Monte-Carlo Simulation

- Besides the traditional ways of data acquisition in laboratory
experiments and field tests the generation of so-called
*synthetic data*via computer simulation has gained increasing importance. - There is a variety of reasons for the increased benefit drawn from
computer simulation used to investigate a wide range of issues,
objects and processes:
- The most prominent reason is the rapidly growing performance of modern computer systems which has extended our computational capabilities in a way that would not have been imaginable even a short time ago.
- Consequently, computer-based data generation is often considerably
*cheaper*and*less time-consuming*than traditional data acquisition in laboratory experiments and field tests. - Moreover, computer experiments can be repeated under constant conditions as frequently as necessary whereas in traditional scientific experiments the investigated object is often damaged or even destroyed.

- A further reason for the value of computer simulations is the fact
- that volume and structure of the analyzed data is often very complex
- and that in this case data processing and evaluation is typically based on mathematical models whose characteristics cannot be (completely) described by analytical formulae.
- Thus, computer simulations of the considered models present a valuable alternative tool for analysis.

- Computer experiments for the investigation of the issues, objects
and processes of scientific interest are based on
*stochastic simulation algorithms*. In this context one also uses the term*Monte-Carlo simulation*summarizing a huge variety of simulation algorithms.*Random number generators*are the basis for Monte-Carlo simulation of single features, quantities and variables.- By these algorithms realizations of random variables can be
generated via the computer. Those are called
*pseudo-random numbers*. - The simulation of random variables is based on so-called
*standard random number generators*providing realizations of random variables that are uniformly distributed on the unit interval . - Certain
*transformation and rejection methods*can be applied to these standard pseudo-random numbers in order to generate pseudo-random numbers for other (more complex) random variables having e.g. binomial, Poisson or normal distributions.

- By these algorithms realizations of random variables can be
generated via the computer. Those are called
- Computer experiments designed to investigate
*high-dimensional random vectors*or the evolution of certain objects in time are based on more sophisticated algorithms from so-called dynamic Monte-Carlo simulation.- In this context
*Markov-Chain-Monte-Carlo-Simulation*(MCMC simulation) is a construction principle for algorithms that are particularly appropriate to simulate time stationary equilibria of objects or processes. - Another example for the application of MCMC simulation is
*statistical image analysis*. - An active field of research that resulted in numerous publications
during the last years are so-called
*coupling algorithms for perfect MCMC simulation*. - These coupling algorithms enable us to simulate time-stationary equilibria of objects and processes in a way that does not only allow approximations but simulations that are ,,perfect'' in a certain sense.

- In this context

- Generation of Pseudo-Random Numbers
- Transformation of Uniformly Distributed Random Numbers
- Simulation Methods Based on Markov Chains
- Error Analysis for MCMC Simulation
- Coupling Algorithms; Perfect MCMC Simulation