# Monte Carlo Simulations

## What is Monte Carlo Simulations?

One of the main motivations to switch from spreadsheet-type tools (such

as Microsoft Excel) to a program like R is for simulation modeling. R

allows us to repeat the same (potentially complex and detailed)

calculations with different random values over and over again.

Within the same software, we can then summarize and plot the results of

these replicated calculations. Monte Carlo methods are used to perform

this type of analysis they randomly sample from a set of values in order

to generate and summarize a distribution of some statistic related to

the sampled quantities.

## Randomness

Random processes are an important aspect of simulation modeling. A

random process is one that produces a different result each time it is

run according to some rules. They are inextricably tied to the concept

of uncertainty, you have no idea what will happen the next time the

process is run.

There are two basic ways to introduce randomness in R

**Random deviattes**

**Resampling**

## Random Deviates

Each person alive at the start of the year has the option of living or

dying at the conclusion of the year. There are two possible endings

here, and each person has an 80% probability of surviving. survive is

the outcome of a binomial random process in which there were n

individuals alive at the start of this year and p is the probability

that any one of them would live to the next year.

In R, we can simulate a binomial random process with p=0.8 and n=100.

```
rbinom(n= 1, size =100,
prob= 0.8)
```

`## [1] 80`

At this time I got 73, but we almost certainly get different number than

this one.

## With a little tinkering, we can also plot it

```
survivors = rbinom(1000,
100, 0.8)
hist(survivors,
col = "skyblue")
```

## We could also used other processes like log normal

The log normal process is another random process. It creates random

numbers using a log of the values that is regularly distributed, with a

mean of log mean and a standard deviation of log sd.

`hist(rlnorm(1000,0,0.1),col="skyblue")`

## Need for sampling

There are several situations in probability, and more broadly in machine

learning, where an analytical solution cannot be calculated immediately.

In Machine Learning, a problem of class imbalance exists. In fact, some

would argue that for most practical probabilistic models, accurate

inference is impossible.

The desired calculation is usually a sum of discrete distributions or an

integral of continuous distributions, and thus is computationally

difficult. For a variety of reasons, such as the huge number of random

variables, the stochastic nature of the domain, noise in the data, a

shortage of observations, and more, the calculation may be intractable.

## Resampling

Using random deviates to generate fresh random numbers is excellent, but

what if we already have a set of numbers to which we want to add

randomness? We can utilize resampling techniques to do this. To sample

size elements from the vector x in R, use the sample() function.

## Resampling of 1 to 10

`sample(x = 1:10, size =5)`

`## [1] 4 3 10 9 2`

## Sample with replacement

`sample(x = c("a","b","c"), size = 10, replace = T)`

`## [1] "a" "a" "a" "b" "a" "b" "a" "c" "b" "b"`

## Sample with set probalilities

`sample(x = c("live","die"),size = 10, replace = T, prob = c(0.8,0.2))`

`## [1] "live" "live" "die" "die" "live" "live" "live" "die" "live" "live"`

## Reproducing Randomness

We may want to receive the same precise random integers each time we run

our script for reproducibility. To do so, we must first set the random

seed, which is the starting point of our computer’s random number

generator.

```
set.seed(1234)
rnorm(1)
```

`## [1] -1.207066`

Let’s try without random seed

`rnorm(1)`

`## [1] 0.2774292`

Each time we get different result.

## Replication

To use Monte Carlo methods, we need to be able to replicate some random

process many times. There are two main ways this is commonly done,

either with `replicate()`

or with `for()`

loops.

The replicate() functions executes same expression many times and

returns the output from each execution. Say we have a vector x, which

represents 40 observations of an animal length(mm).

`x = rnorm(30, 500,40)`

We want to create the mean length sampling distribution “by hand.” We

can take a random sample, determine the mean, and then repeat the

process as many times as necessary.

## Replication with “for” loop

A loop is a command in programming that repeats itself until it reaches

a specified point. R has a few types of loops, repeat(), while(), and

for(). for() loops are among the most common in simuation modeling.For

each value in a vector, a for() loop performs an operation for the

number of times you specify.

For loop syntax

for(var in seq){ expression(var) }

```
for( i in 1:5){
print(i^2)
}
```

```
## [1] 1
## [1] 4
## [1] 9
## [1] 16
## [1] 25
```

```
nt = 100
N= NULL
N[1] = 1000
for(t in 2:nt){
N[t] = (N[t-1]*1.1*rlnorm(1,0,0.1))*(1-0.08)
}
```

Let’s plot it

`plot(N, type= "l", pch = 15, xlab = "Year", ylab = "Abundance")`

## Summarization of simulation

After replicating a calculation many times, we will need to summarize

the results.

## Simulating Based Learning

```
mu = 500
sig = 30
random = rnorm(100,mu,sig)
p = seq(0.01, 0.99, 0.01)
random_q = quantile(random,p)
normal_q = qnorm(p,mu,sig)
plot(normal_q~random_q)
abline(c(0,1))
```

```
q = seq(400,600,10)
random_cdf = ecdf(random)
random_p =random_cdf(q)
normal_p = pnorm(q,mu,sig)
plot(normal_p~q, type= "l", col = "blue")
points(random_p~q,col = "red")
```

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