FFT based covariance estimation in R — Pt. 1
Introduction Consider a vector of $n$ real values1, with entries corresponding to observations on discrete times $t=1,\dots,n$, $$ \boldsymbol x = \begin{pmatrix} x_1 & x_2 & \dots & x_{n} \end{pmatrix}’\in\mathbb R^n. $$ The discrete Fourier transform (DFT) $\{a_j \in \mathbb C\}$ of $\boldsymbol x$ at frequencies $\omega_j =j/n\in[0,2\pi]$, $j=0,1,\dots,n-1$ is defined by $$ \begin{align} a_j = \sum_{t=1}^{n} x_t e^{-i2\pi t\omega_j}. \end{align} $$ The DFT decomposes the time-domain data $\boldsymbol{x}$ into its constituent frequencies, represented by the complex coefficients $a_j$. Each coefficient describes the amplitude and phase of a sinusoidal wave at frequency $\omega_j = j/n$, capturing how much that frequency contributes to the overall data. This transformation reveals the data’s periodic patterns in the frequency domain, laying the groundwork for exploring the periodogram and its connection to the (sample) autocovariance function. ...