.. _bda-functional-data-analysis:

========================================================================
Functional data analysis
========================================================================

**Part 5 · Stage 15 · 🌊 Basis Functions & Gaussian Processes**  ·  Lesson 132 of 144  ·  *advanced*

:doc:`◀ Previous · Latent Gaussian process models <131-latent-gaussian-process-models>`   ·   :doc:`Next · Density estimation and regression ▶ <133-density-estimation-and-regression>`   ·   :doc:`↑ Section <index>`


.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
   AI language model and is provided as a learning aid. Despite careful
   review, it may still contain mistakes, omissions, or out-of-date
   information. Whether you are new to the topic, a team lead, or a senior
   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
   (code, commands, figures, and factual claims) against official
   documentation and primary sources before relying on it.

When each observation is a curve
----------------------------------

Sometimes a single data point is an entire **function**: a growth curve for one child, a daily
temperature profile, a spectrum, a gait cycle. **Functional data analysis** treats these curves as the
unit of analysis, and Bayesian methods bring to it what they bring everywhere — a generative model with
propagated uncertainty over the functions themselves.

Representing functions
------------------------

The observed curves are noisy, irregularly sampled realisations of underlying smooth functions, so the
first step is a **smooth representation**. Two routes, both from this stage: expand each curve in a
**basis** and model its coefficients, or model each curve as a draw from a **Gaussian process**. Either
turns an infinite-dimensional object into something estimable, with the smoothing priors of the earlier
lessons controlling how rough each curve may be.

.. math::

   y_{ij} = f_i(t_{ij}) + \epsilon_{ij}, \qquad
   f_i \sim \mathcal{GP}(\mu, k) \;\;\text{or}\;\; f_i(t) = \sum_k \beta_{ik} B_k(t),

with a **hierarchical** structure across curves: the :math:`f_i` share a common mean function and
covariance, so each individual curve borrows strength from the population — the batching idea, applied to
whole functions.

Analysing the functions
-------------------------

Once represented, the functions become objects to explore. **Functional principal components** find the
dominant modes of variation — the few characteristic ways the curves differ from their mean, a
dimension reduction on function space. Curves can be **registered** (aligned in time to separate
variation in *shape* from variation in *timing*), regressed on covariates (**function-on-scalar**), or
used as predictors (**scalar-on-function**).

.. code-block:: python

   import pymc as pm
   # hierarchical functional model: each curve a GP deviation around a shared mean
   with pm.Model():
       mean_coef = pm.Normal("mean_coef", 0, 1, shape=K)         # population mean curve
       tau = pm.HalfNormal("tau", 1)
       dev = pm.Normal("dev", 0, tau, shape=(n_curves, K))       # per-curve deviations, pooled
       f = (mean_coef + dev) @ B.T                               # individual smooth curves
       pm.Normal("y", f[curve_id, t_idx], pm.HalfNormal("s", 1), observed=y)

Where it sits
---------------

Functional data analysis is the natural **culmination** of the flexible-modelling stage: it combines
basis expansions and Gaussian processes (to represent curves), hierarchical models (to pool across
curves), and dimension reduction (to summarise them) into one framework for data whose fundamental unit
is a function. It closes Part V's parametric-nonlinear thread — from mechanistic ODE models through
splines and Gaussian processes to whole-function data — and hands off to the final stage, where the
flexibility comes not from smooth functions but from **mixtures** and infinite-dimensional
**nonparametric** priors.

.. hint::

   **Related lessons:** :doc:`Gaussian process regression <129-gaussian-process-regression>`  ·  :doc:`Basis selection and shrinkage of coefficients <127-basis-selection-and-shrinkage-of-coefficients>`  ·  :doc:`Hierarchical models for batches of variance components <105-hierarchical-models-for-batches-of-variance-components>`  ·  :doc:`Latent Gaussian process models <131-latent-gaussian-process-models>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/09/functional-data-analysis/ <https://insightful-data-lab.com/2025/12/09/functional-data-analysis/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: advanced