DelayedMatrixStats ports the matrixStats API to work with DelayedMatrix objects from the DelayedArray package. It provides high-performing functions operating on rows and columns of DelayedMatrix objects, including all subclasses such as RleArray (from the DelayedArray package) and HDF5Array (from the HDF5Array) as well as supporting all types of seeds, such as matrix (from the base package) and Matrix (from the Matrix package).
The DelayedArray package allows developers to store array-like data using in-memory or on-disk representations (e.g., in HDF5 files) and provides a common and familiar array-like interface for interacting with these data.
The DelayedMatrixStats package is designed to make life easier for Bioconductor developers wanting to use DelayedArray by providing a rich set of column-wise and row-wise summary functions.
We briefly demonstrate and explain these two features using a simple example. We’ll simulate some (unrealistic) RNA-seq read counts data from 10,000 genes and 20 samples and store it on disk as a HDF5Array:
library(DelayedArray)
x <- do.call(cbind, lapply(1:20, function(j) {
rpois(n = 10000, lambda = sample(20:40, 10000, replace = TRUE))
}))
colnames(x) <- paste0("S", 1:20)
x <- realize(x, "HDF5Array")
x
#> <10000 x 20> HDF5Matrix object of type "integer":
#> S1 S2 S3 S4 ... S17 S18 S19 S20
#> [1,] 26 40 42 31 . 33 28 29 22
#> [2,] 44 29 27 26 . 44 18 40 35
#> [3,] 43 25 27 19 . 34 19 33 21
#> [4,] 33 39 24 37 . 42 25 22 34
#> [5,] 27 21 37 28 . 18 22 38 20
#> ... . . . . . . . . .
#> [9996,] 32 42 27 35 . 17 39 25 23
#> [9997,] 40 27 43 19 . 31 29 31 23
#> [9998,] 30 25 24 25 . 41 28 29 30
#> [9999,] 21 41 33 51 . 30 26 43 30
#> [10000,] 28 23 26 24 . 24 31 23 47Suppose you wish to compute the standard deviation of the read counts for each gene.
You might think to use apply() like in the following:
system.time(row_sds <- apply(x, 1, sd))
#> user system elapsed
#> 268.76 65.59 338.81head(row_sds)
#> [1] 6.194012 8.428648 7.099852 8.021517 9.662053 7.920659This works, but takes quite a while.
Or perhaps you already know that the matrixStats package
provides a rowSds() function:
matrixStats::rowSds(x)
#> Error in rowVars(x, rows = rows, cols = cols, na.rm = na.rm, refine = refine, : Argument 'x' must be a matrix or a vector.Unfortunately (and perhaps unsurprisingly) this doesn’t work. matrixStats is designed for use on in-memory matrix objects. Well, why don’t we just first realize our data in-memory and then use matrixStats
system.time(row_sds <- matrixStats::rowSds(as.matrix(x)))
#> user system elapsed
#> 0.00 0.02 0.01head(row_sds)
#> [1] 6.194012 8.428648 7.099852 8.021517 9.662053 7.920659This works and is many times faster than the apply()-based approach! However,
it rather defeats the purpose of using a HDF5Array for storing the
data since we have to bring all the data into memory at once to compute the
result.
Instead, we can use DelayedMatrixStats::rowSds(), which has the speed
benefits of matrixStats::rowSds()1 In fact, it currently uses matrixStats::rowSds() under the hood. but without having to load the
entire data into memory at once2 In this case, it loads blocks of data row-by-row. The amount of
data loaded into memory at any one time is controlled by the
default block size global setting; see ?DelayedArray::getAutoBlockSize
for details. Notably, if the data are small enough (and the default block size
is large enough) then all the data is loaded as a single block, but this
approach generalizes and still works when the data are too large to be
loaded into memory in one block.:
library(DelayedMatrixStats)
system.time(row_sds <- rowSds(x))
#> user system elapsed
#> 0.03 0.00 0.04head(row_sds)
#> [1] 6.194012 8.428648 7.099852 8.021517 9.662053 7.920659Finally, by using DelayedMatrixStats we can use the same code,
(colMedians(x)) regardless of whether the input is an ordinary matrix or a
DelayedMatrix. This is useful for packages wishing to support both types of
objects, e.g., packages wanting to retain backward compatibility or during a
transition period from matrix-based to DelayeMatrix-based objects.
The initial release of DelayedMatrixStats supports the complete column-wise and row-wise API matrixStats API3 Some of the API is covered via inheritance to functionality in DelayedArray. Please see the matrixStats vignette (available online) for a summary these methods. The following table documents the API coverage and availability of ‘seed-aware’ methods in the current version of DelayedMatrixStats, where:
| Method | Block processing | base::matrix optimized | Matrix::dgCMatrix optimized | Matrix::lgCMatrix optimized | DelayedArray::RleArray (SolidRleArraySeed) optimized | DelayedArray::RleArray (ChunkedRleArraySeed) optimized | HDF5Array::HDF5Matrix optimized | base::data.frame optimized | S4Vectors::DataFrame optimized |
|---|---|---|---|---|---|---|---|---|---|
colAlls() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colAnyMissings() |
❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colAnyNAs() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colAnys() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colAvgsPerRowSet() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCollapse() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCounts() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCummaxs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCummins() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCumprods() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colCumsums() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colIQRDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colIQRs() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colLogSumExps() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colMadDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colMads() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colMaxs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colMeans2() |
✔ | ✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ |
colMedians() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colMins() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colOrderStats() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colProds() |
✔ | ✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ |
colQuantiles() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colRanges() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colRanks() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colSdDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colSds() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colSums2() |
✔ | ✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ |
colTabulates() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colVarDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colVars() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colWeightedMads() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colWeightedMeans() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colWeightedMedians() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
colWeightedSds() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colWeightedVars() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
colsum() |
☑️ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowAlls() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowAnyMissings() |
❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowAnyNAs() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowAnys() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowAvgsPerColSet() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCollapse() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCounts() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCummaxs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCummins() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCumprods() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowCumsums() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowIQRDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowIQRs() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowLogSumExps() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMadDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMads() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMaxs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMeans2() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMedians() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowMins() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowOrderStats() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowProds() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowQuantiles() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowRanges() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowRanks() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowSdDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowSds() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowSums2() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowTabulates() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowVarDiffs() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowVars() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowWeightedMads() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowWeightedMeans() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowWeightedMedians() |
✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowWeightedSds() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowWeightedVars() |
✔ | ✔ | ✔ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ |
rowsum() |
☑️ | ❌ | ✔ | ❌ | ❌ | ❌ | ❌ | ❌ | ❌ |
As well as offering a familiar API, DelayedMatrixStats provides ‘seed-aware’ methods that are optimized for specific types of DelayedMatrix objects.
To illustrate this idea, we will compare two ways of computing the column sums of a DelayedMatrix object:
force_block_processing argumentWe will demonstrate this by computing the column sums matrices with 20,000 rows and 600 columns where the data have different structure and are stored in DelayedMatrix objects with different types of seed:
We use the microbenchmark package to measure running time and the profmem package to measure the total memory allocations of each method.
In each case, the ‘seed-aware’ method is many times faster and allocates substantially lower total memory.
library(DelayedMatrixStats)
library(sparseMatrixStats)
library(microbenchmark)
library(profmem)
set.seed(666)
# -----------------------------------------------------------------------------
# Dense with values in (0, 1)
# Fast, memory-efficient column sums of DelayedMatrix with ordinary matrix seed
#
# Generate some data
dense_matrix <- matrix(runif(20000 * 600),
nrow = 20000,
ncol = 600)
# Benchmark
dm_matrix <- DelayedArray(dense_matrix)
class(seed(dm_matrix))
#> [1] "matrix" "array"dm_matrix
#> <20000 x 600> DelayedMatrix object of type "double":
#> [,1] [,2] [,3] ... [,599] [,600]
#> [1,] 0.7743685 0.6601787 0.4098798 . 0.89118118 0.05776471
#> [2,] 0.1972242 0.8436035 0.9198450 . 0.31799523 0.63099417
#> [3,] 0.9780138 0.2017589 0.4696158 . 0.31783791 0.02830454
#> [4,] 0.2013274 0.8797239 0.6474768 . 0.55217184 0.09678816
#> [5,] 0.3612444 0.8158778 0.5928599 . 0.08530977 0.39224147
#> ... . . . . . .
#> [19996,] 0.19490291 0.07763570 0.56391725 . 0.09703424 0.62659353
#> [19997,] 0.61182993 0.01910121 0.04046034 . 0.59708388 0.88389731
#> [19998,] 0.12932744 0.21155070 0.19344085 . 0.51682032 0.13378223
#> [19999,] 0.18985573 0.41716539 0.35110782 . 0.62939661 0.94601427
#> [20000,] 0.87889047 0.25308041 0.54666920 . 0.81630322 0.73272217microbenchmark(
block_processing = colSums2(dm_matrix, force_block_processing = TRUE),
seed_aware = colSums2(dm_matrix),
times = 10)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> block_processing 97.3300 110.8430 234.47212 129.1594 159.2363 695.8529 10
#> seed_aware 24.4068 25.3364 33.37363 32.6529 39.9316 44.2650 10
#> cld
#> a
#> btotal(profmem(colSums2(dm_matrix, force_block_processing = TRUE)))
#> [1] 96100848total(profmem(colSums2(dm_matrix)))
#> [1] 6064
# -----------------------------------------------------------------------------
# Sparse (60% zero) with values in (0, 1)
# Fast, memory-efficient column sums of DelayedMatrix with ordinary matrix seed
#
# Generate some data
sparse_matrix <- dense_matrix
zero_idx <- sample(length(sparse_matrix), 0.6 * length(sparse_matrix))
sparse_matrix[zero_idx] <- 0
# Benchmark
dm_dgCMatrix <- DelayedArray(Matrix(sparse_matrix, sparse = TRUE))
class(seed(dm_dgCMatrix))
#> [1] "dgCMatrix"
#> attr(,"package")
#> [1] "Matrix"dm_dgCMatrix
#> <20000 x 600> sparse DelayedMatrix object of type "double":
#> [,1] [,2] [,3] ... [,599] [,600]
#> [1,] 0.7743685 0.0000000 0.0000000 . 0.89118118 0.00000000
#> [2,] 0.1972242 0.0000000 0.9198450 . 0.00000000 0.00000000
#> [3,] 0.9780138 0.0000000 0.4696158 . 0.31783791 0.00000000
#> [4,] 0.0000000 0.8797239 0.6474768 . 0.55217184 0.00000000
#> [5,] 0.3612444 0.0000000 0.0000000 . 0.08530977 0.39224147
#> ... . . . . . .
#> [19996,] 0.1949029 0.0776357 0.0000000 . 0.09703424 0.00000000
#> [19997,] 0.0000000 0.0000000 0.0000000 . 0.00000000 0.88389731
#> [19998,] 0.0000000 0.2115507 0.1934408 . 0.00000000 0.00000000
#> [19999,] 0.1898557 0.0000000 0.3511078 . 0.62939661 0.94601427
#> [20000,] 0.8788905 0.2530804 0.0000000 . 0.00000000 0.73272217microbenchmark(
block_processing = colSums2(dm_dgCMatrix, force_block_processing = TRUE),
seed_aware = colSums2(dm_dgCMatrix),
times = 10)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> block_processing 339.4852 340.9495 402.13222 352.7036 367.2005 846.0457 10
#> seed_aware 8.9544 8.9571 10.19116 10.5389 11.1878 11.3479 10
#> cld
#> a
#> btotal(profmem(colSums2(dm_dgCMatrix, force_block_processing = TRUE)))
#> [1] 249641608total(profmem(colSums2(dm_dgCMatrix)))
#> [1] 4848
# -----------------------------------------------------------------------------
# Dense with values in {0, 100} featuring runs of identical values
# Fast, memory-efficient column sums of DelayedMatrix with Rle-based seed
#
# Generate some data
runs <- rep(sample(100, 500000, replace = TRUE), rpois(500000, 100))
runs <- runs[seq_len(20000 * 600)]
runs_matrix <- matrix(runs,
nrow = 20000,
ncol = 600)
# Benchmark
dm_rle <- RleArray(Rle(runs),
dim = c(20000, 600))
class(seed(dm_rle))
#> [1] "SolidRleArraySeed"
#> attr(,"package")
#> [1] "DelayedArray"dm_rle
#> <20000 x 600> RleMatrix object of type "integer":
#> [,1] [,2] [,3] [,4] ... [,597] [,598] [,599] [,600]
#> [1,] 20 65 9 50 . 43 74 85 14
#> [2,] 20 65 9 50 . 43 74 85 14
#> [3,] 20 65 9 50 . 43 74 85 14
#> [4,] 20 65 9 50 . 43 74 85 14
#> [5,] 20 65 9 50 . 43 74 85 14
#> ... . . . . . . . . .
#> [19996,] 65 9 50 16 . 74 85 14 55
#> [19997,] 65 9 50 16 . 74 85 14 55
#> [19998,] 65 9 50 16 . 74 85 14 55
#> [19999,] 65 9 50 16 . 74 85 14 55
#> [20000,] 65 9 50 16 . 74 85 14 55microbenchmark(
block_processing = colSums2(dm_rle, force_block_processing = TRUE),
seed_aware = colSums2(dm_rle),
times = 10)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> block_processing 647.0173 677.0515 721.48616 704.9114 754.8140 849.9562 10
#> seed_aware 6.2375 6.5286 12.49204 6.8695 9.9354 55.2704 10
#> cld
#> a
#> btotal(profmem(colSums2(dm_rle, force_block_processing = TRUE)))
#> [1] 144256800total(profmem(colSums2(dm_rle)))
#> [1] 41048The development of ‘seed-aware’ methods is ongoing work (see the Roadmap), and for now only a few methods and seed-types have a ‘seed-aware’ method.
An extensive set of benchmarks is under development at http://peterhickey.org/BenchmarkingDelayedMatrixStats/.
A key feature of a DelayedArray is the ability to register ‘delayed
operations’. For example, let’s compute sin(dm_matrix):
system.time(sin_dm_matrix <- sin(dm_matrix))
#> user system elapsed
#> 0.02 0.00 0.01This instantaneous because the operation is not actually performed, rather
it is registered and only performed when the object is realized. All methods
in DelayedMatrixStats will correctly realise these delayed
operations before computing the final result. For example, let’s compute
colSums2(sin_dm_matrix) and compare check we get the correct answer:
all.equal(colSums2(sin_dm_matrix), colSums(sin(as.matrix(dm_matrix))))
#> [1] TRUEThe initial version of DelayedMatrixStats provides complete coverage of the matrixStats column-wise and row-wise API4 Some of the API is covered via inheritance to functionality in DelayedArray, allowing package developers to use these functions with DelayedMatrix objects as well as with ordinary matrix objects. This should simplify package development and assist authors to support to their software for large datasets stored in disk-backed data structures such as HDF5Array. Such large datasets are increasingly common with the rise of single-cell genomics.
Future releases of DelayedMatrixStats will improve the
performance of these methods, specifically by developing additional ‘seed-aware’
methods. The plan is to prioritise commonly used methods (e.g.,
colMeans2()/rowMeans2(), colSums2()/rowSums2(), etc.) and the
development of ‘seed-aware’ methods for the HDF5Matrix class. To do so, we
will leverage the beachmat package. Proof-of-concept code
has shown that this can greatly increase the performance when analysing such
disk-backed data.
Importantly, all package developers using methods from DelayedMatrixStats will immediately gain from performance improvements to these low-level routines. By using DelayedMatrixStats, package developers will be able to focus on higher level programming tasks and address important scientific questions and technological challenges in high-throughput biology.
sessionInfo()
#> R version 4.4.0 RC (2024-04-16 r86468 ucrt)
#> Platform: x86_64-w64-mingw32/x64
#> Running under: Windows Server 2022 x64 (build 20348)
#>
#> Matrix products: default
#>
#>
#> locale:
#> [1] LC_COLLATE=C
#> [2] LC_CTYPE=English_United States.utf8
#> [3] LC_MONETARY=English_United States.utf8
#> [4] LC_NUMERIC=C
#> [5] LC_TIME=English_United States.utf8
#>
#> time zone: America/New_York
#> tzcode source: internal
#>
#> attached base packages:
#> [1] stats4 stats graphics grDevices utils datasets methods
#> [8] base
#>
#> other attached packages:
#> [1] profmem_0.6.0 microbenchmark_1.4.10
#> [3] sparseMatrixStats_1.17.0 DelayedMatrixStats_1.27.1
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#>
#> loaded via a namespace (and not attached):
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#> [4] crayon_1.5.2 Rcpp_1.0.12 rhdf5filters_1.17.0
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