Dataset: CD8+ T cell gene expression data

Description

The CD8T dataset provides the gene expression data of fetal CD8+ T cells obtained in a single-cell RNA-seq experiment.

Usage

data(CD8T)

Format

A data frame with 9369 rows (cells) and 1000 columns (genes).

Source

Suo et al., Science (2022).

References

Suo, C., Dann, E., Goh, I., Jardine, L., Kleshchevnikov, V., Park, J.-E., Botting, R. A., et al. "Mapping the developing human immune system across organs." Science 376(6597), eabo0510 (2022).

Monge–Kantorovich ranks (uniform OT via squared distances)

Description

Computes the optimal matching that maps each observation in X to a reference point in U using uniform weights and squared Euclidean cost. Internally uses transport::transport(method = "networkflow", p = 2). In 1D, this reduces to a rank-based matching sort(U)[rank(X, ties.method = "random")].

Usage

MK_rank(X, U)

Arguments

Details

• Rows must match: nrow(X) == nrow(U) (otherwise an error is thrown).

• Columns must match: ncol(X) == ncol(U) (otherwise an error is thrown).

• Weights are uniform (1/n) and the cost matrix is the sum of squared coordinate differences across columns.

• In 1D, ties in X are broken at random via ties.method = "random"; use set.seed() for reproducibility.

Value

If ncol(X) == 1, a numeric vector of length n containing the entries of U reordered to match the ranks of X. Otherwise, a numeric n \times d matrix whose i-th row is the matched row of U corresponding to the i-th row of X.

Dependencies

Requires the transport package.

Examples

# 1D example (set seed for reproducible tie-breaking)
set.seed(1)
x <- rnorm(10)
u <- seq(0, 1, length.out = 10)
MK_rank(x, u)

# 2D example
set.seed(42)
X <- matrix(rnorm(200), ncol = 2)   # 100 x 2
U <- matrix(runif(200),  ncol = 2)  # 100 x 2
R <- MK_rank(X, U)
dim(R)  # 100 2

Coverage-based Dependence Measure with Optional Visualisation

Description

Computes the coverage correlation coefficient between input x and y, as introduced in the arXiv preprint. This coefficient measures the dependence between two random variables or vectors.

Usage

coverage_correlation(
  x,
  y,
  visualise = FALSE,
  method = c("auto", "exact", "approx"),
  M = NULL,
  na.rm = TRUE
)

Arguments

Details

The procedure is as follows:

1. Calculate the rank transformations (r_x, r_y) of the inputs x and y.

2. Construct small cubes (in 2D, squares) of volume n^{-1} centered at each rank-transformed point.

3. Compute the total area of the union of these cubes, intersected with [0,1]^d where d = d_x + d_y.

The coverage correlation coefficient is then calculated based on this union area.

For more details, please refer to the original paper: the arXiv preprint.

The method argument controls how the computation is performed:

• "exact": Computes the exact value.

• "approx": Uses a Monte Carlo approximation with M sample points.

• "auto": Automatically selects a method based on the total number of columns in x and y: if more than 6, "approx" is used (with M = nrow(x)^{1.5} if M is not provided); otherwise, "exact" is used.

Value

A list with four elements:

• stat – The numeric value of the coverage correlation coefficient.

• pval – The p-value, calculated using the exact variance under the null hypothesis of independence between x and y.

• method – A character string indicating the computation method used.

• mc_se – A numeric value. If method "approx" was used mc_se is the standard error of the Monte Carlo approximation, otherwise it is 0.

Examples

set.seed(1)
n <- 100
x <- runif(n)
y <- sin(3*x) + runif(n) * 0.01
coverage_correlation(x, y, visualise = TRUE)

Total volume of union of rectangles

Description

Total volume of union of rectangles

Usage

covered_volume(zmin, zmax)

Arguments

Details

This is a wrapper of the C_covered_volume_partitioned function in C

Value

a numeric value of the volume of the union

Total volume of union of rectangles using Monte Carlo integration

Description

Total volume of union of rectangles using Monte Carlo integration

Usage

covered_volume_mc(zmin_s, zmax_s, M)

Arguments

Details

This is a wrapper of the C_covered_volume_mc function in C

Value

a list of the estimated volume of the union and its standard error

Total volume of union of rectangles using volume hashing

Description

Total volume of union of rectangles using volume hashing

Usage

covered_volume_partitioned(zmin, zmax)

Arguments

Details

This is a wrapper of the C_covered_volume_partitioned function in C

Value

a numeric value of the volume of the union

Plot a collection of axis-aligned rectangles in the unit square

Description

Draws rectangles specified by their xmin, xmax, ymin, and ymax, optionally adding them to an existing plot. When add = FALSE, a fresh [0,1]\times[0,1] plot with a grid and equal aspect ratio is created.

Usage

plot_rectangles(xmin, xmax, ymin, ymax, add = FALSE)

Arguments

Value

Invisibly returns NULL. Use this function for its plotting output, not for a returned value.

Split rectangles by wrapping them around edges of [0,1]^d

Description

Split rectangles by wrapping them around edges of [0,1]^d

Usage

split_rectangles(zmin, zmax)

Arguments

Details

This is a wrapper of the C_split_rectangles function implemented in C

Value

a list of zmin and zmax, describing the bottom-left and top-right coordinates of splitted rectangles

Variance of the the excess vacancy

Description

Exact formula for n times the variance of the excess vacancy. For independent X and Y, the variance of the coverage correlation coefficient is obtained by dividing the returned value by n(1 - e^{-1})^2. check the arXiv preprint for more details

Usage

variance_formula(n, d)

Arguments

Value

variance formula in paper