STATISTICAL COMPLEXITY ANALYSIS

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STATISTICAL COMPLEXITY ANALYSIS

• Dr. Dmitry Nerukh
• Giorgos Karvounis

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What is Complexity?

• Many different definitions.
• A natural system, is converted into a formal
  system that our mind can manipulate and we have a
  model.
  
• Complexity is the property of a real world
  system that is manifest in the inability of any
  one formalism being adequate to capture all its
  properties

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WHY USE COMPLEXITY ANALYSIS (1)

• When a new state of matter emerges from a phase
  transition,
  
• certain pattern formation takes on the this
  newness with respect to other structures .
  
• This process is defined as intrinsic
  emergence.
  
• There is an increase in intrinsic computational
  capability which can be capitalised and
  measured.
  

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WHY USE COMPLEXITY ANALYSIS (2)

• Contemporary physics can measure order (e.g.
  temperature) or ideal randomness (e.g. entropy,
  thermodynamics).
  
• No tools to address problems of innovation, or
  the discovery of patterns
  
• Measuring the computational capabilities of the
  system is the only way to address such
  questions
  
• discovering and quantifying emergence, pattern,
  information process and memory in quantitative
  units.
  
• The term intrinsic computation defines the way
  the system stores information with respect to
  time, transmits it between internal degrees of
  freedom and makes use of it in order to produce
  future behaviour.

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METHODOLOGY (1)

• Complexity estimates how sophisticated are the
  dynamical laws governing the time evolution of
  the system.
  
• We adopted the approach by Crutchfield et. al.
  termed computational mechanics.
  
• We implement ideas from both Shannon entropy and
  KC algorithmic complexity theories, measuring the
  size of the informational description of the
  process.
• This is a direct approach to reveal the
  symmetries possessed by any spatial patterns and
  to estimate the minimum amount of memory
  required to reproduce any configuration
  ensemble.

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BUT WE ARE MODELLERS
HOW DO WE DEAL WITH MODELS??
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METHODOLOGY (2)

• We can reconstruct an algorithmic machine (termed
  as e-machine) that provides the means to build
  the statistically minimal optimally predictive
  model .
• In order to build this machine, we need the
  smallest possible set of predictive states, the
  causal states.
  
• We can state that two predictive states are
  equivalent () if and only if they give rise to
  same future values in terms of conditional
  probabilities

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COMPUTATIONAL IMPLEMENTATION(1)

• The algorithm is based on the use of symbolic
  dynamics generated from symbols assigned to
  discrete time steps.
  
• The crucial part in the implementation of the
  methodology is converting a continuous real
  signal into a sequence of symbols i.e signal
  symbolization of the molecular trajectory. The
  one dimensional case is shown below

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CAUSAL STATE
Consider the following sequence
bla.bla.bla.lab.lba.bla.bla.lab.bal.bla.alb.alb.bl
a.bla
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E-machine

• An ?-machine, the set of causal states and the
  probabilities of the transitions between them,
  provides a direct description of the patterns
  present in the systems internal degrees of
  freedom.

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FINITE STATISTICAL COMPLEXITY

• Finite Statistical Complexity can be defined as
  the minimum amount of evolutionary information
  (or hidden memory) required to statistically
  reproduce a process.

• It expresses the informational size of the
  distribution of the causal states as measured by
  the Shannon Entropy
  

• Statistical Complexity is based on the assumption
  that randomness is statistically simple an ideal
  random process has zero statistical complexity.
  Equally, simple periodic processes give low
  complexity values as well.
• Complex process is the one that lies between
  these extremes and is an amalgam of predictable
  and stochastic mechanisms.
  

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STATISTICAL COMPLEXITY OF A ZWITTERION a
folding event

• We measured the statistical complexity of the
  dynamical trajectories of four significant atoms
  within a zwitterion.
  
• Attain insights regarding complexity and how
  this can be a useful tool to characterise or
  capture the folding event.
  
• Depending on the temperature of the simulation,
  the zwitterion adopts a stable folded
  conformation.
  
• Statistical Complexity Analysis of various atoms
  trajectories at the unfolded configuration and
  compare their values at the folded state.
  

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COMPLEXITY ANALYSIS OF THE EXTENDED STATE
At the extended state, there is no significant
change on the complexity value, as the zwitterion
remains as an extended chain, following basically
the same pattern throughout the process.
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COMPLEXITY ANALYSIS OF THE FOLDED STATE

• In the folding event, there is a considerable
  drop in the complexity value, assigned to the
  transitional stage.
  
• Afterwards, there is a sudden rise in the
  complexity, until all atoms reach the same value,
  assigned to the pattern of the folded state.

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COMPLEXITY ANALYSIS(1)
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COMPLEXITY ANALYSIS (2)

• The essentiality of complexity measurements is
  that we can distinguish those patterns in
  quantitative terms.
  
• Better insight to the mechanisms that underlie
  the formation of this structure and separate the
  more ordered regularities to those that are
  more random

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FUTURE WORK

• Further development of the algorithm in order to
  achieve a better representation of the
  ?-machine.
  
• Apply Statistical Complexity Analysis to a
  larger system such as protein folding and
  polymers phase transitions.
  

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ACKNOWLEDGMENTS

• For this work we are grateful to
• Prof. R. Glen
• The Newton Trust and UNILEVER for their
  financial support.