Autocorrelation

Autocorrelation measures how correlated a series is with a shifted (lagged) version of itself — "does knowing today's value help predict the value lag steps from now?" It's the ordinary correlation coefficient formula, applied to a series against itself instead of two separate variables:

def autocorrelation(series, lag):
    n = len(series)
    mean = sum(series) / n
    numerator = sum((series[i] - mean) * (series[i + lag] - mean) for i in range(n - lag))
    denominator = sum((x - mean) ** 2 for x in series)
    return numerator / denominator

alternating = [1, 2, 1, 2, 1, 2, 1, 2]
print(autocorrelation(alternating, 1))   # strongly negative -- each value is the OPPOSITE of the last

The full ACF (autocorrelation function) computes this at every lag (1, 2, 3, ...) and plots how it decays. A series with no real time structure (pure noise) has autocorrelation near zero at every lag beyond 0. A series with real dependence on its own past has autocorrelation that stays meaningfully nonzero for a while before decaying:

def first_significant_lag(series, threshold):
    for lag in range(1, len(series)):
        if abs(autocorrelation(series, lag)) < threshold:
            return lag
    return -1

This is exactly the diagnostic quant researchers and forecasters use to decide how many past values a model actually needs to look at — feeding a model 20 lags of history is wasted complexity if the ACF shows autocorrelation has already decayed to near-zero by lag 3.