# Fractional State Space Analysis of Economic Systems

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Data and Methodology

## 4. MDS Analysis of World Economies

**C**of item-to-item similarities, where n is the total number of items in an r-dimensional space. In classic MDS,

**C**is symmetric and its main diagonal is composed of “1”. Based on that information, MDS extrapolates an approximate map in a u-dimensional space ($u<r$) with the objects’ locations. MDS uses a function minimization algorithm that evaluates different configurations with the goal of maximizing a goodness-of-fit. By rearranging the items, positions in the u-space, MDS arrives at a configuration that best approximates the reproduced and observed similarities. For low dimensional spaces (e.g., $u=2$ or $u=3$), the resulting points can be displayed in a map. The MDS interpretation is based on the emerging clusters and distances between points in the map, rather than on their absolute coordinates or the geometric form of the locus. We can rotate or translate the MDS map since the distances between points remain identical.

#### 4.1. MDS Analysis Based on the Pearson Correlation

**Figure 1.**The multidimensional scaling (MDS) maps for the Pearson correlation ${\rho}_{ij}$ between the GDP per capita time series 1870–2010 of n = 34 countries: (

**a**) two-dimensional; (

**b**) three-dimensional.

**Figure 2.**Shepard plots for the Pearson correlation ${\rho}_{ij}$ between the GDP per capita time series 1870–2010 of n = 34 countries for representations: (

**a**) two-dimensional; (

**b**) three-dimensional.

**Figure 3.**Stress plot for the Pearson correlation ${\rho}_{ij}$ between the GDP per capita time series 1870–2010 of $n=34$ countries.

#### 4.2. MDS Analysis Based on the Mutual Information

**Figure 4.**The MDS maps for the normalized mutual information ${I}_{N}({y}_{i},{y}_{j})$ between the GDP per capita time series 1870–2010 of n = 34 countries: (

**a**) two-dimensional; (

**b**) three-dimensional.

## 5. State Space Analysis of World Economies

#### 5.1. Classic State Space Portrait

**Figure 6.**The state space portrait (SSP) for the USA GDP per capita time series 1872–2008: (

**a**) two-dimensional map; (

**b**) three-dimensional map.

#### 5.2. Fractional-Order State Space

#### 5.2.1. Determination of the fSSP Based on the Pearson Correlation

**Figure 7.**Maps of ${\rho}_{pq}$ for the GDP per capita of USA during 1870–2010: (

**a**) contour plot; (

**b**) contour plot showing a few isoclines.

**Figure 9.**Locus of points $({\alpha}_{p}^{c},{\alpha}_{q}^{c})$ corresponding to the maximum curvatures of the isoclines that contain the point $(0,1)$, obtained with the Pearson correlation.

#### 5.2.2. Determination of the fSSP Based on the Mutual Information

**Figure 11.**Locus of points $({\alpha}_{p}^{c},{\alpha}_{q}^{c})$ corresponding to the maximum curvatures of the isoclines that contain the point $(0,1)$, obtained with the mutual information, ${I}_{pq}$.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Machado, J.A.T.; Mata, M.E.; Lopes, A.M.
Fractional State Space Analysis of Economic Systems. *Entropy* **2015**, *17*, 5402-5421.
https://doi.org/10.3390/e17085402

**AMA Style**

Machado JAT, Mata ME, Lopes AM.
Fractional State Space Analysis of Economic Systems. *Entropy*. 2015; 17(8):5402-5421.
https://doi.org/10.3390/e17085402

**Chicago/Turabian Style**

Machado, J. A. Tenreiro, Maria Eugénia Mata, and António M. Lopes.
2015. "Fractional State Space Analysis of Economic Systems" *Entropy* 17, no. 8: 5402-5421.
https://doi.org/10.3390/e17085402