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Machine Learning of Chaotic
Systems
Solving Complex and Insoluble Problems via Artificial Intelligence
By Lipa Roitman PhD...
Contents
• Chaos VS Randomness
• Chaotic Processes
• Modeling Chaos- Statistics Approach
• Modeling Chaos- Artificial Inte...
Chaos and Randomness
• Random noise
No known cause, no regularity, no rationality, no repeatability, no pattern
Impossible...
Chaos VS Randomness
• Randomness Examples
Previous coin flips do not predict the next one.
Brownian motion - random walk
G...
• Stationary process: statistical properties: mean value,
variance, moments, and probability distribution do not change ov...
• Real life chaotic processes are neither stationary nor ergodic!
Their statistics have to be constantly monitored since t...
• Astronomy: Three-Body Problem
• Sunspots
• Geology: Earthquakes
• Oceanology: El Niño (Pacific ocean temperature) ,
Tide...
• Fluid flow: luminary vs turbulent
• Candle flame
• Quantum chaos
• Biology: Population growth
• Physiology: Arrhythmia, ...
• Social: fashion trends
• Wars
• Music and speech
• Stock markets, etc.
Chaos in Natural Processes
Chaotic Processes
Chaotic Processes
Three competing paradigms:
Stability
Instability
Sudden and Dramatic Change
Chaotic Systems Properties
What is the pattern?
• Stability: Persistent
trends.
• Memory: What
happens next
depends on pri...
Chaotic Systems Properties
• Instability - “tired trend” - accumulation of small random imbalances,
or of slow systematic ...
• Change: paradigm changes suddenly, seemingly without warning. –
often with reversal of trend
• Fat-Tail: The change coul...
Chaotic Systems Properties
• Cycles of varying lengths.
• Periods of quiet followed by big jumps
• Chaotic patterns are pr...
• Measuring Chaos - Statistically
Modeling Chaos
• Mathematical modeling of chaotic systems is difficult:
Tiny changes in parameters can sometimes lead to extreme changes ...
• The ubiquity of gradual trends and the rarity of the extreme events
resemble the spectral density of a stochastic proces...
Although 1/f noise is widely present in natural and social time series, the source of
such noise is not well and understoo...
In a random autoregressive process the autocorrelation functions
decay exponentially
In chaotic process, they leave a smal...
If one looks at a chaotic process at different degrees of magnification, one
finds they are similar. This self –similarity...
• Chaos Fractals Connection
Modeling
Chaos
• Rescaling Range
• Given a relation
• Scaling the argument x by a constant factor c causes only a
proportionate scaling o...
• In other words:
 Scaling by a constant c simply multiplies the original power-law relation by
the constant c^{-k}. Thus...
• “Power Law Signature”: Logarithms of both f(x) and x, have linear
relationship: straight-line on the log-log plot.
• Res...
• 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 reverting
• negative feedback:
• high noise
• h...
• 1>H>1/2 Chaotic trending process:
Positive feedback
Less noise
Smaller fractional dimension
Fractional Brownian motion, ...
Maximal Lyapunov Exponent
Maximal Lyapunov exponent (MLE) is a measure of sensitivity to
initial conditions, i.e. unpredic...
• Maximal Lyapunov exponent (MLE) is a measure of sensitivity to
initial conditions, a property of chaos
• Hurst exponent ...
Fractal time series are good
approximations of chaotic
processes. They are complex
systems that have similar
properties.
M...
Modeling Chaos with Fractals
Fat-tailed probability
distribution
Memory Effect: Slowly decaying
autocorrelation function
P...
Fractal dimension D and Hurst exponent H each characterize the local
irregularity (D) and global persistence (H).
Thus D a...
Fractal Dimension and Hurst Exponent
• For self-affine processes, the local properties are
reflected in the global ones
• ...
Chaos and Fractals Connection
Fractals have self-similar patterns at different scales.
Fractal dimension
Multi fractal sys...
Random shocks to the process, such as news events. The shocks can
have both temporary and lasting effect
Combination of in...
Modeling Chaos:
 Artificial Intelligence and Machine Learning
Approach
Modeling Chaos - AI Approach
Artificial Intelligence
• Machine Learning Purpose: Generalization
• Find the laws within the data
• Predicting change
• N...
Artificial Intelligence Types
Rules Based AI
Man creates the rules: Expert Systems
The rule-based approach is time consumi...
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 learning models high-level abstractions in data by using multiple
processing layers with complex struct...
Deep learning can automatically select the features
For a simple machine learning, a human has to tell the algorithm which...
“Ultra Deep Learning”
Machine has learned so much, it can not only derive the rules, but
detect when the rules change: det...
Steps in Machine Learning
Provide Framework
Mathematical and Programming Tools
Data preparation
Parameters estimation
Give...
Steps in Machine Learning
• Creating a Model (or Models).
• Fitness Function: What to optimize?
• Example: Make more good ...
Data Preparation
Data preparation
Convert the generally non-stationary data into more-or-less stationary
Remove the cycles...
Parameters Estimation
• Parametric OR Nonparametric?
• Parametric model: fixed number of parameters
• Nonparametric: no as...
Creating a Model
“All Models are Wrong, Some Models are Useful” –
George E. P. Box
Multivariate time series
Multivariate time series modeling is required when the outcome of
one process depends on other pr...
Multivariate time series
To create a model one could use the available knowledge about
interrelationship of the processes,...
Machine Learning
The fitness function is improved through machine learning by varying
the parameters in the model. The goa...
Dimensionality Reduction
• Dimensionality reduction
• Speeds up algorithm execution
• Improves performance
• The less vari...
• Principal Component Analysis is one of
the methods of dimensionality reduction.
• Orthogonally transforms the original d...
• Methods:
• Low Variance Filter.
• High Correlation Filter.
• Pruning the network.
• Adding and replacing inputs.
• Other...
Clustering
• The many examples in the data can be compressed into clusters
according to the similarity through fitting to ...
Time Constraint
• A <insert favorite programming language> programmer knows the
value of everything, but the cost of nothi...
Time Constraint
• Some problems are insoluble or too complex to be completely solved
in reasonable time.
• Compromises are...
Time Complexity (Big O Notation)
Choice of Algorithm
• Which Algorithm?
 Depends on the task
 Depends on time available
 Depends on the precision requir...
Local and Global Minimum
accp1.org/pharmacometrics/theory.htm
Uphill SearchingDownhill Gradient
Searching
Local Search Algorithms
• Local search methods:
• steepest descent or
• best-first criterion,
• stochastic search.
• simul...
A random move altering the state
Assess the fitness of the new state
Compare the fitness to the previous state
Decide whet...
Global Search Algorithms
• Stochastic optimization
• Uphill searching
• Basin hopping
accp1.org/pharmacometrics/theory.htm
Local and Global Minimum
Basin Hopping
The algorithm is iterative with each cycle composed of the following
features
Random perturbation of the coo...
Genetic Algorithms
• Many solutions are in the pool, some good, some not so.
• Each solution is analogous to a chromosome ...
Genetic Algorithms
• Ways to improve gene pool:
• Combination:
• Combine two or more solutions in hope of producing a
bett...
68
Bain-Template
Gene
Pool
Reprod
uceMutate
Select
Genetic
Algorithm
I Know First Predictive Algorithm
• Most financial time series exhibit classical chaotic behavior. The
chaos theory, the c...
I Know First Predictive Algorithm
 The following slides are the method and the
results of applying the algorithm to learn...
The I Know First Algorithm
The results are constantly improving as the algorithm learns from its
successes and failures
Tr...
Synopsis of the Algorithm
The results are constantly improving as the algorithm learns
from its successes and failures
Two indicators:
Signal – Predicted movement
of the asset
Predictability Indicator –
Historical correlation between
the pre...
XOMA returned 61.45% in
1 month from this forecast
Forecast vs. Actual
I Know First Sample
Portfolio
I Know First
beats the
S&P500 by
96.4%
View Full Portfolio
I Know First Live Portfolio 2015 Performance
The Performance
I Know First
beats the
S&P500 by
20.8%
The Performance
The Performance
The Performance
Main Features of the
Algorithm
Identifies The Best Market Opportunities Daily
6 Time Frames
Tracks Over 3,000 Markets
Self...
More Applications Of I Know First Algorithm
• Time Series Forecasting of Multidimensional Chaotic Systems.
• What if? It i...
Machine Learning, Stock Market and Chaos
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Machine Learning, Stock Market and Chaos

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Dr. Roitman discusses the use of Artificial Intelligence to solve complex and insoluble problems. Artificial intelligence approach is in the root of I Know First predictive algorithm.

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Machine Learning, Stock Market and Chaos

  1. 1. Machine Learning of Chaotic Systems Solving Complex and Insoluble Problems via Artificial Intelligence By Lipa Roitman PhD November 1st, 2015
  2. 2. Contents • Chaos VS Randomness • Chaotic Processes • Modeling Chaos- Statistics Approach • Modeling Chaos- Artificial Intelligence and Machine Learning Approach • Steps in Machine Learning • Financial Markets as Chaotic Processes
  3. 3. Chaos and Randomness • Random noise No known cause, no regularity, no rationality, no repeatability, no pattern Impossible to predict
  4. 4. Chaos VS Randomness • Randomness Examples Previous coin flips do not predict the next one. Brownian motion - random walk Gaussian and non-Gaussian Random (white) noise with frequency-independent power spectrum Other modes of random processes.
  5. 5. • 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
  6. 6. • 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
  7. 7. • Astronomy: Three-Body Problem • Sunspots • Geology: Earthquakes • Oceanology: El Niño (Pacific ocean temperature) , Tides • Meteorology: Weather Chaos in Natural Processes
  8. 8. • Fluid flow: luminary vs turbulent • Candle flame • Quantum chaos • Biology: Population growth • Physiology: Arrhythmia, Epilepsy, Diabetis • DNA code • Epidemiology: diseases Chaos in Natural Processes
  9. 9. • Social: fashion trends • Wars • Music and speech • Stock markets, etc. Chaos in Natural Processes
  10. 10. Chaotic Processes Chaotic Processes Three competing paradigms: Stability Instability Sudden and Dramatic Change
  11. 11. Chaotic Systems Properties What is the pattern? • Stability: Persistent trends. • Memory: What happens next depends on prior history. • Predictable: One can predict while the pattern continues.
  12. 12. Chaotic Systems Properties • Instability - “tired trend” - accumulation of small random imbalances, or of slow systematic imbalances that precede large change. • “Sand pile avalanche model” • Predictability is lower
  13. 13. • 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
  14. 14. Chaotic Systems Properties • Cycles of varying lengths. • Periods of quiet followed by big jumps • Chaotic patterns are predictable, but only in terms of probabilities.
  15. 15. • Measuring Chaos - Statistically Modeling Chaos
  16. 16. • 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
  17. 17. • 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^α
  18. 18. Although 1/f noise is widely present in natural and social time series, the source of such noise is not well and understood. 1/f noise 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
  19. 19. In a random autoregressive process the autocorrelation functions decay exponentially In chaotic process, they leave a small persistent residue: “long memory”. Modeling Chaos
  20. 20. 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 fractals Self similarity = Power laws scale invariance fractals (Mandelbrot) Hurst exponent Scale Invariance
  21. 21. • Chaos Fractals Connection Modeling Chaos
  22. 22. • Rescaling Range • Given a relation • Scaling the argument x by a constant factor c causes only a proportionate scaling of the function itself Modeling Chaos
  23. 23. • 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
  24. 24. • “Power Law Signature”: Logarithms of both f(x) and x, have linear relationship: straight-line on the log-log plot. • Rescaled range - The slope of this line gives the Hurst exponent, H. Modeling Chaos
  25. 25. • Hurst exponent can distinguish fractal from random time series, or find the long memory cycles Hurst Exponent H
  26. 26. • H =1/2 Random walk - Brownian motion -Normal Distribution • H < 1/2 mean reverting • negative feedback: • high noise • high fractal dimension Hurst exponent H
  27. 27. • 1>H>1/2 Chaotic trending process: Positive feedback Less noise Smaller fractional dimension Fractional Brownian motion, or 1/f noise Hurst exponent H
  28. 28. 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”.
  29. 29. • Maximal Lyapunov exponent (MLE) is a measure of sensitivity to initial conditions, a property of chaos • Hurst exponent H is a measure of persistency Maximal Lyapunov Exponent
  30. 30. Fractal time series are good approximations of chaotic processes. They are complex systems that have similar properties. Modeling Chaos with Fractals
  31. 31. 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.
  32. 32. 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
  33. 33. Fractal Dimension and Hurst Exponent • For self-affine processes, the local properties are reflected in the global ones • For a self-affine surface in n-dimensional space • D+H=n+1 D: fractal dimension H: Hurst exponent
  34. 34. Chaos and Fractals Connection Fractals have self-similar patterns at different scales. Fractal dimension Multi fractal system - continuous spectrum of exponents - singularity spectrum.
  35. 35. 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
  36. 36. Modeling Chaos:  Artificial Intelligence and Machine Learning Approach Modeling Chaos - AI Approach
  37. 37. Artificial Intelligence • Machine Learning Purpose: Generalization • Find the laws within the data • Predicting change • Number crunching allows finding hidden laws, not obvious to human eye
  38. 38. Artificial Intelligence Types Rules Based AI Man creates the rules: Expert Systems The rule-based approach is time consuming and not very accurate
  39. 39. Supervised learning from examples The examples must be representative of the entire data set. Artificial Intelligence Types
  40. 40. Un-supervised learning Classification: clustering Artificial Intelligence Types
  41. 41. Deep learning Deep learning models high-level abstractions in data by using multiple processing layers with complex structures. Artificial Intelligence Types
  42. 42. 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
  43. 43. “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
  44. 44. Steps in Machine Learning Provide Framework Mathematical and Programming Tools Data preparation Parameters estimation Give examples to learn from: the input (and in some methods the output)
  45. 45. Steps in Machine Learning • Creating a Model (or Models). • Fitness Function: What to optimize? • Example: Make more good predictions than bad ones.
  46. 46. Data Preparation Data preparation Convert the generally non-stationary data into more-or-less stationary Remove the cycles, trends to reduce the uniqueness of each data point
  47. 47. Parameters Estimation • Parametric OR Nonparametric? • Parametric model: fixed number of parameters • Nonparametric: no assumptions about the probability distributions of the variables. • In non-parametric model the number of parameters increases with the amount of training data.
  48. 48. Creating a Model “All Models are Wrong, Some Models are Useful” – George E. P. Box
  49. 49. Multivariate time series Multivariate time series modeling is required when the outcome of one process depends on other processes. Examples are systems of interdependent global and local processes, asset prices, exchange rates, interest rates, and other variables.
  50. 50. Multivariate time series To create a model one could use the available knowledge about interrelationship of the processes, and combine it with unknowns in one or more of the linear or non-linear models. The “fitness” or “error” function is then created, which compares the model with the data.
  51. 51. Machine Learning The fitness function is improved through machine learning by varying the parameters in the model. The goal is to maximize the fitness of the model to the data presented for learning (minimize the error). Different models are screened Part of the data is saved from the learning cycle to be used for testing. The successful model should be able to perform adequately on the test data.
  52. 52. Dimensionality Reduction • Dimensionality reduction • Speeds up algorithm execution • Improves performance • The less variables the better is generality
  53. 53. • Principal Component Analysis is one of the methods of dimensionality reduction. • Orthogonally transforms the original data set into a new set of “principal components” Dimensionality Reduction Methods
  54. 54. • Methods: • Low Variance Filter. • High Correlation Filter. • Pruning the network. • Adding and replacing inputs. • Other methods. Dimensionality Reduction Methods
  55. 55. Clustering • The many examples in the data can be compressed into clusters according to the similarity through fitting to one or more criteria. • Each data member that belongs to a cluster is associated with a number from 0 to 1 that shows the degree of belonging. • Each data member can also belong to multiple clusters with each specific degree of belonging. • Clustering can be a goal in itself, or a part of a general model, that includes the behavior of clusters as a whole.
  56. 56. Time Constraint • A <insert favorite programming language> programmer knows the value of everything, but the cost of nothing. -- Alan J. Perlis
  57. 57. Time Constraint • Some problems are insoluble or too complex to be completely solved in reasonable time. • Compromises are necessary, e.g. speed vs precision vs generality • Time complexity (big O notation) of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input.
  58. 58. Time Complexity (Big O Notation)
  59. 59. Choice of Algorithm • Which Algorithm?  Depends on the task  Depends on time available  Depends on the precision required
  60. 60. Local and Global Minimum accp1.org/pharmacometrics/theory.htm Uphill SearchingDownhill Gradient Searching
  61. 61. Local Search Algorithms • Local search methods: • steepest descent or • best-first criterion, • stochastic search. • simulated annealing, • genetic selection • others
  62. 62. A random move altering the state Assess the fitness of the new state Compare the fitness to the previous state Decide whether to accept the new solution or reject it. Repeat until you have converged on an acceptable answer Simulated Annealing
  63. 63. Global Search Algorithms • Stochastic optimization • Uphill searching • Basin hopping
  64. 64. accp1.org/pharmacometrics/theory.htm Local and Global Minimum
  65. 65. Basin Hopping The algorithm is iterative with each cycle composed of the following features Random perturbation of the coordinates Local minimization Accept or reject the new coordinates based on the minimized function value
  66. 66. Genetic Algorithms • Many solutions are in the pool, some good, some not so. • Each solution is analogous to a chromosome in genetics
  67. 67. Genetic Algorithms • Ways to improve gene pool: • Combination: • Combine two or more solutions in hope of producing a better solution. • Mutation: • -Modify a solution in random places in hope of producing a better solution. • Crossover: • Import a solution from a similar problem • Selection: • Survival of the fittest
  68. 68. 68 Bain-Template Gene Pool Reprod uceMutate Select Genetic Algorithm
  69. 69. I Know First Predictive Algorithm • Most financial time series exhibit classical chaotic behavior. The chaos theory, the classification and predictive capabilities of the machine learning has been applied to forecasting of such time series. • This artificial intelligence approach is in the root of I Know First predictive algorithm.
  70. 70. I Know First Predictive Algorithm  The following slides are the method and the results of applying the algorithm to learn the database of historical time series data.
  71. 71. The I Know First Algorithm The results are constantly improving as the algorithm learns from its successes and failures Tracks and predicts the flow of money from one market or investment channel to another The system is a predictive model based on Artificial Intelligence, Machine Learning, and incorporates elements of Artificial Neural Networks and Genetic Algorithms Artificial Intelligence (AI) Artificial Neural Networks I Know First predicts 2000 Market’s Eeveryday
  72. 72. Synopsis of the Algorithm The results are constantly improving as the algorithm learns from its successes and failures
  73. 73. Two indicators: Signal – Predicted movement of the asset Predictability Indicator – Historical correlation between the prediction and the actual market movement Daily Market Heat map
  74. 74. XOMA returned 61.45% in 1 month from this forecast
  75. 75. Forecast vs. Actual
  76. 76. I Know First Sample Portfolio
  77. 77. I Know First beats the S&P500 by 96.4% View Full Portfolio I Know First Live Portfolio 2015 Performance The Performance
  78. 78. I Know First beats the S&P500 by 20.8% The Performance
  79. 79. The Performance
  80. 80. The Performance
  81. 81. Main Features of the Algorithm Identifies The Best Market Opportunities Daily 6 Time Frames Tracks Over 3,000 Markets Self-Learning Adaptable Always Learning New Patterns Scalable A Decision Support System (DSS) Predictability Indicator Strong Historical Performance – 60.66% gain in 2013 The algorithm becomes more and more accurate with every prediction as it constantly tests multiple models in different market circumstances
  82. 82. More Applications Of I Know First Algorithm • Time Series Forecasting of Multidimensional Chaotic Systems. • What if? It is a Scenario-based Forecasting

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