elliptical-sim

Simulation study for an article about multivariate extreme quantile region estimation. Current simulation settings serve three different purposes:

Performance of an elliptical extreme quantile region estimator is compared to a competing extreme quantile region estimator based on halfspace depth [1]. Relative errors for both estimators are computed for each simulation scenario. However, simulation data is not stored in Github since files are too large. On the other hand, directory summmary-data/ includes some summary statistics for relative errors of each scenario. Also, illustrative figures are included.
Performance of the elliptical extreme quantile region estimator is examined in high dimensional cases. Conservative estimates of the relative errors are stored in the directory high-dim-data/data/. On the other hand, figures about how the relative error fluctuates as dimension changes are stored in the directory high-dim-data/figures/.
An example with a skewed t-distribution is constructed. Here we use the elliptical extreme quantile region estimator for estimation, and thus, the estimated quantile regions have an elliptical shape, even though the true quantile regions are not elliptically shaped. For the skewed t-distribution we repeat each scenario once. Thus, summary statistics are not relevant here, but illustrative figures are in the directory summary-data/figures-skew/.

Requirements

Access to Aalto University Triton computing cluster is required for running the simulations. See this link for details about Triton.

Clone or unzip the repository.

git clone https://github.com/perej1/elliptical-sim.git

Install required packages by running the following R command in the project's root folder (R package renv has to be installed).
```
renv::restore()
```

Running the simulation

In total there are eight scripts.

simulate.R - Performs simulations for selected parameters.
sim-batch.slurm - Performs simulations for sets of different parameters.
summarise.R - Computes summary statistics and produces figures.
functions.R - Includes functions needed for simulations.
gen-arg.R - Generates arguments for simulation settings.
simulate-high-dim.R - Standalone script for performing simulations in high dimensional cases.
test-compute_error.R - Unit tests for the function compute_error.R.
test-compute_error_elliptical.R - Unit tests for the function compute_error_elliptical.R.

With the following command one can run all simulation settings specified by the sim-args.txt (except high dimensional simulations).

sbatch sim-batch.slurm

Instead one can run just one specified scenario (for this Triton is not needed).

Rscript simulate.R --type tdistDeg4 --s 100 --d 3 --n 1000 --p high --k medium --seed 278

Also, for high dimensional simulations triton is not needed. One can simply run the following.

Rcsript simulate-high-dim.R

Acknowledgements

We acknowledge the computational resources provided by the Aalto Science-IT project.

References

[1] Y. He, J. H. Einmahl, Estimation of extreme depth-based quantile regions, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79 (2017) 449–461.