# Optimal discretization and expansion order of arbitrary data

• laxsu19
In summary, the conversation is about finding a solution for splitting a large set of data points and calculating a functional expansion within each subdomain in an optimal manner. The topic of math involved is function approximation, specifically approximation by rational functions and automatic knot placement for spline curve fitting. The goal is to minimize the number of subdomains and expansion orders within each subdomain.
laxsu19
Hi all,

I am trying to figure out 1) What to call my problem so I can better research the literature, and 2) see if anyone here knows of a solution.

Essentially, I have a large set of f(x) vs x points (~20,000) which I need to split into subdomains in x, and within each subdomain calculate a functional expansion of f(x). I want to do this in an optimal manner such that 1) the number of subdomains is minimized - or at least manageable, and 2) the number of expansion orders (probably Legendre) within each subdomain is also minimized.

Does anyone have any idea what 'field' of math this could be considered, and where to begin searching around? Unfortunately, this is just a minor step in what I have to do so I don't want to expend much effort here.

Does your data contain "noise" or is the data simply known values of some precisely defined function? If your data is known values of a precisely defined function, then the general topic to research is "function approximation". For many functions, the simplest approximations (for a given mean square error) are done by using ratios of polynomials. That topic is "approximation by rational functions".

If you can fit each subdomain by a low order polynomial, some buzzwords are automatic knot placement for spline curve fitting. (The "knots" are the points at the end of each subdomain, i.e. the end of each spline segment).

## 1. What is optimal discretization?

Optimal discretization is the process of dividing continuous data into discrete intervals or values in a way that minimizes error and maximizes accuracy.

## 2. How is optimal discretization determined?

Optimal discretization is determined by evaluating the trade-off between discretization error and model complexity. This can be done through various techniques, such as cross-validation or information-theoretic approaches.

## 3. What is the role of expansion order in optimal discretization?

The expansion order, also known as the number of basis functions used, plays an important role in optimal discretization. It determines the complexity of the model and can affect the trade-off between error and complexity. A higher expansion order may result in better accuracy but also increase the risk of overfitting.

## 4. Is there a universal optimal discretization and expansion order for all types of data?

No, the optimal discretization and expansion order may vary depending on the type of data and the specific problem being addressed. It is important to choose the most appropriate approach based on the characteristics of the data and the goals of the analysis.

## 5. What are the benefits of using optimal discretization and expansion order?

Optimal discretization and expansion order can lead to more accurate and reliable results in data analysis. It can also help in reducing the complexity of the model and avoiding overfitting. Additionally, it allows for the identification of important features and relationships within the data.

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