roots (version 1.0)

diffuseMat: Generic diffusion function

Description

Generic diffusion function using automated individualised sigma calculation

Usage

diffuseMat(data, ndims = 20, nsig = 5, removeFirst = TRUE,
  useARPACK = TRUE, distfun = NULL, sigmas = NULL, sqdistmat = NULL)

Arguments

data

Matrix of data with genes in rows and cells in columns.

ndims

Number of dimensions to return

nsig

For automatic sigma calculation

removeFirst

Default TRUE. Removes the first eigenvector

useARPACK

Default TRUE. Uses Arnoldi algorithm for eignvector calculations

distfun

A different distance function that returns the squared distance

sigmas

Manually provide sigma

sqdistmat

Squared distance matrix. Give your own squared distance matrix.

Value

List output containing:

values	Eigenvalues, excluding the first eigenvalue, which should always be 1.
vectors	Matrix of eigen vectors in columns, first eigen vector removed.
nconv	Number of eigen vectors/values that converged.
niter	Iterations taken for Arnoldi algorithm to converge.
nops	Number of operations.
val0	1st eigen value - should be 1. If not be suspicious!
vec0	1st eigen vector - should be \(n^{-\frac{1}{2}}\), where n is the number of cells/samples.
usedARPACK	Predicates use of ARPACK for spectral decomposition.
distfun	Function used to calculate the squared distance.
nn	Number of nearest neighbours used for calculating sigmas.
d2	Matrix of squared distances, returned from distfun.
sigmas	Vector of sigmas. Same length as number of cells if individual
	sigmas were calculated, otherwise a scalar if was supplied.
gaussian	Unnormalised transition matrix after applying Gaussian.
markov	Normalised gaussian matrix.
densityCorrected	Matrix after applying density correction to markov.

Details

Generic diffusion function using automated individualised sigma calculation.

A Gaussian kernel is applied to the chosen distance metric producing an \(n \times n\) square unnormalised symmetric transition matrix, \(A\). Let \(D\) be an \(n \times n\) diagonal matrix with row(column) sums of \(A\) as entries. The density corrected transition matrix will now be:

$$D^{-1} A D^{-1}$$

and can be normalised:

$$B^{-1} D^{-1} A D^{-1}$$

where \(B\) is an \(n \times n\) diagonal matrix with row sums of the density corrected transition matrix as entries. The eigen decomposition of this matrix can be simplified by solving the symmetric system:

$$B^{-\frac{1}{2}} D^{-1} A D^{-1} B^{-\frac{1}{2}} R^\prime = % R^\prime \lambda^\prime$$

where \(R^\prime\) is a matrix of the right eigenvectors that solve the system and \(\lambda^\prime\) is the corresponding eigenvalue diagonal matrix. Now the solution of:

$$B^{-1} D^{-1} A D^{-1} R = R \lambda$$

in terms of \(R^\prime\) and \(B^{-\frac{1}{2}}\) is:

$$B^{-1} D^{-1} A D^{-1} B^{-\frac{1}{2}} R^\prime = % B^{-\frac{1}{2}} R^\prime \lambda^\prime$$

and

$$R = B^{-\frac{1}{2}} R^\prime$$

This \(R\) without the first eigen vector is returned as the diffusion map.

References

Haghverdi, L., Buettner, F., Theis, F.J., 2015. Diffusion maps for high-dimensional single-cell analysis of differentiation data. Bioinformatics 31, 2989<U+2013>2998.

Haghverdi, L., B<U+00FC>ttner, M., Wolf, F.A., Buettner, F., Theis, F.J., 2016. Diffusion pseudotime robustly reconstructs lineage branching. Nat Meth 13, 845<U+2013>848.

Angerer, P., Haghverdi, L., B<U+00FC>ttner, M., Theis, F.J., Marr, C., Buettner, F., 2016. destiny: diffusion maps for large-scale single-cell data in R. Bioinformatics 32, 1241<U+2013>1243.

Examples

# NOT RUN {
xx <- diffuseMat(x)
# }