depgraph: Maximum Variance Adjacency Matrix

In jrzubizarreta/depinf: Statistical Inference for Linear Unbiased Estimators under with Constrained Dependent Observations

Description

Function for finding the adjacency matrix that maximizes the variance of a mean estimate given general constraints on the degree of dependence of the observations.

Usage

	depgraph(v, d, GR = NULL, solver = "glpk", approximate = 1)

Arguments

v	a matrix with the estimated (sample) covariance matrix.
d	a vector with the observed degree; if length(d)=1 then d is the total number of non-zero (off-diagonal) elements of the adjancency matrix A (for solving a more relaxed version of the problem); if length(d)=nrow(v) then d is the total number of non-zero (off-diagonal) elements of each row of A.
GR	a matrix with the known dependencies. The default is GR = NULL.
solver	a string with the optimization solver to be used. The options are: cplex, glpk and gurobi. The default solver is glpk but cplex or gurobi are much faster. Note that cplex and gurobi require a license, which is free for people affiliated to universities, but not otherwise. Between cplex and gurobi, the gurobi interface for R is much easier to install.
approximate	a scalar that determines whether an exact solution is to be found by solving the original integer programming problem (approximate = 0), or whether an approximate solution is to be obtained by solving the relaxed problem via linear programming (approximate = 1). Of course, obtaining the approximate solution is faster but will result in more conservative variance estimates. The default is approximate = 1.

Value

A list with the following objects:

A_max	adjacency matrix that maximizes the variance;
obj_val	objective value of the graph optimization problem at the optimum;
time	time elapsed to find the optimal solution.

Author(s)

Peter M. Aronow <[e-mail]>, Forrest W. Crawford <[e-mail]>, Jose R. Zubizarreta <[e-mail]>.

References

Aronow, P. M., Crawford, F. W., and Zubizarreta, J. R., (2017), "Confidence intervals for linear unbiased estimators under constrained dependence," submitted, X, X-X.

Examples

################################# 
# Data
#################################

# Example with 100 nodes
data(example)

# Total number of observations
n = nrow(dat)

# Observed data 
# Observed dependencies (observed (known) a_ij's; this is A_R in (5) in the paper)
GR = GR
# Observed degrees
d = dat$degree
# Observed outcomes
y = dat$hiv

################################# 
# General case
#################################

# Solve (5) in the paper
# Sample covariance matrix
my = mean(y)
v = outer(y-my, y-my)
# No known dependencies (so in (5) in the paper A_R (the matrix of known dependencies) is the n times n matrix of 0's)
A_max_gral_no_dep = depgraph(v, d, NULL, "glpk", 1)$A_max
# Some known dependencies
A_max_gral_some_dep = depgraph(v, d, GR, "glpk", 1)$A_max

################################# 
# Homoskedastic case
#################################

# Solve (8) in the paper
# No known dependencies (so in (8) in the paper A_R (the matrix of known dependencies) is the n times n matrix of 0's)
v = matrix(1, nrow=n, ncol=n)
A_max_homo_no_dep = depgraph(v, d, NULL, "glpk", 0)$A_max
# Some known dependencies
A_max_homo_some_dep = depgraph(v, d, GR, "glpk", 0)$A_max
	

jrzubizarreta/depinf documentation built on May 20, 2019, 2:07 a.m.