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.

Parameterizations can be smooth functions or piecewise smooth (Du, 2019):

- 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:

## References

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.