Random Number Generators

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Random Number Generators
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Random Number Generators

• Based upon specific mathematical algorithms
• Which are repeatable and sequential

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Random

• Truly Random
• Exhibiting true randomness
• Pseudorandom
• Appearance of randomness but having a specific
  repeatable pattern
• Quasi-random
• Having a set of non-random numbers in a
  randomized order

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Problems

• Difficult to isolate
• Often need to replace current generator
• Require
• Knowledge of current generator
• Sometimes in-depth understanding of random number
  generators themselves
• Large scale tests cause most problems
• Needing sometimes millions or billions of random
  numbers

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Desirable Properties

• When performing Monte Carlo Simulations
• Attributes of each particle should be independent
  of those attributes of any other particle
• Fill the entire attribute space in a manner which
  is consistent with the physics

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Random Number Cycle

• Basis
• sequence of pseudorandom integers
• Some exceptions
• Integers (Fixed)
• Manipulated arithmetically to yield floating
  point (real)
• Can be presented in either Integer or Real
  numbers

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Cycle
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What Does This Show Us?

• Properties of pseudorandom sequences of integers
• The sequence has a finite number of integers
• The sequence gets traversed in a particular order
• The sequence repeats if the period of the
  generator is exceeded

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LCG

• Most commonly used RNG
• Linear Congruential Generator
• Requires initial seed denoted as X0
• Appears random because of Modulo function
• Next random number depends heavily on previous
  X
• Typical of linear, congruential generators
• Restricts period

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Equations - LCG
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Using LCG

• Choosing Correct Input is Key
• LCG (a,c,m,X0)
• LCG (5, 1, 16, 1)
• Yields
• 1,6,15,12,13,2,11,8,9,14,7,4,5,10,3,0,
• 1,6,15,12,13,2,11,8,9,14,
• When the next result depends upon only the
  previous integer, the longest period possible is
  PM
• Odd/Even pattern
• lack of randomness results from using a power of
  two for M

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Cycle LCG(5,1,16,1)
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Table
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Example 2

• LCG(5,0,16,1)
• Yields - 1,5,9,13,1,5,9,13,
• M is a power of 2 (here 24)
• C0
• Maximum period is going to be 2(m-2)
• Correlation (each differ by 4)

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Cycle (5,0,16,1)
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Prime Numbers

• Using a prime as the divisor for the modulus can
  result in a period of m-1 as maximum period
• There are cases of prime moduli that fail bitwise
  testing
• Case-by-case basis

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Seed

• Using the date and time
• Enter the date and time into an equations and
  return an integer then make sure it is odd
• Standard seed for these equations

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Overflow Negative Numbers

• Using large values of a and large values of M are
  needed
• Often 31 bits long
• On 32 bit machines
• AM results in 62 bit number
• Overflow
• Can result in 32nd bit being a negative

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N-Tuple Generalization

• Choose R1 and R2
• Choose Rn and R(n1)
• Then plot this point of interest in a surrounding
  area.
• Plot these points in succession
• The area will be uniformly covered by the LCG in
  a random order
• Covering of only part of the unit or certain
  areas of the unit would prove to be not useful
  for Monte Carlo Methods

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Embarrassingly Parallel'

• Little or no interprocessor communication
• Easy to code

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N Streams

• N Streams
• N independent random numbers
• N independent processes
• Need to find N seeds far away from each other on
  the cycle

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Find Seeds

• Find Seeds
• LCG rule successively applied

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Lagged Fibonacci Generators

• Increasingly popular
• Lags are k and l
• M is power of 2
• With proper choice of k and L
• Period of Generator can be
• (2L)-1 2(m-1)

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LFG

• Computationally simple
• Integer add
• Logical AND
• Decrement of 2 array pointers
• Must keep L words current in memory
• LCG needs only one

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LFG (cont)

• LFG are an attempt to improve LCG
• Similar to Combined LCG
• Take 2 previous numbers in the sequence to
  produce a new number
• Where p and q are the lags
• Some arithmetic computation is performed
• Then mod that answer for the next number

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Monte Carlo Methods
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Overview

• Introduction
• History
• Examples
• Applications
• Real Life practices

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Introduction

• Define Monte Carlo Method
• The Monte Carlo method is a numerical method for
  solving mathematical problems using stochastic
  sampling.
• It performs simulation of any process whose
  development is influenced by random factors, but
  also if the given problem involves no chance, the
  method enables artificial construction of a
  probabilistic model.

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Introduction cont

• Similarly, Monte Carlo methods randomly select
  values to create scenarios of a problem. These
  values are taken from within a fixed range and
  selected to fit a probability distribution e.g.
  bell curve, linear distribution, etc.. This is
  like rolling a dice. The outcome is always within
  the range of 1 to 6 and it follows a linear
  distribution - there is an equal opportunity for
  any number to be the outcome.

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Introduction cont

• MC method is often referred to as the method of
  last resort, as it is apt to consume large
  computing resources
• Characteristics
• consuming vast computing resources
• have historically had to be executed upon the
  fastest computers available at the time
• and employ the most advanced algorithms
• implemented with substantial programming acumen.
  

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Introduction cont

• Major components of Monte Carlo methods
• Probability distribution functions
• Random number generator
• Scoring
• Error estimation
• Variance reduction techniques
• Parallelization and vectorization

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History

• Where does Monte Carlo method come from? When?
  Who?
• The name "Monte Carlo" comes from the city of
  Monte Carlo in the principality of Monaco, famous
  for its gambling house
• Birth date of the Monte Carlo method is 1949,
  when an articale entitled "The Monte Carlo
  Method"( by N. Metropolis and S. Ulam ) appeared.
  
• The American mathematicians J. Neyman and S. Ulam
  are considered its originators.

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History cont
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History cont

• The theoretical foundation of the method had been
  known long before first articles were published.
• Well before 1949 certain problems in statistics
  were sometimes solved by means of random sampling
  
• However, simulation of random variables by hand
  is a laborious process
• Use of the Monte Carlo method as a universal
  numerical technique became practical only with
  the advent of computers and high-quality
  pseudorandom number generators

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History cont

• Buffon's needle problem
• In 1768 Buffon, a French mathematician,
  experimentally determined a value of p by
  casting a needle on a ruled grid
• Lord Rayleigh even delved into this field near
  the turn of the century.
• Fredericks and Levy in 1928 showed how the method
  could be used to solve boundary value problems
• Enrico Fermi in the 1930's used Monte Carlo in
  the calculation of neutron diffusion (involving
  nuclear reactors )

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History cont

• In the 1940's, a formal foundation for the Monte
  Carlo method was developed by von Neumann (PDE)
• Stanislaw Ulam realized the importance of the
  digital computer in the implementation of the
  approach from collaboration results of the work
  on the Manhattan project during World War II

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Examples

• Simple Example to Understand computing the area
  of a plane figure S.
• completely arbitary figure with a curvilinear
  boundary, given graphically or analytically,
  connected or consisting of several pieces
• assume that it is contained completely within the
  unit square.

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Examples cont
Figure S in the unit square, being covered with
sampling points randomly
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Examples cont

• Applying Randomness to the example
• Choose at random N points in the square and
  designate the number of points lying inside S by
  N'. It is geometrically obvious that the area of
  S is approximately equal to the ratio N'/N. The
  greater the N, the greater the accuracy of this
  estimate.

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Examples cont

• Buffon's Needle
• A simple Monte Carlo method for the estimation of
  the value of p, 3.1415926
• Assumptions
• Suppose you have a tabletop with a number of
  parallel lines drawn on it, which are equally
  spaced (say the spacing is 1 inch, for example).
• Suppose you also have a pin or needle, which is
  also an inch long.

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Examples cont

• Dropping needles on the tablet
• The needle crosses or touches one of the lines
• The needle crosses no lines
• Keep dropping this needle over and over on the
  table
• Record the statistics.
• Keep track of both the total number of times that
  the needle is randomly dropped on the table N,
  and the number of times that it crosses a line
  N.

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Examples cont

• Findings
• 2N/Np
• Because, the probability on any given drop of the
  needle that it should cross a line is given by
  2/pi
• After many tries, N/N will approach the
  probability number.

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Applications

• Monte Carlo methods can help in design and
  prediction of behavior of systems in nuclear
  applications and radiation physics
• The use of MC in the area of nuclear power has
  undergone an important evolution. Notable are the
  extensions to compute burnup in reactor cores,
  and full core neutronic simulations.

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Applications cont

• help researchers understand the probability of
  the occurrence of an adverse effect associated
  with exposures to chemicals. Monte Carlo sampling
  simulates the distribution of total exposures, by
  simulating random samples of factors associated
  with each exposure route and accumulating them to
  arrive at an individual total exposure.

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Applications cont

• The use of MC methods to model physical problems
  allows us to examine more complex systems than we
  otherwise can. Solving equations which describe
  the interactions between two atoms is fairly
  simple solving the same equations for hundreds
  or thousands of atoms is impossible. With MC
  methods, a large system can be sampled in a
  number of random configurations, and that data
  can be used to describe the system as a whole.

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Applications cont

• Random numbers generated by the computer are used
  to simulate naturally random processes
• many previously intractable thermodynamic and
  quantum mechanics problems have been solved using
  Monte Carlo techniques

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Real Life Practice

• Quantum Monte Carlo
• The microscopic world is described by quantum
  mechanics. We need to use simulation techniques
  to solve many-body quantum problems.
• Both the wavefunction and expectation values are
  determined by the simulations.
• QMC gives most accurate method for general
  quantum many-body systems.

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Real Life Practice cont

• Weather
• Equipment Productivity
• Soil Conditions
• Projects are often associated with a high degree
  of uncertainty resulting from the unpredictable
  nature of events

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Real Life Practice cont

• Risk Analysis and Risk Management
• Monte Carlo Simulation is a valuable modeling
  tool that generates multiple scenarios depending
  upon the data and the assumptions fed into the
  model.
• Simulation calculates multiple scenarios by
  repeatedly inserting different sampling values
  from probability distribution for the uncertain
  variables into the computerized spread-sheet.
• probability or percentage chance that a
  particular forecast value will fall within a
  certain specified range.

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Why has the Monte Carlo method become so popular?

• Analytic methods tend to be prohibitive (but some
  very difficult problems have finally been solved
  using MC)
• Monte Carlo is somewhat intuitive (and several
  good books have now been written on the subject)
• Computers continue to get faster and cheaper

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Reference

• http//random.mat.sbg.ac.at/links/monte.html
• http//www.ccs.uky.edu/csep/csep.html