Calculator Inputs
Example Data Table
This sample series can be pasted directly into the calculator to test lag behavior, variance structure, and serial dependence.
| Time Index | Observed Value | Comment |
|---|---|---|
| 1 | 12 | Starting value |
| 2 | 15 | Positive rise |
| 3 | 14 | Small pullback |
| 4 | 18 | Stronger growth |
| 5 | 17 | Minor decline |
| 6 | 19 | Recovery point |
| 7 | 21 | Trend continuation |
| 8 | 20 | Short pause |
| 9 | 23 | Renewed strength |
| 10 | 22 | Mild reversion |
| 11 | 24 | Later advance |
| 12 | 26 | Ending peak |
Formula Used
For lag k, autocovariance is computed from centered pairs in the same series.
Where:
- xt = value at time t
- μ = sample mean, zero mean, or custom mean
- dk = n for the biased estimator
- dk = n - k for the unbiased estimator
Autocorrelation is also shown for interpretation:
How to Use This Calculator
- Paste or type the time-series values into the large input box.
- Choose a maximum lag based on the serial depth you want to inspect.
- Select biased or unbiased normalization for the denominator.
- Choose whether the calculator should use the sample mean, zero mean, or a custom mean.
- Set decimal precision and pick a chart style.
- Press Calculate Autocovariance to generate summary metrics, a lag table, and a Plotly graph.
- Use the export buttons to download the output as CSV or PDF.
FAQs
1) What does autocovariance measure?
Autocovariance measures how a series co-moves with its own lagged values. Positive values indicate similar movement across periods, while negative values suggest opposite movement at that lag.
2) Why is lag 0 important?
Lag 0 compares each observation with itself. That makes autocovariance at lag 0 the variance under the chosen mean and denominator convention.
3) What is the difference between biased and unbiased estimators?
The biased version divides by n at every lag. The unbiased version divides by n-k, which compensates for the shrinking number of available pairs at larger lags.
4) How many lags should I test?
A practical choice depends on your sample size and problem type. Use fewer lags for short series, and inspect more lags only when enough observations remain at higher shifts.
5) What does a negative autocovariance mean?
A negative value means above-average observations tend to align with below-average lagged observations. It often signals reversal behavior or oscillation at that specific lag.
6) When should I use a custom mean?
Use a custom mean when your process is centered on a known theoretical level, target, or external benchmark. Otherwise, the sample mean is usually the default choice.
7) How is autocovariance different from autocorrelation?
Autocovariance keeps the original scale of the data. Autocorrelation standardizes that relationship by dividing through lag 0, making lags easier to compare.
8) Can I use this on trending or non-stationary data?
You can, but interpretation becomes harder. Strong trends or changing variance can dominate the results, so detrending or differencing the series may provide cleaner lag insights.