Diagonal scaling

Model scaling can have a significant effect on the optimisation algorithm - a poorly scaled model can cause certain techniques to fail. When two parameters of the model lie on very different numeric scales the model is said to be poorly scaled. For example in model-free analysis the order of magnitude of the order parameters is one whereas for the internal correlation times the order of magnitude is between 1e-12 to 1e-8. Most effected are the trust region algorithms - the multidimensional sphere of trust will either be completely ineffective against the correlation time parameters or severely restrict optimisation in the order parameter dimensions. Again in model-free analyses the significant scaling disparity can even cause failure of optimisation due to amplified effects of machine precision. Therefore the model parameters need to be scaled.

This can be done by supplying the optimisation algorithm with the scaled rather than unscaled parameters. When the chi-squared function, gradient, and Hessian are called the vector is then premultiplied with a diagonal matrix in which the diagonal elements are the scaling factors.

The relax user manual (PDF), created 2024-06-08.