# qpHTF: Hastie Tibshirani Friedman iterative regression algorithm In qpgraph: Estimation of genetic and molecular regulatory networks from high-throughput genomics data

## Description

Performs maximum likelihood estimation of a covariance matrix given the independence constraints from an input undirected graph.

## Usage

 1 qpHTF(S, g, tol = 0.001, verbose = FALSE, R.code.only = FALSE)

## Arguments

 S input matrix, in the context of this package, the sample covariance matrix. g input undirected graph. tol tolerance under which the iterative algorithm stops. verbose show progress on calculations. R.code.only logical; if FALSE then the faster C implementation is used (default); if TRUE then only R code is executed.

## Details

This is an alternative to the Iterative Proportional Fitting (IPF) algorithm (see, Whittaker, 1990, pp. 182-185 and qpIPF) which also adjusts the input matrix to the independence constraints in the input undirected graph. However, differently to the IPF, it works by going through each of the vertices fitting the marginal distribution over the corresponding vertex boundary. It stops when the adjusted matrix at the current iteration differs from the matrix at the previous iteration in less or equal than a given tolerance value. This algorithm is described by Hastie, Tibshirani and Friedman (2009, pg. 634), hence we name it here HTF, and it has the advantage over the IPF that it does not require the list of maximal cliques of the graph which may be exponentially large. In contrast, it requires that the maximum boundary size of the graph is below the number of samples where the input sample covariance matrix S was estimated. For the purpose of exploring qp-graphs that meet such a requirement, one can use the function qpBoundary.

## Value

The input matrix adjusted to the constraints imposed by the input undirected graph, i.e., a maximum likelihood estimate of the sample covariance matrix that includes the independence constraints encoded in the undirected graph.

## Note

Thanks to Giovanni Marchetti for bringing us our attention to this algorithm and sharing an early version of its implementation on the R package ggm.

R. Castelo

## References

Castelo, R. and Roverato, A. A robust procedure for Gaussian graphical model search from microarray data with p larger than n. J. Mach. Learn. Res., 7:2621-2650, 2006.

Hastie, T., Tibshirani, R. and Friedman, J.H. The Elements of Statistical Learning, Springer, 2009.

Tur, I., Roverato, A. and Castelo, R. Mapping eQTL networks with mixed graphical Markov models. Genetics, 198(4):1377-1393, 2014.

Whittaker, J. Graphical Models in Applied Multivariate Statistics. Wiley, 1990.