1 to 11 Point - Polynomial Wave Projection EquationsPrev TopicNext Topic

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  December 7, 2001
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  Aug 4, 2007, 4:53 pm

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  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

  The One Over None
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