# MonteCarloMethod

**The Essay topic**: Monte Carlo Method in Simulations

**Class**:4IT496 Simulation of Systems (WS 2017/2018)

**Author**: Bc. Yauheniya Andreyuk

## Contents

# 1. Introduction

Monte Carlo method is an algorithm that rely on repeated random sampling to obtain numerical results. Essential idea of Monte Carlo is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. ^{[1]} Monte Carlo simulation is a technique that allows to see all the possible outcomes of the decisions and assess the impact of risk, allowing better decision making under uncertainty. The technique is used by professionals in such fields as finance, project management, energy, manufacturing, engineering, research and development, insurance, oil & gas, transportation, and the environment. The simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. It shows the extreme possibilities — the outcomes of going for broke and for the most conservative decision — along with all possible consequences for middle-of-the-road decisions.

# 2. History

Principles of Monte Carlo Simulations the way we know it today were developed at the beginning of the 18th century. The history of Monte Carlo started with the work of a French scientist Georges Louis LeClerc, Comte de Buffon (1707-1788), who created “Buffon’s Needle.” ^{[2]} In his work he used a method of dropping random needles on a lined background to estimate π. He proved that for a needle the same length as the distance between the lines the probability that each time a needle would intersect a line was 2/ π. He tested his theory by throwing baguettes over his shoulder on to a tile floor.

His formula was:

π = 2N/X

Where N is the number of tosses and X is the number of times a needle intersected a line.

The idea of using randomness in a determinative manner was revolutionary due to the fact that simulations beforehand were using a deterministic problem with statistical sampling to estimate the uncertainties in simulations. In the 19th and early 20th centuries, simulation was increasingly used as an experimental means of confirming theory, analyzing data, or supplementing intuition in mathematical statistics.

These were seminal studies, but there is a significant difference between them and typical modern Monte Carlo simulations studying problems that are otherwise intractable, e.g. galaxy formation modeling. The early simulations dealt with previously understood deterministic problems. Modern Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog and solving the problem probabilistically.

These simulations were used in investigating radiation shielding and the distance that neutrons would likely travel through various materials. This form of simulation was used during the Manhattan Project, the American World War II effort to develop nuclear weapons. During their work on the Manhattan Project John von Neumann and Stanislaw Ulam named the method after the Monte Carlo Casino in Monaco.

Since then Monte Carlo simulations have proven to be extremely useful in other fields such as physics, physical chemistry, operations research, business, and medical fields. Monte Carlo simulation is now a much-used scientific tool for problems that are analytically intractable and for which experimentation is too time-consuming, costly, or impractical. Researchers explore complex systems, examine quantities that are hidden in experiments, and easily repeat or modify experiments. But the simulation also has some disadvantages: it doesn't give exact solutions and results are only as good as the model and inputs used; it can require huge computing resources and simulation software, like any software, is prone to bugs. Nowadays there are some disputes that simulations are over used because they remove the need for analytical or experimental approaches and that people take it as a lazy way out. ^{[3]}

# 3. How Monte Carlo simulation works.

Monte Carlo Simulation is used for obtaining a distribution of possible numerical outcome values for a given problem. It performs analysis by building models of possible results by substituting a range of values — a probability distribution — for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values.

**Monte Carlo Simulation has 3 basic elements**:

**1.** A mathematical formula that represents how inputs turn into an output

**2.** A reasonable estimation of the variation of each input.

**3.** An idea of what output performance is acceptable.

The inputs are the possible range of values that the input values will take in the randomization process. The inputs are probability distributions (various types of distributions are listed below) and can vary in type. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis.

**Common Probability Distributions:**^{[4]}

**Normal** – Or “bell curve.” The user simply defines the mean or expected value and a standard deviation to describe the variation about the mean. Values in the middle near the mean are most likely to occur. It is symmetric.

**Lognormal** – Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that don’t go below zero but have unlimited positive potential.

**Uniform** – All values have an equal chance of occurring, and the user simply defines the minimum and maximum.

**Triangular** – The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur.

**PERT**- The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized.

**Discrete** – The user defines specific values that may occur and the likelihood of each.

During a Monte Carlo simulation, values are sampled at random from the input probability distributions. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen.

Monte Carlo simulation provides a number of advantages over deterministic, or “single-point estimate” analysis:

Probabilistic Results. Results show not only what could happen, but how likely each outcome is. Graphical Results. Because of the data a Monte Carlo simulation generates, it’s easy to create graphs of different outcomes and their chances of occurrence. This is important for communicating findings to other stakeholders. Sensitivity Analysis. With just a few cases, deterministic analysis makes it difficult to see which variables impact the outcome the most. In Monte Carlo simulation, it’s easy to see which inputs had the biggest effect on bottom-line results. Scenario Analysis: In deterministic models, it’s very difficult to model different combinations of values for different inputs to see the effects of truly different scenarios. Using Monte Carlo simulation, analysts can see exactly which inputs had which values together when certain outcomes occurred. This is invaluable for pursuing further analysis. Correlation of Inputs. In Monte Carlo simulation, it’s possible to model interdependent relationships between input variables. It’s important for accuracy to represent how, in reality, when some factors goes up, others go up or down accordingly. An enhancement to Monte Carlo simulation is the use of Latin Hypercube sampling, which samples more accurately from the entire range of distribution functions.

# 4. Summary

Monte Carlo simulation is a key tool for studying analytically intractable problems. Its history dates back to the eighteenth century, but it came into its modern form in the push to develop nuclear weapons during World War II.

Often a single ‘event’ in a simulation will require sampling from many different PDFs, some many times, and be tallied in many ways. Each sample will require one or more random numbers, and may require converting a uniform random number into a sample from another distribution using the inversion or acceptance-rejection methods.

## References

- ↑ Monte Carlo method. Wikipedia.org [online]. Available at: https://en.wikipedia.org/wiki/Monte_Carlo_method
- ↑ Buffon's needle. Wikipedia.org [online]. Available at: https://en.wikipedia.org/wiki/Buffon%27s_needle
- ↑ Introduction To Monte Carlo Simulation. NCBI [online]. Available at: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2924739/
- ↑ Monte Carlo Simulation. Palisade.com [online]. Available at: http://www.palisade.com/risk/monte_carlo_simulation.asp