# Time-Reversibility, Causality and Compression-Complexity

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Effort-to-Compress

#### 2.2. Compression-Complexity Causality

#### 2.3. A Novel Temporal Asymmetry Measure

#### 2.3.1. Theoretical Underpinnings of ETC

- ${Y}_{1}$: the given sequence Y as it is;
- ${Y}_{2}$: transformed Y, once the most frequently occurring pair in Y has been substituted with another symbol, i.e., the sequence after the first iteration of the ETC algorithm;
- ${Y}_{i}$: transformed sequence after $i-1$ iterations of the ETC algorithm, where, at each iteration, the most frequently occurring pair in that sequence is being substituted by a new symbol;
- ${Y}_{n+1}$: transformed sequence Y after n iterations, where no more iterations after this are possible, and hence n is the ETC value;
- ${X}_{1}$: most frequently occurring pair in ${Y}_{1}$. In other words, it is the first most dominant shortest pattern (of length 2);
- ${X}_{2}$: most frequently occurring pair in ${Y}_{2}$. In other words, it is the second most dominant shortest pattern (of length 2 in ${Y}_{2}$, but may be of length 2 or 3 in the original sequence, Y);
- ${X}_{i}$: most frequently occurring pair in ${Y}_{i}$;
- ${X}_{n}$: most frequently occurring pair in ${Y}_{n}$. It is the ${n}^{th}$ most dominant shortest pattern.

#### 2.3.2. Compressive Potential

#### 2.3.3. Compressive-Potential-Based Temporal Asymmetry Measure

## 3. Results

#### 3.1. Causality between Coupled, Time-Reversed Processes

#### 3.2. Detection of Temporal Reversibility

#### 3.2.1. Example Cases to Demonstrate the Performance of Compressive Potential

#### 3.2.2. Performance of Compressive-Potential-Based Temporal Asymmetry Measure on Simulations

**Time-reversible**processes that were simulated include:

- Linear Gaussian Process (LGP), that is, Gaussian noise with distribution $\mathcal{N}(0,1)$;
- Autoregressive process of second order, AR(2)$$X\left(t\right)=0.7X(t-1)+0.2X(t-2)+0.03{\u03f5}_{t},$$
- Static nonlinear transformation of a first order Gaussian process, STAR(1)$$\begin{array}{cc}\hfill X\left(t\right)& ={tanh}^{2}\left(Y\left(t\right)\right),\phantom{\rule{0.222222em}{0ex}}\mathrm{where}\hfill \\ \hfill Y\left(t\right)& =0.6Y(t-1)+0.03{\u03f5}_{t},\hfill \end{array}$$

**Time-irreversible**processes that were simulated include:

- Self-Exciting Threshold AR (SETAR(2;2,2)) process with two regimes, each one with second-order delays$$X\left(t\right)=\left(\right)open="\{"\; close>\begin{array}{cc}0.62+1.25X(t-1)-0.43X(t-2)+0.0381{\u03f5}_{t}\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}X(t-2)\le 3.25\hfill \\ 2.25+1.52X(t-1)-1.24X(t-2)+0.0626{\u03f5}_{t}\hfill & \mathrm{otherwise},\hfill \end{array}$$
- Chaotic tent-map process$$X\left(t\right)=\left(\right)open="\{"\; close>\begin{array}{c}2X(t-1),\phantom{\rule{2.em}{0ex}}0\le X(t-1)1/2,\hfill \\ 2-2X(t-1),\phantom{\rule{2.em}{0ex}}1/2\le X(t-1)\le 1.\hfill \end{array}$$

#### 3.2.3. Performance of Compressive-Potential-Based Temporal Asymmetry Measure on Real Data

- Sunspot numbersTime series analysis of sunspot numbers is an active area of research. Sunspots are regions of reduced surface temperature on the Sun’s surface that appear as spots darker than the surrounding areas. They are caused by concentrations of magnetic field flux that inhibit convection. These numbers are important in the study of the solar system as well as the activities of the sun. Monthly and annual data of sunspot activity are typically known to be nonlinear, non-Gaussian and non-stationary with characteristics of chaotic sequences [63,64,65], and, hence, these are treated as irreversible.The sunspot numbers used in this study were obtained from the SILSO website (www.sidc.be/silso/datafiles; accessed on 24 February 2021) as monthly mean measurements (Sunspot data from the World Data Center SILSO, Royal Observatory of Belgium, Brussels). This dataset consists of monthly data recorded starting from January, 1749. It currently comprises data from 3265 months of observation, recorded up to January, 2021. The entire dataset, comprising these 3265 datapoints, was used in our analysis. A subset of this dataset, starting from the beginning of the year 1920 to the end of year 2020, is depicted in Figure 5.
- Digits of the transcendental number $\pi $The decimal expansion of number $\pi $ is known to be non-repeating (transcendental irrational number). Furthermore, it is widely believed that $\pi $ is a normal number (though proof has remained elusive till date). A number is said to be normal (in base b) if, for every positive integer N, all possible strings of length N have a frequency of occurrence ${b}^{-N}$. Equivalently, we can say that a normal number does not prefer a set of patterns in its expansion over others, and, thus, every possible pattern occurs equally often. A recent work by Peter Trueb [66] analyzed the first $22.4$ trillion decimal digits of $\pi $ and found that frequencies of sequences of lengths one, two and three are consistent with the hypothesis of $\pi $ being a normal number in base-10 and base-16. We claim that this would mean that normal numbers are essentially reversible, since the reversed order of the digits would not change the frequency of occurrence in any way (else it would fail to be normal). We consider the decimal expansion ($b=10$) of $\pi $ up to a 1000 digits and check its reversibility using our proposed measure.For the computation of ${A}_{{P}_{C}}$ value for 1. and 2., the time-series taken were symbolized using four bins. The number of bins taken here were fewer compared to that taken for simulated data in order to allow for more patterns to repeat (and, hence, an appropriate computation of ETC to take place) in a shorter length of available data. The parameters for the measure were set as $\tau =500,k=100$. In order to assess the statistical significance of the obtained ${A}_{{P}_{C}}$ for each process taken, surrogate data testing was done by generating a surrogate ensemble of 50 realizations in the same manner as for simulated data (discussed in Section 3.2.2). Figure 6 displays the distribution of ${A}_{{P}_{C}}$ values of surrogate data, as well as a dotted line showing where the ${A}_{{P}_{C}}$ value of the original time series lies for real datasets 1 and 2. ${A}_{{P}_{C}}$ distribution of surrogates for both the processes was found to satisfy normality based on the Anderson–Darling test. For Sunspot numbers, the null hypothesis (of reversibility) was rejected with p-value = 0.04 (Figure 6a). On the other hand, for digits of $\pi $, the null hypothesis was not rejected with p-value = 0.13 (Figure 6b). Hence, the sunspot numbers time-series was correctly determined as being irreversible and digits of $\pi $ were correctly determined as being time-reversible based on the performance of ${A}_{{P}_{C}}$.
- Heart period variabilityThe last set of data taken was of heart interbeat intervals from young and elderly healthy human subjects. This dataset was obtained from “Physionet: Fantasia database” [67] and was originally acquired for the study in [68]. In the study, twenty young (21–34 years old) and twenty elderly (68–85 years old) subjects underwent 120 min of continuous supine resting while watching the movie Fantasia (Disney, 1940) in order to help maintain wakefulness. During this time, continuous ECG data were recorded and digitized by sampling at 250 Hz. The heartbeats were annotated using an automated arrhythmia detection algorithm and the beat annotations were later verified by visual inspection. The occurrence of each "R" peak was noted, and the time series consisting of the time difference between successive peaks was generated. This process was repeated for each of the participants.The interbeat interval variations in the heart are characterized by an asymmetric behavior under time reversal. This is because the heart decelerates faster than it accelerates, thus resulting in heart period variability asymmetry (HPVA) [69,70]. In previous studies, the HPVA has been assessed by the use of different measures applied to the difference between two successive heart period values. These studies include [11,13,14,71]. Furthermore, it has been shown that the HPVA is influenced by pathological conditions such as heart failure [14,72] and mental disorders [73,74], as well as aging [11,75]. In particular, HPVA is found to reduce with aging [11,75].In the analysis here, we estimated the values of ${A}_{{P}_{C}}$ for each of the young and old subjects by using time series obtained by subtracting successive values of heart periods (or RR intervals) taken from the Physionet Fantasia database. The first 2500 observations of heart period were taken from each subject. The number of bins used to obtain the symbolic sequence were set to four, and the parameters for estimation of ${A}_{{P}_{C}}$ were set to $\tau =450,k=500$. Since the ${A}_{{P}_{C}}$ can attain both positive and negative values, and a higher magnitude of the measure implies higher asymmetry, the absolute values of ${A}_{{P}_{C}}$ obtained from young and old subjects were compared. The mean and standard deviation of absolute ${A}_{{P}_{C}}$ for young subjects were found to be 27.96 and 32.99, respectively, and the mean and standard deviation of absolute ${A}_{{P}_{C}}$ for elderly subjects were found to be 11.59 and 15.82, respectively. Further, the Mann–Whitney U (single tailed) test was done to check if the magnitude of ${A}_{{P}_{C}}$ values estimated from young subjects were significantly greater than the magnitude of ${A}_{{P}_{C}}$ values estimated from elderly subjects. The null hypothesis, ${H}_{0}$, was that the median of the magnitude of ${A}_{{P}_{C}}$ values obtained from the young population was less than or equal to the median of the magnitude of ${A}_{{P}_{C}}$ values obtained from the old population. $p<0.05$ was considered statistically significant. It was found that ${H}_{0}$ was rejected in favour of the alternate hypothesis, with the p value being equal to 0.0168. This suggested that the HPV of younger subjects was more irreversible or asymmetric as compared to that of older subjects. This result is in line with the findings of existing studies.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ETC | Effort-to-Compress |

CCC | Compression-Complexity Causality |

${P}_{C}$ | Compressive potential |

${A}_{{P}_{C}}$ | Compressive potential based temporal asymmetry measure |

GC | Granger Causality |

TE | Transfer Entropy |

CMI | Conditional Mutual Information |

CCM | Convergent Cross Mapping |

PI | Predictability Index |

KLD | Kullback Leibler Divergence |

AR | Autoregressive |

LGP | Linear Gaussian Process |

STAR | Static nonlinear transformation of a first order Gaussian process |

SETAR | Self-Exciting Threshold AR |

IAAFT | Iterative Amplitude Adjusted Fourier Transform |

HP | Heart Period |

HPV | Heart Period Variability |

HPVA | Heart Period Variability Asymmetry |

SAP | Systolic Arterial Pressure |

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**Figure 1.**Mean causality values estimated using (

**a**) CCC, (

**b**) TE and (

**c**) GC for coupled time-reversed AR(1) processes, from ${Z}^{\prime}$ to ${Y}^{\prime}$ (solid-line circles, black) and ${Y}^{\prime}$ to ${Z}^{\prime}$ (solid-line crosses, magenta) as the degree of coupling, $\u03f5$ is varied. CCC is invariant to time reversal, while for TE and GC, the dominant direction of causality is seen to be reversed.

**Figure 2.**Mean causality values estimated using (

**a**) CCC and (

**b**) TE for linearly coupled time-reversed tent map processes, from ${Z}^{\prime}$ to ${Y}^{\prime}$ (solid line-circles, black) and ${Y}^{\prime}$ to ${Z}^{\prime}$ (solid line-crosses, magenta) as the degree of coupling, $\u03f5$ is varied. CCC is invariant to time reversal and, in the case of TE, the dominant direction of causality identified is same as for the original processes.

**Figure 3.**Variation in Compressive potential, ${P}_{C}$, with k for time series (

**a**) ${X}_{1}$ (periodic with short patterns), (

**b**) ${X}_{2}$ (periodic with long patterns), (

**c**) ${X}_{3}$ (partly periodic, partly random) and (

**d**) ${X}_{4}$ (completely random), simulated as per Table 1.

**Figure 4.**Compressive-potential-based temporal asymmetry test result on simulated data from processes: (

**a**) LGP, (

**b**) AR(2), (

**c**) STAR(1), (

**d**) SETAR(2;2,2) (

**e**) Tent map. Dashed line indicates ${A}_{{P}_{c}}$ value obtained for original series. Its position is indicated with respect to Gaussian=curve-fitted normalized histogram of surrogate ${A}_{{P}_{C}}$ values that form the null hypothesis of reversible processes. Null hypothesis is not rejected in case of (

**a**)–(

**d**) and rejected in case of (

**e**).

**Figure 5.**Variation in Sunspot numbers with time. Monthly mean total sunspot numbers are plotted from the beginning of the year 1920 to the end of the year 2020. Data source: www.sidc.be/silso/datafiles; accessed on 24 February 2021.

**Figure 6.**Compressive-potential-based temporal asymmetry test result on time series of (

**a**) Sunspot numbers, and (

**b**) Digits of $\pi $. Dashed line indicates ${A}_{{P}_{c}}$ value obtained for original series. Its position is indicated with respect to Gaussian-curve-fitted normalized histogram of surrogate ${A}_{{P}_{C}}$ values that form the null hypothesis of reversible processes. The null hypothesis is rejected in the case of (

**a**) and not rejected in the case of (

**b**).

Time Series | Composed of | ETC |
---|---|---|

${X}_{1}$ | Repeating periodic sequence: $\left[1\phantom{\rule{0.222222em}{0ex}}2\phantom{\rule{0.222222em}{0ex}}3\phantom{\rule{0.222222em}{0ex}}4\right]$ | 3 |

${X}_{2}$ | Repeating periodic sequence: $[1\phantom{\rule{0.222222em}{0ex}}2\phantom{\rule{0.222222em}{0ex}}3\dots 1000]$ | 95 |

${X}_{3}$ | Repeating partly periodic partly random sequence: $[1\phantom{\rule{0.222222em}{0ex}}2\phantom{\rule{0.222222em}{0ex}}3\dots 20]$ | |

followed by 100 random numbers uniformly chosen from between 1 and 20 | 100 | |

${X}_{4}$ | Uniformly randomly distributed real numbers in the range (0,1) | 4110 |

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**MDPI and ACS Style**

Kathpalia, A.; Nagaraj, N.
Time-Reversibility, Causality and Compression-Complexity. *Entropy* **2021**, *23*, 327.
https://doi.org/10.3390/e23030327

**AMA Style**

Kathpalia A, Nagaraj N.
Time-Reversibility, Causality and Compression-Complexity. *Entropy*. 2021; 23(3):327.
https://doi.org/10.3390/e23030327

**Chicago/Turabian Style**

Kathpalia, Aditi, and Nithin Nagaraj.
2021. "Time-Reversibility, Causality and Compression-Complexity" *Entropy* 23, no. 3: 327.
https://doi.org/10.3390/e23030327