Entropy Measures Research Articles

A novel multi-criteria group decision making algorithm for enhancing supply chain efficiency under high uncertainty during crisis based on q-rung orthopair fuzzy information

In recent years, q-rung orthopair fuzzy sets (q-ROFSs) have been becoming gradually more advantageous in handling information with high uncertainty. Several multi-criteria group decision-making (MCGDM) algorithms under q-ROFS information have been proposed in literature and used to solve various real-life problems. However, several shortcomings of some existing MCGDM algorithms under certain circumstances also emerged which limit their applicability. To overcome these challenges and deal with information under high uncertainty more accurately and reasonably, we propose a novel MCGDM algorithm under q-ROF information, i.e., (q-ROF-MCGDM), that retains the advantages of currently available methods, but extend its applicability by introducing a new q-rung orthopair fuzzy weighted averaging aggregation operator (q-ROFWAAO) along with a new entropy measure. The proposed q-ROF-MCGDM algorithm involves the steps: firstly, the input requirement in terms of alternatives, criteria and expert evaluations are required. Secondly, aggregating expert opinions using weights and normalizing decision matrices, and finally, calculating entropy weights and aggregating overall evaluations for final prioritization. Further, new operational laws have been developed and several necessary properties of the proposed q-ROFWAAO are also proved. Moreover, a sensitivity and comparative analysis has been carried out for validity and effectiveness of the proposed q-ROF-MCGDM algorithm. The proposed q-ROF-MCGDM algorithm has been implemented to enhance the efficiency of the United Arab Emirates (UAE) food industry under high uncertainty during the recent crisis. Finally, an evaluation of identifying and prioritizing the most severe food SC disruptions and appropriate mitigation strategies under crisis is provided to demonstrate applicability of the proposed q-ROF-MCGDM algorithm, and the obtained real case results confirm its usefulness. Findings of the study offer valuable insights to both food industry researchers and managers in developing effective recovery strategies, mitigating risks, and improving overall efficiency to ensure the survival of food businesses during the crisis.

Characterizing Complex Spatiotemporal Patterns from Entropy Measures.

In addition to their importance in statistical thermodynamics, probabilistic entropy measurements are crucial for understanding and analyzing complex systems, with diverse applications in time series and one-dimensional profiles. However, extending these methods to two- and three-dimensional data still requires further development. In this study, we present a new method for classifying spatiotemporal processes based on entropy measurements. To test and validate the method, we selected five classes of similar processes related to the evolution of random patterns: (i) white noise; (ii) red noise; (iii) weak turbulence from reaction to diffusion; (iv) hydrodynamic fully developed turbulence; and (v) plasma turbulence from MHD. Considering seven possible ways to measure entropy from a matrix, we present the method as a parameter space composed of the two best separating measures of the five selected classes. The results highlight better combined performance of Shannon permutation entropy (SHp) and a new approach based on Tsallis Spectral Permutation Entropy (Sqs). Notably, our observations reveal the segregation of reaction terms in this SHp×Sqs space, a result that identifies specific sectors for each class of dynamic process, and it can be used to train machine learning models for the automatic classification of complex spatiotemporal patterns.

Assessing resiliency in scale-free supply chain networks: a stress testing approach based on entropy measurements and value-at-risk analysis

Supply chain (SC) resiliency and risk management have garnered increasing attention recently. While several studies have explored the use of scale-free network models to design and optimise SC networks, there remains a lack of a generalised stress-testing method that can be applied to various types and sizes of SCs. To address this, we propose a novel approach to assess the resiliency of scale-free SC networks based on entropy measurements. By considering various sources of disruptions, including demand fluctuations, capacity constraints, and transportation disruptions, we create a stress test that measures the network's resiliency. Our method categorises the SC into plants, ports, and customers and analyses the impact of disruptions on individual nodes within similar categories. We utilise real-world SC logistics data and map it into a scale-free network representation. By calculating the global vulnerability of the entire SC network and the localised impact of node vulnerability, we gain insights into how disruptions propagate and affect interconnected nodes. We use value-at-risk (VaR) and conditional-VaR (CVaR) to identify high-risk nodes, highlighting nodes at risk of extreme performance fluctuations or failures. The analytical results have led to developing six propositions, which serve as references for practitioners and help generalise the findings.

Dendropsophus minutus repertoire complexity and its relationship with environmental traits

ABSTRACT Anurans have several characteristics in their acoustic communication that distinguish them. These characteristics are influenced by body-related, environmental and social variables. However, the complexity of anuran repertoire and the effect of such variables on it are still understudied. Dendropsophus minutus is a Neotropical treefrog that has a wide repertoire composed of three notes, grouped in at least 71 unique combinations. To quantify complexity, we used the entropy measurement, since it measures the randomness of a sequence. Therefore, we could test if repertoire complexity in D. minutus is related with body condition, social context, and environmental variables. We recorded vocalisations of 80 males in three sites and found 44 unique note combinations not described before, increasing the species’ repertoire to 115 combinations. We observed that the species has near maximum entropy for note frequency, but a low entropy for note sequencies, showing a clear a bias for some note combinations. We found a positive relationship between repertoire complexity and air temperature. Thus, based on our results, the complexity of this species’ repertoire is an additional trait influenced by climate, and should be considered in future studies.

Divergent functional connectivity changes associated with white matter hyperintensities

Age-related white matter hyperintensities are a common feature and are known to be negatively associated with structural integrity, functional connectivity, and cognitive performance. However, this has yet to be fully understood mechanistically. We analyzed multiple MRI modalities acquired in 465 non-demented individuals from the Swedish BioFINDER study including 334 cognitively normal and 131 participants with mild cognitive impairment. White matter hyperintensities were automatically quantified using fluid-attenuated inversion recovery MRI and parameters from diffusion tensor imaging were estimated in major white matter fibre tracts. We calculated fMRI resting state-derived functional connectivity within and between predefined cortical regions structurally linked by the white matter tracts. How change in functional connectivity is affected by white matter lesions and related to cognition (in the form of executive function and processing speed) was explored. We examined the functional changes using a measure of sample entropy. As expected hyperintensities were associated with disrupted structural white matter integrity and were linked to reduced functional interregional lobar connectivity, which was related to decreased processing speed and executive function. Simultaneously, hyperintensities were also associated with increased intraregional functional connectivity, but only within the frontal lobe. This phenomenon was also associated with reduced cognitive performance. The increased connectivity was linked to increased entropy (reduced predictability and increased complexity) of the involved voxels’ blood oxygenation level-dependent signal. Our findings expand our previous understanding of the impact of white matter hyperintensities on cognition by indicating novel mechanisms that may be important beyond this particular type of brain lesions.

The General Theory of Scientific Variability for Technological Evolution

The proposed general theory of scientific variability for technological evolution explains one of the drivers of technological change for economic progress in human society. Variability is the predisposition of the elements in systems to assume different values over time and space. In biology, the variability is basic to explaining differences and development in organisms. In economics of technical change, the effects of variability within research fields on evolutionary dynamics of related technologies are unknown. In a broad analogy with the principles of biology, suggested theoretical framework here can clarify a basic driver of technological evolution: the variability within research fields can explain the dynamics of scientific development and technological evolution. The study sees whether statistical evidence supports the hypothesis that the rate of growth of scientific and technological fields can be explained by the level of variability within scientific fields. The validation is based on emerging research fields in quantum technologies: quantum imaging, quantum meteorology, quantum sensing, and quantum optics. Statistical evidence seems in general to support the hypothesis stated that the rate of growth can be explained by the level of scientific variability within research fields, measured with the relative entropy (indicating the dispersion of scientific topics in a research field underlying a specific technology). Nonparametric correlation with Spearman’s rho shows a positive coefficient of 0.80 between entropy measures and rates of growth between scientific and technological fields. The linear model of the relation between rate of growth and scientific variability reveals a coefficient of regression equal to 1.63 (R2 = 0.60). The findings here suggest a general law that variability within research fields positively drives scientific development and technological evolution. In particular, a higher variability within research fields can support a high rate of growth in scientific development and technological evolution. The proposed general theory of scientific variability is especially relevant in turbulent environments of technology-based competition to clarify a basic determinant of technological development to design strategies of technological forecasting and management of promising innovations.

Is EEG Entropy a Useful Measure for Alzheimer's Disease?

The number of individuals diagnosed with Alzheimer's disease (AD) has increased, and it is estimated to continue rising in the coming years. The diagnosis of this disease is challenging due to variations in onset and course, its diverse clinical manifestations, and the indications for measuring deposit biomarkers. Hence, there is a need to develop more precise and less invasive diagnostic tools. Multiple studies have considered using electroencephalography (EEG) entropy measures as an indicator of the onset and course of AD. Entropy is deemed suitable as a potential indicator based on the discovery that variations in its complexity can be associated with specific pathologies such as AD. Following PRISMA guidelines, a literature search was conducted in 4 scientific databases, and 40 articles were analyzed after discarding and filtering. There is a diversity in entropy measures; however, Sample Entropy (SampEn) and Multiscale Entropy (MSE) are the most widely used (21/40). In general, it is found that when comparing patients with controls, patients exhibit lower entropy (20/40) in various areas. Findings of correlation with the level of cognitive decline are less consistent, and with neuropsychiatric symptoms (2/40) or treatment response less explored (2/40), although most studies show lower entropy with greater severity. Machine learning-based studies show good discrimination capacity. There is significant difficulty in comparing multiple studies due to their heterogeneity; however, changes in Multiscale Entropy (MSE) scales or a decrease in entropy levels are considered useful for determining the presence of AD and measuring its severity.

Enhancing Financial Time Series Prediction with Quantum-Enhanced Synthetic Data Generation: A Case Study on the S&P 500 Using a Quantum Wasserstein Generative Adversarial Network Approach with a Gradient Penalty

This study introduces a novel Quantum Wasserstein Generative Adversarial Network approach with a Gradient Penalty (QWGAN-GP) model that leverages a quantum generator alongside a classical discriminator to synthetically generate time series data. This approach aims to accurately replicate the statistical properties of the S&P 500 index. The synthetic data generated by this model were compared to the original series using various metrics, including Wasserstein distance, Dynamic Time Warping (DTW) distance, and entropy measures, among others. The outcomes demonstrate the model’s robustness, with the generated data exhibiting a high degree of fidelity to the statistical characteristics of the original data. Additionally, this study explores the applicability of the synthetic time series in enhancing prediction models. An LSTM (Long-Short Term Memory)-based model was developed to evaluate the impact of incorporating synthetic data on forecasting accuracy, particularly focusing on general trends and extreme market events. The findings reveal that models trained on a mix of synthetic and real data significantly outperform those trained solely on historical data, improving predictive performance.

Abnormal Scanning Patterns Based on Eye Movement Entropy in Early Psychosis

BackgroundRestricted scan path mode is hypothesized to explain abnormal scanning patterns in patients with schizophrenia. Here, we calculated entropy scores (drawing on gaze data to measure the statistical randomness of eye movements) to quantify how strategical and random participants were when processing image stimuli. MethodsEighty-six patients with first-episode schizophrenia (FES), 124 individuals at clinical high risk (CHR) for psychosis, and 115 healthy control participants (HCs) completed an eye-tracking examination while freely viewing 35 static images (each presented for 10 seconds) and cognitive assessments. We compared group differences in the overall entropy score, as well as entropy scores under various conditions. We also investigated the correlations between entropy scores and symptoms and cognitive function. ResultsIncreased overall entropy scores were noted in the FES and CHR groups compared with the HC group, and these differences were already apparent within 0 to 2.5 seconds. In addition, the CHR group exhibited higher entropy than the HC group when viewing low-meaning images. Moreover, the entropy within 0 to 2.5 seconds showed significant correlations with negative symptoms in the FES group, attention/vigilance scores in the CHR group, and speed of processing and attention/vigilance scores across all 3 groups. ConclusionsThe results indicate that individuals with FES and those at CHR scanned pictures more randomly and less strategically than HCs. These patterns also correlated with clinical symptoms and neurocognition. The current study highlights the potential of the eye movement entropy measure as a neurophysiological marker for early psychosis.

Entropy based network anomaly detection

an era of relentless cyber threats, the increasing complexity and volume of data significantly intensify the risk and impact of cybersecurity breaches. As organizations generate and store more data, the potential attack surface grows, providing more opportunities for malicious actors to exploit vulnerabilities. Consequently, there is a growing necessity for more advanced analytical techniques to effectively detect and mitigate these evolving threats. Shannon entropy, introduced by Claude Shannon in 1948, is a fundamental concept in information theory that measures the unpredictability or randomness of information. It serves as a primary tool for identifying unusual patterns within extensive datasets, offering a quantitative approach to detect anomalies This paper explores the application of Shannon’s Entropy to detect and prevent distributed denial-of-service (DDoS) attacks. Unlike traditional motif identification tools, which focus on recurring patterns within data, Shannon entropy provides a broader measure of randomness and can detect subtle variations that may indicate a security breach. By leveraging the entropy measure, cybersecurity systems can identify and respond to abnormal traffic patterns that signify a potential DDoS attack, thereby enhancing the robustness and reliability of data protection mechanisms

Modeling brain information flow dynamics with multidimensional fuzzy inference systems

In this paper, we introduce a pioneering approach to analyzing brain information flow dynamics through the development of a novel hybrid system. The key innovation of our system lies in the integration of n-cell fuzzy functions into a multidimensional fuzzy inference system (MFIS). Fuzzy n-cell numbers, representing a significant advancement in fuzzy mathematics, are central to our methodology, showcasing their capability to handle intricate uncertainties and imprecise data within multidimensional scenarios. Our proposed system aligns seamlessly with neuroscience principles, providing a robust framework to unravel the complexities of brain information flow dynamics. We leverage Transfer Entropy and Granger Causality measures for a quantitative assessment of influence strengths, enabling a systematic exploration of temporal information flow within the brain. Furthermore, the utilization of fuzzy n-cell numbers enhances the system's adaptability to the intricacies of neural processes. The outcomes of our study are presented through a series of visualizations, offering a comprehensive demonstration of the effectiveness and adaptability of the proposed hybrid system. This article aims to contribute to the understanding of brain dynamics by presenting a versatile and complex methodology that combines fuzzy mathematics, neuroscience principles, and advanced analytical techniques.

Preparing quantum statistical ensembles using mid-circuit measurements

We explore the relationship between entropy and quantum measurements and present a variational algorithm for preparing statistical ensembles on quantum computers using mid-circuit measurements. This algorithm optimizes both the entropy and variational parameters describing the state to obtain the minimum free energy of quantum systems in thermal equilibrium with some external heat bath. We demonstrate our algorithm on IBM-Q Lagos.

Designs of graph echo state networks for node classification

Among the Graph Neural Network (GNN) models that address the task of node classification, Graph Echo State Networks (GESN) have proved particularly effective in addressing the challenge of heterophily, i.e. the presence of a significant fraction of inter-class edges in the learning task graph. The effectiveness of GESN is paired with its efficiency, owing to the reservoir computing paradigm. While previous literature has analyzed the design of reservoirs for sequence ESN and GESN for graph-level tasks, the problem of providing effective designs of reservoirs for node-level GESN is so far largely unexplored. In this paper we analyze the impact of different reservoir designs on node classification accuracy and on the quality of node embeddings computed by GESN, focusing both on dense and sparse reservoir layouts. As measures of embedding richness, we adopt both graph topology-dependent metrics previously employed in the analysis of embedding smoothing, and topology-independent metrics from the areas of information theory and numerical analysis. In particular, we propose the application of entropy measures for quantifying information in node embeddings.

Time-resolved coupling between connectome harmonics and subjective experience under the psychedelic DMT.

Exploring the intricate relationship between brain's structure and function, and how this affects subjective experience is a fundamental pursuit in neuroscience. Psychedelic substances offer a unique insight into the influences of specific neurotransmitter systems on perception, cognition and consciousness. Specifically, their impact on brain function propagates across the structural connectome - a network of white matter pathways linking different regions. To comprehensively grasp the effects of psychedelic compounds on brain function, we used a theoretically rigorous framework known as connectome harmonic decomposition. This framework provides a robust method to characterize how brain function intricately depends on the organized network structure of the human connectome. We show that the connectome harmonic repertoire under DMT is reshaped in line with other reported psychedelic compounds - psilocybin, LSD and ketamine. Furthermore, we show that the repertoire entropy of connectome harmonics increases under DMT, as with those other psychedelics. Importantly, we demonstrate for the first time that measures of energy spectrum difference and repertoire entropy of connectome harmonics indexes the intensity of subjective experience of the participants in a time-resolved manner reflecting close coupling between connectome harmonics and subjective experience.

The analysis of diversification properties of stablecoins through the Shannon entropy measure

The common goal for investors is to minimise the risk and maximise the returns on their investments. This is often achieved through diversification, where investors spread their investments across various assets. This study aims to use the MAD-entropy model to minimise the absolute deviation, maximise the mean return, and maximise the Shannon entropy of the portfolio. The MAD model is used because it is a linear programming model, allowing it to resolve large-scale problems and nonnormally distributed data. Entropy is added to the MAD model because it can better diversify the weight of assets in the portfolios. The analysed portfolios consist of cryptocurrencies, stablecoins, and selected world indices such as the SP500 and FTSE obtained from Yahoo Finance. The models found that stablecoins pegged to the US dollar, followed by stablecoins pegged to gold, are better diversifiers for traditional cryptocurrencies and stocks. These results are probably due to their low volatility compared to the other assets. Findings from this study may assist investors since the MAD-Entropy model outperforms the MAD model by providing more significant portfolio mean returns with minimal risk. Therefore, crypto investors can design a well-diversified portfolio using MAD entropy to reduce unsystematic risk. Further research integrating mad entropy with machine learning techniques may improve accuracy and risk management.

Overdistention Accelerates Electrophysiological Changes in Uterine Muscle Towards Labour in Multiple Gestations

Background for the researchPremature birth and its associated complications are one of the biggest global health problems, since there is currently no effective screening method in clinical practice to accurately identify the true Preterm Birth (PTB) from the false threatened ones. Despite the high prevalence of PTB in multiple gestation (MG) women which amounted up to 60%, in the literature there is any work about their uterine myoelectric activities in vivo system. Electrohysterography (EHG) has been emerged as an alternative technique for predicting PTB in single gestation (SG) women. PurposeThe aim of this study was to characterize and compare the uterine myoelectrical activity in vivo system of SG and MG women in regular check-ups, to provide the basis for early detection and prevention of preterm labour in MG. Basic proceduresA prospective observational cohort study was conducted on 31 SG and 18 MG women between the 28th and 32th WoG who underwent regular check-ups in the Polytechnic and University Hospital La Fe (Valencia, Spain). The 30-minute bipolar recording was filtered in the 0.1-4 Hz bandwidth and downsampled to 20 Hz. Signal analysis was performed in 120-second moving windows with 50% overlap, after removing artefacts by a double- blind expert process. A set of 8 temporal, spectral and non-linear parameters were calculated: root mean square (RMS), kurtosis of the Hilbert envelope (KHE), median frequency (MDF), H/L ratio, and sample entropy (SampEn) and bubble entropy (BubbEn) calculated in the whole bandwidth (WBW) and the fast wave high (FWH). The 10th, 50th and 90th percentiles of all windows analysed were calculated to obtain representative values of the recordings. For each parameter and percentile, statistically significant differences between the SG and MG groups and their statistical power (SP) were analysed to determine both the existence of an effect and substantive significance, respectively. Main findingsIn comparison to SG, MG EHG exhibited significant higher impulsiveness and higher predictability than SG which was reflected in the KHE (SP10 = 85.2, p10 < 0.001) and entropy measures (SampEn FWH: SP50 = 62.0, p50 = 0.0.016; SP90 = 52.5, p90 = 0.059. BubbEn FWH: SP50 = 75.2, p50 < 0.001; SP90 = 60.3, p90 = 0.002), suggesting an accelerated evolution of uterine electrophysiological condition. In addition, several EHG parameters were found to significantly correlate with foetal weight such as amplitude (RMS: r90 = 0.311, p90 = 0.006), signal impulsiveness (KHE: r10 = 0.311, p10 = 0.006) and entropy measures (SampEn FWH: r50 = −0.317, p50 = 0.005*; r90 = −0.279, p90 = 0.013*. BubbEn FWH: r50 = −0.370, p50 = 0.001*; r90 = −0.313, p90 = 0.005*), suggesting an electromechanical coupling between uterine overdistension and contractile activity in vivo system. Principal conclusionsIn comparison to SG women, MG showed higher impulsiveness and predictability in early third gestational trimester, as reflected in KHE, SampEn and BubbEn, respectively. We found similar cell excitability between SG and MG women far from delivery. In addition, we confirmed the relationship between uterine overdistension and surface myoelectric activity, revealing the electromechanical coupling pathway in uterine smooth muscle. Therefore, contextualized EHG-biomarkers would provide valuable information for early detection of PTB risk, which would allow clinicians better PTB management through personalised therapeutic interventions.

Controlling Chaos: Periodic Defect Braiding in Active Nematics Confined to a Cardioid.

This work examines self-mixing in active nematics, a class of fluids in which mobile topological defects drive chaotic flows in a system comprised of biological filaments and molecular motors. We present experiments that demonstrate how geometrical confinement can influence the braiding dynamics of the defects. Notably, we show that confinement in cardioid-shaped wells leads to realization of the golden braid, a maximally efficient mixing state of exactly three defects with no defect creation or annihilation. We characterize the golden braid state using different measures of topological entropy and the Lyapunov exponent. In particular, topological entropy measured from the stretching rate of material lines agrees well with an analytical computation from braid theory. Increasing the size of the confining cardioid produces a transition from the golden braid, to the fully chaotic active turbulent state.

The Agreement between Wearable Sensors and Force Plates for the Analysis of Stride Time Variability.

The variability and regularity of stride time may help identify individuals at a greater risk of injury during military load carriage. Wearable sensors could provide a cost-effective, portable solution for recording these measures, but establishing their validity is necessary. This study aimed to determine the agreement of several measures of stride time variability across five wearable sensors (Opal APDM, Vicon Blue Trident, Axivity, Plantiga, Xsens DOT) and force plates during military load carriage. Nineteen Australian Army trainee soldiers (age: 24.8 ± 5.3 years, height: 1.77 ± 0.09 m, body mass: 79.5 ± 15.2 kg, service: 1.7 ± 1.7 years) completed three 12-min walking trials on an instrumented treadmill at 5.5 km/h, carrying 23 kg of an external load. Simultaneously, 512 stride time intervals were identified from treadmill-embedded force plates and each sensor where linear (standard deviation and coefficient of variation) and non-linear (detrended fluctuation analysis and sample entropy) measures were obtained. Sensor and force plate agreement was evaluated using Pearson's r and intraclass correlation coefficients. All sensors had at least moderate agreement (ICC > 0.5) and a strong positive correlation (r > 0.5). These results suggest wearable devices could be employed to quantify linear and non-linear measures of stride time variability during military load carriage.

Analysis of fetal heart rate fluctuations in women diagnosed with preeclampsia during the latent phase of labor

IntroductionFetal heart rate variability (fHRV) is a tool used to investigate the functioning of the fetal autonomic nervous system. Despite the significance of preeclampsia, fHRV during the latent phase of labor has not been extensively studied. This study aimed to evaluate fetal cardiac autonomic activity by using linear and nonlinear indices of fHRV analysis in women diagnosed with preeclampsia without hypertensive treatment during gestation, compared to normotensive women during the latent phase of labor.MethodsA cross-sectional and exploratory study was conducted among pregnant women in the latent phase of labor, forming three study groups: normotensive or control (C, 38.8 ± 1.3 weeks of pregnancy, n = 22), preeclampsia with moderate features (P, 37.6 ± 1.4 weeks of pregnancy n = 10), and preeclampsia with severe features (SP, 36.9 ± 1.2 weeks of pregnancy, n = 12). None of the participants received anti-hypertensive treatment during their pregnancy. Linear and nonlinear features of beat-to-beat fHRV, including temporal, frequency, symbolic dynamics, and entropy measures, were analyzed to compare normotensive and preeclamptic groups.ResultsSignificantly lower values of multiscale entropy (MSE) and short-term complexity index (Ci) were observed in the preeclamptic groups compared to the C group (p &amp;lt; 0.05). Additionally, higher values of SDNN (standard deviation of R-R intervals) and higher values of low-frequency power (LF) were found in the P group compared to the C group.ConclusionOur findings indicate that changes in the complexity of fetal heart rate fluctuations may indicate possible disruptions in the autonomic nervous system of fetuses in groups affected by undiagnosed preeclampsia during pregnancy. Reduced complexity and shifts in fetal autonomic cardiac activity could be associated with preeclampsia’s pathophysiological mechanisms during the latent phase of labor.

Arrow of time and gravitational entropy in collapse

We investigate the status of the gravitational arrow of time in the case of a spherical collapse of a fluid that conducts heat and radiates energy. In particular, we examine the results obtained by W. B. Bonnor in his 1985 paper where he found that the gravitational arrow of time was opposite to the thermodynamic arrow of time. The measure of gravitational epoch function P used by Bonnor was given by the ratio of the Weyl square to the Ricci square. In this paper, we have assumed the measure of gravitational entropy (GE) P 1 to be given by the ratio of the Weyl scalar to the Kretschmann scalar. Our analysis indicates that Bonnor’s result seems to be validated, i.e. the gravitational arrow and the thermodynamic arrow of time point in opposite directions. This strengthens the opinion that the Weyl proposal of GE applies only to the Universe as a whole (provided that we exclude the white holes).