Grid Search

Concept

Theoretically, Grid Search is an exhaustive optimisation method that systematically evaluates every combination of predefined parameter values within a specified search space. Rather than following gradients or stochastic search strategies, Grid Search enumerates all candidate solutions and identifies the combination that optimises a chosen objective function. In health economics, it is primarily used for model calibration, parameter tuning, machine learning applications and sensitivity analysis when the parameter space is sufficiently small to permit complete evaluation.

Mathematically, Grid Search defines a finite discrete set of candidate values for each model parameter and evaluates the objective function at every possible parameter combination. The optimal solution is the parameter vector that minimises or maximises the objective function according to the modelling objective. Because every candidate is evaluated, Grid Search guarantees identification of the best solution within the specified grid but not necessarily the global optimum over the continuous parameter space.

In practice, Grid Search is implemented to calibrate disease models, optimise predictive algorithms and tune machine learning hyperparameters used in health economic analyses. Analysts specify parameter ranges and grid resolution before executing the search. Although computationally expensive for high-dimensional problems, Grid Search remains straightforward to implement and provides a transparent benchmark against which more sophisticated optimisation methods may be compared.

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Purpose

Used to identify the optimal combination of model parameters or hyperparameters by exhaustively evaluating a predefined set of candidate values within a discrete search space.

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Mathematical Formulae

Primary Formula

Grid Search solves:

? = arg min?? ? �? J(?)*

or, for maximisation,

? = arg max?? ? �? J(?)*

where:

• � = finite parameter grid
• J(?) = objective function
• ?* = optimal parameter combination

Supporting Formulae

Total number of parameter combinations:

N = ????? m?

where:

• k = number of parameters
• m? = number of candidate values for parameter i

Related Mathematical Methods

• Exhaustive search
• Hyperparameter optimisation
• Model calibration
• Cross-validation
• Bayesian optimisation
• Genetic algorithms
• Gradient-based optimisation

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Example

A disease simulation model contains three uncertain calibration parameters.

• Transmission rate: 5 candidate values
• Recovery rate: 4 candidate values
• Vaccine effectiveness: 6 candidate values

The total number of parameter combinations is:

N = 5 ? 4 ? 6 = 120

Each of the 120 combinations is evaluated by comparing predicted disease incidence with observed epidemiological data. The parameter combination producing the smallest calibration error is selected for the final health economic model.

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Excel Implementation

Function	Example Formula	Health Economics Application
Solver	Evaluate each parameter combination manually or through iteration	Compare objective function values across candidate parameter sets.
MIN	=MIN(F2:F121)	Identify the lowest calibration error across all grid combinations.
INDEX	=INDEX(A2:E121,MATCH(MIN(F2:F121),F2:F121,0),0)	Retrieve the optimal parameter combination.
SEQUENCE	=SEQUENCE(120)	Generate candidate evaluation indices for exhaustive parameter searches.

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VBA (Optional)

Automate exhaustive evaluation of parameter combinations, record objective function values and identify the optimal solution across the complete search grid.

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Sources

• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
• Nocedal J, Wright SJ. Numerical Optimization. 2nd ed.
• Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning.
• Bishop CM. Pattern Recognition and Machine Learning.