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.. _process_definitions:
Stochastic process definitions
------------------------------
Notations
~~~~~~~~~
In this document, we note:
- :math:`X: \Omega \times\cD \rightarrow \Rset^{\inputDim}` a multivariate
stochastic process of dimension :math:`d`, where
:math:`\omega \in \Omega` is an event, :math:`\cD` is a domain of
:math:`\Rset^n`, :math:`\vect{t}\in \cD` is a multivariate index and
:math:`X(\omega, \vect{t}) \in \Rset^{\inputDim}`;
- :math:`X_{\vect{t}}: \Omega \rightarrow \Rset^{\inputDim}` the random variable
at index :math:`\vect{t} \in \cD` defined by
:math:`X_{\vect{t}}(\omega)=X(\omega, \vect{t})`;
- :math:`X(\omega): \cD \rightarrow \Rset^{\inputDim}` a realization of the
process :math:`X`, for a given :math:`\omega \in \Omega` defined by
:math:`X(\omega)(\vect{t})=X(\omega, \vect{t})`.
If :math:`n=1`, :math:`t` may be interpreted as a time stamp to
recover the classical notation of a stochastic process.
If the process is a second order process, we note:
- :math:`m : \cD \rightarrow \Rset^{\inputDim}` its *mean function*, defined by
:math:`m(\vect{t})=\Expect{X_{\vect{t}}}`,
- :math:`C : \cD \times \cD \rightarrow \cM_{d \times d}(\Rset)` its
*covariance function*, defined by:
.. math::
C(\vect{s}, \vect{t})
& := \Cov{X_{\vect{s}}, X_{\vect{t}}} \\
& \; = \Expect{(X_{\vect{s}} - m(\vect{s}))(X_{\vect{t}} - m(\vect{t}))^t},
- :math:`R : \cD \times \cD \rightarrow \mathcal{M}_{d \times d}(\Rset)`
its *correlation function*, defined for all
:math:`(\vect{s}, \vect{t})`, by :math:`R(\vect{s}, \vect{t})` such
that for all :math:`(i,j)`:
.. math::
R_{ij}(\vect{s}, \vect{t})
& := \Cor{X_{\vect{s}}, X_{\vect{t}}} \\
& \; = \frac{C_{ij}(\vect{s}, \vect{t})}{\sqrt{C_{ij}(\vect{s}, \vect{s}) C_{ij}(\vect{t}, \vect{t})}}.
We recall here some useful definitions.
Spatial (temporal) and Stochastic Mean
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The *spatial mean* of the process :math:`X` is the function
:math:`m: \Omega \rightarrow \Rset^{\inputDim}` defined by:
.. math::
:label: spatMean
\displaystyle m(\omega)=\frac{1}{|\cD|} \int_{\cD} X(\omega)(\vect{t})\, d\vect{t}
If :math:`n=1` and if the mesh is a regular grid
:math:`(t_0, \dots, t_{N-1})`, then the spatial mean corresponds to the
*temporal mean* defined by:
.. math::
:label: tempMean
m(\omega) = \frac{1}{t_{N-1} - t_0} \int_{t_0}^{t_{N-1}}X(\omega)(t) \, dt
The spatial mean is estimated from one realization of the process (see
the use case on Field or Time series).
The *stochastic mean* of the process :math:`X` is the function
:math:`g: \cD \rightarrow \Rset^{\inputDim}` defined by:
.. math::
:label: stocMean
\displaystyle g(\vect{t}) = \Expect{X_{\vect{t}}}
The stochastic mean is estimated from a sample of realizations of the
process (see the use case on the Process sample).
For an *ergodic process*, the stochastic mean and the spatial mean are
equal and constant (equal to the constant vector noted
:math:`\vect{c}`):
.. math::
:label: ergodic
\forall \omega\in \Omega, \, \forall \vect{t} \in \cM, \, m(\omega)= g(\vect{t}) = \vect{c}
Normal process
~~~~~~~~~~~~~~
A stochastic process is *normal* if all its finite
dimensional joint distributions are normal, which means that for all
:math:`k \in \Nset` and :math:`I_k \in \Nset^*`, with
:math:`\mathrm{card} I_k = k`, there exist
:math:`\vect{m}_1,\dots,\vect{m}_k\in\Rset^{\inputDim}` and
:math:`\mat{C}_{1,\dots,k}\in\mathcal{M}_{k\inputDim,k\inputDim}(\Rset)` such that:
.. math::
\Expect{\exp\left\{i\vect{X}_{I_k}^t \vect{U}_{k} \right\}} =
\exp{\left\{i\vect{U}_{k}^t\vect{M}_{k}-\frac{1}{2}\vect{U}_{k}^t\mat{C}_{1,\dots,k}\vect{U}_{k}\right\}}
where
:math:`\vect{X}_{I_k}^t = (X_{\vect{t}_1}^t, \hdots, X_{\vect{t}_k}^t)`,
:math:`\vect{U}_{k}^t = (\vect{u}_{1}^t, \hdots, \vect{u}_{k}^t)` and
:math:`\vect{M}_{k}^t = (\vect{m}_{1}^t, \hdots, \vect{m}_{k}^t)` and
:math:`\mat{C}_{1,\dots,k}` is the symmetric matrix:
.. math::
:label: covMatrix
\mat{C}_{1,\dots,k} = \left(
\begin{array}{cccc}
C(\vect{t}_1, \vect{t}_1) &C(\vect{t}_1, \vect{t}_2) & \hdots & C(\vect{t}_1, \vect{t}_{k}) \\
\hdots & C(\vect{t}_2, \vect{t}_2) & \hdots & C(\vect{t}_2, \vect{t}_{k}) \\
\hdots & \hdots & \hdots & \hdots \\
\hdots & \hdots & \hdots & C(\vect{t}_{k}, \vect{t}_{k})
\end{array}
\right)
A normal process is entirely defined by its mean function :math:`m`
and its covariance function :math:`C` (or correlation function
:math:`R`).
Weak stationarity (second order stationarity)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A process
:math:`X` is *weakly stationary* or *stationary of second order* if
its mean function is constant and its covariance function is invariant
by translation:
.. math::
:label: stat2order
\forall (\vect{s},\vect{t}) \in \cD, & \, m(\vect{t}) = m(\vect{s}) \\
\forall (\vect{s},\vect{t},\vect{h}) \in \cD, & \, C(\vect{s}, \vect{s}+\vect{h}) =C(\vect{t}, \vect{t}+\vect{h})
We note :math:`C^{stat}(\vect{\tau})` for
:math:`C(\vect{s}, \vect{s}+\vect{\tau})` as this quantity does not
depend on :math:`\vect{s}`.
In the continuous case, :math:`\cD` must be equal to
:math:`\Rset^n`\ as it is invariant by any translation. In the
discrete case, :math:`\cD` is a lattice
:math:`\mathcal{L}=(\delta_1 \Zset \times \dots \times \delta_n \Zset)`
where :math:`\forall i, \delta_i >0`.
Stationarity
~~~~~~~~~~~~
A process :math:`X` is *stationary* if its
distribution is invariant by translation: :math:`\forall k \in \Nset`,
:math:`\forall (\vect{t}_1, \dots, \vect{t}_k) \in \cD`,
:math:`\forall \vect{h}\in \Rset^n`, we have:
.. math::
:label: statGen
\forall k \in \Nset, \, \forall (\vect{t}_1, \dots, \vect{t}_k) \in \cD, \, \forall \vect{h}\in \Rset^n, \, (X_{\vect{t}_1}, \dots, X_{\vect{t}_k}) \stackrel{\mathcal{D}}{=} (X_{\vect{t}_1+\vect{h}}, \dots, X_{\vect{t}_k+\vect{h}})
Spectral density function
~~~~~~~~~~~~~~~~~~~~~~~~~
If :math:`X` is a zero-mean weakly
stationary continuous process and if for all :math:`(i,j)`,
:math:`C^{stat}_{i,j} : \Rset^n \rightarrow \Rset^n` is
:math:`\cL^1(\Rset^n)` (ie
:math:`\int_{\Rset^n} |C^{stat}_{i,j}(\vect{\tau})|\, d\vect{\tau}\, < +\infty`),
we define the *bilateral spectral density function*
:math:`S : \Rset^n \rightarrow \cH^+(\inputDim)` where
:math:`\mathcal{H}^+(\inputDim) \in \mathcal{M}_d(\Cset)` is the set of
:math:`\inputDim`-dimensional positive definite hermitian matrices, as the
Fourier transform of the covariance function :math:`C^{stat}`:
.. math::
:label: specdensFunc
\forall \vect{f} \in \Rset^n, \,S(\vect{f}) = \int_{\Rset^n}\exp\left\{ -2i\pi <\vect{f},\vect{\tau}> \right\} C^{stat}(\vect{\tau})\, d\vect{\tau}
Furthermore, if for all :math:`(i,j)`,
:math:`S_{i,j}: \Rset^n \rightarrow \Cset` is :math:`\cL^1(\Cset)` (ie
:math:`\int_{\Rset^n} |S_{i,j}(\vect{f})|\, d\vect{f}\, < +\infty`),
:math:`C^{stat}` may be evaluated from :math:`S` as follows:
.. math::
:label: cspectransform
C^{stat}(\vect{\tau}) = \int_{\Rset^n}\exp\left\{ 2i\pi <\vect{f}, \vect{\tau}> \right\}S(\vect{f})\, d\vect{f}
In the discrete case, the spectral density is defined for a zero-mean
weakly stationary process, where
:math:`\cD=(\delta_1 \Zset \times \dots \times \delta_n \Zset)` with
:math:`\forall i, \delta_i >0` and where the previous integrals are
replaced by sums.
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