# Principal differential analysis with covariates: A simulation study on the effect of the smoothing parameters

#### Abstract

Principal Differential Analysis deals with functional data. The word functional data refers to a collection of curves that are independent and measured on a dense grid of time points in an interval. These time points can be equally or unequally spaced. A differential equation is believed capable of capturing the features of these *n* curves. ^ Ramsay(1996) first introduced Principal Differential Analysis (PDA) as an alternative to the Principal Component Analysis(PCA). PDA finds a linear differential equation that captures features of a collection of curves, in order to have a low dimensional approximation for functional data. PDA is based on the theorem: the span of the functions *f*_{ 1} ,…, *f _{m}* with

*m*derivatives has an annihilating linear differential operator(LDO) of the form

*L*=

*w*

_{0}

*I*+

*w*

_{1}

*D*+ ,…,

*w*

_{ m}_{−1}

*D*

^{m}^{ −1}+

*w*. ^ In PDA the coefficients

_{m}D^{m}*w*

_{0}, … ,

*w*of the LDO are estimated using data for a specified

_{m}*m*. The sum of squared norms of the residuals with penalty is used as the fitting criterion. Here the residual is that part of the data curve that is not annihilated by the LDO. The penalty of Eilers and Marx(1996) is used to impose the smoothness. A low-dimensional approximation of the curve data can be obtained by a linear combination of the null space basis functions. ^ Jin(2006) developed the theory for PDA with covariates when functional data collected from experimental units are described by covariates. This thesis is a Monte-Carlo simulation study to identify the effect of the smoothing parameters for the bias and variance of the estimators in PDA with covariates. Implementations of Jin(2006) were translated from Splus to R and were made to run more efficiently. A hearing data set taken from Wood(2007) was analyzed using PDA with covariates.^

#### Subject Area

Statistics

#### Recommended Citation

Mallawaarachchi, Indika Varuna, "Principal differential analysis with covariates: A simulation study on the effect of the smoothing parameters" (2011). *ETD Collection for University of Texas, El Paso*. AAI1498301.

http://digitalcommons.utep.edu/dissertations/AAI1498301