Question 1.5.1: Fitting Data with the Power Function Find a functional descr...
Fitting Data with the Power Function
Find a functional description of the following data:
| x | 1 | 2 | 3 | 4 |
| y | 5.1 | 19.5 | 46 | 78 |
These data do not lie close to a straight line when plotted on linear or semilog axes. However, they do when plotted on log-log axes. Thus a power function y = b x^{m} can describe the data.
Using the transformations X = log x and Y = log y, we obtain the new data table:
| X = log x | 0 | 0.301 | 0.4771 | 0.6021 |
| Y = log y | 0.7076 | 1.2900 | 1.6628 | 1.8921 |
From this table we obtain
\sum_{i=1}^{4} {X_{i} = 1.3803} \sum_{i=1}^{4} {Y_{i} = 5.5525}
\sum_{i=1}^{4} {X_{i} Y_{i} = 2.3208} \sum_{i=1}^{4} {X^{2}_{i}} = 0.6807
Using X, Y , and B = log b instead of x, y, and b in (1.5.1) and (1.5.2) we obtain
m\sum\limits_{i=1}^{n}{x_i^2} + b \sum\limits_{i=1}^{n}{x_i} = \sum\limits_{i=1}^{n}{y_i x_i} (1.5.1)
m \sum\limits_{i=1}^{n}{x_i} +bn = \sum\limits_{i=1}^{n}{y_i } (1.5.2)
0.6807m + 1.3803B = 2.3208
1.3803m + 4B = 5.5525
The solution is m = 1.9802 and B = 0.7048. This gives b = 10^{B} = 5.068 . Thus, the desired function is y = 5.068 x^{1.9802}.