locan.analysis.radial_distribution

Radial distance distribution analysis.

Radial distance distribution analysis is closely related to the pairwise distance analysis.

The pairwise distance distribution p(r) represents the probability distribution function to find for any localization at :math: r = 0 another localization at distance r.

The radial distribution function (also called pair correlation function) g(r) represents the pairwise distance distribution function p(r) normalized to the region measure m(r) of the circular ring or shell with inner radius r and outer radius :math: r + delta_r. and relative to the expected number of localizations per unit area for complete spatial randomness.

\[g(r) &= p(r) \ (\]

ho * m(r) )

The region measure depends on the coordinates dimension and is

\[ \begin{align}\begin{aligned}m(r) &= r * \delta r \qquad \text{in 1D}\\m(r) &= 2 * \pi * r * \delta r \qquad \text{in 2D}\\m(r) &= 4 * \pi * r^2 * \delta r \qquad \text{in 3D}\end{aligned}\end{align} \]

For a spatial homogeneous Poisson process (i.e. complete spatial randomness, CSR) with intensity \(\rho\) (expected number of points per unit area) the radial distribution function results to \(g(r) = 1\).

See also

PairDistances

References

[1]

Sengupta, P., Jovanovic-Talisman, T., Skoko, D. et al., Probing protein heterogeneity in the plasma membrane using PALM and pair correlation analysis. Nat Methods 8, 969–975 (2011), https://doi.org/10.1038/nmeth.1704

[2]

J. Schnitzbauer, Y. Wang, S. Zhao, M. Bakalar, T. Nuwal, B. Chen, B. Huang, Correlation analysis framework for localization-based superresolution microscopy. Proc. Natl. Acad. Sci. U.S.A. 115 (13) 3219-3224 (2018), https://doi.org/10.1073/pnas.1711314115

[3]

Bohrer, C.H., Yang, X., Thakur, S. et al. A pairwise distance distribution correction (DDC) algorithm to eliminate blinking-caused artifacts in SMLM. Nat Methods 18, 669–677 (2021). https://doi.org/10.1038/s41592-021-01154-y

Classes

RadialDistribution(bins[, pair_distances, meta])	Compute the radial distribution function within data or between data and other_data.
RadialDistributionBatch([bins, meta])	Generate RadialDistribution results from a batch of data.
RadialDistributionBatchResults([radii, data])
RadialDistributionResults([radii, data])