Acceptance-Rejection method The acceptance-rejection method is - PowerPoint PPT Presentation

Acceptance-Rejection Method Acceptance-Rejection method The acceptance-rejection method is usually used when the inverse transform method is not directly applicable or is inefficient. To simulate a sample x of X with pdf f X . Aim: 1 Consider a random variable Y with pdf f Y whose sample can be simulated. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 38 / 46

Acceptance-Rejection Method Acceptance-Rejection method The acceptance-rejection method is usually used when the inverse transform method is not directly applicable or is inefficient. To simulate a sample x of X with pdf f X . Aim: 1 Consider a random variable Y with pdf f Y whose sample can be simulated. 2 Choose a constant c such that f X ( t ) ≤ cf Y ( t ) , ∀ t . The random variable Y is said to majorize X . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 38 / 46

Acceptance-Rejection Method Acceptance-Rejection method The acceptance-rejection method is usually used when the inverse transform method is not directly applicable or is inefficient. To simulate a sample x of X with pdf f X . Aim: 1 Consider a random variable Y with pdf f Y whose sample can be simulated. 2 Choose a constant c such that f X ( t ) ≤ cf Y ( t ) , ∀ t . The random variable Y is said to majorize X . Suppose X and Y are discrete random variables with pmfs 1 p i = Pr { X = i } , q i = Pr { Y = i } . Then c can be chosen as follows: � p j � c = max : j ≥ 1 q j R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 38 / 46

Acceptance-Rejection Method Acceptance-Rejection method The acceptance-rejection method is usually used when the inverse transform method is not directly applicable or is inefficient. To simulate a sample x of X with pdf f X . Aim: 1 Consider a random variable Y with pdf f Y whose sample can be simulated. 2 Choose a constant c such that f X ( t ) ≤ cf Y ( t ) , ∀ t . The random variable Y is said to majorize X . Suppose X and Y are discrete random variables with pmfs 1 p i = Pr { X = i } , q i = Pr { Y = i } . Then c can be chosen as follows: � p j � c = max : j ≥ 1 q j Suppose X and Y are continuous random variables with PDFs f X and f Y . 2 Then c can be chosen as � f X ( x ) � c = max f Y ( x ) : ∀ x . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 38 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · Figure: Acceptance-Rejection method R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 39 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The algorithm for this method is as follows: 1 Generate a random variate y 1 from the distribution of Y . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 40 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The algorithm for this method is as follows: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number variate y 2 between 0 and cf Y ( y 1 ) as follows: R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 40 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The algorithm for this method is as follows: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number variate y 2 between 0 and cf Y ( y 1 ) as follows: Generate a uniform random number r . 1 R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 40 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The algorithm for this method is as follows: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number variate y 2 between 0 and cf Y ( y 1 ) as follows: Generate a uniform random number r . 1 Compute y 2 = 0 + ( cf Y ( y 1 ) − 0) r = cf Y ( y 1 ) r . 2 R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 40 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The algorithm for this method is as follows: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number variate y 2 between 0 and cf Y ( y 1 ) as follows: Generate a uniform random number r . 1 Compute y 2 = 0 + ( cf Y ( y 1 ) − 0) r = cf Y ( y 1 ) r . 2 3 If y 2 < f X ( y 1 ), then x = y 1 , else go to Step 1. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 40 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The above algorithm is equivalent to the following: steps: R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 41 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The above algorithm is equivalent to the following: steps: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number r in [0 , 1]. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 41 / 46

Acceptance-Rejection Method Acceptance-Rejection method · · · The above algorithm is equivalent to the following: steps: 1 Generate a random variate y 1 from the distribution of Y . 2 Generate a uniform random number r in [0 , 1]. 3 If r ≤ f X ( y 1 ) cf Y ( y 1 ), then x = y 1 , else go back to step 1. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 41 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for 1 < k < 5 The variate is generated by the acceptance-rejection method, where the majorizing distribution is Erlang with parameter λ and k , where k = ⌊ k ⌋ . Figure: Gamma random variate for k = 2 . 5 and λ = 1 R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 42 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k < 1 Let X be a gamma random variable with parameters λ, k . 1 Generate two random numbers r 1 and r 2 . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 43 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k < 1 Let X be a gamma random variable with parameters λ, k . 1 Generate two random numbers r 1 and r 2 . 1 1 2 Compute x 1 = r 1 − k 1 , x 2 = r k . 2 R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 43 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k < 1 Let X be a gamma random variable with parameters λ, k . 1 Generate two random numbers r 1 and r 2 . 1 1 2 Compute x 1 = r 1 − k 1 , x 2 = r k . 2 3 Generate an exponential random variate with parameter λ , say, y . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 43 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k < 1 Let X be a gamma random variable with parameters λ, k . 1 Generate two random numbers r 1 and r 2 . 1 1 2 Compute x 1 = r 1 − k 1 , x 2 = r k . 2 3 Generate an exponential random variate with parameter λ , say, y . yx 1 4 Then the sample x of X is x = . x 1 + x 2 R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 43 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k > 5 Let a = ⌊ k ⌋ . Then choose the sample x of X randomly 1 from Erlang with parameters λ, a with probability 1 − ( k − a ). R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 44 / 46

Acceptance-Rejection Method Gamma random variate Gamma random variate for k > 5 Let a = ⌊ k ⌋ . Then choose the sample x of X randomly 1 from Erlang with parameters λ, a with probability 1 − ( k − a ). 2 from Erlang with parameters λ, a + 1 with probability k − a . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 44 / 46

Acceptance-Rejection Method Normal random variate Normal random variate - Polar form (Box-Muller form) 1 Generate two uniform random numbers r 1 and r 2 in [0 , 1]. Define v i = 2 r i − 1 , i = 1 , 2 and let w = v 2 1 + v 2 2 . R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 45 / 46

Acceptance-Rejection Method Normal random variate Normal random variate - Polar form (Box-Muller form) 1 Generate two uniform random numbers r 1 and r 2 in [0 , 1]. Define v i = 2 r i − 1 , i = 1 , 2 and let w = v 2 1 + v 2 2 . 2 If w = 0 or w > 1, go back to step 1. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 45 / 46

Acceptance-Rejection Method Normal random variate Normal random variate - Polar form (Box-Muller form) 1 Generate two uniform random numbers r 1 and r 2 in [0 , 1]. Define v i = 2 r i − 1 , i = 1 , 2 and let w = v 2 1 + v 2 2 . 2 If w = 0 or w > 1, go back to step 1. � − 2 log w 3 Let y = , x 1 = v 1 y and x 2 = v 2 y . Then x 1 and x 2 are IID w N (0 , 1) random variates. R.B. Lenin (rblenin@daiict.ac.in) () Random Variate Generation Autumn 2007 45 / 46