PROJECT BASED LEARNING : PROBABILITY AND RANDOM PROCESSES

Study on Monte-Carlo Method using MATLAB Code

OBJECTIVE OF THE PROJECTTo understand the concept of Monte Carlo Method and its various applications and it rely on repeated and randomsampling to obtain numerical result.

Developing the computational algorithms to solve the problem related to random sampling.

Objective also contains simulation of specific problem in Matlab Software.

TERMINOLOGIESComputational Algorithm.

Random Sampling.

Mathematical Optimization.

Probability Distribution.

Computational algorithm An exactly defined specification of the operations to be carried out on data, by means of which it is possible, using a discrete-operation digital computer, to convert a certain amount of data (input data) into a certain amount of other data (output data) by performing a finite number of operations. A computational algorithm is realized in the form of a computational process, i.e. as a finite sequence of states of a real computer, discretely distributed in time, the real computer unlike an abstract computer having a restricted rate of performance of the operations, a restricted number of digit places to form a number and a restricted storage capacity.

Random samplingOne of the best ways to achieve unbiased results in a study is through random sampling. Random sampling includes choosing subjects from a population through unpredictable means. In its simplest form, subjects all have an equal chance of being selected out of the population being researched.

Mathematical OptimizationInmathematics,computer science, ormanagement science,mathematical optimization(alternatively,optimizationor mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

Probability DistributionInprobability and statistics, aprobability distributionassigns aprobabilityto each of the possible outcomes of a randomexperiment,survey, or procedure ofstatistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be acategorical distribution; experiments whose sample space is encoded by discreterandom variables, where the distribution is aprobability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution is aprobability density function. More complex experiments, such as those involvingstochastic processesdefined incontinuous-time, may demand the use of more generalprobability measures.

INTRODUCTION TO THE PROJECTThis project is a study to understand the concept of Monte-Carlo methods which are the broad class of computational algorithm that rely on repeated randomsampling to obtain numerical results i.e. by running simulations many times over in order to calculate those same probabilities heuristically just like actually playing and recording your results in a real casino situation.They are often used inphysical andmathematical problems and are most suited to be applied when it is impossible to obtain aclosed-form expression or infeasible to apply adeterministic algorithm.Monte Carlo methods are mainly used in three distinct problems:Optimization.

Generation of samples from a probability distribution.

Numerical Integration.

Monte Carlo methods are especially useful for simulating systems with manycoupleddegrees of freedom, such as:

Fluids.

Disordered materials.

Strongly coupled solids.

Cellular structures (cellular Potts model).

Significantuncertaintyin inputs, such as the calculation ofriskin business.

Note - We have mainly studied Monte-Carlo method in category of generation of samples from a probability distribution.Steps to generate a 'Monte-Carlo'

Step1 : Define a domain of possible inputs.Step2 : Generate inputs randomly from aprobability distributionover the domain.Step3 : Perform adeterministiccomputation on the inputs. Step4 : Aggregate the results.

Fig.1: Monte-Carlo method simulation step wise flow chart.

Calculation of Pi Using Monte-Carlo Method using MATLABProblem Statement: Consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is/4, the value of can be approximated using a Monte Carlo method.Solution:There are following steps involved for calculating the value of Pi using Monte-Caro method as specified in the problem statement. High level language, Matlab is used to tackle the problem by applying the algorithms of Monte-Carlo in following steps : Draw a square on the ground, and theninscribea circle within it.

Uniformly scatter some objects of uniform size (grains of rice or sand) over the square.

Count the number of objects inside the circle and the total number of objects.

The ratio of the two counts is an estimate of the ratio of the two areas, which is/4. Multiply the result by 4 to estimate.

If you are playing the dart, it is easy to imagine throwing darts randomly at Figure 2, and it should be apparent that of the total number of darts that hit within the square, the number of darts that hit the shaded part (circle quadrant) is proportional to the area of that part.

Formula used to calculate the value of pi http://upload.wikimedia.org/wikipedia/commons/8/84/Pi_30K.gif Fig.2 : Monte Carlo method applied to approximating the value of. After placing 30000 random points, the estimate foris within 0.07% of the actual value. This happens with an approximate probability of 20%.Explanation :If each dart thrown lands somewhere inside the square, the ratio of "hits" (in the shaded area) to "throws" will be one-fourth the value of pi. If you actually do this experiment, you'll soon realize that it takes a very large number of throws to get a decent value of pi...well over 1,000. To make things easy on ourselves, we can have computers generate randomnumbers.

Simulation Using MatLAB Code

Estimated values of pi for different values of n by applying the monte-carlo algorithm.

Tabulation of different value of pi for different values of n while applying Monte-Carlo algorithm and calculation of approximate error.

S.No.Number of throws, nValue of piApproximate error

110003.1680.028

250003.1200.020

3100003.1084.0336

4200003.1354.0066

5250003.1389.0031

6770003.1440-.002

71000003.1488-.068

APPLICATIONS OF 'MONTE-CARLO' METHOD

PEREGRINE, a Monte Carlo all-particle transport code is developed for calculating the dose deposition in patients receiving radiation therapy.

Simulates the transport of neutrons, photons, electrons and protons through a patient using a geometry derived from a computed tomography (CT) scan.

It will help doctor to choose a treatment that maximized the radiation dose to the tumor and minimizes the dose to normal tissues.

Monte Carlo method is used because it is the most accurate way to simulate radiation transport on a computer.

Quantum Monte Carlo methods are powerful numerical approach to the investigation of Quantum Many-Body system.

The methods are also used in following processes -

Random Number Generations

Nuclear Reactor Design

Traffic Flow

Dow-Jones forecasting

Oil-Well explorations

Economics

Environmental air pollution

Biological Thermodynamics of Aging

Benefits and drawback of Monte -Carlo MethodADVANTAGES

Very flexible. There is virtually no limit to the analysis. Empirical distributions can be handled.

Can generally be easily extended and developed as required.

Easily understood by non-mathematicians.

Enables study of interactions between components.

Enables the inclusion of real-world complications.

Can be used to analyze large and complex real-world situations.

DISADVANTAGES

Usually requires a computer.

Calculations can take much longer than analytical models.

Solutions are not exact, but depend on the number of repeated runs used to produce the output statistics

RESULTS OBTAINEDStudy on Monte-Carlo Method is done.

Simulation was done in Matlab software for Specific Problem: Calculation of pi Using Matlab through Monte Carlo method. Tabulation for different values of throw was also done.

Study on various application of Monte-Carlo Method and on its area was done.

Step was studied for defining computational algorithm.

REFERENCESwww. wikipedia.org/wiki/Monte_Carlo_method

www.mathswork.in

Modeling Derivatives using C++, Justin London, 7 edition, Springer Publications

Financial derivatives, Journal of Derivatives, Volume 26, Pages 111-119, 2012.

Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao Copyright 2009 by John Wiley & Sons, Inc

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