# Functional Principal Component Analysis for Extrapolating Multi-stream Longitudinal Data

## TL;DR

This paper introduces a non-parametric functional principal component analysis method for predicting multi-stream longitudinal data by leveraging historical data and real-time updates, outperforming existing approaches.

## Contribution

It presents a novel Gaussian process-based framework that combines functional PCA with Bayesian updating for accurate real-time predictions of multi-stream data.

## Key findings

- Outperforms state-of-the-art methods in synthetic and real data
- Effectively captures heterogeneity among units
- Achieves high predictive accuracy with real-time data

## Abstract

The advance of modern sensor technologies enables collection of multi-stream longitudinal data where multiple signals from different units are collected in real-time. In this article, we present a non-parametric approach to predict the evolution of multi-stream longitudinal data for an in-service unit through borrowing strength from other historical units. Our approach first decomposes each stream into a linear combination of eigenfunctions and their corresponding functional principal component (FPC) scores. A Gaussian process prior for the FPC scores is then established based on a functional semi-metric that measures similarities between streams of historical units and the in-service unit. Finally, an empirical Bayesian updating strategy is derived to update the established prior using real-time stream data obtained from the in-service unit. Experiments on synthetic and real world data show that the proposed framework outperforms state-of-the-art approaches and can effectively account for heterogeneity as well as achieve high predictive accuracy.