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Fitting a line to N data points – 1PowerPoint Presentation

Fitting a line to N data points – 1

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Fitting a line to N data points – 1

- If we use
then a, b are not independent.

- To make a, b independent, compute:
- Then use:
- Intercept = optimally weighted mean value:
- Variance of intercept:

Fitting a line to N data points – 2

- Slope = optimally weighted mean value:
- Optimal weights:
- Hence get optimal slope and its variance:

Linear regression

- If fitting a straight line, minimize:
- To minimize, set derivatives to zero:
- Note that these are a pair of simultaneous linear equations -- the “normal equations”.

The Normal Equations

- Solve as simultaneous linear equations in matrix form – the “normal equations”:
- In vector-matrix notation:
- Solve using standard matrix-inversion methods (see Press et al for implementation).
- Note that the matrix M is diagonal if:
- In this case we have chosen an orthogonal basis.

General linear regression

- Suppose you wish to fit your data points yi with the sum of several scaled functions of the xi:
- Example: fitting a polynomial:
- Goodness of fit to data xi, yi, i:
- where:
- To minimise 2, then for each k we have an equation:

Normal equations

- Normal equations are constructed as before:
- Or in matrix form:

Uncertainties of the answers

- We want to know the uncertainties of the best-fit values of the parameters aj .
- For a one-parameter fit we’ve seen that:
- By analogy, for a multi-parameter fit the covariance of any pair of parameters is:
- Hence get local quadratic approximation to 2 surface using Hessian matrix H:

The Hessian matrix

- Defined as
- It’s the same matrix M we derived from the normal equations!
- Example: y = ax + b.

a

b

a

Principal axes of 2 ellipsoid- The eigenvectors of H define the principal axes of the 2 ellipsoid.
- H is diagonalised by replacing the coordinates xi with:
- This gives
- And so orthogonalises the parameters.

Principal axes for general linear models

- In the general linear case where we fit K functions Pk with scale factors ak:
- The Hessian matrix has elements:
- Normal equations are
- This gives K-dimensional ellipsoidal surfaces of constant 2 whose principal axes are eigenvectors of the Hessian matrix H.
- Use standard matrix methods to find linear combinations of xi, yi that diagonalise H.

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