L-ratio (l_ratio)

Calculation

This example assumes use of a tetrode.

L-ratio uses 4 principal components (PCs) for each tetrode channel (the first being energy, the square root of the sum of squares of each sample in the waveform, followed by the first 3 PCs of the energy normalised waveform). This yields spikes which are each represented as a point in 16 dimensional space.

Define, for each cluster \(C\), \(D_{i,C}^2\), the squared Mahalanobis distance from the centre of cluster \(C\) for every spike \(i\) in the dataset (similarly to the calculation for isolation distance above). Assume that spikes in the cluster distribute normally in each dimension, so that \(D^2\) for spikes in a cluster will distribute as \(\chi^2\) with 16 degrees of freedom. This yields \(\textrm{CDF}_{\chi^2_{\mathrm{df}}}\), the cumulative distribution function of the \(\chi^2\) distribution. Define for each cluster \(C\), the value \(L(C)\), representing the amount of contamination of the cluster \(C\):

\[L(C) = \sum_{i \notin \mathrm{C}} 1 - \mathrm{CDF}_{\chi^2_{\mathrm{df}}}(D^2_{i, C})\]

\(L\) is then the sum of probabilities that each spike which is not a member of cluster \(C\) should be. Therefore the inverse of this cumulative distribution yields the probability of cluster membership for each spike \(i\). \(L\) is then normalised by the number of spikes \(N_s\) in \(C\) to allow larger clusters to tolerate more contamination. This yields L-ratio, which can be expressed as:

\[L_{\mathrm{ratio}}(C) = \frac{L(C)}{N_s}\]

Expectation and use

Since this metric identifies unit separation, a high value indicates a highly contaminated unit (type I error) ([Schmitzer-Torbert] et al.). [Jackson] et al. suggests that this measure is also correlated with type II errors (although more strongly with type I errors).

A well separated unit should have a low L-ratio ([Schmitzer-Torbert] et al.).

Example code

import spikeinterface.qualitymetrics as sqm

_, l_ratio = sqm.isolation_distance(all_pcs=all_pcs, all_labels=all_labels, this_unit_id=0)

References

spikeinterface.qualitymetrics.pca_metrics.mahalanobis_metrics(all_pcs, all_labels, this_unit_id)

Calculates isolation distance and L-ratio (metrics computed from Mahalanobis distance)

Parameters

all_pcs2d array

The PCs for all spikes, organized as [num_spikes, PCs].

all_labels1d array

The cluster labels for all spikes. Must have length of number of spikes.

this_unit_idint

The ID for the unit to calculate these metrics for.

Returns

isolation_distancefloat

Isolation distance of this unit.

l_ratiofloat

L-ratio for this unit.

References

Based on metrics described in [Schmitzer-Torbert]

Literature

Introduced by [Schmitzer-Torbert] et al.. Early discussion and comparison with isolation distance by [Jackson] et al..