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Appendix A: Linear and Nonlinear Curve Fit Functions

Linear Equations

Linear Group Function Numbers

  1. y=a0+a1*x
  2. y=a0+a1/x
  3. y=a0+a1*sqrt(x)
  4. y=a0+a1*sqrt(x)
  5. y=a0+a1/x^2
  6. y=a0+a1*exp(x)
  7. y=a0+a1*exp(–x)
  8. y =a0+a1*ln(x)
  9. y=a0+a1/ln(x)
  10. y=a0+a1*(ln(x))^2
  11. 1/y=a0+a1*x
  12. 1/y=a0+a1/x
  13. 1/y=a0+a1*sqrt(x)
  14. 1/y=a0+a1/sqrt(x)
  15. 1/y=a0+a1/x^2
  16. 1/y=a0+a1*exp(x)
  17. 1/y=a0+a1*exp(–x)
  18. 1/y=a0+a1*ln(x) 
  19. 1/y=a0+a1/ln(x)
  20. 1/y=a0+a1*(ln(x))^2
  21. y^2=a0+a1*x
  22. y^2=a0+a1/x
  23. y^2=a0+a1*sqrt(x)
  24. y^2=a0+a1/sqrt(x)
  25. y^2=a0+a1/x^2
  26. y^2=a0+a1*exp(x)
  27. y^2=a0+a1*exp(–x)
  28. y^2=a0+a1*ln(x)
  29. y^2=a0+a1/ln(x)
  30. y^2=a0+a1*(ln(x))^2
  31. sqrt(y)=a0+a1*x
  32. sqrt(y)=a0+a1/x
  33. sqrt(y)=a0+a1*sqrt(x)
  34. sqrt(y)=a0+a1/sqrt(x)
  35. sqrt(y)=a0+a1/x^2
  36. sqrt(y)=a0+a1*exp(x)
  37. sqrt(y)=a0+a1*exp(–x)
  38. sqrt(y)=a0+a1*ln(x)
  39. sqrt(y)=a0+a1/ln(x)
  40. sqrt(y)=a0+a1*(ln(x))^2

Exponential Group Function Numbers

  1. ln(y)=a0+a1*x
  2. ln(y)=a0+a1/x 
  3. ln(y)=a0+a1*sqrt(x)
  4. ln(y)=a0+a1/sqrt(x)
  5. ln(y)=a0+a1/x^2
  6. ln(y)=a0+a1*exp(x)
  7. ln(y)=a0+a1*exp(–x)
  8. ln(y)=a0+a1*ln(x)
  9. ln(y)=a0+a1/ln(x)
  10. ln(y)=a0+a1*(ln(x))^2

Power Group Function Numbers

  1. y=a0+a1*x^2
  2. y=a0+a1*x^3
  3. y=a0+a1*x^4
  4. y=a0+a1*x^5
  5. y=a0+a1*x^6
  6. 1/y=a0+a1*x^2
  7. 1/y=a0+a1*x^3
  8. 1/y=a0+a1*x^4
  9. 1/y=a0+a1*x^5
  10. 1/y=a0+a1*x^6
  11. y^2=a0+a1*x^2
  12. y^2=a0+a1*x^3
  13. y^2=a0+a1*x^4
  14. y^2=a0+a1*x^5
  15. y^2=a0+a1*x^6
  16. sqrt(y)=a0+a1*x^2
  17. sqrt(y)=a0+a1*x^3
  18. sqrt(y)=a0+a1*x^4
  19. sqrt(y)=a0+a1*x^5
  20. sqrt(y)=a0+a1*x^6
  21. ln(y)=a0+a1*x^2
  22. ln(y)=a0+a1*x^3
  23. ln(y)=a0+a1*x^4
  24. ln(y)=a0+a1*x^5
  25. ln(y)=a0+a1*x^6   

Polynomial Group Function Numbers

  1. y=a0+a1*x+a2*x^2
  2. y=a0+a1*x+a2*x^2+a3*x^3
  3. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
  4. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
  5. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
  6. 1/y=a0+a1*x+a2*x^2
  7. 1/y=a0+a1*x+a2*x^2+a3*x^3
  8. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
  9. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
  10. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
  11. y^2=a0+a1*x+a2*x^2
  12. y^2=a0+a1*x+a2*x^2+a3*x^3
  13. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
  14. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
  15. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
  16. sqrt(y)=a0+a1*x+a2*x^2
  17. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3
  18. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
  19. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
  20. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
  21. ln(y)=a0+a1*x+a2*x^2
  22. ln(y)=a0+a1*x+a2*x^2+a3*x^3
  23. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
  24. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
  25. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6

Notes:

  • Equations 11-20, 56-60, 81-85: 1/y=f(x) corresponds to fitting data to y=1/(f(x))
    • For example, 1/y = a0+a1ln(x) is the linearized version of the nonlinear equation y = 1/(a0+a1exp(x)).
  • Equations 11-20, 56-60, 81-85: 1/y=f(x) corresponds to fitting data to y=1/(f(x))
    • For example, 1/y = a0+a1ln(x) is the linearized version of the nonlinear equation y = 1/(a0+a1exp(x)).
  • Equations 21-30, 61-65, 86-90: y^2=f(x) corresponds to fitting data to y = sqrt(f(x))
    • For example, y^2 = a0+a1x^2 is the linearized version of the nonlinear equation y = sqrt(a0+a1x^2).
  • Equations 31-40, 66-70, 91-95: sqrt(y)=f(x) corresponds to fitting data to y=(f(x))^2
    • For example, sqrt(y) = a0+a1x+a2x^2+a3x^3 is the linearized version of the nonlinear fitting equation y = (a0+a1x+a2x^2+a3x^3)^2
  • Equations 41-50, 71-75, 96-100: ln(y)=f(x) corresponds to fitting data to y = exp(f(x))
    • For example, ln(y) = a0+a1x is the linearized form of nonlinear equation y = exp(a0+a1x)
    • Note this corresponds to the familiar exponential equation y = a0′exp(a1′x) where a0′ = exp(a0) and a1′ = a1.

Nonlinear Equations

Standard Functions

Power
y=a0+a1*x^a2

Exponential
y=a0+a1*exp(-x/a2)

Peak Functions

Gaussian
y=a0+a1exp(-0.5((x–a2)/a3)^2)

Lorentzian
y=a0+a1/(1+((x–a2)/a3)^2)

Logistic Peak
y=a0+a14n/(1+n)^2 n = exp(-(x–a2)/a3)

Erfc Peak
y=a0+a1*erfc(((x–a2)/a3)^2)

Log-Normal
y=a0+a1exp(-0.5(ln(x/a2)/a3)^2)

Transition Functions

Sigmoidal
y=a0+a1/(1+exp(-(x–a2)/a3))

Cumulative
y=a0+a10.5(1+erf((x–a2)/(sqrt(2)*a3)))

DoseRspLgstc
y=a0+a1/(1+(x/a2)^a3)

pH Activity
y=(a0+a1*10^(x–a2))/(1+10^(x–a2))

1-Site Ligand
y=a0*x/(a1+x)

Photosynthesis Rate
y=a0a1x/(a0+a1*x)

Waveform Functions

SineWave
y=a0+a1sin(2pi*x/a3+a2)

SineWaveSquared
y=a0+a1(sin(2piπ*x/a3+a2))^2

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