Simulation Parameter Sensitivity Analysis and Uncertainty Quantification in Crowd Evacuation

Simulation Parameter Sensitivity Analysis and Uncertainty Quantification in Crowd Evacuation

Problem Description

In crowd evacuation simulation models, small changes in input parameters (such as individual movement speed, decision delay time, density threshold, etc.) can lead to significant differences in output results (such as total evacuation time and congestion level). Parameter sensitivity analysis aims to identify which parameters have the greatest impact on the results, while uncertainty quantification needs to assess how parameter uncertainty propagates to the output results, thereby enhancing the model's reliability and decision support capability.

Solution Process

1. Define Analysis Objectives

First, determine the key output variables (e.g., total evacuation time \(T_{\text{total}}\)), input parameters to be analyzed (such as minimum movement speed \(v_{\min}\), maximum speed \(v_{\max}\), decision time distribution \(\tau\), herd probability \(p\), etc.), and set reasonable ranges for the parameters (usually based on literature or experimental data).

2. Construct Parameter Sample Space

Generate multiple sets of parameter combinations within the parameter ranges using sampling methods (such as Latin Hypercube Sampling) to ensure representative coverage of the parameter space. For example:

  • Sample \(v_{\min}\) uniformly within [0.5, 1.0] m/s, sample \(v_{\max}\) within [1.2, 2.0] m/s, and sample \(\tau\) from a log-normal distribution, etc.

3. Run Simulations and Collect Data

Run the evacuation simulation for each set of parameter combinations and record the output results (e.g., \(T_{\text{total}}\)), forming a "parameter-result" dataset. To reduce random errors, each parameter set should be run multiple times (e.g., 50 times) and the average should be taken.

4. Sensitivity Analysis Methods

  • Local Sensitivity Analysis: Fix other parameters and change only one parameter (e.g., \(v_{\max}\)) while observing the output change, calculating the partial derivative \(\frac{\partial T_{\text{total}}}{\partial v_{\max}}\). However, this method cannot handle interactions between parameters.
  • Global Sensitivity Analysis:
    • Sobol Index Method: Decomposes the output variance into contributions from each parameter and their interactions. The first-order Sobol index \(S_i\) represents the independent influence of parameter \(i\), and the second-order index \(S_{ij}\) represents the interactive influence between parameters \(i\) and \(j\).
    • Morris Screening Method: Quickly identifies important parameters by calculating the "elementary effect" (output change due to incremental changes) for each parameter. Suitable for preliminary screening when there are many parameters.

5. Uncertainty Quantification

  • Monte Carlo Method: Generates a large number of random samples based on the probability distributions of parameters (e.g., \(v_{\max} \sim N(1.5, 0.2)\)), runs simulations, and obtains the probability distribution of outputs (e.g., mean and confidence intervals for \(T_{\text{total}}\)).
  • Surrogate Model (e.g., Gaussian Process Regression): When simulation costs are high, a simplified model is trained with a small amount of data to quickly predict outputs corresponding to parameters. Sensitivity analysis and uncertainty propagation calculations are then performed using the surrogate model.

6. Result Interpretation and Model Optimization

  • Rank parameters based on sensitivity indices to identify key parameters (e.g., decision delay \(\tau\) has a more significant impact on \(T_{\text{total}}\) than speed).
  • For highly sensitive parameters, further calibrate their values (e.g., correct the distribution of \(\tau\) using experimental data) or optimize the model structure (e.g., refine decision rules).
  • Provide confidence intervals for output results through uncertainty quantification (e.g., "with 95% confidence, \(T_{\text{total}} \in [120s, 180s]\)") to assist in risk assessment for emergency planning.

Example Illustration

Assume a shopping mall evacuation model includes parameters such as \(v_{\max}\), herd probability \(p\), and exit recognition distance \(d\). Sobol analysis reveals:

  • The first-order index for \(p\) is \(S_p = 0.6\), indicating that herd behavior is the main factor affecting evacuation time.
  • The interaction index between \(v_{\max}\) and \(d\) is \(S_{v,d} = 0.2\), showing that when the exit recognition distance is small, increasing speed has limited improvement on evacuation.
    Based on this, managers can prioritize implementing guidance measures to reduce herd behavior rather than simply widening passages.