Bifurcation diagrams in estimated parameter space using a pruned extreme learning machine

We propose an algorithm to estimate parameter spaces by using a pruned extreme learning machine, but without using principal component analysis, and we plot bifurcation diagrams in the estimated parameter spaces to visualize changes in system patterns. The estimation of parameter spaces can predict changes in the behavior of a system when its parameters are changed. It can be very helpful to adjust the optimal parameters of unknown systems. In this paper, we estimate the dimension of parameter spaces using the singular values of trained synaptic weights and the parameter spaces based on the method proposed by Bagarinao et al., using a pruned extreme learning machine. We motivate this use of a pruned extreme learning machine through numerical experiments, with the estimation of the dimension of parameter spaces for various systems, and we show that the proposed method can successfully plot bifurcation diagrams in the estimated parameter spaces. In addition, we show the results of a bifurcation diagram in the estimated parameter space for maps that are nonlinear-in-parameters, since Bagarinao et al. limited their method to linear-in-parameters maps. In particular, we estimate the parameter spaces for a mathematical model of induction motor drives and for a model of vegetation biomass in ecosystems, which are both nonlinear-in-parameter maps corresponding to real-world systems.

Figure 1

Structure of (a) ELM and (b) SAP-ELM.

Figure 2

Distribution of the trained synaptic weights.

Figure 3

Mean value and standard deviation of the synaptic weight differences.

Figure 4

Contribution ratio and cumulative contribution ratio for (a) logistic map and (b) Hénon map. The solid and broken lines show the contribution ratio and the cumulative contribution ratio, respectively.

Figure 5

BD of the logistic map in the original parameter space.

Figure 6

Results of BD in the estimated parameter space using various patterns of time-series data sets. Parts (a), (b), and (c) are the results of local range for Experiment 1, 2, and 3, respectively. Parts (d), (e), and (f) are the results of Glocal range for Experiment 1, 2, and 3, respectively.

Figure 7

Two-dimensional BDs of the Hénon map in (a) original and (b) estimated parameter spaces. The yellow points in these figures are those with period 20 or greater, and those that are nonperiodic (chaotic) or divergent to infinity. Largest Lyapunov exponents in (c) original and (d) estimated parameter spaces. The points whose time-series data sets are the divergence to infinity are plotted as the same color as that of the maximum value of the largest Lyapunov exponents in each figure.

Figure 8

Results of the Rössler equations. (a) Contribution ratio and cumulative contribution ratio. The solid and broken lines show the contribution ratio and the cumulative contribution ratio, respectively. BDs in (b) original and (c) estimated parameter space.

Figure 9

Contribution ratio and cumulative contribution ratio of the Rössler equations using three parameters. The solid and broken lines show the contribution ratio and the cumulative contribution ratio, respectively.

Figure 10

Results of the mathematical model of induction motor drives. (a) Contribution ratio and cumulative contribution ratio. The solid and broken lines show the contribution ratio and the cumulative contribution ratio, respectively. BDs in (b) original and (c) estimated parameter space.

Figure 11

Results of time-series predictions for the vegetation biomass with white Gaussian noise. The parameter ${\rho }_1$ is set as follows: (a) 1.5 and (b) 2.1.

Figure 12

BDs of the vegetation biomass in (a) original and (b) estimated parameter spaces.