Scaling%20a%20profile%20to%20fit%20data

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Example
?
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Scaling a profile to fit data

• Using the notation weve just learned,
• So what basis vectors correspond to
  ?
• Answer
• i.e. ?i is the unit of distance on the ith axis
  of data space!

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Length scales

• In the orthogonal system we used for profile
  fitting
• Dot product of 2 basis vectors
• i.e. the basis vectors are orthogonal, but not
  unit length.
• Need to stretch axis i by factor ?i and define
  new basis vectors b

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?? ellipses become circles

• Old basis vectors
• Stretched basis vectors

b2
?? contours are circles
x2 /?2
x
b1
b2
b1
x1 /?1
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Example primordial helium abundance

• Fitting a line to data with error bars in both X
  and Y
• If ?x??y, get misleading point of closest
  approach
• Horizontal stretch by factor ?y/?x

?x
yaxb
?
R
?y
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Defining ?? with errors in both X and Y

• Hence determine minimum distance R
• Hence

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Constructing orthogonal basis functions

• First way diagonalize Hessian matrix.
• Quadratic approximation to ?? surface
• Orthogonal basis vectors are the eigenvectors of
  Hij along the principal axes of the ?? contours.
• Sometimes called Principal Component Analysis
  (PCA), also related to singular-value
  decomposition (see Press et al).

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Gram-Schmidt Orthogonalization 1

• Second way the Gram-Schmidt process.
• 1. Start with N vectors Vi , i1,...N. They must
  be independent, i.e. no two of them parallel.
• 2. Normalize vector 1
• 3. Make v2 perpendicular to e1
• i.e. subtract component of v2 in direction of e1
• 4. Normalize v2

v2
e1
e2
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Gram-Schmidt Orthogonalization 2

• 5. Make v3 perpendicular to e1
• 6. Make v3 perpendicular to e2
• Note v3 is perpendicular to e1 AND e2.
• 7. Normalize v3
• ...and so on, making v4 perpendicular to e1, e2,
  e3 and normalising to get e4
• Repeat for all vectors up to vN to get
  ortho-normal basis e1, e2, ..., eN

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Differences between successive ?? fits

• Fit A B x C x2
• A, B, C are not independent
• 1, x, x2 are not orthogonal
• If Pk(x) is a polynomial of degree k fitted to
  the data, then fit
• AP0(x) BP1(x) - P0(x) C P2(x) - P1(x)
• A, B, C are independent
• the functions Pk(x) - Pk-1(x) are orthogonal

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Periodic signals

• To search a time series of data for a sinusoidal
  oscillation of unknown frequency ?
• Fold data on trial period P?????
• Fit a function of the form

Programming hint Use phiatan2(S,C) if you care
about which quadrant ? ends up in!
Correct ?? good ??, large A
S
Phase
0
1
C
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Periodograms

• Repeat for a large number of ? values
• Plot A(?) vs ? to get a periodogram

A(?)
?
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Fitting a sinusoid to data

• Data ti, xi ?i, i1,...N
• Model
• Parameters X0, C, S, ?
• Model is linear in X0, C, S and nonlinear in ?
• Use an iterative ?? fit to linear parameters at a
  sequence of fixed trial ?.

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• Iterate to convergence
• Error bars