Using PCA to Reduce Commodity Forward Curve Risk Factors

# How PCA is performed in the paper "Markov Models..."


# How PCA is performed in the paper "Markov Models..."












can anyone explain in a bit detail on how PCA is performed in the paper "Markov Models for Commodity Futures: Theory and Practice" by Leif B. G. Andersen. I'm not clear on how the high dimension matrix for Xj is constructed, specifically what is the rows and columns in the matrix? Thanks. link: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1138782

$X_j(t)$ is defined as:

$X_j(t)=lnF(t,t+\delta_j)$, $j=$ 1 month, 2 months,...,48 months. So I think the rows are these 48 $X_j(t)$ variables, but what is the column component? By using PCA, he reduced the dimension of a 48*N matrix to 48*2, and concludes that the number of random drivers is 2.

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/27739

Generally, the PCA approach proceeds as follows.

Consider historical observation times $t_0 < t_1 < \cdots < t_K \le0$. For bucketing times $\delta_1 < \cdots \delta_n$, let $X_j(t_k) = \ln F(t_k, t_k+\delta_j)$. Moreover, let $\eta_j$, for $j=1, \ldots, n$, be a normal random variable with a sample set $\big\{X_j(t_k)-X_j(t_{k-1}\}_{k=1}^K\big\}$. Furthermore, let $\Omega = (\rho_{i, j})_{i,j=1}^n$ be the sample correlation matrix. Observe that \begin{align*} \Omega = \Sigma \Gamma\Sigma^T, \end{align*} where $\Gamma = diag(\gamma_1, \ldots, \gamma_n)$ is a diagonal matrix of eigenvalues $\gamma_i\ (i=1, \ldots, n)$, and the columns of $\Sigma = (\omega_{i, j})_{i,j=1}^n$ contains the orthonormal eigenvectors of $\Omega$; here, we assume that $\gamma_i\ge \gamma_{i+1}$, for $i=1, \ldots, n-1$. Then, in distribution, there are $n$ independenr standard normal random variables $\xi_i$, for $i=1, \ldots, n$, such that \begin{align*} \left(\! \begin{array}{c} \frac{\eta_1-\mu_1}{\sigma_1}\\ \vdots\\ \frac{\eta_n-\mu_n}{\sigma_n} \end{array} \!\right) = \left(\! \begin{array}{ccc} \omega_{1,1} & \cdots & \omega_{1, n}\\ \vdots & \ddots &\vdots\\ \omega_{n,1} & \cdots & \omega_{n, n} \end{array} \!\right) \left(\! \begin{array}{ccc} \sqrt{\gamma_1} & & \\ & \ddots & \\ & & \sqrt{\gamma_n} \end{array} \!\right) \left(\! \begin{array}{c} \xi_1\\ \vdots\\ \xi_n \end{array} \!\right). \end{align*} The PCA approach is to find the integer $p$, where $0< p \le n$, such that \begin{align*} \frac{\sum_{i=1}^p \gamma_i}{\sum_{i=1}^n \gamma_i}\ge l, \end{align*} where $l$ is a certain given level, for example, 95%. In general, $p=2$ is sufficient. We assume that $p=2$ below.

Based on PCA, we assume that \begin{align*} \left(\! \begin{array}{c} \frac{\eta_1-\mu_1}{\sigma_1}\\ \vdots\\ \frac{\eta_n-\mu_n}{\sigma_n} \end{array} \!\right) &\approx \left(\! \begin{array}{cc} \omega_{1,1} & \omega_{1, 2}\\ \vdots & \vdots\\ \omega_{n,1} & \omega_{n, 2} \end{array} \!\right) \left(\! \begin{array}{cc} \sqrt{\gamma_1} & 0 \\ 0 & \sqrt{\gamma_2} \end{array} \!\right) \left(\! \begin{array}{c} \xi_1\\ \xi_2 \end{array} \!\right) \label{pca_1} \\ &=\left(\! \begin{array}{c} \omega_{1,1}\sqrt{\gamma_1}\xi_1 + \omega_{1,2}\sqrt{\gamma_2}\xi_2\\ \vdots\\ \omega_{n,1}\sqrt{\gamma_1}\xi_1 + \omega_{n,2}\sqrt{\gamma_2}\xi_2 \end{array} \!\right). \tag{1} \end{align*}

Motivated by $(1)$, we can assume that, for $i=1, \ldots, n$, \begin{align*} d \ln F(t, t+\delta_i) &= \mu_i dt + \sigma_i^0 \left(\omega_{i, 1}\sqrt{\gamma_1} dW_t^1 + \omega_{i, 2}\sqrt{\gamma_2} dW_t^2\right)\\ &=\mu_i dt + \sigma_i^1 dW_t^1 + \sigma_i^2 dW_t^2, \end{align*} where $\{W_t^1, t\ge 0\}$ and $\{W_t^2, t\ge 0\}$ are two independent standard Brownian motions, or certain other independent normal (e.g., mean-reverting) processes.