Fitting Nonlinear Data in the Absence of a Physical Model Function

Introduction

The purpose of this lab was to linearize the data and determine the parameters of unknown number 35 by linearizing six two-parameter functional relationships and determining which of them best fit the data.  Unknown number 35 was entered into Microsoft Excel and plotted. The functional relationship between X and Y variables of the unknown were investigated by linearizing six common two-parameter functional relationships. Equation 1 is the basic linear form that the functions were put in.

y=mx+b (1) 

The slope (m) of the linear form of the function allowed for the determination of the a and c values.

Procedure

Unknown number 35 data was entered into Microsoft Excel and was plotted. There were six common two-parameter functions given to be tested, one of which would fit the data. The six functions were linearized and the data plotted accordingly. From the linear plot the values of the a and c parameters were determined from the slope and intercept of the best-fit line. The values of the parameter were substituted into the appropriate function and a curve was calculated by the data set.

Detailed procedures can be found in reference 1.

Results 

The raw x and y data of unknown number 35 was can be found in Figure 1.

Figure 1. Raw data of unknown 35. 

Table 1 shows six trial functions in both linear and nonlinear form, along with how the data was plotted, and values of a and c as calculated from the slope of mx+b.

Table 1. Trial functions in linear and nonlinear form, how they were plotted, and their a and c values.

Nonlinear Form	Linear Form	Plot	a	c
y=ae±cx	lny=±cx+lna	x vs. lny	eb	m
y=acx	lny=lna+xlnc	x vs. lny	eb	em
y=axc	lny=lna+clnx	lnx vs. lny	eb	em
y=a+cx2	y=a+cx2	x2 vs. y	1/m	b(1/m)
y=a/(c+x)	1/y=(a/c)+(x/a)	x vs. 1/y	1/m	b/a
y=ax/(1+cx)	1/y=1/ax+c/a	1/x vs. 1/y	1/m2x	b/c

The function y=a/(c+x) was found to best linearize the data in Figure 1. The linearized plot is shown in Figure 2, where the data plotted were the original x values versus the inverse of the y values.

Figure 2. The unknown 35 data plotted in linear form.

From the equation of the best-fit line the parameter a was calculated to be 0.9051 and c was calculated to be 0.2170. The values of a and c were used to calculate a set of y values. These were then plotted with the original y values, which is shown in Figure 3.

Figure 3. The original unknown number 35 data (Series 1) plotted with the calculated y values (Series 2).

Discussion

The function that best linearized the data was y=a/(c+x). The parameter a was calculated to be 0.9051 and the parameter c was calculated to be 0.2170. The calculated y values were consistent with the original y values when plotted in Figure 2.

References

1. General Chemistry Experiments: A Manual for Chemistry 204, 205, and 206, Department of Chemistry, Southern Oregon University: Ashland, OR, 2010