Morris Method (Elementary Effects)

Theoretical Foundations of the Morris Method

The Role of Screening

Definition of the Elementary Effect

Normalize the inputs \( \mathbf{x} = (x_1, \dots, x_k) \) to the unit hypercube and perturb only factor \( i \) by a step \( \Delta \). The resulting rate of change is that factor's Elementary Effect (EE):

$$ EE_i = \frac{f(x_1, \dots, x_i + \Delta, \dots, x_k) - f(\mathbf{x})}{\Delta} $$

This is a finite-difference derivative at one point — but Morris's key idea is to sample EEs at many locations across the input space and use their statistics as a global sensitivity measure. Nonlinearity and interactions invisible to a single local gradient show up as scatter in the EEs.

The Three Statistics μ, μ*, σ and How to Read Them

From \( r \) repetitions you compute, per factor:

Plot the results in the \( \mu^* \)–\( \sigma \) plane: near the origin = negligible (safe to fix); along the horizontal axis = influential and linear; above the diagonal = influential with nonlinearity/interactions. A large gap between \( \mu^* \) and \( \mu \) additionally flags non-monotonic behavior with sign changes.

Sampling Design and Computation

Trajectory Design and Run Count

EEs are collected efficiently via trajectories: starting from a random point, perturb the \( k \) factors one at a time to obtain \( k+1 \) points — one trajectory yields one EE per factor. With \( r \) trajectories, the total number of analyses is

$$ N = r\,(k+1) $$

With \( k=20 \) factors and \( r=15 \) trajectories: 315 runs. Direct Sobol' estimation typically needs on the order of \( N=(k+2)\times 10^3 \) runs — two orders of magnitude more. Inputs are discretized to \( p \) levels per dimension (even; \( p=4 \) is standard) with the step \( \Delta = p/(2(p-1)) \) (= 2/3 for \( p=4 \)).

Choosing r and Confirming Stability

Start with \( r = 10\sim20 \) — but always pair it with a stability check. Compute rank correlation of \( \mu^* \) between the full trajectory set and a half-thinned one, confirming the top factors don't reshuffle. Bootstrap confidence intervals on \( \mu^* \) let you report overlapping factors honestly as "tied." If your goal is only separating the important group from the negligible group, small rank swaps within a group don't matter.

CAE-Specific Preprocessing — Ranges and Normalization

When multiple responses matter (peak stress and first natural frequency, say), rankings usually differ per response. Run Morris per response and only fix factors that are negligible for every response — that is the conservative rule.

Applying It in Practice

Standard Workflow

1. Define factors and ranges — material constants, thicknesses, boundary stiffnesses, loads, friction coefficients; record the basis (standards, measured scatter, process capability)
2. Generate trajectory samples — an input matrix of \( r(k+1) \) rows via SALib or similar
3. Automate the runs — template substitution → job submission → response extraction, fully scripted
4. Compute EE statistics and the μ*–σ plot — triage factors (important / nonlinear-interacting / negligible)
5. Confirm stability — trajectory thinning and bootstrap
6. Hand off — carry only the top factors into Sobol' analysis, robust optimization, or calibration

Handling Failed Runs

Sweep a CAE model across ranges and some runs will fail — mesh generation errors, non-convergence — typically near range edges. Silently discarding them biases the EEs, so decide the policy up front: ① if failures cluster at one factor's extreme, revisit the range (that corner may be unphysical); ② if sporadic, drop the whole trajectory and add a fresh one; ③ if feasibility itself is the interesting output, switch the response to a feasible/infeasible indicator. Report the failure rate and the policy — that's a reproducibility requirement.

Reporting Results

Three artifacts make your screening auditable: the μ*–σ scatter plot with factor labels, a μ* bar chart with bootstrap intervals for the top factors, and a table of fixed factors with justification. A report that says "sensitivity analysis showed thickness and joint stiffness dominate" without these figures is unverifiable — avoid it.

Tools and CAE Integration

Tool Support

Minimal SALib Implementation

Replace run_fem with your solver call and this skeleton becomes production code:

import numpy as np
from SALib.sample import morris as morris_sample
from SALib.analyze import morris as morris_analyze

problem = {
    "num_vars": 3,
    "names": ["E", "thickness", "k_support"],
    "bounds": [[190e9, 210e9], [1.8e-3, 2.2e-3], [1e6, 1e8]],
}
X = morris_sample.sample(problem, N=15, num_levels=4)  # 15 trajectories -> 15*(3+1)=60 runs
Y = np.array([run_fem(x) for x in X])                  # run CAE per row
Si = morris_analyze.analyze(problem, X, Y, num_levels=4)
print(Si["mu_star"], Si["sigma"])

Job Automation Reality

Automating hundreds of runs means scripting four things: ① input-file templating (parameters as placeholders), ② parallel job submission and completion monitoring, ③ response extraction from result files, ④ failure flagging. If any manual step remains, 315 runs will never actually happen — "the whole loop runs unattended" is the de facto prerequisite for adopting Morris.

Research Frontiers

Improved Sampling Designs

Beyond classical random trajectories: Campolongo's trajectory selection maximizing inter-trajectory spread, radial designs (perturbing each factor from a common base point), and Latin-hypercube-based hybrids. Radial designs share components with Sobol' total-index estimators, making the transition from screening to variance-based analysis seamless.

Group Screening for High Dimensions

With dozens to hundreds of factors, group physically related parameters and measure group-level EEs first, then analyze only surviving groups factor-by-factor. The two-stage approach scales roughly logarithmically with factor count — well matched to CAE problems like many-parameter hardening laws or hundreds of spot-weld stiffnesses.

Relation to Metamodels and Total Indices

\( \mu^* \) is known to behave as a cheap proxy for the Sobol' total-effect index, which underpins today's best-practice pipeline: Morris for dimension reduction → Kriging surrogate on the survivors → rigorous Sobol' indices on the surrogate. Over-aggressive screening shows up later as unexplained surrogate residuals, so keep borderline factors in — that's the safe side.

Troubleshooting

Symptoms, Causes, and Fixes

Common Misuse — Over-Interpreting "Sensitivity"

Related reading: Sparse PCE with LAR, Kriging surrogates, Bayesian calibration.