Autocorrelation function
Topic of Dependence along a series, part of Time series · Also in Forecasting
Autocorrelation function
| Type | Topic |
| Taught in | Dependence along a series · Forecasting |
| Part | Time series |
| Questions using it | 9 |
For a weakly stationary series, \rho_k=\gamma_k/\gamma_0, the correlation between observations k steps apart, plotted against k as the ACF. A moving average of order q has \rho_k=0 beyond lag q; an autoregression’s autocorrelation decays instead.
In Dependence along a series
Without some repetition, a series of length T does not identify a different mean at every time and a different covariance for every pair. Strict stationarity says that any block of observations has the same joint distribution as the same block shifted by \tau. The path changes. The distribution does not. Weak stationarity asks only for two moments: a mean \mu that does not depend on time, and a covariance that depends on the two times only through their difference, \gamma_{t-s}=\mathrm{Cov}(X_s,X_t). The lag-zero value \gamma_0=\mathrm{Var}(X_t) is then constant as well. The autocorrelation is \gamma_h/\gamma_0, which lies in [-1,1]. For a Gaussian process, weak stationarity and strict stationarity are the same statement. Otherwise weak is weaker.
In Forecasting
From Forecasting
The autocorrelation and the partial autocorrelation both keep oscillating. An MA process would cut the autocorrelation off after a finite lag. An AR process would cut the partial autocorrelation off. Neither plot dies. That is the seasonality, still in the correlations.
Used in
- Taught in Dependence along a series · Forecasting
- Also used in Markov chains, Gibbs sampling, and Metropolis–Hastings · Forecasting
- Questions that use it (9) — Q5 (answer) · Q9 (answer) · Q10 (answer) · Q19 (answer) · Question 3 (answer) · Question 9 (answer) · Question 16 (answer) · C45 · C49
- All 7 questions on Dependence along a series
See also
Partial autocorrelation · Autoregression and moving averages · Weak stationarity · Correlation · Effective sample size · Seasonality · Thinning a Markov chain · Warm-up is not a convergence proof