# Kalman Filter Based Emulator for

Document Sample

```					    Kalman Filter Based Emulator for
Engineering Models (Title, Part 2)

Results of the Engineering Methodology Working Group

Tom Santner, Susie Bayarri, Bruce Pitman
Gentry White, Peter Reichert

Development Assessment and Utilization of Complex Computer
Models Transitional Workshop
May 15, 2007
Contents

1. Motivation

2. Concept

3. Notation

4. Implementation

5. Example

6. Outlook
Motivation

• Dynamic models are diﬃcult to emulate with Gaussian pro-
cesses due to the large number of outputs in the time domain.

• It seems to be advantageous to try to use knowledge about
the mechanisms in a system (or the structure of the simula-
tor) to construct an emulator.

• For processes that can be reasonably modeled as Non-Linear
Dynamic Systems we can build an emulator based on the
linearization of these systems with the addition of the bias
term to account for the discrepancy between the non-linear
simulator and the linear emulator.
Concept

• Use a simpliﬁed stochastic dynamic model as the basis (i.e.
as a prior) for emulating the time dimension of the simulator.

• Couple the bias terms of this model in parameter space by
Gaussian processes.

Simulator = Emulator + Bias
Non-Linear = Linear + Gaussian Process

• By taking advantage of the knowledge that our simulator is
a Non-Linear Dynamic Model we can use a Dynamic Linear
Model as a “smart” mean for our emulator.
Notation

y M(x, θ M)             Deterministic model = simulator.
x                       Input.
θM                      Model parameters.

˜           ˜
Y M(x, θ M, θ M)        Simpliﬁed probabilistic model.
M M ˜
f
Y  M (y | x, θ , θ )
˜                   Likelihood function of the simpliﬁed model.
˜
θM                      Additional parameters of the simpliﬁed
model.
Notation

Y E(x, θ M)               Probabilistic emulator.
fY E (y | x, θ M)         Probability density function of the prob-
abilistic emulator.
y E(x, θ M)               Deterministic emulator.

θ M = {θ M, ..., θ M }
D      1         nD     Design parameter set.
y D = {y 1, ..., y nD }   Results of the simulator at the design
points.
D = {θ M , y D }
D                  Design data,
Notation
Design data:
D = {θ M, y D} = {θ M, ..., θ M }, {y 1, ..., y nD }
D            1         nD

Simpliﬁed model:
M                               M M  ˜
f
Y   M (y | x, θ )
˜               ,       f
Y   M (y | x, θ , θ )
˜

Emulator:

fY E (y | x, θ M) = f           M (y
˜        | x, θ M, D)
Y
M M ˜                  ˜        ˜
=        f       M (y | x, θ , θ , D)f
˜                        ˜      (θ M | D)dθ M
Y                           θ M|D

y E(x, θ M) = E Y E(x, θ M)
Implementation

Simpliﬁed Model

Use a linear, discrete-time state space model as the simpliﬁed
model to base time development on:
Ξtj = Ftj−1 tj (x[tj−1,tj ], θ M) · Ξtj−1
+gtj−1 tj (x[tj−1,tj ], θ M) + Vtj−1 tj

Ytj = Htj (θ M) · Ξtj
The random variable Vtj−1 tj is used to describe the bias of the
simpliﬁed model.
Implementation

Simpliﬁed Model

x[t0,t1]                                 x[tj-1,tj]              x[tj,tj+1]                                           x[tn -1,tn ]
t            t

gt0 t1                                     gtj-1 tj               gtj tj+1                                                 gtn -1 tn
t       t

V0 t1
t                                     Vj-1 tj
t                      Vj tj+1
t                                           Vn -1tn
t t       t

Ft0 t1                Ft1 t2           Ftj-1 tj                Ftj tj+1                Ftj+1 tj+2           Ftn -1 tn
Ξ t0              Ξ t1                .....                 Ξ tj                   Ξ tj+1                  .....          t       t
Ξ tn
t

Ht1                                        Htj                    Htj+1                                                    Htn
t

Yt1                                       Ytj                    Ytj+1                                                 Ytn
t
Implementation
Emulator

The emulator is derived in four steps:

1. Construct extended state space for design and prediction pa-
rameter sets.

2. Construct (prior) model in extended state space the bias
terms of which are coupled as Gaussian processes.

3. Derive posterior distribution of model conditioned on design
data.

4. Derive prediction of posterior model.
Implementation

Emulator: 1. Extended state space

           
M)
Ξtj (θ 1 


      .
.
.


˜t = 
            
Ξj                
 Ξt (θ n
        M ) 

   j     D 
M)
            
Ξtj (θ
Implementation
Emulator: 2. Extended model

                                                            
Ftj−1 tj (θ M)
1
...
                                                             
                                                             
˜           M, θ M) = 
Ftj−1 tj (θ D                                                                       
Ftj−1 tj (θ M )
                                                             

                                      nD                     

Ftj−1 tj (θ M)

                    
M)
gtj−1 tj (θ 1

          .
.
.


g             M, θ M) = 
˜tj−1   tj (θ D

gtj−1 tj (θ M ) 
                   

               nD 
gtj−1 tj (θ M)
Implementation

Emulator: 2. Extended model
                                                      
K(θ M, θ M)
1    1          ...   K(θ M, θ M )
1    nD
M, θ M)
K(θ 1

      .
.
.              ...        .
.
.             .
.
.


˜t          M, θ M) = 
Var V j−1 tj (θ D

 K(θ M , θ M)                  M , θ M ) K(θ M , θ M ) 
K(θ nD nD

     nD 1            ...                     nD        

K(θ M, θ M)
1          ...   K(θ M , θ M ) K(θ M , θ M)
nD

k 1 (θ M , θ M )
                                          
i     j
M, θ M) = 
K(θ i                              ...                     
j
                                            
                                            
knΞ (θ M, θ M)
i    j
                                       
2
2
kl (θ M, θ M) = σl exp −
i    j                    β k ( θ M ) k − ( θ M )k
i           j

k
Implementation

Emulator: 2. Extended model

Htj (θ M)
                              
1                   0
Htj (θ M, θ M) = 
˜

...            . 
. 
. 
D         
Htj (θ M ) 0
nD
Implementation
Emulator: 3. Updating. Forward iteration

˜     ˜ tj−1 ytj−1 ˜            ˜       ˜ tj−1 ytj−1
E Ξtj | Yt1 (˜t1 ) = Ftj−1 tj · E Ξtj−1 | Yt1 (˜t1 ) + ˜tj−1 tj
g

˜     ˜ tj−1
Var Ξtj | Yt1         ˜              ˜       ˜ tj−1 ˜ t        ˜
= Ftj−1 tj · Var Ξtj−1 | Yt1 · FT tj + Var Vtj−1 tj
j−1

˜     ˜ tj−1 tj−1  ˜       ˜     ˜ tj−1 ytj−1
E Ytj | Yt1 (yt1 ) = Htj · E Ξtj | Yt1 (˜t1 )

˜     ˜ tj−1
Var Ytj | Yt1         ˜         ˜     ˜ tj−1 ˜ t
= Htj · Var Ξtj | Yt1 · HT  j

˜     ˜ tj ytn        ˜     ˜ tj−1 ytj−1
E Ξtj | Yt1 (˜t1t ) = E Ξtj | Yt1 (˜t1 )
−1
˜     ˜ tj−1 ˜ t     ˜     ˜ tj−1
+ Var Ξtj | Yt1 · HT · Var Ytj | Yt1                      ˜     ˜ tj−1 ytj−1
· ytj − E Ytj | Yt1 (˜t1 )
˜
j

˜     ˜ tj
Var Ξtj | Yt1            ˜     ˜ tj−1
= Var Ξtj | Yt1
−1
˜     ˜ tj−1 ˜ t     ˜     ˜ tj−1
− Var Ξtj | Yt1 · HT · Var Ytj | Yt1              ˜         ˜     ˜ tj−1
· Htj · Var Ξtj | Yt1
j
Implementation

Emulator: 3. Updating. Backward iteration

˜     ˜ tn ytn          ˜     ˜ tj
E Ξtj | Yt1 t (˜t1t ) = E Ξtj | Yt1
−1
˜     ˜ tj ˜ t            ˜       ˜ tj
+ Var Ξtj | Yt1 · FT tj+1 · Var Ξtj+1 | Yt1
j

˜       ˜ tn ytn          ˜       ˜ tj ytj
· E Ξtj+1 | Yt1 t (˜t1t ) − E Ξtj+1 | Yt1 (˜t1 )

˜     ˜ tn
Var Ξtj | Yt1 t         ˜     ˜ tj
= Var Ξtj | Yt1
−1
˜     ˜ tj ˜ t            ˜       ˜ tj
− Var Ξtj | Yt1 · FT tj+1 · Var Ξtj+1 | Yt1
j

˜       ˜ tj      ˜       ˜ tn
· Var Ξtj+1 | Yt1 − Var Ξtj+1 | Yt1 t
−1 T
˜     ˜ tj ˜ t            ˜       ˜ tj
· Var Ξtj | Yt1 · FT tj+1 · Var Ξtj+1 | Yt1
j
Implementation

Emulator: 4. Updating. Derivation of results

˜
E[YE] = Htj · E[Ξtj ]
˜

˜                 ˜
Var[YE] = Htj · Var[Ξtj ] · HT
˜         tj

˜
Htj =    0 · · · 0 Htj (θ M)
Example
Projectile Motion

• Newton’s Second Law
ΣF = ma

• As applied to an object propelled straight into the air (no
motion in any direction but up).
ma = Fg + Fa

• There are two forces at work on the object gravity and air
resistance.
• In diﬀerential calculus terms
m¨ + cD sign(z)z 2 + mg = 0
z           ˙ ˙
a single second order diﬀerential equation, we need two initial
conditions to solve, zt0 and zt0 = vt0 . Note that the sign
˙
operator is required as the air resistance slows the descent
after apogee, whereas the force of gravity speeds the descent.

• This can be written in state space (ﬁrst order) representation

˙
z = v
cD
v = −g −
˙           sign(v)v 2
m

• This can be written as a state-space model in discrete time.
Example

Simulator
ztj = ztj−1 + vtj−1 ∆t
2
vtj = vtj−1 − g∆t − cf sign vtj−1 vtj−1 ∆t

cD
x = (g, zt0 , vt0 )   ,   θ M = cf =
m

yM(x, θ M) = zt1 (x, θ M), ..., ztnt (x, θ M)
Example
Simpliﬁed Model - linearize friction term
Fa/m = −cf sign(v)v 2 ≈ −cf b1v + b0

z                       1      ∆t
Ξ=          , F(cf , v∗) =
v                       0 1 − cf b1v∆t

0
g=                           , H=     1 0
−g∆t + cf b0∆t
                   
2
σz,evol
Var V   =             2

σv,evol

˜               2           2
θM = b0, b1, β, σz,evol and σv,evol
Example
Results at time = 30 s

q
simulator

3000
emulator

2500
2000
z [m]

1500
1000                                                                        q

0e+00         1e−04           2e−04                  3e−04       4e−04

cf [1/m]

Results at time = 30 s                                                                        Results at time = 30 s

q                                                                                       q
simulator                                                                              simulator
3000

3000
emulator                                                                               emulator
2500

2500
2000

2000
z [m]

z [m]

q
1500

1500

q

q
1000

1000

q                                                                                     q

0e+00   1e−04           2e−04                  3e−04        4e−04                       0e+00         1e−04           2e−04            3e−04       4e−04

cf [1/m]                                                                                      cf [1/m]
Intermission

1. The presented technique seems to be a promising start for
simple and very fast emulators of dynamic simulators.

2. More work is needed to further increase the eﬃciency by
direct emulator evaluation from the posterior.

3. More steps must follow. In particular estimation of parame-
ters of the simpliﬁed model and combination with inference
from data.
Continuation

2
• In order to estimate the parameters σl , β, b0 and b1 we need
to use the forward ﬁltering backwards sampling algorithm
from Carter and Kohn 1994 in order to sample from the
joint posterior of the states using Gibbs sampling.

• Given this and priors for the parameters the evaluation of the
parameters’ posterior distributions would be straightforward.

• This is the next step in our process, we will also apply the
same methodology shown here to the hydrological model
presented by Peter.
• The additional equations required for implementing the For-
ward Filtering Backwards Sampling Algorithm are

tnt           tnt
E Ξtj | Ξtj+1 , Yt1 (ξ tj+1 , yt1 ) =
tj                    tj
E Ξtj | Yt1 + Var Ξtj | Yt1 · FT tj+1
tj
tj −1                          t
j    t
j
· Var Ξtj+1 | Yt1       · ξ tj+1 − E Ξtj+1 |   Yt1 (yt1 )

tn                 tj
Var Ξtj | Ξtj+1 , Yt1 t = Var  Ξtj | Yt1
T                      tj −1                          tj
· I − Ftj tj+1 · Var Ξtj+1 | Yt1     · Ftj tj+1 Var   Ξtj | Yt1
˜
Additionally we need priors for θM

λ = exp(−4β)
π(λ) ∼ Beta(η, γ)
π(b0, b1) ∝ 1
2
π(σl ) ∼ IG(αl , βl )
Results:

• 5000 iterations

• Heavy Tail density for β

• Slow mixing and convergence for b0

2    2
• Using σ1 = σ2 pretty stable density

• b1 pretty stable density as well
Density of b1                                                      Density of b1

4e−04

0.04
3e−04

0.03
Density

Density
2e−04

0.02
1e−04

0.01
0e+00

0.00
−3000     −2000         −1000     0         1000                    0       50        100        150        200

Density of β

0.4
8

σ2
z
σ2
v

0.3
6
Density

Density

0.2
4

0.1
2

0.0
0

0.2      0.3       0.4       0.5       0.6    0.7                    0   5        10         15         20        25
Trace of b0                                                                                                   Cumu a ve Mean o b
1000

q                                      q
qq
q                          q qq
qq
q
qq
q
qq     q
q
q
q q
q q
q q                        q q
q qq
q qq
q
q
q
q
q
q
q
q
q
q
q
qq     q
q
q qq q
q
qq
q
q
q q                        q qq q
q q
q q qq
q q
q q qq
q
q
qq
q
q q qq qq q
qq q qq q
q
qq
qq qq q q q                         qqq                                      qq
q
q
q
qq q
qq qq q q q
q q
q q q
q
q
qq q qqq
q
qq q q q q
qq qq q q q
q                        qq q q q
q q
q q
q
qq qqq q
q q qq
q
q
qq
q
q
q
q                                             qq qq q q q q
q q
qq qq q qq q
qq qq q qq
q q qq
qq q qq
q qq
qq q
q
qq
q
q
qq
qqqq q
q q qq
qq
qq qqqq                                     q
q
qq
q
q
q q
q                                             qq
qq q q q q q
qq
qq
qq q qq qq
qq
qq
qqq
q
qq qqqq q q q
qq qq q qq qq
q       q q
q qq q                qqqq q q
q q q
qqqq q q
q
q qq
qq q
q q qqq
q q q
q                                        qq
q
qq
qq
q q
q q
q q
q q                                           qq
qq q q q qq q
qq  q
q     qq q q q
qqq q qq qq qq
q q
q qq q
qqq q qq qq q                     qqqqqq q
qqqq qq
qq q qq q
q qqqq
qq qqq
qq q                                     qq
q
qq
q q
q                                             q
qq
qq
q         qq qq q
q
qq qq q
qq qq q
q q q
q qq q
qqq q qqqqqq qq
qq
qqq q q
qqq qq q
qq q q qq                                   qq
q
q
q q
q q
q q
q qq                                         qqq q qq qqq qq
qq
q
qqq q qqq qq qq
q
q qq q qqqqq qq
q
qq         q qq q
qq qqq qq
qq
q qq q
q qq q
qq qqq qq
qqqqqqqq
q qq qqq
qq q q
q qqqqqq
qqqqqqqq
q q qq
q q
qq q q                                    qq
qq
q q
q
q
q qq
q
q
q qq
q                                           q q q q q q q qq
q
q q
q          q q q
q q
q qq q qq qqq qq
q qq q qq qqq qq
q         qq qq qq
q qq qqqq
q q
q q qq q
q qq qqq
qqqqqqqq
q qq qq q
qq qqqqq  q
q                                 qq
qq
q
q
q
q q                                          qq q qq qq qq  q q
q qq q qqq qq qq
q q q qqqq qq
q qq q
q qq qq q q
q qq q qq qqq qq
q              q
q q                q qqqq q
qqqqqqqq
qq q
q
qqqqqqqq
q
qq
q qq q
q qq                                        q q q q q qq q qq
q         q q qq q
q q q q qq
q qq q qqq qq qq                    q qq q
q qqq qq
qqq qqqq
q
q qqq qq
q q
q q q q                                    qq
q

0
q qq
q q                                                    q q q
q q q qq                  qqqqqq qq                                     q
qq q
q qq qq                       q
q             q q q q q qq qq
q      q q q q qq
q qqqqq qq
q q qq
q q
q q q
q q qqq qq
q                    q q
q qqq q
q qqq qq
qq qqqq q
qq q q
q q q qq
q
q
q qq
qq q
q qqq
qq q
qq q q                       q
q
q             qq      q q q q q qq
q qq q q q qq qq
q q q q q qq
q q q qq
q qqq q q qqqq q
q             qq qqq qq
q qq q
qq qqq qq
q q q qq
q                                   q
q
0

q               q       q                     q q q q qq  qq             q qq q qq q                                   q
q qqq
q qq
qqq
q qq
qqqqqq
q
q               q q q
q
q
q
q             q qq q q qqq qq
q
q q qq qq qq
q
q q q q q q q qq
q
q      q q q q qq                qq qqq qq
qq
q
q qq q qq
q
q
q
q qq
qq q                q q qq                q q q q q qq q q
q      q q q q qq
q     q
q q qq
q
q qq q q qqq q q
q
qq q
q      qq
q
qq qq qqq q
q
q
qq                      q q q qq q

eq 1 25000
qqqq
qqq
qqqq
qqq
q
q qq
q
qq
qqq
qq
q qq q
q qq q
q qq q
qqq q
q qq q
q
qq
q      q
q     q q q
q
q qq q q qqqqq q
q
qq q q qq q
q
q     q q q
q q q
q
q q q
q qq q q
q     q q q
q
q
qq q q q q
q
q
q      qq
qq
qq
qq
qq
qq
q qqq
q qqq                   qq
q qq                 q qq          q                           q qq
q                                     qq
qqq q
qqq q
q qq
qqq
qqq
qqqq
q qq
q
q qq
qq
qq
qq
q qq q
q qq q
q qqqq
q qq q
q qqqq
q qqqq
q
q q
qq q
q q
q q
q qqq q              q q q q q q qq q q
qq
qq
qq
q qq q q q q q q
qq
qq     q
q
qq q q q qq   q q
q q
q q qq
q q qq
q
q
q
q
qq
qq
q     q qq
q q
q qq
q qq
q q
qqq
q
qqq
q
q qq
qq
q qq
q qq
qq
q qq
q q
qqq q
qq
q q
q q
qq q
qqq q                 q
q qqq q
q qqqq
qq
q q q
q qqqq
q
q    q
q                    q q
q q q q q q qq
q      q   q q
q
q q qq
q q             q
q        q qq
q q
q q
q qq q
q                                     q
q
qq q
qqq q
qqq q
qq
qq
qqq q                q qq q
qq
q q
q
q qqqqq
q q q
q qq q
q qq
q
q    q
q qq q
q qq qq               q q
q q
qq
q
q          q qq
q qq
q qq
qq
q qq
q qq            q            q
q qqq
q
qq q
q qqq                                     q
q
q
qq q
qq
qq
q                   q qqqqq
q q
q qqq qq
q qqqq
q q    q           q    q
q qq q q q               q qqqq                                    q
q q
qqq
qqq
q
qq q
qq
qq
qq q
q q
q
q
q qq q q
q q qq
q
qq qq q q
q qq
qq qqq q
q               q q q
q q    q
q          q qq
q
qq    q
q qq q q q
q
q
q q q q
q qq q q q
qq q  q                  q qqq
q qq q
qq
q
q
q
q
q
qq q
qq q
q
qqq q
q q
q                          q
qq qqq q
q
qqqq q q
q qq q
q qq
qq qq q q        q q q
q q        qq          q
q qq q q q
q qq q q
q
q     q                  q qqqq
q qqq q                                    q
q
qq q
q
q
q
q
q q                 qq qq q qq
q qqq q
q    qq q
q qqq q
q qq q q
q         q q q      q
q
q
q          q qq q q
q
q q qq q q
q qq q q
q qq q qq                q q qq
qq
qq
q qqq
q
q
q
q
−1000

qq
q                    q
q q q qq
qqqq q q
q
q         q
q          qq          q     q
qq                     qq
q qqq
qq                                      q
q
q q
q q
q q
q q                  qq qqq q
qq
q    q
qq qqq q
qq qq q q
q qqq q
q         q
q q
q          q
q
qq          q qq q q q
q    qqq qq
q qq q
q q q q qq
q
q q qq               qq
qq
qqqq
q
q q
qq
q q                                     q
q
q q
q q  q                q
qq qq q q
q q qq q
qqq q q q
q
q    qq
q         q q
q
q q        q
q
qq
q               qq q qq
qq q qq
q
q q q q qq    q             qq
qq
qq
q q
qq
qq                                      q
q
q
q q  q
q
q              qqqqqq q
qq qq q
q
q
q qq q q
q q
q qqq q q       qq
q          q
qq               qq q qq
q qq qq
q         q           q q
q
qq
qq                                      qq

−500
q
q
q
q
q          q q qqq q qq
q qq
q q q qq
q q qqq q qq
qq       q
qq
qq q
q
q
q q
q          q
q
qq               qq
q qq qq
qq q qq
qq
q qq q qq
q       qq
q
q
q            qq
qq
qq
qq
qq
qq
q
q
q
qq q
b0

q          q q qq q      q     qq
q
qq q
q                           q q
q                      qq
qq                                      q
q
q    q
q           q q qq
q qqq
q q qq
q q qq
q
q q qq
q q qq
q
qq
q
q
q qq q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q q
qq
q
q
q
q qq q qq
q qq q qq
q qq q qq
qq q q
qqq q qq
qq q q qq
qq q qq
q
q           qq
qq
qq
qq q
q
q
q qq
q
q
q
q
q          qqq qq qq qq q
qq
qqq q q
qq
q q q  q     qq
qq qq
q     q
q
qq           q               q qq qq
q q
q q q qq
qq
qq
q qq q qq q            qqq
q qq
q                                     q
q
q
q          qqq q q
qq q
q
qqq q q
q
qq q q q      q
qqq q q qq qq q
q
q
q       q          q               qqq q
qq q
q qqq qq                  q
q qq
qq                                    q
q

um um b0
q          q            q
q     q q                             q
q qqq qq               q qq
q
qqq q
qqq q
qqq q
qq
qqq q
qq q q
q     q     q
qq
qq qq q
qq
q
q q qq
q     q
q q qq q
q                             qqq q
q qqq q
qq
qqq q
qqq q
qqq q               q qq
q q
q q
q                                    q
q
q
qq
q
qq       qq q q
qq
q
qqq q
qqq q
qqq q
q            q
q
q qq                               q qqq q
qq
qqq q
qqq
qqq q
qqq q               q q q
q                                    q
q
q
qq
q        qq q q
qqq q
qqq q
qqq q
qq
qqq q
q qq q
q
q
q q qq q
q q qq q
qq q    q                             qqqqq
qqq qq
qq q
q qqqq
qq
q
q
q
q
qq
qq
qq
qq
qqq      q qq q
qqq q
q
qq
q qq q
q
q
q
q qq q         qq q q
q q qq
qqqqq
q
qqq q
qq
q qqqq
qqqqq
qq q
qq q
q
qq q
qq
q
q
q
q
q
q
q
q
q
q
qq
qq
qq  q q qq q
q q            qq q q                               q qqq
q qq
q qqq
q                    qq                                    q
q
qq q q
qq
q q qqqq q
qq
qq q q q
q q qq q
q q
q q q
q q
q q
q qq
q          q qq q
q
q qq q q
q
q qqqq
qq
qq
q qq q
qqq q
q qq q
qq q
q
q
q
q
q
q
q
q
q
q qq q q q                 qq                                                         q                                     q
qq q q                 q qq q q                                qq
q qq
q qq q
−2000

qqq q q
q q
q q  qq                 q
q qq q                              q qq qq
q qq
q qq q
q                  q
q                                     qq                                                                          qqqq
qqqq
qqqqq
qqqqq
qqq q q
qqq q q
q
qq q q
q q
q q
qq q
q qq q
q
q
qq
q
q
qq
q qq
q q q q
qq q
q
q qq q q   q                         q qqqq
q qq
q q
q qq
qq
q qq q                  q
q
q
q
q q
q
q
q
q                                                                       qq
qq
qq
qq
qq
qq
qqqqq
qqqqqq
qq
qqq qq q
q qq q
q q  q
qq               q qq q q
q qq q q
q qq q q
q       q
q qq
q qq
q qq qq                  qq
q q
q q                                   q
q                                                                     qq
qq
qq
qq
qq
q
q
qq
q
qqq qq q
qq q
q qq q
q qq q
q qq
q q  qq
q               q qq q q
q
qq qq
q qq
q qq
q qq qq                             q qq
q
qq
q
q                   q
qq
q                                     q
q
q                                                      qqq
qq
qqqqq
qqq
qqqq          qq
qq
q
qq
qq qq q
qq q qq
q q qqq
qqqqqqq
q qqqq
q qqq
qq
q qq q q
q qq qq
qq
q
q qq
q qq
qq q q
q qq
qqq q q
q qq
qq q q   q                        q q
q q
q
qq
q
q
q                   qq
qq
qq
qq q
q
q
q
q                                                 qq
qqqqq qqqq
qq qqqq
qq qqqq qq qqq
qq qqqq qqq
qq      qq
qq
qq
qqqqq
qqq
q qq q
q qq
q q
q
qq qqq
q qqqq                     qq qq
q qq
q
qq q q
q
qq qq
q                         q q
q q
q
q
q
qq
qq
qq
qq q
q
q
q
q                                                qq
qq
qq
qq
qq
qq            qqq q
qqq q
qqqq
qqq
qqq
qqq

−1000
q qq
q qqq
q qq
qqqq qq                    qq qq
q
q qq
q                                q q  q                   qq
qq
qq q                                   q                                              q
q
q
q qqq
q qqqq
q qqq
q q
q qqq
q qqq
q                  qqq qq
qq q
q qq
qqq qq
qq qq
q q
q qq                              q q  q
q
qq
qq
qq
qq
qq
qq
q
q
q                                            q
q
q
q
q
q
q qqq
q qqq
q qqq
q qqq
q qqq
qq
q qqq
q qqq
q qqq
q qqq
qq
qqq qq
q q qq
q qq
qqq qq
q qqqqq                                                   qq
qq
qq
qq
qq
q
qq
q
q
q
q                                         qq
q
qq
q
qq
q qq
q qq
q
q qqq
q q                    q qq q q
q      q
q                                                     qq
qq
qq
qq                                      q
q                                        q
q
q
q qq
q qq
q qq
q qq
q q
q qq
q qqqqq                 q q qq
q
qq
q
q
q
q
q
q
qq
q
q
q
qq
q
qq
qq
q
q
q
q                                      q
q
q
q
q
qq
q qq
q qq
q qqq
q qq                    q
q
qq
q
q
q
qq
qqq
q                                                    qq
q
q
qq
q
q
q
q                                    q
qq
q
q qq
q qq
q    q                 q
qq
q
q
q
q
qq    qq
qqq                                                     q
q
q
qq                                      q
q
q                                   qq
q
q q q
qq                      q
qq
qq      q
q
q                                                     q
q
q
q                                       q
q
q                qqq
qq
qq
qq
qqq
q
qq                 q
q
q
−3000

q q
q
q q                     q
qq
q
q
q       q
q
q
q                                                     q
q
qq
q                                       q
q
q             qq q
qq q
qq q
qq q
q q                q
q
qq
qq
qq
qq
q q
q                       q
q
q
q
q
q
q
q
q
q                                                     q
q
q
q
q
q
q
q
q
q           q
qq
q
q     qq
q          q
qqq
q
q                       q
qq       q
q                                                     q
qq
q                                       q
qq         q
q
qq
q       q
qq
q
q         q
q
q        q                                                      q                                                            q        q
qq
q                                                             qq                                        q
q
q        qq
qq
qq
qq
qq
qq
qqq
qqq
qqq
qqq q
qqq q  q
q
q
q
q
q
qq                                        qqq q q
q
qq q
qq q
qqq q
qqqq
qq q
qq            qq q
q q
q q
q qq
q                                                                                                           qq
qq
q
qq                  qq
qq
q
q
q                                                                                                                                 q
q
q
qq

0               5000              10000               15000              20000              25000                          0              5000             10000             15000              20000             25000

Index                                                                                                                    nde
What does this all mean?

• Model can explain variability in either the mean term or the
error term

• Mean term can be derived improved based on knowledge of
the process

• Often times the mean term is easier to estimate than a co-
variance

• Parameter indiﬀerence is problematic and the physical mean-
ing of parameters as well as their numerical properties can
be problematic, so beware.
Where to next?

• Develop the prediction model for new cf

• Improve estimate for b0

• Re-program in MATLAB for the hydrological model

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 5 posted: 6/22/2010 language: English pages: 31
How are you planning on using Docstoc?