Morris Elementary Effects Screening

Summary

The Morris method is a derivative-based screening technique. It computes elementary effects (EE) — finite-difference changes in output from perturbing one factor by a step $\Delta$ along randomly placed trajectories through a discretized grid. Averaging the EEs gives the mean ${\mu }_i$ (overall influence) and standard deviation ${\sigma }_i$ (interaction / nonlinearity). Campolongo’s revised ${\mu }_i^{\ast }$ averages the absolute EEs to avoid cancellation in non-monotonic models. At cost $O(r(p+1))$ model runs it cheaply ranks factors and flags the non-influential ones to fix — but it ranks rather than precisely quantifies.

Overview

Morris falls under derivative-based GSA: sensitivity is based on (averaged) finite-difference derivatives of the output. The paper: “The basic idea of Morris method is built upon calculation of elementary effects (EE) for each input factor by dividing the range of each factor into $p$ levels and considering $\Delta$ as a predetermined multiple of $\frac{1}{(p-1)}$.” It is the canonical screening tool (see Global Sensitivity Analysis - Overview): identify and discard unimportant factors before expensive variance-based quantification.

Main Content

Elementary effect ^elementary-effect

For factor $X_i$ with step $\Delta$ (where $X_i+\Delta \le 1$ on the unit grid):

$$E{E}_i=\frac{F(X_1,\dots ,X_{i-1},X_i+\Delta ,X_{i+1},\dots ,X_k)-F(X_1,\dots ,X_{i-1},X_i,X_{i+1},\dots ,X_k)}{{\Delta }_i}$$

a finite-difference (one-step) derivative of the output with respect to $X_i$ at one point in the input space. Each trajectory of $k+1$ runs yields one $EE$ per factor.

Morris measures ${\mu }_i$ and ${\sigma }_i$

After sampling and computing $E{E}_i$ over $r$ trajectories:

$${\mu }_i=\frac{1}{r}\sum _{i=1}^{r}E{E}_i{\sigma }_i=\sqrt{\frac{1}{r}\sum _{i=1}^r(E{E}_i-\frac{1}{r}\sum _{i=1}^{r}E{E}_i{)}^2}$$

• ${\mu }_i$ — mean elementary effect: overall importance of $X_i$.
• ${\sigma }_i$ — standard deviation of the EEs: “higher values of ${\sigma }_i$ suggest increased interaction between $X_i$ and the other variables or a non-linear effect.”

Revised mean ${\mu }_i^{\ast }$ (Campolongo)

$${\mu }_i^{\ast }=\frac{1}{r}\sum _{i=1}^r\left|E{E}_i\right|$$

Campolongo et al. proposed ${\mu }^{\ast }$ — the mean of the absolute elementary effects — “to mitigate the issue of cancellation of opposite signs in non-monotonic models.” Higher ${\mu }_i^{\ast }$ ⇒ greater influence of $X_i$ on the output. ${\mu }^{\ast }$ is a reliable proxy for the Sobol total-effect index $S_{Ti}$ for ranking purposes (see Variance-Based Sensitivity and Sobol Indices).

Reading the $({\mu }^{\ast },\sigma )$ plane

Plotting factors on $({\mu }^{\ast },\sigma )$ classifies them:

• Low ${\mu }^{\ast }$, low $\sigma$ → negligible factor → fix it (screen out).
• High ${\mu }^{\ast }$, low $\sigma$ → important and (nearly) linear/additive effect.
• High ${\mu }^{\ast }$, high $\sigma$ → important with strong interactions and/or nonlinearity → keep and quantify with Sobol. Related: DGSM (eq. 18) generalizes Morris as $v_i={\int }_{H^n}{\left(\frac{\partial f}{\partial x_i}\right)}^{2}dx$; there is a known link between DGSM and Sobol’s total index.

In the MNIST case study, Morris’s ${\mu }^{\ast }$ and $\sigma$ were among the most reliable importance measures (alongside Sobol $S_T$), and Morris obtained good accuracy “by utilizing the minimum number of most important pixels.” Sampling cost there: 50 trajectories in 4 levels (Table 1) — the cheapest budget of all methods tested.

Examples

A 10-parameter ABM screened with $r=20$ trajectories costs $20\times 11=220$ runs. Results:

• Parameter A: ${\mu }^{\ast }=8.0,\sigma =0.5$ → strong, near-linear main driver → keep.
• Parameter B: ${\mu }^{\ast }=6.5,\sigma =7.0$ → strong but highly interacting/nonlinear → keep, expect large $S_{TB}-S_B$ in Sobol.
• Parameter C: $\mu =0.1$ but ${\mu }^{\ast }=5.0$ → its raw mean nearly cancelled (non-monotone); ${\mu }^{\ast }$ correctly flags it as important — exactly the case ${\mu }^{\ast }$ was designed for.
• Parameters D–J: ${\mu }^{\ast }<0.2,\sigma <0.2$ → fix all six, shrinking the space from 10 to 4 before Sobol quantification (see Sampling and Estimation for Sobol Indices).

Connections

• Global Sensitivity Analysis - Overview — Morris as the screening half of the screen-then-quantify workflow.
• Variance-Based Sensitivity and Sobol Indices — ${\mu }^{\ast }$ ranks like $S_{Ti}$; $\sigma$ flags interactions that Sobol then decomposes.
• Sampling and Estimation for Sobol Indices — trajectory/Saltelli sampling and SALib implement both.
• Local vs Global Sensitivity Analysis — Morris is global (many points, full range) yet derivative-based, bridging OAT and variance methods.
• Population Initialization and Parameter Sensitivity — vault’s local OAT note; Morris extends OAT-style steps to a global screen.

See Also

• Uncertainty Quantification for ABM Calibration; ABM Validation Challenges.
• Morris (1991), “Factorial sampling plans for preliminary computational experiments.”
• Campolongo, Cariboni & Saltelli (2007), “An effective screening design for sensitivity analysis of large models.”