Some Equations for Curve Fitting
This appendix describes a number of equation types that can be used for curve fitting. Some of the equation types can be handled by Excel's Trendline utility for charts; these cases are noted below.
Multiple Regression. Multiple regression fits data to a model that defines y as a function of two or more independent x variables. For example, you might want to fit the yield of a biological fermentation product as a function of temperature (7), pressure of C02 gas (P) in the fermenter and fermentation time using data from a series of fermentation experiments with different conditions of temperature, pressure and time.
Since you can't create a chart with three jcaxes (e.g., T, P and t), you can't use Trendline for multiple regression.
Polynomial Regression. Polynomial regression fits data to a power series such as equation A42:
Figure A41. Polynomial of order 3. The curve follows equation A42 with a = 5,b = \,c = 5 and d= 1.
The Trendline type is Polynomial. The highestorder polynomial that Trendline can use as a fitting function is a regular polynomial of order six, i.e., y = ax6 + bx5 +cx4 + dx + ex2 +fx + g.
LI NEST is not limited to order six, and LI NEST can also fit data using other n n ja i «
polynomials such as y = ax + bx + cx + dx + e. Exponential Decrease.
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Figure A42. Exponential decrease to zero. The curve follows equation A43 with a = 0.1 and b = 0.5. The Trendline equation is shown on the chart.
Data with the behavior shown in Figure A42 can be fitted by the exponential equation y = aeb* (A43)
The sign of b is often negative (as in radioactive decay), giving rise to the decreasing behavior shown in Figure A42.
The linearized form of the equation is In y = bx + In a; the Trendline type is Exponential.
Exponential Growth. If the sign of b in equation A43 is positive, the curvature is upwards, as in Figure A43.
Figure A43. Exponential increase. The curve follows equation A43 with a = 0.1 and b = 0.5. The Trendline equation is shown on the chart.
Figure A43. Exponential increase. The curve follows equation A43 with a = 0.1 and b = 0.5. The Trendline equation is shown on the chart.
Exponential Decrease or Increase Between Limits. If the curve decreases exponentially to a nonzero limit, or rises exponentially to a limiting value as in Figure A44, the form of the equation is y  aebx + c (A44)
Excel's Trendline cannot handle data of this type.
Figure A44. Exponential increase to a limit. The curve follows equation A44 with a = \,b = 0.5 and c = 1.
Figure A44. Exponential increase to a limit. The curve follows equation A44 with a = \,b = 0.5 and c = 1.
The linearized form of the equation is In (y  c) = bx + In a.
Double Exponential Decay to Zero. The sum of two exponentials (equation A45) gives rise to behavior similar to that shown in Figure A45. This type of behavior is observed, for example, in the radioactive decay of a mixture of two nuclides with different halflives, one shortlived and the other relatively longerlived.
Figure A45. Double exponential decay. The curve follows equation A45 with a = 1, b = 2, c = 1 and d = 0.2.
Figure A45. Double exponential decay. The curve follows equation A45 with a = 1, b = 2, c = 1 and d = 0.2.
If the second term is subtracted rather than added, a variety of curve shapes are possible. Figures A46 and A47 illustrate two of the possible behaviors.
Figure A46. Double exponential decay. The curve follows equation A45 with a = 1, b = 0.2, c = 2 and d = 2.
Figure A47. Double exponential decay. The curve follows equation A45 with a = 1, 6 = 2, c = 1 and d = 0.2.
Equation A45 is intrinsically nonlinear (cannot be converted into a linear form).
Power. Data with the behavior shown in Figure A48 can be fitted by equation A46.
Figure A48. Power curve.
The curve follows equation A46 with a = 1.1, b =0.5. The Trendline equation is shown on the chart.
The linearized form of equation A46 is In y = b In x + In a; the Trendline form is Power.
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