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Least squares method

Least squares method. Let adjustable parameters for structure refinement be u j Then if R = S w( hkl ) (|F obs | – |F calc |) 2 = S w D 2 Must get ∂R/∂u i = 0 one eqn/parameter. hkl. hkl. hkl. hkl. Least squares method.

skyler-barnett
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Least squares method

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  1. Least squares method Let adjustable parameters for structure refinement be uj Then if R = S w(hkl)(|Fobs|– |Fcalc|)2 = S w D2 Must get ∂R/∂ui = 0 one eqn/parameter hkl hkl hkl hkl

  2. Least squares method Let adjustable parameters for structure refinement be uj Then if R = S w(hkl)(|Fobs|– |Fcalc|)2 = S w D2 Must get ∂R/∂ui = 0 one eqn/parameter Then S w D ∂|Fc|/∂ui = 0 hkl hkl hkl hkl

  3. Least squares Simple example – again To solve simultaneous linear eqns: a11x1 + a12x2 + … = y1 a21x1 + a22x2 + … = y2 If: Then simultaneous eqns given by A x = y

  4. Least squares a11x1 + a12x2 + … ≈ y1 a21x1 + a22x2 + … ≈ y2 Then: a11x1 + a12x2 + … – y1 = e1 a21x1 + a22x2 + … – y2 = e2 No exact solution as before – but can get best solution by minimizing S ei Suppose: 2 i

  5. Least squares a11x1 + a12x2 + … – y1 = e1 a21x1 + a22x2 + … – y2 = e2 No exact solution as before – but can get best solution by minimizing S ei Also – note that no. observations > no. of variable parameters (n > m) Minimize: 2 i

  6. Least squares Minimize:

  7. Least squares To illustrate calcn, let n, m = 2 (a11x1 + a12x2 – y1)2 = e12 (a21x1 + a22x2 – y2)2 = e22 Take partial derivative wrtx1, set = 0: (a11x1 + a12x2 – y1) a11 = 0 (a21x1 + a22x2 – y2) a21 = 0

  8. Least squares To illustrate calcn, let n, m = 2 (a11x1 + a12x2 – y1)2 = e12 (a21x1 + a22x2 – y2)2 = e22 Take partial derivative wrtx1, set = 0: (a11x1 + a12x2 – y1) a11 = 0 (a21x1 + a22x2 – y2) a21 = 0 (a11 a11) x1 + (a11 a12)x2 = (a11) y1 (a21 a21)x1 + (a21 a22)x2 = (a21) y2 (a11 a11 +a21 a21) x1 + (a11 a12 + a21 a22)x2 = (a11 y1 + a21 y2 )

  9. Least squares (a11 a11 +a21 a21) x1 + (a11 a12 + a21 a22)x2 = (a11 y1 + a21 y2 ) x1 Sai1 + x2 S ai1 ai2 = S ai1 yi 2 2 2 2 i=1 i=1 i=1

  10. Least squares (a11 a11 +a21 a21) x1 + (a11 a12 + a21 a22)x2 = (a11 y1 + a21 y2 ) x1 Sai1 + x2 S ai1 ai2 = S ai1 yi Now consider: 2 2 2 2 i=1 i=1 i=1

  11. Least squares (a11 a11 +a21 a21) x1 + (a11 a12 + a21 a22)x2 = (a11 y1 + a21 y2 ) x1 Sai1 + x2 S ai1 ai2 = S ai1 yi Now consider: ATA 2 2 2 2 i=1 i=1 i=1

  12. Least squares (a11 a11 +a21 a21) x1 + (a11 a12 + a21 a22)x2 = (a11 y1 + a21 y2 ) x1 Sai1 + x2 S ai1 ai2 = S ai1 yi Now consider: ATA And: (AT A) x = (AT y) 2 2 2 2 i=1 i=1 i=1

  13. Least squares In general:

  14. Least squares In general: And: (AT A) x = (AT y)

  15. Least squares In general: (AT A) x = (AT y) x = (AT A)-1 (AT y)

  16. Least squares Again: ƒs are not linear in xi

  17. Least squares Again: ƒs are not linear in xi Expand ƒs in Taylor series

  18. Least squares Again: ƒs are not linear in xi Expand ƒs in Taylor series

  19. Least squares Solve, as before:

  20. Least squares Solve, as before:

  21. Least squares Solve, as before:

  22. Least squares Weighting factors matrix:

  23. Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj

  24. Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj Eqns not exact, so refinement process requires no. of cycles to complete the refinement Add shifts ∆xj to xjo for each new refinement cycle

  25. Least squares How good are final parameters? Use usual procedure to calculate standard deviations, s(xj) no. observations no. parameters

  26. Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously?

  27. Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously? Use correlation matrix: Calc correlation matrix for each refinement cycle Look for strong interactions (rij > + 0.5 or < – 0.5, roughly) If 2 parameters interact, hold one constant

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