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


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


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


Archipelbuurt, 2013-11-30










Network Motifs


Network Motifs


Network Motifs


Fisher Information


Fisher Information 7 days frequency

Thank you very much!

5 10 15 20 25 30 35 40 450.00

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0.0005

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0.04 0.05 0.06k

0.00

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0.06F

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Education

Regime shifts and fisher information: an information theoretic approach to phase transitions