Topic

Mathematical and computational methods

The numerical techniques behind model building and analysis. Matrix algebra underlies cohort calculations, rules such as the trapezoidal and Simpson rules estimate areas under curves, and optimisation methods, from linear programming to genetic algorithms, find the best allocation of a budget or the best fit of a model to data.

Concepts in this topic

• Bayesian OptimisationBayesian optimisation seeks the inputs that optimise a costly function, such as a model's fit to calibration targets, using a surrogate to pick each run.
• Closed-Form Analytical MethodA closed-form analytical method solves a health economic model with a formula, such as a matrix inverse, instead of by simulation or random sampling.
• Constrained OptimisationConstrained optimisation identifies the best attainable value of an objective while requiring the decision variables to satisfy resource, policy, clinical or mathematical restrictions.
• Cubic SplineA cubic spline represents a curve with separate cubic polynomials joined at selected points called knots.
• DeterminantA scalar value calculated from a square matrix that indicates properties including invertibility, scaling and the independence of its rows or columns.
• Dynamic Programming HealthAn optimisation technique solving complex sequential decision problems by breaking them into smaller, nested subproblems solved in relation to one another.
• Euler MethodThe Euler method is a first-order numerical procedure that approximates the solution of an initial-value ordinary differential equation by advancing from the current value along the derivative over discrete steps.
• Finite Difference MethodThe finite difference method replaces derivatives with algebraic differences between function values at neighbouring grid points.
• Gaussian Process RegressionGaussian process regression is a probabilistic method that predicts an unknown function from observed input-output pairs by conditioning a joint Gaussian model specified through a mean and covariance function.
• Gaussian QuadratureGaussian quadrature is a family of numerical integration methods that approximates a definite integral using carefully chosen evaluation points and weights.
• Genetic AlgorithmAn optimisation technique inspired by evolution that iteratively selects and combines the best candidate solutions across generations to improve results.
• Gradient DescentGradient descent is an iterative first-order optimization method that reduces a differentiable objective function by repeatedly moving the parameter vector in the direction opposite its gradient.
• Grid SearchAn optimisation approach evaluating an objective function across a predefined set of candidate parameter combinations arranged in a regular grid.
• Integer ProgrammingInteger programming chooses which indivisible health programmes or medicines to fund to maximise health gain, such as QALYs, within a fixed budget.
• Least Squares ApproximationLeast squares approximation estimates the coefficients of a chosen model by minimizing the sum of squared residuals between observed or target values and the corresponding model values.
• Linear ProgrammingLinear programming identifies the best feasible allocation of limited resources when the objective and constraints can be expressed as linear relationships.
• Linear Programming HealthAn optimisation technique identifying the resource allocation that maximises or minimises a linear objective subject to linear constraints, such as a fixed budget.
• Matrix AlgebraMatrix algebra provides a compact way to represent and manipulate related quantities arranged in rows and columns.
• Matrix InversionMatrix inversion finds a matrix that reverses the effect of a square matrix.
• Matrix MultiplicationMatrix multiplication combines two organised sets of quantities so that the rows of the first matrix are matched with the columns of the second.
• Multi-Objective OptimisationAn optimisation approach used when a decision involves competing objectives, such as maximising benefit while minimising cost, producing a set of trade-off solutions.
• Nelder-Mead MethodA numerical optimisation algorithm searching for a function's minimum using a moving shape, called a simplex, without needing the function's derivative.
• Newton-Raphson MethodThe Newton–Raphson method is an iterative numerical algorithm that uses local derivative information to find a root of an equation.
• Nonlinear ProgrammingNonlinear programming is the optimization of an objective function subject to equality, inequality, or bound constraints when the objective or at least one constraint is nonlinear in the decision variables.
• Numerical IntegrationNumerical integration approximates the area represented by a definite integral when an exact antiderivative is unavailable, inconvenient or evaluated from discrete data.
• Numerical OptimisationNumerical optimisation uses iterative computational methods to find decision or parameter values that minimise or maximise an objective function.
• Particle Swarm OptimisationA population-based optimisation technique inspired by flocking behaviour, in which candidate solutions adjust position based on their own and the group's best results.
• Response Surface MethodologyStatistical techniques for modelling and optimising the relationship between several inputs and an output by fitting a smooth surface to limited results.
• Robust OptimisationAn optimisation approach that selects a feasible decision to perform well under the worst parameter values in a specified uncertainty set, often requiring constraints to hold throughout that set.
• Runge-Kutta MethodA Runge-Kutta method is a one-step numerical scheme for approximating the solution of an ordinary differential equation by combining several derivative evaluations within each integration step.
• Simpson RuleSimpson rule is a closed Newton-Cotes numerical integration formula that approximates a definite integral over pairs of equal subintervals by integrating quadratic interpolants through successive triples of function values.
• Simulated AnnealingAn optimisation technique modelled on slowly cooling metal, occasionally accepting worse solutions during search to avoid becoming trapped in a poor local solution.
• Stochastic ProgrammingAn optimisation framework incorporating parameter uncertainty directly into the process, seeking a solution that performs well across possible parameter values.
• Trapezoidal RuleThe trapezoidal rule approximates a definite integral by replacing a curve with straight-line segments and summing the resulting trapezoid areas.