Split a correlation matrix in blocks as independent as possible. This finds the splitting in blocks that minimizes the sum of squared correlation between these blocks (i.e. everything outside these blocks). In case of equivalent splits, it then minimizes the sum of squared sizes of the blocks.

snp_ldsplit(
corr,
thr_r2,
min_size,
max_size,
max_K = 500,
max_r2 = 0.3,
max_cost = ncol(corr)/200
)

## Arguments

corr

Sparse correlation matrix. Usually, the output of snp_cor().

thr_r2

Threshold under which squared correlations are ignored. This is useful to avoid counting noise, which should give clearer patterns of costs vs. number of blocks. It is therefore possible to have a splitting cost of 0. If this parameter is used, then corr can be computed using the same parameter in snp_cor() (to increase the sparsity of the resulting matrix).

min_size

Minimum number of variants in each block. This is used not to have a disproportionate number of small blocks.

max_size

Maximum number of variants in each block. This is used not to have blocks that are too large, e.g. to limit computational and memory requirements of applications that would use these blocks. For some long-range LD regions, it may be needed to allow for large blocks. You can now provide a vector of values to try.

max_K

Maximum number of blocks to consider. All optimal solutions for K from 1 to max_K will be returned. Some of these K might not have any corresponding solution due to the limitations in size of the blocks. For example, splitting 10,000 variants in blocks with at least 500 and at most 2000 variants implies that there are at least 5 and at most 20 blocks. Then, the choice of K depends on the application, but a simple solution is to choose the largest K for which the cost is lower than some threshold. Default is 500.

max_r2

Maximum squared correlation allowed for one pair of variants in two different blocks. This is used to make sure that strong correlations are not discarded and also to speed up the algorithm. Default is 0.3.

max_cost

Maximum cost reported. Default is ncol(corr) / 200.

## Value

Either NULL when no block splitting satisfies the conditions, or a tibble with seven columns:

• $max_size: Input parameter, useful when providing a vector of values to try. • $n_block: Number of blocks.

• $cost: The sum of squared correlations outside the blocks. • $cost2: The sum of squared sizes of the blocks.

• $perc_kept: Percentage of initial non-zero values kept within the blocks defined. • $all_last: Last index of each block.

• $all_size: Sizes of the blocks. • $block_num: Resulting block numbers for each variant. This is not reported anymore, but can be computed with rep(seq_along(all_size), all_size).

## Examples

if (FALSE) {

# adjust THR_R2 depending on sample size used to compute corr
# use e.g. 0.05 for small sample sizes, and 0.01 for large sample sizes
THR_R2 <- 0.02
m <- ncol(corr)
(SEQ <- round(seq_log(m / 30, m / 5, length.out = 10)))
# replace min_size by e.g. 100 for larger data
(res <- snp_ldsplit(corr, thr_r2 = THR_R2, min_size = 10, max_size = SEQ))

# add the variant block IDs corresponding to each split
res$block_num <- lapply(res$all_size, function(.) rep(seq_along(.), .))

library(ggplot2)
# trade-off cost / number of blocks
qplot(n_block, cost, color = factor(max_size, SEQ), data = res) +
theme_bw(14) +
scale_y_log10() +
theme(legend.position = "top") +
labs(x = "Number of blocks", color = "Maximum block size",
y = "Sum of squared correlations outside blocks")

# trade-off cost / number of non-zero values
qplot(perc_kept, cost, color = factor(max_size, SEQ), data = res) +
theme_bw(14) +
# scale_y_log10() +
theme(legend.position = "top") +
labs(x = "Percentage of non-zero values kept", color = "Maximum block size",
y = "Sum of squared correlations outside blocks")

# trade-off cost / sum of squared sizes
qplot(cost2, cost, color = factor(max_size, SEQ), data = res) +
theme_bw(14) +
scale_y_log10() +
geom_vline(xintercept = 0)+
theme(legend.position = "top") +
labs(x = "Sum of squared blocks", color = "Maximum block size",
y = "Sum of squared correlations outside blocks")

## Pick one solution and visualize blocks
library(dplyr)
all_ind <- res %>%
arrange(cost2 * sqrt(5 + cost)) %>%
print() %>%
slice(1) %>%
pull(all_last)

## Transform sparse representation into (i,j,x) triplets
corrT <- as(corr, "dgTMatrix")
upper <- (corrT@i <= corrT@j & corrT@x^2 >= THR_R2)
df <- data.frame(
i = corrT@i[upper] + 1L,
j = corrT@j[upper] + 1L,
r2 = corrT@x[upper]^2
)
df$y <- (df$j - df\$i) / 2

ggplot(df) +
geom_point(aes(i + y, y, alpha = r2)) +
theme_minimal() +
theme(axis.text.y = element_blank(), axis.ticks.y = element_blank(),
strip.background = element_blank(), strip.text.x = element_blank()) +
scale_alpha_continuous(range = 0:1) +
scale_x_continuous(expand = c(0.02, 0.02), minor_breaks = NULL,
breaks = head(all_ind[[1]], -1) + 0.5) +
facet_wrap(~ cut(i + y, 4), scales = "free", ncol = 1) +
labs(x = "Position", y = NULL)
}