Definition

  • Temporal autocorrelation (also called serial correlation) means that values in a time series are correlated with their own past values.
  • In other words, the value at time $t$ depends (partly) on values at times $t-1, t-2, \dots$.

Mathematical Form

The autocorrelation function (ACF) at lag $k$:

$\rho_k = \frac{\text{Cov}(X_t, X_{t-k})}{\sigma^2}$

  • $X_t$​ = value at time $t$
  • $\sigma^2$ = variance of the series
  • $\rho_k$​ = correlation between series and its lag-$k$ version

Example

  • Stock prices: Today’s price is often close to yesterday’s.
  • Weather: Today’s temperature is correlated with yesterday’s.
  • Website traffic: Traffic on Monday looks similar to previous Mondays.

Why It Matters

  1. Violates i.i.d. assumption
    • Many ML models assume data points are independent.
    • Temporal autocorrelation breaks that — past strongly influences future.
  2. Forecasting
    • ARIMA, SARIMA, and other time-series models explicitly model autocorrelation.
    • Autocorrelation plots (ACF, PACF) help identify model orders (AR, MA terms).
  3. Statistical Testing
    • Durbin-Watson test checks autocorrelation in regression residuals.
    • If residuals are autocorrelated → model is misspecified.

Positive vs. Negative Autocorrelation

  • Positive autocorrelation → if value is high now, it’s likely high next step (momentum).
  • Negative autocorrelation → if value is high now, it’s likely low next step (mean-reversion).

Visualization

  • ACF plot → shows correlation at different lags.
  • Example: Strong spikes at lag 1, 2, 7 → suggests weekly seasonality.

Summary
Temporal autocorrelation = correlation of a time series with its past values.

  • Core property of time-series data.
  • Measured with autocorrelation function (ACF).
  • Critical for forecasting models (ARIMA, SARIMA).
  • Must be checked in residuals to ensure valid regression/forecasting.