# Distances in Probability Space and the Statistical Complexity Setup

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## Abstract

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## 1. Statistical Complexity Measures

#### 1.1. Meaning of the Concept

#### 1.2. Information Measures

#### 1.3. Distances and Statistical Complexity Measure

- (a)
- This is the “natural” choice (the most simple one) for the distance $\mathcal{D}$. We have$${\mathcal{D}}_{E}[{P}_{1},{P}_{2}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\parallel {P}_{1}\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{P}_{2}\parallel}_{E}^{2}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{j=1}^{N}\phantom{\rule{3.33333pt}{0ex}}{\left(\right)}^{\phantom{\rule{3.33333pt}{0ex}}}2\phantom{\rule{4pt}{0ex}}$$This is the disequilibrium form for the complexity measure proposed by López-Ruiz, Mancini and Calbet (LMC-complexity measure [11]). Such straightforward definition of distance has been criticized by Wootters in an illuminating communication [24] because, in using the Euclidean norm, one is ignoring the fact that we are dealing with a space of probability distributions and thus disregarding the stochastic nature of the distribution P.
- (b)
- The concept of “statistical distance” originates in a quantum mechanical context. One uses it primarily to distinguish among different preparations of a given quantum state, and, more generally, to ascertain to what an extent two such states differ from one another. The concomitant considerations being of an intrinsic statistical nature, the concept can be applied to “any” probabilistic space [24]. The main idea underlying this notion of distance is that of adequately taking into account statistical fluctuations inherent to any finite sample. As a result of the associated statistical errors, the observed frequencies of occurrence of the various possible outcomes typically differ somewhat from the actual probabilities, with the result that, in a given fixed number of trials, two preparations are indistinguishable if the difference between the actual probabilities is smaller than the size of a typical fluctuation [24].$${\mathcal{D}}_{W}[{P}_{1},{P}_{2}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{cos}^{-1}\left(\right)open="\{"\; close="\}">\phantom{\rule{3.33333pt}{0ex}}\sum _{j=1}^{N}\phantom{\rule{3.33333pt}{0ex}}{\left(\right)}^{\phantom{\rule{3.33333pt}{0ex}}}1/2\xb7{\left(\right)}^{\phantom{\rule{3.33333pt}{0ex}}}1/2$$

- (c)
- The relative entropy of ${P}_{1}$ with respect to ${P}_{2}$ associated to Shannons measure is the relative Kullbak-Leibler Shannon entropy, that in the discrete case reads$${\mathcal{D}}_{K}[{P}_{1},{P}_{2}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}K\left[{P}_{1}\right|{P}_{2}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{j=1}^{N}\phantom{\rule{3.33333pt}{0ex}}{p}_{j}^{\left(1\right)}log\left(\right)open="("\; close=")">\frac{{p}_{j}^{\left(1\right)}}{{p}_{j}^{\left(2\right)}}\phantom{\rule{4pt}{0ex}}$$Consider now the probability distribution P and the uniform distribution ${P}_{e}$. The distance between these two distributions, in Kullback-Leiber Shannon terms, will be$${\mathcal{D}}_{K}[P,{P}_{e}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}K\left[P\right|{P}_{e}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathrm{S}\left[{P}_{e}\right]-\mathrm{S}\left[P\right]\phantom{\rule{4pt}{0ex}}$$
- (d)
- In general, the entropic difference $\mathrm{S}\left[{P}_{1}\right]-\mathrm{S}\left[{P}_{2}\right]$ does not define an information gain (or divergence) because the difference is not necessarily positive definite. Something else is needed. An important example is provided by Jensen’s divergence, which is a symmetric version of the Kullback-Leibler relative entropy, which in terms of the Shannon entropy can be written as:$$\begin{array}{ccc}\hfill {\mathcal{D}}_{J}[{P}_{1},{P}_{2}]& =& {\mathcal{J}}_{S}[{P}_{1},{P}_{2}]=\{K\left[{P}_{1}\right|{P}_{2}]+K\left[{P}_{2}\right|{P}_{1}]\}/2\hfill \\ \hfill & =& \mathrm{S}\left(\right)open="["\; close="]">\frac{{P}_{1}+{P}_{2}}{2}-\mathrm{S}\left[{P}_{1}\right]/2-\mathrm{S}\left[{P}_{2}\right]/2\phantom{\rule{4pt}{0ex}}\hfill \end{array}$$The Jensen-Shannon divergence verifies the following properties$$\begin{array}{ccc}\hfill \left(\mathit{i}\right)\phantom{\rule{1.em}{0ex}}& {\mathcal{J}}_{S}[{P}_{1},{P}_{2}]& \ge 0\phantom{\rule{4pt}{0ex}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left(\mathit{ii}\right)\phantom{\rule{1.em}{0ex}}& {\mathcal{J}}_{S}[{P}_{1},{P}_{2}]& ={\mathcal{J}}_{S}[{P}_{2},{P}_{1}]\phantom{\rule{4pt}{0ex}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill \left(\mathit{iii}\right)\phantom{\rule{1.em}{0ex}}& {\mathcal{J}}_{S}[{P}_{1},{P}_{2}]& =0\iff {P}_{2}={P}_{1}\phantom{\rule{4pt}{0ex}}\hfill \end{array}$$Moreover, it square root satisfies the triangle inequality$$\left(\mathit{iv}\right)\phantom{\rule{1.em}{0ex}}{\left({\mathcal{J}}_{S}[{P}_{1},{P}_{2}]\right)}^{1/2}+{\left({\mathcal{J}}_{S}[{P}_{2},{P}_{3}]\right)}^{1/2}={\left({\mathcal{J}}_{S}[{P}_{1},{P}_{3}]\right)}^{1/2}\phantom{\rule{4pt}{0ex}}$$

#### 1.4. Time Evolution

#### 1.5. Additional Issues

- (1)
- it is neither an intensive nor an extensive quantity.
- (2)
- it vanishes exponentially in the thermodynamic limit for all one-dimensional, finite range systems.

- (3)
- be able to distinguish among different degrees of periodicity;
- (4)
- vanish only for unity periodicity.

- (i)
- able to grasp essential details of the dynamics (i.e., chaos, intermittency, etc.)
- (ii)
- capable of discerning between different degrees of periodicity, and
- (iii)
- an intensive quantity if Jensen’s divergence is used.

## 2. Methodologies for Selecting PDFs

#### 2.1. PDF Based on Histograms

#### 2.2. PDF Based on Bandt and Pompe’s Methodology

## 3. The Classical Limit of Quantum Mechanics (a Special Semi-Classical Model)

## 4. Results and Discussion

**Figure 1.**Signal vs. time graphs. Subplots 1–9: solutions of the system (21) (semi-quantum signal), for representative fixed values of ${E}_{r}$. Subplot 10: solution of the classical counterpart of the system (21) (classical, $I=0$). We took ${m}_{q}={m}_{cl}={\omega}_{q}=e=1$. Initial conditions: $E=0.6$, $\langle L\rangle \left(0\right)=L\left(0\right)=0,$ and $A\left(0\right)=0$. The uppermost left plot corresponds to the “pure quantum” signal. At the bottom right we plot the classical signal vs. time. The remaining are intermediate situations. All quantities are dimensionless.

- In Figure 2a,b we plot the Normalized Shannon Entropy, $\mathcal{H}$, for PDF histogram and for PDF Bandt and Pompe. Notice in Figure 2a that both definitions tend to different final results for large ${E}_{r}$ values. These are the classical results, i.e., the corresponding entropic values calculated using data obtained from the classical versions of Equations (21). Also note in Figure 2b that the PDF histogram’s entropy does not clearly distinguish between transitional and classical sectors. The Bandt and Pompe entropy does distinguish the three process zones and correctly orders entropic sizes according the the physical criteria expounded above (Figure 2b), i.e., it can be said to appropriately represent our three regions.
- The LMC statistical complexity, ${\mathcal{C}}^{\left(E\right)}$ (Figure 3b) does not distinguish between transitional and classical zones in its PDF histogram version, and neither is able to correctly represent the classical sector. Also, the LMC’s PDF Bandt and Pompe version fails to aptly describe the quantal region (see Figure 3b).
- The MPR statistical complexity with Wooters’ distance (Figure 4b) does distinguish amongst the three zones in its two PDF versions. For the PDF histogram, maximum complexity is attained at ${{E}_{r}}^{cl}$. Its quantal sector representation is marred by the fact that one detects complexity values within it larger than those encountered within the transition region (Figure 4b). For the PDF Bandt and Pompe version the associated representation is for all zones grosso modo correct. One would expect, however, that the classical zone’s complexity should not be so similar to that for the transitional one. Note that this classical zone’s complexity is slightly smaller than the maximal complexity attained at ${E}_{r}^{M}=8.09$ (Figure 4b).
- The MPR (Jensen-Shannon) statistical complexity (Figure 4c,d) is quite similar to its MPR (Wootters) counterpart (Figure 4a,b), and distinguishes quite well amongst the three typical regions for both PDF versions, although the quantal representation in the PDF histogram’s case fails, as was the case with the MPR (Wootters) counterpart (Figure 4b), a facet that is improved by the Bandt and Pompe PDF treatment. Consequently, the symbolic MPR (Jensen-Shannon) statistical complexity exhibits the best overall performance (Figure 4d).

**Figure 2.**Normalized Shannon entropies for two PDF types: “histogram” and Bandt and Pompe are plotted vs. ${E}_{r}$. (a) convergence to classicality. (b) the PDF histogram’s entropy does not clearly distinguish between transition and classical zones while the Bandt and Pompe one does distinguish and appropriately represents our three regions.

**Figure 3.**LMC statistical complexity for histogram and Bandt-Pompe PDF’s vs. ${E}_{r}$. The LMC statistical complexity neither distinguishes the transition from the classical zones in its histogram PDF version nor can correctly represent the classical sector (see (b)). Also, the LMC’s PDF Bandt and Pompe version fails to aptly describe the quantal region.

**Figure 4.**(a)–(b) MPR statistical complexity (Wooters) vs. ${E}_{r}$, where one distinguishes the three zones for the two PDF versions (see (b)). The PDF histogram, MPR statistical complexity fails to describe the quantal region, see (b). The PDF Bandt-Pompe treatment is, for all zones, grosso modo correct. (c)–(d) MPR (Jensen-Shannon) statistical complexity vs. ${E}_{r}$. One distinguishes the three typical regions for both PDF versions. The quantal representation in the PDF histogram’s case fails as is the case of its MPR (Wootters) counterpart (see (d)).

**Figure 5.**Entropy-complexity plane using a PDF histogram (${N}_{bin}=512$), for: (a) LMC statistical complexity, (b) MPR statistical complexity with Wootters distance, (c) MPR statistical complexity with Jensen-Shannon distance. Entropy-complexity plane in the case of the PDF Bandt and Pompe ($D=5$), for: (d) LMC statistical complexity, (e) MPR statistical complexity with Wooters’ distance, (f) MPR statistical complexity with Jensen-Shannon distance. We also display the maximum and minimum possible values of the corresponding statistical complexity (continuous curves) and the signal points (dashed lines).

## 5. Conclusions

- The selection of an appropriate disequilibrium distance form $\mathcal{Q}$.
- The selection of an adequate probability distribution function (PDF).

## Acknowledgment

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**MDPI and ACS Style**

Kowalski, A.M.; Martín, M.T.; Plastino, A.; Rosso, O.A.; Casas, M.
Distances in Probability Space and the Statistical Complexity Setup. *Entropy* **2011**, *13*, 1055-1075.
https://doi.org/10.3390/e13061055

**AMA Style**

Kowalski AM, Martín MT, Plastino A, Rosso OA, Casas M.
Distances in Probability Space and the Statistical Complexity Setup. *Entropy*. 2011; 13(6):1055-1075.
https://doi.org/10.3390/e13061055

**Chicago/Turabian Style**

Kowalski, Andres M., Maria Teresa Martín, Angelo Plastino, Osvaldo A. Rosso, and Montserrat Casas.
2011. "Distances in Probability Space and the Statistical Complexity Setup" *Entropy* 13, no. 6: 1055-1075.
https://doi.org/10.3390/e13061055