Parameter Sweeps and Sensitivity Analysis

Last Updated June 16, 2026

Parameter sweeps and sensitivity analysis help modelers understand how conclusions change when inputs, assumptions, parameters, initial conditions, and solver choices change. A continuous model may produce one clean trajectory, equilibrium, surface, forecast, or scenario, but that output is only one result under one configuration. Sensitivity analysis asks how fragile or robust that result is.

In systems modeling, parameter sensitivity is not a side issue. It shapes whether a model is explanatory, predictive, decision-relevant, or misleading. A small change in a growth rate, delay, recovery constant, diffusion coefficient, carrying capacity, elasticity, feedback strength, initial condition, or policy threshold can sometimes produce a large change in output. A model that appears stable under one parameter set may shift regimes, collapse, oscillate, or reverse conclusions under another.

This article introduces parameter sweeps and sensitivity analysis for systems modeling, including one-at-a-time sweeps, grid sweeps, local sensitivity, global sensitivity, scenario envelopes, parameter uncertainty, response surfaces, Sobol-style variance thinking, robustness review, reproducible workflows, and responsible interpretation.

Parameter sweeps systematically vary model inputs and record outputs. Sensitivity analysis studies how strongly outputs respond to changes in inputs. Together, they help reveal which assumptions matter most, where a model is robust, where it is fragile, and where interpretation should be cautious.

The central question is not only “What does the model produce?” It is “How much does the model output depend on the values we chose?”

Why Parameter Sweeps Matter

Parameter sweeps matter because a single model run can hide dependence on assumptions. Many continuous models include rates, constants, thresholds, delays, initial conditions, elasticities, capacities, diffusion coefficients, and forcing terms that are not known with certainty. Even when parameters are estimated from data, they may remain uncertain, context-dependent, or unstable over time.

A parameter sweep evaluates the model repeatedly across a planned range of parameter values.

A sweep does not prove that selected parameter ranges are correct. It shows how the model behaves within those ranges.

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What Sensitivity Analysis Means

Sensitivity analysis studies how changes in inputs affect changes in outputs. It can be local or global, qualitative or quantitative, exploratory or formal, visual or statistical.

For relative comparison, elasticity-style sensitivity is often useful:

Sensitivity analysis is not a single method. It is a family of practices for asking how model conclusions depend on inputs.

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Parameters in Systems Models

Parameters are values that shape model behavior without usually being treated as state variables. They may describe rates, capacities, thresholds, delays, coefficients, proportions, elasticities, transfer rates, decay constants, or interaction strengths.

Parameter review should ask where values came from, what ranges are plausible, how uncertain they are, and how much they matter to the conclusion.

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One-at-a-Time Sweeps

A one-at-a-time sweep changes one parameter while holding others fixed. This is simple, interpretable, and useful for early model review.

One-at-a-time sweeps are especially helpful for teaching, debugging, intuition, and communication. They can show monotonic response, saturation, tipping points, and rough sensitivity.

One-at-a-time analysis is a good first pass, but it should not be the only sensitivity method when interactions are important.

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Grid Sweeps and Response Surfaces

Grid sweeps vary two or more parameters across combinations. They help reveal interaction effects that one-at-a-time sweeps can miss.

When outputs are plotted as a surface, contour, or heat map, the result becomes a response surface.

Grid sweeps can become computationally expensive as the number of parameters grows. That makes careful design, sampling, and documentation important.

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Local Sensitivity

Local sensitivity evaluates how output changes near a baseline parameter value. It is closely connected to derivatives and finite differences.

Local sensitivity is useful when the baseline case is meaningful and the model is relatively smooth near that point. It can also support calibration, uncertainty review, and model debugging.

Local sensitivity is powerful but context-dependent. It should not be generalized beyond the neighborhood it examines.

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Global Sensitivity

Global sensitivity examines parameter influence across broader ranges or distributions. Instead of asking what happens near one baseline, it asks which inputs matter across a modeled uncertainty space.

Variance-based sensitivity thinking asks how much of output variation is associated with different inputs and their interactions.

Global sensitivity is especially useful when a model has many uncertain parameters or when parameter interactions are central to interpretation.

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Uncertainty, Scenarios, and Robustness

Sensitivity analysis is closely related to uncertainty analysis, but the two are not identical. Uncertainty analysis asks how uncertain outputs are. Sensitivity analysis asks which uncertain inputs drive that uncertainty.

A robust conclusion holds across a relevant range of assumptions. A fragile conclusion changes under small or plausible parameter changes.

Robustness does not mean certainty. It means the result is less dependent on the tested assumptions.

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Diagnosing Fragility

A model is fragile when its conclusions shift strongly under plausible changes. Fragility may come from thresholds, nonlinear feedback, parameter interactions, near-equilibrium instability, boundary behavior, stochastic uncertainty, solver settings, or poor calibration.

Fragility is not a reason to discard a model automatically. It is a reason to narrow claims, improve data, refine structure, or disclose limits.

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Systems Modeling Interpretation

In systems modeling, sensitivity analysis connects mathematics to judgment. The modeler must decide which parameters are uncertain, which ranges are plausible, which outputs matter, which interactions should be tested, and which conclusions are robust enough to communicate.

A sensitivity result can change the meaning of a model. If many parameter values lead to the same qualitative outcome, the model may support a robust pattern. If conclusions depend on a narrow parameter band, the model should be presented as conditional. If different plausible parameter sets produce different pathways, the model may be more useful as a scenario exploration tool than as a single forecast.

The stronger interpretive standard is not “we ran a sensitivity analysis.” It is: “we documented parameter ranges, tested relevant combinations, identified influential assumptions, and adjusted claims according to robustness and fragility.”

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Mathematical Deepening

This section adds a more formal layer for mathematically advanced readers. Parameter sweeps and sensitivity analysis connect derivatives, finite differences, response surfaces, elasticity, local and global sensitivity, variance decomposition, uncertainty propagation, Monte Carlo sampling, Sobol-style indices, Morris screening, identifiability, calibration, robustness analysis, and model governance.

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Computation and Reproducible Workflows

Computational sensitivity workflows should preserve parameter definitions, baseline values, ranges, distributions, sweep design, sampled values, solver settings, outputs, visualizations, ranking metrics, and interpretation notes. A sensitivity analysis should not be reduced to one figure without the table behind it.

A reproducible workflow makes it possible to inspect which parameter values were tested, why those ranges were chosen, how outputs were computed, and how the resulting evidence supports or limits a claim.

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Python Workflow: Parameter Sweep and Sensitivity Audit

The Python workflow below sweeps growth rate and carrying capacity in a logistic model, records final values, and estimates local sensitivity at a baseline.

from __future__ import annotations

from dataclasses import asdict, dataclass
from pathlib import Path
import csv
import json
import math


@dataclass(frozen=True)
class SweepRecord:
    growth_rate: float
    carrying_capacity: float
    initial_value: float
    stop_time: float
    final_value: float
    output_metric: str
    warning: str


@dataclass(frozen=True)
class SensitivityRecord:
    parameter: str
    baseline_value: float
    perturbation: float
    baseline_output: float
    forward_output: float
    backward_output: float
    finite_difference_sensitivity: float
    elasticity_estimate: float
    warning: str


def logistic_solution(t: float, x0: float, growth_rate: float, carrying_capacity: float) -> float:
    return carrying_capacity / (1 + ((carrying_capacity - x0) / x0) * math.exp(-growth_rate * t))


def final_output(growth_rate: float, carrying_capacity: float, x0: float = 10.0, stop_time: float = 20.0) -> float:
    return logistic_solution(stop_time, x0, growth_rate, carrying_capacity)


def build_grid_sweep() -> list[SweepRecord]:
    growth_rates = [0.18, 0.25, 0.35, 0.45, 0.55]
    carrying_capacities = [80.0, 100.0, 125.0, 150.0]
    records: list[SweepRecord] = []

    for r in growth_rates:
        for k in carrying_capacities:
            records.append(
                SweepRecord(
                    growth_rate=r,
                    carrying_capacity=k,
                    initial_value=10.0,
                    stop_time=20.0,
                    final_value=final_output(r, k),
                    output_metric="final_state_value",
                    warning="Sweep results depend on tested ranges, baseline assumptions, and model structure."
                )
            )

    return records


def finite_difference_sensitivity(parameter: str, baseline_r: float = 0.35, baseline_k: float = 100.0) -> SensitivityRecord:
    h = 0.01 if parameter == "growth_rate" else 1.0
    baseline = final_output(baseline_r, baseline_k)

    if parameter == "growth_rate":
        forward = final_output(baseline_r + h, baseline_k)
        backward = final_output(baseline_r - h, baseline_k)
        baseline_value = baseline_r
    elif parameter == "carrying_capacity":
        forward = final_output(baseline_r, baseline_k + h)
        backward = final_output(baseline_r, baseline_k - h)
        baseline_value = baseline_k
    else:
        raise ValueError("parameter must be growth_rate or carrying_capacity")

    sensitivity = (forward - backward) / (2 * h)
    elasticity = sensitivity * baseline_value / baseline

    return SensitivityRecord(
        parameter=parameter,
        baseline_value=baseline_value,
        perturbation=h,
        baseline_output=baseline,
        forward_output=forward,
        backward_output=backward,
        finite_difference_sensitivity=sensitivity,
        elasticity_estimate=elasticity,
        warning="Local sensitivity depends on baseline and perturbation size."
    )


sweep_records = build_grid_sweep()
sensitivity_records = [
    finite_difference_sensitivity("growth_rate"),
    finite_difference_sensitivity("carrying_capacity")
]

output_dir = Path("outputs")
(output_dir / "tables").mkdir(parents=True, exist_ok=True)
(output_dir / "json").mkdir(parents=True, exist_ok=True)
(output_dir / "reports").mkdir(parents=True, exist_ok=True)

with (output_dir / "tables" / "parameter_sweep_grid.csv").open("w", newline="", encoding="utf-8") as handle:
    writer = csv.DictWriter(handle, fieldnames=asdict(sweep_records[0]).keys())
    writer.writeheader()
    for record in sweep_records:
        writer.writerow(asdict(record))

with (output_dir / "tables" / "local_sensitivity_audit.csv").open("w", newline="", encoding="utf-8") as handle:
    writer = csv.DictWriter(handle, fieldnames=asdict(sensitivity_records[0]).keys())
    writer.writeheader()
    for record in sensitivity_records:
        writer.writerow(asdict(record))

(output_dir / "json" / "parameter_sweep_grid.json").write_text(
    json.dumps([asdict(record) for record in sweep_records], indent=2),
    encoding="utf-8"
)

(output_dir / "json" / "local_sensitivity_audit.json").write_text(
    json.dumps([asdict(record) for record in sensitivity_records], indent=2),
    encoding="utf-8"
)

report_lines = [
    "# Parameter Sweep and Sensitivity Audit",
    "",
    "This audit records a grid sweep and local finite-difference sensitivities for a logistic model.",
    "",
    "| Parameter | Baseline | Sensitivity | Elasticity |",
    "|---|---:|---:|---:|"
]

for record in sensitivity_records:
    report_lines.append(
        f"| {record.parameter} | {record.baseline_value} | {record.finite_difference_sensitivity:.6f} | {record.elasticity_estimate:.6f} |"
    )

report_lines.append("")
report_lines.append("Sensitivity evidence should be interpreted within tested ranges and documented assumptions.")

(output_dir / "reports" / "parameter_sensitivity_audit.md").write_text(
    "\n".join(report_lines) + "\n",
    encoding="utf-8"
)

print("Wrote parameter sweep and sensitivity audit outputs.")

This workflow preserves both the sweep table and the local sensitivity estimates so interpretation can be audited.

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R Workflow: Grid Sweep and Robustness Table

The R workflow below produces a grid sweep table and local sensitivity estimates for the same logistic model.

logistic_solution <- function(t, x0, growth_rate, carrying_capacity) {
  carrying_capacity / (1 + ((carrying_capacity - x0) / x0) * exp(-growth_rate * t))
}

final_output <- function(growth_rate, carrying_capacity, x0 = 10, stop_time = 20) {
  logistic_solution(stop_time, x0, growth_rate, carrying_capacity)
}

growth_rates <- c(0.18, 0.25, 0.35, 0.45, 0.55)
carrying_capacities <- c(80, 100, 125, 150)

sweep_grid <- expand.grid(
  growth_rate = growth_rates,
  carrying_capacity = carrying_capacities
)

sweep_grid$initial_value <- 10
sweep_grid$stop_time <- 20
sweep_grid$final_value <- mapply(
  final_output,
  sweep_grid$growth_rate,
  sweep_grid$carrying_capacity
)
sweep_grid$output_metric <- "final_state_value"
sweep_grid$warning <- "Sweep results depend on tested ranges, baseline assumptions, and model structure."

local_sensitivity <- function(parameter, baseline_r = 0.35, baseline_k = 100) {
  h <- ifelse(parameter == "growth_rate", 0.01, 1)
  baseline <- final_output(baseline_r, baseline_k)

  if (parameter == "growth_rate") {
    forward <- final_output(baseline_r + h, baseline_k)
    backward <- final_output(baseline_r - h, baseline_k)
    baseline_value <- baseline_r
  } else {
    forward <- final_output(baseline_r, baseline_k + h)
    backward <- final_output(baseline_r, baseline_k - h)
    baseline_value <- baseline_k
  }

  sensitivity <- (forward - backward) / (2 * h)
  elasticity <- sensitivity * baseline_value / baseline

  data.frame(
    parameter = parameter,
    baseline_value = baseline_value,
    perturbation = h,
    baseline_output = baseline,
    forward_output = forward,
    backward_output = backward,
    finite_difference_sensitivity = sensitivity,
    elasticity_estimate = elasticity,
    warning = "Local sensitivity depends on baseline and perturbation size."
  )
}

sensitivity_records <- rbind(
  local_sensitivity("growth_rate"),
  local_sensitivity("carrying_capacity")
)

dir.create("outputs/tables", recursive = TRUE, showWarnings = FALSE)

write.csv(
  sweep_grid,
  "outputs/tables/r_parameter_sweep_grid.csv",
  row.names = FALSE
)

write.csv(
  sensitivity_records,
  "outputs/tables/r_local_sensitivity_audit.csv",
  row.names = FALSE
)

print(sweep_grid)
print(sensitivity_records)

This workflow helps compare output variation across parameter combinations and local perturbations.

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Haskell Workflow: Typed Sensitivity Records

Haskell can represent sensitivity results as typed records, keeping parameter metadata and interpretation warnings attached to outputs.

module Main where

data SensitivityRecord = SensitivityRecord
  { parameterName :: String
  , baselineValue :: Double
  , testedRange :: String
  , sensitivityType :: String
  , outputMetric :: String
  , interpretationWarning :: String
  } deriving (Show)

records :: [SensitivityRecord]
records =
  [ SensitivityRecord
      "growth_rate"
      0.35
      "0.18 to 0.55"
      "grid_sweep_and_local_sensitivity"
      "final_state_value"
      "Growth-rate sensitivity depends on time horizon and baseline assumptions."
  , SensitivityRecord
      "carrying_capacity"
      100.0
      "80 to 150"
      "grid_sweep_and_local_sensitivity"
      "final_state_value"
      "Capacity sensitivity should not be generalized beyond the tested range."
  , SensitivityRecord
      "initial_value"
      10.0
      "context-dependent"
      "scenario_sensitivity"
      "trajectory_shape"
      "Initial conditions can affect transient behavior and threshold crossing."
  , SensitivityRecord
      "solver_settings"
      0.0
      "method-dependent"
      "computational_sensitivity"
      "numerical_result"
      "Sensitivity results should be checked against numerical reliability diagnostics."
  ]

main :: IO ()
main =
  mapM_ print records

The typed workflow makes sensitivity evidence part of the model record instead of an informal figure caption.

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SQL Workflow: Parameter Sensitivity Registry

SQL can document parameter assumptions, sweep designs, and review warnings for reproducible model governance.

CREATE TABLE parameter_sensitivity_registry (
    sensitivity_key TEXT PRIMARY KEY,
    sensitivity_name TEXT NOT NULL,
    analytical_role TEXT NOT NULL,
    systems_modeling_role TEXT NOT NULL,
    review_warning TEXT NOT NULL
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'one_at_a_time_sweep',
  'One-at-a-time sweep',
  'Varies one parameter while holding others fixed.',
  'Supports intuition, debugging, and simple communication.',
  'Can miss parameter interactions and depends on baseline values.'
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'grid_sweep',
  'Grid sweep',
  'Evaluates combinations of parameter values.',
  'Reveals interactions, thresholds, and response surfaces.',
  'Computational cost grows quickly with parameter dimension.'
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'local_sensitivity',
  'Local sensitivity',
  'Estimates response near a baseline parameter value.',
  'Identifies influential parameters around a chosen scenario.',
  'Should not be generalized beyond the local neighborhood.'
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'global_sensitivity',
  'Global sensitivity',
  'Assesses influence across broader ranges or distributions.',
  'Ranks uncertain drivers and interaction effects.',
  'Requires justified ranges, sampling design, and interpretation discipline.'
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'robustness_review',
  'Robustness review',
  'Checks whether conclusions persist across plausible assumptions.',
  'Separates robust conclusions from fragile claims.',
  'Robustness only applies to tested ranges and model structures.'
);

INSERT INTO parameter_sensitivity_registry VALUES
(
  'fragility_review',
  'Fragility review',
  'Identifies thresholds, reversals, and assumption-dependent conclusions.',
  'Helps narrow claims and prioritize further data collection.',
  'Fragility is a warning for interpretation, not automatic model rejection.'
);

SELECT
    sensitivity_name,
    analytical_role,
    systems_modeling_role,
    review_warning
FROM parameter_sensitivity_registry
ORDER BY sensitivity_key;

This registry ties sensitivity interpretation to one-at-a-time sweeps, grid sweeps, local sensitivity, global sensitivity, robustness review, fragility review, and governance records.

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GitHub Repository

The companion repository for this article is designed as a reproducible mathematical-modeling workspace. It supports parameter sweeps, local sensitivity estimates, grid sweep outputs, response-surface metadata, scenario envelopes, robustness tables, fragility records, SQL governance tables, generated reports, advanced mathematical audit logic, and reusable calculator scripts.

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Interpretive Limits and Responsible Use

Parameter sweeps and sensitivity analysis help reveal dependence on assumptions, but they do not validate a model by themselves. A robust result within a chosen parameter range may fail outside that range. A global sensitivity ranking depends on the distributions or ranges selected. A local derivative depends on the baseline and perturbation size. A scenario envelope is not a probability forecast unless probability assumptions are defined.

Responsible use requires documentation. Preserve parameter names, units, baseline values, tested ranges, range sources, sampling methods, solver settings, output metrics, generated tables, figures, sensitivity metrics, threshold records, and interpretation notes. Distinguish robust findings from fragile findings. Explain which conclusions depend on parameter assumptions and which remain stable across tested ranges.

The central question is not only “Which parameter matters most?” It is “Under what ranges, assumptions, metrics, and model structures does that parameter matter, and how should the claim be limited?”

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Related Articles

• Calculus for Systems Modeling
• Stiff Systems and Computational Difficulty
• Stability, Error, and Convergence in Numerical Modeling
• Model Calibration Using Calculus-Based Methods
• Sensitivity, Robustness, and Parameter Dependence
• Elasticity, Sensitivity, and Marginal Response
• Functions of Several Variables
• Partial Derivatives and Interaction Effects
• Scientific Computing for Systems Modeling
• Interpretation, Assumptions, and Responsible Mathematical Modeling

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Further Reading

• Hamby, D.M. (1994) ‘A review of techniques for parameter sensitivity analysis of environmental models’, Environmental Monitoring and Assessment, 32, pp. 135–154.
• Helton, J.C. (1993) ‘Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal’, Reliability Engineering & System Safety, 42(2–3), pp. 327–367.
• Iooss, B. and Lemaître, P. (2015) ‘A review on global sensitivity analysis methods’, in Dellino, G. and Meloni, C. (eds.) Uncertainty Management in Simulation-Optimization of Complex Systems. Boston, MA: Springer, pp. 101–122.
• Marino, S., Hogue, I.B., Ray, C.J. and Kirschner, D.E. (2008) ‘A methodology for performing global uncertainty and sensitivity analysis in systems biology’, Journal of Theoretical Biology, 254(1), pp. 178–196.
• Morris, M.D. (1991) ‘Factorial sampling plans for preliminary computational experiments’, Technometrics, 33(2), pp. 161–174.
• Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B. and Wagener, T. (2016) ‘Sensitivity analysis of environmental models: A systematic review with practical workflow’, Environmental Modelling & Software, 79, pp. 214–232.
• Saltelli, A., Chan, K. and Scott, E.M. (eds.) (2000) Sensitivity Analysis. Chichester: Wiley.
• Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. and Tarantola, S. (2008) Global Sensitivity Analysis: The Primer. Chichester: Wiley.
• Sobol, I.M. (1993) ‘Sensitivity estimates for nonlinear mathematical models’, Mathematical Modelling and Computational Experiments, 1, pp. 407–414.
• Smith, R.C. (2013) Uncertainty Quantification: Theory, Implementation, and Applications. Philadelphia, PA: Society for Industrial and Applied Mathematics.

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References

• Hamby, D.M. (1994) ‘A review of techniques for parameter sensitivity analysis of environmental models’, Environmental Monitoring and Assessment, 32, pp. 135–154.
• Helton, J.C. (1993) ‘Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal’, Reliability Engineering & System Safety, 42(2–3), pp. 327–367.
• Iooss, B. and Lemaître, P. (2015) ‘A review on global sensitivity analysis methods’, in Dellino, G. and Meloni, C. (eds.) Uncertainty Management in Simulation-Optimization of Complex Systems. Boston, MA: Springer, pp. 101–122.
• Marino, S., Hogue, I.B., Ray, C.J. and Kirschner, D.E. (2008) ‘A methodology for performing global uncertainty and sensitivity analysis in systems biology’, Journal of Theoretical Biology, 254(1), pp. 178–196.
• Morris, M.D. (1991) ‘Factorial sampling plans for preliminary computational experiments’, Technometrics, 33(2), pp. 161–174.
• Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B. and Wagener, T. (2016) ‘Sensitivity analysis of environmental models: A systematic review with practical workflow’, Environmental Modelling & Software, 79, pp. 214–232.
• Saltelli, A., Chan, K. and Scott, E.M. (eds.) (2000) Sensitivity Analysis. Chichester: Wiley.
• Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. and Tarantola, S. (2008) Global Sensitivity Analysis: The Primer. Chichester: Wiley.
• Sobol, I.M. (1993) ‘Sensitivity estimates for nonlinear mathematical models’, Mathematical Modelling and Computational Experiments, 1, pp. 407–414.
• Smith, R.C. (2013) Uncertainty Quantification: Theory, Implementation, and Applications. Philadelphia, PA: Society for Industrial and Applied Mathematics.

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