In the paper written by Doctor Eva Lobach from University of Amsterdam and presented at parapsychology conference in October 2008 (see ref. 4) following definition
is given to presentiment:

Presentiment, also called prestimulus response, involves changes in physiological processes that are related to future stimuli.

Pesentiment is also known as:

• "anomalous prestimulus response".
• Presentiment is an unconscious sensing of an event in the future.
• Presentiment may influence decision making.
• Presentiment can be measured as physiological changes (heartrate, EEG, skin conductance level, etc.).

Doctor Eva Lobach (quoting from her website, see ref. 2) engages in a formal research regarding intuition:

"... Intuition. I am completing a PhD project about Intuition (with Dick Bierman).

Intuition can be defined as "affectively charged judgments that arise through rapid, non-conscious, and holistic associations." (Dane & Pratt, 2007).

Although intuition involves non-conscious processes, the outcomes of these intuitive processes can somehow be evaluated.

I try to find answers to questions like:

1. to what extent are physiological changes (heartrate, EEG, etc.) indicative of intuitive processes?
2. Is intuition helpful in decision making?
3. Can we improve intuition,
4. and if so, how can we improve it?
5. In what types of situations is intuition more likely to happen or be used?
6. In what way is the use of implicit knowledge the same as or different from intuition?

..."

Presentation paper by Doctor Eva Lobach (see ref. 1) outlines a research programe' (past, current and future) and already contains some intriguing results to quote few points from the paper (what the author also calls Presentiment Hypothesis) :

"Psi-effect will be stronger with threatening stimuli"

"Presentiment-effect is due to unconscious psi-augmented decisions of the experimenter."

"Presentiment-effect will be stronger if consciousness is more coherent" 

"Life is a consequence of advanced waves; all living systems reflect retrocausality"

"Presentiment-effect increases when participants’ minds can act like REGs, i.e., remain undetermined."  (REG= Random Event Generator)

Summary:

Paper lays nice foundation to the cut edge research to this fascinating subject, ideas quoted such as " Syntropy (Fantappiè, Vannini): Life is a consequence of advanced
waves; all living systems reflect retrocausality" warrant further investigation in the context of retro-causality.

References:

1. Presentiment Research: Past, Present, and Future by Drs. Eva Lobach University of Amsterdam (note: you need Microsoft Powerpoint software or compatible reader to open this document)

http://home.medewerker.uva.nl/e.lobach/bestanden/Utrecht%20II%202008%20presentiment%20research%20Eva%20Lobach.ppt

2. Drs. Eva Lobach University of Amsterdam website:

http://home.medewerker.uva.nl/e.lobach/

3. The Parapsychology Foundation Lyceum

http://www.pflyceum.org

4. PARAPSYCHOLOGY FOUNDATION INTERNATIONAL CONFERENCES (Utrecht II: Charting the Future of Parapsychology October 15th through October 18th, 2008) Abstracts of Presentations

http://www.pflyceum.org/271.html

Beginning series of exploring or rather augmenting statistical and probability research with my other interest which is parapsychology.

Below talk given by Professor Garret Moddel University of Colorado at Boulder Exploring the Science Underlying Psi Phenomena, on that retrocausation (the present affecting the past) is one of very recent scientific minded explorations into the world of paranormal.

What is really interesting about retro-causality is that it does not violate 2nd law of thermodynamics (key opposing factor to any form of precognition) there is no logical argument that can be posed (at the moment) that would overthrow retrocausality as physically unsound.

I could compare this idea to Hugh Everett "Many World Interpretation" where the quantum wave collapse is interpreted as continuously splitting (or branching) world event lines, in this theory just as is retrocausality, there is no currently known logical and physical argument that would disprove it. By the same token according to scientific philosopher Karl Popper, since such a theory can not be falsified (disproved) it does not fall into category of valid science.

Talk part 1 of 2

Talk part 2 of 2

References:

1. PsiPhen Laboratory University of Colorado at Boulder

http://psiphen.colorado.edu/

2. "Can you still win yesterday’s lottery? or, retrocausation: is it compatible with known physics?," G. Moddel, Annual Meeting of the Society for Scientific Exploration, June 27-29, 2008, Boulder, Colorado (slides)

http://psiphen.colorado.edu/Pubs/ModdelSSE08.pdf

3. Many-worlds interpretation - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Many-worlds_interpretation

Last Edited: September 6, 2009, 10:36 am

Few words of explanation:

1. Main idea for me was to see entropy data together with performance of Padovan numbers.

2. Entropy calculations were done as follows: ("reduced" lottery matrix set was used as per usual)

a) for every game event and each number, count number of frequencies going past 10 draws (thats why data points start from 10 and up for the entropy)
b) calculate probability: number probability = number frequency / 10 * 80 (80 is total numbers in NY P10 game)
c) calculate Shannon entropy = entropy += (number_probability * log(1 / number_probability))

3. Green represents Shannon Entropy and Red Padovan Sequence Numbers matched

Few observations:

1. it actually took some time to measure correct time window (10 in the NY P10 game) in order to arrive with well scaled trend line, believe it to be the correct measure

2. in many instances Shannon Entropy "predicts" (saddle points precede day or two in advance increase in the number of hits from the Padovan sequence) and this is exactly value of this chart.

3. Overall strategy to optimize entry into the game (not to optimize getting exact numbers!) appears to be at the lowest "saddle" points of entropy trend lines that is when uncertainty is at its highest, (this parallels findings in the RPS game! where betting during lowest entropy points brought best payoff!)

Similar chart can be constructed for entire prediction sequence (not just Padovan or some other arbitrary "well behaved" sequence)

Different lottery game require different scale (the above fits well with 20/80 game).

Immediate prediction for NY P10 game is that there will be >=7 hits in the Padovan sequence (even knowing that brings HUGE advantage into the game) timing this event is exactly subject of this blog entry!

Last Edited: August 2, 2009, 3:12 pm

Some explanations follow:

1. Red line shows total number of predictions being made where predictions are coming from a reduced number field that is < 80 using numbers from sorted sequences that have property of being on the "boundary" of next "potential" frequency (potential as it is not known obviously if the number will hit or not, in some respect this is to high degree arbitrary choice of smaller lottery matrix), slope trends upwards as frequency accumulate of course and there is more and more numbers to choose from.

2. Green line represents total number of hits when matched between result of the draw (for that day) and numbers from reduced set.

3. Blue line represents Padovan numbers that matched.

Few interesting points:

1. Padovan numbers settle approximately at 5 hits, regardless of increased number of predictions (red line), it also exhibits interesting oscillatory behavior and to great extent coincides with the green line (spikes in hits) .

Chance for > 5 in Padovan hits increases after approx. > 70 draws, the possibility of 0 hits in Padovans also decreases substantially

Overall, winning this game might be equated to "guessing" the right slope of green and blue line, the chart also shows that there might indeed be a fractal structure beneath of it all.

more research will follow, (computation of fractal dimension, and LLE) as well as trending other games using the above.

Compiled yet another statistics this time for the New York Pick 10 (Keno) game. (Trying to ultimate to devise a "model" to inject entropy, cluster density, Lyapunov threshold and others, I can't use blind "lottery" matrix, need to reduce probability pool substantially even if artificially. Work is ongoing to assemble and research tools in domain from stochastic analysis, chaos theory all the way to catastrophe theory

Data was assembled for the last 132 draws between 03/20/2009 and 07/27/2009 and simulation run to detect optimal time window when a game play using frequency/number transition would yield best performance.

Idea is to detect best possible entry into the game for highest possible hit/total ratio in a reduced number field

In example, if we would enter the game for the day 07/25 , we would capture 16 numbers (19 - duplicates) in the field of 65, and even though this is still high number of possibilities it does offer a glimmer of hope for a better result!

Draw Sequence: 130 -----------------------------------------------------------

Draw 07/25/2009 01 06 07 09 11 21 24 31 35 36 38 41 43 52 55 65 66 71 72 74

Number/Frequency (Sorted by Frequency)

Number     02 63 61 70 43 08 68 77 09 16 23 33 48 07 39 53 64 22 12 26
Frequency 22 22 23 23 24 25 25 25 26 26 26 26 26 27 27 27 27 28 29 29

Number     27 47 76 03 06 17 51 54 04 21 36 46 58 62 24 34 38 44 52 72
Frequency 29 29 29 30 30 30 30 30 31 31 31 31 31 31 32 32 32 32 32 32

Number     78 01 11 45 75 79 30 41 49 55 57 10 50 65 67 74 19 20 25 60
Frequency 32 33 33 33 33 33 34 34 34 34 34 35 35 35 35 35 36 36 36 36

Number     66 69 18 28 35 40 42 59 15 31 05 56 73 13 14 71 32 80 29 37
Frequency 36 36 37 37 37 37 37 37 38 38 39 39 40 41 41 41 42 43 44 46

Predictions made on: 07/25/2009 for the next draw: 

02 63 02 63 61 61 70 43 43 08 68 77 09 16 48 07 39 64 22 12 12 26 27 76 03 06 54 04 21 62 24 34 78 01 11 79 30 41 57 10 50 74 19 20 69 18 28 59 15 31 31 05 56 56 73 13 13 14 71 71 32 80 80 29 37

[65 Numbers]

Matched [For Date]: 07/26/2009 77 09 16 39 26 06 24 01 79 69 15 13 13 71 71 80 80 29 37 [Total = 19]

-------------------------------------------------------------------------------------

Last Edited: July 27, 2009, 10:52 pm

Further expanded chaos toolkit by adding new process to calculate Fractal Dimension in a Time Series.

Below are sample runs using RPS game model (graph 1 and graph 2), scaling property is clearly visible, (scaling property is  common to fractals and an indicator that time series under study is most likely chaotic)

Calculations were performed with the help of software FD3 (see ref. 1) C language was ported to C++ and several tests were conducted to verify validity of the implementation from other sources (including successful calculation of the Fractal Dimension of the coast of Great Britain)

Fractal Dimension, might offer yet another way for entry/exit game strategy, we will at some point take time to model P3 and P4 games to translate individual number probability trajectories and apply to them exactly the same techniques as we are doing with RPS game (RPS is a stochastic game already, but with key important difference being that there is a learning algorithm that recalculates opposite player probability, nevertheless player's choice is still determined by the RNG)

It is highly interesting if: entropy/ fractal dimension/hurst exponent (not yet implemented and verified)/ lyapunov exponent and yet to be modeled: nsb-entropy (hardest but with highest potential) can be made applicable in lottery games. this question is still very much open.

It is also interesting that RPS game exhibits similar Fractal Dimension as Spectrum of Fibonacci Hamiltonian (0.88137) (see ref. 2 and 3)

Graph 1 (500 Iterations)

Graph 2 (1000 Iterations):

References:

1. FD3 Software - written by John Sarraille and Peter DiFalco, using ideas from "A FAST ALGORITHM TO DETERMINE FRACTAL DIMENSION BY BOX COUNTING", by Liebovitch and Toth, Physics Letters A, 141, 386-390 (1989).

2. From Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Fractal_dimension

3. List of fractals by Hausdorff Dimension (Fractal Dimension)

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Last Edited: July 21, 2009, 9:59 pm

First a little background, what is Entropy ?

According to Wiki entry (ref. 1) In information theory, entropy is a measure of the uncertainty associated with a random variable. (there are other types of entropies such as associated with thermodynamics, but they don't apply in our context)

The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". (ref. 2)

Very short way to describe information entropy (H) is to understand it in terms of measure of certainty (maximum certainty H = 1.0) or uncertainty (maximum uncertainty H = 0.0) and of course anything in between (0.0 < H < 1.0) representing degree of (un) certainty in the information content. 

For the purpose of using it in a random strategy game (ie RPS) we want to measure value of using H in timing profitable entry into the game when certainty of the information is at the lowest (H = 0.0) (see graph 1).

These single events (H = 0.0) were measured to be few (in the span of 100 game iterations) and surprisingly result in up-wards of 50% (Win/Total ratio) gain!  when measured against a player that does "blind" bets.

Below, is application (ref. 3) console output representing run using 2 types of strategies (0 strategy (-) and strategy employing entropy measurement with a threshold of 0, that is a bets are made <=> [only and only iff] Entropy H = 0.0

Each run (total of 5) consists of 525 game iterations, notice player using Entropy strategy places significantly less bets than player using no strategy and betting on every single game occurence. Results are consistent and show not only gains but also that no single game resulted in the loss bet.

Strategy without(-)
Total Game Iterations: 525 Super Agent Played: 525 times. Wins: 179 Losses: 170 Ties: 176 Ratio Win/Total: 0.340952

Strategy with   (+) Entropy (sampling time window: 5, Threshold min: 0, max: 0)
Total Game Iterations: 525 Super Agent Played: 49 times. Wins: 35 Losses: 14 Ties: 0 Ratio Win/Total: 0.714286                 [GAIN of 2 x]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 525 Super Agent Played: 525 times. Wins: 167 Losses: 176 Ties: 182 Ratio Win/Total: 0.318095

Strategy with   (+) Entropy (sampling time window: 5, Threshold min: 0, max: 0)
Total Game Iterations: 525 Super Agent Played: 56 times. Wins: 39 Losses: 15 Ties: 2 Ratio Win/Total: 0.696429                  [GAIN of 2 x]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 525 Super Agent Played: 525 times. Wins: 194 Losses: 150 Ties: 181 Ratio Win/Total: 0.369524

Strategy with   (+) Entropy (sampling time window: 5, Threshold min: 0, max: 0)
Total Game Iterations: 525 Super Agent Played: 39 times. Wins: 23 Losses: 16 Ties: 0 Ratio Win/Total: 0.589744                  [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 525 Super Agent Played: 525 times. Wins: 172 Losses: 182 Ties: 171 Ratio Win/Total: 0.327619

Strategy with   (+) Entropy (sampling time window: 5, Threshold min: 0, max: 0)
Total Game Iterations: 525 Super Agent Played: 52 times. Wins: 34 Losses: 17 Ties: 1 Ratio Win/Total: 0.653846                  [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 525 Super Agent Played: 525 times. Wins: 169 Losses: 161 Ties: 195 Ratio Win/Total: 0.321905        [GAIN]

Strategy with   (+) Entropy (sampling time window: 5, Threshold min: 0, max: 0)
Total Game Iterations: 525 Super Agent Played: 21 times. Wins: 13 Losses: 8 Ties: 0 Ratio Win/Total: 0.619048

Press any key to continue . . .

Graph 1 showcases a sample run of a Super Agent making bets against Player ONE using his total Entropy derived from total sequence of his bets

a pseudo code used in calculating his total entropy is: entropy := probability[event type] * log (1 / probability[event type]) for v = R, P, S

Reference:

1. Entropy From Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Entropy_(Information_theory)

2. Claude E. Shannon "A Mathematical Theory of Communication"

http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf

3. Application Source code (Entropy methods added):

http://members.lotterypost.com/edge/programs/src/rps_game.h

http://members.lotterypost.com/edge/programs/src/rps_game.cpp

Last Edited: July 18, 2009, 8:05 pm

We will be adding entropy studies into random numerical/pattern sequences research.

This research sparked my interest as it pertains directly to undersampled data.

Reference 3 points out possible applications of this algorithm:

Dynamical systems

1. Complexity of dynamics
2. Dimensions of strange attractors

as well as

"rare events statistics"

Reference 3, is actually extremely interesting even outside of theme of study we are doing (random games, random sequences), in fact, NSB Entropy is used by bio-physicist to study "emergence" of intelligent behaviour...

References:

1. The nsb-entropy project is devoted to implementation and practical use of the NSB algorithm for estimation of entropy and related information-theoretic quantities
from undersampled discrete data

http://nsb-entropy.sourceforge.net/

2. (Formal paper) Entropy and inference, revisited by Ilya Nemenman, Fariel Shafee, William Bialek

http://xxx.lanl.gov/abs/physics/0108025

3. (Power Point Presentation) A Bayesian Estimator of Entropies in a Severely Undersampled Regime: Theory and Applications to the Neural Code) by Ilya Nemenman

http://www.menem.com/~ilya/wiki/images/2/2a/LANL-D1.pdf

Last Edited: July 16, 2009, 12:13 am

After much research and troubleshooting finally some progress in the analysis of RPS (Rock Paper Scissors) game model.

Why the RPS game is of so much interest to me? Because RPS represents one of the simplest models (two players with three probabilistic states each) in fact RPS game
is used by researches in game theory, statistics and chaos theory to study phenomena in non-linear dynamics.

In the game theory  the simplest goal is to improve payout matrix (as its in the lottery which is a type of probabilistic game, albeit very extreme case of one!)

One thing RPS game has in common with lottery is its random probability distribution between game iterations, of course RPS probability states are only 3 where else in a typical lottery game let say mega lottery, there are 56 different  probability states between each iteration, nevertheless its still of value to study simpler models
(possible reduction of large dimensional probabilistic state game (such as lottery) to simpler model is unknown at this stage, I don't even know if at all possible!)

With RPS game, 3 different trials were run, each iterating 4 times across 500 game runs and each run was divided to one of two bets:

1. blind bets, using RPS game learning algorithm that at best yields > 0.333% improvement over statistical medium

2. using Local Lyapunov Threshold to consistently (and selectively) bet on Local Lyapunov Threshold bands

Both 1 and 2 game outcome were collected and are displayed below (note that consistent improvement has been made of 2 over 1, yielding some credibility to the fact that using Local Lyapunov Threshold can in fact improve game's outcome matrix.

This in fact is really surprising to me, to recall RPS game probabilistic states are chosen from the random numbers! So in short, Lyapunov strategy compensates for the random!

There are many things not done as yet such as introduction of:

1. adaptive threshold (to inject statistical inference to Local Lyapunov Threshold)

2. entropy filtering

Below is output from rps_game.cpp, improved trials (using LLE threshold) were marked witth "GAIN" label, there were 12 trials run resulting in 10 improved (GAIN) payouts with LLE over 2 that did not use LLE Threshold.

----------------------

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1143 Losses: 898 Ties: 459 Ratio Win/Total: 0.4572

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 586 times.   Wins: 274   Losses: 209 Ties: 103 Ratio Win/Total: 0.467577       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1136 Losses: 930 Ties: 434 Ratio Win/Total: 0.4544

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1688 times. Wins: 813   Losses: 590 Ties: 285 Ratio Win/Total: 0.481635       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1194 Losses: 915 Ties: 391 Ratio Win/Total: 0.4776

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 537 times.   Wins: 259   Losses: 199 Ties: 79   Ratio Win/Total: 0.482309       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1120 Losses: 927 Ties: 453 Ratio Win/Total: 0.448

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1654 times. Wins: 753   Losses: 610 Ties: 291 Ratio Win/Total: 0.45526         [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1177 Losses: 943 Ties: 380 Ratio Win/Total: 0.4708

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 617 times.   Wins: 301   Losses: 224 Ties: 92   Ratio Win/Total: 0.487844      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1162 Losses: 933 Ties: 405 Ratio Win/Total: 0.4648

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1693 times. Wins: 828   Losses: 598 Ties: 267 Ratio Win/Total: 0.489073      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1110 Losses: 936 Ties: 454 Ratio Win/Total: 0.444

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 596 times.   Wins: 304   Losses: 195 Ties: 97   Ratio Win/Total: 0.510067      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1217 Losses: 869 Ties: 414 Ratio Win/Total: 0.4868

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1568 times. Wins: 785   Losses: 535 Ties: 248 Ratio Win/Total: 0.500638      [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1113 Losses: 940 Ties: 447 Ratio Win/Total: 0.4452

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 568 times.   Wins: 270   Losses: 189 Ties: 109 Ratio Win/Total: 0.475352      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1189 Losses: 898 Ties: 413 Ratio Win/Total: 0.4756

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1738 times. Wins: 810   Losses: 626 Ties: 302 Ratio Win/Total: 0.466053      [LOSS]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1247 Losses: 883 Ties: 370 Ratio Win/Total: 0.4988

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 568 times.   Wins: 272   Losses: 202 Ties: 94   Ratio Win/Total: 0.478873      [LOSS]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1159 Losses: 904 Ties: 437 Ratio Win/Total: 0.4636

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1497 times. Wins: 709   Losses: 531 Ties: 257 Ratio Win/Total: 0.473614      [GAIN]

Press any key to continue . . .

Last Edited: July 12, 2009, 5:56 pm

Happy to report that tools development to compute Lyapunov exponents spectrum data is now complete:

3 different methods now yield the same numerical results and are in agreement with other sources

Henon map (discrete dynamical system) was used to test validity of software algorithm models.

Also RPS (Rock Paper Scissors game) computations were re-done yielding correct Lyapunov Exponent spectrum for the game, verifying that the game can be modeled by chaotic attractor (just the same as Henon map) despite its random probabilistic trajectories phase space.

In the summary following 3 different algorithms were used:

1. Wolf method for system of differential equations (see ref. 1 appendix A and ref. 2)

2. Wolf method for discrete time series (without ODE model)  (see ref. 1 appendix B)

3. Sano Sawada method (see ref 3)

In all above cases Lyapunov exponents converge and oscillate around .42, in the full agreement with quoted sources.

In addition a new source of algorithms and application tools can be found in ref. 4

Hope is that we can use these tools to run thru some discretized form of lottery data , let it be frequency/number transition or some other reformulated time series.

As the tool-set is expanding we will be adding computation algorithms and models for Local Lyapunov Exponents, Entropy and Entropy Filtering.

Attempt is to first verify that we follow to best degree possible existing body of research and agree on the numerical results before tackling on lottery number distributions. 

References:

1. Determining Lyapunov exponents from a time series by Wolf, Alan; Swift, Jack B.; Swinney, Harry L.; Vastano, John A. [Physica D: Nonlinear Phenomena, Volume 16, Issue 3, p. 285-317.] 

http://adsabs.harvard.edu/abs/1985PhyD...16..285W

2. Numerical Calculation of Largest Lyapunov Exponent by J. C. Sprott

http://sprott.physics.wisc.edu/chaos/lyapexp.htm

3. Measurement of the Lyapunov Spectrum from a Chaotic Time Series by M. Sano and Y. Sawada Research Institute of Electrical Communication, Tohoku University, Sendai 980, Japan

http://prola.aps.org/abstract/PRL/v55/i10/p1082_1

4. TISEAN - Nonlinear Time Series Analysis by Rainer Hegger, Holger Kantz and Thomas Schreiber 

("TISEAN is a software project for the analysis of time series with methods based on the theory of nonlinear deterministic dynamical systems, or chaos theory, if you prefer. It has grown out of the work of our groups during the last few years.")

http://www.mpipks-dresden.mpg.de/~tisean

Last Edited: July 5, 2009, 11:25 am

Compiled yet another statistics this time for the New York Lottery game.

Data was assembled for the last 100 draws between 07/19/2008 and 07/01/2009 (ref. 1) and simulation run to detect optimal time window when a game play using frequency/number transition would yield best performance.

One surprising fact was detected: not a single set of complete (6) hits was found, despite prediction sets reaching upwards of 34 numbers! (see ref. 2) (i.e prediction made with 34 numbers on 05/16/2009 for the upcoming draw 05/20/2009 yielded only 1 number, so this in fact is the opposite result!

However the task at hand is to determine optimal "Time Window" in this game, and so search is narrowed down to 5 hits while minimizing prediction set size.

Just as in the power ball game, draw # 45 has yielded first 5 hits in the predictions and it was accomplished with 25 numbers (one duplicate due to an overlap) below

Draw Sequence: 45 -----------------------------------------------------------

Number/Frequency (Sorted by Frequency)

Number     22 27 42 56 57 58 59 20 09 16 23 34 44 46 50 01 10 11 19 29
Frequency 00 00 00 00 00 00 00 01 02 02 02 02 02 02 02 03 03 03 03 03

Number     30 35 41 02 04 06 07 08 12 15 18 21 25 31 37 38 45 53 54 28
Frequency 03 03 03 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 05

Number     33 43 47 48 49 51 03 05 13 24 36 39 52 55 14 17 26 32 40
Frequency 05 05 05 05 05 05 06 06 06 06 06 06 06 06 07 07 07 09 09

Predictions made on: 12/20/2008 for the next draw: 22 27 59 20 09 09 16 23 50 01 10 41 02 04 54 28 33 51 03 05 55 14 17 26 32 40 [26 Numbers]

Matched [For Date]: 12/24/2008 22 41 02 17 32 [Total = 5]

-------------------------------------------------------------------------------------

In the summary, the elusive 6th number (to complete the jackpot) has not been reached using this method, the type of "attractor" present in this game is of a very different kind than in games with < 59.

References:

1. http://members.lotterypost.com/edge/programs/data/past_nyp6_numbers.txt  New York Lottery draw results (time period between 07/01/2009 & 07/05/2008)

2. http://members.lotterypost.com/edge/programs/doc/nyp6_large.txt                   large (full output including frequency tables) containing frequency/number predictions

Last Edited: July 4, 2009, 7:04 pm

Data sample was taken between: 07/19/2008 and 07/01/2009 (total of 100 draws)

Looked for highest number of hits, while keeping prediction set small

Prediction set starts from a small set of 7 and ends up 32 numbers (100th draw), this is easily understood as the initial frequency/number transition boundaries (from which predictions are derived from) are mostly 0.

On the opposite spectrum however, starting from approximately 59 drawing, there are already 34 numbers being predicted, due to high count of frequency/number transition boundaries and thats basically to the fact that this is a point where the game has/is entered a statistical equilibrium.

What is interesting is, that larger prediction set does not guarantee higher number of hits! (see ref. 1)

I tend to think that the best moment to enter the game is on the right threshold, in the mega that seems to be 56th draw (so if each number had to be drawn at least once and only once, it would do so on every draw until 56th)

Powerball has larger game set (59 draws) and theoretically i would assume the same would hold, set the "cut off" threshold at 59 past draws, but this has not worked very well.

In fact , we are either too early or too late, and numbers jump around frequency/number transition boundaries, believe this is a wrong threshold for this game and correct one could be to look at no higher than 45 draws (in the data sample prediction set for this draw happened to be 25 numbers (one repeat due to boundary overlay)

Draw Sequence: 45 -----------------------------------------------------------

Draw 12/20/2008 03 19 32 54 14 +24

Number/Frequency (Sorted by Number)

Number      01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
Frequency  03 04 06 04 06 04 04 04 02 03 03 04 06 07 04 02 07 04 03 01

Number      21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Frequency  04 00 02 06 04 07 00 05 03 03 04 09 05 02 03 06 04 04 06 09

Number      41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Frequency  03 00 05 02 04 02 05 05 05 02 05 06 04 04 06 00 00 00 00

Number/Frequency (Sorted by Frequency)

Number      22 27 42 56 57 58 59 20 09 16 23 34 44 46 50 01 10 11 19 29
Frequency  00 00 00 00 00 00 00 01 02 02 02 02 02 02 02 03 03 03 03 03

Number      30 35 41 02 04 06 07 08 12 15 18 21 25 31 37 38 45 53 54 28
Frequency  03 03 03 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 05

Number      33 43 47 48 49 51 03 05 13 24 36 39 52 55 14 17 26 32 40
Frequency  05 05 05 05 05 05 06 06 06 06 06 06 06 06 07 07 07 09 09

Predictions made on: 12/20/2008 for the next draw: 22 27 59 20 09 09 16 23 50 01 10 41 02 04 54 28 33 51 03 05 55 14 17 26 32 40 [26 Numbers]

Matched [For Date]: 12/24/2008 22 41 02 17 32 [Total = 5]

References:

1. http://members.lotterypost.com/edge/programs/doc/powerball_large.txt  large (full output including frequency tables) containing frequency/number predictions

Last Edited: July 4, 2009, 12:48 pm

Added new tools to first re-interpret and confirm findings from the research paper (ref. 1 and 2) and second to have a way to compute Lyapunov exponents from any (including lottery) data/time series.

In 1984 a paper titled "DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES" by Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO
outlined a new algorithm for Lyapunov Exponent Spectrum calculations.

Excerpt from the abstract:

"
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic."

Subsequently, Fortran code was ported to C++ and incooperated into Chaos Toolkit. More tests are needed to verify correctness of its implementation.

And although implementation details are complex, Lyapunov exponents are well known and serve in analyzing non-linear data series random/chaotic properties.

Using the new toolkit and RPS game model (ref. 2) following two graphs below were generated in order to demonstrate that the RPS Game exhibits chaos (Average Lyapunov Exponents > 0)

Graph 1. Probability Values for Player One RPS Game

Graph 2. Lyapunov Exponents in RPS Game

References:

1. DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO Department of Physics, University of Texas, Austin, Texas 78712, USA

2. Chaotic time series prediction for the game, Rock-Paper-Scissors by Franco Salvetti, Paolo Patelli and Simone Nicolo

http://www.francosalvetti.com/FrancoSalvetti_in_press.pdf

Complete source code files:

Last Edited: June 29, 2009, 12:25 am

Recent NJ Cash 5 Lottery Hit (number selections derived by the frequency/number distributions and their visualization via a heat map they generated)

Winning Numbers for Saturday , June 27, 2009 were:

09 13 14 19 40

We had:

13 19 40

Ticket Scan below:

Last Edited: June 27, 2009, 9:08 pm

Recent Mega Lottery Hit (number selections derived by the frequency/number distributions and their visualization via a heat map they generated)

Winning Numbers for Friday , June 26, 2009 were:

11 15 17 29 46 +16

We had:

11 29 +16

Ticket Scan below:

Last Edited: June 27, 2009, 9:18 am