An Illustrative Guide to Multivariable and Vector Calculus by Stanley J. Miklavcic

Author:Stanley J. Miklavcic
Language: eng
Format: epub, pdf
ISBN: 9783030334598
Publisher: Springer International Publishing

Fitting a straight line to observations

Suppose we believe that a variable y is in some way dependent on a variable x by the relation . In situations such as this it is implicitly assumed that the independent variable is deterministic, that is, given, and not subject to error, while the dependent variable is subject to observation or measurement errors.

To find the dependency relationship, we select a sequence of values of the independent variable, and measure the corresponding values of the dependent variable.

Because observed measurements always have some error associated with them, the observations won’t necessarily lie exactly on the straight line, but may fall above or below the line, as in Figure 3.10 above.

Problem: How to determine the “line of best fit” through the “noisy” discrete experimental observations. (The values and become estimates of the true parameters k and b in the relation which we believe connects x with y.)

Solution: We choose the line parameters k and b so that the sum of the squares of the differences between the observations and the fitted values is a minimum.

That is, we construct the function of k and b from the known data and the desired model and seek its minimum to give us the optimal k and b values. We apply the techniques of the preceding chapter to get the critical points of S by solving the two equations

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