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Machine Learning of Chaotic SystemsSolving Complex and Insoluble Problems via Artificial Intelligence

By Lipa Roitman PhDNovember 1st, 2015


ContentsChaos VS RandomnessChaotic ProcessesModeling Chaos- Statistics ApproachModeling Chaos- Artificial Intelligence and Machine Learning ApproachSteps in Machine LearningFinancial Markets as Chaotic Processes


Chaos and RandomnessRandom noise No known cause, no regularity, no rationality, no repeatability, no pattern Impossible to predict

Message:Randomness is unpredictableChaos can be predictable


Chaos VS RandomnessRandomness ExamplesPrevious coin flips do not predict the next one.Brownian motion - random walkGaussian and non-Gaussian Random (white) noise with frequency-independent power spectrumOther modes of random processes.

How to tell Chaos from Randomness in Time Series dataExamples of random processesNone of these can be predictedFuture does no depend on the past


Stationary process: statistical properties: mean value, variance,moments, and probability distribution do not change over time.

Stationary ergodic process: the process has constant statistical properties with time, AND its global statistical properties can be reliably derived from a long enough sample of the process.

Chaos VS Randomness

While the probability distribution is known, still, what comes next can not be predicted. Its completely random. Future does no depend on the past


Real life chaotic processes are neither stationary nor ergodic!

Their statistics have to be constantly monitored since they drift with time.

A nonparametric analysis is needed when the probability distribution of the system is not normal.

Chaos VS Randomness

Chaotic time series dont have fixed statistical properties. They change wit h time, sometimes abruptly..


Astronomy: Three-Body ProblemSunspotsGeology: EarthquakesOceanology: El Nio (Pacific ocean temperature) , TidesMeteorology: WeatherChaos in Natural Processes

Fluid flow: luminary vs turbulent Candle flameQuantum chaosBiology: Population growthPhysiology: Arrhythmia, Epilepsy, DiabetisDNA codeEpidemiology: diseasesChaos in Natural Processes

Social: fashion trendsWarsMusic and speechStock markets, etc.

Chaos in Natural Processes

Chaotic ProcessesChaotic Processes Three competing paradigms: Stability InstabilitySudden and Dramatic Change

Chaotic Systems Properties11Slide

What is the pattern?

Stability: Persistent trends. Memory: What happens next depends on prior history. Predictable: One can predict while the pattern continues.

Chaotic Systems PropertiesInstability - tired trend - accumulation of small random imbalances, or of slow systematic imbalances that precede large change.

Sand pile avalanche model

Predictability is lower

Change: paradigm changes suddenly, seemingly without warning. often with reversal of trend

Fat-Tail: The change could be much stronger from what is expected in the normal Gaussian distribution.

Black Swan Events

Chaotic Systems Properties

Chaotic Systems PropertiesCycles of varying lengths. Periods of quiet followed by big jumpsChaotic patterns are predictable, but only in terms of probabilities.

Measuring Chaos - Statistically

Modeling Chaos

Mathematical modeling of chaotic systems is difficult: Tiny changes in parameters can sometimes lead to extreme changes in the outcome.There is no certainty, only probability.Modeling Chaos

The ubiquity of gradual trends and the rarity of the extreme events resemble the spectral density of a stochastic process, having the form

In this 1/f noise model the magnitude of the signal (event) is inversely proportional to its frequency.

Modeling Chaos S(f)=1/f^

Although 1/f noise is widely present in natural and social time series, the source of such noise is not well and understood.

1/fnoise is an intermediate between the white noise with no correlation in time and random walk (Brownian motion) noise with no correlation between increments.

In most real chaotic processes the random (white) frequency-independent noise overlaps the 1/f noise.

Modeling Chaos

In a random autoregressive process the autocorrelation functions decay exponentially

In chaotic process, they leave a small persistent residue: long memory.Modeling Chaos

If one looks at a chaotic process at different degrees of magnification, one finds they are similar. This self similarity brings us to a subject of fractalsSelf similarity = Power laws scale invariance fractals (Mandelbrot)Hurst exponent

Scale Invariance

Chaos Fractals Connection

Modeling Chaos

Rescaling RangeGiven a relation

Scaling the argument x by a constant factor c causes only a proportionate scaling of the function itselfModeling Chaos

In other words:

Scaling by a constant c simply multiplies the original power-law relation by the constant c^{-k}. Thus Self-Similarity

Modeling Chaos

Power Law Signature: Logarithms of both f(x) and x, have linear relationship: straight-line on the log-log plot.Rescaled range - Theslopeof this line gives theHurst exponent, H.

Modeling Chaos

Hurst exponent can distinguish fractal from random time series, or find the long memory cycles

Hurst Exponent H

H =1/2 Random walk - Brownian motion -Normal Distribution

H < 1/2 mean revertingnegative feedback:high noisehigh fractal dimension

Hurst exponent H

1>H>1/2 Chaotic trending process:

Positive feedback Less noise Smaller fractional dimension Fractional Brownian motion, or 1/f noise

Hurst exponent H

Maximal Lyapunov Exponent Maximal Lyapunov exponent (MLE) is a measure of sensitivity to initial conditions, i.e. unpredictability. Positive MLE: chaos The inverse of Lyapunov exponent: predictability: 1/MLE Large MLE: shorter half-life of signal, faster loss of predictive power.

Maximal Lyapunov exponent (MLE) is a measure of sensitivity to initial conditions, a property of chaos Hurst exponent H is a measure of persistencyMaximal Lyapunov Exponent

Fractal time series are good approximations of chaotic processes. They are complex systems that have similar properties.Modeling Chaos with Fractals

Modeling Chaos with Fractals Fat-tailed probability distribution Memory Effect: Slowly decaying autocorrelation function Power spectrum of 1/f type Modeled with fractal dimension and the Hurst parameter Global or local self-similarity.

Fractal dimension D and Hurst exponent H each characterize the local irregularity (D) and global persistence (H).

Thus D and H are the fractal analogues of variance and mean, which are not constant in the chaotic time series.

Fractal Dimension and Hurst Exponent

Fractal Dimension and Hurst ExponentFor self-affine processes, the local properties are reflected in the global onesFor a self-affine surface in n-dimensional spaceD+H=n+1 D: fractal dimensionH: Hurst exponent

Chaos and Fractals Connection Fractals have self-similar patterns at different scales.

Fractal dimension

Multi fractal system - continuous spectrum of exponents - singularity spectrum.

Random shocks to the process, such as news events. The shocks can have both temporary and lasting effect

Combination of interdependent autoregressive processes, each with its own statistical properties.Two Reasons For 1/F Noise

Modeling Chaos: Artificial Intelligence and Machine Learning Approach

Modeling Chaos - AI Approach

Artificial IntelligenceMachine Learning Purpose: GeneralizationFind the laws within the dataPredicting changeNumber crunching allows finding hidden laws, not obvious to human eye

Artificial Intelligence Types

Rules Based AI

Man creates the rules: Expert Systems

The rule-based approach is time consuming and not very accurate

Supervised learning from examples

The examples must be representative of the entire data set.Artificial Intelligence Types

Un-supervised learning

Classification: clustering

Artificial Intelligence Types

Deep learning

Deep learningmodels high-level abstractions in data by using multiple processing layers with complex structures.Artificial Intelligence Types

Deep learning can automatically select the features For a simple machine learning, a human has to tell the algorithm which combination of features to consider

Deep learning finds the relationships on its own

No human involvement

Artificial Intelligence Types

Ultra Deep Learning

Machine has learned so much, it can not only derive the rules, but detect when the rules change: detect the change in paradigms.

Combines the supervised, un-supervised types and rule based machine learning into a more intelligent system.

Artificial Intelligence Types

Steps in Machine LearningProvide FrameworkMathematical and Programming Tools

Data preparation

Parameters estimation

Give examples to learn from: the input (and in some methods the output)

Steps in Machine LearningCreating a Model (or Models).Fitness Function: What to optimize?Example: Make more good predictions than bad ones.

Data Preparation

Data preparationConvert the genera


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