SMART: Support-vector Margin Algoritm for Reliability esTimation

Description

Calculate a failure probability with SMART method. This should not be used by itself but only through S2MART.

Usage

SMART(
  dimension,
  lsf,
  N1 = 10000,
  N2 = 50000,
  N3 = 2e+05,
  Nu = 50,
  lambda1 = 7,
  lambda2 = 3.5,
  lambda3 = 1,
  tune_cost = c(1, 10, 100, 1000),
  tune_gamma = c(0.5, 0.2, 0.1, 0.05, 0.02, 0.01),
  clusterInMargin = TRUE,
  alpha_margin = 1,
  k1 = round(6 * (dimension/2)^(0.2)),
  k2 = round(12 * (dimension/2)^(0.2)),
  k3 = k2 + 16,
  X = NULL,
  y = NULL,
  failure = 0,
  limit_fun_MH = NULL,
  sampling_strategy = "MH",
  seeds = NULL,
  seeds_eval = NULL,
  burnin = 20,
  thinning = 4,
  plot = FALSE,
  limited_plot = FALSE,
  add = FALSE,
  output_dir = NULL,
  z_MH = NULL,
  z_lsf = NULL,
  verbose = 0
)

Value

An object of class list containing the failure probability and some more outputs as described below:

proba: The estimated failure probability.
cov: The coefficient of variation of the Monte-Carlo probability estimate.
Ncall: The total number of calls to the limit_state_function.
X: The final learning database, ie. all points where lsf has been calculated.
y: The value of the limit_state_function on the learning database.
meta_fun: The metamodel approximation of the limit_state_function. A call output is a list containing the value and the standard deviation.
meta_model: The final metamodel.
points: Points in the failure domain according to the metamodel.
meta_eval: Evaluation of the metamodel on these points.
z_meta: If plot==TRUE, the evaluation of the metamodel on the plot grid.

Arguments

dimension: the dimension of the input space
lsf: the limit-state function
N1: Number of samples for the (L)ocalisation step
N2: Number of samples for the (S)tabilisation step
N3: Number of samples for the (C)onvergence step
Nu: Size of the first Design of Experiments
lambda1: Relaxing parameter for MH algorithm at step L
lambda2: Relaxing parameter for MH algorithm at step S
lambda3: Relaxing parameter for MH algorithm at step C
tune_cost: Input for tuning cost paramter of the SVM
tune_gamma: Input for tuning gamma parameter of the SVM
clusterInMargin: Enforce selected clusterised points to be in margin
alpha_margin: a real value defining the margin. While 1 is the ‘real’ margin for a SVM, one can decide here to stretch it a bit.
k1: Rank of the first iteration of step S
k2: Rank of the first iteration of step C
k3: Rank of the last iteration of step C
X: Coordinates of alredy known points
y: Value of the LSF on these points
failure: Failure threshold
limit_fun_MH: Define an area of exclusion with a limit function
sampling_strategy: Either MH for Metropolis-Hastings of AR for accept-reject
seeds: If some points are already known to be in the subdomain defined by limit_fun_MH
seeds_eval: Value of the metamodel on these points
burnin: Burnin parameter for MH
thinning: Thinning parameter for MH
plot: Set to TRUE for a full plot, ie. refresh at each iteration
limited_plot: Set to TRUE for a final plot with final DOE, metamodel and LSF
add: If plots are to be added to the current device
output_dir: If plots are to be saved in jpeg in a given directory
z_MH: For plots, if the limit_fun_MH has already been evaluated on the grid
z_lsf: For plots, if LSF has already been evaluated on the grid
verbose: Either 0 for almost no output, 1 for medium size output and 2 for all outputs

Author

Clement WALTER clementwalter@icloud.com

Details

SMART is a reliability method proposed by J.-M. Bourinet et al. It makes uses of a SVM-based metamodel to approximate the limit state function and calculates the failure probability with a crude Monte-Carlo method using the metamodel-based limit state function. As SVM is a classification method, it makes use of limit state function values to create two classes : greater and lower than the failure threshold. Then the border is taken as a surogate of the limit state function.

Concerning the refinement strategy, it distinguishes 3 stages, known as Localisation, Stalibilsation and Convergence stages. The first one is proposed to reduce the margin as much as possible, the second one focuses on switching points while the last one works on the final Monte-Carlo population and is designed to insure a strong margin; see F. Deheeger PhD thesis for more information.

References

J.-M. Bourinet, F. Deheeger, M. Lemaire:
Assessing small failure probabilities by combined Subset Simulation and Support Vector Machines
Structural Safety (2011)
F. Deheeger:
Couplage mecano-fiabiliste : 2SMART - methodologie d'apprentissage stochastique en fiabilite
PhD. Thesis, Universite Blaise Pascal - Clermont II, 2008