# Functional principal component analysis models

### Why does Kiuse FPCA models?

One goal of Ki is to discover phenotypic variation in growth trajectories. Functional Principal Component Analysis (FPCA) is a method for investigating the dominant modes of variation in functional data. In the context of growth modeling, FPCA uses non-parametric functions to characterize both the population mean growth trajectory and a collection of uncorrelated functions that describe principal directions of subject-specific deviation from the mean growth trajectory. For example, principal components could identify higher or lower initial growth velocities, or earlier or later growth deceleration patterns. By using a non-parametric approach, FPCA doesn’t impose a functional form on the data. Instead, FPCA allows the data to determine common axes of variations in growth trajectories.

### WHAT IS FUNCTIONAL DATA ANALYSIS?

Information about curves, surfaces, or anything else that varies over a continuum, such as growth over time is called functional data.

Time series data are an example of functional data.[2] Functional data are often characterized by:

• High-frequency measurements
• An underlying “smooth” but complex process
• Repeated observations
• Multi-dimensionality

Functional data analysis (FDA) is a set of statistical tools that enables a more accurate summary and analysis of these types of data.[3]

### WHAT IS FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS?

Principal Component Analysis (PCA) is a statistical procedure used to investigate and characterize dominant modes of variation in multivariate data, called principal components, or principal modes of variation.[3] PCA is used across disciplines as a form of dimensionality reduction.[3]

Analogously, Functional Principal Component Analysis (FPCA) is a method for investigating and characterizing the dominant modes of variation in functional data.

The visualization below (Figure 1) shows an example of FPCA inputs and output.[1]

###### FIGURE 1. An example of FPCA taken from Zhang et al.[1] In the top row of figures, the lighter curves show individually fit growth curves for height-for-age z-score (HAZ) from birth to 2-years. The mean curves are shown in the bolded curve. In contrast, the bottom figures show the two leading functional principal components of HAZ. One can see that the cross-sectional mean provides a poor summary of these HAZ curves compared to FPCA.

Flexible, data-driven approach for modeling growth trajectories and charactering patterns of variation in growth without imposing parametric functional form.[1, 4]

• Best suited for data measured at a high frequency, although can be used with sparsely collected measures.
• Can be difficult to interpret non-parametric functions which characterize functional variation.
• Current methods do not easily allow for inclusion of covariate effects.
• Modeling and simulation of trajectories beyond the range of observed data, with respect to time, is not advisable.

### KIUTILIZATION OF FPCA MODEL TO DESCRIBE

#### Longitudinal LAZ measures

Semi-parametric model to describe growth length for age z-score (LAZ), where the only predictor is time (age).

Model is fit separately for each study.

#### Longitudinal growth measures

Semi-parametric model to separately describe length, weight, and head circumference for ages 0-1 year, where the only predictor is time (age).

#### Longitudinal fetal growth measures

Describes fetal growth trajectories for all main ultrasound measures for gestational ages 14-43 weeks, where the only predictor is time (age).