FDA

Notes

What is functional data?

Functional data are data where each observation is a function, typically a smooth curve, surface, or anything that varies continuously over a domain such as time, space, or frequency.

key assumption of functional data is smoothness

Key assumption is smoothness:

yij = xi(tij) + ij with t in a continuum (usually time), and xi(t) smooth Functional data = the functions xi(t).

Neccessities for functional data? (5)

• must believably derive from a smooth process

• process should not be easily parameterizable (should not be able to write down a formula)

• enough data to resolve the essential features of the process (peaks, zero-crossings, speed... will depend on application)

• some repetition in the process

• do not need equally-spaced or perfect measurements

Longitudinal vs Functional data

• Because of the intense recording, functional data have been customarily modeled with a nonparametric approach.

  • Often smoothness of the functions is assumed.

• Longitudinal data have traditionally been modeled by a parametric approach, such as a linear mixed-effects model. However, it may not be easy to spot the pattern due to sparsity of and noise in the longitudinal data.

• Both longitudinal and functional data may be observed with noise (measurement errors).

• A strength of the FDA approach is its ability to handle noise.

Diescrete to Functional Data

• Allow evaluation of record at any time point (especially if observation times are not the same across records).

• Evaluate rates of change.

• Reduce noise.

• Allow registration onto a common time-scale.

From Diescrete to Functional Data

Basis Expansion

Fourier Basis Examples

• Φ(t) = (1,sin(ωt), cos(ωt))

• Φ(t) = (1,sin(ωt), cos(ωt),sin(2ωt), cos(2ωt),sin(3ωt), cos(3ωt))

• Φ(t) = (1,sin(ωt), cos(ωt), . . . ,sin(6ωt), cos(6ωt))

Fourier Basis Advantages

• Only alternative to monomial bases until the middle of the 20th century

• Excellent computational properties, especially if the observations are equally spaced.

• Natural for describing periodic data, such as the annual weather cycle

  BUT functions are periodic; this can be a problem if the data are, for example, growth curves. Fourier basis is still the first choice in many fields, such as signal analysis, even when the data are not periodic.

Splines

• Splines are polynomial segments joined end-to-end

• Segments are constrained to be smooth at the join

• The points at which the segments join are called knots

• The order m (order = degree+1) of the polynomial segments and

• the location of the knots define the system.

• Bsplines are a particularly useful means of incorporating the constraints.

Properties of B-Splines

• Number of basis functions:

order + number interior knots

• Derivatives up to m − 2 are continuous.

• B-spline basis functions are positive over at most m adjacent intervals → fast computation for even thousands of basis functions.

• Sum of all B-splines in a basis is always 1; can fit any polynomial of order m.

• Most popular choice is order 4, implying continuous second derivatives. Second derivatives have straight-line segments.

B-SplineS: Choosing knots and order

• The order of the spline should be at least k + 2 if you are interested in k derivatives.

• Knots are often equally spaced (a useful default)

• But there are two important rules:

  • Place more knots where you know there is strong curvature, and fewer where the function changes slowly.

  • Be sure there is at least one data point in every interval.

Other Basis

The fda library in R also allows the following bases:

• Constant φ(t) = 1, the simplest of all.

• Power t λ1 ,t λ2 ,t λ3 , . . ., powers are distinct but not necessarily integers or positive.

• Exponential e λ1t , e λ2t , e λ3t , . . .

Other possible bases include

• Wavelets especially for sharp, local features

• Empirical we will investigate functional Principal Components

• Designer see our section on dynamic models: tailoring a basis to data (if you know something about the data) can be much more efficient.

Choosing the Number of Basis Functions

Choosing the Number of Basis Functions Tradeoff

Trade off: Too many basis functions over-fits the data and reflect errors of measurement

Too few basis functions fails to capture interesting features of the curves.

Bias and Variance Trade-Off

Mean Squared Error

Cross Validation

Least Squares

Linear Regression on Basis Functions

Smoothing Penalties

What do we mean by Smoothness?

Some things are fairly clearly smooth:

• constants

• straight lines

What we really want to do is eliminate small “wiggles” in the data

The D Operator

The Roughness of Derivatives

The Smoothing Spline Theorem

Computing the Smoothing Spline

Calculating the Penalized Fit

More General Smoothing Penalties

A Very General Notion

Linear Smooths and Degrees of Freedom

Choosing the Smoothing Parameter

Generalized Cross Validation

Understanding the Distribution of Collections of Functions

Variance

Mechanics of PCA

Functional PCA

Re-Interpretation

Why Orthogonality?

PCA and Karhunen-Loève

Computing FPCA

Displays of PCA

Varimax Rotations

Defining New Inner Products

fPCA with Multivariate Functions

Smoothing and fPCA

Including Derivatives

A New Measure of Size

Size and Orthogonality

Scalar to Function: Identification

Scalar to Function: Smoothing

Scalar to Function: Calculating

Scalar to Function: Confidence Intervals

Multivariate and Mixed Functional Linear Regression

Multivariate and Mixed Functional Linear Regression Calculations

Principal Components Regression

Functional PCR

Problems of Inference

Permutation Tests

Diagramatic representation of permutation test

Functional Linear Regression and Permutation F-Tests

Permutation t-Tests

Max t

Functional Response Models

Response is a set of curves

yi(t) i = 1,..., n.

Covariates may be group labels scalar values functions

Partitioning Effects

Permutation Test

Functional Response Models – scalar covariate

Functional Covariates: Concurrent Linear Model

Mechanics

Smoothing and Confidence Intervals

Confidence Intervals

Functional Response Models in General

Estimating a Coefficient Function

Estimating B

Interpretation

Some Useful Restrictions