riPEER {mdpeer}

Graph-constrained regression with penalty term being a linear combination of graph-based and ridge penalty terms

Description

Graph-constrained regression with penalty term being a linear combination of graph-based and ridge penalty terms.

See Details for model description and optimization problem formulation.

Usage

riPEER(Q, y, Z, X = NULL, optim.metod = "rootSolve",
  rootSolve.x0 = c(1e-05, 1e-05), rootSolve.Q0.x0 = 1e-05, sbplx.x0 = c(1,
  1), sbplx.lambda.lo = c(10^(-5), 10^(-5)), sbplx.lambda.up = c(1e+06,
  1e+06), compute.boot.CI = FALSE, boot.R = 1000, boot.conf = 0.95,
  boot.set.seed = TRUE, boot.parallel = "multicore", boot.ncpus = 4,
  verbose = TRUE)

Arguments

Q

graph-originated penalty matrix (p \times p); typically: a graph Laplacian matrix

y

response values matrix (n \times 1)

Z

design matrix (n \times p) modeled as random effects variables (to be penalized in regression modeling); assumed to be already standarized

X

design matrix (n \times k) modeled as fixed effects variables (not to be penalized in regression modeling); if does not contain columns of 1s, such column will be added to be treated as intercept in a model

optim.metod

optimization method used to optimize \lambda = (\lambda_Q, \lambda_R)

• "rootSolve" (default) - optimizes by finding roots of non-linear equations by the Newton-Raphson method; from rootSolve package

• "sbplx" - optimizes with the use of Subplex Algorithm: 'Subplex is a variant of Nelder-Mead that uses Nelder-Mead on a sequence of subspaces'; from nloptr package

rootSolve.x0

vector containing initial guesses for \lambda = (\lambda_Q, \lambda_R) used in "rootSolve" algorithm

rootSolve.Q0.x0

vector containing initial guess for \lambda_R used in "rootSolve" algorithm

sbplx.x0

vector containing initial guesses for \lambda = (\lambda_Q, \lambda_R) used in "sbplx" algorithm

sbplx.lambda.lo

vector containing minimum values of \lambda = (\lambda_Q, \lambda_R) grid search in "sbplx" algorithm

sbplx.lambda.up

vector containing mximum values of \lambda = (\lambda_Q, \lambda_R) grid search in "sbplx" algorithm

compute.boot.CI

logical whether or not compute bootstrap confidence intervals for b regression coefficient estimates

boot.R

number of bootstrap replications used in bootstrap confidence intervals computation

boot.conf

confidence level assumed in bootstrap confidence intervals computation

boot.set.seed

logical whether or not set seed in bootstrap confidence intervals computation

boot.parallel

value of parallel argument in boot function in bootstrap confidence intervals computation

boot.ncpus

value of ncpus argument in boot function in bootstrap confidence intervals computation

verbose

logical whether or not set verbose mode (print out function execution messages)

Details

Estimates coefficients of linear model of the formula:

y = X\beta + Zb + \varepsilon

where:

• y - response,

• X - data matrix,

• Z - data matrix,

• \beta - regression coefficients, not penalized in estimation process,

• b - regression coefficients, penalized in estimation process and for whom there is, possibly a prior graph of similarity / graph of connections available.

The method uses a penalty being a linear combination of a graph-based and ridge penalty terms:

\beta_{est}, b_{est}= arg \; min_{\beta,b} \{ (y - X\beta - Zb)^T(y - X\beta - Zb) + \lambda_Qb^TQb + \lambda_Rb^Tb \}

where:

• Q - a graph-originated penalty matrix; typically: a graph Laplacian matrix,

• \lambda_Q - regularization parameter for a graph-based penalty term

• \lambda_R - regularization parameter for ridge penalty term

The two regularization parameters, \lambda_Q and \lambda_R, are estimated as ML estimators from equivalent Linear Mixed Model optimizaton problem formulation (see: References).

• Graph-originated penalty term allows imposing similarity between coefficients based on graph information given.

• Ridge-originated penalty term facilitates parameters estimation: it reduces computational issues arising from singularity in a graph-originated penalty matrix and yields plausible results in situations when graph information is not informative.

Bootstrap confidence intervals computation is available (not set as a default option).

Value

b.est

vector of b coefficient estimates

beta.est

vector of \beta coefficient estimates

lambda.Q

\lambda_Q regularization parameter value

lambda.R

\lambda_R regularization parameter value

lambda.2

lambda.R/lambda.Q value

boot.CI

data frame with two columns, lower and upper, containing, respectively, values of lower and upper bootstrap confidence intervals for b regression coefficient estimates

obj.fn.val

optimization problem objective function value

References

Karas, M., Brzyski, D., Dzemidzic, M., J., Kareken, D.A., Randolph, T.W., Harezlak, J. (2017). Brain connectivity-informed regularization methods for regression. doi: https://doi.org/10.1101/117945

Examples

set.seed(1234)
n <- 200
p1 <- 10
p2 <- 90
p <- p1 + p2
# Define graph adjacency matrix
A <- matrix(rep(0, p*p), nrow = p, ncol = p)
A[1:p1, 1:p1] <- 1
A[(p1+1):p, (p1+1):p] <- 1
L <- Adj2Lap(A)
# Define Q penalty matrix as graph Laplacian matrix normalized)
Q <- L2L.normalized(L)
# Define Z,X design matrices and aoutcome y
Z <- matrix(rnorm(n*p), nrow = n, ncol = p)
b.true <- c(rep(1, p1), rep(0, p2))
X <- matrix(rnorm(n*3), nrow = n, ncol = 3)
beta.true <- runif(3)
intercept <- 0
eta <- intercept + Z %*% b.true + X %*% beta.true
R2 <- 0.5
sd.eps <- sqrt(var(eta) * (1 - R2) / R2)
error <- rnorm(n, sd = sd.eps)
y <- eta + error

## Not run: 
riPEER.out <- riPEER(Q, y, Z, X)
plt.df <- data.frame(x = 1:p, y = riPEER.out$b.est)
ggplot(plt.df, aes(x = x, y = y, group = 1)) + geom_line() + labs("b estimates")

## End(Not run)

## Not run: 
# riPEER with 0.95 bootstrap confidence intervals computation
riPEER.out <- riPEER(Q, y, Z, X, compute.boot.CI = TRUE, boot.R = 500)
plt.df <- data.frame(x = 1:p, 
                     y = riPEER.out$b.est, 
                     lo = riPEER.out$boot.CI[,1], 
                     up =  riPEER.out$boot.CI[,2])
ggplot(plt.df, aes(x = x, y = y, group = 1)) + geom_line() +  
  geom_ribbon(aes(ymin=lo, ymax=up), alpha = 0.3)

## End(Not run)