If you are given a set of data you might want to fit a curve to the data.

Suppose you want to fit a curve of the form

(*)

to the data. Here is the output, is the input and & are functions/formulae in terms of the input .Finally and are constants.

We define the curve of best fit as the one that minimises the sum of the squared errors:

.

Each pair of values gives rise to one curve of the form (*). Each pair of values gives rise to a sum of squared errors . Therefore we can use the calculus of two variables to find the values of and such that is minimised.

It turns out that the curve of best fit is found by solving the simultaneous equations

where the sum is over the data points . We call these the normal equations for the curve(s) .

These equations are found by starting at the curve

multiplying across by the multiple of and the multiple of and summing:

For the example of fitting a line

so we multiply everything by and sum and we also multiply everything by the multiple of : one; and sum:

So yes the normal equations on page 101 belong with the normal equations on page 102. Remember though that you have

curve normal equations for that curve

Now to do a sum of the form . Suppose we call the constant by . We want to find

.

Now the sum is over the data points so really what we have is

This means “add up the pattern from up to “… add up s:

.

Regards,

J.P.

I am going over the notes and have a question, On p101 the formula’s at the top of the page which are called normal equations, what is the relationship between them and are they a part of the normal equations on p102 part 3.2.2 and on page 104 where does the sum of C be got from?

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