Refinements and Error Estimates

Hamilton, Statistics in Physical Science , The Ronald Press Co., New York, 1964. 

The residuals (residual = the observed value minus the value calculated from the estimates of the parameters) constitute the vector V, i.e. 

If the observations are uncorrelated, is a diagonal matrix with elements 

Since the theoretical Intensity values depend non-linearly on the structural parameters (r, u, k), an expansion 

A least-squares calculation carried out in this way will not necessarily converge. Reasonable initial values for the parameters may usually be obtained from the experimental RD function and from related molecules. In the first trials one should usually not refine too many parameters. It is sometimes useful to apply fudge factors , i.e. to take the shift in the parameter as 

---

Table 2 gives a list of sources of errors which will now be discussed. Some of these have already been mentioned in the previous section. An extensive discussion especially of systematic errors has recently been given by Kuchitsu, so the corresponding section in this paper is written rather summarily. The section dealing with least>squares refinement and error estimates is in more detail. 

---