---
tag: KEARMKNZ1Q
kind: concept
name: Functional data analysis
msc: [62R10, 62H25]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_data_analysis
same_as:
  wikidata: Q5508814
  wikipedia: Functional data analysis
---

## Summary

Functional data analysis is the branch of statistics in which each observation is a whole function, such as a curve, a surface or an image.
The functions are modelled as random elements of an infinite-dimensional space, usually the [[Hilbert space|HM2HHE3J51]] $L^2$ of square-integrable functions.
Its basic tools are the mean function, the covariance operator and functional [[principal component analysis|ABXD13M340]], which rests on the [[Karhunen–Loève theorem|5CN7KY8MWP]].
It also has regression models in which the predictor or the response is a function.

## Explanations

### L1 human

Record the height of a hundred children at every age from birth to eighteen.
Each child gives a growth curve, and it is natural to treat each curve as one data point.
Functional data analysis computes the average curve, describes the main ways in which curves differ from it, and relates curves to other variables.
For example, the first principal component of growth curves may describe children who are taller at every age, and the second may describe the timing of the growth spurt.