Calculate Pearson residuals with a dispersion adjustment, to identify cells that deviate significantly from what would be expected under independence. The normalization divides by the standard deviation of the difference, which is adjusted by the dispersion parameter θ.
This normalization is designed for detection of highly variable features and dimension reduction and clustering.
$$\LARGE z_{i,j} = \frac{x_{i,j} - \mu_{i,j}}{\sqrt{\mu_{i,j} + \mu_{i,j}^2 / \theta}} $$
$$\LARGE \mu_{i,j} = \frac{r_i \cdot c_j}{N} $$
Where:
(\(x_{i,j}\)) is the raw count for feature \(i\) in sample \(j\)
(\(\mu_{i,j}\)) is the expected value under the model
(\(r_i\)) is \(\sum_j x_{i,j}\)
(\(c_j\)) is \(\sum_i x_{i,j}\)
(\(N\)) is \(\sum_{i,j} x_{i,j}\)
(\(\theta\)) is a dispersion parameter
(\(z_{i,j}\)) is the Pearson residual clipped to the range \([-\sqrt{n}, \sqrt{n}]\) where \(n\) is the number of columns. This is done to prevent extreme values from dominating the analysis.
Scaling is not recommended after this normalization since it is already transforming the data to z-score-like values with a dispersion adjustment. It is also not recommended to use this with DGE analysis.
theta | dispersion parameter expressed as \(\theta\) in the above formula |
Lause, J., Berens, P. & Kobak, D. Analytic Pearson residuals for normalization of single-cell RNA-seq UMI data. Genome Biol 22, 258 (2021). https://doi.org/10.1186/s13059-021-02451-7
Other normalization parameters:
norm_arcsinh,
norm_default,
norm_l2,
norm_library,
norm_log,
norm_osmfish,
norm_quantile,
norm_tfidf