Dynamical models are widely used to describe chemical, physical and biochemical processes. The main challenge for this class of problems is the identification of kinetic parameters from given measurement data, the so called parameter estimation. However, parameters of such models are never exactly determined, due to measurement noise and the limited amount of data, but remain uncertain. This uncertainty can be captured by a probability density over the parameter space. Unfortunately, studying this probability density is often computationally demanding as this requires the repeated simulation of the underlying model. In this talk we will present a novel method for analysis of such probability densities using networks of radial basis functions. A particular characteristic of radial basis function approximation schemes is meshless nature, which allows for the free choice of sampling nodes. We will show that root lattices have optimality properties and propose a novel algorithm for the generation of lattices restricted to superlevel-sets. Furthermore we introduce an adaptive method for the generation of nodes based on interacting particles. Numerical examples show that our method can yield an expected L2 approximation error that is several orders of magnitude lower compared to classical approximations. This allows a drastic reduction of sampling points, which in turn facilitates the analysis of uncertainty for problems with high computational complexity.

1. Workshop on Numerical Methods for Optimal Control and Inverse Problems
Garching, 02/04/2014
Uncertainty Analysis for Dynamical
Systems using Radial Basis Functions
Fabian Froehlich, Jan Hasenauer, Fabian Theis
Institute for Computational Biology, Helmholtz Center Munich

2. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
In General:
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

3. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

4. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
ML(✓i) = p(✓i|D) =
Z
p(✓|D)d✓j6=i
Marginal Likelihood:
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

5. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
ML(✓i) = p(✓i|D) =
Z
p(✓|D)d✓j6=i
Marginal Likelihood:
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
• no closed form expression
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

6. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
ML(✓i) = p(✓i|D) =
Z
p(✓|D)d✓j6=i
Marginal Likelihood:
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
• no closed form expression
• numerically evaluable (expensive)
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

7. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
ML(✓i) = p(✓i|D) =
Z
p(✓|D)d✓j6=i
Marginal Likelihood:
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
• no closed form expression
• numerically evaluable (expensive)
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

8. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Problem Statement
2
Likelihood Function:
y(tk; ✓)
Dataset:
Solution to
the dynamical system:
D = {yj(tk), jk}
ny,nt
j,k=1
ML(✓i) = p(✓i|D) =
Z
p(✓|D)d✓j6=i
Marginal Likelihood:
Profile Likelihood:
PL✓i (c) = max
✓
L(✓) s.t. : ✓i = c
In General:
• no closed form expression
• numerically evaluable (expensive)
How can we compute
marginals?
L(✓) =
ntY
j=1
ny
Y
k=1
1
jk
p
2⇡
exp
(¯yj(tk) y(tk; ✓))
2
2 2
jk
!

9. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation

10. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative

11. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
nonnegative

12. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
nonnegative
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
real-valued

13. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Radial Basis Function
(✓, ⇠(k)
) = ✏(k✓kM,⇠(k) )
k✓kM,⇠ =
q
(✓ ⇠)T M 1(✓ ⇠),
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
nonnegative
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
real-valued

14. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Radial Basis Function
(✓, ⇠(k)
) = ✏(k✓kM,⇠(k) )
k✓kM,⇠ =
q
(✓ ⇠)T M 1(✓ ⇠),
−0.4 −0.2 0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
⇠
1
✏
✏(r) = exp( ✏2
r2
)
Gaussian Radial Basis
Allows analytic computation
of marginals
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
nonnegative
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
real-valued
θ

15. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 3
Kernel Based Approximation
Radial Basis Function
(✓, ⇠(k)
) = ✏(k✓kM,⇠(k) )
k✓kM,⇠ =
q
(✓ ⇠)T M 1(✓ ⇠),
−0.4 −0.2 0 0.2 0.4
0
0.2
0.4
0.6
0.8
1
⇠
1
✏
✏(r) = exp( ✏2
r2
)
Gaussian Radial Basis
Allows analytic computation
of marginals
Translates Integration Problem
into Approximation Problem!
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
nonnegative
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
nonnegative
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
real-valued
θ

16. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 4
Regularity Restriction on Centers
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)

17. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 4
Regularity Restriction on Centers
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
samples of L(𝜃)
θ1
θ2
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)

18. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 4
Regularity Restriction on Centers
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
samples of L(𝜃)
θ1
θ2
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
θ1
strong regularity
θ2
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)

19. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 4
Regularity Restriction on Centers
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
samples of L(𝜃)
θ1
θ2
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
θ1
strong regularity
θ2
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
weak regularity
θ1
θ2

20. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 4
Regularity Restriction on Centers
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
samples of L(𝜃)
θ1
θ2
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
θ1
strong regularity
θ2
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
weak regularity
θ1
θ2
How to generate the
centers?

21. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 5
00.20.4 −1 0 1 2 3
0
0.5
1
1.5
k1
−5
0
5
k2
0
0.5
1
1.5
2
prob. density
prob.density
θ1
θ2
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 1: Analytical Model
L =
4
5
N(µ(1)
, ⌃(1)
) +
1
5
N(µ(2)
, ⌃(2)
)
⌃(1)
=

0.1 0.25
0.25 1
, ⌃(2)
=

0.01 0.01
0.01 0.5
µ(1)
=

1
1
, µ(2)
=

0.5
1.5

22. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 5
00.20.4 −1 0 1 2 3
0
0.5
1
1.5
k1
−5
0
5
k2
0
0.5
1
1.5
2
prob. density
prob.density
θ1
θ2
• nonlinear correlation
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 1: Analytical Model
L =
4
5
N(µ(1)
, ⌃(1)
) +
1
5
N(µ(2)
, ⌃(2)
)
⌃(1)
=

0.1 0.25
0.25 1
, ⌃(2)
=

0.01 0.01
0.01 0.5
µ(1)
=

1
1
, µ(2)
=

0.5
1.5

23. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 5
00.20.4 −1 0 1 2 3
0
0.5
1
1.5
k1
−5
0
5
k2
0
0.5
1
1.5
2
prob. density
prob.density
θ1
θ2
• nonlinear correlation
• bimodal
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 1: Analytical Model
L =
4
5
N(µ(1)
, ⌃(1)
) +
1
5
N(µ(2)
, ⌃(2)
)
⌃(1)
=

0.1 0.25
0.25 1
, ⌃(2)
=

0.01 0.01
0.01 0.5
µ(1)
=

1
1
, µ(2)
=

0.5
1.5

24. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 5
00.20.4 −1 0 1 2 3
0
0.5
1
1.5
k1
−5
0
5
k2
0
0.5
1
1.5
2
prob. density
prob.density
θ1
θ2
• nonlinear correlation
• bimodal
• exact error analysis
possible
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 1: Analytical Model
L =
4
5
N(µ(1)
, ⌃(1)
) +
1
5
N(µ(2)
, ⌃(2)
)
⌃(1)
=

0.1 0.25
0.25 1
, ⌃(2)
=

0.01 0.01
0.01 0.5
µ(1)
=

1
1
, µ(2)
=

0.5
1.5

25. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 6
Generation of Centers (Samples)
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Generate Markov Chain that
samples by Metropolis-
Hastings Algorithm
(requires at least 1-5
evaluations of L(𝜃) per
sample)
θ2
θ1θ2

26. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 6
Generation of Centers (Samples)
Kernel Density
Estimation
L(✓) t
1
NX
k=1
(✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Generate Markov Chain that
samples by Metropolis-
Hastings Algorithm
(requires at least 1-5
evaluations of L(𝜃) per
sample)
θ2
θ1θ2

27. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 7
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Generation of Centers (Lattice)
⌅ = {⇠|⇠ = Mz, z 2 Zn
L(⇠) > }
θ2
θ1
• Different choices of lattice
bases are possible
depending on
dimensionality of
problem.
• It is possible to show
optimality properties of
certain root lattices for
interpolation using radial
basis functions.

28. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 7
Moving Least Squares
Approximation
L(✓) t
1
NX
k=1
L(⇠(k)
) (✓, ⇠(k)
)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Generation of Centers (Lattice)
⌅ = {⇠|⇠ = Mz, z 2 Zn
L(⇠) > }
θ2
θ1
• Different choices of lattice
bases are possible
depending on
dimensionality of
problem.
• It is possible to show
optimality properties of
certain root lattices for
interpolation using radial
basis functions.

29. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 8
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
Generation of Centers (Particles)
0 0.5 1 1.5 2
0
0.02
0.04
0.06
0.08
0.1
r
V
1
(r)
r
V(r)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Reboux et. al. 2012, Journal of Computational Physics
θ2
θ1
Repulsion Attraction
E =
X
p
X
q
D2
pqV
✓
k⇠q
⇠p
k
Dpq
◆
Dpq = min(Dp, Dq)
Dp = min
k⇠q ⇠pkr⇤Dp
˜D(⇠q
)
˜D(⇠) =
D0
p
1 + krL(⇠)k
minimize:
requires initialisation

30. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 8
Radial Basis Function
Approximation
L(✓) t
1
NX
k=1
wk(L(⇠(k)
)) (✓, ⇠(k)
)
Generation of Centers (Particles)
0 0.5 1 1.5 2
0
0.02
0.04
0.06
0.08
0.1
r
V
1
(r)
r
V(r)
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Reboux et. al. 2012, Journal of Computational Physics
θ2
θ1
Repulsion Attraction
E =
X
p
X
q
D2
pqV
✓
k⇠q
⇠p
k
Dpq
◆
Dpq = min(Dp, Dq)
Dp = min
k⇠q ⇠pkr⇤Dp
˜D(⇠q
)
˜D(⇠) =
D0
p
1 + krL(⇠)k
minimize:
requires initialisation

31. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Error Analysis (Example 1)
9

32. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
Error Analysis (Example 1)
9
function evaluations
L∞-errorinmarginals
10
1
10
2
10
3
10
4
10
5
10
−2
10
−1
10
0
10
1
KDE on samples
MLS on lattice
RBF on lattice
MLS on samples
RBF on samples
Kernel Density Estimation on samples
Radial Basis Function on lattice
Moving Least Squares on lattice
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)

33. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
10
1
10
2
10
3
10
4
10
−3
10
−2
10
−1
10
0
10
1
Function Evaluations
L∞-errorinmarginals
KDE on samples
RBF on particles with high-res. init.
RBF on particles with mid-res. init.
RBF on particles with low-res. init.
RBF on lattice
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
function evaluations
L∞-errorinmarginals
Kernel Density Estimation on samples
Radial Basis Function on lattice
Moving Least Squares on lattice
Radial Basis Function on samples (low res. init.)
Radial Basis Function on samples (mid res. init)
Radial Basis Function on samples (high res. init)
Error Analysis (Example 1)
9
function evaluations
L∞-errorinmarginals
10
1
10
2
10
3
10
4
10
5
10
−2
10
−1
10
0
10
1
KDE on samples
MLS on lattice
RBF on lattice
MLS on samples
RBF on samples
Kernel Density Estimation on samples
Radial Basis Function on lattice
Moving Least Squares on lattice
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)

34. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
10
1
10
2
10
3
10
4
10
−3
10
−2
10
−1
10
0
10
1
Function Evaluations
L∞-errorinmarginals
KDE on samples
RBF on particles with high-res. init.
RBF on particles with mid-res. init.
RBF on particles with low-res. init.
RBF on lattice
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
function evaluations
L∞-errorinmarginals
Kernel Density Estimation on samples
Radial Basis Function on lattice
Moving Least Squares on lattice
Radial Basis Function on samples (low res. init.)
Radial Basis Function on samples (mid res. init)
Radial Basis Function on samples (high res. init)
Error Analysis (Example 1)
9
function evaluations
L∞-errorinmarginals
10
1
10
2
10
3
10
4
10
5
10
−2
10
−1
10
0
10
1
KDE on samples
MLS on lattice
RBF on lattice
MLS on samples
RBF on samples
Kernel Density Estimation on samples
Radial Basis Function on lattice
Moving Least Squares on lattice
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Significantly less points required to
reach the same approximation quality!

35. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 10
A B
𝜃
𝜃2
1
0 1 2 3 4 5
0
0.5
1
1.5
Time
Concentration
A
B
time
concentration
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 2: Conversion Reaction
˙A = ✓1A + ✓2B A(0) = 1
˙B = +✓1A ✓2B B(0) = 0
D = {A(tk), k}5
k=1

36. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 11
Error in Marginals
RBF KDE
L 0.029 0.17
l 0.014 1.3411
N
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
KDE Marginal 437 pts.
RBF Marginal 437 pts.
−2 0 2
0
0.1
0.2
0.3
0.4
log(k )−1
−2
0
2
00.51
log(k)+1
0
1
2
0.5
prob. density
prob.density
KDE Marginal 10 pts.6
Error Analysis (Example 2)

37. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 11
Error in Marginals
RBF KDE
L 0.029 0.17
l 0.014 1.3411
N
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
KDE Marginal 437 pts.
RBF Marginal 437 pts.
−2 0 2
0
0.1
0.2
0.3
0.4
log(k )−1
−2
0
2
00.51
log(k)+1
0
1
2
0.5
prob. density
prob.density
KDE Marginal 10 pts.6
Error Analysis (Example 2)
5-10 fold higher accuracy
with same number of
function evaluations

38. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 12
off on
on
off
kr
mRNA (r)
r
protein (p)
p
kp
DNA (D)
Kazeroonian et. al. 2013, Proceedings of the 10th Workshop on Computational Systems Biology
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 2: Stochastic Gene Expression
• Chemical Master Equation
(Chapman-Kolmogorov Equation)
with 2562 dimensional state space
and 4 dimensional parameter
space
• 1 function evaluation takes from
several seconds up to minutes

39. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 12
off on
on
off
kr
mRNA (r)
r
protein (p)
p
kp
DNA (D)
Kazeroonian et. al. 2013, Proceedings of the 10th Workshop on Computational Systems Biology
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Example 2: Stochastic Gene Expression
• Chemical Master Equation
(Chapman-Kolmogorov Equation)
with 2562 dimensional state space
and 4 dimensional parameter
space
• 1 function evaluation takes from
several seconds up to minutes

40. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 13
−1 0 1 2 3
−5
0
5
log(θ1)
log(θ2)
Error Analysis (Example 3)
0 1 2 3 4 5
0
5
10
15
20
25
30
time t
Proteinmoleculenumber
time
proteinmoleculenumber
1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
30
35
40
log(km)
probabilitydensityML(𝜃)
log(km)
−3.4 −3.2 −3 −2.8 −2.6
0
2
4
6
8
10
12
14
log(τof f)
probabilitydensity
log(𝞃off)
ML(𝜃)
1 1.2 1.4 1.6 1.8
0
10
20
30
40
50
60
log(kp)
probabilitydensity
log(kp)
ML(𝜃)
−3.4 −3.2 −3 −2.8 −2.6
0
5
10
15
20
25
30
log(τon)
probabilitydensity
KDE on 104
pts.
KDE on 6369 pts.
MLS on 6369 pts.
ML(𝜃)
log(𝞃on)

41. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions
• Significant improvement for low dimensional problems,
comparable performance for high dimensional problems.
• Approximated density can be also be used to compute
profiles with high efficiency
14
Conclusions/Outlook

42. Fabian Froehlich Uncertainty Analysis for Dynamical Systems using Radial Basis Functions 15
Institute for Computational Biology
Helmholtz Zentrum München
Atefeh Kazeroonian
Sabrina Hock
Jan Hasenauer
Frank Filbir
Fabian Theis
M12 Mathematische Modellierung
biologischer Systeme
Technische Universität München
Jan Hasenauer
Frank Filbir
Fabian Theis
M15 Applied Numerical Analysis
Technische Universität München
Frank Filbir
Acknowledgements
Thank you for your attention!