Fig. 6

Typical single-junction-switch Metropolis generation series for Central London (United Kingdom): the foreground purple generation series plot, for different effective scaling exponents \({\widetilde{\lambda }}=2\lambda \upsilon\), typical simulations starting from a self-fit output; the background light-grey generation series plot, according to the same modus operandi, typical sequences of self-fit outputs. Each horizontal solid line drawn along a generation series plot represents its in-equilibrium mean value, the accompanying horizontal dashed lines indicate the associated standard-deviation bounds. Coloured lines, Metropolis generation series; greyed lines, sequences of self-fit outputs. The equilibria are assumed reached after the \(5000{\,{\mathrm{th}}}\) generation. The following annealing parameters were used to algebraically cool down to the desired \({\widetilde{\lambda }}:\) \({\widetilde{\lambda }}_{0}=1.495\), \(\epsilon ={10^{-3}}\) and \(m=500\). ( The starting cooling value \({\widetilde{\lambda }}_{0}\) was chosen by hand so that the associated Metropolis equilibrium approaches on average self-fit outputs—as illustrated in (d); the special case \({\widetilde{\lambda }}=\infty\) in (h) corresponds to a full ‘simulated annealing’ process (Press et al 2007; Galassi et al 2009). )