In order to describe a nonparametric function or use it for estimation, you first need to approximate it with a parametric function (or set of functions) — a process called parameterization (Sun & Sun, 2015).
In calculus, you can only work with functions: equations that have one unique input matched with each output. If your curve, surface, or other construct doesn’t meet the requirements of being a function, you can estimate the shape using a known function or set of functions. The zigzag shape in the above image is an example: The zigzag isn’t a function and we can’t work with it. But the curve on the right is a function, and we can use that curve to estimate the behavior of the zigzag.
- Smooth parameterizations have continuous first derivatives on an interval (with the exception of the endpoints),
- Piecewise smooth means that each subinterval is smooth.
How to Parameterize a Function
The answer to this question isn’t simple, because there are an infinite number of ways to parameterize any particular function (Du 2019).
The number of parameters is the number of “free variables.”
- Just one parameter is needed to parameterize a curve,
- Two parameters are needed to parameterize a two-dimensional surface,
- Three parameters are needed for solids.
A circle, which cannot be expressed as a single function, can be split into two curves. Each curve can be parameterized by either a sine function or cosine function (or possibly other trigonometric functions). Watch this short video on how to parameterize a curve for a quick example:
Du, X. (2019). Parametric Equations and Vectors. Retrieved August 21, 2020 from: https://www.andrew.cmu.edu/user/xiongfed/notes/1.15%20Parametric%20Equations%20and%20Vectors.pdf
Rogawski, J. (2007). Multivariable Calculus. W.H. Freeman.
Sun, N. & Sun, A. (2015). Model Calibration and Parameter Estimation. Springer New York.