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Example

x 2. x 2. x 1. e 2. x 1. e 1. Example. . Scaling a profile to fit data. Using the notation we’ve just learned, So what basis vectors correspond to ? Answer: i.e.  i is the unit of distance on the ith axis of data space! . Length scales.

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Example

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  1. x2 x2 x1 e2 x1 e1 Example 

  2. Scaling a profile to fit data • Using the notation we’ve just learned, • So what basis vectors correspond to ? • Answer: • i.e. i is the unit of distance on the ith axis of data space!

  3. 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:

  4.  contours are ellipses x2 x e2 e1 x1  ellipses become circles • Old basis vectors: • Stretched basis vectors: b2  contours are circles x2 /2 x b1 b2 b1 x1 /1

  5. He/H O/H x y=ax+b y 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 y=a’x’+b  R y

  6. Defining  with errors in both X and Y • Hence determine minimum distance R: • Hence:

  7. 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).

  8. v2 v1 v3 v1 e1 v2 v2’ e1 Gram-Schmidt Orthogonalization – 1 • Second way: the Gram-Schmidt process. • 1. Start with N vectors Vi , i=1,...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

  9. v3’ v3 e1 v3’ v3” e2 v3” e3 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

  10. 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) + B[P1(x) - P0(x)] + C [P2(x) - P1(x)] • A, B, C are independent • the functions Pk(x) - Pk-1(x) are orthogonal

  11. S A  C Wrong : bad , small A S C Phase 0 1 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 phi=atan2(–S,C) if you care about which quadrant  ends up in! Correct : good , large A S Phase 0 1 C

  12. S S C C Periodograms • Repeat for a large number of  values • Plot A() vs  to get a periodogram: A() 

  13. Fitting a sinusoid to data • Data: ti, xi ± i, i=1,...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 .

  14. Iterate to convergence: • Error bars:

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