Brownian motion · Your observations

What does your recording actually identify?

Bring an explicitly calibrated trajectory CSV, inspect the observations, and ask whether the selected observation model admits them before interpreting an estimate.

Return to the synthetic inference exercise and derivation →

A small, explicit data format

The following is a format illustration, not experimental evidence. A row is a position; successive rows for the same track produce non-overlapping displacements. Choose time and position units in the form rather than relying on a filename or coordinate magnitude.

track,time,x,y
A,0,0,0
A,1,0.4,-0.2
A,2,0.1,0.3

Choose ideal independent increments or camera-aware disjoint frame pairs. The camera method includes an explicitly known Gaussian localization noise scale and uniform exposure, rather than silently ignoring them. Pairs stay within the same track; unmatched final frames remain in the observations and are listed in the result and export. Unknown noise, selection bias and unsupported observation models still receive no interval.

From your observations to a conditional estimate

Local CSV · reference host calculation

Data stay in this browser tab unless you explicitly download an export. This tool does not upload or persist the observations, and they never appear in a share link. Importing a file does not verify that it is a measurement or that the physical model applies.

1 · Supply positions, not overlapping displacements

Local CSV file (maximum 256 KiB)

Or paste CSV text

Required columns: time,x. Optional: y,z,track; z requires y. Use decimal numbers and elapsed timestamps. Tracks may be interleaved, but times must increase within each track. No observations are sorted, resampled, deduplicated or silently dropped. At most 64 tracks and 10000 displacement coordinates are admitted. SI exports can be re-imported with seconds and metres selected; conflicting units are rejected.

Time unit

Position unit

2 · Choose and declare an observation model

Observation model

I am explicitly assuming independent, isotropic Gaussian increments.For multiple tracks, I am assuming the same drift and diffusion coefficient for all particles.

These are declarations, not findings from the CSV. Different-sized particles, confinement, correlated motion or tracking errors can invalidate them. Leave unknown measurements blank. Entering zero is an explicit idealization, not a way to correct camera data.

The ideal model does not admit nonzero noise or exposure. Select the camera model for an explicitly known Gaussian noise scale and uniform exposure; it never treats adjacent noisy displacements as independent.

Position-localization standard deviation (nm; blank = unknown)

Camera exposure (ms; blank = unknown)

Were observations selected, censored or motion-filtered?

Estimator

Conditional interval coverage (50–99.9%)

3 · Optional modern-SI consistency comparisonI have an independently measured particle radius, not one inferred from these displacements.

Diffusion alone cannot determine molecular number independently of particle radius. With modern SI constants, the optional inverse result is a consistency check, not an independent count of molecules. Its interval holds all declared physical inputs and the calibration exact.

An interval is not an authenticity certificate

Conditional coverage describes a procedure under a model, not a posterior probability for a realized interval. The camera method models the declared noise and exposure while holding their values, timing and spatial calibration exact. Uncertainty in those inputs, selection, confinement, correlated tracking errors and different particle properties are not included. A successful calculation does not establish that the recording satisfies the model. A negative estimate or an empty physical confidence set is retained as a diagnostic, never silently replaced by a plausible positive answer.

Explore how a camera changes the inference →

Return to the paper’s inverse argument →