Attributes

filename

The source of the data. This may be the empty string if the data is simulation data.

x,y,dy

The data values. x is the measurement points of data to be fitted. x must be sorted. y is the measured value dy is the measurement uncertainty.

dx

Resolution at the the measured points. The resolution may be 0, constant, or defined for each data point. dx is the 1-sigma width of the Gaussian resolution function at point x. Note that dx_FWHM = sqrt(8 ln 2) dx_sigma, so scale dx appropriately.

fit_x,fit_dx,fit_y,fit_dy

The points used in evaluating the residuals.

calc_x

The points at which to evaluate the theory function. This may be different from the measured points for a number of reasons, such as a resolution function which suggests over or under sampling of the points (see below). By default calc_x is x, but it can be set explicitly by the user.

calc_y, fx

The value of the function at the theory points, and the value of the function after resolution has been applied. These values are computed by a call to residuals.

Notes on calc_x

The contribution of Q to a resolution of width dQo at point Qo is:

p(Q) = 1/sqrt(2 pi dQo**2) exp ( (Q-Qo)**2/(2 dQo**2) )

We are approximating the convolution at Qo using a numerical approximation to the integral over the measured points, with the integral is limited to p(Q_i)/p(0)>=0.001.

Sometimes the function we are convoluting is rapidly changing. That means the correct convolution should uniformly sample across the entire width of the Gaussian. This is not possible at the end points unless you calculate the theory function beyond what is strictly needed for the data. For a given dQ and step size, you need enough steps that the following is true:

(n*step)**2 > -2 dQ**2 * ln 0.001

The choice of sampling density is particularly important near critical points where the shape of the function changes. In reflectometry, the function goes from flat below the critical edge to O(Q**4) above. In one particular model, calculating every 0.005 rather than every 0.02 changed a value above the critical edge by 15%. In a fitting program, this would lead to a somewhat larger estimate of the critical edge for this sample.

Sometimes the theory function is oscillating more rapidly than the instrument can resolve. This happens for example in reflectivity measurements involving thick layers. In these systems, the theory function should be oversampled around the measured points Q. With a single thick layer, oversampling can be limited to just one period 2 pi/d. With multiple thick layers, oscillations will show interference patterns and it will be necessary to oversample uniformly through the entire width of the resolution. If this is not accommodated, then aliasing effects make it difficult to compute the correct model.

x,y or x,y,dy or x,dx,y,dy

R = (y-f(x;p)) / sigma_y
dR/dp1 = -1/sigma_y df(x;p)/dp1
dR/dp2 = -1/sigma_y df(x;p)/dp2
...