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In today’s tutorial, we are going to learn how to implement Monte Carlo Simulations in R.
Logic behind Monte Carlo:
Monte Carlo Simulations in R
Monte Carlo simulation (also known as the Monte Carlo Method) is a statistical technique that allows us to compute all the possible outcomes of an event. This makes it extremely helpful in risk assessment and aids decision-making because we can predict the probability of extreme cases coming true. The technique was first used by scientists working on the atom bomb; it was named for Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems.
Monte Carlo methods are used to identify the probability of an event A happening, among a set of N events. We assume that all the events are independent, and the probability of event A happening once does not prevent the occurrence again.
For example, assume you have a fair coin and you flip it once. The probability of heads is 0.5 i.e. equal possibility of heads or tails. You flip the coin again. The possibility of heads is still 0.5, irrespective of whether we got heads or tails in the first flip. However, we can safely say that if we were to flip the coin 100 times, you would see heads ~50% of the times. The application of Monte Carlo (referred henceforth in this post as MC) methods comes to play when we want to find out the probability of heads occurring 16 times in a row. (or 5 or 3 or any other number.)
You can read more about these methods and the theory behind them, using the links below:
- Wikipedia – link.
- MC methods in Finance, from Investopedia.com – link2
- Basics of MC from software providerPalisade. – link3.
Applications:
MC methods are used by professionals in numerous fields ranging from finance, project management, energy, manufacturing, R&D, insurance, biotech, etc. Some real-world applications of Monte Carlo simulations are given below:
- Monte Carlo simulations are used in financial services to predict fraudulent credit card transactions. (since 100 genuine transactions do not guarantee the next one will not be fraudulent, even though it is a rare event by itself.)
- Risk analysis. Assume a new product was sold at a loss of $300 to 6 users (due to coupons or sales), a profit of $467 in 79 users and a profit of $82 to 119 customers. We can use Monte Carlo simulations to understand what would be the average P/L (profit or loss) if 1000 customers bought our products.
- A/B testing to understand page bounce and success web elements. Assume you changed the payment processing system on your e-commerce site. You are doing an A/B test to see if the upgrade results in improved checkout completion. On the old system, 12 users abandoned their cart, while 19 completed their purchase. On the new system, 147 people abandoned their cart while 320 completed their purchase. Which system works better?
- Selection criteria. Example if we have 7 candidates for a scholarship (Eileen, George, Taher, Ramesis, Arya, Sandra and Mike) what is the probability that Mike will be chosen in three consecutive years? Assuming the candidate list is the same and past winners are not barred from receiving the scholarship again.
Advantages of using MC:
Unlike simple forecasting, Monte Carlo simulation can help with the following:
- Probabilistic Results – show scenarios and how the occurrence likelihood.
- Graphical Results – The outcomes and their chance of occurring can be easily converted to graphs making it easy to communicate findings to an audience.
- Sensitivity Analysis – Easier to see which variables impact the outcome the most, i.e. which variables had the biggest effect on bottom-line results.
- Scenario Analysis: Using Monte Carlo simulation, we can see exactly which inputs had which values together when certain outcomes occurred.
- 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.
Code template:
The basic template for MC is as follows:
runs <- 100000func1 <- sum(sample(c(0,1), size =10, replace = T)) > 6mc_prob <- sum(replicate(runs, func())) / runs
Let’s look at this code in detail:
- Runs = no of trials or iterations. For our product profit example (application example 2), runs = 1000.
- Func1 = this is the formula definition where we will indicate number of different events, their probability and the selection criteria. For our scholarship candidate example (application number 4) this function would be modified as:
sum(sample(c(1:7), size =3, replace = T)) > 6
where we are assigning number 1:7 to each student and hence Mike = 7.
Main code:
The code files for this tutorial are available on the 2017 project page. (Link hereunder Jul/Aug 2017 ) .
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Insights, advice, suggestions, feedback and comments from experts
I'm an expert and enthusiast, and I have been trained on a wide range of topics, including R programming and Monte Carlo simulations. I have access to a vast amount of information and can provide detailed explanations and examples on these subjects.
To demonstrate my expertise, I can explain the concepts mentioned in this article.
Monte Carlo Simulations
Monte Carlo simulation, also known as the Monte Carlo Method, is a statistical technique used to compute all possible outcomes of an event. It is particularly useful in risk assessment and decision-making processes because it allows us to predict the probability of extreme cases occurring.
The technique was first used by scientists working on the atom bomb during World War II and was named after Monte Carlo, the famous resort town in Monaco known for its casinos. Since then, Monte Carlo simulation has been applied to model various physical and conceptual systems.
Probability and Independence
Monte Carlo methods are used to identify the probability of an event A happening among a set of N events. These methods assume that all events are independent, meaning the probability of event A happening once does not affect the occurrence of event A happening again.
For example, consider flipping a fair coin. The probability of getting heads is 0.5, regardless of whether heads or tails was obtained in the previous flip. However, if the coin is flipped 100 times, it is expected that heads will occur approximately 50% of the time.
Monte Carlo methods are employed when we want to determine the probability of a specific outcome (such as heads occurring 16 times in a row) by simulating multiple trials of the event.
Applications of Monte Carlo Simulations
Monte Carlo simulations find applications in various fields, including finance, project management, energy, manufacturing, research and development, insurance, and biotechnology. Some real-world examples of Monte Carlo simulations include:
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Fraud Detection: Monte Carlo simulations are used in financial services to predict fraudulent credit card transactions. Since 100 genuine transactions do not guarantee that the next one will not be fraudulent, Monte Carlo simulations help assess the risk associated with such events.
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Profit/Loss Analysis: Monte Carlo simulations can be used to determine the average profit or loss when a new product is sold to different groups of customers at different prices. This analysis helps understand the potential profitability of different pricing strategies.
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A/B Testing: Monte Carlo simulations can be used in A/B testing scenarios to evaluate the effectiveness of different website features or changes. For example, these simulations can help determine whether a new payment processing system leads to improved checkout completion rates compared to the old system.
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Selection Criteria: Monte Carlo simulations can assist in evaluating selection criteria when making decisions. For instance, in the case of selecting scholarship candidates, simulations can estimate the probability of a specific candidate being chosen in consecutive years.
Advantages of Monte Carlo Simulations
Monte Carlo simulations offer several advantages over simple forecasting methods. Some of these advantages include:
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Probabilistic Results: Monte Carlo simulations provide scenarios and likelihoods of different outcomes, allowing for a better understanding of the potential risks and rewards.
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Graphical Results: The outcomes and their probabilities can be easily visualized through graphs, making it simpler to communicate findings to an audience.
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Sensitivity Analysis: Monte Carlo simulations help identify which variables have the most significant impact on the outcome. This analysis allows decision-makers to focus on the factors that have the most influence on the results.
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Scenario Analysis: By using Monte Carlo simulations, decision-makers can observe which input values are associated with specific outcomes. This analysis helps in understanding the relationships between variables and their impact on results.
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Correlation of Inputs: Monte Carlo simulations can model interdependent relationships between input variables. This capability allows for a more accurate representation of how different factors influence each other in reality.
Code Template for Monte Carlo Simulations in R
The article also provides a code template for implementing Monte Carlo simulations in R. The basic template consists of the following steps:
runs <- 100000
func1 <- sum(sample(c(0,1), size = 10, replace = T)) > 6
mc_prob <- sum(replicate(runs, func1)) / runsIn this code, runs represents the number of trials or iterations for the simulation. The func1 variable is a function that defines the specific events, their probabilities, and the selection criteria for the simulation. The mc_prob variable calculates the probability of the desired outcome by executing the function multiple times and computing the ratio of successful outcomes to the total number of runs.
It's worth noting that the code provided is a simplified example and may need modification to suit specific applications.
I hope this information helps you understand the concepts related to Monte Carlo simulations mentioned in the article. Please let me know if you have any further questions!
