Nonextensive Statistical Mechanics

7.4 Computer Sciences 275Fig. 7.74 Influence of (q V , q A ) on the average computer time needed for finding the global minimumwith a given precision. For the present problem the optimal choice consists in q V ≃ 2.8 andq A < 1. Further details in [625].We see therefore a strong influence of q V on the cooling allowed speed, whichcan ultimately benefit (decrease) quite strongly the necessary computational time.To test the method, a toy model has been studied. The cost function is shown inFig. 7.72, and typical runs are shown in Fig. 7.73. The influence of (q V , q A )isdepicted in Fig. 7.74.Another toy model [626] is to use the D = 4 cost functionE(x 1 , x 2 , x 3 , x 4 ) =4∑4∑(xi 2 − 8) 2 + 5 x i , (7.30)i=1i=1which has 15 local minima and one global minimum. Typical results can be seen inFigs. 7.75 and 7.76.An extension of these ideas has been advanced which leads to a q-generalizationof the so-called Pivot method [628, 629]. Typical results are indicated in Figs. 7.77and 7.78. The first use of the GSA in quantum systems was done in [630]. Sincethose early times a large number of algorithmic methods have been implemented,for chemical, neural network and other purposes, inspired by q-statistics [631–640,642–727].7.4.2 Analysis of Time Series and SignalsConcepts of q-statistics have inspired several methods for processing time series andsignals, such as electroencephalograms (EEG), electrocardiograms (ECG), and variousothers. It has been possible to focus on some specific features of epilepsy (in humansand turtles), Alzheimer disease, and other complex circumstances [728–754].The analysis of the tonic–clonic transition of some types of epilepsy constitutes a