The optimisation space

The optimisation of the parameters of an arbitrary model is dependent on a function f which takes the current parameter values θ∈$\mathbb {R}$ⁿ and returns a single real value f (θ)∈$\mathbb {R}$ corresponding to position θ in the n-dimensional space. For it is that single value which is minimised as

$$\hat{\theta} \min_{\theta}^{}$$ (14.1)

where $\hat{\theta}$ is the parameter vector which is equal to the argument which minimises the function f (θ). In most analyses in relax, f (θ) is the chi-squared equation

$$\sum_{{i=1}}^{n} {\frac{{(y_i - y_i(\theta))^2}}{{\sigma_i^2}}}$$ (14.2)

where i is the summation index over all data, y_i is the experimental data, y_i(θ) is the back calculated data, and σ_i is the experimental error.