and sampling a distribution
in [Mathematica]
|
From: Bill Rowe on 26 Jun 2010 03:09 On 6/25/10 at 7:27 AM, stone(a)geology.washington.edu (John Stone) wrote: >I am trying to use RandomReal[ ] to sample from bins of different >widths that span the interval 0 - 1. The bin widths represent the >weights I'm assigning to a family of trial solutions in an >optimization problem. The aim is to sample the solutions in >proportion to their weights using a uniform distribution of random >numbers generated by RandomReal[ ]. >For a simple example, however, suppose there are 10 equally weighted >solutions. My selection process would use some code that looks >like: >weights = Table[0.1, {10}]; >bins = Accumulate[weights]; >Select[bins, (# >= RandomReal[] &)][[1]] Rather than RandomReal you should be using RandomChoice. Specifically, RandomChoice[weights->bins,10] will return a list of 10 values with the desired distribution. This can be seen by doing: Histogram[RandomChoice[weights -> bins, 1000]] and note with equal weights and equally spaced bins of size 0.1, the following is equivalent RandomInteger[{1,10}]/10//N >Assuming the result of RandomReal[ ] is uniformly distributed, I >expected this to return 0.1 as frequently as it returns 0.5 or 1, No, this isn't correct. The value 0.1 will be returned whenever RandomReal returns a value greater than or equal to 0.1 but less than 0.2 which should happen 10% of the time. But the value 1 will be returned only if RandomReal returns the value 1 which will happens with probability near 0. Now consider what happens when RandomReal returns a value greater than 0.1 but less than 0.3. This will occur ~20% of the time. And your selection criteria will return 0.2 as the first value in the list of selected values. That is 0.1 occurs with probability 10%, 0.2 occurs with probability 20% an 1 occurs with very low probability (near 0). So, it is clear the distribution with this selection criteria cannot be flat as you were expecting. I haven't worked out the probability for the other values in the list. I think the above is sufficient to show the selection criteria you have used will not return uniform deviates.
From: Bill Rowe on 27 Jun 2010 04:55 On 6/26/10 at 3:09 AM, readnews(a)sbcglobal.net (Bill Rowe) wrote: >On 6/25/10 at 7:27 AM, stone(a)geology.washington.edu (John Stone) >wrote: > >>I am trying to use RandomReal[ ] to sample from bins of different >>widths that span the interval 0 - 1. The bin widths represent the >>weights I'm assigning to a family of trial solutions in an >>optimization problem. The aim is to sample the solutions in >>proportion to their weights using a uniform distribution of random >>numbers generated by RandomReal[ ]. >>For a simple example, however, suppose there are 10 equally >>weighted solutions. My selection process would use some code that >>looks like: >weights = Table[0.1, {10}]; bins = Accumulate[weights]; Select[bins, >(# >= RandomReal[] &)][[1]] >Rather than RandomReal you should be using RandomChoice. >Specifically, >RandomChoice[weights->bins,10] >will return a list of 10 values with the desired distribution. This >can be seen by doing: >Histogram[RandomChoice[weights -> bins, 1000]] >and note with equal weights and equally spaced bins of size 0.1, the >following is equivalent >RandomInteger[{1,10}]/10//N Up to this point my response was fine. RandomChoice is the thing to use when you want random selection from a pre-defined list of things with various weights. But the explanation I gave for why the code didn't work as expected is simply wrong. Peter Pain correctly pointed out something I should have immediately realized. RandomReal generate a new random value for each comparison made. And it is this characteristic that causes the distribution to differ from uniform. A simple demonstration that this is the case is to look at the length of the lists returned that start with 0.1. That is: In[12]:= Union[ Length /@ Cases[Table[Select[bins, (# >= RandomReal[] &)], {1000}], {0.1, __}]] Out[12]= {3,4,5,6,7,8,9} If there were only one random value selected whenever the selection was done, clearly the length of the lists with a given starting value would be constant. The idea of using Select to create the distribution can be made to work as follows: With[a = RandomReal[], Select[bins, (# >= a) &]][[1]] Repeating the demonstration above using this code yields: In[13]:= Union[ Length /@ Cases[Table[ With[{a = RandomReal[]}, Select[bins, (# >= a) &]], {1000}], {0.1, __}]] Out[13]= {10} showing every list returned that starts with the value 0.1 contains all ten values. But while this corrects the issue, this code will execute slower than code using RandomChoice will.
|
Pages: 1 Prev: RandomReal[] and sampling a distribution Next: Sum of terms --> list |