# Common Functional Principal Components

##### Posted on (Update: )

This post is based on Benko, M., Härdle, W., & Kneip, A. (2009). Common functional principal components. The Annals of Statistics, 37(1), 1–34.

## One sample inference

First focus on one sample of i.i.d. smooth random functions $X_1,\ldots, X_n\in L^2[0, 1]$.

The space of $L^2[0,1]$.If $f\in L^2[0,1]$, then $f$ is Lebesgue integrable, and satisfies $\Vert f\Vert_{L^2[0,1]}^2 = \int_0^1 f^2(x)dx < \infty$.Since this norm does not distinguish between functions that differ only on a set of zero Lebesgue measure, we are implicitly identifying all such functions.The space $L^2[0, 1]$ is a Hilbert space when equipped with the inner product $\langle f,g\rangle_{L^2[0, 1]} =\int_0^2 f(x)g(x)dx$.

Assume a well-defined mean function $\mu= \E(X_i)$, and the existence of a continuous covariance function $\sigma(t, s) =\E[(X_i(t)-\mu(t))(X_i(s)-\mu(s))].$

The covariance operator $\Gamma$ of $X_i$ is given by

\[(\Gamma v)(t) = \int \sigma(t, s)v(s)ds,\quad v\in L^2[0, 1]\,.\]The Karhunen–Loève decomposition gives

\[X_i = \mu + \sum_{r=1}^\infty \beta_{ri}\gamma_r,\quad i=1,\ldots,n,\]where $\beta_{ri} = \langle X_i-\mu, \gamma_r\rangle$ are uncorrelated (scalar) factor loadings with $\E(\beta_{ri}) = 0, \E(\beta_{ri}^2)=\lambda_r$ and $\E(\beta_{ri}\beta_{ki})=0$ for $r\neq k$.

An important application is that the first $L$ principal components provide a “best basis” for approximating the sample functions in terms of the integrated square error. For any choice of $L$ orthonormal basis functions $v_1,\ldots,v_L$, the mean integrated square error

\[\rho(v_1,\ldots, v_L) = \E\left(\left\Vert X_i-\mu - \sum_{r=1}^L\langle X_i-\mu, v_r\rangle v_r\right\Vert^2\right)\]is minimized by $v_r = \lambda_r$.

The approach of estimating principal components is motivated by the well-known duality relation between row and column spaces of a data matrix.

Let $\X$ be a $n\times p$ data matrix,

\[u_k = \frac{1}{\sqrt \lambda_k}\X^Tv_k\quad \text{and}\quad v_k = \frac{1}{\sqrt \lambda_k}\X u_k\,,\]

- The matrices $\X^T\X$ and $\X\X^T$ have the same nonzero eigenvalues $\lambda_1,\ldots,\lambda_r$, where $r = \rank(\X)$.
- The eigenvectors of $\X^T\X$ can be calculated from the eigenvectors of $\X\X^T$ and vice versa:

- The coordinates representing the variables (columns) of $\X$ in a $q$-dimensional subspace can be easily calculated by $w_k=\sqrt{\lambda_k}u_k$.

## Two sample inference

For $X_1^{(1)},\ldots, X_{n_1}^{(1)}$ and $X_1^{(2)},\ldots, X_{n_2}^{(2)}$ be two independent samples of smooth functions. **The problem of interest** is to test in how far the distributions of these random functions coincide, which can be translated into testing equality of the different components of these decompositions

Suppose that $\lambda_{r-1}^{(p)} > \lambda_r^{(p)} > \lambda_{r+1}^{(p)}, p = 1,2$ for all $r \le r_0$ components to be considered. Without restriction, additionally assume that signs are such that $\langle \gamma_r^{(1)}, \gamma_r^{(2)}\rangle \ge 0$, as well as $\langle \hat\gamma_r^{(1)}, \hat\gamma_r^{(2)}\rangle \ge 0$.

Consider

\[H_{0_1}: \mu^{(2)} = \mu^{(2)}\quad \text{and} \quad H_{0_{2,r}}: \gamma_r^{(1)} = \gamma_r^{(2)}, r\le r_0\,.\]If $H_{0_{2,r}}$ is accepted, one may additionally want to test hypotheses about the distributions of $\beta_{ri}^{(p)},p=1,2$. If $X_i^{(p)}$ are Gaussian, then $\beta_{ri}^{(p)}\sim N(0,\lambda_r)$, then consider

\[H_{0_{3,r}}: \lambda_r^{(1)} = \lambda_r^{(2)}, \quad r=1,2,\ldots\]The corresponding test statistics are

\[\begin{align*} D_1 &\triangleq \Vert \hat\mu^{(1)} - \hat\mu^{(2)}\Vert\\ D_{2,r} &\triangleq \Vert \hat\gamma_r^{(1)} - \hat\gamma_r^{(2)}\Vert\\ D_{3,r} &\triangleq \vert \hat\gamma_r^{(1)} - \hat\gamma_r^{(2)}\vert\,. \end{align*}\,,\]and the respective null-hypothesis has to be rejected if $D_1\ge \Delta_{1;1-\alpha}, D_{2,r}\ge \Delta_{2,r;1-\alpha}$ or $D_{3,r}\ge \Delta_{3,r;1-\alpha}$, where $\Delta_{1;1-\alpha}$ denotes the critical values of the distributions of

\[\begin{align*} \Delta_1 &\triangleq \Vert \hat\mu^{(1)} - \mu^{(1)} - (\hat\mu^{(2)} - \mu^{(2)}) \Vert^2 \end{align*}\]and the critical value is approximated by the bootstrap distribution of

\[\begin{align*} \Delta_1^* &\triangleq \Vert \hat\mu^{(1)*} - \hat\mu^{(1)} - (\hat\mu^{(2)*} - \hat\mu^{(2)}) \Vert^2\,. \end{align*}\]Similarly for $\Delta_{2,r}$ and $\Delta_{3,r}$.

Even if for $r \le L$ the equality of eigenfunctions is rejected, we may be interested in the question of whether at least the $L$-dimensional eigenspaces generated by the first $L$ eigenfunctions are identical. Let $\cE_L^{(1)}$ and $\cE_{L}^{(2)}$ be the $L$-dimensional linear function spaces generated by the eigenfunctions respectively. Then test

\[H_{0_{4,L}}: \cE_L^{(1)} = \cE_L^{(2)}\,,\]which can be translated into the condition that

\[\sum_{r=1}^L \gamma_r^{(1)}(t)\gamma_r^{(1)}(t) = \sum_{r=1}^L\gamma_r^{(2)}(t)\gamma_r^{(2)}(s)\quad \text{for all }t,s\in [0, 1]\,,\]and a test statistic is

\[D_{4,L}\triangleq \int\int \left\{ \sum_{r=1}^L \hat\gamma_r^{(1)}(t)\hat\gamma_r^{(1)}(s) - \hat\gamma_r^{(2)}(t)\hat\gamma_r^{(2)}(s) \right\}^2dtds\,,\]whose critical value is also determined by the bootstrap samples.

## Implied volatility analysis

European call and put options are derivatives written on an underlying asset with price process $S_i$, which yield the pay-off $\max(S_I-K, 0)$ and $\max(K-S_I, 0)$, respectively. Here

- $i$: the current day
- $I$: the expiration day
- $K$: the strike price
- $\tau = I-i$: time to maturity

The BS pricing formula for a Call option is

\[C_i(S_i, K, \tau, r, σ) = S_i\Phi(d_1) - Ke^{-r\tau}\Phi(d_2)\,,\]where $d_1 = \frac{\log(S_i/K) + (r+\sigma^2/2)\tau}{σ\sqrt{\tau}}$, and $d_2 = d_1 - σ\sqrt\tau$, $r$ is the risk-free interest rate, $\sigma$ is the (unknown and constant) volatility parameter. The Put option price $P_i$ can be obtained from the put-call parity $P_i=C_i-S_i+e^{-\tau r}K$.

The **implied volatility (IV)** $\tilde \sigma$ is defined as the volatility $σ$, for which the BS price $C_i$ equals the price $\tilde C_i$ observed on the market.

For a single asset, obtain at each time point $i$ and for each maturity $\tau$ a IV function $\tilde \sigma_i^\tau(K)$. For given parameters $S_i, r, K, \tau$ the mapping from prices to IVs is a one-to-one mapping.

Goal: study the dynamics of the IV functions for different maturities. More specifically, the aim is to construct low dimensional factor models based on the truncated Karhunen–Loève expansions for the log-returns of the IV functions with different maturities and compare these factor models using the above methodology.

Data: log-IV-returns for two maturity groups

- 1M group with maturity $\tau = 0.12$
- 3M group with maturity $\tau = 0.36$

Conclusion:

- the first factor functions are not identical in the factor model for both maturity groups
- none of the hypotheses for $L=2$ and $L=3$ is rejected at significance level $\alpha = 0.05$.
- in the functional sense, no significant reason to reject the hypothesis of common eigenspaces for these two maturity groups.