The function generates uniformly distributed random numbers. It accepts which is the sample size as its input.
LATIN HYPERCUBE SAMPLING VS. IMPORTANCE SAMPLING CODE
The following is the R code for a Monte Carlo Simulation. We will compare the shape of the sample histograms to the shape of this probability density function. First let us show the graph of the density function of. Using both methods, we will get a sample of 100, 1000 and 10,000 readings and plot the histogram. Let’s say that we would like to sample from the normal distribution with mean 10 and variance 3 (represented by ). Hence we obtain the Latin Hypercube sample and that has an associated probability density function. Similarly is sampled randomly from, which is converted to which lies in. Thus is sampled randomly from, which is converted to which lies in. Hence the interval is split into 4 sub-interval, and. The next diagram displays a simple example in which we would like to have a sample of 4 values from a distribution whose domain is. We generate a uniformly-distributed random variable from each of these intervals and obtain. In general, the interval is split into equal segments. This is carried out by using the Analytical Inversion Method. In this particular case, we would obtain one number from each interval: and. Moreover the number of segment, that is, the sample size, could be any positive integer. In general this idea works even when the domain is unbounded (for example, in the case of the normal distribution where the values can take could be anything from to ). It shows an example of a probability density function whose domain is and the area is split into 4 segments. We take a random value from each portion. Since the total area under the probability density function is 1, each portion would have an area of.
The idea is to split the total area under the probability density function into portions that have equal area. We would like to sample values from this distribution using LHS.
Latin Hypercube SamplingĬonsider a random variable with probability density function and cumulative distribution function. For more information, one can have a look here at the article that compares the Analytical Inversion Method to an alternative method known as the Accept-Reject Method. The idea of generating a uniformly-distributed random value and converting it using in order to follow the desired distribution is called the Analytical Inversion Method. This process is repeated until we have gathered values for our sample. Note that by using mathematical theory, it can be shown that has its probability density function equal to. The value where is the inverse of the cumulative distribution function, is our first sample point, which we call. We generate a random number between 0 and 1 and call it (that is is uniformly-distributed on the interval ). We would like to generate a random sample of values from this distribution. Suppose that we have a random variable with a probability density function and cumulative distribution function. The two sampling methods are then extended to and demonstrated on bivariate cases, for which the rate of convergence is also analysed. The methods are compared with each other in terms of convergence. Both methods are explained and the R code for generating samples is provided for each method. We will see something similar when simulating using MCS and LHS.
The advantage of stratified sampling over simple random sampling is that even though it is not purely random, it requires a smaller sample size to attain the same precision of the simple random sampling. In random sampling the 20 people are chosen randomly (without the use of any structured method) and in stratified sampling, 4 people are chosen randomly from each of the 5 districts. Suppose we want to pick 20 people from a city which has 5 districts. In order to give a rough idea, MC simulation can be compared to simple random sampling whereas Latin Hypercube Sampling can be compared to stratified sampling. In MCS we obtain a sample in a purely random fashion whereas in LHS we obtain a pseudo-random sample, that is a sample that mimics a random structure. Monte Carlo Sampling (MCS) and Latin Hypercube Sampling (LHS) are two methods of sampling from a given probability distribution.
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