stochastic simulations
DESCRIPTION
Stochastic Simulations. Six degrees of Kevin Bacon. Outline. Announcements: Homework II: due Today. by 5, by e-mail Discuss on Friday. Homework III: on web Cookie Challenge Monte Carlo Simulations Random Numbers Example--Small Worlds. Homework III. - PowerPoint PPT PresentationTRANSCRIPT
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Stochastic Simulations
Six degrees of Kevin Bacon
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Outline• Announcements:
– Homework II: due Today. by 5, by e-mail• Discuss on Friday.
– Homework III: on web
• Cookie Challenge• Monte Carlo Simulations• Random Numbers• Example--Small Worlds
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Homework III
• For HW IV--you will create your own programming assignment.– Develop a solution (function or two) to a
particular problem– Pick something relevant to you!
• For HW III– Define your problem– Will give me a chance to comment
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Monte Carlo
• Monte Carlo methods refer to any procedure that uses random numbers
• Monte Carlo methods are inherently statistical (probabilistic)
• Used in every field– Galaxy formation– Population model– Economics– Computer algorithms
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Monte Carlo Example
• Have a computer model which computes price of corn in Omaha using rainfall.– You have a forecast of rainfall for next few months from
NWS– Forecast is rain +/- SE
• How can you incorporate uncertainty of rainfall into your forecast of prices?– Want $ +/- SE
Modelrain1
rain2
rain3
$1
$2
$3
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Monte Carlo Example
• 1. Create several random forecast rainfall series– mean of the series is the forecast– SE of series is the forecast SE
• 2. Compute prices• 3. Calculate SE of prices.
Modelrain1
rain2
rain3
$1
$2
$3
Modelrain1
rain2
rain3
$1
$2
$3
Modelrain1
rain2
rain3
$1
$2
$3
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Random Numbers
• Computers are deterministic– Therefore, computers generate “pseudo-random” numbers
• Matlab’s random numbers are “good”– “The uniform random number generator in MATLAB 5 uses
a lagged Fibonacci generator, with a cache of 32 floating point numbers, combined with a shift register random integer generator.”
– http://www.mathworks.com/support/solutions/data/8542.shtml
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Random functions
• rand(m,n) produces m-by-n matrix of uniformly distributed random numbers [0,1]
• randn(m,n) produces random numbers normally distributed with mean=0 and std=1
• randperm(n) is a random permutation of integers [1:n]– I=randperm(n); B=A(I,:) scrambles the rows of A
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Seeds
• Random number generators are usually recurrence equations:– r(n)=F(r(n-1))
• Must provide an initial value r(0)– Matlab’s random functions are seeded at startup, but
THE SEED IS THE SAME EVERY TIME!– Initialize seed with rand('state', sum(100*clock) )– How would you ensure rand is always random?
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Monte Carlo Example
• 1. Create several random forecast rainfall series– rain, rainerr--n-by-1 vectors of rain forecasts and SE– P=randn(n,p)– randrain=P*(rainerr*ones(1,p))+(rain*ones(1,p))
• 2. Compute prices– for j=1:p;prices(:,j)=Model(randrain(:,j));end
• 3. Calculate SE of prices.– priceerr=std(prices,2)/sqrt(p);– pricemn=mean(prices,2);
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It’s a Small, Small World
• Watts & Strogatz (1998) Nature, 393:440-442
• Complicated systems can be viewed as graphs– describe how components are connected
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Example: Six Degrees of Kevin Bacon
• Components (vertices) are actors• Connections (edges) are movies• Hypothesis: 6 or fewer links separate
Kevin Bacon from all other actors.– “Oracle of Bacon” at
http://www.cs.virginia.edu/oracle/
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Sir Ian McKellen
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Andy Serkis
Lord of the Rings
GI Jane
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Viggo Mortensen
Example: Kevin Bacon & Gollum
A Few Good Men
Jack Nicholson
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Demi Moore
Kevin
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Other Systems
• Power Grid• Food Webs• Nervous system of Caenorhabditis
elegans• Goal is to learn about these systems by
studying their graphs• Many of these systems are “Small
Worlds”--only a few links separate any two points
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Watts & Strogatz
• Can organize graphs on a spectrum from ordered to random
• How do graph properties change across this spectrum?– L=mean path length (# links between points)– C=cluster coefficient (“lumpiness”)
• Used a Monte-Carlo approach--created lots of graphs along spectrum and computed L and C
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Watts & Strogatz
• Creating the graphs• n=# of vertices, k=number of
edges/vertex• Start with a regular ring lattice
and change edges at random with probability p
• For every p, compute stats for many graphs
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Small Worlds in Matlab
• G=createlattice(n,k,p)– creates a lattice--represented as a sparse matrix
• [L,C]=latticestats(G)– computes the path length and clustering stats
• [L,C]=SmallWorldsEx(n,k,P,N)– Creates N graphs for every P(j) and saves the mean
stats in L(j) and C(j)
• plotlattice(G)– Plots a lattice