Renormalization of Cellular Automata and Self-Similarity

Edlund, Erik; Jacobi, Martin Nilsson

doi:10.1007/s10955-010-9974-z

Cited by 8 publications

(17 citation statements)

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“…By reducing the complexity and focusing on the key ingredient underlying the forecast skill, namely the time to the bubble, the method of ghosts of singularities seems to be less sensitive to idiosyncratic noise realisations, thus providing more robust forecasts. This can be interpreted as a kind of effective coarse-graining of the dynamical equations, a process that, when done intelligently, has been shown in the past to improve predictability [46][47][48][49]. …”

Section: Application Of Ghosts Of Finite-time Singularitiesmentioning

confidence: 99%

Predicting Financial Market Crashes Using Ghost Singularities

2017

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We analyse the behaviour of a non-linear model of coupled stock and bond prices exhibiting periodically collapsing bubbles. By using the formalism of dynamical system theory, we explain what drives the bubbles and how foreshocks or aftershocks are generated. A dynamical phase space representation of that system coupled with standard multiplicative noise rationalises the log-periodic power law singularity pattern documented in many historical financial bubbles. The notion of 'ghosts of finite-time singularities' is introduced and used to estimate the end of an evolving bubble, using finite-time singularities of an approximate normal form near the bifurcation point. We test the forecasting skill of this method on different stochastic price realisations and compare with Monte Carlo simulations of the full system. Remarkably, the approximate normal form is significantly more precise and less biased. Moreover, the method of ghosts of singularities is less sensitive to the noise realisation, thus providing more robust forecasts.

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Section: Application Of Ghosts Of Finite-time Singularitiesmentioning

confidence: 99%

Predicting Financial Market Crashes Using Ghost Singularities

2017

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“…Right panel: the same run, coarse-grained in both time and space. The system is now simplified by a factor of nine; we preserve approximate relationships, and rough, probabilistic logics of evolution.phenomenon, for the case of cellular automata, have been investigated in elegant detail by Edlund and Jacobi [15].Spatial coarse-graining is not the only way to simplify a system, and in many cases may not be appropriate. When we move from the physical to the biological or social sciences we find systems that are fundamentally long-range in nature, or have mechanisms that tie together distant locations.…”

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confidence: 99%

“…phenomenon, for the case of cellular automata, have been investigated in elegant detail by Edlund and Jacobi [15].…”

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confidence: 99%

Major Transitions in Political Order

DeDeo

2017

From Matter to Life

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We present three major transitions that occur on the way to the elaborate and diverse societies of the modern era. Our account links the worlds of social animals such as pigtail macaques and monk parakeets to examples from human history, including 18th Century London and the contemporary online phenomenon of Wikipedia. From the first awareness and use of group-level social facts to the emergence of norms and their self-assembly into normative bundles, each transition represents a new relationship between the individual and the group. At the center of this relationship is the use of coarse-grained information gained via lossy compression. The role of top-down causation in the origin of society parallels that conjectured to occur in the origin and evolution of life itself.[T]hey then threw me upon the bed, and one of them (I think it was Mary Smith) kneeled on my breast, and with one hand held my throat; Mary Junque felt for my money; by my struggling about, they did not get it at that time; then they called another woman in . . . when she came in, they said cut him! cut him! -evidence of Benjamin Leethorp in the trial of Mary Junque and Mary Smith for grand larceny, Old Bailey Criminal Court, London, England; 4 April 1779 [1] Unless we are historians, the 18th Century world of Junque, Smith and Leethorp is almost impossible to imagine. In stealing from Leethorp, the two women put themselves at risk not only of imprisonment, but of indentured servitude in the colonies and even death. Leethorp, for his part, begins his evidence by explaining to the jury how he was seeking a different brothel than the one in which he was throttled, stripped, and robbed. Junque and Smith were without benefit of legal counsel and Smith's witnesses, unaware of the trial date, did not appear. The court condemned them to branding and a year's imprisonment in less than five hundred words. The indictment, formally for a non-violent offence, was one of hundreds of its kind that decade marked by assault, knives, and (sometimes) freely flowing blood.In the risks they ran and the things they were ashamed of, the minds of the three are alien to us; in its casual violence, so was the society that enclosed them. Yet this world, gradually, continuously, evolved into one far less tolerant of violence and yet far more protective of an individual's rights-into the world, in other words, of most readers of this volume. How witnesses, victims, and defendants spoke about both facts and norms in the law courts of London shifted, decade by decade, over the course of a hundred and fifty years [2]. This shift in speech paralleled a similar decline in how people behaved towards each other on the street, as the state came, increasingly, to manage its monopoly on violence-part of what is known as the "civilizing process" [3].These changes took place in the decentralized common-law courts, among hundreds of thousands of interacting victims and defendants. Acts of Parliament, sensational crimes, the invention of the criminal defence lawyer-these changed ...

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“…1. This is a natural progression from the work of Edlund and Nilsson Jacobi (2010) who present a concise and systematic approach to renormalisation of such one-dimensional CA, which itself builds on previous work labelled as application of a dynamically driven renormalisation group (Toméand et al 1997;De Oliveira and Satulovsky 1997). It is known that in applying some scale transformation to a CA, we do not guarantee the existence of an exact coarse grained dynamics, one which exactly incorporates all possible microscopic transitions.…”

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confidence: 99%

“…In such instances it is natural to choose the best fit by way of least-squares. The important contribution of Edlund and Nilsson Jacobi (2010) has been to show that what previously was referred to as the dynamically driven renormalisation group amounts to weighting this least-squares fit by the steady-state probability distribution such that error is concentrated on the transitions between relatively uncommon states.…”

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confidence: 99%

Renormalisation of 2D Cellular Automata with an Absorbing State

Weaver

Prügel-Bennett

2015

J Stat Phys

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We describe a real-space renormalisation scheme for non-equilibrium probabilistic cellular automata (PCA) models, and apply it to a two-dimensional binary PCA. An exact renormalisation scheme is rare, and therefore we provide a method for computing the stationary probability distribution of states for such models with which to weight the renormalisation, effectively minimising the error in the scale transformation. While a mean-field approximation is trivial, we use the principle of maximum entropy to incorporate nearest-neighbour spin-correlations in the steady-state probability distribution. In doing so we find the fixed point of the renormalisation is modified by the steady-state approximation order.

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Renormalization of Cellular Automata and Self-Similarity

Cited by 8 publications

References 8 publications

Predicting Financial Market Crashes Using Ghost Singularities

Predicting Financial Market Crashes Using Ghost Singularities

Major Transitions in Political Order

Renormalisation of 2D Cellular Automata with an Absorbing State

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