diffuseMat: Generic diffusion function

In wjawaid/roots: Reconstructing Ordered Ontogenic Trajectories

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.

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^{-0.5} * D^{-1} * A * D^{-1} * B^{-0.5} = R^\prime * L^\prime

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

B^{-1} * D^{-1} * A * D^{-1} * R = R * L

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

B^{-1} * D^{-1} * A * D^{-1} * B^{-0.5} = B^{-0.5} * R' * L'

and

R = B^{-0.5} * R'

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

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 1/sqrt(n), 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.

Author(s)

Wajid Jawaid

References

Haghverdi, L., Buettner, F., Theis, F.J., 2015. Diffusion maps for high-dimensional single-cell analysis of differentiation data. Bioinformatics 31, 2989<e2><80><93>2998.

Haghverdi, L., B<c3><bc>ttner, M., Wolf, F.A., Buettner, F., Theis, F.J., 2016. Diffusion pseudotime robustly reconstructs lineage branching. Nat Meth 13, 845<e2><80><93>848.

Angerer, P., Haghverdi, L., B<c3><bc>ttner, M., Theis, F.J., Marr, C., Buettner, F., 2016. destiny: diffusion maps for large-scale single-cell data in R. Bioinformatics 32, 1241<e2><80><93>1243.

Examples

## Not run: 
xx <- diffuseMat(x)

## End(Not run)

wjawaid/roots documentation built on May 20, 2019, 11:37 a.m.