The Basics interpolation Model fitting
Simulating the Ionosphere Anna Scaife Cavendish Astrophysics University of Cambridge.
MCCT SKADS Summer School on Third Generation Calibration
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Outline
1
The Basics
2
interpolation
3
Model fitting
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Let’s start with the Z–Jones...
Z–Jones Z =
iφ e iono 0
0
eiφiono
φiono = −25λTEC TEC = ....
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Let’s start with the Z–Jones...
Z–Jones Z =
iφ e iono 0
0
eiφiono
φiono = −25λTEC TEC = ....
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Let’s start with the Z–Jones...
Z–Jones Z =
iφ e iono 0
0
eiφiono
φiono = −25λTEC TEC = ....
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
TEC 1 TECU = 1016 m−2 Z
∞
ne d` + const
TEC = 0
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
It seems that we should be able to predict the behaviour of the ionosphere... ...but, in fact we are limited to interpolation of scattered measurements. Counter-intuitively, simulating the ionosphere is in fact an inverse problem. Fortunately there are lots of ways of approaching inverse problems.
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Interpolation
Let’s look at interpolating between measurements... DEFINITION: Approximating measurements at intermediate scales/positions from scattered measurements. We have sparse measurements i.e. We have under-sampled data
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Interpolation
Let’s look at interpolating between measurements... DEFINITION: Approximating measurements at intermediate scales/positions from scattered measurements. We have sparse measurements i.e. We have under-sampled data
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Interpolation schemes
1
Kriging
2
Triangulation based
3
Natural neighbour
4
Splining
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Kriging
Semivariance γ(h) =
1 (f (x) − f (x + h))2 2
A typical model: 3 h s( 2 ( a ) − 12 ( ha )3 ), 0 ≤ h ≤ a γ= s, h>a
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Triangulation Given three measured points we can interpolate to any point within the triangle using: Delaunay triangulation f (x, y ) =
3 X
φi (x, y )fi
i=1
φi (x) is our basis function In a simple case we can use linear equations: f (x, y ) = c1 x + c2 y + c3 We can just solve Ac = f, where f = (f1 , f2 , f3 )T and A = {(xi , yi , 1)}1≤i≤3 . Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Natural neighbour Voronoi tesselation Voronoi tesselation divides the data into cells defined by the positions of the measurements. We use the interpolation point to define a new Voronoi cell. The value of this cell can be evaluated as the weighted sum of the contributions from its overlapping cells.
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Splining
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Bayes Theorem p(D|M, Θ) = posterior =
L(M, Θ|D)π(Θ) E
likelihood × prior Evidence
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Bayes Theorem p(D|M, Θ) = posterior =
L(M, Θ|D)π(Θ) E
likelihood × prior Evidence
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Maximum likelihood
We maximise the likelihood of the DATA w.r.t the MODEL. When we perform a χ2 test we are in fact calculating a Gaussian ML. N(Di , σ) ∝ exp
Anna Scaife
−(Di − Mi )2 2σ 2
Z–Jones
The Basics interpolation Model fitting
Maximum A Posteriori If we know something about our parameters then we can utilise that prior information. Example Say we are fitting a spectral index... S = Aν −α
ln L = −0.5
(di − mi )2 σi2
+ ln π
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Maximum A Posteriori If we know something about our parameters then we can utilise that prior information. Example Say we are fitting a spectral index... S = Aν −α
ln L = −0.5
(di − mi )2 σi2
+ ln π
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Model comparison
What if we have more than one model...? The Evidence: Z E = L(Θ)π(Θ)d D Θ The model selection ratio: R=
Pr(D|H1 )Pr(H1 ) Z1 Pr(H1 ) Pr(H1 |D) = = Pr(H2 |D) Pr(D|H1 )Pr(H1 ) Z2 Pr(H2 )
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Worked Example Let’s look at Kriging again... Kriging variance γ(h) =
1 (f (x) − f (x + h))2 2
TEC(x0 ) =
n X
λi TEC(xi )
i=1
∆=2
n X
λi γ(xi , x0 ) −
i=1
n n X X i=1 j=1
Γij λ = Γi0 Anna Scaife
Z–Jones
λi λj γ(xi , x0 )
The Basics interpolation Model fitting
Worked Example Let’s look at Kriging again... Kriging variance γ(h) =
1 (f (x) − f (x + h))2 2
TEC(x0 ) =
n X
λi TEC(xi )
i=1
∆=2
n X
λi γ(xi , x0 ) −
i=1
n n X X i=1 j=1
Γij λ = Γi0 Anna Scaife
Z–Jones
λi λj γ(xi , x0 )
The Basics interpolation Model fitting
Worked Example Let’s look at Kriging again... Kriging variance γ(h) =
1 (f (x) − f (x + h))2 2
TEC(x0 ) =
n X
λi TEC(xi )
i=1
∆=2
n X
λi γ(xi , x0 ) −
i=1
n n X X i=1 j=1
Γij λ = Γi0 Anna Scaife
Z–Jones
λi λj γ(xi , x0 )
The Basics interpolation Model fitting
We need a model for γ We can calculate γ directly from the data: γ(xi − xj ) =
1 (TEC(xi ) − TEC(xj ))2 2
But we need an analytic form for γ So we have to pick a model... Typical models would be: γ1 = a + b ∗ h γ2 = a + b ∗ hα
Anna Scaife
Z–Jones
The Basics interpolation Model fitting
Anna Scaife
Z–Jones
For Further Reading I
S. Dodelson. Modern Cosmology. Elsevier, 2003. W. Hu & S. Dodelson Cosmic Microwave Background Anisotropies Ann. Rev. Astron. and Astrophys., 2002 R. Subrahmanyan & R. Ekers CMB observations using the SKA SKA Memo Series, 26
Anna Scaife
Z–Jones
