# Autocorrelation

Autocorrelation is the correlation of a signal with a delayed copy of itself. We provide ACF and PACF functions for time series analysis.

## Partial Autocorrelation (PACF)#

The partial autocorrelation function (PACF) describes the relationship between the current and lagged values of a times series while accounting for intermediate lags. In other words, the PACF at lag k is the autocorrelation between $X_{t}$ and $X_{t−k}$ that is not accounted for by lags 1 through k−1.

We compute this iteratively using a QR decomposition.

### Returns#

• Partial autocorrelation coefficients
• Bounds: 2 standard deviations
• Vector of lags

## Autocorrelation (ACF)#

This is serial correlation of a time series with its own incremental lags.

$$R(\tau) = \int_{\infty}^{\infty}h(t + \tau)h(t)dt$$

Even though it can be calculated naively by iterating incrementally and obtaining a series of regressions, we use a result in applied mathematics known as the Wiener-Kinchin Theorem as a shortcut. This consists of using an inverse Fourier transform of the power spectrum to calculate the autocorrelation function.

$$R(\tau) = \int_{-\infty}^{\infty}dfH(f)H^{*}e^{-2 \pi i f} = \int_{-\infty}^{\infty}dfP(f)e^{-2 \pi i f}$$

### Returns#

• Autocorrelation coefficients
• Bounds: 2 standard deviations
• Vector of lags