LambdaGridGraphical {sharp}

Grid of penalty parameters (graphical model)

Description

Generates a relevant grid of penalty parameter values for penalised graphical models.

Usage

LambdaGridGraphical(
  xdata,
  pk = NULL,
  lambda_other_blocks = 0.1,
  K = 100,
  tau = 0.5,
  n_cat = 3,
  implementation = PenalisedGraphical,
  start = "cold",
  scale = TRUE,
  resampling = "subsampling",
  PFER_method = "MB",
  PFER_thr = Inf,
  FDP_thr = Inf,
  Lambda_cardinal = 50,
  lambda_max = NULL,
  lambda_path_factor = 0.001,
  max_density = 0.5,
  ...
)

Arguments

xdata

data matrix with observations as rows and variables as columns. For multi-block stability selection, the variables in data have to be ordered by group.

pk

optional vector encoding the grouping structure. Only used for multi-block stability selection where pk indicates the number of variables in each group. If pk=NULL, single-block stability selection is performed.

lambda_other_blocks

optional vector of parameters controlling the level of sparsity in neighbour blocks for the multi-block procedure. To use jointly a specific set of parameters for each block, lambda_other_blocks must be set to NULL (not recommended). Only used for multi-block stability selection, i.e. if length(pk)>1.

K

number of resampling iterations.

tau

subsample size. Only used if resampling="subsampling" and cpss=FALSE.

n_cat

computation options for the stability score. Default is NULL to use the score based on a z test. Other possible values are 2 or 3 to use the score based on the negative log-likelihood.

implementation

function to use for graphical modelling. If implementation=PenalisedGraphical, the algorithm implemented in glassoFast is used for regularised estimation of a conditional independence graph. Alternatively, a user-defined function can be provided.

start

character string indicating if the algorithm should be initialised at the estimated (inverse) covariance with previous penalty parameters (start="warm") or not (start="cold"). Using start="warm" can speed-up the computations, but could lead to convergence issues (in particular with small Lambda_cardinal). Only used for implementation=PenalisedGraphical (see argument "start" in glassoFast).

scale

logical indicating if the correlation (scale=TRUE) or covariance (scale=FALSE) matrix should be used as input of glassoFast if implementation=PenalisedGraphical. Otherwise, this argument must be used in the function provided in implementation.

resampling

resampling approach. Possible values are: "subsampling" for sampling without replacement of a proportion tau of the observations, or "bootstrap" for sampling with replacement generating a resampled dataset with as many observations as in the full sample. Alternatively, this argument can be a function to use for resampling. This function must use arguments named data and tau and return the IDs of observations to be included in the resampled dataset.

PFER_method

method used to compute the upper-bound of the expected number of False Positives (or Per Family Error Rate, PFER). If PFER_method="MB", the method proposed by Meinshausen and Bühlmann (2010) is used. If PFER_method="SS", the method proposed by Shah and Samworth (2013) under the assumption of unimodality is used.

PFER_thr

threshold in PFER for constrained calibration by error control. If PFER_thr=Inf and FDP_thr=Inf, unconstrained calibration is used (the default).

FDP_thr

threshold in the expected proportion of falsely selected features (or False Discovery Proportion) for constrained calibration by error control. If PFER_thr=Inf and FDP_thr=Inf, unconstrained calibration is used (the default).

Lambda_cardinal

number of values in the grid of parameters controlling the level of sparsity in the underlying algorithm.

lambda_max

optional maximum value for the grid in penalty parameters. If lambda_max=NULL, the maximum value is set to the maximum covariance in absolute value. Only used if implementation=PenalisedGraphical.

lambda_path_factor

multiplicative factor used to define the minimum value in the grid.

max_density

threshold on the density. The grid is defined such that the density of the estimated graph does not exceed max_density.

...

additional parameters passed to the functions provided in implementation or resampling.

Value

A matrix of lambda values with length(pk) columns and Lambda_cardinal rows.

See Also

Other lambda grid functions: LambdaGridRegression(), LambdaSequence()

Examples

# Single-block simulation
set.seed(1)
simul <- SimulateGraphical()

# Generating grid of 10 values
Lambda <- LambdaGridGraphical(xdata = simul$data, Lambda_cardinal = 10)

# Ensuring PFER < 5
Lambda <- LambdaGridGraphical(xdata = simul$data, Lambda_cardinal = 10, PFER_thr = 5)

# Multi-block simulation
set.seed(1)
simul <- SimulateGraphical(pk = c(10, 10))

# Multi-block grid
Lambda <- LambdaGridGraphical(xdata = simul$data, pk = c(10, 10), Lambda_cardinal = 10)

# Denser neighbouring blocks
Lambda <- LambdaGridGraphical(
  xdata = simul$data, pk = c(10, 10),
  Lambda_cardinal = 10, lambda_other_blocks = 0
)

# Using different neighbour penalties
Lambda <- LambdaGridGraphical(
  xdata = simul$data, pk = c(10, 10),
  Lambda_cardinal = 10, lambda_other_blocks = c(0.1, 0, 0.1)
)
stab <- GraphicalModel(
  xdata = simul$data, pk = c(10, 10),
  Lambda = Lambda, lambda_other_blocks = c(0.1, 0, 0.1)
)
stab$Lambda

# Visiting from empty to full graphs with max_density=1
Lambda <- LambdaGridGraphical(
  xdata = simul$data, pk = c(10, 10),
  Lambda_cardinal = 10, max_density = 1
)
bigblocks <- BlockMatrix(pk = c(10, 10))
bigblocks_vect <- bigblocks[upper.tri(bigblocks)]
N_blocks <- unname(table(bigblocks_vect))
N_blocks # max number of edges per block
stab <- GraphicalModel(xdata = simul$data, pk = c(10, 10), Lambda = Lambda)
apply(stab$Q, 2, max, na.rm = TRUE) # max average number of edges from underlying algo