edge's Blog

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 . . .

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

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

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

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:

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:

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:

Found this incredibly fascinating paper (ref. 1) (I usually do research during the week, given time and energy, and try to verify theoretical results on the weekend!)

One of key observations right away is the fact that one of the authors (Paolo Patelli) works at T-13 Complex System Group, Theoretical Division, CNLS Los Alamos National Laboratory which gives it a strong level of credibility and correctness.

Paper main motivation is to study popular but basic game of Rock-Paper-Scissors, and via using ideas from Chaos Theory (Local Lyapunov Exponent LLE and entropy) provide a player with an advantage (which accounts to the same thing as predicting game's future outcome).

Although this particular game is of limited rule type, nevertheless its temporal behavior enters chaos in its probability space.

From our end, we are interested in adding yet another tool, namely LLE (Local Lyapunov Exponent) and Entropy and Entropy Filtering and exploring possibility of its use in the lottery game systems (neither of these concepts were known to the author of this blog previously)

We can think of an idea where 2 abstract players are arranged to engage in a competition play, drawing from a pool of lottery numbers, one employing LLE the other entropy and subject this game to the same type of analysis.

Paper ends with the following statement :

"Our preliminary results are encouraging and show that by accepting intermittency in playing it is possible to decide the best moment to play and consequently to improve the performance of a simple and nai¨ve strategy. Filtering based on the Entropy appears more effective than that based on the LLE, and embedding the trajectory in order to compute the Entropy slightly degrades performance while increasing robustness. The effectiveness of a method for automatically adapting the threshold is proven to be effective. "

The best moment to play is one of the key parameters in win/loose strategy in games based on probabilistic theory such as found in lottery games.

References:

1. 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

2. T-13 Complex System Group, Theoretical Division

http://t13web.lanl.gov/

Re-encoded Mathematica notebook (ref. 1) into C++ classes, calculations were checked by hand and Henon attractor has emerged together with the exact Correlation Integral in the agreement with Fractal Dimension calculations from other sources. (approximately ~ 1.26) [Best Fit: Y = -1.36378 + 1.2656 X]

Correlation Integral is of the form Log[Corr] = D Log [R] and its importance lies in detecting chaos/fractal property in a data series.

Henon attractor has specific fractal dimension (approx 1.26) and exhibits chaotic orbits around an attractor (see below) these orbits are invariant in a sense that they exhibit certain pseudo periodicity (pseudo, as the points will not re-generate in the exact position in the number plane but rather do so on ever enlarging scale)

Next in this project is Lyapunov exponent, which measures how fast this "scale" expansion happens, if the rate is slow, the orbits can be of stable/periodic or chaotic type
(they can also start to "escape" to infinity or "converge" to point).

Hurst exponent will tell us how "smooth" trajectories are (and is related to fractal dimension), we already have its first manifestation in "Correlation Integral" in the Henon attractor, it i useful insofar as to be able to approximate one data series with another (ie Henon map with some n-dimensional embedding added)

Below are various images and output displays related to the above, much still has to be done, but serious progress has been made.

Best fit (linear) Best Fit: Y = -1.36378 + 1.2656 X
in the full agreement with independent calculation of Hurst exponent for Henon map attractor (~ 2.26)

Lyapunov exponents for various (random) sequences

1 periodic -0.103898
3 chaotic 0.565543
4 chaotic 0.259456
5 chaotic 0.8249
8 chaotic 0.465392
9 chaotic 0.381847
10 periodic -0.0727799
11 chaotic 0.669459
12 periodic -0.178313
13 periodic -0.0211324
14 chaotic 0.802993
15 chaotic 0.469818
16 chaotic 0.196232
17 chaotic 0.976126

Results (88 seconds)

Infinite: 0 (0.0%)
Point : 4 (22.2%)
Stable : 0 (0.0%)
Periodic: 4 (22.2%)
Chaotic : 10 (55.6%)


Attractor images (only stable (0 found in this particular trial), periodic and chaotic images are displayed (infinite and point have no visual representation possible)

 

References:

1.  Testing Chaos and Fractal Properties in Economic Time Series

http://www.internationalmathematicasymposium.org/IMS99/paper25/ims99paper25.pdf

2. Numerical "best fit" linear estimation to determine Correlation Integral were performed using GSL - GNU Scientific Library:

 http://www.gnu.org/software/gsl/

3. Images showcasing attractors and their Lyapunov exponents were generated using ideas and sample application found at:

http://www.technocosm.org/chaos/attr-part2.html

Found an excellent paper (ref. 1) that demonstrates methods to analyze discrete set of data (such as lottery data) without any reference to differential equations.

Usually chaos and fractal behavior is analyzed using both DE and/or PDE (differential equations and partial differential equations).

From the the time series point of view where one is seeking to locate fractal properties (or even to determine if data points exhibit fractal properties at all)  there is a need to find ways to approximate non-linear equations with set of discrete points. Such methods are clearly outlined in this paper.

Author uses well known non-linear Henon map function, transforms it to 2 dimensional representation of discretized data points and then builds so called Henon attractor.

Subsequently, via re-parametrization, a one dimensional (1 coordinate) approximate Hennon attractor in a two dimensional phase space is reconstructed.

Next step is an embedding dimension method is introduced and equations form a method to study phase space approximations (there is no limit how many embedding dimensions one can use).

The whole idea is to take any discrete data points and via above methods to approximate chaotic orbits present in the Henon phase space via estimation of Hurst (Fractal Dimension) and Lyapunov exponent for predictive power of the data points generated thru iterative process of various re-parametrization values and embedding dimensions.

This is quite exciting from our point of view as this is exactly type of applied chaos analysis thats needed for a discrete data series in form of past lottery draw, if carefully executed it will provide knowledge (to the best degree of approximation) if the methods of fractal and chaos theory can be applied at all in this context.

There is a strong possibility that this want be the case, ie Hurst exponent defaults to value as present in random walk (0 correlations) and or Lyapunov exponent yields 0 predictability, but in any case this needs to be explored and evaluated.

One drawback of the paper also is that calculations are performed using Mathematica software, and what I really need is set of c++ libraries.

In any case paper is one of best around on the subject of pragmatic approach to time series analysis using methods in chaos and fractal theory.

References:

1. Testing Chaos and Fractal Properties in Economic Time Series

http://www.internationalmathematicasymposium.org/IMS99/paper25/ims99paper25.pdf

Planning to start estimating these exponents using various data points in lottery time series (starting with mega lottery).

Hurst exponent (H) determines the rate of chaos

Lyapunov exponent (L) determines the rate of predictability

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

Will codify both exponents as C++ classes that are generic enough to analyze any data points in lottery time series (ie frequency/number transitions but also others)

If a Hurst exponents is evaluated to a fractal dimension in a lottery time series (key in all this endavour) we will be able to find an attractor and via Lyapunov exponent and multi-fractal and L-variable fractal analysis (the superfractal) try to approximate (an eventually predict) its chaotic "orbits"

For a nice and short explanation on both exponents see ref.1

References:

1. Energy Time Series and Chaos: http://www.iqnet.cz/dostal/CHA2.htm

Took a sample of past mega lottery results (from 06/17/2008 to present) and ran it thru modified version of superfractal.cpp to analyze how well predictions perform when made from numbers selected from those that occur on the frequency/number boundries.

Initially, frequency number transitions will be far to few to yield any useful data, soon enough a certain "spike" occurs (after around 40 draws or so) and some draws start to match upwards of 3 numbers (from 25) ie: Matched [For Date]: 11/21/2008 56 56 01 52 52 [Total = 5] below: (3 because of duplication which could be telling by itself, but there are other examples with higher number of hits)

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

Draw 11/18/2008 10 15 22 52 53 +12 $0.00

Number/Frequency (Sorted by Frequency)

Number     04 37 54 56 03 08 26 31 33 40 45 07 10 11 12 18 20 21 27 44
Frequency 00 00 01 01 02 02 02 02 02 02 02 03 03 03 03 03 03 03 03 03

Number     46 47 48 49 05 06 24 28 30 38 50 53 55 01 15 16 17 25 29 32
Frequency 03 03 03 03 04 04 04 04 04 04 04 04 04 05 05 05 05 05 05 05

Number     34 35 43 02 09 13 19 22 23 36 39 41 51 42 52 14
Frequency 05 05 05 06 06 06 06 06 06 06 06 06 06 07 08 09

Predictions made on: 11/18/2008 for the next draw: 04 37 37 54 56 56 03 08 45 07 10 49 05 06 55 01 15 43 02 09 51 42 52 52 14 [25 Numbers]

Matched [For Date]: 11/21/2008 56 56 01 52 52 [Total = 5]

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

It is my intent to elaborate further on this data, it would be nice to have done some reduction in the game such as limiting pool of numbers to 25 as in the example above.

Fractal dimension of such frequency/number transition is of big interest as well, to some extent I see these frequency boundries as fundamental to random evolution in this (and other games).

 

For full data files (output from superfactal.cpp) see

References:

1. https://members.lotterypost.com/edge/programs/doc/mega_small.txt  small (truncated output without frequency tables) containing frequency/number predictions

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

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 12, 2009 were:

06 11 20 32 44 +38

We had:

06 11 32

Ticket Scan below:

Added one more tool, the "heat map" - here using a 2 dimensional rendering of the mega lottery frequency-number transition using last 100 draws

heat maps are useful when three data points are involved, in our example:

X - Frequency

Y - Frequency

Z - Number Occurence

Heat Maps assume X = Y, and intensity of color in our example indicates number of hits (Count)

Latest Draw     06/05/2009 5 20 38 41 52 +20 $0.00

Frequency       Count

03                     01
04                     04
05                     03
06                     03
07                     06
08                     08
09                     08
10                     06
11                     05
12                     07
13                     01
14                     02
15                     02

Note: below the white space indicate lowest frequency: 3 and 13 (highest frequency bright green 8 and 9), from my end I am constantly refining how the data is presented.  see below references.

above image was produced by the superfractal.cpp using gnuplot (ref. 1) and gnuplot C++ interface (ref. 2)

Reference:

1. Gnuplot homepage:

http://www.gnuplot.info/

2. gnuplot-cpp  C++ interface to Gnuplot via POSIX pipes

http://code.google.com/p/gnuplot-cpp/

Appears that the mega lottery draws give rise to formation to Sierpinski Triangle Fractal

Using IFS iterative deterministic algorithm and 5, 10 and 100 iterations, application started to produce the attractor see my home page in ref. 1

Contours and shapes of the triangles are becoming to form at various densities. Its these densities that are of interest and further study.

In this particular run, 5 x 100 past mega lottery numbers  form the seed to choose from 3 vertices ( n = p Modulus (remainder) 3, where p is a lottery number, and n yields random vertix)

Triangle Formation after 100 mega lottery draws (between 06/17/2008 and 06/05/2009)

Reference:

1. Chaos Game (Sierpinski Triangle Formation) using past lottery data as the random seed. (images produced by superfractal.cpp using opengl and glut libraries for image processing)

https://members.lotterypost.com/edge/index.htm