Autocorrelation Calculator

For a series x_t with length N, the autocovariance at lag k is computed as:

• C(k) = (1 / (N−k)) Σ x_t x_t+k (unbiased), or
• C(k) = (1 / N) Σ x_t x_t+k (biased).

If mean removal is enabled, the calculator first transforms the data: x_t ← x_t − μ.

When normalization is enabled, autocorrelation is: ρ(k) = C(k) / C(0), giving ρ(0)=1.

1. Paste your numeric series into the data box.
2. Set a max lag that is smaller than the sample count.
3. Enable mean removal for drifting or offset signals.
4. Select an estimator: unbiased for accuracy at large lags.
5. Enable normalization to compare signals on a common scale.
6. Click Calculate, then export CSV or PDF if needed.

1) Why autocorrelation matters in physics

Autocorrelation quantifies how strongly a signal relates to its earlier values across discrete lags. In experiments and simulations, it reveals memory, relaxation, and hidden periodicity. A rapidly decaying curve suggests fast mixing, while a slow decay indicates persistent structure.

2) Autocovariance versus autocorrelation

Autocovariance C(k) preserves physical units because it is based on products of the series. Autocorrelation ρ(k) normalizes by C(0) so values typically lie between −1 and 1, enabling direct comparisons between datasets with different amplitudes.

3) Mean removal and stationarity

Many signals include an offset or slow drift. Subtracting the mean improves interpretability by focusing on fluctuations. If the underlying process is approximately stationary, the de-meaned autocorrelation highlights intrinsic dynamics rather than baseline shifts. For constant series, mean removal prevents misleading near-zero denominators in normalization.

4) Choosing max lag and sampling interval

Lag k maps to a physical time using the sampling interval Δt, so the lag axis becomes kΔt. A practical max lag is often 10–30% of the sample count. Larger lags reduce the number of overlapping pairs, increasing noise in C(k).

5) Biased and unbiased estimators

This calculator supports two common normalizations for autocovariance: dividing by N (biased) or by N−k (unbiased). The unbiased estimator better corrects shrinkage at large lags, but it can amplify variance when data are short. If your goal is stable trend comparison, the biased option can be smoother.

6) Interpreting shapes and common patterns

Oscillating autocorrelation often indicates a characteristic frequency, such as vibrations or driven responses. Exponential-like decay is typical of single-time-scale relaxation, while multi-stage decay suggests multiple modes. A negative lobe can appear in anti-persistent signals, feedback systems, or alternating dynamics.

7) Integrated autocorrelation time and uncertainty

The integrated autocorrelation time (IAT) summarizes correlation persistence and influences effective sample size. The calculator uses a simple positive-sequence cutoff: it sums positive ρ(k) until the first non-positive lag. Higher IAT implies fewer independent samples and larger statistical uncertainty for averages.

8) Practical workflow and quality checks

Start with mean removal and normalized output. Verify that ρ(0)=1 and inspect early-lag behavior first. If late-lag values fluctuate wildly, reduce max lag or gather more data. For noisy measurements, compare multiple runs and look for consistent decay scales rather than exact point values.

1) What data formats can I paste?

You can paste numbers separated by spaces, commas, semicolons, or new lines. Scientific notation is accepted. Any non-numeric tokens are ignored during parsing.

2) Why is mean removal recommended?

Removing the mean centers the series on fluctuations, reducing the influence of offsets and slow drift. This usually yields a more interpretable correlation curve for stationary or near-stationary processes.

3) When should I use the unbiased estimator?

Use the unbiased option when you care about accurate large-lag estimates and have enough data. It divides by N−k, reducing systematic shrinkage at long lags, but it can increase noise for short series.

4) What does a negative autocorrelation mean?

Negative values indicate anti-persistence: high values tend to be followed by low values, and vice versa. This can occur in oscillations, feedback-controlled systems, or alternating-step dynamics.

5) Why do late lags look noisy?

As lag increases, fewer overlapping pairs contribute to the estimate, so statistical variance grows. Reducing max lag or increasing sample count typically improves stability.

6) How should I pick max lag?

A common starting point is 10–30% of the series length. Increase it only if you need to capture slow relaxation. If the tail becomes erratic, the lag range is likely too large for the data length.

7) What does the integrated autocorrelation time indicate?

IAT summarizes how long correlations persist. A larger IAT means fewer effectively independent samples, so averages converge more slowly and uncertainty is higher. Treat it as an approximate diagnostic, especially for short datasets.