Calculator Inputs
Formula Used
Phase space reconstruction converts a single measured signal into vectors using time-delay embedding. The reconstructed state at index i is:
X(i) = [ x(i), x(i+τ), x(i+2τ), …, x(i+(m−1)τ) ]
- x(i) is the scalar time series sample at index i.
- τ is the delay in samples, and τ·Δt is the delay time.
- m is the embedding dimension, the vector length.
- The number of vectors is N' = N − (m−1)τ.
How to Use This Calculator
- Paste your samples as numbers separated by commas, spaces, or new lines.
- Set Δt to your sampling interval, or keep 1.
- Choose τ as a delay in samples to spread coordinates.
- Select m to control how many delayed coordinates you use.
- Optionally remove the mean and apply normalization for stability.
- Click Calculate to build the reconstructed vectors table.
- Use Download CSV or Download PDF for sharing.
Practical guidance: try several τ values and increase m gradually. Stable shapes across choices suggest a robust reconstruction.
Example Data Table
This example shows a short signal and a common embedding setup.
| Sample index | x(i) |
|---|---|
| 0 | 0.12 |
| 1 | 0.20 |
| 2 | 0.27 |
| 3 | 0.21 |
| 4 | 0.10 |
| 5 | -0.02 |
| 6 | -0.08 |
| 7 | -0.05 |
| 8 | 0.04 |
| 9 | 0.15 |
With m = 3 and τ = 2, the first vector is [x(0), x(2), x(4)].
Professional Guide to Phase Space Reconstruction
1) What phase space reconstruction achieves
Many experiments record only one observable, such as voltage, displacement, or intensity. Yet the underlying system evolves in a multi-dimensional state space. Time-delay embedding rebuilds a proxy state by stacking delayed samples, producing vectors that can reveal cycles, chaos, switching, and regime changes.
2) Core parameters: delay and embedding dimension
The delay τ sets how far apart coordinates are in time. If τ is too small, coordinates are redundant and the reconstructed trajectory collapses near a diagonal. If τ is too large, coordinates become nearly unrelated and the geometry can fragment. The embedding dimension m controls how much state information is captured.
3) Useful data rules for sample size
The calculator reports the usable vector count N' = N − (m−1)τ. As m or τ increases, N' drops linearly. A practical workflow is to start with modest settings (for example m = 3 to 6 and τ between 1 and 20 samples), then increase m only when you still have enough vectors to populate the attractor densely.
4) Correlation check at the chosen delay
A quick lag-correlation value at τ helps you see whether the delay is separating coordinates. Values close to 1 often mean τ is too small. Values near 0 indicate strong decorrelation. Many analysts choose τ near the first strong correlation drop, then confirm by inspecting the reconstructed geometry.
5) Theiler window and neighbor bias
When you compute nearest-neighbor statistics, adjacent points in time can appear as “neighbors” even if they are not informative about the attractor’s folding. Theiler window w excludes candidates within ±w indices, reducing serial-correlation bias. Larger w is common for high sampling rates or smooth trajectories.
6) Detrending and normalization for comparability
Removing the mean centers the trajectory and can improve stability when offsets dominate. Z-score scaling standardizes variance, while min–max scaling maps values to 0–1. These choices are helpful when comparing datasets collected under different gains, sensors, or operating conditions.
7) Interpreting reconstructed tables and exports
Each table row is one reconstructed state vector X(i). Plotting columns against each other (for example x(t) vs x(t+τ)) often shows loops for periodic motion and more complex structures for chaotic signals. Use CSV export for downstream analysis, and PDF export for reporting reproducible parameter settings.
8) Common pitfalls and quality checks
Noise, missing samples, and non-stationarity can distort embeddings. If geometry changes drastically across nearby τ or m choices, increase data length, apply filtering, or analyze shorter stationary windows. Avoid extreme m that leaves too few vectors. Compare multiple trials to confirm consistent structure.
FAQs
1) What data format should I paste?
Paste numbers separated by commas, spaces, tabs, or new lines. Scientific notation is accepted. Non-numeric tokens are ignored, so keep the series clean for best results.
2) How do I choose a good delay τ?
Start where the lag correlation noticeably drops from 1. Test a small range around that τ and pick the value that gives a stable, well-spread geometry in 2D or 3D projections.
3) How large should the embedding dimension m be?
Increase m gradually until the reconstructed shape stops changing much. Ensure you still have enough vectors N' for dense coverage; very large m can leave too little data.
4) What is the Theiler window and when should I use it?
It excludes neighbor candidates close in time, reducing bias from serial correlation. Use it when sampling is dense or trajectories are smooth, especially for neighbor-based diagnostics.
5) Should I detrend and normalize every time?
Not always. Detrend helps when offsets dominate. Normalize when comparing signals with different scales. If absolute units matter, keep raw scaling and document the parameters.
6) Why did the calculator say “not enough samples”?
Your chosen m and τ reduce usable vectors to N' = N − (m−1)τ. Lower m or τ, or provide a longer series so the reconstruction has enough rows to compute.
7) What should I do after exporting the CSV?
Use the columns as coordinates for plots, recurrence analysis, Lyapunov estimates, or clustering. Keep the same τ, m, and preprocessing settings across datasets to compare dynamics fairly.