Functional Data Analysis
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Definition
Functional data is multivariate data with an ordering on the dimensions.
Key assumption: smoothness
\[y_{ij} = x_i(t_{ij}) + \epsilon_{ij}\]with $t$ in a continuum (usually time), and $x_i(t)$ smooth.
Functional data = the functions $x_i(t)$.
Interests
- Representations of distribution of functions
- mean
- variation
- covariation
- Relationships of functional data to
- covariates
- responses
- other functions
- Relationships between derivatives of functions
- Timing of events in functions