Category: Mathematical modeling

Uncertainty quantification
Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in both computational and real world applications. It tries to determine how likely certa
Lotka–Volterra equations
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in w
Linear partial information
Linear partial information (LPI) is a method of making decisions based on insufficient or fuzzy information. LPI was introduced in 1970 by Polish–Swiss mathematician Edward Kofler (1911–2007) to simpl
Space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation
Open energy system models
Open energy system models are energy system models that are open source. However, some of them may use third party proprietary software as part of their workflows to input, process, or output data. Pr
Complex system
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as
Mixed-mating model
The mixed-mating model is a mathematical model that describes the mating system of a plant population in terms of degree of self-fertilisation. It is a fairly simplistic model, employing several simpl
Applications of sensitivity analysis to environmental sciences
Sensitivity analysis studies the relationship between the output of a model and its input variables or assumptions. Historically, the need for a role of sensitivity analysis in modelling, and many app
Radiation law for human mobility
The radiation law is way of modeling human mobility (geographic mobility, human migration) and it gives better empirical predictions than the gravity model of migration which is widely used in this su
Simulink is a MATLAB-based graphical programming environment for modeling, simulating and analyzing multidomain dynamical systems. Its primary interface is a graphical block diagramming tool and a cus
Cebeci–Smith model
The Cebeci–Smith model is a 0-equation eddy viscosity model used in computational fluid dynamics analysis of turbulent boundary layer flows. The model gives eddy viscosity, , as a function of the loca
Vector field reconstruction
Vector field reconstruction is a method of creating a vector field from experimental or computer generated data, usually with the goal of finding a differential equation model of the system. A differe
Computational model
A computational model uses computer programs to simulate and study complex systems using an algorithmic or mechanistic approach and is widely used in a diverse range of fields spanning from physics, c
PottersWheel is a MATLAB toolbox for mathematical modeling of time-dependent dynamical systems that can be expressed as chemical reaction networks or ordinary differential equations (ODEs). It allows
Press–Schechter formalism
The Press–Schechter formalism is a mathematical model for predicting the number of objects (such as galaxies, galaxy clusters or dark matter halos) of a certain mass within a given volume of the Unive
Classical control theory
Classical control theory is a branch of control theory that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback, using the Laplace transform as a b
Flail space model
The flail space model (FSM) is a model of how a car passenger moves in a vehicle that collides with a roadside feature such as a guardrail or a crash cushion. Its principal purpose is to assess the po
Generalised logistic function
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is
Statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model rep
Empirical modelling
Empirical modelling refers to any kind of (computer) modelling based on empirical observations rather than on mathematically describable relationships of the system modelled.
Microscale and macroscale models
Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. Microscale and macro
Applications of sensitivity analysis in epidemiology
Sensitivity analysis studies the relation between the uncertainty in a model-based the inference and the uncertainties in the model assumptions. Sensitivity analysis can play an important role in epid
In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict
Calculus of voting
Calculus of voting refers to any mathematical model which predicts voting behaviour by an electorate, including such features as participation rate. A calculus of voting represents a hypothesized deci
Boolean delay equation
A Boolean Delay Equation (BDE) is an evolution rule for the state of dynamical variables whose values may be represented by a finite discrete numbers os states, such as 0 and 1. As a novel type of sem
Apparent infection rate
Apparent infection rate is an estimate of the rate of progress of a disease, based on proportional measures of the extent of infection at different times. Firstly, a proportional measure of the extent
PCLake is a dynamic, mathematical model used to study eutrophication effects in shallow lakes and ponds. PCLake models explicitly the most important biotic groups and their interrelations, within the
WRF-SFIRE is a coupled atmosphere-wildfire model, which combines the Weather Research and Forecasting Model (WRF) with a fire-spread model, implemented by the level-set method. A version from 2010 was
Secondary electrospray ionization
Secondary electro-spray ionization (SESI) is an ambient ionization technique for the analysis of trace concentrations of vapors, where a nano-electrospray produces charging agents that collide with th
Breath gas analysis
Breath gas analysis is a method for gaining information on the clinical state of an individual by monitoring volatile organic compounds (VOCs) present in the exhaled breath. Exhaled breath is naturall
Extended Mathematical Programming
Algebraic modeling languages like AIMMS, AMPL, GAMS, MPL and others have been developed to facilitate the description of a problem in mathematical terms and to link the abstract formulation with data-
Phase-field models on graphs
Phase-field models on graphs are a discrete analogue to phase-field models, defined on a graph. They are used in image analysis (for feature identification) and for the segmentation of social networks
Vflo is a commercially available, physics-based distributed hydrologic model generated by Vieux & Associates, Inc. Vflo uses radar rainfall data for hydrologic input to simulate distributed runoff. Vf
Compartmental neuron models
Compartmental modelling of dendrites deals with multi-compartment modelling of the dendrites, to make the understanding of the electrical behavior of complex dendrites easier. Basically, compartmental
VisSim is a visual block diagram program for simulation of dynamical systems and model-based design of embedded systems, with its own visual language. It is developed by Visual Solutions of Westford,
Iterative rational Krylov algorithm
The iterative rational Krylov algorithm (IRKA), is an iterative algorithm, useful for model order reduction (MOR) of single-input single-output (SISO) linear time-invariant dynamical systems. At each
Boolean model of information retrieval
The (standard) Boolean model of information retrieval (BIR) is a classical information retrieval (IR) model and, at the same time, the first and most-adopted one. It is used by many IR systems to this
LINGO (mathematical modeling language)
LINGO is a mathematical modeling language designed for formulating and solving optimization problems, including linear, integer, and nonlinear programming problems.
State-space representation
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or differe
MAgPIE is a non-linear, recursive, dynamic-optimization, global land and water-use model with a cost-minimization objective function.MAgPIE was developed and is employed by the land-use group working
Wildfire modeling
In computational science, wildfire modeling is concerned with numerical simulation of wildland fires in order to understand and predict fire behavior. Wildfire modeling can ultimately aid wildland fir
Arditi–Ginzburg equations
The Arditi–Ginzburg equations describes ratio dependent predator–prey dynamics. Where N is the population of a prey species and P that of a predator, the population dynamics are described by the follo
Microscopic traffic flow model
Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units
Mathematical exposure modeling
Mathematical exposure modeling is an indirect method of determining exposure, particularly for human exposure to environmental contaminants. It is useful when direct measurement of pollutant concentra
Info-metrics is an interdisciplinary approach to scientific modeling, inference and efficient information processing. It is the science of modeling, reasoning, and drawing inferences under conditions
Quantum clock model
The quantum clock model is a quantum lattice model. It is a generalisation of the transverse-field Ising model . It is defined on a lattice with states on each site. The Hamiltonian of this model is H
Minimum-distance estimation
Minimum-distance estimation (MDE) is a conceptual method for fitting a statistical model to data, usually the empirical distribution. Often-used estimators such as ordinary least squares can be though
Logan plot
A Logan plot (or Logan graphical analysis) is a graphical analysis technique based on the compartment model that uses linear regression to analyze pharmacokinetics of tracers involving reversible upta
Multi-compartment model
A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Sometimes, the physical system that we t
OptimJ is an extension for Java with language support for writing optimization models and abstractions for bulk data processing. The extensions and the proprietary product implementing the extensions
Sensitivity analysis
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in it
Dielectric breakdown model
Dielectric breakdown model (DBM) is a macroscopic mathematical model combining the diffusion-limited aggregation model with electric field. It was developed by Niemeyer, Pietronero, and Weismann in 19
JuMP is an algebraic modeling language and a collection of supporting packages for mathematical optimization embedded in the Julia programming language. JuMP is used by companies, government agencies,
Electoral Calculus
Electoral Calculus is a political forecasting web site which attempts to predict future United Kingdom general election results. It considers national factors but excludes local issues.
The multislice algorithm is a method for the simulation of the elastic interaction of an electron beam with matter, including all multiple scattering effects. The method is reviewed in the book by Cow
Energy modeling
Energy modeling or energy system modeling is the process of building computer models of energy systems in order to analyze them. Such models often employ scenario analysis to investigate different ass
Pontifex (project)
PONTIFEX (Planning Of Non-specific Transportation by an Intelligent Fleet EXpert) was a mid-1980s project that introduced a novel approach to complex aircraft fleet scheduling, partially funded by the
Soil production function
Soil production function refers to the rate of bedrock weathering into soil as a function of soil thickness. A general model suggested that the rate of physical weathering of bedrock (de/dt) can be re
Linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the non
Grey box model
In mathematics, statistics, and computational modelling, a grey box model combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information
Applications of sensitivity analysis to model calibration
Sensitivity analysis has important applications in model calibration. One application of sensitivity analysis addresses the question of "What's important to model or system development?" One can seek
Color model
A color model is an abstract mathematical model describing the way colors can be represented as tuples of numbers, typically as three or four values or color components. When this model is associated
Deterministic simulation
In mathematical modeling, deterministic simulations contain no random variables and no degree of randomness, and consist mostly of equations, for example difference equations. These simulations have k
Global cascades model
Global cascades models are a class of models aiming to model large and rare cascades that are triggered by exogenous perturbations which are relatively small compared with the size of the system. The
Historical dynamics
Historical dynamics broadly includes the scientific modeling of history. This might also be termed computer modeling of history, historical simulation, or simulation of history - allowing for an exten
Maas–Hoffman model
The Maas–Hoffman model is a mathematical tool to characterize the relation between crop production and soil salinity. It describes the crop response by a broken line of which the first part is horizon
The Chemical Basis of Morphogenesis
"The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from
Baldwin–Lomax model
The Baldwin–Lomax model is a 0-equation turbulence model used in computational fluid dynamics analysis of turbulent boundary layer flows.
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in
Forward problem of electrocardiology
The forward problem of electrocardiology is a computational and mathematical approach to study the electrical activity of the heart through the body surface. The principal aim of this study is to comp
History of network traffic models
Design of robust and reliable networks and network services relies on an understanding of the traffic characteristics of the network. Throughout history, different models of network traffic have been
Water Protection Zone
A Water Protection Zone is a statutory regulation imposed under Schedule 11 to the Water Resources Act 1991. The power was subsequently subsumed into The Water Resources Act (Amendment) (England and W
Variance-based sensitivity analysis
Variance-based sensitivity analysis (often referred to as the Sobol method or Sobol indices, after Ilya M. Sobol) is a form of global sensitivity analysis. Working within a probabilistic framework, it
Chemical reaction model
Chemical reaction models transform physical knowledge into a mathematical formulation that can be utilized in computational simulation of practical problems in chemical engineering. Computer simulatio
AIMMS (acronym for Advanced Interactive Multidimensional Modeling System) is a prescriptive analytics software company with offices in the Netherlands, United States, China and Singapore. It has two m
Head injury criterion
The head injury criterion (HIC) is a measure of the likelihood of head injury arising from an impact. The HIC can be used to assess safety related to vehicles, personal protective gear, and sport equi
Macroscopic traffic flow model
A Macroscopic traffic flow model is a mathematical traffic model that formulates the relationships among traffic flow characteristics like density, flow, mean speed of a traffic stream, etc.. Such mod
Resource selection function
Resource selection functions (RSFs) are a class of functions that are used in spatial ecology to assess which habitat characteristics are important to a specific population or species of animal, by as
Multiscale modeling
Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of
Propagation graph
Propagation graphs are a mathematical modelling method for radio propagation channels. A propagation graph is a signal flow graph in which vertices represent transmitters, receivers or scatterers. Edg
Analytica (software)
Analytica is a visual software developed by Lumina Decision Systems for creating, analyzing and communicating quantitative decision models. It combines hierarchical influence diagrams for visual creat
Equation-free modeling
Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse sc
Open Energy Modelling Initiative
The Open Energy Modelling Initiative (openmod) is a grassroots community of energy system modellers from universities and research institutes across Europe and elsewhere. The initiative promotes the u
Reaction–diffusion system
Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substanc
Actuarial Society of South Africa HIV/AIDS models
The Actuarial Society of South Africa HIV/AIDS models, also known as ASSA AIDS models, are a series of mathematical models developed to assist the actuarial profession and the in assessing and address
Predictive intake modelling
Predictive intake modelling uses mathematical modelling strategies to estimate intake of food, personal care products, and their formulations.
Quantitative models of the action potential
In neurophysiology, several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e.,
Fractional-order system
In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer o
Von Bertalanffy function
The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy. It is a special case of the generalised log
Patlak plot
A Patlak plot (sometimes called Gjedde–Patlak plot, Patlak–Rutland plot, or Patlak analysis) is a graphical analysis technique based on the compartment model that uses linear regression to identify an
AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and s
Modelling biological systems
Modelling biological systems is a significant task of systems biology and mathematical biology. Computational systems biology aims to develop and use efficient algorithms, data structures, visualizati
Van Genuchten–Gupta model
The van Genuchten–Gupta model is an inverted S-curve applicable to crop yield and soil salinity relations.
Bueno-Orovio–Cherry–Fenton model
The Bueno-Orovio–Cherry–Fenton model, also simply called Bueno-Orovio model, is a minimal ionic model for human ventricular cells. It belongs to the category of phenomenological models, because of its
Quasispecies model
The quasispecies model is a description of the process of the Darwinian evolution of certain self-replicating entities within the framework of physical chemistry. A quasispecies is a large group or "c
Seshat (project)
The Seshat: Global History Databank (named after Seshat, the ancient Egyptian goddess of wisdom, knowledge, and writing) is an international scientific research project of the nonprofit Evolution Inst
Price's model
Price's model (named after the physicist Derek J. de Solla Price) is a mathematical model for the growth of citation networks. It was the first model which generalized the Simon model to be used for n
Theta model
The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to model neurons in the animal Aplysia, and later used in various fields of computational neuro
Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy
Cumulative accuracy profile
A cumulative accuracy profile (CAP) is a concept utilized in data science to visualize discrimination power. The CAP of a model represents the cumulative number of positive outcomes along the y-axis v
Effective selfing model
The effective selfing model is a mathematical model that describes the mating system of a plant population in terms of the degree of self-fertilisation present.
Superiority and inferiority ranking method
The superiority and inferiority ranking method (or SIR method) is a multi-criteria decision making model (MCDA) which can handle real data and provides six different preference structures for the syst
Landscape evolution model
A landscape evolution model is a physically-based numerical model that simulates changing terrain over the course of time. The change in, or evolution of, terrain, can be due to: glacial or fluvial er
Elementary effects method
The elementary effects (EE) method is the most used screening method in sensitivity analysis. EE is applied to identify non-influential inputs for a computationally costly mathematical model or for a
Bidomain model
The bidomain model is a mathematical model to define the electrical activity of the heart. It consists in a continuum (volume-average) approach in which the cardiac mictrostructure is defined in terms
Bounded growth
Bounded growth occurs when the growth rate of a mathematical function is constantly increasing at a decreasing rate. Asymptotically, bounded growth approaches a fixed value. This contrasts with expone
Excitable medium
An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time
Backtesting is a term used in modeling to refer to testing a predictive model on historical data. Backtesting is a type of retrodiction, and a special type of cross-validation applied to previous time
Calculation of glass properties
The calculation of glass properties (glass modeling) is used to predict glass properties of interest or glass behavior under certain conditions (e.g., during production) without experimental investiga
Movable cellular automaton
The movable cellular automaton (MCA) method is a method in computational solid mechanics based on the discrete concept. It provides advantages both of classical cellular automaton and discrete element
Quadratic integrate and fire
The quadratic integrate and fire (QIF) model is a biological neuron model and a type of integrate-and-fire neuron which describes action potentials in neurons. In contrast to physiologically accurate
Viral dynamics
Viral dynamics is a field of applied mathematics concerned with describing the progression of viral infections within a host organism. It employs a family of mathematical models that describe changes
Surrogate model
A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require e
Proper generalized decomposition
The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems (BVPs), that is, partial differential equations constrained by a set of boundary conditi
Akaike information criterion
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC es
Autowave reverberator
In the theory of autowave phenomena an autowave reverberator is an autowave vortex in a two-dimensional active medium. A reverberator appears a result of a rupture in the front of a plane autowave. Su
Malthusian equilibrium
A population is in Malthusian equilibrium when all of its production is used only for subsistence. Malthusian equilibrium is a locally stable and a dynamic equilibrium.
Turing pattern
The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and
SAMPL, which stands for "Stochastic AMPL", is an algebraic modeling language resulting by expanding the well-known language AMPL with extended syntax and keywords. It is designed specifically for repr
Phase-field model
A phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fi
Model order reduction
Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with
Automated efficiency model
An automated efficiency model (AEM) is a mathematical model that estimates a real estate property’s efficiency by using details specific to the property which are available publicly and/or housing cha
Malthusian growth model
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function
Two-fluid model
Two-fluid model is a macroscopic traffic flow model to represent traffic in a town/city or metropolitan area, put forward in the 1970s by Ilya Prigogine and Robert Herman. There is also a two-fluid mo
Analysis (PL: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathema
Exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the q
Applications of sensitivity analysis to multi-criteria decision making
A sensitivity analysis may reveal surprising insights in multi-criteria decision making (MCDM) studies aimed to select the best alternative among a number of competing alternatives. This is an importa
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used
Variational multiscale method
The variational multiscale method (VMS) is a technique used for deriving models and numerical methods for multiscale phenomena. The VMS framework has been mainly applied to design stabilized finite el
Traffic model
A traffic model is a mathematical model of real-world traffic, usually, but not restricted to, road traffic. Traffic modeling draws heavily on theoretical foundations like network theory and certain t
Gradient-enhanced kriging
Gradient-enhanced kriging (GEK) is a surrogate modeling technique used in engineering. A surrogate model (alternatively known as a metamodel, response surface or emulator) is a prediction of the outpu
Linear seismic inversion
Inverse modeling is a mathematical technique where the objective is to determine the physical properties of the subsurface of an earth region that has produced a given seismogram. Cooke and Schneider