Examples

In the following, some simple usage examples are given.

Basics

We will start with some simple basics that are necessary for all following examples.

Besides the actual rearrangement_algorithm package, we will use the scipy.stats module to specify the marginal distributions.

The .ppf method of a distribution implements the quantile function, which we will heavily use in the following examples.

Basic Rearrange

import numpy as np
import rearrangement_algorithm as ra

# create a matrix (10x3) where each columns contains numbers 0 to .9
X = np.tile(np.linspace(0, 1, 10), (3, 1)).T
print(X)

# the row sums are varying a lot
print(np.sum(X, axis=1))

# rearrange it
X_ra = ra.basic_rearrange(X, min)
print(X_ra)

# the row sums are now very balanced
print(np.sum(X_ra, axis=1))

Comonotonic Rearrangement

The first example is to generate a comonotonic rearrangement of random variables with given marginals. For the following example, we will use \(X_1\sim\exp(1)\), \(X_2\sim\mathcal{N}(0, 1)\), and \(X_3\sim\mathcal{U}[0, 1]\).

The important function to create the comonotonic rearrangement is rearrangement_algorithm.create_comonotonic_ra().

from scipy import stats
import rearrangement_algorithm as ra

# create the random variables
X1 = stats.expon()  # exponential distribution
X2 = stats.norm()  # normal distribution
X3 = stats.uniform()  # uniform distribution

# quantile functions are given by the .ppf method in scipy
qf = [X1.ppf, X2.ppf, X3.ppf]

# create rearrangement
level = 0.5
X_ra_lower, X_ra_upper = ra.create_comonotonic_ra(level, qf, 10)

# all columns are ordered in increasing order
print(X_ra_lower)
print(X_ra_upper)

Bounds on the Value-at-Risk (VaR)

One relevant use case of the rearrangement algorithm is the numerical approximation of bounds on the value-at-risk (VaR) of dependent risks.

The following example illustrates the use of the rearrangement_algorithm.bounds_VaR() function using the following example from the paper “Computation of sharp bounds on the distribution of a function of dependent risks” (Puccetti, Rüschendorf, 2012).

Given three Pareto(2)-distributed random variables, we calcuate the lower bound on the VaR (based on the sum of the three random variables). For comparison, the numbers can be found in the above paper (Table 1).

from scipy import stats
import rearrangement_algorithm as ra

# create the quantile functions
qf = [stats.pareto(2, loc=-1).ppf]*3

# set the parameters
alpha = 0.5102
num_steps = 500

# calcuate the bounds on the VaR
lower_under, lower_over = ra.bounds_VaR(1.-alpha, qf, method="lower",
                                        num_steps=num_steps)
VaR_under = lower_under[0]
VaR_over = lower_over[0]

# the expected value is around 0.5
expected = 0.5
print("{:.5f} <= {:.3f} <= {:.5f}".format(VaR_under, expected, VaR_over))

Bounds on the Expected Value

TODO

Bounds on the Survival Probability

TODO