📈 Temporal autocorrelation (Serial Correlation)

Temporal autocorrelation (Serial Correlation)#

Correlation of a time series with its own past values.

Important

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What it is#

Temporal autocorrelation (or serial correlation) is when the values of a time series correlate with their own past — the value at time \(t\) depends partly on \(t-1, t-2, \dots\). It is the defining property of time-series data and the most common way the IID assumption breaks.

The measure#

The autocorrelation function (ACF) at lag \(k\) is

\[\rho_k = \frac{\operatorname{Cov}(X_t, X_{t-k})}{\sigma^2},\]

the correlation between the series and its own lag-\(k\) copy, where \(\sigma^2\) is the series variance. Stock prices (today near yesterday), temperature and weekly website traffic all show it.

Why it matters#

First, it violates IID — past strongly influences future, so models that assume independence are wrong. Second, it drives forecasting: ARIMA and SARIMA explicitly model autocorrelation, and ACF/PACF plots reveal the AR and MA orders. Third, it is a diagnostic: the Durbin-Watson test checks regression residuals, and autocorrelated residuals signal a misspecified model.

Positive, negative, and reading the plot#

Positive autocorrelation means high tends to follow high (momentum); negative means high tends to follow low (mean-reversion). On an ACF plot, strong spikes at lags 1, 2 and 7 would suggest weekly seasonality.

Theme: Signal Processing & Time Series · All terminology

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Mind map — connected ideas

IID (Independent and Identically Distributed) · Time Series · Blocked Splits (Single Holdout) · Bayesian Time Series · Signal Processing · Sliding Window (Rolling Window) Cross-Validation

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ARIMA (AutoRegressive Integrated Moving Average) · Bayesian Time Series · Forecast Error · Forecasting Benchmarks · Forecasting Competitions · Log-Space · Low-pass Filtering · LSTM — Long Short-Term Memory Networks · M-Competitions (Makridakis Competitions) · Naïve Baseline Forecast · Prophet — Time Series Forecasting by Facebook (Meta) · Seasonal Lag · Seasonality · Signal Processing