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1 to 11 Point - Polynomial Wave Projection EquationsPrev TopicNext Topic
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Native American Eagle SunMN
United States
Member #21
December 7, 2001
4,812 Posts
Offline1 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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