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Chapter 15: Multivariate Statistical Modeling:
transform of the cross-covariance functions, the cross-spectra. Given a stationary time series, the goal of statistical modeling is to describe the stochastic process by means of a model containing a few parameters which provide
information about the low order moments of the stochastic process.
For univariate time series, the statistical models normally used are autoregressive (AR) or moving average (MA) processes. The covariance functions
and the power spectra of univariate time series generated by AR or MA processes are functions of the process parameters. Since low order processes are
described by a few parameters, the second moment information can be easily
derived from an univariate time series. In the example of a time series generated by a first order autoregressive (AR(1)] process with parameter a, the
covariance function at lag .6. is proportional to a A •
The situation becomes more complicated for a multivariate time series.
Climate time series are usually multivariate with high dimensions. In the
case of observations, the dimension is of the order of 10 2 to 10 3 , whereas
the dimension of General Circulation Model outputs is normally larger than
10 3 . The second moment information of such a multivariate time series is not
only difficult to derive, but also difficult to interpret. For an rn-dimensional
time series generated by an AR(1) process, one needs to estimate no longer
just one number, as discussed before, but an m x m matrix. Even if one
could interpret cross-covariance function at lag .6. which now is a matrix, it
does not guarantee that one could also interpret the cross-covariance function
at lag .6. + 1. It will be shown later that because of the nonlinearity of
matrix multiplication, there is a strong limitation in deriving second moment
information from parameters of multivariate AR or MA processes.
Besides the random characters, a climate time series displays also deterministic features. Among a myriad of processes, frequently only a few stand
out. They represent the dominant dynamics of the system and are responsible for the time evolution of our time series. This fact suggests that a climate
time series can be considered to be generated separately by a deterministic
process which represents the process and a stochastic one which represents all
other processes. It is therefore reasonable to build problem-specific models
which are, to our knowledge, suitable for the dynamical processes. This kind
of statistical modeling was first proposed by Hasselmann (1988).
The assumption made for Hasselmann's statistical models is that the climate phase space can be splitted into two subspaces called "signal" and
"noise" subspaces. This notion has been introduced in Chapter 13. In our
context, it is assumed that the deterministic processes operate in the "signal"
subspace whereas the other remaining ones operate in the "noise" subspace.
Furthermore, the "signal" subspace can be spanned by a few characteristic
spatial patterns. The coefficients of these patterns are described by a set
of equations which represent the dynamics of the processes and control the
temporal evolution of the "signal" subspace. Thus, depending on the problem considered, the model can take any (linear or non-linear) form. A linear
Chapter 15: Multivariate Statistical Modeling:
transform of the cross-covariance functions, the cross-spectra. Given a stationary time series, the goal of statistical modeling is to describe the stochastic process by means of a model containing a few parameters which provide
information about the low order moments of the stochastic process.
For univariate time series, the statistical models normally used are autoregressive (AR) or moving average (MA) processes. The covariance functions
and the power spectra of univariate time series generated by AR or MA processes are functions of the process parameters. Since low order processes are
described by a few parameters, the second moment information can be easily
derived from an univariate time series. In the example of a time series generated by a first order autoregressive (AR(1)] process with parameter a, the
covariance function at lag .6. is proportional to a A •
The situation becomes more complicated for a multivariate time series.
Climate time series are usually multivariate with high dimensions. In the
case of observations, the dimension is of the order of 10 2 to 10 3 , whereas
the dimension of General Circulation Model outputs is normally larger than
10 3 . The second moment information of such a multivariate time series is not
only difficult to derive, but also difficult to interpret. For an rn-dimensional
time series generated by an AR(1) process, one needs to estimate no longer
just one number, as discussed before, but an m x m matrix. Even if one
could interpret cross-covariance function at lag .6. which now is a matrix, it
does not guarantee that one could also interpret the cross-covariance function
at lag .6. + 1. It will be shown later that because of the nonlinearity of
matrix multiplication, there is a strong limitation in deriving second moment
information from parameters of multivariate AR or MA processes.
Besides the random characters, a climate time series displays also deterministic features. Among a myriad of processes, frequently only a few stand
out. They represent the dominant dynamics of the system and are responsible for the time evolution of our time series. This fact suggests that a climate
time series can be considered to be generated separately by a deterministic
process which represents the process and a stochastic one which represents all
other processes. It is therefore reasonable to build problem-specific models
which are, to our knowledge, suitable for the dynamical processes. This kind
of statistical modeling was first proposed by Hasselmann (1988).
The assumption made for Hasselmann's statistical models is that the climate phase space can be splitted into two subspaces called "signal" and
"noise" subspaces. This notion has been introduced in Chapter 13. In our
context, it is assumed that the deterministic processes operate in the "signal"
subspace whereas the other remaining ones operate in the "noise" subspace.
Furthermore, the "signal" subspace can be spanned by a few characteristic
spatial patterns. The coefficients of these patterns are described by a set
of equations which represent the dynamics of the processes and control the
temporal evolution of the "signal" subspace. Thus, depending on the problem considered, the model can take any (linear or non-linear) form. A linear