Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Non-uniform decay of predictability in
stochastic oscillatory models
N.Barlas2
K. Josič2
S. Lapin1
I. Timofeyev2
1 Department
of Mathematics
Washington State University
2 Department
of Mathematics
University of Houston
Jan.23.08 / Applied Math Seminar
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Outline
1
Introduction
2
Relative entropy for SDEs
Definition of Relative Entropy
Properties of Relative entropy
3
The stochastic linear oscillator
The decay of relative entropy
Return of skill for marginal entropies
4
Nonuniform decay of R(t) in general systems
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Introduction
The weather prediction is extremely sensitive to the
specification of initial conditions
Detailed forecasts are impossible beyond 2 weeks limit
Due to this atmospheric uncertainty statistical prediction
involving ensemble of possible projections has become
common
The ensemble spread may vary considerably meaning that
some predictions are more reliable then others
Need measure to indicate the usefulness of forecast
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Definition of relative entropy.
Due to uncertainty the values of initial conditions are given
by certain probability distribution p.
This distribution evolves in time and approaches an
equilibrium distribution q
Need to measure usefulness of a prediction, i.e. how much
information is added by availability of prediction
The relative entropy gives the information loss sustained by
assuming climatology when prediction is available.
Z
p(~x , t)
d ~x .
R(p(~x , t), q(~x )) = R(t) = p(~x , t) log
q(~x )
q(~x ) the invariant (climatological) distribution,
p(~x , t) the probability density corresponding to the
ensemble.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Properties of Relative entropy.
Relative entropy satisfies three important properties:
Invariant under well behaved non–linear transformations of
state variables.
Non–negative.
Declines monotonically in time for Markov processes.
The fact that relative entropy decreases in time can be
interpreted as a decline in the utility of a prediction, or skill.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Relative entropy for Gaussian distribution
R=

1
log
2
det(σq2 )
det(σp2 )
!


+ Tr(σp2 (σq2 )−1 ) + (µp )T (σq2 )−1 (µp ) −n .
{z
}
|
signal term
For the Gaussian case the relative entropy can be decomposed
into
the signal component =⇒ the difference in the means of
the two distributions,
the dispersion =⇒ the difference in the variances of the
two distributions.
The signal and dispersion terms are analogous to the Anomaly
Correlation Coefficient and Root Mean Squared Error,
respectively.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Fokker-Planck equation
X ∂ 1 X ∂2 ∂p(~x , t)
=−
Ai (~x )p(~x , t) +
Bij (~x )p(~x , t) .
∂t
∂xi
2
∂xi ∂xj
i,j
i
A(~x , t) is the drift vector,
B(~x , t) is the diffusion matrix,
the distribution p(~x , 0) provides the initial data for the FP
equation.
assume that the system has a unique equilibrium solution
q(~x ).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Recall the definition of Relative entropy:
Z
p(~x , t)
d ~x .
R(t) = p(~x , t) log
q(~x )
Using the definition we obtain
Z
dR
∂p(~x , t)
∂q(~x ) p(~x , t)
= d ~x
(log p(~x , t) + 1 − log q(~x )) −
.
dt
∂t
∂t
q(~x )
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Definition of Relative Entropy
Properties of Relative entropy
Drift and diffusion contributions to dR/dt:
XZ
dR
p(~x , t)
∂
.
=
−Ai p(~x , t) log
d ~x
dt drift
∂xi
q(~x )
i
Z
p(~x , t)
p(~x , t)
dR
∂
1X
∂
log
log
=−
d ~x p(~x , t)Bij
.
dt diff
2
∂xi
∂xj
q(~x )
q(~x )
i,j
It can be shown that
dR
=0
dt drift
and
Barlas, Josič, Lapin, Timofeyev,
dR
≤ 0.
dt diff
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
The stochastic linear oscillator
Several low-dimensional models of El Niño/Southern Oscillation
(ENSO) have been proposed in recent years. One common
feature among them is the oscillatory behavior of solutions. As
a prototype behavior we consider a simple two-dimensional
model described in T. Kestin et all. "Time frequency variability
of ENSO and stochastic simulations"
dx1 = αx1 dt + βx2 dt,
dx2 = γx1 dt + δx2 dt + εdW ,
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
The Fokker-Planck equation is
X ∂ 1 X ∂2 ∂p(~x , t)
=−
Ai (~x )p(~x , t) +
Bij (~x )p(~x , t) ,
∂t
∂xi
2
∂xi ∂xj
i,j
i
with
A=
α β
γ δ
and
B=
0 0
0 ε
.
This equation can be solved analytically assuming a
deterministic initial condition p(~x , 0) = δ~x 0 (~x ) where
~x 0 = (x10 , x20 )
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
The solution at time t is a Gaussian with mean
x̄1
x1 (0)
At
=e
,
x̄2
x2 (0)
and covariance matrix
Z t
0
T
0
0 0
< x1 , x1 > < x1 , x2 >
dt 0 eA(t−t )
=
eA (t−t ) .
2
< x2 , x1 > < x2 , x2 >
0
ε
0
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
2
12
10
8
x2 0
R(t) 6
4
2
-2
-1
0
1
x1
5
10
15
20
25
30
t
Figure: Left: The equilibrium distribution q(~x ) for the linear oscillator.
Right: Average relative entropy. Parameters:
α = 0.4573, β = 0.2435, γ = δ = −1.0852, and ε = 0.1.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
For the stochastic linear oscillator the diffusion part of the
corresponding Fokker-Planck equation reduces to
2
Z
dR
∂
ε
(log(p(~x , t)/q(~x ))) .
=−
d ~x p(~x , t)
dt diff
2
∂y
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
For the stochastic linear oscillator the diffusion part of the
corresponding Fokker-Planck equation reduces to

2
Z
Z
dR
∂ log
ε
∂ log p(~x , t)

− 2 d ~x p(~x , t)
= −  d ~x p(~x , t)
dt diff
2
∂y
∂
{z
|
{z
} |
I2
I1
+
Z
|
d ~x p(~x , t)
∂ log q(~x )
∂y
{z
I3
Barlas, Josič, Lapin, Timofeyev,

2 

 ≡ I1 + I2 + I3 .

}
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
1
t
60
5
50
10
15
20
25
-1
40
R(t)
-2
dR/dt
I3
30
-3
-4
20
-5
0
5
10
15
t
20
25
30
-6
Figure: Left: Relative entropy for a particular initial condition
(x10 , x20 ) = (1, 1) (solid), and the contribution to the relative
entropy
due to the signal term (dashed). Right: Behavior of dR
dt diff (solid)
0
and I3 term (dashed) in time for an initial condition (x1 , x20 ) = (1, 1).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
30
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
6
6
6
Transient
Distribution
4
x2 2
4
4
x2 2
x2 2
Transient
Distribution
Equilibrium
Distribution
0
0
0
Equilibrium
Distribution
Equilibrium
Distribution
Transient
Distribution
-2
-2
-3
-2
-1
x1
0
-2
-3
1
-2
-1
x1
1
0
x2
-3
-1
x1
-2
x2
1.2
1
1
1
0.8
0.8
0.8
0.2
0.2
0.2
-1
0.4
0.4
0.4
-2
0.6
0.6
0.6
-3
1
0
x2
1
x1
-3
-2
-1
1
x1
-3
-2
-1
1
Figure: Probability density functions of the transient distribution
(dashed line) and the equilibrium distribution (solid line) for times
t=17.5 (left), t=25 (center) and t=35 (right).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
x1
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
Return of skill for marginal entropies
The marginal relative entropies do not necessarily decay in
time.
Consider conditional relative entropies
Rx2 |x1 (t) = R(p(x2 |x1 , t), q(x2 |x1 ))
Z
Z
p(x2 |x1 , t)
=
p(x1 , t) p(x2 |x1 , t) log
dx1 dx2 .
q(x2 |x1 )
p(x2 |x1 , t) =⇒ the conditional distribution of x2 at time t
given x1
Rx1 |x2 (t) =⇒ excess information provided by the marginal
distribution p(x2 |x1 , t) over q(x2 |x1 ).
R(t) = Rx2 |x1 (t) + Rx1 (t).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The decay of relative entropy
Return of skill for marginal entropies
7
50
6
Mean{x1,2 }
5
Rx1 (t)
4
R x2 (t)
3
R x1 |x2 (t)
2
R x2 |x1 (t)
30
20
10
1
0
40
5
10
15
20
25
30
t
0
5
10
15
t
20
25
Figure: Left: means of the variables |x1 | (solid line) and |x2 | (dashed
line). Right: marginal relative entropies Rx1 (t) (solid line) and Rx2 (t)
(dashed line) and Rx1 |x2 (t) (solid-star line) and Rx2 |x1 (t) (dashed-star
line) for an initial condition (x10 , x20 ) = (1, 1).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
30
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
Nonuniform decay of R(t) in general systems
The nonuniform decay of relative entropy occurs whenever
the main mass of the distribution p(~x , t) approaches, and
then diverges from the main mass of the stationary
distribution q(~x ).
Oscillations in the marginal relative entropies occur when
such divergence occurs in the marginal distributions.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
Hopf bifurcation
Supercritical Hopf bifurcation:
dx1 = µx1 dt − cωx2 dt + Θx1 (x12 + c 2 x22 )dt + εdW1 ,
dx2 =
1
ωx1 + cµx2 + cΘx2 (x12 + c 2 x22 ) dt + εdW2 .
c
In the absence of white noise this system has a stable periodic
orbit
r
r
1
µ
µ
x2 (t) =
− sin(ωt+φ0 ).
x1 (t) = − cos(ωt+φ0 ),
Θ
c
Θ
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
20
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
20
Equilibrium
distribution
10
10
Transient
distribution
x2 0
Equilibrium
distribution
x2 0
−10
−10
−20
−2
−20
−2
Transient
distribution
−1
0
x1
1
2
6
−1
0
x1
1
2
4.5
4
5
3
4
R x 1 (t)
R x 2 (t)
R(t)
3
2
2
1
1
0
1
2
3
Barlas,
Josič,
Timofeyev,
4
5
6 Lapin,
7
8
9
10
0.5
Predictability
0
1
2in stochastic
3
4
5oscillatory
6
7 models
8
9
10
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
Duffing equations
The Duffing equations.
dx1 = x2 dt + εdW1 ,
dx2 = (x1 − x13 − γx2 + βx12 x2 )dt + εdW2 ,
For ε = 0 and parameters γ = 0.4 and β = 0.497 this system
has an attracting double homoclinic cycle to the saddle point at
the origin.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
1
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
1
Position of
initial conditions
x2 0
x2 0
−1
1.5
0
x1
1.5
−1
-1.5
0
x1
Figure: Left: Homoclinic loop for Duffing equation with ε = 0. Right:
Contour plot of probability density function for ε = 0.01.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
1.5
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
Recall: the decay in relative entropy is due to the (dR/dt)diff
term. For the Duffing model the diffusion term becomes
Z
X ∂ ε
p(x1 , x2 ) 2
dR
.
=−
dx1 dx2 p(x1 , x2 )
log
dt diff
2
∂xi
q(x1 , x2 )
i=1,2
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
1
x2
x2
0
-1
1
1
0
x2 0
-1
-1.5
0
1.5
xx11
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
-1.5
0
-1
1.5
-1.5
0
x1
1.5
x1
4
3
3
2
R x 1x(t)
R x 2 (t)
R(t) 2
1
1
0
5
10
15
t
20
25
30
0
5
10
15
t
20
25
30
Figure: Top: Probability density function for time=5 (left), time=10
(middle) and time=15 (right). Bottom Left: Full relative entropy R(t),
Bottom Right: marginal relative entropies Rx1 (t) (solid) and Rx2 (t)
(dashed) in simulations of Duffing equation with initial ensemble
centered at (x10 , x20 ) = (0.25, 0.25) and ε = 0.01.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Decay of relative information for nonlinear oscillators
Stochastically Perturbed Duffing Equation
For the stochastic Duffing equation the behavior of the dR
dt diff
is more complicated thanin the case of linear oscillator.
Namely, the value of dR
dt diff depends not only on the means,
but also on all terms involving variances of the invariant and
transient distributions.
1.2
Var{x1,2 }
Mean{x1,2 }
1.4
0
-0.8
0
5
10
15
t
20
25
30
0.6
0
0
5
10
15
t
20
25
30
Figure: Left: mean in x1 (dashed line), mean in x2 (solid line), Right:
variance in x1 (dashed line), variance in x2 (solid line) in simulations
of the stochastic Duffing equation with ε = 0.01. (Horizontal lines
show equilibrium variances).
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
Summary
The averaged predictability decays exponentially in time
for all three considered models.
However, for particular ensemble simulation full relative
entropy decays non-monotonically and there is a return of
skill for the marginal entropies.
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models
Introduction
Relative entropy for SDEs
The stochastic linear oscillator
Nonuniform decay of R(t) in general systems
Summary
The End
THANK YOU
Barlas, Josič, Lapin, Timofeyev,
Predictability in stochastic oscillatory models