Introduction
Stationary and isotropic random fields
Examples
Notes on stationary random processes and fields
Hans-Jörg Starkloff
Westsächsische Hochschule Zwickau
DFG SPP 1324 - Mathematische Methoden zur Extraktion quantifizierbarer Information
aus komplexen Systemen
Seminar Stochastische Galerkin Methoden
TU Bergakademie Freiberg 15.09.2009
Hans-Jörg Starkloff
Notes on stationary random processes and fields
1
Introduction
Stationary and isotropic random fields
Examples
Introduction
Stationary and isotropic random fields
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Examples
Exponentially correlated random processes
Matern class
Hans-Jörg Starkloff
Notes on stationary random processes and fields
2
Introduction
Stationary and isotropic random fields
Examples
Introductory remarks
I
aim of the talk: give some basic facts about stationary random
processes and random fields and related concepts
I
stationary processes constitute a class of important and well
investigated random processes often used in applications
I
basic property: probabilistic behaviour or certain probabilistic
characteristics do not change for certain transformations
I
coefficients and/or inhomogeneous terms in random differential
equations are often assumed to be stationary, e.g. κ = κ(x, ω)
in uncertain (random) subsurface flow problem
−∇ · (κ ∇u) = f
u=0
Hans-Jörg Starkloff
in D ⊂ Rd
on ∂D
Notes on stationary random processes and fields
3
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Strictly stationary random fields I
I
Let (Ω, A, P) be a fixed probability space.
I
Let (X, SX ) be a measurable space.
An X-valued random process (X (t); t ∈ T = Rd ) is called a
strictly stationary random process or field, if all finite dimensional
distributions are translation invariant, i.e., it holds




n
n
\
\
P
X (tj ) ∈ Bj  = P 
X (tj + t) ∈ Bj 
j=1
j=1
for all n ∈ N, t ∈ T, tj ∈ T, Bj ∈ SX , j = 1, . . . , n.
I
The parameter set T can also be N, Z, a linear space or group,
. . . , or a subset thereof; in this case the additional restriction
tj + t ∈ T (j = 1, . . . , n) must be included in the definition.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
4
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Strictly stationary random fields II
I
Here value sets are X = R or X = C
and parameter sets are T = Rd (d ∈ N) or their subsets.
I
other names:
• T = N, T = Z stationary sequence
• T = R stationary process
• T = Rd (d ∈ N, d > 1) homogeneous field
”stationary in the strong sense” or ”in the narrow sense”
I
important results: ergodic theorems (coincidence of spatial and
ensemble means; laws of large numbers)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
5
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Strictly isotropic random fields
I
An X-valued random field (X (t); t ∈ T = Rd ) is called a strictly
isotropic random field (with respect to 0), if all finite dimensional
distributions are rotation invariant, i.e., it holds




n
n
\
\
X (V tj ) ∈ Bj 
P
X (tj ) ∈ Bj  = P 
j=1
j=1
for all n ∈ N, tj ∈ T, Bj ∈ SX , j = 1, . . . , n and rotation
(orthogonal) matrices V .
I
Very often strictly isotropic random fields are considered which
are also strictly stationary (and they are called strictly isotropic
random fields simply).
Hans-Jörg Starkloff
Notes on stationary random processes and fields
6
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Moments of strictly stationary random fields I
I
If for a strictly stationary random field moments exist they are
invariant with respect to translations.
I
mean value
m(t1 ) := E{X (t1 )} = E{X (t1 + t)} = m(t1 + t) = m = const.
I
covariance function (real case)
KX (t1 , t2 ) := Cov{X (t1 ), X (t2 )} = E{[X (t1 ) − m(t1 )][X (t2 ) − m(t2 )]}
= KX (t1 + t, t2 + t) = KX (t1 − t2 , 0) = K (t1 − t2 )
”autocovariance function” in S TEIN
I
analogous: mixed second order moment function (real case)
CX (t1 , t2 ) := E{X (t1 )X (t2 )} = C(t1 − t2 )
Hans-Jörg Starkloff
Notes on stationary random processes and fields
7
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Moments of strictly stationary random fields II
I
correlation function
RX (t1 , t2 ) := Corr{X (t1 ), X (t2 )} = p
Cov{X (t1 ), X (t2 )}
p
Var{X (t1 )} Var{X (t2 )}
= RX (t1 + t, t2 + t) = RX (t1 − t2 , 0) = R(t1 − t2 )
I
variance function (real case)
Var{X (t1 )} = E{[X (t1 ) − m(t1 )]2 } = E{X (t1 )2 } − m(t1 )2
= Var{X (t1 + t)} = K (0) = const. ≥ 0
I
analogous in the complex case with
Cov{X (t1 ), X (t2 )} := E{[X (t1 ) − m(t1 )][X (t2 ) − m(t2 )]}
= E{X (t1 )X (t2 )} − m(t1 )m(t2 )
Var{X (t)} := E{|X (t) − m(t)|2 } = E{|X (t)|2 } − |m(t)|2
Hans-Jörg Starkloff
Notes on stationary random processes and fields
8
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Weakly stationary and isotropic random fields I
I
A scalar-valued random field (X (t); t ∈ T = Rd ) is called weakly
stationary, if the first two moments exist and they are translation
invariant, i.e., it holds
E{X (t)} = m(t) = m = const.
Cov{X (t1 ), X (t2 )} = K (t1 − t2 )
I
I
A scalar-valued random field (X (t); t ∈ T = Rd ) is called weakly
isotropic (with respect to 0), if the first two moments exist and
they are rotation invariant.
A scalar-valued random field (X (t); t ∈ T = Rd ) is weakly
stationary and isotropic iff it holds with Euclidean norm | · |
E{X (t)} = m(t) = m = const.
Cov{X (t1 ), X (t2 )} = K (|t1 − t2 |).
Hans-Jörg Starkloff
Notes on stationary random processes and fields
9
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Weakly stationary and isotropic random fields II
I
other name: wide sense stationary or isotropic random fields
I
there exist strictly stationary/isotropic random fields, which are
not weakly stationary/isotropic (if moments do not exist)
I
there exist weakly stationary/isotropic random fields, which are
not strictly stationary/isotropic
simple example: (X (t); t ∈ Rd ) with independent random
variables X (t) for t ∈ Rd , equal means and variances, but
different distributions
I
for some classes both concepts coincide, e.g. normal random
fields, lognormal random fields (as exponentials of normal
random fields)
I
in the following we consider mainly weakly stationary and
isotropic random fields
Hans-Jörg Starkloff
Notes on stationary random processes and fields
10
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Mean square calculus
I
I
I
I
different convergence concepts for random variables ⇒
different regularity or smoothness types for random processes
for weakly stationary/isotropic random fields important are
pathwise regularity and mean square regularity
mean square properties of random processes are related to
properties of the mean value function and the covariance
function (or mixed second order moment function)
mean square limit (limit in quadratic mean) of a sequence of
random variables
2
Xn →L X
I
(n → ∞)
⇔
lim E{|Xn − X |2 } = 0
n→∞
it follows convergence in probability (stochastic convergence),
almost sure convergence in general does not follow
Hans-Jörg Starkloff
Notes on stationary random processes and fields
11
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Mean square continuity I
I
given: metric space T ; scalar valued random process
(X (t); t ∈ T) with E{|X (t)|2 } < ∞ ∀ t ∈ T
I
random process (X (t); t ∈ T) is m.s. continuous at t0 ∈ T
⇔
2
X (tn ) →L X (t0 ) for tn → t0
⇔
mixed second order moment function
T × T 3 (s, t) 7→ E{X (s)X (t)} is continuous at (t0 , t0 )
⇔
mean value function T 3 t 7→ E{X (t)} is continuous at t0
and covariance function T × T 3 (s, t) 7→ Cov{X (s), X (t)} is
continuous at (t0 , t0 )
Hans-Jörg Starkloff
Notes on stationary random processes and fields
12
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Mean square continuity II
I
for weakly stationary random processes:
(X (t); t ∈ T ⊂ Rd ) is mean square continuous at t0 ∈ T
⇔
(X (t); t ∈ T) is mean square continuous at any t ∈ T
⇔
autocovariance function K (t) is continuous at t = 0
⇔
autocovariance function K (t) is continuous at
t ∈ T ∀ t ∈ T (and is uniform continuous)
(or mixed second order moment function or correlation function)
I
mean square continuity does not imply continuity of paths
I
If a weakly stationary process is not mean square continuous,
then there does not exist a modification which is measurable.
Measurable random process: mapping
T × Ω 3 (t, ω) 7→ X (t, ω) is B(T) ⊗ A−measurable.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
13
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Mean square differentiability I
I
given: open interval T ⊂ R ; scalar-valued random process
(X (t); t ∈ T) with E{|X (t)|2 } < ∞ ∀ t ∈ T
I
random process (X (t); t ∈ T) is m.s. differentiable at t0 ∈ T
X (t0 + h) − X (t0 ) L2 0
⇔
→ X (t0 ) for h → 0
h
⇔
for mixed second order moment function
T × T 3 (s, t) 7→ C(s, t) := E{X (s)X (t)} exists at (t0 , t0 )
∂ 2 C(s,t)
∂s ∂t
C(s+h,t+h0 )−C(s+h,t)−C(s,t+h0 )+C(s,t)
h h0
h,h →0
= lim
0
⇔
mean value function T 3 t 7→ m(t) is differentiable at t0
and for covariance function T × T 3 (s, t) 7→ K (s, t) exists
∂ 2 K (s, t)
generalized second derivative
at (t0 , t0 )
∂s ∂t
Hans-Jörg Starkloff
Notes on stationary random processes and fields
14
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Mean square differentiability II
I
for weakly stationary random processes:
(X (t); t ∈ T ⊂ R) is mean square differentiable at t0 ∈ T
⇔
(X (t); t ∈ T) is mean square differentiable at any t ∈ T
2
⇔
for autocovariance function K (t) exists d dtK2(t) at t = 0
(or for mixed second order moment or correlation function)
I
I
I
autocovariance function of m.s. derivative (X 0 (t); t ∈ T ⊂ R)
d2 K (t)
(this is again a weakly stationary process) equals −
dt 2
mean square differentiability does not imply differentiability of
paths but there is a modification with continuous paths
analogous for mean square partial derivatives in case of
T ⊂ Rd (d ∈ N, d > 1)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
15
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Preliminary remarks to positive definite functions
I
autocovariance functions of weakly stationary random processes
are positive definite functions
I
this concept is useful for the investigation of many properties of
weakly stationary random processes
I
positive definite functions play a role also in functional analysis,
theory of integral equations, approximation theory, . . .
I
in stochastics mainly two types:
• autocovariance functions and mixed second order moment
functions of weakly stationary processes
• characteristic functions of real random variables or random
vectors
Hans-Jörg Starkloff
Notes on stationary random processes and fields
16
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Positive definite kernels
I
given: nonempty set T; function T × T 3 (s, t) 7→ K (s, t) ∈ C
I
This function is called a positive definite kernel if for arbitrary
n ∈ N, t1 , . . . , tn ∈ T and complex numbers z1 , . . . , zn it holds
n
X
K (tk , tl )zk zl ≥ 0.
(1)
k ,l=1
K (tk , tj ) k ,l=1,...,n are
I
equivalent condition : all matrices
non-negative definite
I
for real-valued functions equivalent condition : K (s, t) = K (t, s)
and equation (1) holds for real numbers z1 , . . . , zn
I
e.g. K (s, t) = s − t, t, s ∈ R is not positive definite although (1)
holds for real numbers z1 , . . . , zn
Hans-Jörg Starkloff
Notes on stationary random processes and fields
17
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Positive definite kernels and random processes
I
given: nonempty set T; positive definite kernel (s, t) 7→ K (s, t)
I
Then there exists a random process (X (t); t ∈ T) with
covariance function K (s, t) on an appropriate probability space.
(The same holds for the mixed second order moment function.)
It can be prescribed also an arbitrary mean value function
T 3 t 7→ m(t).
One can take a Gaussian process for instance.
I
For other distributions of random processes there may be
restrictions to the allowed covariance functions.
I
On the other side, any covariance function and any mixed
second order moment function of a random process with finite
second order moments is a positive definite kernel.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
18
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Properties of positive definite kernels I
I
K (t, t) ≥ 0
∀t ∈ T
I
K (s, t) = K (t, s) ∀ (s, t) ∈ T × T
I
|K (s, t)|2 ≤ K (s, s)K (t, t) ∀ (s, t) ∈ T × T
I
|K (t1 , t3 ) − K (t2 , t3 )| ≤ K (t3 , t3 ) [K (t1 , t1 ) + K (t2 , t2 ) − 2<K (t1 , t2 )]
∀ t1 , t2 , t3 ∈ T
I
If K is a positive definite kernel, then so are the functions K
and <K .
I
If K1 , K2 , . . . , Kn are positive definite kernels and
n
X
ci ≥ 0, i = 1, . . . , n, then K (s, t) :=
ci Ki (s, t) is a positive
i=1
definite kernel.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
19
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Properties of positive definite kernels II
I
If each Kn , n ∈ N, is a positive definite kernel and the limit
K (s, t) := lim Kn (s, t) exists pointwise, then K (s, t) is a
n→∞
I
I
I
positive definite kernel.
The product of positive definite kernels is a positive definite
kernel.
If K (s, t) is a positive definite kernel, then so are the functions
exp(K (s, t)), exp(K (s, t)n ) (n ∈ N), sinh(K (s, t)), cosh(K (s, t)).
A function T × T 3 (s, t) 7→ K (s, t) ∈ C is a positive definite
kernel if and
only if there exists a family (fi ; i ∈ I) of functions
X
on T with
|fi (t)|2 < ∞ ∀ t ∈ T, such that it holds
i∈I
K (s, t) =
X
fi (s)fi (t).
i∈I
Hans-Jörg Starkloff
Notes on stationary random processes and fields
20
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Positive definite functions
I
given: nonempty set T ⊂ Rd ; function T 3 t 7→ K (t) ∈ C
I
This function is called a positive definite function if for arbitrary
n ∈ N, t1 , . . . , tn ∈ T and complex numbers z1 , . . . , zn it holds
n
X
K (tk − tl )zk zl ≥ 0.
(2)
k ,l=1
K (tk − tj ) k ,l=1,...,n are
I
equivalent condition : all matrices
non-negative definite
I
for real-valued functions equivalent condition : K (t) = K (−t)
and equation (2) holds for real numbers z1 , . . . , zn
Hans-Jörg Starkloff
Notes on stationary random processes and fields
21
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Positive definite functions and weakly stationary
random processes
I
given: positive definite function K (t) on Rd
I
Then there exists a weakly stationary random process
(X (t); t ∈ Rd ) with autocovariance function K (t) on an
appropriate probability space.
(The same holds for the mixed second order moment function.)
One can take a Gaussian process for instance.
I
For other distributions of random processes there may be
restrictions to the allowed covariance functions.
I
On the other side, any autocovariance function and any mixed
second order moment function of a weakly stationary random
process is a positive definite function.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
22
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Properties of positive definite functions I
I
K (0) ≥ 0
I
K (t) = K (−t) ∀ t ∈ Rd
I
|K (t)| ≤ K (0) ∀ t ∈ Rd
I
|K (t1 ) − K (t2 )| ≤ 2K (0) [K (0) − <K (t2 − t1 )]
I
If K is positive definite, then so are the functions K and <K .
I
If K1 , K2 , . . . , Kn are positive definite and ci ≥ 0, i = 1, . . . , n,
n
X
then K (t) :=
ci Ki (t) is positive definite.
I
If each Kn is positive definite, and K (t) := lim Kn (t) exists
∀ t1 , t2 ∈ Rd
i=1
n→∞
pointwise, then K (t) is positive definite.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
23
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Properties of positive definite functions II
I
The product of positive definite functions is positive definite.
I
If K (t) is a positive definite function, then so are the functions
exp(K (t)), cosh(K (t)), exp(i(t, x))K (t) (x ∈ Rd ), . . ..
I
d
If K (t) is a positive definite function
Z on R for which the
Fourier transform exists, K̂ (α) :=
K (t) exp(−i(t, α)) dt,
Rd
α ∈ Rd , then it holds K̂ (α) ≥ 0, α ∈ Rd .
I
I
Conversely, a function with non-negative Fourier transform is a
positive definite function.
Z
If K is positive definite with
|K (t)| dt < ∞, then it holds
Rd
Z
K (t) dt ≥ 0.
Rd
Hans-Jörg Starkloff
Notes on stationary random processes and fields
24
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Introduction
Stationary and isotropic random fields
Examples
Characteristic functions of random variables
I
I
given: real random variable ξ with distribution function
Fξ (x) = P(ξ ≤ x) and probability density fξ (x) if exists
characteristic function of ξ : complex-valued function of a real
variable
Z
n o Z
ϕξ (u) = E eiuξ =
eiux dFξ (x) =
eiux fξ (x) dx
R
I
I
I
I
R
it determines uniquely the distribution of the random variable
it is a uniform continuous positive definite function on R;
ϕξ (0) = 1
it is real if and only if the distribution of ξ is symmetric
smoothness of ϕξ (u) at u = 0 is related to the existence of
moments of the random variable; (∃ ϕ(2n) (0) ⇒ E{ξ 2n } < ∞, . . . )
Hans-Jörg Starkloff
Notes on stationary random processes and fields
25
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Characteristic functions of random vectors
I
I
analogous for real random vectors ξ = (ξ1 , . . . , ξd )T with joint
distribution function Fξ (x) = P(ξ ≤ x), x ∈ Rd , and probability
density fξ (x) if exists
characteristic function of random vector ξ : complex-valued
function on Rd
Z
n
o Z
i(u,x)
i(u,ξ)
e
dFξ (x) =
ei(u,x) fξ (x) dx
ϕξ (u) = E e
=
Rd
I
I
Rd
with Euclidean scalar product (u, x)
it is a uniform continuous positive definite function on Rd
For each continuous positive definite function ϕ on Rd with
ϕ(0) = 1 there exists a corresponding uniquely defined
probability distribution on Rd , such that it is the characteristic
function of this distribution.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
26
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Characteristic functions and random processes
I
I
I
I
I
I
given: continuous positive definite function K on Rd with
K (0) = 1
consider random vector ξ in Rd with characteristic function
K (u) = E{exp(i(u, ξ))}, u ∈ Rd
consider a random variable η with uniform distribution on
[0, 2π) which is independent of ξ
then the complex-valued weakly (and strictly) stationary random
process X (t) := ei((ξ,t)+η) , t ∈ Rd , has as its autocovariance
function the function K
analogous: real random process X (t) := 2 cos((ξ, t) + η) for
random vector ξ with real characteristic function
all paths of these weakly stationary random processes are
infinitely often differentiable (although not all these processes
are mean square differentiable)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
27
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Spectral representation of positive definite functions I
Theorem (B OCHNER -K HINCHIN)
For a given continuous positive definite function K on Rd there
exists a uniquely defined finite measure F on (Rd , B(Rd )) with
F (Rd ) = K (0), such that it holds for all t ∈ Rd
Z
K (t) =
ei(t,α) F (dα).
Rd
The measure is called the spectral measure of the weakly stationary
random process if K is the covariance function of the random
process.
If the measure is absolutely continuous with respect to the Lebesgue
measure on (Rd , B(Rd )) the density is called the spectral density.
Hans-Jörg Starkloff
Notes on stationary random processes and fields
28
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Spectral representation of positive definite functions II
Z
I
|K (t)| dt < ∞, the spectral density exists and it is
If
Rd
continuous and bounded on Rd .
I
Linear transformations of weakly stationary random processes
can be desribed with corresponding operations on the spectral
functions or densities.
I
Let (X (t); t ∈ R) be a weakly stationary random process on R
with existing spectral density f . Then the process is mean
square differentiable, iff the function R 3 α 7→ α2 f (α) is
Lebesgue-integrable on R. The spectral density of the mean
square derivative is the function α 7→ α2 f (α).
I
Analogous for higher order derivatives and partial derivatives of
weakly homogeneous random fields on Rd .
Hans-Jörg Starkloff
Notes on stationary random processes and fields
29
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Introduction
Stationary and isotropic random fields
Examples
Example
I
Example:
α1 , . . . , αn ∈ R; random variables ξ1 , . . . , ξn with
E{ξk } = 0, E{ξk ξl } = δkl fk ≥ 0
n
X
ξk eitαk , t ∈ R, is weakly stationary with
Then X (t) :=
k =1
autocovariance function
K (t) =
n
X
itαk
fk e
k =1
with F (B) =
X
Z
=
eitα F (dα)
R
fk , B ∈ B(R).
k :αk ∈B
(superposition of harmonic functions with random amplitudes)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
30
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Measurable isotropic autocovariance functions
I
A measurable covariance function of a wide sense stationary and
isotropic random field on Rd with d ∈ N, d ≥ 2 has the form
K (t) = Kc (t) + a1{0} (t)
I
I
with a continuous
isotropic autocovariance function Kc ,
1 t =0
1{0} (t) =
and a ≥ 0.
0 t 6= 0
The second term corresponds to the so called nugget effect.
as already was mentioned:
measurable modification
Hans-Jörg Starkloff
for a > 0 there does not exist a
Notes on stationary random processes and fields
31
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Weakly stationary and isotropic fields I
I
A function K (t) : R+ → R is an isotropic autocovariance
function of a mean square continuous wide sense stationary and
isotropic random field on Rd if and only it can be represented as
K (t) = 2
I
I
I
d−2
2
Z ∞ J d−2 (st)
d
2
Γ
G(ds),
d−2
2
0
(st) 2
0 ≤ t < ∞,
where G is a finite measure on [0, ∞) with G ([0, ∞)) = K (0).
Z ∞
d = 2 K (t) =
J0 (st) G(ds)
0
Z ∞
sin(st)
d = 3 K (t) = 2
G(ds)
st
0
Jν (x) Bessel function of first kind
Hans-Jörg Starkloff
Notes on stationary random processes and fields
32
Introduction
Stationary and isotropic random fields
Examples
Definitions and basic properties
Mean square properties
Positive definite functions
Spectral representation
Weakly stationary and isotropic fields II
I
differential equation for Bessel functions
x 2 Jν00 (x) + xJν0 (x) + (x 2 − ν 2 )Jν (x) = 0
I
series representation
ν+2k
∞
X
(−1)k x2
Jν (x) =
k !Γ(ν + k + 1)
k =0
I
different representations of covariance functions useful for
investigation of properties
Hans-Jörg Starkloff
Notes on stationary random processes and fields
33
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Exponentially correlated random processes I
I
I
I
I
basic object: measurable second order random process
(Xt ; t ∈ T ⊂ R) with properties
• m(t) := E{Xt } = 0 ∀ t ∈ T
• K (s, t) := E{Xs Xt } = σ 2 exp(−a|t − s|) ∀ t, s ∈ T
with parameters σ 2 > 0, a > 0 (⇒ weakly stationary)
Z ∞
1
−a|τ |
=
e
dτ is often called correlation length
a
0
covariance function K (s, t) is continuous ⇒ ∃ always a
measurable modification for stochastic processes with such a
covariance function
Z T
moreover, due to
R(t, t) dt < ∞ there exist a modification
0
with paths in the space L2 ([0, T ], dt)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
34
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Exponentially correlated random processes II
I
the corresponding spectral function and spectral density of such
a stationary process (on R) are
α
1 1
+ arctan
2 Zπ
a
∞
a
1
exp(−iαt)r (τ ) dτ =
f (α) =
2π −∞
π(a2 + α2 )
F (α) =
I
such processes are mean square continuous but not mean
square differentiable (because the autocovariance function is
continuous but not 2 times differentiable at τ = 0)
Hans-Jörg Starkloff
Notes on stationary random processes and fields
35
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Exponentially correlated random processes III
I
a Gaussian exponentially correlated stationary random process
is also strictly stationary and has a modification with continuous
paths: application of Kolmogorov’s theorem, using
E{|Xt − Xs |4 } ≤ 2a2 e2T |t − s|2
I
s, t ∈ [0, T ]
but there exist exponentially correlated random processes for
which there does not exist a modification with continuous paths !
Hans-Jörg Starkloff
Notes on stationary random processes and fields
36
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Ornstein-Uhlenbeck process I
I
I
centered Gaussian process (Xt ; t ∈ [0, ∞)) with covariance
function E{Xt Xs } = σ 2 exp(−a|t − s|)
is stationary solution of the stochastic differential equation
√
dXt = −a Xt dt + 2a σ dWt
or as integral eq.
Z t
√
Xt = X0 − a
Xs ds + 2a σ Wt
0
I
or solution of this stochastic differential equation with initial
condition X0 ∼ N (0, σ 2 ) indpendent of the standard Wiener
process (Wt ; t ∈ [0, ∞))
heuristic writing with white noise Ẇt :
√
Ẋt + a Xt = 2a σ Ẇt
Hans-Jörg Starkloff
Notes on stationary random processes and fields
37
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Ornstein-Uhlenbeck process II
I
properties of paths e.g. from representations
−at
Xt = X0 e
Xt = X0 e−at
√
Z
t
2a σ
ea(s−t) Ws ds +
0
Z t
√
+ 2a σ
ea(s−t) dWs
+a
√
2a σ Wt
0
I
time discretization on grid tk := kh, k ∈ N0 (h > 0) leads to
approximate random difference equation
√
Yk +1 = (1 − a h) Yk + 2a h σ ξk
with i.i.d. N (0, 1) r.v. (ξk ; k ∈ N) and Y0 := X0
Hans-Jörg Starkloff
Notes on stationary random processes and fields
38
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Matérn class of isotropic autocovariance functions I
I
I
I
S TEIN recommend to use Matérn class of isotropic
autocovariance functions
parameters can be chosen such that different mean square
smoothness properties can be achieved
for weakly stationary random process on R:
spectral density : with c > 0, a > 0, ν > 0
f (α) =
(a2
c
,
+ α2 )ν+1/2
α∈R
autocovariance function
√
c π
K (t) =
(a|t|)ν Kν (a|t|),
2ν−1 Γ ν + 21 a2ν
Hans-Jörg Starkloff
Notes on stationary random processes and fields
t ∈R
39
Introduction
Stationary and isotropic random fields
Examples
Exponentially correlated random processes
Matern class
Matérn class of isotropic autocovariance functions II
I
Kν is a modified Bessel function
I
parameter
ν defines mean square smoothness :
Z
if
α2m f (α) dα < ∞ then the process is m times mean square
R
differentiable
I
for values ν = m + 21 , m ∈ N0 rational spectral density, e.g.
1
2
3
ν=
2
ν=
I
⇒
⇒
cπ −a|t|
e
a
cπ
K (t) =
(1 + a|t|)e−a|t|
2a3
K (t) =
extends also to the case of random fields on Rd with d ≥ 2
Hans-Jörg Starkloff
Notes on stationary random processes and fields
40