Handbook of Spatial Statistics
Chapter 2: Continuous Parameter Stochastic
Process Theory
by Gneiting and Guttorp
Marcela Alfaro Córdoba
August 25, 2016
NCSU Department of Statistics
Continuous Parameter Stochastic Process Theory
• Topics to cover:
• Spatial stochastic process: definition and notation.
• Stationary and intrinsically stationary process, nugget effect.
• Bochner’s theorem, isotropic covariance functions, smoothness
properties and examples.
• Prediction theory for second-order stationary processes.
1
Spatial Stochastic Process
Spatial Stochastic Processes {Y (s) : s ∈ D ⊆ R}, where:
Y (s) = Y (s, ω)
is a collection of random variables with a well-defined joint
distribution.
2
Spatial Stochastic Process
Spatial Stochastic Processes {Y (s) : s ∈ D ⊆ R}, where:
Y (s) = Y (s, ω)
is a collection of random variables with a well-defined joint
distribution.
• If we restrict attention to any fixed, finite set of spatial
locations [s1 , ...., sn ] ⊂ D, then (Y (s1 ), ..., Y (sn ))T is a
random vector, whose multivariate distribution reflects the
spatial dependencies in the variable of interest.
2
Spatial Stochastic Process
Spatial Stochastic Processes {Y (s) : s ∈ D ⊆ R}, where:
Y (s) = Y (s, ω)
is a collection of random variables with a well-defined joint
distribution.
• If we restrict attention to any fixed, finite set of spatial
locations [s1 , ...., sn ] ⊂ D, then (Y (s1 ), ..., Y (sn ))T is a
random vector, whose multivariate distribution reflects the
spatial dependencies in the variable of interest.
• If we fix any elementary event ω ∈ Ω then
{Y (s, ω) : s ∈ D ⊆ R} and
(y1 , ..., yn )T = (Y (s1 , ω), ..., Y (sn , ω))T are realizations of the
spatial stochastic process and the induced random vector.
2
Spatial Stochastic Process
Kolmogorov existence theorem the stochastic process model is
valid if the family of the finite-dimensional joint distributions is
consistent under reordering of the sites and marginalization.
• F (y1 , ...., yn ; s1 , ...., sn ) = P(Y (s1 ) ≤ y1 , ..., Y (sn ) ≤ yn )
• A Gaussian Process is a process where the finite-dimensional
distributions are multivariate normal. In this particular case,
the consistency conditions of the Kolmogorov existence
theorem reduce to the usual requirements that covariance
matrices are nonnegative definite.
3
Stationary and Intrinsically Stationary Process
A spatial stochastic process is strictly stationary if the finite
dimensional joint distributions are invariant under spatial shifts.
This means:
F (y1 , ...., yn ; s1 + h, ...., sn + h) = F (y1 , ...., yn ; s1 , ...., sn )
4
Stationary and Intrinsically Stationary Process
A spatial stochastic process is strictly stationary if the finite
dimensional joint distributions are invariant under spatial shifts.
This means:
F (y1 , ...., yn ; s1 + h, ...., sn + h) = F (y1 , ...., yn ; s1 , ...., sn )
• A process is weakly stationary or second-order stationary if
E(Y (s)) = E(Y (s + h)) = µ
and
Cov (Y (s), Y (s + h)) = Cov (Y (0), Y (h)) = C (h)
where the function C (h), h ∈ Rd , is the covariance function.
4
Stationary and Intrinsically Stationary Process
A spatial stochastic process is strictly stationary if the finite
dimensional joint distributions are invariant under spatial shifts.
This means:
F (y1 , ...., yn ; s1 + h, ...., sn + h) = F (y1 , ...., yn ; s1 , ...., sn )
• A process is weakly stationary or second-order stationary if
E(Y (s)) = E(Y (s + h)) = µ
and
Cov (Y (s), Y (s + h)) = Cov (Y (0), Y (h)) = C (h)
where the function C (h), h ∈ Rd , is the covariance function.
• All Gaussian processes are second-order stationary:
http://teaching.stat.ncsu.edu/shiny/bjreich/
Nonstationary/.
4
Stationary and Intrinsically Stationary Process
A semivariogram is defined as:
1
γ(h) = var (Y (s + h) − Y (s)) = C (0) − C (h)
2
5
Stationary and Intrinsically Stationary Process
A semivariogram is defined as:
1
γ(h) = var (Y (s + h) − Y (s)) = C (0) − C (h)
2
• The variogram can be used in some cases where a covariance
function does not exist.
5
Stationary and Intrinsically Stationary Process
A semivariogram is defined as:
1
γ(h) = var (Y (s + h) − Y (s)) = C (0) − C (h)
2
• The variogram can be used in some cases where a covariance
function does not exist.
• An intrinsically stationary process are such that certain spatial
increments are second-order stationary, so that a generalized
covariance function can be defined.
5
Nugget effect
A spatial stochastic process can be decomposed as:
Y (s) = µ(s) + ν(s) + (s)
| {z } |{z} |{z} |{z}
1
2
3
4
1. spatial stochastic process
2. µ(s) = E(Y (s)) mean function (deterministic and smooth)
3. process with mean 0 and continuous realizations
4. field of spatially uncorrelated mean 0 error and covariance:
(
σ 2 ≥ 0, h = 0
Cov ((s), (s + h)) =
0,
h 6= 0
also referred to as Nugget effect.
6
Nugget effect
Source: ”Model Averaging for Semivariogram Model Parameters”. By Asim Biswas
and Bing Cheng Si
7
Nugget effect
Source: ”Handbook of Spatial Statistics”, Chapter 2
8
Bochner’s Theorem
{Y (s) : s ∈ Rd } is a second-order stationary spatial stochastic
process with covariance function C . Given any finite set of spatial
locations s1 , ..., sn ∈ Rd , the covariance matrix of the finite
dimensional joint distribution is:


C (0)
C (s1 − s2 ) · · · C (s1 − sn )


C (0)
· · · C (s2 − sn ) 
 C (s2 − s1 )


..
..
..
..


.
.
.
.


C (sn − s1 ) C (sn − s2 ) · · ·
C (0)
which needs to be valid (that is, nonnegative definite) covariance
matrix.
9
Bochner’s Theorem
A real-valued continuous function C is positive definite if and only
if it is the Fourier transform of a symmetric, nonnegative measure
F on Rd , that is, if and only if:
Z
Z
T
exp(ih x) dF(x) =
cos(hT x) dF(x).
C (h) =
Rd
Rd
This is the spectral representation of the covariance function. In
most cases, the spectral measure, F, has a Lebesque density f ,
called the spectral density, so that
Z
Z
T
C (h) =
exp(ih x)f (x) dx =
cos(hT x)f (x) dx.
Rd
Rd
10
Isotropic Covariance Functions
An Isotropic covariance function C (h) depends on the spatial
separation vector, h, only through its Euclidean length, ||h||.
Without loss of generality we may assume C (0) = 1, then we can
write:
C (h) = ϕ(||h||),
h∈R
for some continuous function ϕ : [0, ∞) → R with ϕ(0) = 1. Let
Φd denote the class of the continuous functions ϕ that generate
valid isotropic covariance function in Rd . It follows:
\
Φ1 ⊇ Φ2 · · ·
and
Φd ↓ Φ∞ =
Φd
d≥1
An isotropic correlation that is valid in one dimension, might not
be valid for higher dimensions, but the members of the class Φ∞
are valid in all dimensions.
11
Isotropic Covariance Functions
• A function ϕ : [0, ∞) → R belongs to the class Φd if and only
if is of the form:
Z
ϕ(t) =
Ωd (rt)dF0 (r )
[0,∞)
where F0 is a probability measure on the positive half-axis,
often referred to as the radial spectral measure, and:
(d−2)/2
2
Ωd (t) = Γ(d/2)
J(d−2)/2 (t)
t
• The members of the class Φd are scale mixtures of a generator
Ωd , and as such they have lower bound inf t≥0 Ωd (t).
12
Isotropic Covariance Functions
Source: ”Handbook of Spatial Statistics”, Chapter 2
• It can be difficult to check whether or not a function ϕ
belongs to the class Φd and generates a valid isotropic
covariance function.
13
Smoothness Properties
A spatial stochastic process {Y (s) : s ∈ Rd } is called mean square
continuous if E(Y (s) − Y (s + h))2 → 0 as ||h|| → 0. For a
second-order stationary process,
E(Y (s) − Y (s + h))2 = 2(C (0) − C (h)),
mean square continuity is equivalent to the covariance function
being continuous at the origin.
For isotropic process, the properties of the member ϕ of the class
Φd translate into properties of the associated Gaussian spatial
process Rd . In particular, the behavior of ϕ(t) at the origin
determines the smoothness of the sample paths.
1 − ϕ(t) ∼ t α
as
t↓0
14
Smoothness Properties
Source: ”Handbook of Spatial Statistics”, Chapter 2
15
Examples of Isotropic Covariance Functions
Table 1: Isotropic Covariance Function families that belong to Φ∞
Class
Matérn
Class
Powered
Exponential
Family
Cauchy
Family
Formula
ϕ(t) =
21−ν t ν
Γ(ν) ( θ ) Kν (t/θ)
ϕ(t) = exp (−( θt )α )
ϕ(t) = (1 + ( θt )α )−β/α
Characteristics
(ν > 0, θ > 0)
are smoothness
and scale parameters
(0 < α ≤ 2, θ > 0)
are smoothness
and scale parameters
(0 < α ≤ 2, β > 0, θ > 0)
are smoothness
and scale parameters
16
Examples of Isotropic Covariance Functions
• All the aforementioned families generate valid isotropic
covariance functions in all spatial dimensions, they admit the
representation:
Z
ϕ(t) =
exp(−r 2 t 2 )dF (r )
[0,∞)
and thus they are strictly positive and strictly decreasing.
• In practice, these assumptions might be too restrictive.
Examples: hole effect, compact support.
17
Prediction Theory for Second-Order Stationary Processes
The most common problem in spatial statistics is to predict, or
interpolate, the value of the process at a location s ∈ R where no
observation has been made. The optimal predictor Ŷ (s) is the
conditional expectation given the observations:
Ŷ (s) = E(Y (s)|Y (s1 ) = y1 , ...., Y (sn ) = yn ).
Gaussian case ordinary kriging predictor:


Y (s1 ) − µ


..
Ŷ (s) = µ + (C (s − s1 ), · · · , C (s − sn ))[C (si − sj )]−1 

.
Y (sn ) − µ
or


C (s − s1 )


..
Ŷ (s) = µ + (Y (s1 ) − µ, · · · , Y (sn ) − µ)[C (si − sj )]−1 

.
C (s − sn )
18