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Morris method

The Morris method is a screening technique for global sensitivity analysis in computational models, developed by Max D. Morris in 1991 to identify influential input parameters among many in deterministic systems.^[1] It employs a one-factor-at-a-time (OAT) approach, perturbing individual inputs sequentially across multiple randomized trajectories in the input space to capture both main effects and potential interactions or nonlinearities.^[2] The procedure begins by scaling the input domain to a k-dimensional unit hypercube discretized into a p-level grid (commonly p=4 or p=8), with perturbations \Delta as multiples of $1/(p-1).^[2] Elementary effects are then calculated for each input x_i at selected base points \mathbf{x} as finite differences: d_i(\mathbf{x}) = \frac{y(\mathbf{x} + \Delta \cdot \mathbf{e}_i) - y(\mathbf{x})}{\Delta}, where y is the model output and \mathbf{e}_i is the i-th unit vector.^[2] For r trajectories (typically r=10 to $20), r effects per input are generated, yielding distributions summarized by the mean \mu_i (or absolute mean \mu_i^* to mitigate sign cancellation) and standard deviation \sigma_i; large \mu_i^* signals overall importance, while elevated \sigma_i highlights sensitivity to interactions or nonlinearity.^[3] This method's efficiency—requiring approximately r(k+1) model evaluations for k inputs—makes it ideal for initial screening in high-dimensional models where full variance-based analysis is prohibitive.^[4] It assumes no input sparsity, monotonicity, or additivity, enabling robust application across diverse domains including environmental simulations, engineering optimization, and biomedical modeling for uncertainty quantification.^[5]

Introduction

Definition and Purpose

The Morris method is a global sensitivity analysis technique that utilizes randomized one-factor-at-a-time (OAT) designs to assess the influence of individual input factors on the output of deterministic computational models. It computes elementary effects—finite differences in model output resulting from small perturbations to a single input while holding others fixed—at multiple sampled points across the input space, providing a distribution of effects for each factor. This approach allows for the evaluation of factor importance without assuming additivity, monotonicity, or specific functional forms of the model.^[1] The primary purpose of the Morris method is to screen and rank input factors by their overall influence in preliminary computational experiments, particularly for complex models with a moderate to large number of inputs. By analyzing the mean and standard deviation of elementary effects, it identifies non-influential parameters that can be fixed or simplified to reduce model complexity, while also flagging factors that exhibit nonlinear behavior or interactions with others. This makes it valuable in applied statistics and modeling fields, such as environmental simulations and engineering design, where understanding key drivers is essential before more detailed analyses.^[1] Compared to variance-based global sensitivity methods like Sobol indices, the Morris method is notably efficient for high-dimensional problems, requiring only on the order of r(k+1) model evaluations—where r is the number of sampling trajectories and k is the number of inputs—versus thousands or more for Sobol approaches, enabling rapid screening even for computationally expensive models.

Historical Development

The Morris method was introduced by Max D. Morris in 1991 in his seminal paper "Factorial Sampling Plans for Preliminary Computational Experiments," published in Technometrics, where it was presented as an efficient one-at-a-time (OAT) screening technique for preliminary identification of influential input factors in complex computational models with many variables.^[1] Key refinements emerged in subsequent years to enhance the method's robustness, particularly in handling nonlinearities and interactions. In 2007, Francesca Campolongo, Jessica Cariboni, and Andrea Saltelli developed an improved screening design that introduced the μ* measure, which mitigates the cancellation of opposing elementary effects in the original mean metric, providing a more reliable indicator of parameter importance.^[6] This update built directly on Morris's framework while improving its discriminatory power for large-scale models. The method's integration into comprehensive sensitivity analysis practices was advanced by Andrea Saltelli and colleagues in their 2004 book Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, which positioned the Morris approach as a computationally efficient complement to variance-based global methods, emphasizing its role in model screening and uncertainty propagation.^[7] During the 1990s and 2000s, the Morris method gained widespread adoption in environmental and engineering applications, valued for its low computational cost in screening models with dozens of parameters, such as those simulating ecological systems or hydrological processes.^[8] By 2025, it has become a standard feature in open-source and commercial software, including Python's SALib library for global sensitivity analysis and MATLAB's SAFE Toolbox, enabling seamless implementation across interdisciplinary research.^[9]

Background Concepts

Sensitivity Analysis Overview

Sensitivity analysis is a technique used to quantify the relationship between uncertainties in model inputs and the resulting variations in model outputs, providing insights into how input parameters influence system behavior.^[10] This approach is fundamental for understanding uncertainty propagation, validating model structures, and supporting informed decision-making across disciplines such as engineering and environmental science.^[10] In these fields, it enables practitioners to trace how input uncertainties—often represented by probability distributions—affect predictions, such as in hydrological models or structural reliability assessments.^[11] Sensitivity analysis methods are broadly categorized into local and global types. Local methods evaluate sensitivity by examining the effect of small changes in inputs around a specific nominal point, typically using partial derivatives to approximate the output response.^[12] In contrast, global methods assess sensitivity across the entire input space, accounting for parameter interactions, nonlinear effects, and full input distributions, which makes them suitable for complex, nonlinear models.^[10] Within global approaches, variance-based techniques, such as those developed by Sobol, decompose the total output variance to attribute contributions from individual inputs and their interactions.^[11] Screening methods, exemplified by one-at-a-time (OAT) approaches, focus on rapidly identifying influential parameters by varying inputs sequentially while holding others constant.^[13] The importance of sensitivity analysis lies in its ability to identify key input parameters that drive output uncertainty, thereby reducing model complexity and enhancing robustness against input variations.^[12] By prioritizing data collection efforts on critical factors and revealing model deficiencies, it aids in verification and calibration processes.^[10] Prerequisites for conducting sensitivity analysis include a basic understanding of probability theory to define input distributions and appropriate metrics for evaluating model outputs, ensuring that analyses reflect realistic uncertainty scenarios.^[11]

One-at-a-Time vs. Global Methods

One-at-a-time (OAT) methods in sensitivity analysis involve systematically perturbing a single input factor while holding all others constant at nominal values, allowing for the direct assessment of individual effects on model outputs. These approaches are computationally inexpensive, typically requiring a number of model evaluations linear in the number of inputs (e.g., 2k evaluations for k inputs in basic designs), and are exemplified by traditional factorial designs or simple derivative-based analyses. However, OAT methods are limited in their ability to detect interactions between factors or nonlinear behaviors unless perturbations are repeated extensively across the input space, often leading to incomplete insights in complex models.^[14] In contrast, global sensitivity analysis methods explore the entire range of input factors simultaneously, accounting for their distributions, variances, and interactions to provide a more comprehensive evaluation of model behavior. Techniques such as variance-based approaches, including Sobol indices, decompose output variance into contributions from individual factors and their interactions using strategies like Monte Carlo sampling, which can require thousands of model runs (e.g., N(k+2) evaluations where N is often 1000 or more for reliable estimates). While these methods capture higher-order effects and non-monotonic relationships effectively, their high computational demand makes them impractical for initial screening in models with many inputs.^[14] The Morris method occupies a niche as an OAT-based global screening technique, achieving broader coverage than traditional OAT by generating multiple randomized trajectories across the input space to approximate overall factor influences and detect potential nonlinearities or interactions. This design requires r(k+1) evaluations (with r trajectories), offering a balance of efficiency and informativeness suitable for identifying key factors before applying more resource-intensive global methods like Sobol analysis. Nonetheless, it may overlook subtle higher-order interactions, positioning it primarily as a preliminary tool rather than a substitute for full global assessment.^[1]^[14]

Mathematical Formulation

Elementary Effects

The elementary effect in the Morris method quantifies the sensitivity of a model's output to a small perturbation in a single input factor, while keeping all other inputs constant. For a model y = f(\mathbf{x}), where \mathbf{x} = (x_1, \dots, x_k) is the vector of k input factors and y is the scalar output, the elementary effect EE_i(\mathbf{x}) for the i-th input factor x_i is defined as

EE_i(\mathbf{x}) = \frac{f(\mathbf{x} + \Delta \mathbf{e}_i) - f(\mathbf{x})}{\Delta},

where \Delta is a finite increment, \mathbf{e}_i is the i-th unit vector in \mathbb{R}^k, and \mathbf{x} + \Delta \mathbf{e}_i must remain within the input domain \Omega.^[2] This one-at-a-time perturbation captures the local gradient-like change in the output attributable to x_i, serving as the foundational unit for global sensitivity assessment.^[2] The perturbation size \Delta is typically set as \Delta = \frac{p}{2(p-1)} for even p, which is a multiple of the grid spacing \frac{1}{p-1}, ensuring that the perturbed point stays on the grid.^[2] The input domain \Omega is often discretized as a k-dimensional p-level grid within the unit hypercube [0, 1]^k, with factor values at \{0, \frac{1}{p-1}, \dots, 1\}, to approximate continuous inputs or directly represent discrete ones.^[2] Elementary effects are computed at multiple randomly selected base points \mathbf{x} across \Omega, generating a sample from the distribution F_i of EE_i values for each factor i; this sampling, often repeated r times per factor, accounts for nonlinearity and interactions by exploring variability in the effects across the input space.^[2] The method assumes that inputs can be scaled to the unit hypercube and discretized on the grid for evaluation, making it applicable to both continuous distributions (via grid approximation) and inherently discrete factors.^[2] While the original formulation addresses scalar-valued models, the elementary effect concept extends naturally to vector-valued outputs by applying the definition component-wise to each output dimension.

Sensitivity Measures

In the Morris method, the elementary effects computed for each input factor are aggregated into sensitivity indices to quantify and rank the importance of factors in influencing the model output. These indices provide a screening tool for identifying influential factors, particularly in high-dimensional models where computational efficiency is crucial. The primary measures are derived from the distribution of elementary effects EE_i(\mathbf{x}) for factor i, estimated empirically from multiple trajectories in the input space.^[2] The mean elementary effect, denoted \mu_i = E[EE_i(\mathbf{x})], represents the expected change in output per unit change in input x_i, averaged over the input space. It serves as a measure of the overall influence of factor i, with larger absolute values indicating greater importance. However, \mu_i can suffer from sign cancellation when positive and negative effects offset each other, particularly in nonlinear or non-monotonic models, potentially underestimating a factor's significance. In practice, \mu_i is approximated from r samples as \mu_i \approx \frac{1}{r} \sum_{j=1}^r EE_i^{(j)}(\mathbf{x}^{(j)}).^[2] The standard deviation, \sigma_i = \sqrt{\text{Var}[EE_i(\mathbf{x})]}, quantifies the variability of the elementary effects for factor i. A high \sigma_i suggests the presence of nonlinearities or interactions with other factors, as it captures how the effect changes across different base points in the input space. This measure complements \mu_i by highlighting factors with complex behaviors beyond simple additive effects. Empirically, \sigma_i is estimated as the sample standard deviation of the r elementary effects.^[2] To address the limitation of sign cancellation in \mu_i, an improved mean absolute effect is used: \mu_i^* = E[|EE_i(\mathbf{x})|]. This index focuses on the magnitude of effects, providing a more robust ranking of factor importance without interference from opposing signs. It is particularly effective for screening in models with potential non-monotonic responses. The empirical estimate is \mu_i^* = \frac{1}{r} \sum_{j=1}^r |EE_i^{(j)}(\mathbf{x}^{(j)})|.^[15] These measures are often visualized by plotting \mu_i^* against \sigma_i for all factors, enabling classification of their roles. Factors with high \mu_i^* and low \sigma_i are typically linear and additive, exerting consistent influence. In contrast, those with high values of both \mu_i^* and \sigma_i indicate nonlinear or interactive effects, warranting further detailed analysis. This graphical approach facilitates qualitative interpretation and prioritization in sensitivity studies.^[15]

Implementation Procedure

Sampling Design

The input factor space in the Morris method is scaled to the unit hypercube [0, 1]^k for k factors and discretized into a p-level grid \Omega = \{0, 1/(p-1), 2/(p-1), \dots, (p-1)/(p-1)\}^k, where p is the number of discrete levels (typically p = 4 or $8 to balance resolution and computational cost). This grid structure approximates continuous inputs via quasi-Monte Carlo-like spacing, enabling efficient evaluation on a finite set of points while maintaining coverage of the parameter space. Sampling proceeds by constructing r independent trajectories, each comprising k+1 points to compute elementary effects for all factors. A trajectory starts at a randomly selected base point \mathbf{x}^{(0)} from the grid \Omega. From this base, the subsequent k points are generated by sequentially perturbing one factor at a time by a step size \Delta, where \Delta = p / [2(p-1)] when p is even to ensure perturbed values stay on the grid and provide symmetric exploration around the base point. The order in which factors are perturbed is determined by a random permutation, and the direction of each perturbation (either +\Delta or -\Delta) is randomly assigned to avoid bias in the direction of change. This randomization process—selecting the starting point, permutation of factors, and perturbation directions independently for each trajectory—ensures an unbiased and representative sampling of the input space. The design requires a total of r(k+1) model evaluations, with r typically ranging from 5 to 20 depending on desired precision and available resources, resulting in computational demands that scale linearly with k. This efficiency makes the Morris method particularly suitable for screening studies involving hundreds of factors, where full factorial designs would be prohibitive.

Index Computation

Once the sampling design has been generated, consisting of r trajectories each with k+1 points in the k-dimensional input space, the model f is evaluated at all r(k+1) points to obtain the corresponding output values y^{(j)}_l for the l-th point in the j-th trajectory, where j = 1, \dots, r.^[2] For each trajectory j, the k elementary effects EE_i^{(j)} for input factor i = 1, \dots, k are then extracted by differencing consecutive output pairs along the chain: specifically, EE_i^{(j)} = \frac{y^{(j)}_{i+1} - y^{(j)}_i}{\Delta}, where \Delta is the predefined increment size, yielding one effect per factor per trajectory.^[2] These r elementary effects per factor are aggregated to compute the sensitivity indices. The mean effect \mu_i is the sample average \mu_i = \frac{1}{r} \sum_{j=1}^r EE_i^{(j)}, capturing the overall influence of factor i; the standard deviation \sigma_i is \sigma_i = \sqrt{ \frac{1}{r} \sum_{j=1}^r (EE_i^{(j)} - \mu_i)^2 }, indicating nonlinearity or interactions; and the mean absolute effect \mu_i^* = \frac{1}{r} \sum_{j=1}^r |EE_i^{(j)}| addresses potential sign cancellations in non-monotonic responses.^[2] Factors are typically ranked by \mu_i^* to identify influential ones, with thresholds such as \mu_i^* > 0.1 sometimes applied to screen for significance relative to model scale, though this depends on context.^[2] For models with vector-valued outputs, the indices \mu_i, \sigma_i, and \mu_i^* are computed separately for each output dimension, allowing factor importance to be assessed per response variable; if sample size permits, statistical tests like bootstrapping can evaluate index significance.^[16]

Variations and Extensions

Original Morris Method

The original Morris method, introduced by Max D. Morris in 1991, serves as a foundational screening approach in global sensitivity analysis for identifying influential input factors in complex computational models.^[2] It constructs multiple random trajectories in the standardized input space, computing elementary effects along each to derive two primary sensitivity indices: the mean effect μ, which quantifies the average influence of a factor, and the standard deviation σ, which screens for nonlinearity or interactions by measuring the variability of those effects.^[2] In practice, the method typically employs r=10-20 trajectories across a p=8-level grid, resulting in a total of r(k+1) model evaluations for k input factors, balancing computational cost with reliable factor ranking.^[2] Central to the method are the elementary effects, defined for each input factor x_i at a point \mathbf{x} in the k-dimensional unit hypercube as

d_i(\mathbf{x}) = \frac{y(\mathbf{x} + \Delta \mathbf{e}_i) - y(\mathbf{x})}{\Delta},

where \mathbf{e}_i is the i-th unit vector and Δ is the perturbation size.^[2] The original formulation sets \Delta = \frac{p}{2(p-1)} (for even p) to enable symmetric perturbations relative to grid points, with inputs assumed to follow uniform distributions over [0,1] to facilitate grid-based sampling.^[2] Trajectories are generated by sequentially varying one factor at a time from a random starting point, ensuring each trajectory samples k+1 points while approximating the distributions of elementary effects F_i for computing μ and σ as their sample mean and standard deviation, respectively.^[2] Interactions are not explicitly modeled but are implicitly detected through elevated σ values indicating non-additive behavior.^[2] Despite its efficiency, the original method has notable limitations. The mean μ can suffer from sign cancellation, where positive and negative elementary effects offset each other, leading to underestimation of a factor's importance even when its effects are large in magnitude.^[2] Furthermore, reliance on randomly selected trajectories risks inefficient exploration of the input space, as overlaps between trajectories may occur, particularly for higher-dimensional problems, reducing the diversity of sampled points.^[2]

Improved Measures and Trajectories

One significant enhancement to the original Morris method involves the introduction of the sensitivity measure \mu^*, defined as the mean of the absolute values of the elementary effects, which addresses the issue of effect cancellation in models with nonlinear or non-monotonic responses by focusing on the magnitude rather than the signed average.^[6] This measure, proposed by Campolongo et al. in 2007, improves the reliability of factor ranking and is often combined with improved trajectory designs that enhance exploration of the input space, ensuring more uniform coverage compared to random trajectories.^[6] Further refinements include optimized trajectory generation by using Latin Hypercube Sampling (LHS) or Sobol sequences to select starting points, followed by an algorithm that chooses a reduced number of trajectories (typically r = 5 to $10) to maximize spatial spread while preserving estimates of mean and standard deviation of effects.^[6] This optimization reduces computational demands without sacrificing accuracy in identifying influential factors, and it extends naturally to handling groups of factors by perturbing all members simultaneously.^[6] Extensions of the method address specific challenges, such as group sensitivity analysis for correlated inputs, where factors are bundled into groups (e.g., representing interdependent parameters) and perturbed collectively to assess joint effects, as demonstrated in applications to grouped inputs in benchmark functions.^[6] Hybrid approaches integrate the Morris method with Fourier Amplitude Sensitivity Test (FAST) techniques for sequential screening and quantification, using Morris for initial factor identification and FAST for detailed variance decomposition, which has been shown effective in building performance models by confirming key parameters with fewer overall evaluations.^[17] Adaptations for dynamic models incorporate metrics like Dynamic Time Warping (DTW) to compute elementary effects on time-series outputs, capturing time-varying sensitivities across multiple dimensions without relying on output approximations, as applied to ordinary differential equation systems like microbial kinetics models.^[18] Recent developments as of 2025 include second-order Morris methods to quantify higher-order interactions and machine learning-based approaches to compute elementary effects more efficiently in high-dimensional settings.^[19]^[20] These improvements are incorporated into open-source libraries such as Python's SALib, which supports multiple Morris variants including optimized sampling, group analysis, and automated trajectory selection via LHS or Sobol sequences, facilitating efficient implementation in high-dimensional systems modeling.^[21]

Applications and Limitations

Real-World Examples

The Morris method has been applied in environmental modeling to assess parameter importance in groundwater flow simulations using tools like MODFLOW. In a study coupling MODFLOW with the modular three-dimensional multispecies transport model (MT3DMS), the method screened factors influencing pollutant removal efficiency in aquifers, identifying hydraulic conductivity and porosity as the most influential parameters for contaminant transport dynamics.^[22] This approach, originating in the 1990s and echoed in later works such as a 2020 study on arid regions, has highlighted soil porosity's role in hydrological sensitivity analyses.^[23] In engineering applications, particularly automotive design, the Morris method facilitates screening of variables in finite element crash simulations to enhance vehicle safety. For example, an analysis of occupant restraint systems employed the elementary effects variant to evaluate sensitivities of kinematic and kinetic responses to material properties and structural parameters, revealing key crashworthiness factors like belt tension and airbag deployment timing.^[24] Recent implementations have reduced computational demands while prioritizing design optimizations in nonlinear dynamic models.^[25] Beyond traditional domains, the method has informed epidemiology during the COVID-19 pandemic by ranking intervention impacts in transmission models. An elementary effects analysis of SARS-CoV-2 spread demonstrated that social distancing and mask usage exerted the strongest effects on reducing infection rates, guiding policy prioritization over less influential factors like self-isolation compliance.^[26] In machine learning, InterpretML integrates Morris sensitivity analysis for feature selection in interpretable models, enabling global assessment of input importance in black-box systems like gradient boosting classifiers.^[3] In biomedical applications, a 2018 study applied the Morris method with r=10 trajectories to a physiologically based pharmacokinetic (PBPK) model for acetaminophen, screening parameters and identifying influential factors such as absorption rate and renal clearance, which helped reduce model complexity by fixing non-influential parameters.^[27] This demonstration underscored the method's efficiency in biomedical simulations for drug disposition. The Morris method has also been used in renewable energy modeling, such as screening parameters in solar photovoltaic system simulations to identify key factors affecting energy yield under uncertain weather conditions.^[28]

Advantages and Drawbacks

The Morris method offers several key advantages in sensitivity analysis, particularly for preliminary screening of input factors in computational models. Its computational efficiency stands out, as the number of required model evaluations scales linearly with the number of factors k, typically requiring only r(k + 1) runs where r is the number of replications (often 10–20), enabling analysis with as few as 100 runs for 50 factors in practice.^[2]^[29] Unlike many global methods, it imposes no strong distributional assumptions on the model or inputs, making it robust for diverse applications without presupposing linearity or additivity.^[2] The method's sensitivity measures, such as the mean \mu and standard deviation \sigma of elementary effects, facilitate easy interpretation: \mu identifies average effects, while \sigma detects nonlinearities or interactions, providing qualitative insights into factor importance with minimal implementation effort.^[30] Despite these strengths, the Morris method has notable drawbacks that limit its scope. It primarily captures first-order effects and qualitative rankings, offering limited precision for quantifying variance contributions or total effects, and often fails to distinguish between nonlinearities and higher-order interactions.^[29]^[30] For continuous input spaces, the method's reliance on discrete grid levels can introduce discretization bias, potentially altering results for smooth functions.^[28] Additionally, it lacks built-in uncertainty propagation for the sensitivity indices themselves, as the sampling does not converge to population statistics with finite runs, reducing reliability in noisy or highly variable models.^[28] In comparisons to other global sensitivity analysis techniques, the Morris method excels in speed but sacrifices comprehensiveness. It is substantially faster than variance-based methods like Sobol indices, which may demand 10,000 or more runs for similar factor counts due to the need for extensive Monte Carlo sampling, making Morris ideal as a preliminary screening tool before proceeding to more intensive approaches such as E-FAST.^[31] However, this efficiency comes at the cost of lower accuracy for detailed effect decomposition, positioning it best for initial factor identification rather than final quantification.^[32] From a 2025 perspective, the Morris method remains relevant for its simplicity in resource-constrained settings but is increasingly augmented with machine learning techniques, such as deep learning surrogates for computing elementary effects in large-scale models, to address scalability issues.^[20] Critiques highlight its potential oversimplification in complex systems, where undetected high-order interactions can lead to incomplete insights, prompting hybrid uses with advanced variance-based or interaction-focused extensions.^[33]

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