An introduction to functional data analysis and a principal component approach for testing the equality of mean curves

Abstract

We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.

Appendix: Technical lemmas

The results in this section are taken from [52], where their proofs are also given.

Suppose in this section that $\mathbf{Z}_1,\ldots ,\mathbf{Z}_k$ are independent normal random vectors in $\mathbb {R}^d$ so that $E \mathbf{Z}_i = \mathbf{0}$ for all $1\le i\le$, and

$$\begin{aligned} E{\mathbf{Z}_i \mathbf{Z}_i^T} = {\varvec{\Sigma }}_i,\;\; 1\le i \le k. \end{aligned}$$

Define

$$\begin{aligned} {\varvec{\zeta }}=\left( \sum _{\ell =1}^k c_{\ell } {\varvec{\Sigma }}_{\ell }^{-1}\right) ^{-1} \sum _{\ell =1}^k c_{\ell }^{1/2} {\varvec{\Sigma }}_{\ell }^{-1}\mathbf{Z}_\ell , \end{aligned}$$

where $c_i, 1\le i\le k$ satisfy

$$\begin{aligned} \sum _{i=1}^k c_i =1,\;\;\hbox {and}\;\;\;c_i > 0\hbox { for all } 1\le i \le k. \end{aligned}$$

We recall that $\chi ^2(r)$ stands for a $\chi ^2$ random variable with $r$ degrees of freedom.

Lemma 9.1

If

$$\begin{aligned} T=\sum _{\ell =1}^k (\mathbf{Z}_\ell - c_{\ell }^{1/2} {\varvec{\zeta }})^T {\varvec{\Sigma }}_\ell ^{-1}(\mathbf{Z}_\ell - c_{\ell }^{1/2} {\varvec{\zeta }}), \end{aligned}$$

then

$$\begin{aligned} T \mathop {=}\limits ^{{\mathcal {D}}} \chi ^2(d(k-1)). \end{aligned}$$

Lemma 9.2

Let $Y(t)\in L^2$ with $EY(t)=0, E\Vert Y\Vert ^2<\infty$ and $H(t,s)=EY(t)Y(s)$ be a strictly positive function and $\{\psi _i, 1\le i <\infty \}$ be orthonormal functions. Then for any $1\le d <\infty$ the matrix $\mathbf{C}=\{E\langle Y,\psi _i\rangle \langle Y,\psi _j\rangle , 1\le i,j\le d\}$ is nonsingular.

Lemma 9.3

We assume $m\ge 1, g_1, g_2, \ldots ,g_m\in L^2, b_1, b_2, b_m$ are non-negative numbers and $U(t,s)$ is a symmetric positive definite function with eigenvalues $\gamma _1>\gamma _2> \cdots >\gamma _\ell >\gamma _{\ell +1}\ge \cdots 0$ and corresponding orthonormal eigenfunctions $\psi _1, \psi _2, \ldots$ Let

$$\begin{aligned} U^*(t,s)=\sum _{i=1}^mb_ig_i(t)g_i(s)+U(t,s), \end{aligned}$$

with eigenvalues $\gamma _1^*\ge \gamma _2^*\ge \ldots \ge 0$ and corresponding orthonormal eigenfunctions $\psi _1^*, \psi _2^*, \ldots .$ If

$$\begin{aligned} \max _{1\le i\le m}b_i\Vert g_i\Vert ^2>{\uplambda }_\ell , \end{aligned}$$

then with some $j=1,2, \ldots , \ell$ and $i=1,2,\ldots , m$ we have that

$$\begin{aligned} \langle \psi _j^*, g_i\rangle \ne 0. \end{aligned}$$

(9.1)

We assume that

$$\begin{aligned} \psi _1, \psi _2, \ldots ,\psi _m\;\;\hbox {are orthonormal functions} \end{aligned}$$

(9.2)

and

$$\begin{aligned} e_1>e_2>\ldots >e_m>0. \end{aligned}$$

(9.3)

Let ${\mathcal {A}}_0=\hbox {span}(\psi _1, \psi _2, \ldots ,\psi _m)$. We recall that $\bar{\mathcal {B}}$ denotes the orthogonal complement of the set ${\mathcal {B}}$. Assume that

$$\begin{aligned} D\;\;\hbox {is symmetric, square integrable on}\;\;[0,1]^2\;\;\hbox {and non negative definite}. \end{aligned}$$

(9.4)

We say that $D$ has regular maxima of order $n$ with respect to ${\mathcal {A}}_0$ if there are $r_1>r_2>\cdots >r_n$ and orthonormal function $g_1, g_2, \ldots , g_n$ such that

$$\begin{aligned} r_i= & {} \sup _{g\in \bar{{\mathcal {A}}}_{i-1}: \Vert g\Vert =1}\int \!\int g(t)D(t,s)g(s)dtds\nonumber \\&=\int \!\int g_i(t)D(t,s)g_i(s)dtds,\;\;1\le i \le n, \end{aligned}$$

with ${\mathcal {A}}_{i}=\hbox {span}(\psi _1, \ldots , \psi _m, g_1, \ldots ,g_{i}),1\le i \le n-1.$ The functions $g_1, \ldots , g_n$ are unique up to signs. Let

$$\begin{aligned} D_M(t,s)=M\sum _{i=1}^me_i\psi _i(t)\psi _i(s)+D(t,s), \;\;0\le t,s\le 1. \end{aligned}$$

Since $D_M$ is symmetric, non negative definite there are ${\uplambda }_{1,M}\ge {\uplambda }_{2,M}\ge \dots \ge 0$ and orthonormal functions $f_{1,M}, f_{2,M}, \ldots$ such that

$$\begin{aligned} {\uplambda }_{i,M}f_{i,M}(t)=\int D_M(t,s)f_{i,M}(s)ds. \end{aligned}$$

Lemma 9.4

If (9.2)–(9.4) hold and $D$ has regular maxima of order $n$ with respect to ${\mathcal {A}}_0$, then, as $M\rightarrow \infty$ we have

$$\begin{aligned}&\max _{1\le i \le m}\Vert f_{i,M}-{c}_i\psi _i\Vert =o(1), \end{aligned}$$

(9.5)

$$\begin{aligned}&\max _{1\le i\le m}|{\uplambda }_{i,M}/M-e_i|=o(1) \end{aligned}$$

(9.6)

and

$$\begin{aligned}&\max _{m<i\le m+n}\Vert f_{i,M}-{c}_ig_{i-m}\Vert =o(1), \end{aligned}$$

(9.7)

$$\begin{aligned}&\max _{m< i\le n}|{\uplambda }_{i,M}-e_{i-m}|=o(1), \end{aligned}$$

(9.8)

where the values of $c_1=c_{1,M}, c_2=c_{2,M}, \ldots ,c_{m+n}=c_{m+n,M}$ are $1$ or $-1.$