
Re: problem on recordbreaking values in probability
Posted:
Mar 11, 2013 2:07 AM


On Sun, 10 Mar 2013 21:47:27 0400, David Bernier wrote: > On 03/10/2013 08:52 PM, David Bernier wrote: >> On 03/01/2013 08:41 AM, David Bernier wrote: >>> On 02/27/2013 10:24 PM, David Bernier wrote: >>>> On 02/27/2013 04:05 PM, James Waldby wrote: >>>>> On Wed, 27 Feb 2013 07:10:08 0500, David Bernier wrote: >>>>>> On 02/27/2013 05:49 AM, David Bernier wrote: >>>>>>> On 02/27/2013 05:31 AM, David Bernier wrote: >>>>>>>> I used Marsaglia's 64bit SUPER KISS pseudorandom number generator >>>>>>>> to simulate uniform r.v.s on [0, 1] that are independent, as >>>>>>>> X_1, X_2, X_3, ad infinitum >>>>>>>> >>>>>>>> For each go, (or sequence) I define its 1st recordbreaking value >>>>>>>> as R(1) as X_1, its 2nd recordbreaking value R(2) as the >>>>>>>> value taken by X_n for the smallest n with X_n > X_1, and in general >>>>> [ R(k+1) is the value taken by X_n for the smallest n with X_n > R(k)] >>>>> ... [snip] >>>>> [etc] >>>>> >>>>> It would be useful to report the number of trials each simulation >>>>> took to find its 20th RBV. If a simulation takes m trials, the [snip] >>>>> In following, let L(n) = Pr(n'th item of n is lowest). (Distribution >>>>> of the lowest item should be similar to distribution of 1(highest >>>>> item).) I suppose that L(n) = 1/n and that the expected value of the >>>>> number of recordlowvalues (RLV's) in m trials is sum{i=1 to m}(1/i), >>>>> or about H_m, the m'th harmonic number, which can be approximated by >>>>> log(m) + gamma, with gamma = EulerMascheroni constant, about 0.5772. [snip] >>> In the literature, a remarkable article, which may have >>> appeared in the Am. Math. Monthly, can be found by >>> searching for: >>> Breaking Records and Breaking Boards. Ned Glick ... >> I did long simulations for 12th RecordBreaking Values. >> >> With MatLab, I constructed a histogram of the natural >> logarithms of the 76,000 values: >> >> < http://img521.imageshack.us/img521/7702/records12log.jpg > . ... > S_12 is number of trials (steps) taken to find the 12th > RecordBreaking Value. On Average, log(S_12) is close > to 12  gamma (gamma is the EulerMascheroni constant). > > A number of 76,000 sequences were generated, each being > continued until the 12th RecordBreaking Value for > that sequence was found. There is such variance from > one sample S_12 to another that I prefer the > quantities log(S_12) , for the histograms. > > Occasionally, an unusually high record is attained > in the 1st, 2nd, ... or 11th RecordBreaking Value. > That makes breaking the record all the more difficult. > In the simulations, the computer would pass (say) three > hours or more on the same sequence, with no new output > to the file for three or more hours.
You may already have done so, but if not and if you are going to run more simulations, consider (a) profiling the code, and (b) trying different compilation options. (a) allows you to find out which lines of code use most of the hours of CPU time, so you can try alternate ways of coding them. Under (b), starting with the same random seed in each case, try optimization options O1, O2, O3, Ofast, timing the execution and also verifying the same results. From my interpretation of URL below, those 4 optimization options are all you need to try. <http://gcc.gnu.org/onlinedocs/gcc/OptimizeOptions.html>
 jiw

