Quantifying Monte Carlo uncertainty in ensemble Kaiman filter

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Quantifying Monte Carlo uncertainty in ensemble Kaiman filter

n × m = 1000

n × m = 200

n

m

n × m

n

m

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

∇ · κ

µ ∇ p

= λφ ∂p

∂t ,

κ

µ

λ

φ

p

∂p

∂t − ∇ · (κ ∇ p) = f,

p

κ = κ(x, y)

κ

f(x, y) =

 

 

 

 

 

 

 



1/2 (x, y) = (0, 0) 1 (x, y) = ( − 0.9, 0.9) 2 (x, y) = (0.9, − 0.9)

− 1 (x, y) = (0.9, 0.9)

− 1 (x, y) = ( − 0.9, − 0.9) 0

665

1248

κ(x, y)

1.0 × 10 − 10

κ(x, y)

{ x i } n i=1

n (x) = 1 n

X n i=1

I (x i ≤ x) ,

I ( · )

{e x i } n i=1

{ x i } n i=1

{e x i } n i=1

D n = sup

x

n (x) −

f n (x) ,

n

f n

D n

x

{ x i } n i=1

{e x i } n i=1

0.05

x i

x e i

0 2 4 6 8 0

0.2 0.4 0.6 0.8 1

x

F(x)

A priori CDF

2 3 4 5 6 7 8

0 0.2 0.4 0.6 0.8 1

x

F(x)

A posteriori CDF

10(10 − 1)/2 = 45

0.05/45 = 0.0011

3.96 × 10 − 16

665

1248

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

1.0 × 10 ⁻ ¹⁰

{ x i } ⁿ i=1

{e x i } ⁿ i=1

{ x i } ⁿ i=1

{e x i } ⁿ i=1

ⁿ (x) −

f _n (x) ,

f _n

{ x i } ⁿ i=1

{e x i } ⁿ i=1

3.96 × 10 ⁻ ¹⁶

(x) − t ^∗ s(x)

(x) + t ^∗ s(x)

t ^∗

(x) ² .

^∞ (x) −

ⁿ (x) | ,

(x) = (z ^∗ ) ² s ² (x)

(t ^∗ ) ² × m ,

z ^∗

t ^∗ ≈ z ^∗

= Z ^−∞