#### CS3414 Afterclass Notes --- 24 May, 2002

**Fitting Data** (parts of Chapters 4, 7, 10)
- Introduction

- Idea:
given a discrete set of data points
(x
_{i},y_{i}), for i = 1, 2, ..., m;
find a function g(x) that either:
- matches exactly (
*interpolates*) the given data, or
- approximates the given date in some sense (e.g., in
the
*least squares* sense).

- Method of undetermined coefficients: a generic approach to
finding g(x) that satisfies certain properties (like interpolation).

- Choose a set of basis functions to be used to represent g(x).
We will write g(x) as a linear combination of these basis
functions, i.e.,

g(x) = a_{1} b_{1}(x) +
a_{2} b_{2}(x) + ... +
a_{n} b_{n}(x)

- Enforce n constraints or conditions that you want g(x) to
satisfy. The goal is to derive n linear equations in the
n unknowns a
_{1}, ..., a_{n}. For example,
in the case of interpolation, we would want n=m and
g(x_{i}) = y_{i}, for i = 1, ..., n.

- What's left to talk about ?
- What are good choices for the basis functions (with respect
to cost/efficiency and accuracy)?
- Which is more important---cost to construct g(x) or cost to
evaluate g(x)?
- Is it easy to add new data points to my model?
- What if I only want to approximately match the data?