Spatio-temporal stochastic models withembedded deterministic dynamics

Krzysztof Podgorski

28th April 2011

krys@maths.lth.se

What do we want to do?

• In search: a mathematical model X (p, t) that accounts bothfor motion and stochastic variability.

krys@maths.lth.se

What do we want to do?

• In search: a mathematical model X (p, t) that accounts bothfor motion and stochastic variability.

krys@maths.lth.se

What was done? or How to move around stochastics?

• Stochastic differential equations:

L(X , ∂X/∂p, ∂X/∂t,p, t) = dW (p, t)

• Dispersion relations encoded in spectrum:

X (p, t) =∑ω

s(ω) · cos(ω2

gp + ωt

)

krys@maths.lth.se

What was done? or How to move around stochastics?

• Stochastic differential equations:

L(X , ∂X/∂p, ∂X/∂t,p, t) = dW (p, t)

• Dispersion relations encoded in spectrum:

X (p, t) =∑ω

s(ω) · cos(ω2

gp + ωt

)

krys@maths.lth.se

What do we do here?

Essentially, we combine together well-established components:

• Deterministic Model- Physical phenomenon- Deterministic flow

• Stochastic Model- Covariance structures- ’Static’ stochastic flow

• Stochastic-Deterministic Model- Embedding deterministic flow into static stochastic flow- Dynamical stochastic flow

krys@maths.lth.se

What do we do here?

Essentially, we combine together well-established components:

• Deterministic Model- Physical phenomenon- Deterministic flow

• Stochastic Model- Covariance structures- ’Static’ stochastic flow

• Stochastic-Deterministic Model- Embedding deterministic flow into static stochastic flow- Dynamical stochastic flow

krys@maths.lth.se

What do we do here?

Essentially, we combine together well-established components:

• Deterministic Model- Physical phenomenon- Deterministic flow

• Stochastic Model- Covariance structures- ’Static’ stochastic flow

• Stochastic-Deterministic Model- Embedding deterministic flow into static stochastic flow- Dynamical stochastic flow

krys@maths.lth.se

What do we do here?

Essentially, we combine together well-established components:

• Deterministic Model- Physical phenomenon- Deterministic flow

• Stochastic Model- Covariance structures- ’Static’ stochastic flow

• Stochastic-Deterministic Model- Embedding deterministic flow into static stochastic flow- Dynamical stochastic flow

krys@maths.lth.se

Motivation

A need for models that account for:• Variability in time - non-trivial, physically meaningful dynamics• Stochastic variability

One approach is to have a differential equation that is driven by astochastic noise.We propose a reverse:

Stochastic field driven by deterministic flow.

krys@maths.lth.se

Motivation

A need for models that account for:• Variability in time - non-trivial, physically meaningful dynamics• Stochastic variability

One approach is to have a differential equation that is driven by astochastic noise.We propose a reverse:

Stochastic field driven by deterministic flow.

krys@maths.lth.se

Motivation

A need for models that account for:• Variability in time - non-trivial, physically meaningful dynamics• Stochastic variability

One approach is to have a differential equation that is driven by astochastic noise.We propose a reverse:

Stochastic field driven by deterministic flow.

krys@maths.lth.se

Motivation

A need for models that account for:• Variability in time - non-trivial, physically meaningful dynamics• Stochastic variability

One approach is to have a differential equation that is driven by astochastic noise.We propose a reverse:

Stochastic field driven by deterministic flow.

krys@maths.lth.se

Motivation

A need for models that account for:• Variability in time - non-trivial, physically meaningful dynamics• Stochastic variability

One approach is to have a differential equation that is driven by astochastic noise.We propose a reverse:

Stochastic field driven by deterministic flow.

krys@maths.lth.se

Is a stochastic field moving?

krys@maths.lth.se

Is a stochastic field moving?

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krys@maths.lth.se

Measuring velocities of a moving surface

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krys@maths.lth.se

Stochastic Velocity Field

• One of possible definitions of velocity on random surfaces

v(x , t) = −Xt(x , t)

Xx(x , t)= − ∂X (x , t)t

∂X (x , t)x,

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krys@maths.lth.se

Spatial Static Model

The starting point is the following spectral representation of astationary process

X (p)d=

∫Rn

exp(ip · ω)√

S(ω) dB(ω),

Non-stationary extension:

X (p)d=

∫Rn

exp(ip · ω)√

Sp(ω) dB(ω).

The covariance of X is given by

rS(p,p′) =

∫Rn

exp(i(p− p′) · ω)√

Sp(ω)Sp′(ω) dω,

and is non-stationary.

krys@maths.lth.se

Spatial Static Model

The starting point is the following spectral representation of astationary process

X (p)d=

∫Rn

exp(ip · ω)√

S(ω) dB(ω),

Non-stationary extension:

X (p)d=

∫Rn

exp(ip · ω)√

Sp(ω) dB(ω).

The covariance of X is given by

rS(p,p′) =

∫Rn

exp(i(p− p′) · ω)√

Sp(ω)Sp′(ω) dω,

and is non-stationary.

krys@maths.lth.se

Spatial Static Model

The starting point is the following spectral representation of astationary process

X (p)d=

∫Rn

exp(ip · ω)√

S(ω) dB(ω),

Non-stationary extension:

X (p)d=

∫Rn

exp(ip · ω)√

Sp(ω) dB(ω).

The covariance of X is given by

rS(p,p′) =

∫Rn

exp(i(p− p′) · ω)√

Sp(ω)Sp′(ω) dω,

and is non-stationary.

krys@maths.lth.se

Example

Take

Sp(ω) =s2(p)Ln(p)

2πn/2 exp(−L2(p)|ω|2/2

),

The non-stationary covariance of X is given by

rS(p,p′) = s(p)s(p′)(

2L(p)L(p′)L2(p) + L2(p′)

)− n2

exp(−(p− p′)TΣ(p− p′)

2

)where Σ = Σ(p,p′) = 2/

(L2(p) + L2(p′)

)· I.

krys@maths.lth.se

Example

Take

Sp(ω) =s2(p)Ln(p)

2πn/2 exp(−L2(p)|ω|2/2

),

The non-stationary covariance of X is given by

rS(p,p′) = s(p)s(p′)(

2L(p)L(p′)L2(p) + L2(p′)

)− n2

exp(−(p− p′)TΣ(p− p′)

2

)where Σ = Σ(p,p′) = 2/

(L2(p) + L2(p′)

)· I.

krys@maths.lth.se

Spatio-temporal static fields

Assuming that Φ(·; ds) is a Gaussian field-valued measure that isuniquely characterized by the time dependent spatial covariancesrS(p,p′; s), we can define

X (p, t) =

∫f (t, s; p) Φ(p; ds),

which will have the covariance

r(p,p′; t, t ′) =

∫f (t, s; p)f (t ′, s; p′) · rS(p,p′; s) ds

krys@maths.lth.se

Spatio-temporal static fields

Assuming that Φ(·; ds) is a Gaussian field-valued measure that isuniquely characterized by the time dependent spatial covariancesrS(p,p′; s), we can define

X (p, t) =

∫f (t, s; p) Φ(p; ds),

which will have the covariance

r(p,p′; t, t ′) =

∫f (t, s; p)f (t ′, s; p′) · rS(p,p′; s) ds

krys@maths.lth.se

Example: Temporal Ornstein-Uhlenbeck field

Taking f (t) = e−λt1[0,∞)(t) gives

X (p, t) =

∫ t

−∞e−λ(t−s) Φ(p; ds)

and if rS(p,p′; s) = rS(p,p′), its covariance is given by

r(p,p′; t) = rS(p,p′) · 12λ

e−λ|t|.

This example corresponds to the autoregression model of order one

X (p, t) = ρX (p, t −∆t) +√

1− ρ2 Φt(p),

Taking a space dependent λ(p) gives

r(p,p′; t) =rS(p,p′)

λ(p) + λ(p′)

{e−λ(p′)·t ; if t > 0,e−λ(p)·t ; if t < 0.

krys@maths.lth.se

Example: Temporal Ornstein-Uhlenbeck field

Taking f (t) = e−λt1[0,∞)(t) gives

X (p, t) =

∫ t

−∞e−λ(t−s) Φ(p; ds)

and if rS(p,p′; s) = rS(p,p′), its covariance is given by

r(p,p′; t) = rS(p,p′) · 12λ

e−λ|t|.

This example corresponds to the autoregression model of order one

X (p, t) = ρX (p, t −∆t) +√

1− ρ2 Φt(p),

Taking a space dependent λ(p) gives

r(p,p′; t) =rS(p,p′)

λ(p) + λ(p′)

{e−λ(p′)·t ; if t > 0,e−λ(p)·t ; if t < 0.

krys@maths.lth.se

Example: Temporal Ornstein-Uhlenbeck field

Taking f (t) = e−λt1[0,∞)(t) gives

X (p, t) =

∫ t

−∞e−λ(t−s) Φ(p; ds)

and if rS(p,p′; s) = rS(p,p′), its covariance is given by

r(p,p′; t) = rS(p,p′) · 12λ

e−λ|t|.

This example corresponds to the autoregression model of order one

X (p, t) = ρX (p, t −∆t) +√

1− ρ2 Φt(p),

Taking a space dependent λ(p) gives

r(p,p′; t) =rS(p,p′)

λ(p) + λ(p′)

{e−λ(p′)·t ; if t > 0,e−λ(p)·t ; if t < 0.

krys@maths.lth.se

Example: Temporal Ornstein-Uhlenbeck field

Taking f (t) = e−λt1[0,∞)(t) gives

X (p, t) =

∫ t

−∞e−λ(t−s) Φ(p; ds)

and if rS(p,p′; s) = rS(p,p′), its covariance is given by

r(p,p′; t) = rS(p,p′) · 12λ

e−λ|t|.

This example corresponds to the autoregression model of order one

X (p, t) = ρX (p, t −∆t) +√

1− ρ2 Φt(p),

Taking a space dependent λ(p) gives

r(p,p′; t) =rS(p,p′)

λ(p) + λ(p′)

{e−λ(p′)·t ; if t > 0,e−λ(p)·t ; if t < 0.

krys@maths.lth.se

Is there non-trivial dynamics?

• If

r(p,p′; t, t ′) =

∫ ∞−∞

f (t, s) · f (t ′, s) · rS(p,p′; s) ds.

then the dynamics of the field is trivial (i.e. velocities arecentered at zero).

• Thus the field

X (p, t) =

∫f (t, s) Φ(p; ds),

does not exhibit any organized motion.

krys@maths.lth.se

Is there non-trivial dynamics?

• If

r(p,p′; t, t ′) =

∫ ∞−∞

f (t, s) · f (t ′, s) · rS(p,p′; s) ds.

then the dynamics of the field is trivial (i.e. velocities arecentered at zero).

• Thus the field

X (p, t) =

∫f (t, s) Φ(p; ds),

does not exhibit any organized motion.

krys@maths.lth.se

Feeding deterministic flow into stochastic field

• Flow ψt,h(p) obtained from a velocity field v(p, t) satisfyingthe transport equation

ψt,h(p) = p+

∫ t+h

tv(ψt,u−t(p), u) du = p+

∫ h

0v(ψt,s(p), t+s) ds,

Construction of the stochastic field at a fixed location p and afixed time t:

Y (p, t) =

∫ ∞−∞

f (t, s) Φ(ψt,s−t(p); ds)

krys@maths.lth.se

There is a method in the madness

TheoremLet the spatial covariance rS of the innovations Φ(x , y ; dt) beisotropic (i.e. invariant under rotation), then the distribution ofrandom velocities on the surface of Y (p, t) has its center at thevalue of the deterministic velocity field v(x , y , t).

In other words, stochastic dynamics follows the one represented bythe underlying deterministic field.

krys@maths.lth.se

There is a method in the madness

TheoremLet the spatial covariance rS of the innovations Φ(x , y ; dt) beisotropic (i.e. invariant under rotation), then the distribution ofrandom velocities on the surface of Y (p, t) has its center at thevalue of the deterministic velocity field v(x , y , t).

In other words, stochastic dynamics follows the one represented bythe underlying deterministic field.

krys@maths.lth.se

Example: Dynamics driven by shallow water equations

∂u∂t

= −g∂h∂x− u

∂u∂x− v

∂u∂y

+ fv

∂v∂t

= −g∂h∂y− u

∂v∂x− v

∂v∂y− fu

∂h∂t

= −∂(hu)

∂x− ∂(hv)

∂y

• where: u, v denote horizontal velocities, h surface elevation• Used to model: Tsunamis, flows in rivers, internal waves,Jupiter’s Atmopshere

krys@maths.lth.se

Example cont.: Linearization in one dimension

After linearization and reduction to one dimension.• PDE with constant coefficients:

∂u∂t

= −u0∂u∂x− g

∂h∂x

∂h∂t

= −u0∂h∂x− h0

∂u∂x

,

whereu0, h0, g are constants

krys@maths.lth.se

1-dim SWE simulation

FILM

krys@maths.lth.se

Theorem – figures

Figure: Stochastically distorted flow (left) and deterministic shallowwater flow (right).

krys@maths.lth.se

A very incomplete and terribly biased bibliography

• Baxevani, A., Borget, C., Rychlik, I. (2008) Spatial Models for the Variability of the SignificantWave Height on the World Oceans.

• Baxevani, A., Podgórski, K., Rychlik, I. (2003) Velocities for moving random surfaces,• Baxevani, A., Podgórski, K., Rychlik, I. (2011) Dynamically evolving Gaussian spatial fields.• Cox, D.R., Isham, V.S. (1988) A simple spatial-temporal model of rainfall.• Gupta, V.K., Waymire, E. (1987) On Taylor’s hypothesis and dissipation in rainfall.• Longuet-Higgins, M. S (1957) The statistical analysis of a random, moving surface.• Porcu, E., Mateu, J., and Christakos, G., (2009) Quasi-arithmetic means of covariance functions

with potential applications to space-time data.• Schlather, M. (2009) Some covariance models based on normal scale mixtures.• Stein, M.L. (2005) Nonstationary spatial covariance functions• Wegener, J. (2010) Ph.D. Thesis, Lund University.

krys@maths.lth.se

Final Slide

What does Mother Earth think of our methods?

krys@maths.lth.se

Final Slide

What does Mother Earth think of our methods?

We asked...

krys@maths.lth.se

Final Slide

krys@maths.lth.se