286
Chapter 15: Multivariate Statistical Modeling:
15.3.1 The System Matrix A and its POPs
• The Principal Oscillation Patterns (POPs) pi are the eigenvectors of
the system matrix A.
• The POP coefficient Zi is expansion coefficient with zi = p~.T X, where
p~ is the adjoint pattern of the eigenvector pi 3.
• If Xis generated by (15.1), the POP coefficient is then generated by an
univariate AR(l )-process:
(15.10)
with >'i being the j-th eigenvalue of A. Ri = p~.T N is independent of
Z{.
Because A is not symmetric, so me or all of its eigenvalues >'i and eigenvectors p can be complex. Since A is a real matrix, if pi is a complex eigenvector
of A with complex eigenvalue >'i then the complex conjugate pi. is also an
eigenvector of A with eigenvalue >'i.
If A has m different eigenvalues, which is normally the case for a climate
time series, X may be uniquely expressed in terms of its eigenvectors. The
transformation from the real vector X to the generally complex vector Z is
defined by:
PZ or
(15.11)
Z = P"lx
where P and PAare complex m x m matrices. The columns of P are the
eigenvectors of A and those ofP Aare the corresponding adjoint vectors. One
has P·TpA = p*lp = I.
15.3.2 Cross-Spectrum Matrix in POP-Basis:
Its Matrix Formulation
Defining the diagonal matrix
A = (>'1 ... >'m)
with the eigenvalues of A in the main diagonal, one has
A = PAP"l
A~ = PA~p·l
(15.12)
U sing the transformation of (15.11), the expression of cross-covariance matrix
of X (15.8), the eigenvector decomposition of A (15.12), the cross-covariance
matrix of Z is given by
3The eigenvectoq,i and its adjoint vector P~ satisfy (p~)'Tpj = 6',j.
Chapter 15: Multivariate Statistical Modeling:
15.3.1 The System Matrix A and its POPs
• The Principal Oscillation Patterns (POPs) pi are the eigenvectors of
the system matrix A.
• The POP coefficient Zi is expansion coefficient with zi = p~.T X, where
p~ is the adjoint pattern of the eigenvector pi 3.
• If Xis generated by (15.1), the POP coefficient is then generated by an
univariate AR(l )-process:
(15.10)
with >'i being the j-th eigenvalue of A. Ri = p~.T N is independent of
Z{.
Because A is not symmetric, so me or all of its eigenvalues >'i and eigenvectors p can be complex. Since A is a real matrix, if pi is a complex eigenvector
of A with complex eigenvalue >'i then the complex conjugate pi. is also an
eigenvector of A with eigenvalue >'i.
If A has m different eigenvalues, which is normally the case for a climate
time series, X may be uniquely expressed in terms of its eigenvectors. The
transformation from the real vector X to the generally complex vector Z is
defined by:
PZ or
(15.11)
Z = P"lx
where P and PAare complex m x m matrices. The columns of P are the
eigenvectors of A and those ofP Aare the corresponding adjoint vectors. One
has P·TpA = p*lp = I.
15.3.2 Cross-Spectrum Matrix in POP-Basis:
Its Matrix Formulation
Defining the diagonal matrix
A = (>'1 ... >'m)
with the eigenvalues of A in the main diagonal, one has
A = PAP"l
A~ = PA~p·l
(15.12)
U sing the transformation of (15.11), the expression of cross-covariance matrix
of X (15.8), the eigenvector decomposition of A (15.12), the cross-covariance
matrix of Z is given by
3The eigenvectoq,i and its adjoint vector P~ satisfy (p~)'Tpj = 6',j.