SlicedDistributions.jl

A julia package for robust estimation and sampling of sliced distributions [1, 2]

Sliced distributions enable the characterization of, possibly dependent, multivariate data by injecting the physical space into a higher dimensional feature space using a polynomial mapping. SlicedDistributions implements:

• Sliced-Normals
• Sliced-Exponentials
• sampling from the sliced distributions using transitional markov chain Monte Carlo (TMCMC)

The distributions are fully compliant with the Distributions.jl interface of a ContinuousMultivariateDistribution.

Note, that the sliced-Normals are estimated by choosing a square polynomial basis and restricting the decision space of the sliced-Exponentials, see [2, Section VII].

Example

Consider a data sequence obtained from the time-response of the Van-der-Pol oscillator.

To begin fitting a sliced distribution to the data we start by loading the necessary packages and read the input data.

using DelimitedFiles
using SlicedDistributions

δ = readdlm("demo/data/vanderpol.csv", ',')

Next, we define the additional input parameters and estimate a SlicedNormal distribution from the data. The data points are expected to be the columns of the input matrix.

d = 3 # maximum degree of the polynomial mapping into the feature space
b = 10000 # the number of samples used to estimate the normalizing constant

# the (optional) support of the sliced distribution given as lower and upper bounds of a hypercube
lb = [-2.5, -3]
ub = [2.5, 3.5]

sn, lh = SlicedNormal(δ, d, b, lb, ub)

If the support is omitted, the minimum and maximum values of the data will be inferred as the support automatically. To estimate a sliced-exponential distribution simply replace SlicedNormal with SlicedExponential. The next plot presents the probability density function (PDF) of the estimated distribution.

With the distribution in place we can obtain 2000 random samples through rand.

samples = rand(sn, 2000)

Internally this will use TMCMC to sample from the PDF. See the next Figure for a scatter plot of the samples.

References

[1] L. G. Crespo, B. K. Colbert, S. P. Kenny, and D. P. Giesy, ‘On the quantification of aleatory and epistemic uncertainty using Sliced-Normal distributions’, Systems & Control Letters, vol. 134, p. 104560, Dec. 2019, doi: 10.1016/j.sysconle.2019.104560.

[2] L. G. Crespo, B. K. Colbert, T. Slager, and S. P. Kenny, ‘Robust Estimation of Sliced-Exponential Distributions’, in 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA: IEEE, Dec. 2021, pp. 6742–6748. doi: 10.1109/CDC45484.2021.9683584.