11.2 Uncertainty modeling and propagation

Uncertainty modeling and propagation are crucial aspects of geospatial engineering. They help us understand and quantify the limitations of our data and analyses, ensuring more reliable decision-making. By recognizing different types of uncertainty and applying various methods to represent and propagate it, we can improve the accuracy of our models.

Effective uncertainty communication is key to building trust and supporting informed choices. Through visual representations, metrics, and decision support systems, we can convey uncertainty information to stakeholders in a clear and meaningful way. This approach promotes transparency and enables better use of geospatial data in real-world applications.

Types of uncertainty

• Uncertainty is a fundamental concept in geospatial engineering that refers to the lack of complete knowledge or information about a system or process
• Understanding the different types of uncertainty is crucial for accurately modeling and analyzing geospatial data and making informed decisions based on the results

Epistemic uncertainty

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• Arises from incomplete knowledge or limited understanding of a system or process
• Can be reduced by gathering more information or improving the accuracy of measurements and observations
• Examples include uncertainty in the parameters of a spatial interpolation model or uncertainty in the classification of land cover types from remote sensing imagery

Aleatory uncertainty

• Represents the inherent randomness or variability in a system or process that cannot be reduced by gathering more information
• Also known as stochastic uncertainty or irreducible uncertainty
• Examples include the uncertainty in the occurrence of natural hazards like earthquakes or floods, or the uncertainty in the spatial distribution of soil properties within a field

Ontological uncertainty

• Relates to the ambiguity or vagueness in the definition or conceptualization of a system or process
• Arises from the lack of a clear or universally accepted framework for describing or modeling a phenomenon
• Examples include the uncertainty in defining the boundaries of a geographic region or the uncertainty in classifying land use types based on different criteria

Representing uncertainty

• Uncertainty in geospatial data and models can be represented using various mathematical and computational frameworks
• The choice of representation depends on the nature of the uncertainty, the available data, and the intended use of the uncertainty information

Probability theory

• Uses probability distributions to represent the likelihood of different outcomes or states of a system
• Suitable for representing aleatory uncertainty and epistemic uncertainty when sufficient data is available to estimate the probability distributions
• Examples include representing the uncertainty in the location of a point feature using a bivariate normal distribution or representing the uncertainty in the value of a spatial variable using a Gaussian random field

Fuzzy set theory

• Uses membership functions to represent the degree of belonging of an element to a set, allowing for gradual transitions between sets
• Suitable for representing ontological uncertainty and vagueness in the definition of categories or classes
• Examples include representing the uncertainty in the classification of land cover types using fuzzy membership functions or representing the uncertainty in the delineation of soil types using fuzzy boundaries

Possibility theory

• Uses possibility distributions to represent the plausibility of different outcomes or states of a system, without requiring a precise probability assignment
• Suitable for representing epistemic uncertainty when limited information is available to estimate probability distributions
• Examples include representing the uncertainty in the parameters of a spatial interpolation model using possibility distributions or representing the uncertainty in the location of a feature using a possibility map

Belief functions

• Uses belief and plausibility measures to represent the evidence or support for different propositions or hypotheses about a system
• Suitable for representing uncertainty when information from multiple sources needs to be combined and when conflicting evidence may exist
• Examples include representing the uncertainty in the classification of land cover types using belief functions derived from multiple remote sensing data sources or representing the uncertainty in the assessment of natural hazard risk using belief functions based on expert opinions and historical data

Uncertainty propagation methods

• Uncertainty propagation refers to the process of quantifying how uncertainty in the inputs of a model or system affects the uncertainty in the outputs
• Various methods exist for propagating uncertainty through geospatial models and analyses, depending on the complexity of the model and the nature of the uncertainty

Monte Carlo simulation

• Involves generating multiple realizations of the input variables based on their probability distributions and running the model or analysis for each realization
• The resulting distribution of the output variables represents the propagated uncertainty
• Suitable for complex, non-linear models and can handle various types of probability distributions
• Example: Propagating uncertainty in the parameters of a spatial interpolation model by generating multiple realizations of the parameters and running the interpolation for each realization

Taylor series expansion

• Approximates the output of a model or analysis using a Taylor series expansion around the mean values of the input variables
• The coefficients of the Taylor series represent the sensitivity of the output to changes in the input variables
• Suitable for linear or mildly non-linear models and requires the computation of partial derivatives
• Example: Propagating uncertainty in the inputs of a spatial regression model by computing the Taylor series expansion of the regression coefficients

Unscented transformation

• Selects a set of deterministic points (sigma points) around the mean of the input variables and propagates these points through the model or analysis
• The weighted average and covariance of the transformed points represent the mean and uncertainty of the output
• Suitable for non-linear models and can capture higher-order moments of the output distribution
• Example: Propagating uncertainty in the parameters of a spatial dynamic model by selecting sigma points and running the model for each point

Polynomial chaos expansion

• Represents the output of a model or analysis as a polynomial expansion of the input variables, where the basis functions are orthogonal polynomials chosen based on the probability distributions of the inputs
• The coefficients of the polynomial expansion are computed using a spectral projection or regression approach
• Suitable for complex, non-linear models and can provide a compact representation of the output uncertainty
• Example: Propagating uncertainty in the inputs of a spatial environmental model by constructing a polynomial chaos expansion of the model outputs

Sensitivity analysis

• Sensitivity analysis aims to quantify how changes in the inputs of a model or system affect the outputs, and to identify the most influential input variables
• Sensitivity analysis is closely related to uncertainty analysis and helps prioritize efforts for reducing uncertainty and improving model reliability

Local vs global sensitivity

• Local sensitivity analysis evaluates the impact of small changes in the input variables around a nominal point, typically using partial derivatives or finite differences
• Global sensitivity analysis considers the entire range of possible values for the input variables and assesses their relative contributions to the output variability
• Local sensitivity is easier to compute but may not capture non-linear or interaction effects, while global sensitivity provides a more comprehensive understanding of the model behavior

Variance-based methods

• Decompose the total variance of the model output into contributions from individual input variables and their interactions
• The main sensitivity measures are the first-order and total-order Sobol' indices, which represent the main effect and the total effect (including interactions) of each input variable
• Variance-based methods require a large number of model evaluations but provide a robust and model-independent assessment of sensitivity
• Example: Computing the Sobol' indices for the inputs of a spatial interpolation model to identify the most influential parameters

Screening methods

• Aim to identify the most important input variables with a relatively small number of model evaluations, typically using a factorial or fractional factorial design
• Screening methods are based on the assumption that only a few input variables have a significant effect on the output, while the others can be fixed at their nominal values
• Examples of screening methods include Morris' elementary effects method and the Cotter's method
• Example: Applying Morris' method to a spatial environmental model to identify the key input variables that affect the model outputs

Metamodel-based methods

• Construct a simplified model (metamodel) that approximates the relationship between the inputs and outputs of the original model, using techniques such as regression, neural networks, or Gaussian processes
• Sensitivity analysis is then performed on the metamodel, which is computationally cheaper to evaluate than the original model
• Metamodel-based methods can be used in combination with variance-based or screening methods to reduce the computational cost of sensitivity analysis
• Example: Building a Gaussian process metamodel of a spatial dynamic model and using it to compute the Sobol' indices for the model inputs

Bayesian inference

• Bayesian inference is a statistical approach that combines prior knowledge or beliefs about a system with observed data to update the knowledge or beliefs and quantify uncertainty
• In geospatial engineering, Bayesian inference is used for parameter estimation, model selection, and uncertainty quantification in various applications, such as spatial interpolation, remote sensing image classification, and spatial data fusion

Bayes' theorem

• States that the posterior probability of a hypothesis (or a set of parameters) given the observed data is proportional to the product of the prior probability of the hypothesis and the likelihood of the data given the hypothesis
• Provides a formal framework for updating beliefs or knowledge based on new evidence
• Example: Updating the probability distribution of the parameters of a spatial interpolation model based on observed data using Bayes' theorem

Prior vs posterior distributions

• The prior distribution represents the initial knowledge or beliefs about the hypothesis or parameters before observing the data
• The posterior distribution represents the updated knowledge or beliefs after combining the prior distribution with the observed data through Bayes' theorem
• The choice of prior distribution can have a significant impact on the posterior distribution, especially when the observed data is limited or noisy

Conjugate priors

• A conjugate prior is a family of prior distributions that, when combined with the likelihood function, results in a posterior distribution from the same family
• Conjugate priors simplify the computation of the posterior distribution and allow for analytical solutions in some cases
• Examples of conjugate priors include the Beta distribution for the parameter of a Bernoulli likelihood and the Gamma distribution for the parameter of a Poisson likelihood

Markov Chain Monte Carlo (MCMC)

• MCMC is a class of algorithms for sampling from the posterior distribution when analytical solutions are not available or when the posterior distribution is complex or high-dimensional
• MCMC algorithms, such as the Metropolis-Hastings algorithm and the Gibbs sampler, generate a Markov chain that converges to the posterior distribution
• The samples from the Markov chain can be used to estimate the posterior probabilities, credible intervals, and other quantities of interest
• Example: Using MCMC to sample from the posterior distribution of the parameters of a spatial regression model and estimate the uncertainty in the regression coefficients

Uncertainty quantification in GIS

• Uncertainty quantification is the process of characterizing, propagating, and communicating uncertainty in geospatial data, models, and analyses
• In GIS, uncertainty arises from various sources, such as measurement errors, data processing, spatial interpolation, and data integration

Spatial interpolation uncertainty

• Spatial interpolation methods, such as kriging and inverse distance weighting, estimate the values of a spatial variable at unsampled locations based on observed values at sampled locations
• Uncertainty in spatial interpolation arises from the choice of interpolation method, the parameters of the method, and the spatial configuration and density of the sampled locations
• Methods for quantifying spatial interpolation uncertainty include conditional simulation, Bayesian kriging, and Gaussian process regression
• Example: Using conditional simulation to generate multiple realizations of a spatially interpolated surface and compute the uncertainty in the estimated values

Remote sensing data uncertainty

• Remote sensing data, such as satellite imagery and LiDAR point clouds, are subject to various sources of uncertainty, including sensor noise, atmospheric effects, and geometric distortions
• Uncertainty in remote sensing data can propagate to derived products, such as land cover maps and digital elevation models
• Methods for quantifying remote sensing data uncertainty include error propagation, Monte Carlo simulation, and Bayesian inference
• Example: Applying Monte Carlo simulation to propagate the uncertainty in the spectral reflectance values of a satellite image through a land cover classification model

Uncertainty in spatial decision-making

• Spatial decision-making, such as land use planning and natural resource management, often involves multiple criteria, stakeholders, and sources of uncertainty
• Uncertainty in spatial decision-making can arise from the data, the decision model, and the preferences and values of the decision-makers
• Methods for incorporating uncertainty in spatial decision-making include sensitivity analysis, scenario analysis, and multi-criteria decision analysis under uncertainty
• Example: Using sensitivity analysis to assess the robustness of a land use allocation decision to uncertainties in the input data and decision criteria

Visualizing uncertainty in maps

• Visualizing uncertainty in maps is important for communicating the reliability and limitations of geospatial data and analyses to users and decision-makers
• Techniques for visualizing uncertainty in maps include error bars, confidence intervals, probability surfaces, and fuzzy boundaries
• The choice of visualization technique depends on the type of uncertainty, the intended audience, and the purpose of the map
• Example: Using a color gradient to represent the probability of a spatial variable exceeding a certain threshold on a map

Uncertainty in spatial data fusion

• Spatial data fusion involves combining data from multiple sources, such as remote sensing, field surveys, and GIS databases, to create a consistent and comprehensive representation of a geographic area
• Uncertainty in spatial data fusion arises from the differences in the accuracy, resolution, and completeness of the input data sources, as well as from the data integration process itself

Data quality assessment

• Assessing the quality of the input data sources is a critical step in spatial data fusion, as it helps identify the sources and magnitude of uncertainty in each dataset
• Data quality indicators, such as positional accuracy, attribute accuracy, temporal accuracy, and logical consistency, can be used to quantify the uncertainty in the input data sources
• Statistical methods, such as error propagation and Monte Carlo simulation, can be used to estimate the overall uncertainty in the fused dataset based on the uncertainties in the input data sources
• Example: Computing the positional accuracy of a fused land cover map based on the positional accuracies of the input remote sensing and field survey datasets

Conflation techniques

• Conflation techniques, such as feature matching and rubber sheeting, are used to align and integrate spatial features from different data sources
• Uncertainty in conflation arises from the errors in the feature matching process and the distortions introduced by the geometric transformations
• Methods for quantifying uncertainty in conflation include error propagation, Monte Carlo simulation, and sensitivity analysis
• Example: Using Monte Carlo simulation to propagate the uncertainty in the feature matching process through a rubber sheeting transformation and estimate the uncertainty in the conflated dataset

Uncertainty-aware data integration

• Uncertainty-aware data integration methods explicitly account for the uncertainties in the input data sources and the data integration process when combining spatial data
• Examples of uncertainty-aware data integration methods include Bayesian data fusion, Dempster-Shafer theory, and fuzzy set theory
• These methods allow for the representation and propagation of uncertainty in the fused dataset, and provide a measure of the reliability and completeness of the integrated information
• Example: Applying Dempster-Shafer theory to combine land cover classifications from multiple remote sensing data sources, taking into account the uncertainties in each classification

Uncertainty propagation in data fusion

• Uncertainty propagation in data fusion involves quantifying how the uncertainties in the input data sources and the data integration process affect the uncertainty in the fused dataset
• Methods for propagating uncertainty in data fusion include error propagation, Monte Carlo simulation, and sensitivity analysis
• The choice of uncertainty propagation method depends on the complexity of the data fusion model, the nature of the uncertainties, and the computational resources available
• Example: Using error propagation to estimate the uncertainty in a fused digital elevation model based on the uncertainties in the input LiDAR and photogrammetric datasets

Uncertainty communication

• Uncertainty communication involves effectively conveying information about the sources, magnitude, and implications of uncertainty in geospatial data, models, and analyses to users and decision-makers
• Effective uncertainty communication is essential for building trust, supporting informed decision-making, and promoting the appropriate use of geospatial information

Visual representation of uncertainty

• Visual representation of uncertainty involves using graphical techniques, such as maps, charts, and diagrams, to communicate uncertainty information to users
• Examples of visual techniques for representing uncertainty include error bars, confidence intervals, probability surfaces, and fuzzy boundaries
• The choice of visual technique depends on the type of uncertainty, the intended audience, and the purpose of the communication
• Example: Using a graduated color scheme to represent the probability of a spatial variable exceeding a certain threshold on a map

Uncertainty metrics and indices

• Uncertainty metrics and indices are quantitative measures that summarize the magnitude and variability of uncertainty in geospatial data, models, and analyses
• Examples of uncertainty metrics and indices include the coefficient of variation, the entropy, the fuzzy membership value, and the probability of exceedance
• These metrics and indices can be used to compare the uncertainty across different datasets, models, or scenarios, and to communicate the overall reliability and precision of the geospatial information
• Example: Computing the coefficient of variation of a spatially interpolated surface to communicate the relative uncertainty in the estimated values

Uncertainty in decision support systems

• Decision support systems are computer-based tools that assist users in making decisions based on geospatial data and analyses
• Uncertainty in decision support systems arises from the uncertainties in the input data, the decision model, and the user preferences and values
• Methods for incorporating uncertainty in decision support systems include sensitivity analysis, scenario analysis, and multi-criteria decision analysis under uncertainty
• Example: Using sensitivity analysis to assess the robustness of a land use allocation decision to uncertainties in the input data and decision criteria in a GIS-based decision support system

Communicating uncertainty to stakeholders

• Communicating uncertainty to stakeholders, such as policymakers, resource managers, and the general public, is important for building trust, promoting transparency, and supporting informed decision-making
• Effective communication of uncertainty requires understanding the needs, knowledge, and values of the stakeholders, and using clear, concise, and non-technical language
• Strategies for communicating uncertainty to stakeholders include using visual aids, providing context and examples, and engaging in two-way dialogue
• Example: Using a series of maps and charts to communicate the uncertainty in a flood risk assessment to a group of community stakeholders, and soliciting their feedback and concerns