# Iterative proportional fitting algorithm

### Description

Performs maximum likelihood estimation of a covariance matrix given the independence constraints from an input list of (maximal) cliques.

### Usage

1 |

### Arguments

`vv` |
input matrix, in the context of this package, the sample covariance matrix. |

`clqlst` |
list of maximal cliques obtained from an undirected graph
by using the function |

`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

The Iterative proportional fitting algorithm (see, Whittaker, 1990, pp. 182-185) adjusts the input matrix to the independence constraints in the undirected graph from where the input list of cliques belongs to, by going through each of the cliques fitting the marginal distribution over the clique for the fixed conditional distribution of the clique. 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.

### Value

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

### Author(s)

R. Castelo and A. Roverato

### 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.

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.

### See Also

`qpGetCliques`

`qpPAC`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
require(graph)
require(mvtnorm)
nVar <- 50 ## number of variables
nObs <- 100 ## number of observations to simulate
set.seed(123)
g <- randomEGraph(as.character(1:nVar), p=0.15)
Sigma <- qpG2Sigma(g, rho=0.5)
X <- rmvnorm(nObs, sigma=as.matrix(Sigma))
## MLE of the sample covariance matrix
S <- cov(X)
## more efficient MLE of the sample covariance matrix using IPF
clqs <- qpGetCliques(g, verbose=FALSE)
S_ipf <- qpIPF(S, clqs)
## get the adjacency matrix and put the diagonal to one
A <- as(g, "matrix")
diag(A) <- 1
## entries in S and S_ipf for present edges in g should coincide
max(abs(S_ipf[A==1] - S[A==1]))
## entries in the inverse of S_ipf for missing edges in g should be zero
max(solve(S_ipf)[A==0])
``` |