Sensitivity Analysis#

In this tutorial we demonstrate how to perform sensitivity analysis as part of the AutoEmulate workflow. The tutorial covers:

  1. Setting up an example simulation: here we use our “FlowProblem” simulator. This is a cardiovascular modelling example, simulating a blood vessel divided into 10 compartments. This allows for the study of the pressure and flow rate at various points in the tube. See “The Flow Problem” below for more details.

  2. Running the simulation for 100 sets of parameters sampled from the parameter space.

  3. Using Autoemulate to find the best emulator for this simulation

  4. Performing sensitivity analysis.

The Flow Problem

In the field of cardiovascular modeling, capturing the dynamics of blood flow and the associated pressures and volumes within the vascular system is crucial for understanding heart function and disease. This simulator simulates a vessel divided to 10 compartments.

Parameters#

The simulation parameters include :

  1. R (Resistance): Represents the resistance to blood flow in blood vessels, akin to the hydraulic resistance caused by vessel diameter and blood viscosity (Analogous to electrical resistor).

  2. L (Inductance): Represents the inertial effects of blood flow, capturing how blood resists changes in its velocity (Analogous to electrical inductor).

  3. C (Capacitance): Represents the compliance or elasticity of blood vessels, primarily large arteries, which store and release blood volume with changes in pressure (Analogous to a capacitor).

Boundary conditions#

  1. Neumann boundary condition : Specifies the derivative of the variable at the boundary.

  2. Dirichlet Boundary condition : Specifies the value of the variable directly at the boundary.

The setup#

The input flow rate in each compartment is \(Q_i(t)\) for the \(i^{th}\) compartment and the output flow rate is \(Q_{i+1}(t)\).

\(Q_0(t) = \begin{cases} A \cdot \sin^2\left(\frac{\pi}{t_d} t\right), & \text{if } 0 \leq t < T, \\ 0, & \text{otherwise}. \end{cases}\)

Where:

  • \(Q_0(t)\) is the input pulse function (flow rate) at time t

  • A is the amplitude of the pulse

  • \(t_d\) is the pulse duration.

Solve#

Pressure in Each Compartment (\(P_i\)): This determines how the pressure in each compartment evolves over time, based on the inflow (\(Q_i(t)\)) and the outflow (\(Q_{i+1}(t)\)). where \(i\) is the number of compartment.

Circuit Diagram

\(\frac{dP_i}{dt} = \frac{1}{C_n} \left( Q_i(t) - Q_{i+1}(t) \right)\) where, \(C_n = \frac{C}{n_\text{comp}}\)

Flow rate equation (\(Q_i\)): This governs how the flow in each compartment changes over time, depending on the pressures in the neighboring compartments and the resistance and inertance properties of each compartment.

\(\frac{dQ_i}{dt} = \frac{1}{L_n} \left( P_i - P_{10} - R_n Q_i(t) \right)\), where \(L_n = \frac{L}{n_\text{comp}}, \quad R_n = \frac{R}{n_\text{comp}}\)

from autoemulate.core.compare import AutoEmulate
from autoemulate.core.sensitivity_analysis import SensitivityAnalysis
from autoemulate.simulations.flow_problem import FlowProblem
import warnings
warnings.filterwarnings("ignore")
figsize = (9, 5)

Set up the simulation parameters and ranges:

parameters_range = {
    "T": (0.5, 2.0), # Cardiac cycle period (s)
    "td": (0.1, 0.5), # Pulse duration (s)
    "amp": (100.0, 1000.0), # Amplitude (e.g., pressure or flow rate)
    "dt": (0.0001, 0.01), # Time step (s)
    "C": (20.0, 60.0), # Compliance (unit varies based on context)
    "R": (0.01, 0.1), # Resistance (unit varies based on context)
    "L": (0.001, 0.005), # Inductance (unit varies based on context)
    "R_o": (0.01, 0.05), # Outflow resistance (unit varies based on context)
    "p_o": (5.0, 15.0) # Initial pressure (unit varies based on context)
}
output_names = ["pressure"]

simulator = FlowProblem(
    parameters_range=parameters_range,
    output_names=output_names,
    log_level="error"
)

Run the simulation for 100 sets of parameters sampled from the parameter space:

x = simulator.sample_inputs(100)
y = simulator.forward_batch(x)
print(x.shape, y.shape)
torch.Size([100, 9]) torch.Size([100, 1])

Use AutoEmulate to find the best emulator for this simulation:

ae = AutoEmulate(x, y, models=["MLP", "GP"], log_level="error")  # remove models argument to use all models
best = ae.best_result()
print(best.model_name)
GaussianProcess

Sensitivity Analysis#

  1. Define the problem by creating a dictionary which contains the names and the boundaries of the parameters

  2. Evaluate the contribution of each parameter via the Sobol and Morris methods.

problem = {
    'num_vars': simulator.in_dim,
    'names': simulator.param_names,
    'bounds': simulator.param_bounds,
    'output_names': simulator.output_names,
}
sa = SensitivityAnalysis(best.model, problem=problem)

Sobol metrics:

  • \(S_1\): First-order sensitivity index.

  • \(S_2\): Second-order sensitivity index.

  • \(S_t\): Total sensitivity index.

Sobol interpretation:

  • \(S_1\) values sum to ≤ 1.0 (exact fraction of variance explained)

  • \(S_t - S_1\) = interaction effects involving that parameter

  • Large \(S_t - S_1\) gap indicates strong interactions

sobol_df = sa.run("sobol")
sobol_df
output parameter index value confidence
0 pressure T S1 0.000120 0.001968
1 pressure td S1 0.018781 0.013407
2 pressure amp S1 0.916440 0.070222
3 pressure dt S1 -0.000367 0.000966
4 pressure C S1 0.006899 0.008514
5 pressure R S1 0.039339 0.017882
6 pressure L S1 0.000084 0.002646
7 pressure R_o S1 -0.000814 0.000966
8 pressure p_o S1 -0.000100 0.001134
0 pressure T ST 0.000564 0.000059
1 pressure td ST 0.025728 0.003260
2 pressure amp ST 0.930258 0.070896
3 pressure dt ST 0.000112 0.000015
4 pressure C ST 0.010364 0.001373
5 pressure R ST 0.048261 0.005370
6 pressure L ST 0.001014 0.000128
7 pressure R_o ST 0.000133 0.000016
8 pressure p_o ST 0.000147 0.000020
0 pressure (T, td) S2 0.000757 0.002910
1 pressure (T, amp) S2 0.000763 0.003893
2 pressure (T, dt) S2 0.000658 0.002896
3 pressure (T, C) S2 0.000639 0.002872
4 pressure (T, R) S2 0.000644 0.002956
5 pressure (T, L) S2 0.000598 0.002892
6 pressure (T, R_o) S2 0.000659 0.002899
7 pressure (T, p_o) S2 0.000626 0.002902
8 pressure (td, amp) S2 0.005556 0.023175
9 pressure (td, dt) S2 -0.000234 0.018591
10 pressure (td, C) S2 0.000537 0.018622
11 pressure (td, R) S2 0.000697 0.017992
12 pressure (td, L) S2 0.000090 0.018410
13 pressure (td, R_o) S2 -0.000137 0.018447
14 pressure (td, p_o) S2 -0.000118 0.018463
15 pressure (amp, dt) S2 0.001226 0.077647
16 pressure (amp, C) S2 0.002876 0.077000
17 pressure (amp, R) S2 0.007408 0.079094
18 pressure (amp, L) S2 0.001319 0.077640
19 pressure (amp, R_o) S2 0.001445 0.077829
20 pressure (amp, p_o) S2 0.001219 0.077694
21 pressure (dt, C) S2 0.000490 0.001525
22 pressure (dt, R) S2 0.000537 0.001497
23 pressure (dt, L) S2 0.000528 0.001524
24 pressure (dt, R_o) S2 0.000532 0.001528
25 pressure (dt, p_o) S2 0.000525 0.001527
26 pressure (C, R) S2 0.001237 0.013498
27 pressure (C, L) S2 0.000882 0.013187
28 pressure (C, R_o) S2 0.000963 0.013206
29 pressure (C, p_o) S2 0.000789 0.013227
30 pressure (R, L) S2 0.000097 0.027411
31 pressure (R, R_o) S2 0.000093 0.027414
32 pressure (R, p_o) S2 -0.000014 0.027439
33 pressure (L, R_o) S2 0.000394 0.004375
34 pressure (L, p_o) S2 0.000404 0.004368
35 pressure (R_o, p_o) S2 0.001031 0.001634
sa.plot_sobol(sobol_df, index="ST", figsize=figsize) 
../../_images/13684472a2109b3e8d5f64696797b13eb4ae1cfaf490f1a7bcc8d5e099551cdd.png

You can also save the plot directly to a file by passing the fname argument to the plotting function.

sa.plot_sobol(sobol_df, index="ST", figsize=figsize, fname="sobol_plot.png") 

Morris Interpretation:

  • High \(\mu^*\), Low \(\sigma\): Important parameter with linear/monotonic effects

  • High \(\mu^*\), High \(\sigma\): Important parameter with non-linear effects or interactions

  • Low \(\mu^*\), High \(\sigma\): Parameter involved in interactions but not individually important

  • Low \(\mu^*\), Low \(\sigma\): Unimportant parameter

morris_df = sa.run("morris")
morris_df
output parameter mu mu_star sigma mu_star_conf
0 pressure T -16.407669 16.623108 10.210532 0.665426
1 pressure td 101.856438 102.388161 82.367050 4.924413
2 pressure amp 682.548828 682.548828 98.052559 6.204664
3 pressure dt -5.668179 8.004286 8.242211 0.438115
4 pressure C -69.087364 69.087364 49.286430 2.508434
5 pressure R -147.406586 147.410828 78.211418 4.523494
6 pressure L 12.250652 15.570147 14.271719 0.609076
7 pressure R_o -0.455672 7.134204 8.811731 0.320605
8 pressure p_o -0.984186 8.483861 10.890879 0.403473
sa.plot_morris(morris_df, figsize=figsize)
../../_images/3b50e65160a015988dda345e8d19b54bb69deb4260b45a61b594e12cdc70055e.png