Programming Exercise Eight (and last)
Objective
This assignment provides an example in the use of one-dimensional arrays and introduces the
concept of regression analysis, which is used to estimate a relationship between two variables.
Mathematical Background
If several measurements are made on pairs of experimental data {(xi,yi), i = 1,...,N}, we can use a
technique, known as regression analysis, to determine an approximate equation of a straight line
that gives a best fit to the data. The equation of this best-fit line is written as follows.
^
y = a + b x
In this equation, we use the symbol ^
y instead of y to indicate that the predicted value found from
the equation, ^
y = a + b x is an approximate result. For a given data point, (xi,yi), the value of yi
represents the actual data and we would obtain the predicted value of y, at the point x = xi from
the equation ^
yi = a + b xi. The difference between the measured and predicted value is |yi - ^
yi|.
Fitted Line
y
x
indicates data points
y
i
x
i
i
y
ˆ
In the chart at the left, the data points are indicated by
the small ellipses. The coordinates of one of a typical
data point are shown by the dotted lines indicating the
coordinates xi and yi. The solid line is the fitted
regression line, ^
y = a + b x. The point where the dotted
line at x = xi crosses the regression line has the
coordinates (xi,^
yi). In this particular example the value of
^
yi is less than the value of yi. There is a large scatter of
data points about the regression line in this example.
The example plot above might represent calibration data on an instrument. The x values would
denote the instrument reading and the y values would indicate the true value of the quantity being
measured. Once the calibration tests were completed, it would be useful to have a simple
equation to relate the instrument reading (x) to the actual quantity being measured(y).
In addition to finding the values of a and b that give the best-fit line, we would also like to have
some measure of how well the line fits the data. Two different goodness-of-fit measures, the
standard error and the coefficient of variation are presented below in the equations section.
Equations used
The equations used to calculate a and b can be found by an analysis which minimizes the
distances between the actual data points, yi, and the fitted points, ^
yi = a + b xi. The results of this
analysis are shown below. The equations to compute the intercept, a, and the slope, b, in terms of
the entire set of data, {xi,yi}, use the following the definitions of mean values:
N
ii
N
iix
N
xandy
N
y
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