Y=1/($E$1*SQRT($E$2-x)+$E$3+$E$4*SQRT(SQRT($E$5-x)))
You could say these constants came out of a least-squares fit program with an assumed solution including a term varying in x^1/2, a term varying in x^1/4, and a bias term.
E1 is magnitude scaling factor for the ½ power term (centered on E2)
E4 is the scaling factor for the ¼ power term (centered on E5).
E4 is a constant (bias) term.
Maybe Greg has a better way. To get more help on physical interpretation from the folks on the forum I would say you have to give more details of the physical problem (maybe that's something that you don't want to do to protect business secrets?).
Unfortunately I went the opposite direction. I spent (wasted?) some time messing around to add more terms to Greg’s model to reduce the residuals. I did get good reduction in residuals at a cost of substantial increase in complexity. This is probably not what you’re looking for but since I’ve done it, here it is:
The following columns are in my spreadsheet
X, Y = Original data
Ygreg = Greg’s fit * Note - Greg’s fit constants changed when I reoptimized with new terms
Rgreg = Greg’s residual = Y – Ygreg *Same Note
Fpete = my fit to greg’s residual
Ypete = Ygreg +Fpete = estimate of Y
Rpete = Y - Ypete
Rpete/Y = Fractional residuals
I had some difficulties minimizing errors on both the top and bottom end. What finally resolved the problem was to optimize sum of squares of FRACTIONAL residuals , rather than residuals themselves. Probably if I tried this approach from the beginning I could have reduced the complexity. I didn’t go back and try to clean it up. What is left is a complex formula which I’m sure could be simplified or done better. But this is what I have for now.
** The largest residual magnitude (Rpete) is 0.0075
** The largest fractional residual magnitude (Rpete/Y) is 1.7% over the entire range up to 0.9995. That worst performance occurs at 0.999. Below 0.9965 all fractional residuals are less than 1% of Y
If one wanted, one could start with my solution and look at the residuals and attempt to add terms to minimize those residuals (that’s what I did to Greg’s solution).
The spreadsheet is here:
(scroll to bottom of sheet 1 to see summary statistics)
The formulas are
Ygreg=1/($B$3*SQRT($B$4-C3)+$B$5+$B$6*SQRT(SQRT($B$7-C3)))
Ypete = Ygreg+$B$8/($B$11+(C3-0.995)^2) + $B$9*(1-C3)*EXP($B$10*(1-C3)) + $B$12*(0.5-C3)^2*(ATAN(1000*(C3-0.5))+PI()/2)+$B$13*EXP($B$15*C3)
where cells are as follows:
C3 is x (copy down as relative cell reference)
B3 50.63746184 (remaining items used as absolute reference)
B4 1.000239763
B5 0.823257421
B6 -0.006540995
B7 3.654165646
B8 0.015665365
B9 772.441016
B10 -2859.484503
B11 1.23573751
B12 0.008220401
B13 -0.002908095
B14 -10
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