Results from Diehard tests on a binary file with 230=1073741824 Bytes generated by csprngAlvo Algorithm. x0=0; b0=1 and c=0.5

 

              NOTE: Most of the tests in DIEHARD return a p-value, which              

       should be uniform on [0,1) if the input file contains truly             

       independent random bits.   Those p-values are obtained by               

       p=F(X), where F is the assumed distribution of the sample               

       random variable X---often normal. But that assumed F is just            

       an asymptotic approximation, for which the fit will be worst            

       in the tails. Thus you should not be surprised with                     

       occasional p-values near 0 or 1, such as .0012 or .9983.                 

       When a bit stream really FAILS BIG, you will get p's of 0 or            

       1 to six or more places.  By all means, do not, as a                    

       Statistician might, think that a p < .025 or p> .975 means               

       that the RNG has "failed the test at the .05 level".  Such              

       p's happen among the hundreds that DIEHARD produces, even               

       with good RNG's.  So keep in mind that " p happens".                    

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::            This is the BIRTHDAY SPACINGS TEST                 ::       

     :: Choose m birthdays in a year of n days.  List the spacings    ::       

     :: between the birthdays.  If j is the number of values that     ::       

     :: occur more than once in that list, then j is asymptotically   ::       

     :: Poisson distributed with mean m^3/(4n).  Experience shows n   ::       

     :: must be quite large, say n>=2^18, for comparing the results   ::       

     :: to the Poisson distribution with that mean.  This test uses   ::       

     :: n=2^24 and m=2^9,  so that the underlying distribution for j  ::       

     :: is taken to be Poisson with lambda=2^27/(2^26)=2.  A sample   ::       

     :: of 500 j's is taken, and a chi-square goodness of fit test    ::       

     :: provides a p value.  The first test uses bits 1-24 (counting  ::       

     :: from the left) from integers in the specified file.           ::       

     ::   Then the file is closed and reopened. Next, bits 2-25 are   ::       

     :: used to provide birthdays, then 3-26 and so on to bits 9-32.  ::       

     :: Each set of bits provides a p-value, and the nine p-values    ::        

     :: provide a sample for a KSTEST.                                ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

 BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA=  2.0000

           Results for rnd.txt        

                   For a sample of size 500:     mean  

           rnd.txt         using bits  1 to 24   1.932

  duplicate       number       number

  spacings       observed     expected

        0          79.       67.668

        1         123.      135.335

        2         143.      135.335

        3          91.       90.224

        4          46.       45.112

        5          11.       18.045

  6 to INF          7.        8.282

 Chisquare with  6 d.o.f. =     6.43 p-value=  .623122

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  2 to 25   2.036

  duplicate       number       number

  spacings       observed     expected

        0          64.       67.668

        1         128.      135.335

        2         141.      135.335

        3          97.       90.224

        4          44.       45.112

        5          18.       18.045

  6 to INF          8.        8.282

 Chisquare with  6 d.o.f. =     1.38 p-value=  .032909

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  3 to 26   1.942

  duplicate       number       number

  spacings       observed     expected

        0          68.       67.668

        1         140.      135.335

        2         147.      135.335

        3          83.       90.224

        4          34.       45.112

        5          20.       18.045

  6 to INF          8.        8.282

 Chisquare with  6 d.o.f. =     4.70 p-value=  .417783

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  4 to 27   1.918

  duplicate       number       number

  spacings       observed     expected

        0          82.       67.668

        1         129.      135.335

        2         135.      135.335

        3          91.       90.224

        4          38.       45.112

        5          19.       18.045

  6 to INF          6.        8.282

 Chisquare with  6 d.o.f. =     5.14 p-value=  .474033

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  5 to 28   1.968

  duplicate       number       number

  spacings       observed     expected

        0          60.       67.668

        1         145.      135.335

        2         141.      135.335

        3          90.       90.224

        4          42.       45.112

        5          16.       18.045

  6 to INF          6.        8.282

 Chisquare with  6 d.o.f. =     2.87 p-value=  .175233

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  6 to 29   1.912

  duplicate       number       number

  spacings       observed     expected

        0          57.       67.668

        1         155.      135.335

        2         146.      135.335

        3          86.       90.224

        4          36.       45.112

        5          15.       18.045

  6 to INF          5.        8.282

 Chisquare with  6 d.o.f. =     9.23 p-value=  .839046

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  7 to 30   2.000

  duplicate       number       number

  spacings       observed     expected

        0          76.       67.668

        1         125.      135.335

        2         129.      135.335

        3          99.       90.224

        4          44.       45.112

        5          21.       18.045

  6 to INF          6.        8.282

 Chisquare with  6 d.o.f. =     4.11 p-value=  .337624

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  8 to 31   2.066

  duplicate       number       number

  spacings       observed     expected

        0          63.       67.668

        1         121.      135.335

        2         145.      135.335

        3          98.       90.224

        4          45.       45.112

        5          21.       18.045

  6 to INF          7.        8.282

 Chisquare with  6 d.o.f. =     3.88 p-value=  .307568

  :::::::::::::::::::::::::::::::::::::::::

                   For a sample of size 500:     mean  

           rnd.txt         using bits  9 to 32   1.966

  duplicate       number       number

  spacings       observed     expected

        0          72.       67.668

        1         125.      135.335

        2         147.      135.335

        3          92.       90.224

        4          43.       45.112

        5          13.       18.045

  6 to INF          8.        8.282

 Chisquare with  6 d.o.f. =     3.63 p-value=  .272837

  :::::::::::::::::::::::::::::::::::::::::

   The 9 p-values were

        .623122   .032909   .417783   .474033   .175233

        .839046   .337624   .307568   .272837

  A KSTEST for the 9 p-values yields  .621526

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::            THE OVERLAPPING 5-PERMUTATION TEST                 ::       

     :: This is the OPERM5 test.  It looks at a sequence of one mill- ::       

     :: ion 32-bit random integers.  Each set of five consecutive     ::       

     :: integers can be in one of 120 states, for the 5! possible or- ::       

     :: derings of five numbers.  Thus the 5th, 6th, 7th,...numbers   ::       

     :: each provide a state. As many thousands of state transitions  ::       

     :: are observed,  cumulative counts are made of the number of    ::       

     :: occurences of each state.  Then the quadratic form in the     ::       

     :: weak inverse of the 120x120 covariance matrix yields a test   ::       

     :: equivalent to the likelihood ratio test that the 120 cell     ::       

     :: counts came from the specified (asymptotically) normal dis-   ::       

     :: tribution with the specified 120x120 covariance matrix (with  ::        

     :: rank 99).  This version uses 1,000,000 integers, twice.       ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

           OPERM5 test for file rnd.txt       

     For a sample of 1,000,000 consecutive 5-tuples,

 chisquare for 99 degrees of freedom=101.323; p-value= .583625

           OPERM5 test for file rnd.txt       

     For a sample of 1,000,000 consecutive 5-tuples,

 chisquare for 99 degrees of freedom= 99.349; p-value= .528754

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     :: This is the BINARY RANK TEST for 31x31 matrices. The leftmost ::       

     :: 31 bits of 31 random integers from the test sequence are used ::       

     :: to form a 31x31 binary matrix over the field {0,1}. The rank  ::       

     :: is determined. That rank can be from 0 to 31, but ranks< 28   ::       

     :: are rare, and their counts are pooled with those for rank 28. ::       

     :: Ranks are found for 40,000 such random matrices and a chisqua-::       

     :: re test is performed on counts for ranks 31,30,29 and <=28.   ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

    Binary rank test for rnd.txt       

         Rank test for 31x31 binary matrices:

        rows from leftmost 31 bits of each 32-bit integer

      rank   observed  expected (o-e)^2/e  sum

        28       219     211.4   .271909     .272

        29      5103    5134.0   .187307     .459

        30     23277   23103.0  1.309770    1.769

        31     11401   11551.5  1.961438    3.730

  chisquare= 3.730 for 3 d. of f.; p-value= .734632

--------------------------------------------------------------

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     :: This is the BINARY RANK TEST for 32x32 matrices. A random 32x ::       

     :: 32 binary matrix is formed, each row a 32-bit random integer. ::       

     :: The rank is determined. That rank can be from 0 to 32, ranks  ::       

     :: less than 29 are rare, and their counts are pooled with those ::       

     :: for rank 29.  Ranks are found for 40,000 such random matrices ::       

     :: and a chisquare test is performed on counts for ranks  32,31, ::       

     :: 30 and <=29.                                                  ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

    Binary rank test for rnd.txt       

         Rank test for 32x32 binary matrices:

        rows from leftmost 32 bits of each 32-bit integer

      rank   observed  expected (o-e)^2/e  sum

        29       186     211.4  3.055915    3.056

        30      5207    5134.0  1.037688    4.094

        31     23108   23103.0   .001062    4.095

        32     11499   11551.5   .238827    4.333

  chisquare= 4.333 for 3 d. of f.; p-value= .791053

--------------------------------------------------------------

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     :: This is the BINARY RANK TEST for 6x8 matrices.  From each of  ::       

     :: six random 32-bit integers from the generator under test, a   ::       

     :: specified byte is chosen, and the resulting six bytes form a  ::       

     :: 6x8 binary matrix whose rank is determined.  That rank can be ::       

     :: from 0 to 6, but ranks 0,1,2,3 are rare; their counts are     ::       

     :: pooled with those for rank 4. Ranks are found for 100,000     ::       

     :: random matrices, and a chi-square test is performed on        ::       

     :: counts for ranks 6,5 and <=4.                                 ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

         Binary Rank Test for rnd.txt       

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  1 to  8

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          903       944.3       1.806       1.806

          r =5        21794     21743.9        .115       1.922

          r =6        77303     77311.8        .001       1.923

                        p=1-exp(-SUM/2)= .61765

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  2 to  9

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          954       944.3        .100        .100

          r =5        21707     21743.9        .063        .162

          r =6        77339     77311.8        .010        .172

                        p=1-exp(-SUM/2)= .08231

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  3 to 10

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          958       944.3        .199        .199

          r =5        21914     21743.9       1.331       1.529

          r =6        77128     77311.8        .437       1.966

                        p=1-exp(-SUM/2)= .62588

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  4 to 11

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          918       944.3        .733        .733

          r =5        21831     21743.9        .349       1.081

          r =6        77251     77311.8        .048       1.129

                        p=1-exp(-SUM/2)= .43143

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  5 to 12

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          949       944.3        .023        .023

          r =5        21696     21743.9        .106        .129

          r =6        77355     77311.8        .024        .153

                        p=1-exp(-SUM/2)= .07366

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  6 to 13

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4         1006       944.3       4.031       4.031

          r =5        21847     21743.9        .489       4.520

          r =6        77147     77311.8        .351       4.871

                        p=1-exp(-SUM/2)= .91246

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  7 to 14

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          995       944.3       2.722       2.722

          r =5        21839     21743.9        .416       3.138

          r =6        77166     77311.8        .275       3.413

                        p=1-exp(-SUM/2)= .81849

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  8 to 15

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          945       944.3        .001        .001

          r =5        21658     21743.9        .339        .340

          r =6        77397     77311.8        .094        .434

                        p=1-exp(-SUM/2)= .19497

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits  9 to 16

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          962       944.3        .332        .332

          r =5        21668     21743.9        .265        .597

          r =6        77370     77311.8        .044        .640

                        p=1-exp(-SUM/2)= .27402

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 10 to 17

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          979       944.3       1.275       1.275

          r =5        21742     21743.9        .000       1.275

          r =6        77279     77311.8        .014       1.289

                        p=1-exp(-SUM/2)= .47510

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 11 to 18

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          983       944.3       1.586       1.586

          r =5        21622     21743.9        .683       2.269

          r =6        77395     77311.8        .090       2.359

                        p=1-exp(-SUM/2)= .69254

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 12 to 19

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          935       944.3        .092        .092

          r =5        21698     21743.9        .097        .189

          r =6        77367     77311.8        .039        .228

                        p=1-exp(-SUM/2)= .10770

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 13 to 20

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          972       944.3        .812        .812

          r =5        21673     21743.9        .231       1.044

          r =6        77355     77311.8        .024       1.068

                        p=1-exp(-SUM/2)= .41368

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 14 to 21

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          973       944.3        .872        .872

          r =5        21661     21743.9        .316       1.188

          r =6        77366     77311.8        .038       1.226

                        p=1-exp(-SUM/2)= .45835

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 15 to 22

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          906       944.3       1.554       1.554

          r =5        21671     21743.9        .244       1.798

          r =6        77423     77311.8        .160       1.958

                        p=1-exp(-SUM/2)= .62429

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 16 to 23

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          975       944.3        .998        .998

          r =5        21849     21743.9        .508       1.506

          r =6        77176     77311.8        .239       1.745

                        p=1-exp(-SUM/2)= .58200

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 17 to 24

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          979       944.3       1.275       1.275

          r =5        21937     21743.9       1.715       2.990

          r =6        77084     77311.8        .671       3.661

                        p=1-exp(-SUM/2)= .83968

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 18 to 25

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          966       944.3        .499        .499

          r =5        21827     21743.9        .318        .816

          r =6        77207     77311.8        .142        .958

                        p=1-exp(-SUM/2)= .38068

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 19 to 26

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          938       944.3        .042        .042

          r =5        21870     21743.9        .731        .773

          r =6        77192     77311.8        .186        .959

                        p=1-exp(-SUM/2)= .38090

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 20 to 27

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          980       944.3       1.350       1.350

          r =5        21776     21743.9        .047       1.397

          r =6        77244     77311.8        .059       1.456

                        p=1-exp(-SUM/2)= .51723

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 21 to 28

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          956       944.3        .145        .145

          r =5        21985     21743.9       2.673       2.818

          r =6        77059     77311.8        .827       3.645

                        p=1-exp(-SUM/2)= .83837

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 22 to 29

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          898       944.3       2.270       2.270

          r =5        21825     21743.9        .302       2.573

          r =6        77277     77311.8        .016       2.588

                        p=1-exp(-SUM/2)= .72588

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt        

     b-rank test for bits 23 to 30

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          931       944.3        .187        .187

          r =5        21856     21743.9        .578        .765

          r =6        77213     77311.8        .126        .892

                        p=1-exp(-SUM/2)= .35967

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 24 to 31

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          955       944.3        .121        .121

          r =5        21737     21743.9        .002        .123

          r =6        77308     77311.8        .000        .124

                        p=1-exp(-SUM/2)= .05993

        Rank of a 6x8 binary matrix,

     rows formed from eight bits of the RNG rnd.txt       

     b-rank test for bits 25 to 32

                     OBSERVED   EXPECTED     (O-E)^2/E      SUM

          r<=4          917       944.3        .789        .789

          r =5        21726     21743.9        .015        .804

          r =6        77357     77311.8        .026        .830

                        p=1-exp(-SUM/2)= .33982

   TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices

 These should be 25 uniform [0,1] random variables:

     .617652     .082315     .625884     .431434     .073664

     .912465     .818489     .194971     .274021     .475102

     .692544     .107705     .413684     .458345     .624287

     .582001     .839675     .380680     .380905     .517228

     .838374     .725884     .359674     .059925     .339818

   brank test summary for rnd.txt       

       The KS test for those 25 supposed UNI's yields

                    KS p-value= .359077

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::                   THE BITSTREAM TEST                          ::       

     :: The file under test is viewed as a stream of bits. Call them  ::       

     :: b1,b2,... .  Consider an alphabet with two "letters", 0 and 1 ::       

     :: and think of the stream of bits as a succession of 20-letter  ::       

     :: "words", overlapping.  Thus the first word is b1b2...b20, the ::        

     :: second is b2b3...b21, and so on.  The bitstream test counts   ::       

     :: the number of missing 20-letter (20-bit) words in a string of ::       

     :: 2^21 overlapping 20-letter words.  There are 2^20 possible 20 ::       

     :: letter words.  For a truly random string of 2^21+19 bits, the ::       

     :: number of missing words j should be (very close to) normally  ::       

     :: distributed with mean 141,909 and sigma 428.  Thus            ::       

     ::  (j-141909)/428 should be a standard normal variate (z score) ::       

     :: that leads to a uniform [0,1) p value.  The test is repeated  ::       

     :: twenty times.                                                 ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

 THE OVERLAPPING 20-tuples BITSTREAM  TEST, 20 BITS PER WORD, N words

   This test uses N=2^21 and samples the bitstream 20 times.

  No. missing words should average  141909. with sigma=428.

---------------------------------------------------------

 tst no  1:  141722 missing words,    -.44 sigmas from mean, p-value= .33081

 tst no  2:  142151 missing words,     .56 sigmas from mean, p-value= .71385

 tst no  3:  142561 missing words,    1.52 sigmas from mean, p-value= .93607

 tst no  4:  141270 missing words,   -1.49 sigmas from mean, p-value= .06762

 tst no  5:  141841 missing words,    -.16 sigmas from mean, p-value= .43658

 tst no  6:  141976 missing words,     .16 sigmas from mean, p-value= .56189

 tst no  7:  140931 missing words,   -2.29 sigmas from mean, p-value= .01113

 tst no  8:  142613 missing words,    1.64 sigmas from mean, p-value= .94992

 tst no  9:  141312 missing words,   -1.40 sigmas from mean, p-value= .08141

 tst no 10:  142065 missing words,     .36 sigmas from mean, p-value= .64197

 tst no 11:  142296 missing words,     .90 sigmas from mean, p-value= .81685

 tst no 12:  142221 missing words,     .73 sigmas from mean, p-value= .76676

 tst no 13:  141641 missing words,    -.63 sigmas from mean, p-value= .26535

 tst no 14:  141419 missing words,   -1.15 sigmas from mean, p-value= .12597

 tst no 15:  142296 missing words,     .90 sigmas from mean, p-value= .81685

 tst no 16:  141861 missing words,    -.11 sigmas from mean, p-value= .45505

 tst no 17:  141809 missing words,    -.23 sigmas from mean, p-value= .40733

 tst no 18:  142587 missing words,    1.58 sigmas from mean, p-value= .94333

 tst no 19:  142030 missing words,     .28 sigmas from mean, p-value= .61101

 tst no 20:  142020 missing words,     .26 sigmas from mean, p-value= .60202

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::             The tests OPSO, OQSO and DNA                      ::        

     ::         OPSO means Overlapping-Pairs-Sparse-Occupancy         ::       

     :: The OPSO test considers 2-letter words from an alphabet of    ::       

     :: 1024 letters.  Each letter is determined by a specified ten   ::       

     :: bits from a 32-bit integer in the sequence to be tested. OPSO ::       

     :: generates  2^21 (overlapping) 2-letter words  (from 2^21+1    ::       

     :: "keystrokes")  and counts the number of missing words---that  ::       

     :: is 2-letter words which do not appear in the entire sequence. ::       

     :: That count should be very close to normally distributed with  ::       

     :: mean 141,909, sigma 290. Thus (missingwrds-141909)/290 should ::       

     :: be a standard normal variable. The OPSO test takes 32 bits at ::       

     :: a time from the test file and uses a designated set of ten    ::       

     :: consecutive bits. It then restarts the file for the next de-  ::       

     :: signated 10 bits, and so on.                                  ::       

     ::                                                               ::       

     ::     OQSO means Overlapping-Quadruples-Sparse-Occupancy        ::       

     ::   The test OQSO is similar, except that it considers 4-letter ::       

     :: words from an alphabet of 32 letters, each letter determined  ::       

     :: by a designated string of 5 consecutive bits from the test    ::       

     :: file, elements of which are assumed 32-bit random integers.   ::        

     :: The mean number of missing words in a sequence of 2^21 four-  ::       

     :: letter words,  (2^21+3 "keystrokes"), is again 141909, with   ::       

     :: sigma = 295.  The mean is based on theory; sigma comes from   ::       

     :: extensive simulation.                                         ::       

     ::                                                               ::       

     ::    The DNA test considers an alphabet of 4 letters::  C,G,A,T,::       

     :: determined by two designated bits in the sequence of random   ::       

     :: integers being tested.  It considers 10-letter words, so that ::       

     :: as in OPSO and OQSO, there are 2^20 possible words, and the   ::       

     :: mean number of missing words from a string of 2^21  (over-    ::       

     :: lapping)  10-letter  words (2^21+9 "keystrokes") is 141909.   ::       

     :: The standard deviation sigma=339 was determined as for OQSO   ::       

     :: by simulation.  (Sigma for OPSO, 290, is the true value (to   ::       

     :: three places), not determined by simulation.                  ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

 OPSO test for generator rnd.txt       

  Output: No. missing words (mw), equiv normal variate (z), p-value (p)

                                                           mw     z     p

    OPSO for rnd.txt         using bits 23 to 32        142342  1.492  .9321

    OPSO for rnd.txt         using bits 22 to 31        142060   .520  .6983

    OPSO for rnd.txt         using bits 21 to 30        142083   .599  .7254

    OPSO for rnd.txt         using bits 20 to 29        141647  -.905  .1828

    OPSO for rnd.txt         using bits 19 to 28        141377 -1.836  .0332

    OPSO for rnd.txt         using bits 18 to 27        142398  1.685  .9540

    OPSO for rnd.txt         using bits 17 to 26        142085   .606  .7277

    OPSO for rnd.txt         using bits 16 to 25        141998   .306  .6201

    OPSO for rnd.txt         using bits 15 to 24        142310  1.382  .9165

    OPSO for rnd.txt         using bits 14 to 23        142345  1.502  .9335

    OPSO for rnd.txt         using bits 13 to 22        141709  -.691  .2448

    OPSO for rnd.txt         using bits 12 to 21        141750  -.549  .2914

    OPSO for rnd.txt         using bits 11 to 20        142353  1.530  .9370

    OPSO for rnd.txt         using bits 10 to 19        142036   .437  .6689

    OPSO for rnd.txt         using bits  9 to 18        141792  -.405  .3429

    OPSO for rnd.txt         using bits  8 to 17        141979   .240  .5949

    OPSO for rnd.txt         using bits  7 to 16        141900  -.032  .4872

    OPSO for rnd.txt         using bits  6 to 15        142011   .351  .6371

    OPSO for rnd.txt         using bits  5 to 14        141767  -.491  .3118

    OPSO for rnd.txt         using bits  4 to 13        142002   .320  .6253

    OPSO for rnd.txt         using bits  3 to 12        141236 -2.322  .0101

    OPSO for rnd.txt         using bits  2 to 11        141670  -.825  .2046

    OPSO for rnd.txt         using bits  1 to 10        141827  -.284  .3882

 OQSO test for generator rnd.txt       

  Output: No. missing words (mw), equiv normal variate (z), p-value (p)

                                                           mw     z     p

    OQSO for rnd.txt         using bits 28 to 32        142129   .745  .7718

    OQSO for rnd.txt         using bits 27 to 31        142235  1.104  .8652

    OQSO for rnd.txt         using bits 26 to 30        141740  -.574  .2830

    OQSO for rnd.txt         using bits 25 to 29        142191   .955  .8302

    OQSO for rnd.txt         using bits 24 to 28        141051 -2.910  .0018

    OQSO for rnd.txt         using bits 23 to 27        142622  2.416  .9922

    OQSO for rnd.txt         using bits 22 to 26        142285  1.273  .8986

    OQSO for rnd.txt         using bits 21 to 25        141995   .290  .6142

    OQSO for rnd.txt         using bits 20 to 24        142070   .545  .7070

    OQSO for rnd.txt         using bits 19 to 23        141844  -.221  .4124

    OQSO for rnd.txt         using bits 18 to 22        141796  -.384  .3504

    OQSO for rnd.txt         using bits 17 to 21        141469 -1.493  .0678

    OQSO for rnd.txt         using bits 16 to 20        141810  -.337  .3682

    OQSO for rnd.txt         using bits 15 to 19        141633  -.937  .1745

    OQSO for rnd.txt         using bits 14 to 18        141940   .104  .5414

    OQSO for rnd.txt         using bits 13 to 17        141705  -.693  .2443

    OQSO for rnd.txt         using bits 12 to 16        141700  -.710  .2390

    OQSO for rnd.txt         using bits 11 to 15        141721  -.638  .2616

    OQSO for rnd.txt         using bits 10 to 14        142076   .565  .7140

    OQSO for rnd.txt         using bits  9 to 13        141715  -.659  .2550

    OQSO for rnd.txt         using bits  8 to 12        142142   .789  .7849

    OQSO for rnd.txt         using bits  7 to 11        142262  1.195  .8841

    OQSO for rnd.txt         using bits  6 to 10        141984   .253  .5999

    OQSO for rnd.txt         using bits  5 to  9        142331  1.429  .9236

    OQSO for rnd.txt         using bits  4 to  8        141596 -1.062  .1441

    OQSO for rnd.txt         using bits  3 to  7        141548 -1.225  .1103

    OQSO for rnd.txt         using bits  2 to  6        142226  1.073  .8585

    OQSO for rnd.txt         using bits  1 to  5        141847  -.211  .4163

  DNA test for generator rnd.txt       

  Output: No. missing words (mw), equiv normal variate (z), p-value (p)

                                                           mw     z     p

     DNA for rnd.txt         using bits 31 to 32        142103   .571  .7161

     DNA for rnd.txt         using bits 30 to 31        142706  2.350  .9906

     DNA for rnd.txt         using bits 29 to 30        141566 -1.013  .1556

     DNA for rnd.txt         using bits 28 to 29        141827  -.243  .4041

     DNA for rnd.txt         using bits 27 to 28        141897  -.036  .4855

     DNA for rnd.txt         using bits 26 to 27        142208   .881  .8109

     DNA for rnd.txt         using bits 25 to 26        141913   .011  .5043

     DNA for rnd.txt         using bits 24 to 25        141773  -.402  .3438

     DNA for rnd.txt         using bits 23 to 24        141607  -.892  .1862

     DNA for rnd.txt         using bits 22 to 23        142140   .680  .7519

     DNA for rnd.txt         using bits 21 to 22        141634  -.812  .2083

     DNA for rnd.txt         using bits 20 to 21        141791  -.349  .3635

     DNA for rnd.txt         using bits 19 to 20        141827  -.243  .4041

     DNA for rnd.txt         using bits 18 to 19        142064   .456  .6759

     DNA for rnd.txt         using bits 17 to 18        141359 -1.623  .0523

     DNA for rnd.txt         using bits 16 to 17        142101   .565  .7141

     DNA for rnd.txt         using bits 15 to 16        141268 -1.892  .0293

     DNA for rnd.txt         using bits 14 to 15        142469  1.651  .9506

     DNA for rnd.txt         using bits 13 to 14        141710  -.588  .2783

     DNA for rnd.txt         using bits 12 to 13        141925   .046  .5184

     DNA for rnd.txt         using bits 11 to 12        141800  -.323  .3735

     DNA for rnd.txt         using bits 10 to 11        141841  -.202  .4201

     DNA for rnd.txt         using bits  9 to 10        141833  -.225  .4109

     DNA for rnd.txt         using bits  8 to  9        141485 -1.252  .1053

     DNA for rnd.txt         using bits  7 to  8        142971  3.132  .9991

     DNA for rnd.txt         using bits  6 to  7        142121   .624  .7338

     DNA for rnd.txt         using bits  5 to  6        142223   .925  .8226

     DNA for rnd.txt         using bits  4 to  5        141473 -1.287  .0990

     DNA for rnd.txt         using bits  3 to  4        141676  -.688  .2456

     DNA for rnd.txt         using bits  2 to  3        141902  -.022  .4914

     DNA for rnd.txt         using bits  1 to  2        141919   .029  .5114

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::     This is the COUNT-THE-1's TEST on a stream of bytes.      ::       

     :: Consider the file under test as a stream of bytes (four per   ::       

     :: 32 bit integer).  Each byte can contain from 0 to 8 1's,      ::       

     :: with probabilities 1,8,28,56,70,56,28,8,1 over 256.  Now let  ::       

     :: the stream of bytes provide a string of overlapping  5-letter ::       

     :: words, each "letter" taking values A,B,C,D,E. The letters are ::       

     :: determined by the number of 1's in a byte::  0,1,or 2 yield A,::       

     :: 3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus ::       

     :: we have a monkey at a typewriter hitting five keys with vari- ::       

     :: ous probabilities (37,56,70,56,37 over 256).  There are 5^5   ::       

     :: possible 5-letter words, and from a string of 256,000 (over-  ::       

     :: lapping) 5-letter words, counts are made on the frequencies   ::       

     :: for each word.   The quadratic form in the weak inverse of    ::       

     :: the covariance matrix of the cell counts provides a chisquare ::       

     :: test::  Q5-Q4, the difference of the naive Pearson sums of    ::       

     :: (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts.    ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

   Test results for rnd.txt       

 Chi-square with 5^5-5^4=2500 d.of f. for sample size:2560000

                               chisquare  equiv normal  p-value

  Results fo COUNT-THE-1's in successive bytes:

 byte stream for rnd.txt          2510.60       .150      .559565

 byte stream for rnd.txt          2423.96     -1.075      .141105

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::     This is the COUNT-THE-1's TEST for specific bytes.        ::       

     :: Consider the file under test as a stream of 32-bit integers.  ::        

     :: From each integer, a specific byte is chosen , say the left-  ::       

     :: most::  bits 1 to 8. Each byte can contain from 0 to 8 1's,   ::       

     :: with probabilitie 1,8,28,56,70,56,28,8,1 over 256.  Now let   ::       

     :: the specified bytes from successive integers provide a string ::       

     :: of (overlapping) 5-letter words, each "letter" taking values  ::       

     :: A,B,C,D,E. The letters are determined  by the number of 1's,  ::       

     :: in that byte::  0,1,or 2 ---> A, 3 ---> B, 4 ---> C, 5 ---> D,::       

     :: and  6,7 or 8 ---> E.  Thus we have a monkey at a typewriter  ::       

     :: hitting five keys with with various probabilities::  37,56,70,::       

     :: 56,37 over 256. There are 5^5 possible 5-letter words, and    ::       

     :: from a string of 256,000 (overlapping) 5-letter words, counts ::       

     :: are made on the frequencies for each word. The quadratic form ::       

     :: in the weak inverse of the covariance matrix of the cell      ::       

     :: counts provides a chisquare test::  Q5-Q4, the difference of  ::       

     :: the naive Pearson  sums of (OBS-EXP)^2/EXP on counts for 5-   ::       

     :: and 4-letter cell counts.                                     ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

 Chi-square with 5^5-5^4=2500 d.of f. for sample size: 256000

                      chisquare  equiv normal  p value

  Results for COUNT-THE-1's in specified bytes:

           bits  1 to  8  2592.28      1.305      .904063

           bits  2 to  9  2385.97     -1.613      .053410

           bits  3 to 10  2381.46     -1.676      .046836

           bits  4 to 11  2513.47       .191      .575555

           bits  5 to 12  2469.02      -.438      .330668

           bits  6 to 13  2490.08      -.140      .444230

           bits  7 to 14  2613.26      1.602      .945392

           bits  8 to 15  2443.88      -.794      .213704

           bits  9 to 16  2547.57       .673      .749460

           bits 10 to 17  2425.99     -1.047      .147641

           bits 11 to 18  2517.10       .242      .595527

           bits 12 to 19  2524.45       .346      .635249

           bits 13 to 20  2444.18      -.789      .214944

           bits 14 to 21  2577.33      1.094      .862942

           bits 15 to 22  2491.84      -.115      .454045

           bits 16 to 23  2475.96      -.340      .366926

           bits 17 to 24  2509.01       .127      .550699

           bits 18 to 25  2528.35       .401      .655744

           bits 19 to 26  2466.35      -.476      .317091

           bits 20 to 27  2465.30      -.491      .311794

           bits 21 to 28  2552.47       .742      .770969

           bits 22 to 29  2457.93      -.595      .275916

           bits 23 to 30  2510.86       .154      .561045

           bits 24 to 31  2529.91       .423      .663826

           bits 25 to 32  2544.70       .632      .736354

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::               THIS IS A PARKING LOT TEST                      ::       

     :: In a square of side 100, randomly "park" a car---a circle of  ::       

     :: radius 1.   Then try to park a 2nd, a 3rd, and so on, each    ::       

     :: time parking "by ear".  That is, if an attempt to park a car  ::       

     :: causes a crash with one already parked, try again at a new    ::       

     :: random location. (To avoid path problems, consider parking    ::       

     :: helicopters rather than cars.)   Each attempt leads to either ::       

     :: a crash or a success, the latter followed by an increment to  ::       

     :: the list of cars already parked. If we plot n:  the number of ::       

     :: attempts, versus k::  the number successfully parked, we get a::       

     :: curve that should be similar to those provided by a perfect   ::       

     :: random number generator.  Theory for the behavior of such a   ::       

     :: random curve seems beyond reach, and as graphics displays are ::       

     :: not available for this battery of tests, a simple characteriz ::       

     :: ation of the random experiment is used: k, the number of cars ::       

     :: successfully parked after n=12,000 attempts. Simulation shows ::       

     :: that k should average 3523 with sigma 21.9 and is very close  ::       

     :: to normally distributed.  Thus (k-3523)/21.9 should be a st-  ::       

     :: andard normal variable, which, converted to a uniform varia-  ::       

     :: ble, provides input to a KSTEST based on a sample of 10.      ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

           CDPARK: result of ten tests on file rnd.txt       

            Of 12,000 tries, the average no. of successes

                 should be 3523 with sigma=21.9

            Successes: 3519    z-score:  -.183 p-value: .427537

            Successes: 3506    z-score:  -.776 p-value: .218799

            Successes: 3534    z-score:   .502 p-value: .692266

            Successes: 3535    z-score:   .548 p-value: .708135

            Successes: 3556    z-score:  1.507 p-value: .934075

            Successes: 3497    z-score: -1.187 p-value: .117571

            Successes: 3536    z-score:   .594 p-value: .723613

            Successes: 3538    z-score:   .685 p-value: .753306

            Successes: 3548    z-score:  1.142 p-value: .873180

            Successes: 3518    z-score:  -.228 p-value: .409702

 

           square size   avg. no.  parked   sample sigma

             100.            3528.700       17.467

            KSTEST for the above 10: p=  .438300

 

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

 

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::               THE MINIMUM DISTANCE TEST                       ::       

     :: It does this 100 times::   choose n=8000 random points in a   ::       

     :: square of side 10000.  Find d, the minimum distance between   ::       

     :: the (n^2-n)/2 pairs of points.  If the points are truly inde- ::       

     :: pendent uniform, then d^2, the square of the minimum distance ::       

     :: should be (very close to) exponentially distributed with mean ::       

     :: .995 .  Thus 1-exp(-d^2/.995) should be uniform on [0,1) and  ::       

     :: a KSTEST on the resulting 100 values serves as a test of uni- ::       

     :: formity for random points in the square. Test numbers=0 mod 5 ::       

     :: are printed but the KSTEST is based on the full set of 100    ::       

     :: random choices of 8000 points in the 10000x10000 square.      ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

               This is the MINIMUM DISTANCE test

              for random integers in the file rnd.txt       

     Sample no.    d^2     avg     equiv uni           

           5     .0874    .5851     .084103

          10     .0902    .9722     .086626

          15     .3670   1.2155     .308472

          20     .5384   1.1275     .417923

          25     .9195   1.0329     .603102

          30     .8928    .9910     .592332

          35     .0259   1.0519     .025697

          40    4.3788   1.1148     .987732

          45     .5976   1.0936     .451527

          50     .1089   1.0156     .103663

          55    1.3435   1.0464     .740822

          60     .8943    .9859     .592945

          65     .0116   1.0144     .011612

          70     .4537   1.0299     .366150

          75     .2247    .9702     .202182

          80     .1575    .9635     .146420

          85     .2743    .9818     .240940

          90     .3900    .9534     .324299

          95     .3696    .9363     .310284

         100     .1236    .9498     .116854

     MINIMUM DISTANCE TEST for rnd.txt       

          Result of KS test on 20 transformed mindist^2's:

                                  p-value= .583890

 

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     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::              THE 3DSPHERES TEST                               ::       

     :: Choose  4000 random points in a cube of edge 1000.  At each   ::       

     :: point, center a sphere large enough to reach the next closest ::       

     :: point. Then the volume of the smallest such sphere is (very   ::       

     :: close to) exponentially distributed with mean 120pi/3.  Thus  ::       

     :: the radius cubed is exponential with mean 30. (The mean is    ::       

     :: obtained by extensive simulation).  The 3DSPHERES test gener- ::       

     :: ates 4000 such spheres 20 times.  Each min radius cubed leads ::       

     :: to a uniform variable by means of 1-exp(-r^3/30.), then a     ::       

     ::  KSTEST is done on the 20 p-values.                           ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

               The 3DSPHERES test for file rnd.txt       

 sample no:  1     r^3=  53.024     p-value= .82923

 sample no:  2     r^3=  57.427     p-value= .85255

 sample no:  3     r^3=  55.101     p-value= .84066

 sample no:  4     r^3=  13.891     p-value= .37062

 sample no:  5     r^3=  45.576     p-value= .78112

 sample no:  6     r^3=  46.709     p-value= .78922

 sample no:  7     r^3=  38.730     p-value= .72500

 sample no:  8     r^3=  31.979     p-value= .65560

 sample no:  9     r^3=  34.304     p-value= .68129

 sample no: 10     r^3=  24.730     p-value= .56147

 sample no: 11     r^3=  35.756     p-value= .69635

 sample no: 12     r^3=  27.478     p-value= .59986

 sample no: 13     r^3=  25.481     p-value= .57231

 sample no: 14     r^3=  21.136     p-value= .50567

 sample no: 15     r^3=  76.281     p-value= .92135

 sample no: 16     r^3=    .249     p-value= .00828

 sample no: 17     r^3=  48.421     p-value= .80092

 sample no: 18     r^3=    .699     p-value= .02302

 sample no: 19     r^3=  29.867     p-value= .63049

 sample no: 20     r^3=   9.605     p-value= .27398

  A KS test is applied to those 20 p-values.

---------------------------------------------------------

       3DSPHERES test for file rnd.txt              p-value= .937251

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     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

     ::      This is the SQEEZE test                                  ::       

     ::  Random integers are floated to get uniforms on [0,1). Start- ::       

     ::  ing with k=2^31=2147483647, the test finds j, the number of  ::       

     ::  iterations necessary to reduce k to 1, using the reduction   ::       

     ::  k=ceiling(k*U), with U provided by floating integers from    ::       

     ::  the file being tested.  Such j's are found 100,000 times,    ::       

     ::  then counts for the number of times j was <=6,7,...,47,>=48  ::       

     ::  are used to provide a chi-square test for cell frequencies.  ::       

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::       

            RESULTS OF SQUEEZE TEST FOR rnd.txt