Mean Difference and Standard Deviation
1 - Set of Wave Values
{W0, W1, W2, ... Wn-2, Wn-1, Wn}
2 - Approximate Projection Set
First difference calculation.
S0 = {y0, y1, y2, y3, y4, y5} = {W0, W1, W2, W3, W4, W5}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dy0 = y6 - W6 First difference between approximate and actual.
Second difference calculation.
S1 = {y0, y1, y2, y3, y4, y5} = {W1, W2, W3, W4, W5, W6}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dy1 = y6 - W7 Second difference between approximate and actual.
Third difference calculation.
S2 = {y0, y1, y2, y3, y4, y5} = {W2, W3, W4, W5, W6, W7}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dy2 = y6 - W8 Third difference between approximate and actual.
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Continue advancing subset through the set of Wave Values.
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Second to last difference calculation.
Sn-8 = {y0, y1, y2, y3, y4, y5} = {Wn-8, Wn-7, Wn-6, Wn-5, Wn-4, Wn-3}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dyn-8 = y6 - Wn-2 Second to last difference between approximate and actual.
Next to last difference calculation.
Sn-7 = {y0, y1, y2, y3, y4, y5} = {Wn-7, Wn-6, Wn-5, Wn-4, Wn-3, Wn-2}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dyn-7 = y6 - Wn-1 Next to last difference between approximate and actual.
Last difference calculation.
Sn-6 = {y0, y1, y2, y3, y4, y5} = {Wn-6, Wn-5, Wn-4, Wn-3, Wn-2, Wn-1}
Plug values into following 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
y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120
Dyn-6 = y6 - Wn Next to last difference between approximate and actual.
Dy = {Dy0, Dy1, Dy2, ... Dyn-8, Dyn-7, Dyn-6} Set of differences between approximate and actual
The set of differences will always be 6 less than the number of the Wave Data set count. For that reason, there needs to be at least 10 or more points in the Wave to calculate the mean and standard deviation.
3 - Mean Difference
m = ( [0 to n - 6] å Dyi ) / (n - 6)
m = (Dy0 + Dy1 + Dy2 + ... + Dyn-8 + Dyn-7 + Dyn-6) / (n - 6)
4 - Standard Deviation of Difference
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s = Ö( [0 to n - 6] å (Dyi - m)²) / (n - 6) |
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s = Ö((Dy0 - m)² + (Dy1 - m)² + (Dy2 - m)² + ... + (Dyn-8 - m)² + (Dyn-7 - m)² + (Dyn-6 - m)²) / (n - 6) |