Elementary Effects

OAT (one-at-a-time) trajectories through the input space. Screening: which factors matter, cheaply.

When to use: You have many factors (10–100+) and a limited computational budget. Morris elementary effects tell you which factors are influential and which can be fixed without the cost of a full Sobol’ analysis. Typical budget: r \times (d + 1) model evaluations, with r = 10–50. Compare this to Sobol’, which costs N \times (d + 2) with N typically 1024–16384.

Morris 1991

Morris (1991) Technometrics 33(2), 161–174. [bib]

The method constructs r random trajectories through a p-level grid on the unit hypercube [0, 1]^d. Each trajectory is a sequence of d + 1 points where consecutive points differ in exactly one coordinate (the OAT property). At each step, a single factor x_i is perturbed by a fixed step \Delta, and the resulting change in output defines the elementary effect for that factor at that base point:

EE_i = \frac{f(x_1, \ldots, x_i + \Delta, \ldots, x_d) - f(\mathbf{x})}{\Delta}

The step size follows Saltelli’s convention:

\Delta = \frac{p}{2(p - 1)}

For p = 4 (the default), \Delta = 2/3. Base points are drawn from the lower half of the grid so that adding \Delta stays on a grid point.

Each trajectory visits each factor exactly once. With r trajectories, every factor accumulates r elementary effects. Three statistics are computed per factor i:

\mu_i = \frac{1}{r} \sum_{k=1}^{r} EE_i^{(k)}

\mu_i^* = \frac{1}{r} \sum_{k=1}^{r} \lvert EE_i^{(k)} \rvert

\sigma_i = \sqrt{\frac{1}{r - 1} \sum_{k=1}^{r} (EE_i^{(k)} - \mu_i)^2}

\mu_i^* was introduced by Campolongo et al. (2007) [bib] to fix a cancellation problem with the signed mean \mu_i: for non-monotonic factors, positive and negative elementary effects average toward zero even when the factor is influential. \mu_i^* uses absolute values and is the modern standard for screening.

\sigma_i indicates the degree of non-linearity or interaction effects. A factor with high \mu_i^* and low \sigma_i has a strong, approximately linear, additive effect. A factor with high \sigma_i relative to \mu_i^* participates in interactions or has a strongly non-linear effect.

Sampling

use salib::samplers::build_morris_trajectories;
use salib::*;

let mut rng = RngState::from_seed([0u8; 32]);

// d=8 factors, r=20 trajectories, p=4 grid levels
let trajectories = build_morris_trajectories(
    8, 20, 4, &mut rng
).unwrap();
// Total cost: 20 * (8 + 1) = 180 model evaluations.

Grid levels (p) must be even (Saltelli convention). p = 4 is the standard choice; p = 8 gives finer resolution at the cost of a denser grid but does not change the number of model evaluations.

Estimation

use salib::estimators::estimate_morris_effects;

let effects = estimate_morris_effects(&trajectories, |x| {
    // Ishigami function
    x[0].sin()
        + 7.0 * x[1].sin().powi(2)
        + 0.1 * x[2].powi(4) * x[0].sin()
}).unwrap();

println!("{effects}");
// Prints μ, μ*, σ per factor.

Verify against the additive-linear test function Y = \sum_{i=1}^{d} i \cdot x_i (d = 8, r = 50, p = 4, seed [0u8; 32]):

Factor	\mu_i	\mu_i^*	\sigma_i	Analytic
0	1.0000	1.0000	0.0000	\mu = 1, \sigma = 0
1	2.0000	2.0000	0.0000	\mu = 2, \sigma = 0
…	…	…	…	…
7	8.0000	8.0000	0.0000	\mu = 8, \sigma = 0

For a purely linear function, elementary effects are constant across all trajectories: EE_i = b_i regardless of the base point. The estimator recovers each coefficient exactly (to floating-point precision), and \sigma_i = 0. This is the degenerate case. For non-linear functions, \sigma_i > 0 and \mu_i converges at rate O(1/\sqrt{r}).

Caveat: The identity \mu_i^* \geq |\mu_i| always holds (mean of absolute values is at least the absolute value of the mean). If your results violate this, there is a bug.

Grouped Morris

Campolongo et al. (2007) Env. Mod. Soft. 22(10), 1509–1518. [bib]

When inputs are naturally grouped (all parameters of a subsystem, all coefficients of one physical process), grouped Morris perturbs all factors in a group simultaneously rather than one at a time. Instead of d + 1 points per trajectory, there are n_{\text{groups}} + 1 points. Each step moves every factor in the selected group by \Delta.

The elementary effect for a group g is:

EE_g = \frac{f(\mathbf{x}_{\text{after}}) - f(\mathbf{x}_{\text{before}})}{\Delta}

where the step perturbs all |g| factors in the group. The same \mu, \mu^*, \sigma statistics are computed per group.

Total cost: r \times (n_{\text{groups}} + 1) model evaluations. When n_{\text{groups}} \ll d, this is substantially cheaper than ungrouped Morris.

Sampling

use salib::samplers::build_grouped_morris_trajectories;
use salib::*;

let groups = vec![
    Group { name: "subsystem_A".into(), factor_indices: vec![0, 1] },
    Group { name: "subsystem_B".into(), factor_indices: vec![2, 3] },
];

let mut rng = RngState::from_seed([0u8; 32]);

// d=4 total factors, r=50 trajectories, p=4 levels
let trajectories = build_grouped_morris_trajectories(
    &groups, 4, 50, 4, &mut rng
).unwrap();
// Total cost: 50 * (2 + 1) = 150 evaluations (vs 50 * 5 = 250 ungrouped).

Estimation

use salib::estimators::estimate_grouped_morris_effects;

let effects = estimate_grouped_morris_effects(
    &trajectories,
    &groups,
    |x| x[0] + x[1] + 3.0 * x[2] + 3.0 * x[3],
).unwrap();

// Per-group statistics
let grouped_mu_star = effects.grouped_mu_star
    .as_ref().unwrap();
let group_names = effects.group_names
    .as_ref().unwrap();
for (name, ms) in group_names.iter().zip(grouped_mu_star) {
    println!("{name}: μ* = {ms:.4}");
}

The output MorrisEffects contains both per-factor and per-group statistics. Per-factor effects in grouped Morris record the group-level elementary effect for each member factor (individual factor contributions cannot be separated when factors move simultaneously).

Verify: With singleton groups (one factor per group), grouped Morris produces the same \mu^* values as ungrouped Morris under the same seed.

Caveat: Grouped Morris answers “does this subsystem matter?” not “which factor within the subsystem matters?” If a group screens as important, follow up with an ungrouped analysis on its member factors.

Interpreting the (\mu^*, \sigma) plane

The standard screening decision: factors with \mu_i^* below some threshold are non-influential and can be excluded from a subsequent full analysis (e.g., Sobol’).