Probability in Physics by Andy Lawrence

Author:Andy Lawrence
Language: eng
Format: epub
ISBN: 9783030045449
Publisher: Springer International Publishing

Each of the terms is simply a number made of a sum over the data values and their errors. The equations now read

The standard way to solve such a pair of linear equations is by the method of determinants. The solution is

The procedure then is that first we combine the data values to form the sums , then we combine those values with the equations above, and this gives us the “least squares” solution for the parameters a and b.

9.10 Testing the Fit Using

The procedure above gives the best fit, but is the fit statistically acceptable? To test the absolute quality of the fit, we need to have values for the errors on each data point, , so that we can calculate , as in equation (9.7). Because we have estimated two parameters from the data, the remaining degrees of freedom is . We can assign a probability to the fit, , based on the distribution for that value of , and then possibly reject the fit if or whatever, depending on our chosen confidence level. As with our previous use of , it is standard for fit tests to consider only a one-tailed test—a bad fit will always give a large . If the model is wrong, this will not make the data points cluster nearer to the regression line than they ought to.

A value that is unexpectedly large may occur (i) simply by chance, (ii) because the model is wrong, or (iii) because the errors have been underestimated. The third possibility should always be considered!

A value that is unexpectedly small may occur (i) simply by chance, (ii) because the errors have been overestimated. Although the more common failing is underestimating your errors, overestimating errors does happen more often than you might think...

Fig. 9.6Regression analysis on three simulated datasets. In each case the middle line shows the true input relationship, and the other two lines show the result from regressing y on x and x on y respectively. See text for further details

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