Approx FPTD for Motion Models

Code belonging to the paper

Marcel Reith-Braun, Jakob Thumm, Florian Pfaff, und Uwe D. Hanebeck, "Approximate First Passage Time Distributions for Gaussian Motion and Transportation Models," (submitted to Fusion), 2023.

The repo contains methods to compute approximate first passage time distributions (FPTD) for Gaussian processes with an increasing trend, such as motion and transportation models for which it is valid to assume that the increasing trend in the mean is primarily resposible for the first passage. This may include, e.g., models such as the constant velocity (CV) model, and the constant acceleration (CA) model.

The repo contains examples for three different process, the Wiener process with drift, for which the analytical solution is known, the CV model and the CA model. The methods are compared with Monte Carlo simulations of the corresponding discrete-time process model.

Note that the code is intended to support two-dimensional processes, where one is also interested in the distribution of y at the first passage time. The CV and CA models are therefore defined as 2D processes, where the process in x is independent of the process in y.

Prerequisites

See dockerfile/Dockerfile for a list of required packages. Use

docker build -t tensorflow/approx_fptd:2.8.0-gpu /path/to/repo/dockerfile

to build a docker image with all required packages.

Run with CPU (Docker, Linux)

docker run -u $(id -u):$(id -g) -it --rm -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix -v /path/to/repo:/mnt tensorflow/approx_fptd:2.8.0-gpu

Run with GPU (Docker, Linux)

docker run -u $(id -u):$(id -g) --gpus all -it --rm -e DISPLAY=$DISPLAY -v /tmp/.X11-unix:/tmp/.X11-unix -v /path/to/repo:/mnt tensorflow/approx_fptd:2.8.0-gpu

Executable scripts

wiener_process.py

• Methods and simulations for a first passage time problem with a Wiener process with drift, for which the analytical solution is known.

cv_process.py

• Methods and simulations for a first passage time problem with a CV model.

ca_process.py

• Methods and simulations for a first passage time problem with a CA model.

cv_experiments.py

• Runs predefined experiments with CV models. The experiments are defined via a hard-coded config dictionary in the file.

ca_experimets.py

• Runs predefined experiments with CA models. The experiments are defined via a hard-coded config dictionary in the file.

Example Run Commands

• Get a list and explanations of all possible flags.

     python3 /mnt/cv_process.py --help

  Similar for all other executables.

• Run a single experiment with a CV model with predefined settings (see the main function of cv_experiments.py) and save the plots to /mnt/results/.

     python3 /mnt/cv_process.py --save_results --result_dir /mnt/results/

  Similar for ca_process.py and wiener_process.py.

• Run a single experiment with a CV model with predefined settings (see the main function of cv_experiments.py), save the plots to /mnt/results/, and save stdout to /mnt/results/stdout_cv_model.txt.

     python3 /mnt/cv_process.py --save_results --result_dir /mnt/results/ > /mnt/results/stdout_cv_model.txt

  Similar for ca_process.py and wiener_process.py.

• Run predefined experiments with CV models, save the plots to /mnt/results/cv_experiments/, and save stdout to /mnt/results/cv_experiments/stdout_cv_models.txt, but do not show the plots.

     python3 /mnt/cv_experiments.py --save_results --result_dir /mnt/results/cv_experimenents/ --no_show > /mnt/results/cv_experimenents/stdout_cv_model.txt

  Similar for ca_experiments.py. Experiment configs and chosen experiments can be adjusted in cv_experiments.py and ca_experiments.py, respectively.

• Run a single experiment with a CV model with predefined settings, measure the computational times (averaged over ten runs) and do not save or show the plots (measured times will be printed to stdout).

     python3 /mnt/cv_process.py --measure_computational_times --no_show

  Similar for ca_process.py and wiener_process.py.

Example Results

First Passage Time

CV model with a power spectral density of 93 640 mm²s^-3 (experiment Long_Track_Sw10_denorm in cv_experiments.py).

Wiener process with drift with sigma of 10 (default of wiener_process.py).

Example Tracks

Some example tracks for the CV model with a power spectral density of 93 640 mm²s^-3.

Additional Notes

Additional plot options can be set in hitting_time_uncertainty_utils.py, e.g., the size of the figures or whether to omit the headers of the plot (--for_paper-mode).

Cite as

Marcel Reith-Braun, Jakob Thumm, Florian Pfaff, and Uwe D. Hanebeck, "Approximate First Passage Time Distributions for Gaussian Motion and Transportation Models," 2023.

TODO: Adjust citation and provide bibtex.