatial patterns and
  time coefficients which account for a fraction of
  the correlation between two processes analyzed
  jointly.
• CCA projects the data on the subspace of maximum
  data correlation
• Given two data matrices, X and Y, of zero-mean
  spatio-temporal series x(si,t) and y(si,t), with
  between-sets spatial covariance matrix Cxy and
  spatial covariance matrices Cxx and Cyy, the
  goal is to find the two directions of maximal
  data correlazione i.e. the directions wxk and
  wyk such that the linear combination axkXwxk and
  aykYwyk have maximum correlation.
• The spatial patterns Wx and Wy can be obtained by
  generalized eigenvalue decomposition (6) where

The eigenvalue lk represents the correlation
coefficients between axk and ayk.
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Redundancy Analysis

• Given two processes, if the aim is to predict one
  process as well as possible in the least square
  error sense, the spatial patterns must be chosen
  so that this error measure is minimized or
  equivalently the redundancy index is maximized.
• RA projects the data on the subspace of minimum
  prediction error
• Given two data matrices, X and Y, of zero-mean
  spatio-temporal series x(si,t) and y(si,t), with
  between-sets spatial covariance matrix Cxy and
  spatial covariance matrix, of X, Cxx, the
  spatial patterns Wx and Wy can be obtained by
  generalized eigenvalue decomposition (6) where

The eigenvalue lk represents the regression
coefficient of ayk on axk.
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State-Space Models for Temporal Predictions

• Several prediction models can be represented in a
  unified formulation using the state-space
  representation

State equation
Measurement equation

• where vt is the state vector, F is the transition
  matrix, H is the measurement matrix and ut and et
  are zero mean Gaussian error terms.
• State-Space Model for a Single Field
• One possibility is to assume the measurement
  matrix H known and equal to Wx(K). In this case,
  the state equation models the temporal evolution
  of the expansion coefficients ax(K)(t). (Mardia
  et. Al. 1998 Wikle and Cressie 1999)

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State-Space Model for Coupled Fields

• symmetric case both variables are equally
  important such that none of them can be
  considered as a predictor for the other.
• The state and measurement equations could be
  defined as follows

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State-Space Model for Coupled Fields

• asymmetric case it is explicitly recognized that
  one variable can be fruitfully used to predict
  the other variable. In this case, the state-space
  model can be derived by the following regression
  model where, assuming Y as the processes to be
  predicted, we have

Both for RA and CCA, B is a diagonal matrix.
Thus, in this case, since the expansion
coefficients are zero-mean random variables, the
state and measurement equations could be defined
as follows
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Conclusions

• Oscillation patterns could be useful in exploring
  the spatial variability of the fields
• Parsimonious dynamic models can be defined by
  only using a few spatial patterns
• State-Space models can be used to model the
  temporal evolution of the spatial patterns. An
  alternative procedure to State-Space models could
  be parsimonious VAR models in the expansion
  coefficients domain.
• Different approach can be used to estimate
  spatial covariance and cross-covariance
  structures (i.e. MOM, geostatistical approach)
• PLS, CCA and RA can also be used when only one
  variable is observed to take in account the
  temporal dependence
• Further work is needed ..

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Intercomparison of methods space-time series of
PM10 and NOX

• DATA daily mean concentrations of PM10 and NOX
  observed at 15 monitoring stations located in the
  Lombardy monitoring network (Italy). All the data
  range from 1st January to 13th December 2004 and
  constitute 348 temporal observations

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Comparative measure of explained covariance and
goodness of reconstruction

• Cumulative squared covariance fraction (CSCFK)
  indicates how successful each method has been in
  explaining the observed covariance matrix using a
  reduced number K of spatial patterns to
  reconstruct the data matrices X and Y

where the Frobenius matrix norm of a matrix C is
given by

• The cumulative squared covariance fraction has
  been applied to
• the NOX-PM10 between-set covariance matrix Cxy
• the NOX covariance matrix Cxx
• the PM10 covariance matrix Cyy

Root mean squared error (RMSEK) indicates how
successful each method has been in reconstructing
the observed data matrix using a reduced number K
of spatial patterns to reconstruct the data
matrices X and Y
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Cumulative squared covariance fraction (CSCFK)
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Root Mean Squared Error (RMSEK)
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PCA PM10
Correlation map for PM10 and NOx with ay1
First spatial pattern for PM10 wy1
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First spatial pattern for PM10 wy1
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First spatial pattern for NOx wx1
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Second spatial pattern for PM10 wy2
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Second spatial pattern for NOx wx2