Module pda 

Linear differential operators and principal differential analysis.

This module provides two complementary tools for working with linear ordinary differential equations (ODEs) in functional data analysis:

• Lfd: A linear differential operator that can be applied to a set of observed curves, forming Lx = D^m x + β_{m-1}(t)·D^{m-1}x + … + β₀(t)·x.
• principal_differential_analysis: Estimates the coefficient functions of a linear ODE from a collection of observed solution curves.

Relationship to the R fda package

The Lfd struct corresponds to the Lfd object in the R fda package (see https://rdrr.io/cran/fda/man/Lfd.html). The PDA estimator corresponds to pda.fd and recovers the weight functions β₀(t), …, β_{m-1}(t) of the order-m ODE by independent least squares at each grid point (see https://arxiv.org/abs/2406.18484).

Examples

use fdars_core::pda::{Lfd, PdaResult, principal_differential_analysis};
use fdars_core::matrix::FdMatrix;
use std::f64::consts::PI;

// Build a harmonic-oscillator dataset: x_i(t) = cos(2π t)
let omega = 2.0 * PI;
let n_pts = 101;
let argvals: Vec<f64> = (0..n_pts).map(|i| i as f64 / (n_pts - 1) as f64).collect();
let n_curves = 5;
let mut data = FdMatrix::zeros(n_curves, n_pts);
for i in 0..n_curves {
    let a = (i + 1) as f64;
    for (j, &t) in argvals.iter().enumerate() {
        data[(i, j)] = a * (omega * t).cos();
    }
}

// Apply the identity-like Lfd (m=1, β₀ = 0) to verify shape invariance.
let lfd = Lfd { coefs: vec![vec![0.0]] };
let lx = lfd.apply(&data, &argvals).unwrap();
assert_eq!(lx.shape(), data.shape());

Structs

Lfd

A linear differential operator of the form Lx(t) = D^m x(t) + β_{m-1}(t)·D^{m-1}x(t) + … + β₀(t)·x(t).

PdaResult

Result of principal_differential_analysis.

Functions

principal_differential_analysis

Estimate the coefficient functions of a linear ODE from observed solution curves.