Normalization Functions • dominatR

Data

The data used in this article comes from only from the following source:

A SummarizedExperiment object from the library airway

In this case we are using a single object given that it can be manipulated as a SummarizedExperiment or data.frame

### summarized experiment
data("airway")
se <-airway

Normalization Methods

Normalization is critical for correcting technical biases and enabling meaningful biological comparisons.

The package contains different normalization methods, some of them include a log_transf attribute that calculates the log2(x+1) of the normalized value if set to TRUE:

cpm_normalization
minmax_normalization
quantile_normalization
rpkm_normalization
tpm_normalization

Let’s explore the usage of each normalization method on the airway data.

Min-Max Normalization

Min-Max normalization is a linear transformation technique that rescales each gene’s expression values to a specified range (typically [0, 1]). This normalization method is useful when you want to bring the data onto the same scale.

Function Purpose:

· Rescales each column to fit within a range [new_min, new_max].

· Preserves the relative structure of values within each column.

· Useful when different assays or samples have varying scales.

Example 1: Normalize a matrix

# Prepare input matrix
count_mat <- assay(se)

# Apply min-max normalization
se_minmax <- minmax_normalization(count_mat, new_min = 0, new_max = 1)

# Inspect structure
dim(se_minmax)
#> [1] 63677     8
summary(as.vector(se_minmax))
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#> 0.000e+00 0.000e+00 0.000e+00 9.679e-04 2.739e-05 1.000e+00
head(se_minmax[, 1:5])
#>                   SRR1039508   SRR1039509   SRR1039512   SRR1039513
#> ENSG00000000003 0.0022792424 0.0017523136 1.699217e-03 0.0014897144
#> ENSG00000000005 0.0000000000 0.0000000000 0.000000e+00 0.0000000000
#> ENSG00000000419 0.0015676086 0.0020143784 1.208721e-03 0.0013327102
#> ENSG00000000457 0.0008727585 0.0008253084 5.119062e-04 0.0005988068
#> ENSG00000000460 0.0002014058 0.0002151278 7.785646e-05 0.0001277941
#> ENSG00000000938 0.0000000000 0.0000000000 3.892823e-06 0.0000000000
#>                   SRR1039516
#> ENSG00000000003 2.860799e-03
#> ENSG00000000005 0.000000e+00
#> ENSG00000000419 1.475649e-03
#> ENSG00000000457 6.159013e-04
#> ENSG00000000460 1.960829e-04
#> ENSG00000000938 2.513883e-06

You can set new_min = 10 and new_max = 20 if your downstream application prefers values in a different scale:

df_scaled <- minmax_normalization(count_mat, new_min = 10, new_max = 20)
head(df_scaled)  # All columns now range from 10 to 20
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003   10.02279   10.01752   10.01699   10.01490   10.02861
#> ENSG00000000005   10.00000   10.00000   10.00000   10.00000   10.00000
#> ENSG00000000419   10.01568   10.02014   10.01209   10.01333   10.01476
#> ENSG00000000457   10.00873   10.00825   10.00512   10.00599   10.00616
#> ENSG00000000460   10.00201   10.00215   10.00078   10.00128   10.00196
#> ENSG00000000938   10.00000   10.00000   10.00004   10.00000   10.00003
#>                 SRR1039517 SRR1039520 SRR1039521
#> ENSG00000000003   10.02607   10.02033   10.01536
#> ENSG00000000005   10.00000   10.00000   10.00000
#> ENSG00000000419   10.01990   10.01101   10.01364
#> ENSG00000000457   10.00824   10.00615   10.00615
#> ENSG00000000460   10.00157   10.00201   10.00161
#> ENSG00000000938   10.00000   10.00000   10.00000

Example 2: Normalize a SummarizedExperiment

se <- se

# Option A: Overwrite the default assay
se1 <- minmax_normalization(se)
head(assay(se1))
#>                   SRR1039508   SRR1039509   SRR1039512   SRR1039513
#> ENSG00000000003 0.0022792424 0.0017523136 1.699217e-03 0.0014897144
#> ENSG00000000005 0.0000000000 0.0000000000 0.000000e+00 0.0000000000
#> ENSG00000000419 0.0015676086 0.0020143784 1.208721e-03 0.0013327102
#> ENSG00000000457 0.0008727585 0.0008253084 5.119062e-04 0.0005988068
#> ENSG00000000460 0.0002014058 0.0002151278 7.785646e-05 0.0001277941
#> ENSG00000000938 0.0000000000 0.0000000000 3.892823e-06 0.0000000000
#>                   SRR1039516   SRR1039517   SRR1039520   SRR1039521
#> ENSG00000000003 2.860799e-03 0.0026074678 0.0020325525 0.0015356158
#> ENSG00000000005 0.000000e+00 0.0000000000 0.0000000000 0.0000000000
#> ENSG00000000419 1.475649e-03 0.0019898441 0.0011007460 0.0013637987
#> ENSG00000000457 6.159013e-04 0.0008243284 0.0006150451 0.0006147833
#> ENSG00000000460 1.960829e-04 0.0001568963 0.0002006156 0.0001610786
#> ENSG00000000938 2.513883e-06 0.0000000000 0.0000000000 0.0000000000

# Option B: Write to a new assay slot
se2 <- minmax_normalization(se, new_assay_name = "minmax_counts")

By using the option new_assay_name it is possible to store the normalized data in a new assay in the summarizedexperiment object keeping the count matrix intact. If no name is provided upon normalization, then the function will overwrite the count matrix

Quantile Normalization

Quantile normalization makes the distribution of values across all samples identical. This technique adjusts the data so that the rank distributions of the data across samples are equal.

Example 1: Normalize a matrix

count_mat <- assay(se)

se_quantile <- quantile_normalization(count_mat)

## Check result
dim((se_quantile))
#> [1] 63677     8
summary(as.vector(se_quantile))
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#>      0.00      0.00      0.00    344.36      9.62 361483.12
head(se_quantile[1:5, 1:5])
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003    690.875    504.750    773.875     613.75   1010.000
#> ENSG00000000005      0.000      0.000      0.000       0.00      0.000
#> ENSG00000000419    468.875    582.375    550.625     552.00    516.875
#> ENSG00000000457    257.375    241.375    225.250     254.00    213.125
#> ENSG00000000460     58.000     65.250     31.500      53.75     67.750

Example 2: Normalize a summarized experiment

## Apply quantile normalization to a SummarizedExperiment
se_quantile <- quantile_normalization(se)

## Check result
dim(assay(se_quantile))
#> [1] 63677     8
summary(as.vector(assay(se_quantile)))
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#>      0.00      0.00      0.00    344.36      9.62 361483.12
head(assay(se_quantile)[1:5, 1:5])
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003    690.875    504.750    773.875     613.75   1010.000
#> ENSG00000000005      0.000      0.000      0.000       0.00      0.000
#> ENSG00000000419    468.875    582.375    550.625     552.00    516.875
#> ENSG00000000457    257.375    241.375    225.250     254.00    213.125
#> ENSG00000000460     58.000     65.250     31.500      53.75     67.750

CPM Normalization

The cpm_normalization() function rescales raw count data such that each column sums to one million. This makes count data comparable across samples of different sequencing depths.

Example 1: Normalize a data.frame

df <- assay(se)
# Normalize without log2-transform
df_cpm <- cpm_normalization(df, log_trans = FALSE)
head(df_cpm[, 1:5])
#>                 SRR1039508 SRR1039509  SRR1039512 SRR1039513  SRR1039516
#> ENSG00000000003  32.900521  23.817776 34.43970525  26.906868 46.54699807
#> ENSG00000000005   0.000000   0.000000  0.00000000   0.000000  0.00000000
#> ENSG00000000419  22.628193  27.379809 24.49834703  24.071095 24.00974329
#> ENSG00000000457  12.598138  11.217747 10.37530639  10.815506 10.02110240
#> ENSG00000000460   2.907263   2.924057  1.57799337   2.308187  3.19039178
#> ENSG00000000938   0.000000   0.000000  0.07889967   0.000000  0.04090246

# Normalize with log2-transform
df_cpm_log <- cpm_normalization(df, log_trans = TRUE)
head(df_cpm_log[, 1:5])
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003   5.083236   4.633302  5.1472947   4.802548 5.57128235
#> ENSG00000000005   0.000000   0.000000  0.0000000   0.000000 0.00000000
#> ENSG00000000419   4.562437   4.826793  4.6723318   4.647953 4.64441834
#> ENSG00000000457   3.765337   3.610906  3.5078335   3.562609 3.46219663
#> ENSG00000000460   1.966158   1.972346  1.3662486   1.726041 2.06708514
#> ENSG00000000938   0.000000   0.000000  0.1095607   0.000000 0.05783488

Example 2: Normalize a SummarizedExperiment


# Apply in-place normalization (overwrite assay)
se1 <- cpm_normalization(se, log_trans = FALSE)
head(assay(se1))
#>                 SRR1039508 SRR1039509  SRR1039512 SRR1039513  SRR1039516
#> ENSG00000000003  32.900521  23.817776 34.43970525  26.906868 46.54699807
#> ENSG00000000005   0.000000   0.000000  0.00000000   0.000000  0.00000000
#> ENSG00000000419  22.628193  27.379809 24.49834703  24.071095 24.00974329
#> ENSG00000000457  12.598138  11.217747 10.37530639  10.815506 10.02110240
#> ENSG00000000460   2.907263   2.924057  1.57799337   2.308187  3.19039178
#> ENSG00000000938   0.000000   0.000000  0.07889967   0.000000  0.04090246
#>                 SRR1039517 SRR1039520 SRR1039521
#> ENSG00000000003  33.973415  40.259015  27.026857
#> ENSG00000000005   0.000000   0.000000   0.000000
#> ENSG00000000419  25.926226  21.802609  24.002873
#> ENSG00000000457  10.740401  12.182273  10.820193
#> ENSG00000000460   2.044246   3.973617   2.834985
#> ENSG00000000938   0.000000   0.000000   0.000000

# Save to a new assay slot
se2 <- cpm_normalization(se, log_trans = TRUE, new_assay_name = 
                            "cpm_logged")
head(assay(se2, "cpm_logged"))
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003   5.083236   4.633302  5.1472947   4.802548 5.57128235
#> ENSG00000000005   0.000000   0.000000  0.0000000   0.000000 0.00000000
#> ENSG00000000419   4.562437   4.826793  4.6723318   4.647953 4.64441834
#> ENSG00000000457   3.765337   3.610906  3.5078335   3.562609 3.46219663
#> ENSG00000000460   1.966158   1.972346  1.3662486   1.726041 2.06708514
#> ENSG00000000938   0.000000   0.000000  0.1095607   0.000000 0.05783488
#>                 SRR1039517 SRR1039520 SRR1039521
#> ENSG00000000003   5.128187   5.366637   4.808738
#> ENSG00000000005   0.000000   0.000000   0.000000
#> ENSG00000000419   4.750940   4.511127   4.644022
#> ENSG00000000457   3.553410   3.720527   3.563182
#> ENSG00000000460   1.606085   2.314295   1.939221
#> ENSG00000000938   0.000000   0.000000   0.000000

RPKM Normalization

Reads per kilobase per million (RPKM) normalization adjusts for both gene length and sequencing depth, making it particularly useful for RNA-Seq data. RPKM helps compare gene expression levels across genes of different lengths.

Example 1: Normalize a data.frame

df <- assay(se)
length <- rowData(se)$gene_seq_end - rowData(se)$gene_seq_start
df_rpkm <- rpkm_normalization(df, gene_length = length)

head(df_rpkm[, 1:5])
#>                 SRR1039508 SRR1039509  SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003 2.90614973 2.10385794 3.042108051 2.37672181 4.11156241
#> ENSG00000000005 0.00000000 0.00000000 0.000000000 0.00000000 0.00000000
#> ENSG00000000419 0.95525977 1.15585145 1.034209179 1.01617253 1.01358254
#> ENSG00000000457 0.28224164 0.25131614 0.232442566 0.24230454 0.22450718
#> ENSG00000000460 0.01514389 0.01523137 0.008219743 0.01202331 0.01661870
#> ENSG00000000938 0.00000000 0.00000000 0.003398943 0.00000000 0.00176205

Example 2: Normalize a SummarizedExperiment

## Gene length needed
rowData(se)$gene_length <- rowData(se)$gene_seq_end - rowData(se)$gene_seq_start

## Apply RPKM normalization
se_rpkm <- rpkm_normalization(se, gene_length, log_trans = TRUE)

## Check the result
dim(assay(se_rpkm)) 
#> [1] 63677     8
head(assay(se_rpkm)[1:5, 1:5])
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003 1.96574725 1.63406253 2.01510789 1.75562333 2.35376434
#> ENSG00000000005 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
#> ENSG00000000419 0.96736029 1.10825777 1.02446804 1.01161910 1.00976461
#> ENSG00000000457 0.35866816 0.32344632 0.30152042 0.31301889 0.29220123
#> ENSG00000000460 0.02168423 0.02180855 0.01181011 0.01724252 0.02377868

TPM Normalization

Transcripts per million normalization.

Example 1: Normalize a data.frame

df <- assay(se)
length <- sample(c(400:800), nrow(df), replace = TRUE)
df_tpm <- tpm_normalization(df, gene_length = length)

head(df_tpm[, 1:5])
#>                 SRR1039508 SRR1039509  SRR1039512 SRR1039513  SRR1039516
#> ENSG00000000003  33.390121  24.122006 34.96289802  27.202754 47.35784714
#> ENSG00000000005   0.000000   0.000000  0.00000000   0.000000  0.00000000
#> ENSG00000000419  17.352628  20.952835 18.79251602  18.388476 18.45814018
#> ENSG00000000457   9.547481   8.483696  7.86531621   8.165143  7.61347687
#> ENSG00000000460   3.784083   3.798038  2.05453877   2.992829  4.16299091
#> ENSG00000000938   0.000000   0.000000  0.07636036   0.000000  0.03967295

Example 2: Normalize a SummarizedExperiment

## Gene length needed
rowData(se)$gene_length <- rowData(se)$gene_seq_end - rowData(se)$gene_seq_start

## Apply RPKM normalization
se_tpm <- tpm_normalization(se, gene_length, log_trans = TRUE)

## Check the result
dim(assay(se_tpm)) 
#> [1] 63677     8
head(assay(se_tpm)[1:5, 1:5])
#>                 SRR1039508 SRR1039509 SRR1039512 SRR1039513 SRR1039516
#> ENSG00000000003  4.4294986  3.9577411 4.55434051  4.2249660  4.8647104
#> ENSG00000000005  0.0000000  0.0000000 0.00000000  0.0000000  0.0000000
#> ENSG00000000419  2.9549906  3.1678707 3.11233806  3.0989031  2.9884095
#> ENSG00000000457  1.5828547  1.4524115 1.44301228  1.4877425  1.3427320
#> ENSG00000000460  0.1467549  0.1443756 0.08513068  0.1237197  0.1553897

Session Info

#> R version 4.5.1 (2025-06-13)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.3 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8       
#>  [4] LC_COLLATE=C.UTF-8     LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8   
#>  [7] LC_PAPER=C.UTF-8       LC_NAME=C              LC_ADDRESS=C          
#> [10] LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats4    stats     graphics  grDevices utils     datasets  methods  
#> [8] base     
#> 
#> other attached packages:
#>  [1] ggplot2_4.0.0               dominatRData_0.99.1        
#>  [3] airway_1.28.0               SummarizedExperiment_1.38.1
#>  [5] Biobase_2.68.0              GenomicRanges_1.60.0       
#>  [7] GenomeInfoDb_1.44.3         IRanges_2.42.0             
#>  [9] S4Vectors_0.46.0            BiocGenerics_0.54.1        
#> [11] generics_0.1.4              MatrixGenerics_1.20.0      
#> [13] matrixStats_1.5.0           dominatR_0.99.4            
#> [15] knitr_1.50                 
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6            xfun_0.53               bslib_0.9.0            
#>  [4] lattice_0.22-7          vctrs_0.6.5             tools_4.5.1            
#>  [7] tibble_3.3.0            pkgconfig_2.0.3         Matrix_1.7-3           
#> [10] ggnewscale_0.5.2        RColorBrewer_1.1-3      S7_0.2.0               
#> [13] desc_1.4.3              lifecycle_1.0.4         GenomeInfoDbData_1.2.14
#> [16] compiler_4.5.1          farver_2.1.2            textshaping_1.0.4      
#> [19] geomtextpath_0.2.0      ggforce_0.5.0           htmltools_0.5.8.1      
#> [22] sass_0.4.10             yaml_2.3.10             pillar_1.11.1          
#> [25] pkgdown_2.1.3           crayon_1.5.3            jquerylib_0.1.4        
#> [28] MASS_7.3-65             DelayedArray_0.34.1     cachem_1.1.0           
#> [31] abind_1.4-8             tidyselect_1.2.1        digest_0.6.37          
#> [34] dplyr_1.1.4             polyclip_1.10-7         fastmap_1.2.0          
#> [37] grid_4.5.1              cli_3.6.5               SparseArray_1.8.1      
#> [40] magrittr_2.0.4          S4Arrays_1.8.1          withr_3.0.2            
#> [43] UCSC.utils_1.4.0        scales_1.4.0            rmarkdown_2.30         
#> [46] XVector_0.48.0          httr_1.4.7              ragg_1.5.0             
#> [49] evaluate_1.0.5          rlang_1.1.6             glue_1.8.0             
#> [52] tweenr_2.0.3            jsonlite_2.0.0          R6_2.6.1               
#> [55] systemfonts_1.3.1       fs_1.6.6