Vernier Tech Info Library TIL #1675

Question

How do LabQuest Curve fits work?

Answer

Labquest (version 1.0.0) has five curve fit forms, whose coefficients are computed in one of two ways.

*** Linear form ***
Linear: y = mx +b
The m and x coefficients of the Linear form are computed using the linear fit algorithm taken from "An Introduction to Error Analysis" (pp 156-157) It is an implementation of the least squares approach. The root mean square error (RMSE,) uncertainties for each coefficient and the correlation are also computed. The correlation calculation is taken from the same book (p 180.)

*** Non-linear forms ***
Proportional: y = Ax
Quadratic: y = Ax^2+Bx+C
Power: y = Ax^B
Natural Exponent: y = Aexp(-Cx)+B

The coefficients of the non-linear forms are computed using the iterative, best-fit values approach of the Levenburg-Marquadt method taken from "Numerical Recipes in C" (pp 681-688.) The root mean square error (RMSE) and uncertainties for each coefficient are also computed.

*** Natural Exponent starting coefficient adjustment ***
The starting coefficients for the Natural Exponent are adjusted prior to finding the best-fit coefficients. The adjustment makes a few assumptions and creates some ranged, random coefficients. Therefore, multiple fit attempts can produce different, best-fit coefficient results.

*** Details on the adjustment of the Natural Exponent coefficients ***
Form: A * exp (-Cx) +B
Assumption: The x column contains ascending values.

Let:
x = the list of x-column values (where x[0] is the first value and x[n] the last)
y = the list of y-column values
x_range = | x[n] - x[0] | (absolute range)

If x_range >10; then:
x_range = 1/x_range

The C coefficient is very roughly inversely proportional to the x_range. The goal is to start with a reasonable value in the exponent, based on the x_range value.
C = random value between 0 and +\- x_range. The sign is based on another assumption. If x[n] > x[0], it is assumed positive, else negative. This coefficient is the most critical in terms of providing the curve fit algorithm a good starting point for reaching a best-fit.

If the C coefficient is assumed to be a positive value, the exponential term is negative. So as x->infinity: exp (-Cx)->0 (zero); Therefore: as x->infinity: y = B
B = y[n] (for positive C)
B = y[0] (for negative C)

An assumption about the sign of the A coefficient is difficult to make. However, we can give it a range. This makes a further assumption that there is indeed an x = 0 value, which provides a y-intercept. At x = 0: y = A+B
A = random value between (y[n]-y[0]) and (y[[0]+y[n])

Created by: ihonohan on August 29 2007
Last updated by: ihonohan on September 15 2008