1 / 10

Fitting a line to N data points – 1

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:

koren
Download Presentation

Fitting a line to N data points – 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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:

  2. Fitting a line to N data points – 2 • Slope = optimally weighted mean value: • Optimal weights: • Hence get optimal slope and its variance:

  3. 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”.

  4. 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.

  5. 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:

  6. Normal equations • Normal equations are constructed as before: • Or in matrix form:

  7. 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:

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

  9. 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.

  10. 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.

More Related