In the `mosbi`

package, similarities between biclusters are
computed using different possible similarity metrics.

This vignette gives an overview about the implemented metrics.

`library(mosbi)`

The following similarity metrics are currently implemented:

Bray-Curtis similarity (Wikipedia)

Jaccard index (Wikipedia)

overlap coefficient (Wikipedia)

Fowlkesâ€“Mallows index (Wikipedia)

```
# Bray-Curtis similarity
bray_curtis <- function(s1, s2, overlap) {
return(((2 * overlap) / (s1 + s2)))
}
# Jaccard index
jaccard <- function(s1, s2, overlap) {
return(((overlap) / (s1 + s2 - overlap)))
}
# overlap coefficient
overlap <- function(s1, s2, overlap) {
return((overlap / min(s1, s2)))
}
# Fowlkesâ€“Mallows index
folkes_mallows <- function(s1, s2, overlap) {
tp <- choose(overlap, 2)
fp <- choose(s1 - overlap, 2)
fn <- choose(s2 - overlap, 2)
return(sqrt((tp / (tp + fp)) * (tp / (tp + fn))))
}
```

The behavior of the similarity metrics will be evaluated for two scenarios:

Two biclusters of the same size with an increasing overlap.

Two biclusters of different sizes (One twice as big as the other) with an increasing overlap.

```
# Scenario 1 - two biclusters of the same size
size1_1 <- rep(1000, 1000)
size2_1 <- rep(1000, 1000)
overlap_1 <- seq(1, 1000)
# Scenario 2 - two biclusters one of size 500, the other of size 1000
size1_2 <- rep(1000, 500)
size2_2 <- rep(500, 500)
overlap_2 <- seq(1, 500)
```

Two biclusters of the same size:

```
plot(overlap_1, bray_curtis(size1_1, size2_1, overlap_1),
col = "red", type = "l", xlab = "Overlap", ylab = "Similarity",
ylim = c(0, 1)
)
lines(overlap_1, jaccard(size1_1, size2_1, overlap_1), col = "blue")
lines(overlap_1, overlap(size1_1, size2_1, overlap_1), col = "green", lty = 2)
lines(overlap_1, folkes_mallows(size1_1, size2_1, overlap_1), col = "orange")
legend(
x = .8, legend = c("Bray-Curtis", "Jaccard", "Overlap", "Fowlkesâ€“Mallows"),
col = c("red", "blue", "green", "orange"),
lty = 1, cex = 0.8, title = "Similarity metrics"
)
```

Two biclusters of different sizes:

```
plot(overlap_2, bray_curtis(size1_2, size2_2, overlap_2),
col = "red", type = "l", xlab = "Overlap", ylab = "Similarity",
ylim = c(0, 1)
)
lines(overlap_2, jaccard(size1_2, size2_2, overlap_2), col = "blue")
lines(overlap_2, overlap(size1_2, size2_2, overlap_2), col = "green")
lines(overlap_2, folkes_mallows(size1_2, size2_2, overlap_2), col = "orange")
legend(
x = .8, legend = c("Bray-Curtis", "Jaccard", "Overlap", "Fowlkesâ€“Mallows"),
col = c("red", "blue", "green", "orange"),
lty = 1, cex = 0.8, title = "Similarity metrics"
)
```

```
sessionInfo()
#> R version 4.3.1 (2023-06-16)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 22.04.3 LTS
#>
#> Matrix products: default
#> BLAS: /home/biocbuild/bbs-3.18-bioc/R/lib/libRblas.so
#> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
#>
#> locale:
#> [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
#> [3] LC_TIME=en_GB LC_COLLATE=C
#> [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
#> [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
#> [9] LC_ADDRESS=C LC_TELEPHONE=C
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
#>
#> time zone: America/New_York
#> tzcode source: system (glibc)
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] mosbi_1.8.0 BiocStyle_2.30.0
#>
#> loaded via a namespace (and not attached):
#> [1] tidyr_1.3.0 generics_0.1.3 sass_0.4.7
#> [4] utf8_1.2.4 class_7.3-22 lattice_0.22-5
#> [7] digest_0.6.33 magrittr_2.0.3 evaluate_0.22
#> [10] grid_4.3.1 RColorBrewer_1.1-3 bookdown_0.36
#> [13] fastmap_1.1.1 jsonlite_1.8.7 BiocManager_1.30.22
#> [16] purrr_1.0.2 fansi_1.0.5 scales_1.2.1
#> [19] modeltools_0.2-23 jquerylib_0.1.4 cli_3.6.1
#> [22] isa2_0.3.6 rlang_1.1.1 Biobase_2.62.0
#> [25] munsell_0.5.0 cachem_1.0.8 yaml_2.3.7
#> [28] tools_4.3.1 parallel_4.3.1 biclust_2.0.3.1
#> [31] dplyr_1.1.3 colorspace_2.1-0 ggplot2_3.4.4
#> [34] BiocGenerics_0.48.0 vctrs_0.6.4 R6_2.5.1
#> [37] stats4_4.3.1 lifecycle_1.0.3 magick_2.8.1
#> [40] QUBIC_1.30.0 MASS_7.3-60 pkgconfig_2.0.3
#> [43] RcppParallel_5.1.7 bslib_0.5.1 pillar_1.9.0
#> [46] gtable_0.3.4 glue_1.6.2 Rcpp_1.0.11
#> [49] tidyselect_1.2.0 xfun_0.40 tibble_3.2.1
#> [52] flexclust_1.4-1 knitr_1.44 fabia_2.48.0
#> [55] igraph_1.5.1 htmltools_0.5.6.1 rmarkdown_2.25
#> [58] BH_1.81.0-1 compiler_4.3.1 additivityTests_1.1-4.1
```