11.1 Sensitivity analysis
6 min read•august 14, 2024
Sensitivity analysis is a crucial tool in mathematical modeling. It helps determine how changes in input variables affect the model's output, allowing us to identify which factors are most important. This process builds confidence in the model and guides future refinements.
By examining the model's sensitivity to different parameters, we can simplify it, prioritize data collection, and identify tipping points. This leads to more robust models that better support decision-making and risk assessment in real-world applications.
Sensitivity Analysis in Modeling
Purpose and Importance of Sensitivity Analysis
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- Determines how different values of an independent variable affect a dependent variable under a given set of assumptions
- Predicts the outcome of a decision given a certain range of variables
- Ascertains how the model depends upon the information fed into it, its structure, and the framing assumptions made to build it
- Allows the modeler to determine the level of confidence in the model as a basis for decision-making
- Helps build confidence in the model by studying the uncertainties often associated with parameters in models
- Many parameters in system dynamics models represent quantities that are very difficult or impossible to measure accurately in the real world
- Determines what level of accuracy is necessary for a parameter to make the model sufficiently useful and valid
- If tests reveal that the model is insensitive to a parameter, it may be possible to use an estimate rather than a more precise value
- Provides a basis for model simplification by fixing model parameters or removing variables that have little impact on model performance
- Allows for the development of a simpler model that still maintains a high level of accuracy
Benefits of Sensitivity Analysis
- Prioritizes data collection efforts by identifying the most influential parameters
- Resources can be allocated to collecting more accurate data for parameters with the greatest impact on model output
- Guides model development and refinement through an iterative process
- Each iteration informs decisions about parameter values, model structure, and data collection needs, leading to a more robust and reliable model over time
- Identifies tipping points or thresholds in the model where small changes in input parameters lead to large changes in model output
- Understanding these thresholds is important for decision-making and risk assessment
- Facilitates communication and understanding of the model by focusing on the most important parameters and processes
- Helps stakeholders grasp the key drivers of model behavior and the implications for decision-making
Sensitivity Analysis Methods
One-Factor-at-a-Time (OFAT) Analysis
- Involves changing one parameter at a time while keeping others at their baseline (nominal) values
- Parameter is returned to its nominal value before repeating the process for each of the other inputs
- Provides a representation of the effect of a single parameter on the model output
- Useful for distinguishing between model inputs and determining which inputs have the greatest effect on the model output
- Considered a method because it only examines the effect of a single parameter at a time
- Does not take into account the simultaneous variation of input parameters
- Works best if the model is linear
Global Sensitivity Analysis
- Examines the effect of all the input parameters varying simultaneously
- Includes the joint effect of inputs on the output
- Apportions the output uncertainty to the uncertainty in each input variable
- (FAST) is a commonly used approach
- Estimates the expected value and variance of the output variable through the multidimensional integral of the model output using a transformation into a one-dimensional integral
- Sobol sensitivity analysis is a variance-based method
- Decomposes the variance of the model output into fractions attributable to inputs or sets of inputs
- Estimates sensitivity across the whole input space, dealing with nonlinear responses and interactions
Other Sensitivity Analysis Methods
- Local methods such as
- Uses of the output functions with respect to the input variables
- Sampling-based methods such as Monte Carlo filtering
- Involves running the model multiple times with randomly selected input values and analyzing the distribution of the output values
- Screening methods such as ()
- Identifies important input factors using a relatively small number of model evaluations
- Useful for models with a large number of input factors
Interpreting Sensitivity Analysis Results
Identifying Influential Parameters
- The change in model output for a given change in an input parameter measures the sensitivity of the model to that parameter (OFAT)
- The greater the change in output for a given change in input, the more sensitive the model is to that parameter
- Sensitivity indices provide a measure of the contribution of each input parameter to the variance of the model output (global methods)
- Higher sensitivity indices indicate greater influence of the parameter on the model output
Visualizing Sensitivity Analysis Results
- Tornado plots are commonly used for OFAT analysis
- Shows the change in model output for a given change in each input parameter, with the parameters ranked from most to least influential
- Pie charts or bar graphs can be used to display global sensitivity indices
- Illustrates the relative contribution of each input parameter to the variance of the model output
- Scatter plots can show the relationship between input parameters and model output
- Helps identify any nonlinear relationships or interactions between parameters
Reducing Uncertainty and Improving Model Accuracy
- Identifying the most influential parameters guides efforts to reduce uncertainty in the model
- Focusing on improving the accuracy of the most important parameters through additional data collection or refinement of the model structure
- Allocating resources to collecting more accurate data for the most influential parameters
- Ensures that efforts to improve model accuracy are targeted and efficient
- Understanding the relationships between input parameters and model output
- Informs decisions about model structure, complexity, and the need for additional data collection
Sensitivity Analysis for Model Refinement
Model Simplification
- Identifying parameters that have little or no influence on the model output
- These parameters can potentially be fixed at their nominal values or removed from the model entirely
- Simplifying the model structure by removing or aggregating processes that have little impact on model behavior
- Results in a simpler model that is easier to understand, communicate, and maintain
- Iteratively applying sensitivity analysis to guide model simplification
- Each iteration identifies opportunities for further simplification while ensuring that model accuracy is maintained
Guiding Model Development
- Using sensitivity analysis to inform decisions about model structure and complexity
- If certain parameters or processes have little influence on the model output, they may be candidates for removal or simplification
- Identifying tipping points or thresholds in the model
- Understanding these thresholds is important for decision-making and risk assessment
- Helps focus model development efforts on accurately representing these critical points
- Iteratively applying sensitivity analysis to guide model development
- Each iteration provides insights into model behavior, parameter importance, and data needs
- Leads to a more robust, reliable, and fit-for-purpose model over time
Communicating Model Results
- Focusing on the most influential parameters and processes when communicating model results
- Helps stakeholders understand the key drivers of model behavior and the implications for decision-making
- Using visualizations of sensitivity analysis results to convey the relative importance of different parameters
- Tornado plots, pie charts, and bar graphs can be effective communication tools
- Providing a clear rationale for model simplification and refinement decisions based on sensitivity analysis
- Helps build trust and confidence in the model among stakeholders
- Demonstrates a rigorous and transparent approach to model development and testing
Key Terms to Review (20)
Differential Sensitivity Analysis: Differential sensitivity analysis is a technique used to determine how the uncertainty in the output of a model can be apportioned to different sources of uncertainty in its inputs. This approach allows researchers and analysts to identify which variables have the most influence on a model's outcomes and to assess the robustness of those outcomes under varying conditions. By examining how small changes in input parameters affect results, this analysis provides insights into which factors are critical for decision-making and can guide further investigation.
Elasticity: Elasticity measures how responsive a variable is to changes in another variable. In mathematical modeling, it often refers to the sensitivity of an output variable when there are changes in input parameters, highlighting how small changes can lead to significant impacts on results or behaviors within a system.
Elementary Effects Method: The elementary effects method is a sensitivity analysis technique used to evaluate how changes in input parameters of a mathematical model affect its output. This method focuses on varying one parameter at a time while keeping others fixed, allowing for a clear assessment of the influence of each parameter on the model's behavior. It provides insights into the significance of parameters and helps identify which ones have the most substantial impact on outputs.
Engineering design: Engineering design is a systematic approach to problem-solving that involves defining a need, generating concepts, and developing solutions to meet specific requirements. It combines technical knowledge, creativity, and practical considerations to create effective products, systems, or processes. In the context of sensitivity analysis, engineering design seeks to understand how variations in input parameters affect system performance, allowing designers to optimize solutions under uncertain conditions.
Financial modeling: Financial modeling is the process of creating a numerical representation of a financial situation or scenario, often using spreadsheets to forecast future financial performance. This practice connects various components like revenue projections, expenses, and investment returns, allowing for analysis and decision-making. It plays a crucial role in assessing the viability of projects, understanding risk, and evaluating potential outcomes in financial contexts.
Fourier Amplitude Sensitivity Test: The Fourier Amplitude Sensitivity Test (FAST) is a method used in sensitivity analysis to quantify how variations in input parameters affect the output of a model. This technique involves transforming the input parameters into their Fourier series representation, which allows for efficient computation of sensitivity indices. By analyzing the amplitude of these Fourier components, researchers can identify which inputs have the most significant impact on model outputs and how uncertainties propagate through the system.
Global sensitivity analysis: Global sensitivity analysis is a statistical method used to determine how variations in model input parameters influence the output of a mathematical model. This approach considers all input variables simultaneously and assesses their impact on the model's behavior across a specified range, providing a comprehensive understanding of which factors are most influential. By applying this technique, researchers can identify critical parameters that drive model predictions, aiding in decision-making and model refinement.
Linear models: Linear models are mathematical representations of relationships between variables that assume a straight-line relationship. They are characterized by a linear equation of the form $$y = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. This simple structure allows for easy analysis and interpretation, making linear models essential in sensitivity analysis, result visualization, case studies with modeling software, and various types of mathematical models.
Local sensitivity analysis: Local sensitivity analysis is a technique used to determine how the variation in the output of a model can be attributed to different variations in its input parameters. This approach helps identify which parameters have the most significant impact on the model's results, allowing for better decision-making and model refinement. It is particularly useful for understanding the robustness of a model in scenarios where input parameters may fluctuate within a certain range.
Monte Carlo Simulation: Monte Carlo Simulation is a computational technique that uses random sampling to obtain numerical results, often used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. This method is widely applicable, providing insights into various scenarios by simulating the impact of uncertainty and variability, making it valuable in decision-making and optimization problems.
Morris Method: The Morris Method is a global sensitivity analysis technique used to assess the influence of input variables on the output of a mathematical model. This method generates a series of sample points in the input space and evaluates the model's response, allowing for the identification of which inputs have the most significant impact on the output while considering interactions between variables.
Nonlinear models: Nonlinear models are mathematical representations in which the relationship between variables is not a straight line; this means that the change in one variable does not produce a proportional change in another. These models are essential in capturing complex behaviors and interactions in various fields, allowing for more accurate predictions and analyses. Nonlinear relationships can lead to phenomena such as bifurcations and chaos, making it critical to assess how changes in input can affect outputs, visualize results effectively, and understand the nature of different types of models.
One-factor-at-a-time analysis: One-factor-at-a-time analysis is a method used in sensitivity analysis that involves changing one input variable at a time while keeping all other variables constant to observe the effect on the output of a model. This approach helps to identify how sensitive the model's outcomes are to individual factors, allowing for a clearer understanding of which inputs have the most influence. By isolating one variable, it simplifies the analysis but may not capture interactions between multiple variables.
Parameter variation: Parameter variation refers to the changes in the values of parameters within a mathematical model and how these changes affect the outcomes or predictions of that model. Understanding parameter variation is crucial for assessing the robustness and reliability of a model, as small changes in input parameters can lead to significant differences in results, particularly in complex systems. This concept is closely tied to sensitivity analysis, where the aim is to determine how variations in parameters impact model behavior.
Partial Derivatives: Partial derivatives are mathematical tools used to measure how a function changes as one of its variables changes while keeping the other variables constant. They play a crucial role in understanding the sensitivity of multivariable functions, allowing analysts to identify how individual inputs impact outputs in scenarios like optimization and modeling.
Perturbation method: The perturbation method is a mathematical technique used to find an approximate solution to a problem that cannot be solved exactly, by introducing a small change or 'perturbation' to the known solution of a related problem. This approach is often used in sensitivity analysis to determine how small changes in input parameters can affect the output of a model, highlighting the model's responsiveness and stability.
Robustness: Robustness refers to the ability of a mathematical model to maintain its performance and reliability under varying conditions or assumptions. It highlights the model's strength in producing consistent and valid results despite uncertainties in input parameters, environmental changes, or structural alterations. This quality is crucial for ensuring that models can be effectively used in real-world applications, where various factors may impact their performance.
Sensitivity coefficient: The sensitivity coefficient is a numerical measure that quantifies how sensitive a model's output is to changes in its input parameters. It plays a critical role in sensitivity analysis, helping to identify which parameters have the most significant impact on the results of a mathematical model, thereby guiding decision-making and optimization.
Sobol' Indices: Sobol' indices are a set of quantitative measures used in sensitivity analysis to determine the contribution of individual input variables to the uncertainty in model outputs. They help identify which inputs have the most significant impact on the output variability, allowing for more informed decision-making and model optimization. By calculating these indices, analysts can prioritize efforts in model refinement and understand the relationships between inputs and outputs more effectively.
Uncertainty Propagation: Uncertainty propagation refers to the process of determining how uncertainties in input variables affect the uncertainty in the output of a model or system. It is crucial for understanding the reliability of predictions made by mathematical models, especially when those models involve complex calculations or multiple uncertain parameters. By analyzing how errors and uncertainties propagate through a model, one can identify which variables have the most significant impact on the outcomes and thus prioritize efforts for improving measurement and accuracy.