regime shifts and fisher information: an information theoretic approach to phase transitions
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Regime shifts and Fisher information: an information theoretic approach to phase transitions
Omri Har-ShemeshPhD CandidateComputational Science LabUniversity of Amsterdam
Name: OmriFourth year PhD studentInterested in using tools from statistical mechanics to describe phase transitions in complex systems.1
Epileptic seizures
https://www.kaggle.com/c/seizure-detection
NYSE Phaseshttp://www.wsj.com/articles/SB10001424052748704858404576134372454343538
Evolution of an ER Graph
A. Barabasi, Network Science, Chapter 3, page 58
Argue that Information Geometry, specifically Fisher information is a good framework to unify these phenomena
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Phases of water
These are three phases of water. We all know these different phasesIf we keep pressure constant, we move between the phases by changing the temperature.Below 0 we have ice, between 0 and 100 we have liquid and above 100 we have vapourWhat is the difference between them?How can we quantify the difference?Are they always different?Lets look at a simplified microscopic description of the different phases
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Phases of water
https://web.stanford.edu/~peastman/statmech/phasetransitions.htmlSolidLiquidGas
Solid phase crystal pattern that repeats itself.Low energy and low entropyMolecules form hydrogen bonds which are stableLiquid phaseMolecules move more or less freelySome hydrogen bonds remainMedium energy and entropy configurationGas phaseMolecules move completely freelyPractically no hydrogen bonds between moleculesOccupy large volume per moleculeHigh energy and high entropyDifferences are statistical in nature. Solid phase: hardly any uncertainty about position and velocity of molecules,6
Phase Diagram
Allow pressure to change story is more complicatedExplain the graph bottom axis, left axis, bulk are phases, lines phase transitions, triple point and critical point
What to remember water can have different phases, these phases depend on external parameters, in this case, temperature and pressure.
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Realistic Phase Diagram
Ising Phase Diagram
A. Barabasi, Network Science, Chapter 3, page 44
Second example: 2d Ising model, critical point, arrangement of spins,
What can we learn from these examples?9
SummarySame system can be in different phasesExternal parameters: Temperature, Pressure,Phase diagramPhases differ in statistical properties
Fisher Information
Convinced about statistical propertiesInvert the logic define phases as having very different statistical propertiesAdvantages instead of looking at the Gibbs distribution, look at general statistics of observables under change of paramtersLines of phase diagram borders between very different statistical propertiesGet lines from Fisher information roughly speaking fisher tells us how much we can know about parameters of a distributionFor example:11
Example: Normal Distribution
What can we know about the mean from these samples?What about the variance?In both we know more about both the mean and the variance on the left than on the right
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Ising Phase Diagram
A. Barabasi, Network Science, Chapter 3, page 44
Prescription
Gray-Scott RD System
Explain the equations. Two stable trivial states (with no diffusion).Look at the two parameters F and k which are ()Many different patterns in different areas of the parameter space.One of the first to study the patterns appearing in the 2D system is Pearson (where the picture credit belongs)Characterize the different regions of parameter space as different phases of the GS system and the transitions between them as phase transitions.
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Pattern evolution
Self replicating spotsChaotic spotsOther spotsCorral
Probabilistic Description
Fisher Information Phase Map
http://uva.computationalscience.nl/omri/gs/poster.html
FF Component of the Fisher information matrix18
One dimensional transitions
One dimensional transitions
http://uva.computationalscience.nl/omri/gs/poster.html
And now for something completely different
After finishing GS looking for a project to measure phases in real dataSpoke to Jurjen which convinced me that his data might have these phasesWhat phases are there? Non-interacting phase (one car per CP+-)Multiple possibilities per RFIDCompetition / LearningSaturation21
The DataRFIDCharging PointCP LocationStart Charging DatetimeStop Charging DatetimekWh charged
A LOT!!!
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The Data (The Hague)
The Data (The Hague)
Prescription
Idea 1: Locality/Loyalty Coefficient (LoCo)
Idea 1: Locality/Loyalty Coefficient (LoCo)
Idea 2: Hopping probabilities per CP
Idea 2: Hopping probabilities per CP
Archipelbuurt, 2013-11-30
Idea 2: Hopping probabilities per CP
Archipelbuurt, 2014-04-30
Idea 2: Hopping probabilities per CP
Archipelbuurt, 2014-09-30
Idea 2: Hopping probabilities per CP
Archipelbuurt, 2015-02-30
Idea 2: Hopping probabilities per CP
Archipelbuurt, 2016-01-30
Idea 2: Hopping probabilities per CP
Network Motifs
Idea 2: Hopping probabilities per CP
Network Motifs
Idea 2: Hopping probabilities per CP
Network Motifs
Idea 2: Hopping probabilities per CP
Fisher Information
Idea 2: Hopping probabilities per CP
Fisher Information 7 days frequency
Thank you very much!
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0.04 0.05 0.06k
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0.06F
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