stress_moment {SWIM}

Stressing Moments

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed model components (random variables) fulfill the moment constraints. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_moment(
  x,
  f,
  k,
  m,
  normalise = TRUE,
  show = FALSE,
  names = NULL,
  log = FALSE,
  ...
)

Arguments

x

A vector, matrix or data frame containing realisations of random variables. Columns of x correspond to random variables; OR
A SWIM object, where x corresponds to the underlying data of the SWIM object.

f

A function, or list of functions, that, applied to x, constitute the moment constraints.

k

A vector or list of vectors, same length as f, indicating which columns of x each function in f operates on.
When f is a list, k[[i]] corresponds to the input variables of f[[i]].

m

Numeric vector, same length as f, containing the stressed moments of f(x). Must be in the range of f(x).

normalise

Logical. If true, values of f(x) are linearly scaled to the unit interval.

show

Logical. If true, print the result of the call to nleqslv.

names

Character vector, the names of stressed models.

log

Boolean, the option to print weights' statistics.

...

Additional arguments to be passed to nleqslv.

Details

The moment constraints are given by E^Q( f(x) ) = m, where E^Q denotes the expectation under the stressed model. stress_moment solves the subsequent set of equations with respect to theta, using nleqslv from package nleqslv:

E^Q( f(x) ) = E( f(x) * exp(theta * f(x)) ) = m.

There is no guarantee that the set of equations has a solution, or that the solution is unique. SWIM will return a warning if the termination code provided by nleqslv is different from 1 (convergence has been achieved). It is recommended to check the result of the call to nleqslv using the "show" argument. The user is referred to the nleqslv documentation for further details.

Normalising the data may help avoiding numerical issues when the range of values is wide.

Value

A SWIM object containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is a vector of scenario weights;

• type = "moment";

• specs, a list, each component corresponds to a different stress and contains f, k and m.

See SWIM for details.

The function call will print a message containing the termination code returned by the call to nleqslv and a table with the required and achieved moment, and the absolute and relative error.

References

Pesenti SM, Millossovich P, Tsanakas A (2019). “Reverse sensitivity testing: What does it take to break the model?” European Journal of Operational Research, 274(2), 654–670.

Pesenti S BAMPTA (2020). “Scenario Weights for Importance Measurement (SWIM) - An R package for sensitivity analysis.” Annals of Actuarial Science 15.2 (2021): 458-483. Available at SSRN: https://www.ssrn.com/abstract=3515274.

Csiszar I (1975). “I-divergence geometry of probability distributions and minimization problems.” The Annals of Probability, 146–158.

See Also

See stress_mean for stressing means and stress_mean_sd for stressing mean and standard deviation jointly.

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

set.seed(0)
x <- data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2),
  "beta" = rbeta(1000, shape1 = 2, shape2 = 2)))

## stressing covariance of columns 1, 2 while leaving the means unchanged
res1 <- stress_moment(x = x,
  f = list(function(x)x, function(x)x, function(x)x[1] * x[2]),
  k = list(1, 2, c(1, 2)), m = c(0, 2, 0.5),
  method = "Newton", control = list(maxit = 1000, ftol = 1E-10))
## means under the stressed model
summary(res1)
apply(x, 2, stats::weighted.mean, w = get_weights(res1))
## covariance of columns 1,2 under the stressed model
stats::weighted.mean(x[, 1] * x[, 2], w = get_weights(res1))

## stressing jointly the tail probabilities of columns 1, 3
res2 <- stress_moment(x = x,
  f = list(function(x)(x > 1.5), function(x)(x > 0.9)),
  k = list(1, 3), m = c(0.9, 0.9))
summary(res2)
## probabilities under the stressed model
mean((x[, 1] > 1.5) * get_weights(res2))
mean((x[, 3] > 0.9) * get_weights(res2))