Temporal autocorrelation (Serial Correlation)

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 $t-1, t-2, \dots$.

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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

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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.

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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.

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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).

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Visualization

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

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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.