National Physical Laboratory

Mathematical and Statistical Modelling


A mathematical model is used to describe the behaviour of a physical system in terms of equations involving variables and parameters representing aspects of the system. For example, Newton's law of motion F = ma relates the acceleration a that an object of mass m experiences when subjected to a force F. Models are important in metrology as they allow a value of one variable to be estimated from measurements of other variables. For example, in a mass spectrometer, the mass of a particle is estimated using the above law from measurements of F and a.


A mathematical model describes the relationships between the quantities that affect the behaviour of a system. The model equation constrains the possible combinations of variable values that are thought to be possible. For example, the equation L(t) = L0 + c(t-t0) models the length L(t) of a metal rod as a function of temperature t in terms of its length L0 at temperature t0 and its coefficient c of thermal expansion. Once L0 and c are prescribed, the model assigns a unique L(t) to each t.

A model allows us to determine the values of one set of variables once the values of others are known. These values are usually determined by solving sets of mathematical equations derived from those used to build the model. Depending on the type of model, these equations may represent a system of linear or nonlinear algebraic equations or differential equations. More often than not, some of the variable values are derived from measurements and have associated uncertainties. This means that the solutions of the equations that give the values of the remaining variables will also have associated uncertainties that need to be evaluated in order to assess their quality. The method of determining the solution values will also need to take into account the fact that some of the input information is not exact.

Modelling involves:

  • Model building: physical, empirical, algebraic and continuous models
  • Solution of linear and nonlinear algebraic equations: numerical linear algebra, least-squares techniques
  • Solution of linear and nonlinear differential equations: finite element method, boundary element method
  • Numerical analysis: efficient and numerically accurate solution algorithms
  • Data approximation: fitting a model to data
  • Optimisation: determining best estimates of model parameters, either to explain data or to achieve optimal functionality of the modelled system
  • Uncertainty evaluation: propagation of uncertainties in model inputs to model outputs
  • Model validation: deciding whether a model explains satisfactorily system behaviour on the basis of measurement results and other knowledge
  • Scientific software development: encoding algorithms in library level software, validation and verification of software


NPL has a highly experienced team of technical experts in the area of mathematical and statistical modelling. We are part of a world-class physical metrology organisation, with experience and expertise in this field that is unique in the UK. We offer a range of services including advice, consultancy and training in the areas of modelling, experimental data analysis, and software development.


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