# Multidimensional Scaling Visualization Using Parametric Similarity Indices

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Tools

#### 2.1. Parametric Similarity Indices

#### 2.1.1. Generalized Correlation

_{i}, Y

_{i}) and (X

_{j}, Y

_{j}) are independent bivariate vectors with i ≠ j, (i, j) = 1, 2, …, n. The function g

_{q}(z) in Equation (1) is given by:

_{q}∈ [−1, 1] with lower and upper values corresponding to the Kendall, ρ

_{K}, and Pearson, ρ

_{P}, correlation coefficients, respectively:

#### 2.1.2. Minkowski Distance

_{1}, x

_{2}, …,x

_{n}) and Y = (y

_{1}, y

_{2}, …, y

_{n}) in ${\mathrm{\mathbb{R}}}^{n}$, the q-order Minkowski distance is expressed by:

#### 2.1.3. Entropy

_{i}) = − ln p

_{i}, of an event with probability of occurrence p

_{i}, where $\sum _{i}{p}_{i}=1$.

^{q}(·) is the fractional derivative of order q and Γ (·) and ψ (·) represent the gamma and digamma functions, respectively.

#### 2.2. Multidimensional Scaling

_{ij}stands for the reproduced distances between items i and j, (i, j) = {0, 1, 2, …, h − 1}, δ

_{ij}represents the observed distances and f(·) indicates some type of transformation. The smaller the value of $\mathcal{S}$, the better is the fit between d

_{ij}and δ

_{ij}.

_{ij}distances, for a particular value m, versus the δ

_{ij}distances. Therefore, a narrow scatter around the 45-degree line indicates a good fit between d

_{ij}and δ

_{ij}.

## 3. MDS Analysis and Visualization of Complex Systems

#### 3.1. Illustrative Example

_{k}(t), kT/h ≤ t < (k + 1)T/h, k = {0, 1, 2, …, h − 1}. For a PSI, we calculate p similarity matrices, M

_{q}, h × h dimensional, where q ∈ {q

_{1}, q

_{2},…, q

_{p}}. Matrices M

_{q}feed the MDS algorithm, which generates p intermediate maps of “points” (i.e., one map per value q). The charts are then processed by means of Procrustes analysis in order to obtain a single global plot of “shapes” where the “points” of the original maps are optimally “superimposed”.

- During the time period 1800–1930, the GDP had a moderate growth with small oscillations over time. In the MDS maps, we observe small “shapes”, representing Windows 1 up to 26, appearing in sequence and close to each other;
- In the time period 1930–1935, corresponding to the worst years of the Great Depression, the GDP had a strong oscillation, increasing (with small fluctuation) during the subsequent five-year period, 1935–1940. Such behavior is observed in the MDS maps as two moderate length “jumps” from Window 26 towards Window 27 and then towards Window 28;
- Periods 1940–1945, 1945–1950 and 1950–1955 had contrasting behavior, characterized by growth and recession, mostly determined by World War II. In the MDS maps, we observe large “jumps” across Windows 29 up to 31;
- For the time period 1955–1960, GDP growth became slower and then recovered during the period 1960–1965. This corresponds to large “jumps” across Windows 31, 32 and 33 in the MDS;
- From year 1960 onward, we observe a GDP moderate growth trend, with oscillations over time. This behavior translates into moderate, and similar, length “jumps” between shapes in the MDS maps.

#### 3.2. MDS Based on the Generalized Correlation Index

_{i}(t) and x

_{j}(t).

_{q}reveals difficulties with time flow discretization, since we obtain a large number of chaotic “jumps” between “shapes”.

_{q}leads to a poor time discrimination between the data.

_{q}. For the BR data, and for all PSI, the results are similar. Figure 5 represents the p Shepard diagrams superimposed in a single chart. We can observe a scatter of points distributed around the 45-degree line, which means a good fit of the observed distances to the dissimilarities. As expected, for m = 3, we get slightly better results than for m = 2. Figure 6 depicts the superposition of the stress diagrams, revealing that, for the p intermediate MDS maps, a three-dimensional space describes the data well.

#### 3.3. MDS Based on Minkowski Distance

_{q}is able to discriminate time. Here, we consider ‘good time discrimination’ as the ability to depict charts with a small content of discontinuities over time evolution. For the DJ, we observe three distinct clusters of “shapes”: $\mathcal{A}=\{1,2,3,4,5,6,7,8,9,10\}$, $\mathcal{B}=\{11,12,13,14,15,16,17,18,19,20,21,22,23,24,25\}$ and $\mathcal{C}=\{26,27\}$. For $\mathcal{A}$ and $\mathcal{C}$, the time-windows that are closer to each other in time appear closer to each other in the MDS map. For cluster $\mathcal{C}$, the time behavior is more “random”.

#### 3.4. MDS Based on Entropy Measures

_{i}(t) and x

_{j}(t):

_{q}, the entropy-based indices reveal difficulties with time discrimination, giving rise to a large number of “jumps” between “shapes”. For the DJ time-series, windows $\mathcal{A}=\left\{1,18,19,21\right\}$ and $\mathcal{B}=\left\{14\right\}$ have different behavior, being located far apart from the remaining. Regarding the BR data, windows $\mathcal{A}=\left\{4,9\right\}$ and $\mathcal{B}=\left\{27\right\}$ unveil different characteristics of those exhibited by the other time periods.

## 4. Conclusions

## References

- Lillo, F.; Mantegna, R.N. Power-law relaxation in a complex system: Omori law after a financial market crash. Phys. Rev. E
**2003**, 68, 016119. [Google Scholar] - Calzetta, E. Chaos, decoherence and quantum cosmology. Class. Quantum Grav.
**2012**, 29, 143001. [Google Scholar] - Klimenko, A. What is mixing and can it be complex? Phys. Scripta
**2013**, 2013, 014047. [Google Scholar] - Harris, J.A.; Hobbs, R.J.; Higgs, E.; Aronson, J. Ecological restoration and global climate change. Restor. Ecol
**2006**, 14, 170–176. [Google Scholar] - Neil Adger, W.; Arnell, N.W.; Tompkins, E.L. Successful adaptation to climate change across scales. Global Environ. Change
**2005**, 15, 77–86. [Google Scholar] - Johnson, N.F.; Jefferies, P.; Hui, P.M. Financial market complexity; Oxford University Press: New York, NY, USA,, 2003. [Google Scholar]
- Mategna, R.; Stanley, H. An introduction to econophysics; Cambridge University: Cambridge, UK, 2000. [Google Scholar]
- Amaral, L.A.N.; Scala, A.; Barthélémy, M.; Stanley, H.E. Classes of small-world networks. Proc. Natl. Acad. Sci.
**2000**, 97, 11149–11152. [Google Scholar] - Sornette, D.; Pisarenko, V. Fractal plate tectonics. Geophys. Res. Lett.
**2003**, 30, 1105. [Google Scholar] - Rind, D. Complexity and climate. Science
**1999**, 284, 105–107. [Google Scholar] - Lopes, A.M.; Tenreiro Machado, J. Analysis of temperature time-series: Embedding dynamics into the MDS method. Comm. Nonlinear Sci. Numer. Simul.
**2014**, 19, 851–871. [Google Scholar] - Glunt, W.; Hayden, T.; Raydan, M. Molecular conformations from distance matrices. J. Comp. Chem.
**1993**, 14, 114–120. [Google Scholar] - Tenenbaum, J.B.; De Silva, V.; Langford, J.C. A global geometric framework for nonlinear dimensionality reduction. Science
**2000**, 290, 2319–2323. [Google Scholar] - Martínez-Torres, M.R.; García, F.B.; Marín, S.T.; Vázquez, S.G. A digital signal processing teaching methodology using concept-mapping techniques. IEEE Trans. Educ.
**2005**, 48, 422–429. [Google Scholar] - Tzagarakis, C.; Jerde, T.A.; Lewis, S.M.; Uğurbil, K.; Georgopoulos, A.P. Cerebral cortical mechanisms of copying geometrical shapes: A multidimensional scaling analysis of fMRI patterns of activation. Exp. Brain Res.
**2009**, 194, 369–380. [Google Scholar] - Polzella, D.J.; Reid, G.B. Multidimensional scaling analysis of simulated air combat maneuvering performance data. Aviat. Space Environ. Med.
**1989**. [Google Scholar] - Costa, A.M.; Machado, J.T.; Quelhas, M.D. Histogram-based DNA analysis for the visualization of chromosome, genome and species information. Bioinformatics
**2011**, 27, 1207–1214. [Google Scholar] - Machado, J.T.; Duarte, G.M.; Duarte, F.B. Identifying economic periods and crisis with the multidimensional scaling. Nonlinear Dyn
**2011**, 63, 611–622. [Google Scholar] - Oñate, J.J.; Pou, A. Temperature variations in Spain since 1901: a preliminary analysis. Int. J. Climatol.
**1996**, 16, 805–815. [Google Scholar] - Stephenson, D.B.; Doblas-Reyes, F.J. Statistical methods for interpreting Monte Carlo ensemble forecasts. Tellus A
**2000**, 52, 300–322. [Google Scholar] - Chinchilli, V.M.; Phillips, B.R.; Mauger, D.T.; Szefler, S.J. A general class of correlation coefficients for the 2× 2 crossover design. Biom. J.
**2005**, 47, 644–653. [Google Scholar] - Deza, M.M.; Deza, E. Encyclopedia of Distances; Springer: Berlin, Germany, 2009. [Google Scholar]
- Balasis, G.; Daglis, I.A.; Papadimitriou, C.; Anastasiadis, A.; Sandberg, I.; Eftaxias, K. Quantifying dynamical complexity of magnetic storms and solar flares via nonextensive Tsallis entropy. Entropy
**2011**, 13, 1865–1881. [Google Scholar] - Levada, A. Learning from Complex Systems: On the Roles of Entropy and Fisher Information in Pairwise Isotropic Gaussian Markov Random Fields. Entropy
**2014**, 16, 1002–1036. [Google Scholar] - Seely, A.J.; Newman, K.D.; Herry, C.L. Fractal Structure and Entropy Production within the Central Nervous System. Entropy
**2014**, 16, 4497–4520. [Google Scholar] - Khinchin, A.I. Mathematical Foundations of Information Theory; Courier Dover Publications: New York, USA, 1957; Volume 434. [Google Scholar]
- Ubriaco, M.R. Entropies based on fractional calculus. Phys. Lett. A
**2009**, 373, 2516–2519. [Google Scholar] - Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Models and Numerical Methods; World Scientific: Singapore, Singapore, 2012; Volume 3. [Google Scholar]
- Kenneth, M.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models; World Scientific: Singapore, Singapore, 2010. [Google Scholar]
- Machado, J.T.; Fractional, Order. Generalized Information. Entropy
**2014**, 16, 2350–2361. [Google Scholar] - Valério, D.; Trujillo, J.J.; Rivero, M.; Machado, J.T.; Baleanu, D. Fractional calculus: A survey of useful formulas. Eur. Phys. J. Spec. Top.
**2013**, 222, 1827–1846. [Google Scholar] - Shepard, R.N. The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika
**1962**, 27, 125–140. [Google Scholar] - Shepard, R.N. The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika
**1962**, 27, 219–246. [Google Scholar] - Kruskal, J.B.; Wish, M. Multidimensional Scaling; Sage: New York, NY, USA, 1978; Volume 11. [Google Scholar]
- Borg, I.; Groenen, P.J. Modern Multidimensional Scaling: Theory and Applications; Springer: Berlin, Germany, 2005. [Google Scholar]
- Kruskal, J.B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika
**1964**, 29, 1–27. [Google Scholar] - Machado, J.T. Relativistic time effects in financial dynamics. Nonlinear Dyn
**2014**, 75, 735–744. [Google Scholar] - Machado, J.T. Complex dynamics of financial indices. Nonlinear Dyn
**2013**, 74, 287–296. [Google Scholar] - The Maddison-Project 2013 version. Available online: http://www.ggdc.net/maddison/maddison-project/home.htm accessed on 25 March 2015.

**Figure 2.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The Minkowski distance, d

_{q}, is used with the U.S. GDP per capita time-series.

**Figure 3.**Multidimensional scaling (MDS) maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The generalized correlation index, ρ

_{q}, is used with the Dow Jones (DJ) data.

**Figure 4.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The generalized correlation index, ρ

_{q}, is used with the Brent Spot (BR) data.

**Figure 5.**Superposition of the p MDS Shepard diagrams: (

**a**) m = 2; (

**b**) m = 3. The generalized correlation index, ρ

_{q}, is used with the DJ data.

**Figure 6.**Superposition of the p MDS stress diagrams. The generalized correlation index, ρ

_{q}, is used with the DJ data.

**Figure 7.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The Minkowski distance, d

_{q}, is used with the DJ data.

**Figure 8.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The Minkowski distance, d

_{q}, is used with the BR data.

**Figure 9.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The generalized entropy index, ${S}_{q}^{(G)}$, is used with the DJ data.

**Figure 10.**MDS maps of “shapes” and the flow of time, obtained after Procrustes analysis: (

**a**) m = 2; (

**b**) m = 3. The generalized entropy index, ${S}_{q}^{(G)}$, is used with the BR data.

**Figure 11.**Locus of $\mathcal{T}$ versus Σ for the DJ time-series and h = 27: (

**a**) generalized correlation; (

**b**) Minkwoski distance; (

**c**) generalized entropy.

**Figure 12.**Locus of $\mathcal{T}$ versus Σ for the DJ time-series: (

**a**) h = 2×27 and ρ

_{q}; (

**b**) h = 3×27 and ρ

_{q}; (

**c**) h = 2×27 and d

_{q}; (

**d**) h = 3×27 and d

_{q}; (

**e**) h = 2×27 and ${S}_{q}^{(G)}$; (

**f**) h = 3×27 and ${S}_{q}^{(G)}$.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tenreiro Machado, J.A.; Lopes, A.M.; Galhano, A.M. Multidimensional Scaling Visualization Using Parametric Similarity Indices. *Entropy* **2015**, *17*, 1775-1794.
https://doi.org/10.3390/e17041775

**AMA Style**

Tenreiro Machado JA, Lopes AM, Galhano AM. Multidimensional Scaling Visualization Using Parametric Similarity Indices. *Entropy*. 2015; 17(4):1775-1794.
https://doi.org/10.3390/e17041775

**Chicago/Turabian Style**

Tenreiro Machado, J. A., António M. Lopes, and Alexandra M. Galhano. 2015. "Multidimensional Scaling Visualization Using Parametric Similarity Indices" *Entropy* 17, no. 4: 1775-1794.
https://doi.org/10.3390/e17041775