**6.2 Complexity analysis using multiscale permutation entropy (***MPE***) method**

The multiscale permutation method involves two steps. A "coarse graining" is applied first to a time series *Xi*, *i* ¼ 1, *N* to construct a consecutive coarse-grained time series. The coarse-grained process involves averaging a successively increasing number of data points in non-over lapping windows. The elements of each of the coarse grained time series *y* ð Þ*s <sup>j</sup>* is computed as,

*Chaotic Dynamics and Complexity in Real and Physical Systems DOI: http://dx.doi.org/10.5772/intechopen.96573*

$$\mathcal{Y}\_j^{(\boldsymbol{\varsigma})} = \frac{1}{s} \sum\_{i=(\ \boldsymbol{j}-1)\boldsymbol{r}+1}^{\boldsymbol{j}\boldsymbol{r}} \mathbf{X}\_i \tag{24}$$

where 1≤*i* ≤ *N=s* and *s* defines the scale factor. Each time series length is of size that is an integral multiple of *N=s*. For *s* ¼ 1, the coarse-grained time series is just the original time series.

The second step involves the computation of permutation entropy [47] for each of the coarse-grained time series. For a coarse-grained time series *y <sup>j</sup>* we first consider the series of vector of length *<sup>m</sup>*, and obtain *Sm*ð Þ¼ *<sup>n</sup> yn*, *yn*þ1, <sup>⋯</sup>, *yn*þ*m*�<sup>1</sup> � �, <sup>1</sup>≤*<sup>n</sup>* <sup>≤</sup>ð Þ� *<sup>N</sup>=<sup>s</sup> <sup>m</sup>* <sup>þ</sup> 1. Subsequently, *Sm*ð Þ *<sup>n</sup>* is arranged in an increasing order viz., *yn*<sup>þ</sup> *<sup>j</sup>* <sup>1</sup>þ<sup>1</sup> <sup>≤</sup>*yn*<sup>þ</sup> *<sup>j</sup>* <sup>2</sup>þ<sup>1</sup> ≤⋯≤*yn*<sup>þ</sup> *<sup>j</sup> <sup>n</sup>*þ1 h i. For *<sup>m</sup>* different numbers, there will be *<sup>m</sup>*! possible order patterns/structures Π which are termed as permutations. If *f*ð Þ Π denotes the frequency of order pattern Π, then the relative frequency and hence the probability *p* ¼ *f*ð Þ Π *=*ð Þ *N=s* � *m* þ 1 . The permutation entropy *H m*ð Þ therefore is given by

$$H(m) = -\sum\_{\Pi=1}^{m!} p\left(\Pi\right) \ln\left(\Pi\right). \tag{25}$$

The maximum value of *H m*ð Þ is *log m*ð Þ! thus showing all permutations to have equal probability. Also, the time series is termed as regular if minimum value of *H m*ð Þ is zero. Therefore *H m*ð Þ the permutation entropy provide a quantitative measure of dynamical complexity of a time series as it refers to its local structures. It may be noted that the permutation entropy depends on the chosen value of *m*. For *m* < 3 , there will be very few distinct states and the foregoing scheme does not work satisfactorily. In the present analysis we have considered sufficiently large time series and chosen *m* ¼ 6 to estimate the complexity measure *MPE*.

For the financial model, Eq. (23), we have simulated the permutation entropy as a function of the scale *s* for *m* ¼ 6. The simulated results have been shown in **Figure 23a** where we observe a saturation behavior with increased value of the scale factor *s*. It is also observed that the permutation entropy at any scale *s* for the interest rate *X* time series is higher than the investment demand time series *Y* which

**Figure 23.** *Multiscale permutation of (a) Financial model time series, (b) Rossler model.*

The representation of two dimensional phase portrait ð Þ *X*, *Y* and the attractor are shown in **Figure 22**. Obviously they represent chaotic dynamics of the temporal

**6.2 Complexity analysis using multiscale permutation entropy (***MPE***) method**

The multiscale permutation method involves two steps. A "coarse graining" is applied first to a time series *Xi*, *i* ¼ 1, *N* to construct a consecutive coarse-grained time series. The coarse-grained process involves averaging a successively increasing number of data points in non-over lapping windows. The elements of each of the

*<sup>j</sup>* is computed as,

behavior shown in **Figure 22**.

**Figure 21.**

**Figure 22.**

**52**

*Temporal evolution of Finance model.*

coarse grained time series *y*

ð Þ*s*

*(a) Phase portrait of Finance model, (b) 3-D attractor of the Finance model.*

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

is further higher than that of the price index *Z* time series. Therefore we may conclude that the complexity measure relation for the considered financial model time series can be expressed as *MPEX* > *MPEY* > *MPEZ*. The behavior of the complexity measure of the considered finance model has been found to be quite similar to that of chaotic Rossler attractor [with parameters *a* ¼ 0*:*15; *b* ¼ 0*:*20; *c* ¼ 10*:*0] (**Figure 23b**).

The simulation results for the multi-scale permutation entropy, *MPE*, presented for the financial model and the Rossler chaotic model exhibit long term correlation of the respective time series of a dynamical variable. Such inference is made in view of the increasing trend of *MPE* with scale factor *s* for a given *m*. In case of a standard financial model, the efficacy of such model could be made on comparing the *MPE* trend of resulting simulated time series for interest rate (*X*), investment demand (*Y*) and that of price index (*Z*) with the availability of the real time series data for the corresponding dynamical variables. Finally, we introduce the idea of generation of time series of a nonlinear chaotic dynamical system, say a Lorenz system, using artificial neural network.
