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Numerical Analysis - Simulation - Hanyang University Jong-Il Park

Modeling Modeling and fitting unmodeled World modeled World modeling observed data model parameters fitting

Modeling - complexity Eg. Computer graphics

Simulation ◦ Data generation Simulation model Synthetic data ※varying fitting ◦ Data generation Random number generation Uniform distribution Gaussian distribution NR in C chap. 7

General procedure of random number generation 1. Determine the probability density function(pdf) - uniform, Gaussian, Poisson, Gamma,... 2. Generate a RN with uniform distribution eg. Call ran1() in NR in C. 3. Generate a RN with an arbitrary pdf using the RN of 2. - Transformation method, Rejection method... eg. Call gasdev() in NR in C for Gaussian distribution

Generating an arbitrary pdf(I) Transformation method

Generating an arbitrary pdf(II) Rejection method

Synthetic data generation Flow diagram of synthetic data generation Mix Random number generation pdf of x yideal y Contaminated data = practical data model noise a Random number generation Determined x pdf of noise pdf of x : uniform, Gaussian, ... ; probability Determined x : x1,x2,...,xN given yideal : no noise y : contaminated data pdf of noise : uniform, Gaussian, ... Mixer : additive, multiplicative, ...

Homework #6 Programming on Random Number Generation: [Due: Nov. 5] Programming on Random Number Generation: (1) Uniform distribution in [a,b], (2) Gaussian distribution with mean=m, standard deviation=s. - Generate 1000 samples and draw a histogram for each distribution (a=-4, b=1, m=0.5, s=1.5). - Repeat the same job with varying the number of samples. (eg. 100, 10000, 100000) - Discuss the shape of the histograms in terms of the number of samples. Refer to Ch. 7, NR in C.