Jensen–Shannon Divergence Calculator

These example distributions represent two normalized spectra across four bins.

Try pasting these values to verify your workflow.

The Jensen–Shannon divergence compares two distributions by measuring how far each is from their mixture:

• M = αP + (1−α)Q
• JSD(P,Q) = 0.5·KL(P||M) + 0.5·KL(Q||M)
• KL(P||M) = Σ Pᵢ · log(Pᵢ / Mᵢ)

If you use base 2, JSD is in bits. Base e gives nats. A common derived metric is the Jensen–Shannon distance sqrt(JSD).

1. Choose manual input or upload a CSV.
2. Enter nonnegative values for P and Q with equal length.
3. Enable normalization if your values are not already sums of one.
4. Set ε to a small value if your data contains exact zeros.
5. Pick a log base and adjust precision as needed.
6. Press Calculate to see results above the form.
7. Use Download CSV or Download PDF to export tables.

Jensen–Shannon divergence in physics data comparison

When two physical processes look similar, you still need a defensible number to compare them. Jensen–Shannon divergence (JSD) turns paired distributions into a bounded, symmetric measure of difference. It works well for binned spectra, histograms from particle counts, probability vectors from state estimation, and normalized power distributions in signals. This calculator supports smoothing, normalization, and selectable log bases for consistent reporting.

1) Turning measurements into distributions

Start by defining bins and building nonnegative vectors: counts per detector channel, intensity per wavelength band, or occupancy per energy level. A practical range is 32 to 2048 bins, depending on resolution. Normalize so the vector sums to one before comparing runs.

2) Spectroscopy and imaging comparisons

For Raman, absorption, or fluorescence spectra, convert each spectrum into a probability-like shape by dividing by total area. JSD then measures shape change, not absolute brightness. This helps compare samples recorded with different exposure times while preserving relative peak balance.

3) Statistical mechanics and thermal data

Distributions appear in velocity histograms, energy populations, or microstate probabilities. JSD can compare an observed histogram to a model prediction, such as a Maxwell–Boltzmann fit, without over-weighting rare bins. It remains finite when you use small smoothing for empty bins.

4) Dynamics and change detection

In time-resolved experiments, compute distributions over sliding windows and track JSD between consecutive windows. A rising JSD can indicate regime shifts, such as a phase transition onset, mode hopping in lasers, or a change in particle source conditions. The contribution table pinpoints which bins drive the shift.

5) Interpreting the numeric scale

JSD equals 0 for identical distributions. With the standard symmetric choice and base 2, the maximum is 1 bit, making comparisons across studies easier. With base e, the maximum is ln(2) which is about 0.693. The square-root form, JS distance, behaves like a metric and is often easier to threshold.

6) Choosing the mixture weight α

Using α = 0.5 treats P and Q equally and is the most common setting. If one distribution is a trusted reference, you can bias the mixture by choosing α closer to 1 for P or closer to 0 for Q. Keep α fixed across a study to avoid hiding trends.

7) Zeros, noise floors, and smoothing ε

Count data often contain exact zeros, especially with high bin counts. JSD tolerates zeros better than KL, but smoothing still improves numerical stability. Add a small ε such as 1e-12 for normalized vectors, or a small pseudo-count like 0.5 for raw counts before normalization, then re-normalize.

8) Reporting results and reproducibility

To make results reproducible, report binning choices, whether normalization was applied, the log base, α, and ε. Export the metrics and contributions to CSV for lab notebooks, and use the PDF export for quick sharing. When comparing many runs, keep precision consistent across tables.

1) Do P and Q need to sum to one?

No, but it is strongly recommended. If your data are counts or intensities, enable normalization so JSD compares shapes rather than total magnitude. If you disable normalization, interpret results as divergence between unscaled vectors.

2) What is a good value for ε?

For already-normalized distributions, start with ε = 1e-12 to 1e-9. For raw counts, a small pseudo-count approach is often better: add ε to each bin, then normalize. Increase ε only if you still see infinities.

3) Which log base should I choose?

Base 2 yields results in bits and caps the standard JSD at 1, which is convenient for reporting. Base e yields nats and caps the standard JSD at ln(2). Choose one base and keep it consistent across comparisons.

4) What does the contribution table tell me?

It breaks the total JSD into per-bin contributions. Large contributions identify where the two distributions disagree most, such as shifted peaks, missing bands, or altered tails. This is useful for diagnosing instrument drift or sample changes.

5) Can I compare distributions with different lengths?

Not directly. You must define a common set of bins so P and Q have the same length. For spectra, resample both onto the same wavelength grid. For histograms, use identical bin edges before exporting or pasting values.

6) Is JS distance always between 0 and 1?

JS distance is the square root of JSD, so its range depends on the chosen log base. With base 2 and the standard symmetric setting, JS distance stays between 0 and 1. With other bases, the maximum is sqrt(ln(2)) and similar.

7) How should I use thresholds in practice?

Thresholds are application-specific. As a starting point with base 2, JSD below about 0.01 often indicates very similar shapes, while values above 0.1 suggest meaningful differences. Validate thresholds using repeated measurements and known controls.