Fitting to a Proportionality

When data are assumed to fit a proportionality, only one parameter needs to be extracted from the data namely the slope. $$ y = mx$$ There are two methods to do this analytically: one is just a weighted average of the slopes derived from each data point, and the second is using the least squares (maximum likelihood) methodology.

Weighted average

The pairs of points to be fit are $(x_i,y_i)$ where $i = 1,...,N$. Here we will treat the case where only the uncertainties in $y_i$ are significant and the uncertainty in the ith $y$ value is $\delta y_i$. Each ratio $m_i = \frac{y_i}{x_i}$ is an estimate of the proportionality constant. The relative error of $m_i$ is equal to the relative error in $y_i$ because we are here assuming that any uncertainty in $x_i$ is negligible. Thus the error in each slope estimate $\delta m_i$ can be written as: $$\delta m_i = \frac{y_i}{x_i}\frac{\delta y_i}{y_i} = \frac{\delta y_i}{x_i}$$ To determine the best value of the proportionality constant, $m$, that is consistant with the data one must average the estimates derived from each data point weighted by the inverse of the squared uncertainty: $$ m = \frac{ \displaystyle\sum_{i=1}^N \frac{y_i}{x_i}\frac{x_i^2}{\delta y_i^2} }{\displaystyle\sum_{i=1}^N \frac{x_i^2}{\delta y_i^2}}= \frac{ \displaystyle\sum_{i=1}^N \frac{x_i y_i}{\delta y_i^2} }{\displaystyle\sum_{i=1}^N \frac{x_i^2}{\delta y_i^2}}$$ If all the $\delta y_i$ are the same then this becomes $$ m = \frac{ \displaystyle\sum_{i=1}^N {x_i y_i} }{\displaystyle\sum_{i=1}^N{x_i^2}}$$

Least Squares Approach

This method finds the value of $m$ which gives a minimum of the reduced $\chi_r^2$. $$\chi^2_r = \displaystyle\sum_{i=1}^N \frac{(y_i - m x_i)^2}{\delta y_i^2}$$ The reduced $\chi_r^2$ quantifies how much the fitted function misses the experimental points in comparison with the uncertainties. In the first instance we will assume that all the $\delta y_i$s are the same. Therefore, minimizing the reduced $\chi_r^2$ is equivalent to minimizing the $\chi^2$. Differentiate the $\chi^2$, expand and solve for $m$. $$\frac{d\chi^2}{dm} = \frac{d}{dm} \left[ \displaystyle\sum_{i=1}^N (y_i^2 -2mx_iy_i + m^2x_i^2) \right] = 0$$ Performing the derivative $$ \displaystyle\sum_{i=1}^N (0-2 x_i y_i +2m x_i^2) = 0$$ Solving for $m$ gives the result we got in the last section: $$ m = \frac{ \displaystyle\sum_{i=1}^N {x_i y_i} }{\displaystyle\sum_{i=1}^N{x_i^2}}$$

---

Potatoes

Persons	Camp R	Camp S
	(lbs)	(lbs)
100	979.2	454.2
200	1686	905.4
300	2549	1360
400	3188	1790
500		2123
	Camp T
8	68.7
20	127.5
48	221.7
80	350.7
100	430.4
120	501.4

---

Return to Main Page
©2015 N. Alberding, Simon Fraser University.