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1 to 11 Point - Polynomial Wave Projection Equations

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The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
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Posted: August 4, 2007, 4:53 pm - IP Logged

1 to 11 Point - Polynomial Wave Projection Equations

Reference - 5th Degree Polynomial Wave Projection, Bidirectional Mean Averaging and The Wave Matrix, Example Process for Finding Variable Coefficients and Solved Projection Equations, Example of Finding Powerball Number Linear Regression BMA and Wave Matrix

The different point projections are an expansion process of finding the Variable Coefficient Equations and the Solved Projection Equations. The Cofactors in the Coefficient Equations are solved through an algorithmic process. You can see the process for this example of 1 to 11 points as an Excel sheet, link: Excel Sheet Example of 1 to 11 Point Algorithmic Cofactors Process

Each Projection is based on a Known Point Set, n, and is Related to a given (n - 1)th Degree Polynomial Equation. The Variable Coefficient Equations are solved for that given (n - 1)th Degree Polynomial Equation and are then plugged into the Solved (n + 1)th Point Projection Equation. Point Projections 3 through 11 are curve fittings for the most probable wave projections. As the number of Point Projections increases, the Projection process can handle higher frequency data. As the number of Point Projections decreases, the process is more tuned to low frequency data. Use the links above to get a feel for where each Point Projection can best do the work.

If you're having a problem seeing the on LP, go here, link: 1 to 11 Point - Polynomial Wave Projection Equations

 

1 Point Projection

 
Known Point Set
  {y0}
 
Related 0th Degree Polynomial Equation
y =  a0 x0
 
Variable Coefficient Equation

a0 =

 1y0 

 
Solved 2nd Point Projection

y1 =

 (1a0) / 1 




2 Point Projection

 
Known Point Set
  {y0, y1}
 
Related 1st Degree Polynomial Equation
y =  a0 x0 + a1 x1
 
Variable Coefficient Equations

a0 =

 2y0 - 1y1 

a1 =

 -1y0 + 1y1 

 
Solved 3rd Point Projection

y2 =

 (1a0 + 3a1) / 1 



3 Point Projection

 
Known Point Set
  {y0, y1, y2}
 
Related 2nd Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2
 
Variable Coefficient Equations

a0 =

 6y0 - 6y1 + 2y2 

a1 =

 -5y0 + 8y1 - 3y2 

a2 =

 1y0 - 2y1 + 1y2 

 
Solved 4th Point Projection

y3 =

 (1a0 + 4a1 + 16a2) / 2 




4 Point Projection

 
Known Point Set
  {y0, y1, y2, y3}
 
Related 3rd Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3
 
Variable Coefficient Equations

a0 =

 24y0 - 36y1 + 24y2 - 6y3 

a1 =

 -26y0 + 57y1 - 42y2 + 11y3 

a2 =

 9y0 - 24y1 + 21y2 - 6y3 

a3 =

 -1y0 + 3y1 - 3y2 + 1y3 

 
Solved 5th Point Projection

y4 =

 (1a0 + 5a1 + 25a2 + 125a3) / 6 



5 Point Projection

 
Known Point Set
  {y0, y1, y2, y3, y4}
 
Related 4th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4
 
Variable Coefficient Equations

a0 =

 120y0 - 240y1 + 240y2 - 120y3 + 24y4 

a1 =

 -154y0 + 428y1 - 468y2 + 244y3 - 50y4 

a2 =

 71y0 - 236y1 + 294y2 - 164y3 + 35y4 

a3 =

 -14y0 + 52y1 - 72y2 + 44y3 - 10y4 

a4 =

 1y0 - 4y1 + 6y2 - 4y3 + 1y4 

 
Solved 6th Point Projection

y5 =

 (1a0 + 6a1 + 36a2 + 216a3 + 1296a4) / 24



6 Point Projection

 
Known Point Set
  {y0, y1, y2, y3, y4, y5}
 
Related 5th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5
 
Variable Coefficient Equations

a0 =

 720y0 - 1800y1 + 2400y2 - 1800y3 + 720y4 - 120y5

a1 =

 -1044y0 + 3510y1 - 5080y2 + 3960y3 - 1620y4 + 274y5

a2 =

 580y0 - 2305y1 + 3720y2 - 3070y3 + 1300y4 - 225y5

a3 =

 -155y0 + 685y1 - 1210y2 + 1070y3 - 475y4 + 85y5

a4 =

 20y0 - 95y1 + 180y2 - 170y3 + 80y4 - 15y5

a5 =

 -1y0 + 5y1 - 10y2 + 10y3 - 5y4 + 1y5

 
Solved 7th Point Projection

y6 =

 (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120 



7 Point Projection

 
Known Point Set
  {y0, y1, y2, y3, y4, y5, y6}
 
Related 6th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6
 
Variable Coefficient Equations

a0 =

 5040y0 - 15120y1 + 25200y2 - 25200y3 + 15120y4 - 5040y5 + 720y6 

a1 =

 -8028y0 + 31644y1 - 56940y2 + 59040y3 - 36180y4 + 12228y5 - 1764y6 

a2 =

 5104y0 - 23574y1 + 46680y2 - 50900y3 + 32160y4 - 11094y5 + 1624y6 

a3 =

 -1665y0 + 8520y1 - 18285y2 + 21120y3 - 13875y4 + 4920y5 - 735y6 

a4 =

 295y0 - 1620y1 + 3705y2 - 4520y3 + 3105y4 - 1140y5 + 175y6 

a5 =

 -27y0 + 156y1 - 375y2 + 480y3 - 345y4 + 132y5 - 21y6 

a6 =

 1y0 - 6y1 + 15y2 - 20y3 + 15y4 - 6y5 + 1y6 

 
Solved 8th Point Projection

y7 =

 (1a0 + 8a1 + 64a2 + 512a3 + 4096a4 + 32768a5 + 262144a6) / 720



8 Point Projection

 
Known Point Set
  {y0, y1, y2, y3, y4, y5, y6, y7}
 
Related 7th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7
 
Variable Coefficient Equations

a0 =

 40320y0 - 141120y1 + 282240y2 - 352800y3 + 282240y4 - 141120y5 + 40320y6 - 5040y7

 

a1 =

 -69264y0 + 312984y1 - 673008y2 + 870660y3 - 710640y4 + 360024y5 - 103824y6 + 13068y7

 

a2 =

 48860y0 - 256942y1 + 602532y2 - 815920y3 + 684740y4 - 353430y5 + 103292y6 - 13132y7

 

a3 =

 -18424y0 + 107023y1 - 270144y2 + 384755y3 - 334040y4 + 176589y5 - 52528y6 + 6769y7

 

a4 =

 4025y0 - 25060y1 + 67095y2 - 100240y3 + 90335y4 - 49140y5 + 14945y6 - 1960y7

 

a5 =

 -511y0 + 3346y1 - 9387y2 + 14630y3 - 13685y4 + 7686y5 - 2401y6 + 322y7 

 

a6 =

 35y0 - 238y1 + 693y2 - 1120y3 + 1085y4 - 630y5 + 203y6 - 28y7 

 

a7 =

 -1y0 + 7y1 - 21y2 + 35y3 - 35y4 + 21y5 - 7y6 + 1y7 

 
Solved 9th Point Projection

y8 =

 (1a0 + 9a1 + 81a2 + 729a3 + 6561a4 + 59049a5 + 531441a6 + 4782969a7) / 5040 




9 Point Projection
 
Known Point Set
  {y0, y1, y2, y3, y4, y5, y6, y7, y8}
 
Related 8th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8
 
Variable Coefficient Equations

a0 =

 362880y0 - 1451520y1 + 3386880y2 - 5080320y3 + 5080320y4 - 3386880y5 + 1451520y6 - 362880y7 + 40320y8 

 

a1 =

 -663696y0 + 3380544y1 - 8452416y2 + 13101984y3 - 13356000y4 + 9016896y5 - 3898944y6 + 981216y7 - 109584y8 

 

a2 =

 509004y0 - 3000528y1 + 8127728y2 - 13142304y3 + 13746600y4 - 9442384y5 + 4133808y6 - 1050048y7 + 118124y8 

 

a3 =

 -214676y0 + 1394456y1 - 4045104y2 + 6845944y3 - 7382200y4 + 5180616y5 - 2304176y6 + 592424y7 - 67284y8 

 

a4 =

 54649y0 - 380072y1 + 1165332y2 - 2059064y3 + 2294110y4 - 1650264y5 + 748132y6 - 195272y7 + 22449y8 

 

a5 =

 -8624y0 + 63056y1 - 202104y2 + 371056y3 - 427000y4 + 315504y5 - 146216y6 + 38864y7 - 4536y8 

 

a6 =

 826y0 - 6272y1 + 20832y2 - 39536y3 + 46900y4 - 35616y5 + 16912y6 - 4592y7 + 546y8 

 

a7 =

 -44y0 + 344y1 - 1176y2 + 2296y3 - 2800y4 + 2184y5 - 1064y6 + 296y7 - 36y8 

 

a8 =

 1y0 - 8y1 + 28y2 - 56y3 + 70y4 - 56y5 + 28y6 - 8y7 + 1y8 

 
Solved 10th Point Projection

y9 =

 (1a0 + 10a1 + 100a2 + 1000a3 + 10000a4 + 100000a5 + 1000000a6 + 10000000a7 + 100000000a8) / 40320



10 point Projection

 
Known Point Set
  {y0, y1, y2, y3, y4, y5, y6, y7, y8, y9}
 
Related 9th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8 + a9 x9
 
Variable Coefficient Equation

a0 =

 3628800y0 - 16329600y1 + 43545600y2 - 76204800y3 + 91445760y4 - 76204800y5 + 43545600y6 - 16329600y7 + 3628800y8 - 362880y9 

 

a1 =

 -6999840y0 + 39664080y1 - 113028480y2 + 204150240y3 - 249552576y4 + 210500640y5 - 121322880y6 + 45787680y7 - 10225440y8 + 1026576y9 

 

a2 =

 5753736y0 - 37559052y1 + 115366752y2 - 216787536y3 + 271479600y4 - 232741656y5 + 135711072y6 - 51667632y7 + 11617416y8 - 1172700y9 

 

a3 =

 -2655764y0 + 19063224y1 - 62458416y2 + 122402616y3 - 157623480y4 + 137809224y5 - 81526704y6 + 31384296y7 - 7118676y8 + 723680y9 

 

a4 =

 761166y0 - 5844573y1 + 20183688y2 - 41154876y3 + 54581940y4 - 48787326y5 + 29356488y6 - 11453148y7 + 2625966y8 - 269325y9 

 

a5 =

 -140889y0 + 1136961y1 - 4096764y2 + 8654436y3 - 11815398y4 + 10811934y5 - 6630876y6 + 2627604y7 - 610281y8 + 63273y9 

 

a6 =

 16884y0 - 141498y1 + 527688y2 - 1149624y3 + 1612800y4 - 1511244y5 + 946008y6 - 381528y7 + 89964y8 - 9450y9 

 

a7 =

 -1266y0 + 10926y1 - 41904y2 + 93744y3 - 134820y4 + 129276y5 - 82656y6 + 33984y7 - 8154y8 + 870y9 

 

a8 =

 54y0 - 477y1 + 1872y2 - 4284y3 + 6300y4 - 6174y5 + 4032y6 - 1692y7 + 414y8 - 45y9 

 

a9 =

 -1y0 + 9y1 - 36y2 + 84y3 - 126y4 + 126y5 - 84y6 + 36y7 - 9y8 + 1y9 

 
Solved 11th Point Projection

y10 =

 (1a0 + 11a1 + 121a2 + 1331a3 + 14641a4 + 161051a5 + 1771561a6 + 19487171a7 + 214358881a8 + 2357947691a9) / 362880



11 point projection
 
Known Point Set
  {y0, y1, y2, y3, y4, y5, y6, y7, y8, y9, y10}
 
Related 10th Degree Polynomial Equation
y =  a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8 + a9 x9 + a10 x10
 
Variable Coefficient Equations

a0 =

 47174400y0 - 214099200y1 + 606009600y2 - 1197504000y3 + 1676505600y4 - 1676505600y5 + 1197504000y6 - 598752000y7 + 199584000y8 - 39916800y9 + 3628800y10

 

a1 =

 -95142240y0 + 531957600y1 - 1623088800y2 + 3316939200y3 - 4727540160y4 + 4783423680y5 - 3445243200y6 + 1733313600y7 - 580543200y8 + 116552160y9 - 10628640y10

 

a2 =

 79362936y0 - 521270280y1 + 1736650440y2 - 3698304480y3 + 5393046960y4 - 5541317712y5 + 4035361680y6 - 2047105440y7 + 690085080y8 - 139262760y9 + 12753576y10

 

a3 =

 -36781540y0 + 278356040y1 - 1004826060y2 + 2233166160y3 - 3342229800y4 + 3497286240y5 - 2581262040y6 + 1322982960y7 - 449614260y8 + 91331800y9 - 8409500y10

 

a4 =

 11028590y0 - 92615030y1 + 355598730y2 - 821580360y3 + 1263374700y4 - 1348939620y5 + 1011120180y6 - 524563080y7 + 180021510y8 - 36862550y9 + 3416930y10 

 

a5 =

 -2310945y0 + 20390160y1 - 81560115y2 + 194790960y3 - 307585530y4 + 335437200y5 - 255740310y6 + 134522640y7 - 46695285y8 + 9653280y9 - 902055y10 

 

a6 =

 326613y0 - 2992710y1 + 12376665y2 - 30429000y3 + 49260330y4 - 54871236y5 + 42592410y6 - 22748040y7 + 7999425y8 - 1672230y9 + 157773y10 

 

a7 =

 -30810y0 + 290760y1 - 1235790y2 + 3115440y3 - 5159700y4 + 5866560y5 - 4638060y6 + 2517840y7 - 898290y8 + 190200y9 - 18150y10 

 

a8 =

 1860y0 - 17970y1 + 78120y2 - 201240y3 + 340200y4 - 394380y5 + 317520y6 - 175320y7 + 63540y8 - 13650y9 + 1320y10 

 

a9 =

 -65y0 + 640y1 - 2835y2 + 7440y3 - 12810y4 + 15120y5 - 12390y6 + 6960y7 - 2565y8 + 560y9 - 55y10 

 

a10 =

 1y0 - 10y1 + 45y2 - 120y3 + 210y4 - 252y5 + 210y6 - 120y7 + 45y8 - 10y9 + 1y10 

 
Solved 12th Point Projection

y11 =

 (1a0 + 12a1 + 144a2 + 1728a3 + 20736a4 + 248832a5 + 2985984a6 + 35831808a7 + 429981696a8 + 5159780352a9 + 61917364224a10) / 3628800

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