## WAPP+: Parameter Errors -- details

As discussed elsewhere, in simple cases (and approximately in many other cases) χ2 is a simple quadratic form in the parameters, for example:

= a A2 + b AB + c B2 + d A + e B + f

so the constant χ2 curves are ellipses. Parameter error determination depends on understanding the region where χ2 is near its minimum value, for example:

χ2(a,b) ≤ χ2min + 1

In particular we need to find the maximum reach of these ellipses, for example δA and δB shown below:

If we shift the origin for the parameters a and b to be at the center of the ellipse, using:

Δb = B - Bcenter

the quadratic form can be simply expressed as a matrix equation:

We then generalize that a bit to include N re-origined parameters αi put together into a column vector.

The symmetric matrix M is called the curvature matrix. In those cases where χ2 is a simple quadratic form, this result is exact. More generally we can view this equation as the first non-zero term in a Taylor expansion of Δχ2 at the minimum. M is then closely related to the matrix of second partial derivatives of χ2 (which is known as the Hessian). We will soon have use for the inverse of the matrix M, which is known as the covariance matrix.

The points we seek (the extreme reach of these ellipses) may be identified by gradient of χ2 (wrt the parameters αi): at the extreme points the gradient points directly along the corresponding coordinate axis as shown below:

Note that because all the ellipses are self-similar, the extreme points all lie along a ray (shown in dotted red above).