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第98行： − − − In addition, there are some other quantitative theories of emergence. There are mainly two methods that are widely discussed. One is to understand emergence from the process from disorder to order. Moez Mnif and Christian Müller-Schloer <ref>Mnif, M.; Müller-Schloer, C. Quantitative emergence. In Organic Computing—A Paradigm Shift for Complex Systems; Springer: Basel, Switzerland, 2011; pp. 39–52. </ref> use Shannon entropy to measure order and disorder. In the self-organization process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, the defect of this method is that it depends on the abstract observation level and the initial conditions of the system. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference <ref>Fisch, D.; Jänicke, M.; Sick, B.; Müller-Schloer, C. Quantitative emergence–A refined approach based on divergence measures. In Proceedings of the 2010 Fourth IEEE International Conference on Self-Adaptive and Self-Organizing Systems, Budapest, Hungary, 27 September–1 October 2010; IEEE Computer Society: Washington, DC, USA, 2010; pp. 94–103. </ref> suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. But this method has disadvantages such as large computational complexity and low estimation accuracy. To solve these problems, reference <ref>Fisch, D.; Fisch, D.; Jänicke, M.; Kalkowski, E.; Sick, B. Techniques for knowledge acquisition in dynamically changing environments. ACM Trans. Auton. Adapt. Syst. (TAAS) 2012, 7, 1–25. [CrossRef] </ref> further proposes an approximate method for estimating density using Gaussian mixture models and introduces Mahalanobis distance to characterize the difference between data and Gaussian components, thus obtaining better results. In addition, Holzer, de Meer et al. <ref>Holzer, R.; De Meer, H.; Bettstetter, C. On autonomy and emergence in self-organizing systems. In International Workshop on Self-Organizing Systems, Proceedings of the Third International Workshop, IWSOS 2008, Vienna, Austria, 10–12 December 2008; Springer: Berlin/Heidelberg, Germany, 2008; pp. 157–169.</ref><ref>Holzer, R.; de Meer, H. Methods for approximations of quantitative measures in self-organizing systems. In Proceedings of the Self-Organizing Systems: 5th International Workshop, IWSOS 2011, Karlsruhe, Germany, 23–24 February 2011; Proceedings 5; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1–15.</ref> proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Then, we can measure emergence according to the ratio between the Shannon entropy measure of all communications between agents and the sum of Shannon entropies as separate sources. −
→Other theories for quantitatively characterizing emergence
==== Other theories for quantitatively characterizing emergence ====
==== Other theories for quantitatively characterizing emergence ====
In addition, there are some other quantitative theories of emergence. There are mainly two methods that are widely discussed. One is to understand [[emergence]] from the process from disorder to order. Moez Mnif and Christian Müller-Schloer <ref>Mnif, M.; Müller-Schloer, C. Quantitative emergence. In Organic Computing—A Paradigm Shift for Complex Systems; Springer: Basel, Switzerland, 2011; pp. 39–52. </ref> use [[Shannon entropy]] to measure order and disorder. In the [[self-organization]] process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, the defect of this method is that it depends on the abstract observation level and the initial conditions of the system. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference <ref>Fisch, D.; Jänicke, M.; Sick, B.; Müller-Schloer, C. Quantitative emergence–A refined approach based on divergence measures. In Proceedings of the 2010 Fourth IEEE International Conference on Self-Adaptive and Self-Organizing Systems, Budapest, Hungary, 27 September–1 October 2010; IEEE Computer Society: Washington, DC, USA, 2010; pp. 94–103. </ref> suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. But this method has disadvantages such as large computational complexity and low estimation accuracy. To solve these problems, reference <ref>Fisch, D.; Fisch, D.; Jänicke, M.; Kalkowski, E.; Sick, B. Techniques for knowledge acquisition in dynamically changing environments. ACM Trans. Auton. Adapt. Syst. (TAAS) 2012, 7, 1–25. [CrossRef] </ref> further proposes an approximate method for estimating density using [[Gaussian mixture models]] and introduces [[Mahalanobis distance]] to characterize the difference between data and Gaussian components, thus obtaining better results. In addition, Holzer, de Meer et al. <ref>Holzer, R.; De Meer, H.; Bettstetter, C. On autonomy and emergence in self-organizing systems. In International Workshop on Self-Organizing Systems, Proceedings of the Third International Workshop, IWSOS 2008, Vienna, Austria, 10–12 December 2008; Springer: Berlin/Heidelberg, Germany, 2008; pp. 157–169.</ref><ref>Holzer, R.; de Meer, H. Methods for approximations of quantitative measures in self-organizing systems. In Proceedings of the Self-Organizing Systems: 5th International Workshop, IWSOS 2011, Karlsruhe, Germany, 23–24 February 2011; Proceedings 5; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1–15.</ref> proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Then, we can measure emergence according to the ratio between the Shannon entropy measure of all communications between agents and the sum of Shannon entropies as separate sources.
In addition, there are some other quantitative theories of emergence. There are mainly two methods that are widely discussed. One is to understand [[emergence]] from the process from disorder to order. Moez Mnif and Christian Müller-Schloer <ref>Mnif, M.; Müller-Schloer, C. Quantitative emergence. In Organic Computing—A Paradigm Shift for Complex Systems; Springer: Basel, Switzerland, 2011; pp. 39–52. </ref> use [[Shannon entropy]] to measure order and disorder. In the [[self-organization]] process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, the defect of this method is that it depends on the abstract observation level and the initial conditions of the system. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference <ref>Fisch, D.; Jänicke, M.; Sick, B.; Müller-Schloer, C. Quantitative emergence–A refined approach based on divergence measures. In Proceedings of the 2010 Fourth IEEE International Conference on Self-Adaptive and Self-Organizing Systems, Budapest, Hungary, 27 September–1 October 2010; IEEE Computer Society: Washington, DC, USA, 2010; pp. 94–103. </ref> suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. But this method has disadvantages such as large computational complexity and low estimation accuracy. To solve these problems, reference <ref>Fisch, D.; Fisch, D.; Jänicke, M.; Kalkowski, E.; Sick, B. Techniques for knowledge acquisition in dynamically changing environments. ACM Trans. Auton. Adapt. Syst. (TAAS) 2012, 7, 1–25. [CrossRef] </ref> further proposes an approximate method for estimating density using [[Gaussian mixture models]] and introduces [[Mahalanobis distance]] to characterize the difference between data and Gaussian components, thus obtaining better results. In addition, Holzer, de Meer et al. <ref>Holzer, R.; De Meer, H.; Bettstetter, C. On autonomy and emergence in self-organizing systems. In International Workshop on Self-Organizing Systems, Proceedings of the Third International Workshop, IWSOS 2008, Vienna, Austria, 10–12 December 2008; Springer: Berlin/Heidelberg, Germany, 2008; pp. 157–169.</ref><ref>Holzer, R.; de Meer, H. Methods for approximations of quantitative measures in self-organizing systems. In Proceedings of the Self-Organizing Systems: 5th International Workshop, IWSOS 2011, Karlsruhe, Germany, 23–24 February 2011; Proceedings 5; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1–15.</ref> proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Then, we can measure emergence according to the ratio between the Shannon entropy measure of all communications between agents and the sum of Shannon entropies as separate sources.
Another method is to understand emergence from the perspective of "the whole is greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This method defines emergence from interaction rules and the states of agents rather than statistically measuring from the totality of the entire system. Specifically, this measure consists of subtracting two terms. The first term describes the collective state of the entire system, while the second term represents the sum of the individual states of all components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.
Another method is to understand emergence from the perspective of "the whole is greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This method defines emergence from interaction rules and the states of agents rather than statistically measuring from the totality of the entire system. Specifically, this measure consists of subtracting two terms. The first term describes the collective state of the entire system, while the second term represents the sum of the individual states of all components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.
=== Causal emergence theory based on effective information ===
=== Causal emergence theory based on effective information ===