Stochastic process definitions

In this document, we note:

  • X: \Omega \times\cD \rightarrow \Rset^d a multivariate stochastic process of dimension d, where \omega \in \Omega is an event, \cD is a domain of \Rset^n, \vect{t}\in \cD is a multivariate index and X(\omega, \vect{t}) \in \Rset^d;

  • X_{\vect{t}}: \Omega \rightarrow \Rset^d the random variable at index \vect{t} \in \cD defined by X_{\vect{t}}(\omega)=X(\omega, \vect{t});

  • X(\omega): \cD \rightarrow \Rset^d a realization of the process X, for a given \omega \in \Omega defined by X(\omega)(\vect{t})=X(\omega, \vect{t}).

If n=1, 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:

  • m : \cD \rightarrow \Rset^d its mean function, defined by m(\vect{t})=\Expect{X_{\vect{t}}},

  • C : \cD \times \cD \rightarrow \cM_{d \times d}(\Rset) its covariance function, defined by C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))(X_{\vect{t}}-m(\vect{t}))^t},

  • R : \cD \times \cD \rightarrow \mathcal{M}_{d \times d}(\Rset) its correlation function, defined for all (\vect{s}, \vect{t}), by R(\vect{s}, \vect{t}) such that for all (i,j), R_{ij}(\vect{s}, \vect{t})=C_{ij}(\vect{s}, \vect{t})/\sqrt{C_{ii}(\vect{s}, \vect{t})C_{jj}(\vect{s}, \vect{t})}.

We recall here some useful definitions.

Spatial (temporal) and Stochastic Mean

The spatial mean of the process X is the function m: \Omega \rightarrow \Rset^d defined by:

(1)\displaystyle m(\omega)=\frac{1}{|\cD|} \int_{\cD} X(\omega)(\vect{t})\, d\vect{t}

If n=1 and if the mesh is a regular grid (t_0, \dots, t_{N-1}), then the spatial mean corresponds to the temporal mean defined by:

(2)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 X is the function g: \cD \rightarrow \Rset^d defined by:

(3)\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 \vect{c}):

(4)\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 k \in \Nset and I_k \in \Nset^*, with \mathrm{card} I_k = k, there exist \vect{m}_1,\dots,\vect{m}_k\in\Rset^d and \mat{C}_{1,\dots,k}\in\mathcal{M}_{kd,kd}(\Rset) such that:

\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 \vect{X}_{I_k}^t = (X_{\vect{t}_1}^t, \hdots, X_{\vect{t}_k}^t), \vect{U}_{k}^t = (\vect{u}_{1}^t, \hdots, \vect{u}_{k}^t) and \vect{M}_{k}^t = (\vect{m}_{1}^t, \hdots, \vect{m}_{k}^t) and \mat{C}_{1,\dots,k} is the symmetric matrix:

(5)\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 m and its covariance function C (or correlation function R).

Weak stationarity (second order stationarity)

A process X is weakly stationary or stationary of second order if its mean function is constant and its covariance function is invariant by translation:

(6)\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 C^{stat}(\vect{\tau}) for C(\vect{s}, \vect{s}+\vect{\tau}) as this quantity does not depend on \vect{s}. In the continuous case, \cD must be equal to \Rset^nas it is invariant by any translation. In the discrete case, \cD is a lattice \mathcal{L}=(\delta_1 \Zset \times \dots \times \delta_n \Zset) where \forall i, \delta_i >0.

Stationarity

A process X is stationary if its distribution is invariant by translation: \forall k \in \Nset, \forall (\vect{t}_1, \dots, \vect{t}_k) \in \cD, \forall \vect{h}\in \Rset^n, we have:

(7)\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 X is a zero-mean weakly stationary continuous process and if for all (i,j), C^{stat}_{i,j} : \Rset^n \rightarrow \Rset^n is \cL^1(\Rset^n) (ie \int_{\Rset^n} |C^{stat}_{i,j}(\vect{\tau})|\, d\vect{\tau}\, < +\infty), we define the bilateral spectral density function S : \Rset^n \rightarrow \cH^+(d) where \mathcal{H}^+(d) \in \mathcal{M}_d(\Cset) is the set of d-dimensional positive definite hermitian matrices, as the Fourier transform of the covariance function C^{stat}:

(8)\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 (i,j), S_{i,j}: \Rset^n \rightarrow \Cset is \cL^1(\Cset) (ie \int_{\Rset^n} |S_{i,j}(\vect{f})|\, d\vect{f}\, < +\infty), C^{stat} may be evaluated from S as follows:

(9)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 \cD=(\delta_1 \Zset \times \dots \times \delta_n \Zset) with \forall i, \delta_i >0 and where the previous integrals are replaced by sums.