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_main.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_main.py

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

ca_process_main.py

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

cv_experiments_main.py

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

ca_experimets_main.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_main.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_main.py) and save the plots to /mnt/results/.

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

  Similar for ca_process_main.py and wiener_process_main.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_main.py --save_results --result_dir /mnt/results/ > /mnt/results/stdout_cv_model.txt

  Similar for ca_process_main.py and wiener_process_main.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_main.py --save_results --result_dir /mnt/results/cv_experimenents/ --no_show > /mnt/results/cv_experimenents/stdout_cv_model.txt

  Similar for ca_experiments_main.py. Experiment configs and chosen experiments can be adjusted in cv_experiments_main.py and ca_experiments_main.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_main.py --measure_computational_times --no_show

  Similar for ca_process_main.py and wiener_process_main.py.

Import the Repo as a Module & Build the Distributions

The distributions can be imported from the module Approx_FPDT_for_Motion_Models (for this, make sure you added /path/to/repo to your $PYTHONPATH).

Example:

from Approx_FPDT_for_Motion_Models import GaussTaylorCVHittingTimeDistribution

# define hyperparameters for the distribution...
S_w = 10  # power spectral density (PSD)
x_L = np.array([0.3, 6.2, 0.5, 0.2])  # initial state (x, x_dot, y, y_dot)
C_L = np.array([[2E-7, 2E-5, 0, 0], [2E-5, 6E-3, 0, 0], [0, 0, 2E-7, 2E-5], [0, 0, 2E-5, 6E-3]])  # initial covariance
x_predTo = 0.6458623971412047  # boundary to hit (x-coordinate)
t_L = 0  # initial time
cv_temporal_point_predictor = lambda pos_l, v_l, x_predTo: (x_predTo - pos_l[..., 0]) / v_l[..., 0]  
# constant-velocity prediction
  
# build the distribution...
htd = GaussTaylorCVHittingTimeDistribution(x_L, C_L, S_w, x_predTo, t_L,
                                         point_predictor=cv_temporal_point_predictor)
# evaluate the distribution, e.g., 
htd.cdf(0.05)   
# or, for a larger sample_size, 
htd.cdf(np.array([0.04, 0.05, 0.06, 0.07]))       

The distributions support a batch size.

Example:

from Approx_FPDT_for_Motion_Models import GaussTaylorCVHittingTimeDistribution

# define hyperparameters for the distribution...
S_w = np.stack([S_w, 10**2 * S_w])   # the process at the second batch has a PSD of 100 times the PSD of the first batch 
x_L = np.stack([x_L, x_L])
C_L = np.stack([C_L, C_L])
x_predTo = np.stack([x_predTo, x_predTo])
  
# build the distribution...
htd = GaussTaylorCVHittingTimeDistribution(x_L, C_L, S_w, x_predTo, t_L,
                                         point_predictor=cv_temporal_point_predictor)
# evaluate the distribution, e.g., 
htd.cdf(0.05) >> array([0.07154484, 0.44075421])
# or, for a larger sample_size, 
cdf_values = htd.cdf(np.array([[0.04], [0.05], [0.06], [0.07]]))     
cdf_values.shape >> (4, 2)      
cdf_values = htd.cdf(np.array([[0.04, 0.01], [0.05, 0.04], [0.06, 0.07], [0.07, 0.1]]))    
cdf_values.shape >> (4, 2)      

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 evaluators.hitting_evaluator.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.