Using a discrete Monte-Carlo technique in a multi-variable model

In summary, when trying to get an average of a function using random sampling, it is recommended to discard already-used combinations for more accurate results and a smaller standard deviation. However, this may lead to less random inputs and there is a mathematical relationship between the number of samples and the likelihood of having no repeats.
  • #1
snowjoke
15
0
If I have a large amount of data I can sample, with a several discrete variables, and I need to get an average of some function of that data, but it's too computationally intensive to do exhaustively...

I want to do some sampling of the possible outcomes. I guess random sampling (Monte-Carlo technique) is the way forward, but my question is, theoretically, when selecting random numbers for the inputs, should I discard combinations that have already been used?

For large samples it probably won't in practice make any difference, but say I had I have 3 variables a, b and c, each of which can be (0,1,2,3,4,5,6,7,8,9), and I want to compute 100 outputs. Say I've already used {a = 1, b = 8, c = 4}, in theory should I check whether I've used this combination already?

Intuitively it seems like I'll get a more accurate result if I discard already-used inputs, but then the inputs won't be random.
 
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  • #2
Discarding already used combinations will lead to a better (smaller) standard deviation for the estimated average.
 
  • #3
Interestingly, the chance of having no repeats in sampling k from n is approximately exp(-k^2/2n). I wonder if there is a simple expression for the expected number of unique samples.
 

1. What is a discrete Monte-Carlo technique?

A discrete Monte-Carlo technique is a statistical method used to simulate random processes by generating a large number of possible outcomes and calculating the probabilities of each outcome. It involves dividing the variables into discrete intervals and then using a random number generator to determine the outcome of each interval. This technique is commonly used in modeling complex systems with multiple variables.

2. How does a discrete Monte-Carlo technique work?

The technique works by generating a large number of random numbers, which are then used to determine the values of each variable in the model. These values are then used to calculate the overall outcome of the model. By repeating this process many times, the technique can provide a more accurate representation of the possible outcomes of the system being modeled.

3. What are the advantages of using a discrete Monte-Carlo technique in a multi-variable model?

One advantage of using this technique is that it can handle complex systems with multiple variables that have complex relationships. It also allows for the consideration of uncertainty and variability in the model, which can lead to more accurate results. Additionally, the technique is flexible and can be applied to a wide range of problems.

4. What are the limitations of using a discrete Monte-Carlo technique?

One limitation is that it can be computationally intensive, requiring a large number of simulations to be run to achieve accurate results. It also assumes that the variables are independent, which may not always be the case in real-world systems. Additionally, the technique may not be suitable for systems with a large number of variables or complex interactions between variables.

5. How can the results of a discrete Monte-Carlo simulation be interpreted?

The results of a simulation can be interpreted by analyzing the distribution of outcomes and determining the probabilities of different outcomes occurring. Monte-Carlo techniques can also be used to identify the most important variables in a model and how they contribute to the overall outcome. Sensitivity analysis can also be performed to understand how changes in input variables affect the output of the model.

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