 # Sensitivity Analysis

## Sensitivity Analysis

The study of how a change in the output of a model (numerical or otherwise) may be allocated, qualitatively or statistically, to distinct sources of variance is known as sensitivity analysis.

A mathematical model is described as a set of equations, input factors, parameters, and variables that are used to characterize the process under consideration. Many causes of uncertainty exist in input, including measurement mistakes, a lack of knowledge, and a poor or incomplete understanding of the driving forces and systems.

This limits our confidence in the model’s reaction or output. Furthermore, models may have to deal with the system’s inherent unpredictability, such as the occurrence of stochastic occurrences. A good modeling practice demands the modeler to offer an evaluation of the model’s confidence, perhaps analyzing the uncertainties associated with the modeling process and the model’s output. Uncertainty and Sensitivity Analysis are useful methods for characterizing a model’s uncertainty.

### Types of Sensitivity Analysis

There are primarily two types of sensitivity analysis, which are

Local Sensitivity Analysis
Global Sensitivity Analysis

### Local Sensitivity Analysis

This type is based on derivatives (numerical or analytical). The word local denotes that the derivatives are taken at a single location. This approach is appropriate for basic cost functions, but it is not appropriate for complicated models, such as models with discontinuities, which do not necessarily have derivatives.

The partial derivative of the cost function with respect to those parameters is equivalent to the sensitivity of the cost function with respect to those parameters.

Local sensitivity analysis is a one-at-a-time (OAT) technique that investigates the impact of one parameter on the cost function at a time while keeping the other parameters constant.

### Global Sensitivity Analysis

The second type of sensitivity analysis is global sensitivity analysis, which is frequently carried out using Monte Carlo techniques. To explore the design space, this method employs a global collection of samples.

### Uses of Sensitivity Analysis

Sensitivity Analysis can be used to make this determination.

1. The model’s similarity to the process under study.

2. The accuracy with which the model is defined.

3. Factors that have the greatest impact on output variability.

4. The area in the space of input components with the greatest model variation.

5. Optimal – or unstable – areas in the factor space for use in a later calibration study.

6. Interactions between variables.

7. Sensitivity Analysis is widely used in financial applications, risk analysis, signal processing, neutral networks, and any other field where models are created.

### Importance of Sensitivity Analysis

Sensitivity analysis has a number of advantages for decision-makers. For starters, it serves as an in-depth examination of all the factors. Predictions may be considerably more trustworthy since it is more detailed. Second, it enables decision-makers to identify areas where they might improve in the future.

### Method of Sensitivity Analysis

There are various methods for doing uncertainty (UA) and sensitivity analysis (SA). The most frequent type of sensitivity analysis is a sampling-based analysis. The model is performed repeatedly for combinations of values sampled from the distribution (assumed known) of the input components in a sampling-based sensitivity. Other model-independent approaches are based on the breakdown of the variance of the model output.

In general, UA and SA are conducted concurrently by running the model repeatedly for a given set of factor values selected from a probability distribution.

The following steps can be listed:

1. Define the goal function and choose the relevant input.

2. Assign a distribution function to the factors you’ve chosen.

3. Using an acceptable design, generate a matrix of inputs with that distribution(s).

4. Evaluate the model and compute the target function distribution.

5. Choose a technique for determining the impact or relative relevance of each input element on the target function.

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