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

Simulating the Ionosphere - GitHub

Sep 30, 2009 - DEFINITION: Approximating measurements at intermediate scales/positions from scattered measurements. We have sparse measurements.

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