Score mean profiles with AutoGeneS-style objectives

Computes the two objectives used by AutoGeneS for signature quality. For columns \(\boldsymbol{\mu}_{\cdot j},\boldsymbol{\mu}_{\cdot k}\) of \(\boldsymbol{\mu}\), the cosine (angle similarity, akin to Pearson correlation of centred, unit-norm vectors but here without mean-centring—only \(\ell_2\) normalisation) is $$ \cos(\boldsymbol{\mu}_{\cdot j},\boldsymbol{\mu}_{\cdot k}) = \frac{ \boldsymbol{\mu}_{\cdot j}^{\mathsf{T}} \boldsymbol{\mu}_{\cdot k} }{ \|\boldsymbol{\mu}_{\cdot j}\|_2\, \|\boldsymbol{\mu}_{\cdot k}\|_2 }, $$ and the Euclidean separation is $$ \|\boldsymbol{\mu}_{\cdot j}-\boldsymbol{\mu}_{\cdot k}\|_2 = \sqrt{ \|\boldsymbol{\mu}_{\cdot j}\|_2^{2} +\|\boldsymbol{\mu}_{\cdot k}\|_2^{2} -2\, \boldsymbol{\mu}_{\cdot j}^{\mathsf{T}} \boldsymbol{\mu}_{\cdot k} }. $$ The returned scores are the mean absolute pairwise cosine (to minimise) and the sum of pairwise Euclidean distances (to maximise).

Usage

compute_mean_profile_objectives(mean_signature_matrix)

Arguments

mean_signature_matrix

Numeric matrix \(\boldsymbol{\mu}\in\mathcal{M}_{G\times J}\) (columns = cell types).

Value

A named list with mean_abs_cosine and sum_euclidean_distance.

Examples

compute_mean_profile_objectives(matrix(c(20, 22, 22, 20), 2))
#> $mean_abs_cosine
#> [1] 0.9954751
#> 
#> $sum_euclidean_distance
#> [1] 2.828427
#>