Calculator
Formula used
Average mutual information (AMI) between a signal and its delayed copy is estimated here using a binned (histogram) approximation. For a chosen delay τ, form paired samples (x(t), x(t+τ)) for all valid t.
The mutual information is: I(τ) = Σᵢ Σⱼ pᵢⱼ(τ) · log( pᵢⱼ(τ) / (pᵢ · pⱼ) ) where pᵢ and pⱼ are marginal probabilities and pᵢⱼ(τ) is the joint probability from the 2D histogram.
Note: Histogram-based AMI is sensitive to bin count and sample size; consider testing multiple bin settings for robustness.
How to use this calculator
- Paste your time series values into the data box.
- Set τ to compute AMI at a specific delay.
- Choose a maximum delay to generate the AMI curve.
- Adjust the number of histogram bins for your sample size.
- Press Calculate, then export results as CSV or PDF.
Example data table
| Index t | Sample x(t) |
|---|---|
| 1 | 0.12 |
| 2 | 0.18 |
| 3 | 0.34 |
| 4 | 0.55 |
| 5 | 0.40 |
| 6 | 0.28 |
Try bins = 12, τ from 1 to 10, and compare where the curve first dips.
Average mutual information in time series
Average mutual information (AMI) quantifies how much knowing one observation reduces uncertainty about another. In physics, AMI is widely used when analyzing nonlinear dynamics, turbulence, coupled oscillators, and other systems where dependence can be nonlinear and not captured by correlation alone.
1) What AMI measures
AMI compares paired samples x(t) and x(t+τ) and estimates shared structure across a delay. A value near zero suggests near-independence at that delay, while larger values indicate stronger dependence. Because AMI uses probability distributions, it can detect relationships that remain invisible to linear metrics.
2) Why delay selection matters
Many reconstruction workflows, such as phase-space embedding, require an informative delay. If τ is too small, coordinates are redundant and trajectories cluster. If τ is too large, coordinates become weakly related and structure breaks down. A practical rule is selecting the first local minimum of the AMI curve across delays.
3) Histogram estimator and data needs
This calculator uses a 2D histogram to approximate the joint distribution and marginal distributions. The number of bins controls resolution: more bins can capture fine structure but requires more samples for stability. With short datasets, fewer bins reduce noise and prevent empty-bin artifacts that can bias the estimate.
4) Units and logarithm base
The logarithm base sets the information units. Base 2 reports bits, natural logs report nats, and base 10 reports digits. The numeric value changes by a constant scaling factor across bases, but the curve shape versus τ remains comparable.
5) Interpreting the AMI curve
The curve typically starts high at small delays and decreases as the delayed copy becomes less informative. In many experimental signals, the curve exhibits a dip and then flattens. The first local minimum often balances redundancy and independence, helping downstream embedding and prediction tasks.
6) Sensitivity checks
For reliable conclusions, compute the curve with a few bin choices (for example 8, 12, 16) and compare the recommended delay. If the suggested delay shifts dramatically, increase sample count, smooth the signal, or reduce bins. Stable recommendations across settings indicate a robust delay.
7) Practical physics examples
AMI is helpful for detecting memory in stochastic processes, identifying coupling in driven systems, and selecting delays for reconstructing attractors in chaotic dynamics. It is commonly paired with methods like false nearest neighbors to set embedding dimension after choosing τ.
8) Reporting results
When documenting your analysis, report sample count, bin count, log base, the scanned delay range, and the selected delay rule. This calculator exports both the summary metrics and the full τ-to-AMI table, making your workflow reproducible.
FAQs
1) What is the difference between AMI and autocorrelation?
Autocorrelation measures linear dependence, while AMI measures information shared through any relationship, including nonlinear. AMI can reveal structured dependence even when correlation is near zero.
2) How many samples do I need for stable AMI?
More is better, especially with higher bin counts. As a practical guide, aim for hundreds of paired samples for smooth curves. For short series, reduce bins and keep the delay range modest.
3) Which delay should I choose for embedding?
A common choice is the first local minimum of the AMI curve. It often provides a balance between redundancy and independence for phase-space reconstruction. Always confirm stability across a couple of bin settings.
4) Why does the suggested delay sometimes not appear?
Some signals produce a monotonic or nearly flat AMI curve in the scanned range. In that case, expand the maximum delay, reduce bins, or use an alternate rule such as a fixed fraction of the initial value.
5) Does changing the log base change the best delay?
The base rescales values but does not change relative ordering of delays. The curve shape stays the same, so the location of a local minimum typically remains unchanged.
6) What bin count should I start with?
Start with 8 to 16 bins for many practical datasets. If the curve is noisy, reduce bins. If you have abundant data and expect fine structure, increase bins gradually and compare results.
7) Can I use this with discrete symbols instead of real values?
Yes, but histogram binning is designed for continuous values. For symbolic data, map symbols to numeric codes and set bins to match the number of states, or consider a dedicated discrete estimator.