The chi-squared function - χ2(θ)

For the minimisation of the model-free models a chain of calculations, each based on a different theory, is required. At the highest level the equation which is actually minimised is the chi-squared function

χ2(θ) = $\displaystyle \sum_{{i=1}}^{n}$$\displaystyle {\frac{{(\mathrm{R}_i- \mathrm{R}_i(\theta))^2}}{{\sigma_i^2}}}$, (7.1)

where the index i is the summation index ranging over all the experimentally collected relaxation data of all spins used in the analysis; Ri belongs to the relaxation data set R for an individual spin, a collection of spins, or the entire macromolecule and includes the R1, R2, and NOE data at all field strengths; Ri(θ) is the back-calculated relaxation value belonging to the set R(θ); θ is the model parameter vector which when minimised is denoted by $\hat{\theta}$; and σi is the experimental error.

The significance of the chi-squared equation (7.1) is that the function returns a single value which is then minimised by the optimisation algorithm to find the model-free parameter values of the given model.

The relax user manual (PDF), created 2020-08-26.