Least Squares - Considerations

Thomas J. Kennedy

Contents:

So far… we have discussed two examples for Discrete Least Squares Approximation:

Linear - $\mathbb{P}^1$ - Polynomial of Degree 1
Quadratic - $\mathbb{P}^2$ - Polynomial of Degree 2

1 Behavior of Data

For these examples… I have selected linear or quadratic approximation functions seemingly arbitrarily. However, selecting an appropriate type of approximation function is more nuanced. In general we need to:

plot the data
understand the behavior of the data
sometimes split the domain

Our focus is on the methodology of computing approximation functions (i.e., the method itself). While we will discuss how to measure error… the selection of appropriate basis functions will not be a point of emphasis.