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# The Use of the Monte Carlo Method in Sensitivity Analysis and Its Advantages

### Key Takeaways

●     Gain a greater understanding of the Monte Carlo method.

●     Learn about the advantages of the Monte Carlo method of sensitivity analysis. Sensitivity analyses have their distinct merits.

An analysis, by nature, is an examination. In the field of mathematics, the process incorporates proof of a mathematical proposition by accepting the result and determining a compelling statement by a succession of reversible steps. There are any number of possible outcomes with random number configurations involved, but good product design still requires risk assessment in and outside of standard deviation.

However, in the area of electronics, it is typically a comprehensive examination of anything intricate in an effort to understand its nature, thus, determining its vital features. This is equally true of a Monte Carlo sensitivity analysis. An MC simulation tries to work with repeated random sampling and random variable definitions to yield common probability distributions, confidence interval in an overall computerized mathematical technique for statistical analysis of expected value

The Sensitivity Analysis

A sensitivity analysis is an analysis we use to determine how various sources or input values of an individual variable affect a specific dependent variable under an allotted group of theories or assumptions. In general, a sensitivity analysis studies how different sources of uncertainty in a mathematical model impact a model's overall uncertainty.

For example, suppose we want to analyze a Class-D amplifier's performance based on its input level (sensitivity). In that case, we could examine the output changes that occur each time we adjust its input level. In its simplest form, this encompasses a sensitivity analysis.

Overall, we utilize sensitivity analyses in the following capacities:

• To categorize the level at which variations in the input values for an allotted variable affect a mathematical model's results.

• To detect the most crucial data we collect for studies to evaluate a project or design's ROI (return on investment).

• To help engineers create more robust and reliable designs by assessing points of uncertainty in a design's structure.

## Sensitivity Analysis and Circuit Behavior

When focusing on component deviations utilizing sensitivity analysis, we must first understand that components do not technically possess fixed or exact values in a real-world scenario. These values include voltage, current, resistance, inductance, and capacitance, to name a few. Therefore, this influences designs and design strategy, i.e., engineers design within a tolerance rather than an exact value.

Although these values are not exact, we still require a way to assess or examine these component values in terms of their behavior within a circuit's confines. However, in circumstances such as these, sensitivity analysis is ideally suited to the task. Even though we technically view these types of examinations as sensitivity analyses, they are more functional in nature.

Those in the field of electronics understand that there is a myriad of contributors to a component's behavioral variations within a circuit. Unfortunately, this is also why sensitivity analysis does not readily identify the correlation between the sensitivity one observes in a circuit's output to a particular source of variance in a circuit. Therefore, it is necessary to incorporate it with other analysis methods to accurately diagnose the physical variable's effects on a circuit's behavior.

## The Advantages of the Monte Carlo Simulation or Method

A Monte Carlo simulation (method) affords several advantages over single-point estimate or deterministic analysis:

• It provides probabilistic results: This means its results show what can happen and the likelihood of each outcome.

• It provides graphical results: Due to the data a Monte Carlo simulation produces, it affords effortless generation of graphs of various outcomes and their chances of occurring. As you might imagine, this can be beneficial when conveying your results to others.

• It excels at sensitivity analysis: Typically, deterministic analysis makes it difficult to see which variables impact the outcome the most. However, in the Monte Carlo simulation, it is effortless to see which inputs significantly affect overall results.

• It excels at scenario analysis: In deterministic models, it is difficult to model different combinations of values for distinct inputs to see the various scenarios' effects. However, with the Monte Carlo simulation, analysts can precisely see which inputs have which values or parameters together when specific outcomes occur. As you can imagine, this is an invaluable insight that affords further analysis pursuits.

• Better correlation of device or system inputs: The Monte Carlo simulation affords the ability to model symbiotic interactions among input variables. The need for accuracy in situations such as these is vital because one must correlate when specific values or factors change (increase/decrease) and subsequently cause other aspects to change (increase/decrease).

## Monte Carlo Method in Sensitivity Analysis

In today's electronics and engineering field, the use of mathematical tools for the purpose of sensitivity analysis is commonplace, and we place these tools into two categories:

• Deterministic: A deterministic model is devoid of elements of randomness. Therefore, each time you operate the model with the same original conditions, you will obtain the same results.

• Probabilistic: A probabilistic model does include elements of randomness. Therefore, each time you operate the model, you typically receive different results, even with the exact original conditions.

As stated earlier, sensitivity analysis alone does not effectively address the variants in input parameters since they consider them unrelated. For example, systems that utilize feedback for their functionality, although the correlation is evident, make such analysis methods ineffective.

Although we can address component value variants as independent circuits with increased complexity (i.e., utilizes feedback), analyzing systems in this manner can lead to inaccurate results that translate into system instability in your designs.  A more effective method of analyzing systems such as this typically involves probabilistic numerical methods. Moreover, the technique that allows you to easily see which inputs had the most significant effect on overall results is the Monte Carlo method or simulation.

## The Monte Carlo Method Continued

The Monte Carlo method is a computational mathematical technique that affords the ability to account for quantitative analysis risk. Professionals utilize the method in a vast array of areas such as energy, engineering, electronics, manufacturing, and PCBA.

Monte Carlo simulations provide designers with an array of potential results and the probability that they will happen for any action choice. Historically, its introduction dates back to World War II during the first atomic bomb invention.

Functionally, Monte Carlo simulations execute risk analysis by building models of possible results by substituting a range of values (probability distribution) for any factor with inherent uncertainty. Next, it calculates results repeatedly, utilizing different sets of arbitrary values from the probability functions.

In general, a Monte Carlo simulation can be thousands or even tens of thousands of recalculations before completion. Of course, this depends on how many uncertainties and the specified ranges for each uncertainty. In summary, a Monte Carlo simulation generates allocations of probable result values.

## Monte Carlo Methods in Sensitivity Analysis Applications

Monte Carlo methods are in widespread use in engineering for the express purpose of quantitative probabilistic and sensitivity analysis within process design. This need corresponds to the circuit complexities one encounters in the area of engineering. These circuit behaviors include non-linear, mixed or co-linear, and interactive, stemming from process simulations. The following are some of the primary applications of the Monte Carlo method in use in sensitivity analysis:

• Monte Carlo methods are applied in microelectronics engineering to analyze correlated and uncorrelated variations in analog and digital integrated circuits.

• In geometallurgy, Monte Carlo methods provide quantitative risk analysis and reinforce the design of mineral processing flowsheets.

### Note: Geometallurgy refers to a practice of merging geostatistics and metallurgy. This specifically focuses on extractive metallurgy to generate a spatially based predictive model like in mineral processing plants.

• With fluid dynamics, more specifically rarefied gas dynamics, we solve the Boltzmann equation for finite Knudsen number fluid flow with the Monte Carlo method (direct simulation). We, in turn, combine this method with highly efficient computational algorithms.

### Note: The Boltzmann equation (Ludwig Boltzmann 1872) statistically explains a thermodynamic system's behavior when not in a state of equilibrium.

• In Robotics (autonomous), Monte Carlo localization determines the position of a robot. We typically apply this to stochastic filters like the particle filter or Kalman filter. Furthermore, these two filters are the central components of the simultaneous localization and mapping (SLAM) algorithm.

• Designs in telecommunications: In today's high demand for data, a wireless network needs to perform under various conditions and high demands. Therefore, the Monte Carlo method provides a representation of these demands and users to evaluate a network design's accuracy. This, of course, provides a designer with the opportunity to optimize their network to ensure it will meet all demands.

Monte Carlo in Sensitivity Analysis is a much more realistic way of describing uncertainty in a risk analysis variable. It accomplishes this by utilizing probability distributions, thus, variables can have occurrences of different probabilities of different outcomes. Stochastic modeling, like the Monte Carlo method in sensitivity analysis, forecasts the probability of various outcomes under different conditions.

It’s better to be safe than sorry by investing in a PCB Design and Analysis software with a Monte Carlo analysis tool. The PSpice Simulator gives you the flexibility to set up the Monte Carlo simulation to accurately predict the variables in question.