Using CRC32('5a58d67fa172d0e6d21972b8caac5e19' + n) for n = 0 .. 1000000000000
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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=1-F(X), where F is the assumed distribution of the sample | |
random variable X---often normal. But that assumed F is often just | |
an asymptotic approximation, for which the fit will be worst | |
in the tails. Thus you should not be surprised with occasion- | |
al 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 RNGs. | |
So keep in mind that "p happens" | |
Enter the name of the file to be tested. | |
This must be a form="unformatted",access="direct" binary | |
file of about 10-12 million bytes. Enter file name: | |
HERE ARE YOUR CHOICES: | |
1 Birthday Spacings | |
2 Overlapping Permutations | |
3 Ranks of 31x31 and 32x32 matrices | |
4 Ranks of 6x8 Matrices | |
5 Monkey Tests on 20-bit Words | |
6 Monkey Tests OPSO,OQSO,DNA | |
7 Count the 1`s in a Stream of Bytes | |
8 Count the 1`s in Specific Bytes | |
9 Parking Lot Test | |
10 Minimum Distance Test | |
11 Random Spheres Test | |
12 The Sqeeze Test | |
13 Overlapping Sums Test | |
14 Runs Test | |
15 The Craps Test | |
16 All of the above | |
To choose any particular tests, enter corresponding numbers. | |
Enter 16 for all tests. If you want to perform all but a few | |
tests, enter corresponding numbers preceded by "-" sign. | |
Tests are executed in the order they are entered. | |
Enter your choices. | |
Segmentation fault | |
deebster@bernie:~/code/die.c$ ./diehard | |
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=1-F(X), where F is the assumed distribution of the sample | |
random variable X---often normal. But that assumed F is often just | |
an asymptotic approximation, for which the fit will be worst | |
in the tails. Thus you should not be surprised with occasion- | |
al 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 RNGs. | |
So keep in mind that "p happens" | |
Enter the name of the file to be tested. | |
This must be a form="unformatted",access="direct" binary | |
file of about 10-12 million bytes. Enter file name: | |
out.bin | |
HERE ARE YOUR CHOICES: | |
1 Birthday Spacings | |
2 Overlapping Permutations | |
3 Ranks of 31x31 and 32x32 matrices | |
4 Ranks of 6x8 Matrices | |
5 Monkey Tests on 20-bit Words | |
6 Monkey Tests OPSO,OQSO,DNA | |
7 Count the 1`s in a Stream of Bytes | |
8 Count the 1`s in Specific Bytes | |
9 Parking Lot Test | |
10 Minimum Distance Test | |
11 Random Spheres Test | |
12 The Sqeeze Test | |
13 Overlapping Sums Test | |
14 Runs Test | |
15 The Craps Test | |
16 All of the above | |
To choose any particular tests, enter corresponding numbers. | |
Enter 16 for all tests. If you want to perform all but a few | |
tests, enter corresponding numbers preceded by "-" sign. | |
Tests are executed in the order they are entered. | |
Enter your choices. | |
16 | |
|-------------------------------------------------------------| | |
| This is the BIRTHDAY SPACINGS TEST | | |
|Choose m birthdays in a "year" of n days. List the spacings | | |
|between the birthdays. Let j be 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=2^10, so that the underlying distribution for j | | |
|is taken to be Poisson with lambda=2^30/(2^26)=16. A sample | | |
|of 200 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, then bits 2-25 of the same inte-| | |
|gers are used to provide birthdays, 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. | | |
|------------------------------------------------------------ | | |
RESULTS OF BIRTHDAY SPACINGS TEST FOR out.bin | |
(no_bdays=1024, no_days/yr=2^24, lambda=16.00, sample size=500) | |
Bits used mean chisqr p-value | |
1 to 24 776.21 21905.9731 0.000000 | |
2 to 25 783.02 21905.9731 0.000000 | |
3 to 26 807.18 21905.9731 0.000000 | |
4 to 27 794.72 21905.9731 0.000000 | |
5 to 28 780.60 21905.9731 0.000000 | |
6 to 29 775.00 21905.9731 0.000000 | |
7 to 30 796.81 21905.9731 0.000000 | |
8 to 31 793.80 21905.9731 0.000000 | |
9 to 32 794.29 21905.9731 0.000000 | |
degree of freedoms is: 17 | |
--------------------------------------------------------------- | |
p-value for KStest on those 9 p-values: 0.000000 | |
|-------------------------------------------------------------| | |
| 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 | |
(For samples of 1,000,000 consecutive 5-tuples) | |
sample 1 | |
chisquare=1845650.758241 with df=99; p-value= 0.000000 | |
_______________________________________________________________ | |
sample 2 | |
chisquare=1850079.660671 with df=99; p-value= 0.000000 | |
_______________________________________________________________ | |
|-------------------------------------------------------------| | |
|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 chisqu-| | |
|are test is performed on counts for ranks 31,30,28 and <=28. | | |
|-------------------------------------------------------------| | |
Rank test for binary matrices (31x31) from out.bin | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=28 40000 211.4 7488157.097 7488157.097 | |
r=29 0 5134.0 5134.011 7493291.107 | |
r=30 0 23103.0 23103.048 7516394.155 | |
r=31 0 11551.5 11551.524 7527945.678 | |
chi-square = 7527945.678 with df = 3; p-value = 0.000 | |
-------------------------------------------------------------- | |
|-------------------------------------------------------------| | |
|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. | | |
|-------------------------------------------------------------| | |
Rank test for binary matrices (32x32) from out.bin | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=29 40000 211.4 7488157.097 7488157.097 | |
r=30 0 5134.0 5134.011 7493291.107 | |
r=31 0 23103.0 23103.048 7516394.155 | |
r=32 0 11551.5 11551.524 7527945.678 | |
chi-square = 7527945.678 with df = 3; p-value = 0.000 | |
-------------------------------------------------------------- | |
|-------------------------------------------------------------| | |
|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 (0,...,4) (pooled together). | | |
|-------------------------------------------------------------| | |
Rank test for binary matrices (6x8) from out.bin | |
bits 1 to 8 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6248 944.3 29788.450 29788.450 | |
r=5 93752 21743.9 238465.338 268253.788 | |
r=6 0 77311.8 77311.800 345565.588 | |
chi-square = 345565.588 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 2 to 9 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6249 944.3 29799.685 29799.685 | |
r=5 93751 21743.9 238458.715 268258.399 | |
r=6 0 77311.8 77311.800 345570.199 | |
chi-square = 345570.199 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 3 to 10 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6237 944.3 29665.015 29665.015 | |
r=5 93763 21743.9 238538.200 268203.215 | |
r=6 0 77311.8 77311.800 345515.015 | |
chi-square = 345515.015 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 4 to 11 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6304 944.3 30420.824 30420.824 | |
r=5 93696 21743.9 238094.578 268515.402 | |
r=6 0 77311.8 77311.800 345827.202 | |
chi-square = 345827.202 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 5 to 12 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 7785 944.3 49555.413 49555.413 | |
r=5 92215 21743.9 228393.983 277949.396 | |
r=6 0 77311.8 77311.800 355261.196 | |
chi-square = 355261.196 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 6 to 13 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 7766 944.3 49280.516 49280.516 | |
r=5 92234 21743.9 228517.156 277797.672 | |
r=6 0 77311.8 77311.800 355109.472 | |
chi-square = 355109.472 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 7 to 14 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6292 944.3 30284.756 30284.756 | |
r=5 93708 21743.9 238174.002 268458.759 | |
r=6 0 77311.8 77311.800 345770.559 | |
chi-square = 345770.559 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 8 to 15 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6242 944.3 29721.090 29721.090 | |
r=5 93758 21743.9 238505.080 268226.170 | |
r=6 0 77311.8 77311.800 345537.970 | |
chi-square = 345537.970 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 9 to 16 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6232 944.3 29608.992 29608.992 | |
r=5 93768 21743.9 238571.323 268180.315 | |
r=6 0 77311.8 77311.800 345492.115 | |
chi-square = 345492.115 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 10 to 17 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6243 944.3 29732.311 29732.311 | |
r=5 93757 21743.9 238498.456 268230.767 | |
r=6 0 77311.8 77311.800 345542.567 | |
chi-square = 345542.567 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 11 to 18 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6271 944.3 30047.371 30047.371 | |
r=5 93729 21743.9 238313.027 268360.398 | |
r=6 0 77311.8 77311.800 345672.198 | |
chi-square = 345672.198 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 12 to 19 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6301 944.3 30386.778 30386.778 | |
r=5 93699 21743.9 238114.433 268501.211 | |
r=6 0 77311.8 77311.800 345813.011 | |
chi-square = 345813.011 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 13 to 20 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6234 944.3 29631.395 29631.395 | |
r=5 93766 21743.9 238558.073 268189.468 | |
r=6 0 77311.8 77311.800 345501.268 | |
chi-square = 345501.268 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 14 to 21 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6252 944.3 29833.400 29833.400 | |
r=5 93748 21743.9 238438.846 268272.245 | |
r=6 0 77311.8 77311.800 345584.045 | |
chi-square = 345584.045 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 15 to 22 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6239 944.3 29687.438 29687.438 | |
r=5 93761 21743.9 238524.951 268212.390 | |
r=6 0 77311.8 77311.800 345524.190 | |
chi-square = 345524.190 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 16 to 23 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6259 944.3 29912.142 29912.142 | |
r=5 93741 21743.9 238392.487 268304.630 | |
r=6 0 77311.8 77311.800 345616.430 | |
chi-square = 345616.430 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 17 to 24 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6213 944.3 29396.590 29396.590 | |
r=5 93787 21743.9 238697.210 268093.799 | |
r=6 0 77311.8 77311.800 345405.599 | |
chi-square = 345405.599 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 18 to 25 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6228 944.3 29564.212 29564.212 | |
r=5 93772 21743.9 238597.822 268162.035 | |
r=6 0 77311.8 77311.800 345473.835 | |
chi-square = 345473.835 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 19 to 26 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6229 944.3 29575.404 29575.404 | |
r=5 93771 21743.9 238591.197 268166.601 | |
r=6 0 77311.8 77311.800 345478.401 | |
chi-square = 345478.401 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 20 to 27 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6225 944.3 29530.650 29530.650 | |
r=5 93775 21743.9 238617.698 268148.348 | |
r=6 0 77311.8 77311.800 345460.148 | |
chi-square = 345460.148 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 21 to 28 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 7756 944.3 49136.140 49136.140 | |
r=5 92244 21743.9 228581.998 277718.138 | |
r=6 0 77311.8 77311.800 355029.938 | |
chi-square = 355029.938 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 22 to 29 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 7741 944.3 48919.973 48919.973 | |
r=5 92259 21743.9 228679.277 277599.250 | |
r=6 0 77311.8 77311.800 354911.050 | |
chi-square = 354911.050 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 23 to 30 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6277 944.3 30115.100 30115.100 | |
r=5 93723 21743.9 238273.301 268388.402 | |
r=6 0 77311.8 77311.800 345700.202 | |
chi-square = 345700.202 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 24 to 31 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6234 944.3 29631.395 29631.395 | |
r=5 93766 21743.9 238558.073 268189.468 | |
r=6 0 77311.8 77311.800 345501.268 | |
chi-square = 345501.268 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
bits 25 to 32 | |
RANK OBSERVED EXPECTED (O-E)^2/E SUM | |
r<=4 6247 944.3 29777.218 29777.218 | |
r=5 93753 21743.9 238471.961 268249.180 | |
r=6 0 77311.8 77311.800 345560.980 | |
chi-square = 345560.980 with df = 2; p-value = 0.000 | |
-------------------------------------------------------------- | |
TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices | |
These should be 25 uniform [0,1] random variates: | |
0.000000 0.000000 0.000000 0.000000 0.000000 | |
0.000000 0.000000 0.000000 0.000000 0.000000 | |
0.000000 0.000000 0.000000 0.000000 0.000000 | |
0.000000 0.000000 0.000000 0.000000 0.000000 | |
0.000000 0.000000 0.000000 0.000000 0.000000 | |
The KS test for those 25 supposed UNI's yields | |
KS p-value = 0.000000 | |
|-------------------------------------------------------------| | |
| 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 for out.bin | |
(20 bits/word, 2097152 words 20 bitstreams. No. missing words | |
should average 141909.33 with sigma=428.00.) | |
---------------------------------------------------------------- | |
BITSTREAM test results for out.bin. | |
Bitstream No. missing words z-score p-value | |
1 257601 270.31 0.000000 | |
2 295921 359.84 0.000000 | |
3 224863 193.82 0.000000 | |
4 237996 224.50 0.000000 | |
5 300889 371.45 0.000000 | |
6 218786 179.62 0.000000 | |
7 218946 179.99 0.000000 | |
8 228877 203.20 0.000000 | |
9 211807 163.31 0.000000 | |
10 242312 234.59 0.000000 | |
11 223517 190.67 0.000000 | |
12 211558 162.73 0.000000 | |
13 254730 263.60 0.000000 | |
14 212331 164.54 0.000000 | |
15 214337 169.22 0.000000 | |
16 253500 260.73 0.000000 | |
17 278531 319.21 0.000000 | |
18 219422 181.10 0.000000 | |
19 218135 178.10 0.000000 | |
20 264951 287.48 0.000000 | |
---------------------------------------------------------------- | |
|-------------------------------------------------------------| | |
| 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. | | |
|------------------------------------------------------------ | | |
OPSO test for file out.bin | |
Bits used No. missing words z-score p-value | |
23 to 32 1034780 3078.8644 0.000000 | |
22 to 31 1034639 3078.3782 0.000000 | |
21 to 30 1034944 3079.4299 0.000000 | |
20 to 29 1034474 3077.8092 0.000000 | |
19 to 28 1034500 3077.8989 0.000000 | |
18 to 27 1036398 3084.4437 0.000000 | |
17 to 26 1034777 3078.8540 0.000000 | |
16 to 25 1034818 3078.9954 0.000000 | |
15 to 24 1034689 3078.5506 0.000000 | |
14 to 23 1034607 3078.2678 0.000000 | |
13 to 22 1034537 3078.0264 0.000000 | |
12 to 21 1034503 3077.9092 0.000000 | |
11 to 20 1034610 3078.2782 0.000000 | |
10 to 19 1034492 3077.8713 0.000000 | |
9 to 18 1034477 3077.8196 0.000000 | |
8 to 17 1034523 3077.9782 0.000000 | |
7 to 16 1034780 3078.8644 0.000000 | |
6 to 15 1034639 3078.3782 0.000000 | |
5 to 14 1034944 3079.4299 0.000000 | |
4 to 13 1034474 3077.8092 0.000000 | |
3 to 12 1034500 3077.8989 0.000000 | |
2 to 11 1036398 3084.4437 0.000000 | |
1 to 10 1034777 3078.8540 0.000000 | |
----------------------------------------------------------------- | |
|------------------------------------------------------------ | | |
| 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. | | |
|------------------------------------------------------------ | | |
OQSO test for file out.bin | |
Bits used No. missing words z-score p-value | |
28 to 32 1047243 3068.9277 0.000000 | |
27 to 31 1047192 3068.7548 0.000000 | |
26 to 30 1047342 3069.2633 0.000000 | |
25 to 29 1047126 3068.5311 0.000000 | |
24 to 28 1047009 3068.1345 0.000000 | |
23 to 27 1047213 3068.8260 0.000000 | |
22 to 26 1047300 3069.1209 0.000000 | |
21 to 25 1047141 3068.5819 0.000000 | |
20 to 24 1047297 3069.1107 0.000000 | |
19 to 23 1047264 3068.9989 0.000000 | |
18 to 22 1047513 3069.8429 0.000000 | |
17 to 21 1047345 3069.2735 0.000000 | |
16 to 20 1047345 3069.2735 0.000000 | |
15 to 19 1047125 3068.5277 0.000000 | |
14 to 18 1047245 3068.9345 0.000000 | |
13 to 17 1047123 3068.5209 0.000000 | |
12 to 16 1047243 3068.9277 0.000000 | |
11 to 15 1047192 3068.7548 0.000000 | |
10 to 14 1047342 3069.2633 0.000000 | |
9 to 13 1047126 3068.5311 0.000000 | |
8 to 12 1047009 3068.1345 0.000000 | |
7 to 11 1047213 3068.8260 0.000000 | |
6 to 10 1047300 3069.1209 0.000000 | |
5 to 9 1047141 3068.5819 0.000000 | |
4 to 8 1047297 3069.1107 0.000000 | |
3 to 7 1047264 3068.9989 0.000000 | |
2 to 6 1047513 3069.8429 0.000000 | |
1 to 5 1047345 3069.2735 0.000000 | |
----------------------------------------------------------------- | |
|------------------------------------------------------------ | | |
| 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. | | |
|------------------------------------------------------------ | | |
DNA test for file out.bin | |
Bits used No. missing words z-score p-value | |
31 to 32 1048460 2674.1908 0.000000 | |
30 to 31 1048444 2674.1436 0.000000 | |
29 to 30 1048448 2674.1554 0.000000 | |
28 to 29 1048448 2674.1554 0.000000 | |
27 to 28 1048448 2674.1554 0.000000 | |
26 to 27 1048444 2674.1436 0.000000 | |
25 to 26 1048444 2674.1436 0.000000 | |
24 to 25 1048448 2674.1554 0.000000 | |
23 to 24 1048444 2674.1436 0.000000 | |
22 to 23 1048444 2674.1436 0.000000 | |
21 to 22 1048448 2674.1554 0.000000 | |
20 to 21 1048448 2674.1554 0.000000 | |
19 to 20 1048448 2674.1554 0.000000 | |
18 to 19 1048448 2674.1554 0.000000 | |
17 to 18 1048444 2674.1436 0.000000 | |
16 to 17 1048460 2674.1908 0.000000 | |
15 to 16 1048460 2674.1908 0.000000 | |
14 to 15 1048444 2674.1436 0.000000 | |
13 to 14 1048448 2674.1554 0.000000 | |
12 to 13 1048448 2674.1554 0.000000 | |
11 to 12 1048448 2674.1554 0.000000 | |
10 to 11 1048444 2674.1436 0.000000 | |
9 to 10 1048444 2674.1436 0.000000 | |
8 to 9 1048448 2674.1554 0.000000 | |
7 to 8 1048444 2674.1436 0.000000 | |
6 to 7 1048444 2674.1436 0.000000 | |
5 to 6 1048448 2674.1554 0.000000 | |
4 to 5 1048448 2674.1554 0.000000 | |
3 to 4 1048448 2674.1554 0.000000 | |
2 to 3 1048448 2674.1554 0.000000 | |
1 to 2 1048444 2674.1436 0.000000 | |
----------------------------------------------------------------- | |
|-------------------------------------------------------------| | |
| 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 result for the byte stream from out.bin | |
(Degrees of freedom: 5^4-5^3=2500; sample size: 2560000) | |
chisquare z-score p-value | |
531837.94 7485.969 0.000000 | |
|-------------------------------------------------------------| | |
| 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. | | |
|-------------------------------------------------------------| | |
Test results for specific bytes from out.bin | |
(Degrees of freedom: 5^4-5^3=2500; sample size: 256000) | |
bits used chisquare z-score p-value | |
1 to 8 443487.89 6236.511 0.000000 | |
2 to 9 585000.75 8237.805 0.000000 | |
3 to 10 463217.32 6515.527 0.000000 | |
4 to 11 544391.37 7663.501 0.000000 | |
5 to 12 472057.35 6640.544 0.000000 | |
6 to 13 478001.49 6724.607 0.000000 | |
7 to 14 446485.58 6278.904 0.000000 | |
8 to 15 450302.02 6332.877 0.000000 | |
9 to 16 444177.82 6246.268 0.000000 | |
10 to 17 625034.82 8803.972 0.000000 | |
11 to 18 941859.02 13284.543 0.000000 | |
12 to 19 998469.58 14085.137 0.000000 | |
13 to 20 714760.26 10072.881 0.000000 | |
14 to 21 564968.26 7954.502 0.000000 | |
15 to 22 1697183.13 23966.439 0.000000 | |
16 to 23 511671.26 7200.769 0.000000 | |
17 to 24 442834.00 6227.263 0.000000 | |
18 to 25 586102.43 8253.385 0.000000 | |
19 to 26 459497.91 6462.926 0.000000 | |
20 to 27 544108.11 7659.495 0.000000 | |
21 to 28 471147.42 6627.675 0.000000 | |
22 to 29 477746.44 6721.000 0.000000 | |
23 to 30 446438.58 6278.240 0.000000 | |
24 to 31 450303.65 6332.900 0.000000 | |
25 to 32 443768.44 6240.478 0.000000 | |
|-------------------------------------------------------------| | |
| 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 10 tests on file out.bin | |
(Of 12000 tries, the average no. of successes should be | |
3523.0 with sigma=21.9) | |
No. succeses z-score p-value | |
427 -141.3699 1.000000 | |
270 -148.5388 1.000000 | |
410 -142.1461 1.000000 | |
276 -148.2648 1.000000 | |
278 -148.1735 1.000000 | |
276 -148.2648 1.000000 | |
272 -148.4475 1.000000 | |
416 -141.8721 1.000000 | |
433 -141.0959 1.000000 | |
436 -140.9589 1.000000 | |
Square side=100, avg. no. parked=349.40 sample std.=75.36 | |
p-value of the KSTEST for those 10 p-values: 0.000000 | |
|-------------------------------------------------------------| | |
| 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 file out.bin | |
Sample no. d^2 mean equiv uni | |
5 0.4191 704.7770 0.343750 | |
10 0.1862 352.4909 0.170652 | |
15 0.4191 235.1181 0.343750 | |
20 0.4191 176.4108 0.343750 | |
25 0.0000 141.1789 0.000000 | |
30 0.4191 117.6972 0.343750 | |
35 0.1862 100.9019 0.170652 | |
40 0.4191 88.3416 0.343750 | |
45 0.4191 78.5579 0.343750 | |
50 0.0000 70.7273 0.000000 | |
55 0.4191 64.3238 0.343750 | |
60 0.1862 58.9767 0.170652 | |
65 0.4191 54.4722 0.343750 | |
70 0.4191 50.6020 0.343750 | |
75 0.0000 47.2453 0.000000 | |
80 0.4191 44.3105 0.343750 | |
85 0.1862 41.7160 0.170652 | |
90 0.4191 39.4192 0.343750 | |
95 0.4191 37.3597 0.343750 | |
100 0.0000 35.5043 0.000000 | |
-------------------------------------------------------------- | |
Result of KS test on 100 transformed mindist^2's: p-value=0.000000 | |
|-------------------------------------------------------------| | |
| 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 out.bin | |
sample no r^3 equiv. uni. | |
1 89.229 0.948917 | |
2 135.744 0.989163 | |
3 93.768 0.956090 | |
4 46.105 0.784939 | |
5 96.727 0.960214 | |
6 127.389 0.985683 | |
7 111.865 0.975979 | |
8 105.081 0.969884 | |
9 62.460 0.875320 | |
10 90.388 0.950853 | |
11 57.682 0.853794 | |
12 54.221 0.835914 | |
13 103.556 0.968313 | |
14 50.021 0.811255 | |
15 110.796 0.975107 | |
16 92.110 0.953594 | |
17 33.500 0.672626 | |
18 71.398 0.907443 | |
19 93.957 0.956365 | |
20 101.907 0.966523 | |
-------------------------------------------------------------- | |
p-value for KS test on those 20 p-values: 0.000000 | |
|-------------------------------------------------------------| | |
| This is the SQUEEZE 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 out.bin | |
Table of standardized frequency counts | |
(obs-exp)^2/exp for j=(1,..,6), 7,...,47,(48,...) | |
-0.1 -1.2 13.5 40.6 38.8 58.0 | |
63.7 54.5 14.5 -5.8 -32.5 -59.8 | |
-65.0 -58.4 22.6 109.9 147.6 -7.1 | |
-42.4 -66.6 -66.3 -61.4 -49.3 -11.1 | |
66.9 67.3 17.0 11.0 -6.2 -18.4 | |
-18.4 -14.8 -11.7 -9.2 -7.2 270.6 | |
149.5 130.6 45.5 27.0 -1.3 -1.0 | |
-1.1 | |
Chi-square with 42 degrees of freedom:203641.581471 | |
z-score=22214.535561, p-value=0.000000 | |
_____________________________________________________________ | |
|-------------------------------------------------------------| | |
| The OVERLAPPING SUMS test | | |
|Integers are floated to get a sequence U(1),U(2),... of uni- | | |
|form [0,1) variables. Then overlapping sums, | | |
| S(1)=U(1)+...+U(100), S2=U(2)+...+U(101),... are formed. | | |
|The S''s are virtually normal with a certain covariance mat- | | |
|rix. A linear transformation of the S''s converts them to a | | |
|sequence of independent standard normals, which are converted| | |
|to uniform variables for a KSTEST. | | |
|-------------------------------------------------------------| | |
Results of the OSUM test for out.bin | |
Test no p-value | |
1 0.000000 | |
2 0.000000 | |
3 0.000000 | |
4 0.000000 | |
5 0.000000 | |
6 0.000000 | |
7 0.000000 | |
8 0.000000 | |
9 0.000000 | |
10 0.000000 | |
_____________________________________________________________ | |
p-value for 10 kstests on 100 kstests:0.000000 | |
|-------------------------------------------------------------| | |
| This is the RUNS test. It counts runs up, and runs down,| | |
|in a sequence of uniform [0,1) variables, obtained by float- | | |
|ing the 32-bit integers in the specified file. This example | | |
|shows how runs are counted: .123,.357,.789,.425,.224,.416,.95| | |
|contains an up-run of length 3, a down-run of length 2 and an| | |
|up-run of (at least) 2, depending on the next values. The | | |
|covariance matrices for the runs-up and runs-down are well | | |
|known, leading to chisquare tests for quadratic forms in the | | |
|weak inverses of the covariance matrices. Runs are counted | | |
|for sequences of length 10,000. This is done ten times. Then| | |
|another three sets of ten. | | |
|-------------------------------------------------------------| | |
The RUNS test for file out.bin | |
(Up and down runs in a sequence of 10000 numbers) | |
Set 1 | |
runs up; ks test for 10 p's: 0.000000 | |
runs down; ks test for 10 p's: 0.000000 | |
Set 2 | |
runs up; ks test for 10 p's: 0.000000 | |
runs down; ks test for 10 p's: 0.000000 | |
|-------------------------------------------------------------| | |
|This the CRAPS TEST. It plays 200,000 games of craps, counts| | |
|the number of wins and the number of throws necessary to end | | |
|each game. The number of wins should be (very close to) a | | |
|normal with mean 200000p and variance 200000p(1-p), and | | |
|p=244/495. Throws necessary to complete the game can vary | | |
|from 1 to infinity, but counts for all>21 are lumped with 21.| | |
|A chi-square test is made on the no.-of-throws cell counts. | | |
|Each 32-bit integer from the test file provides the value for| | |
|the throw of a die, by floating to [0,1), multiplying by 6 | | |
|and taking 1 plus the integer part of the result. | | |
|-------------------------------------------------------------| | |
RESULTS OF CRAPS TEST FOR out.bin | |
No. of wins: Observed Expected | |
144792 98585.858586 | |
z-score=206.661, pvalue=0.00000 | |
Analysis of Throws-per-Game: | |
Throws Observed Expected Chisq Sum of (O-E)^2/E | |
1 77556 66666.7 1778.664 1778.664 | |
2 13850 37654.3 15048.623 16827.287 | |
3 13973 26954.7 6252.163 23079.450 | |
4 2054 19313.5 15423.906 38503.357 | |
5 2436 13851.4 9407.831 47911.187 | |
6 5251 9943.5 2214.498 50125.686 | |
7 1265 7145.0 4838.989 54964.675 | |
8 790 5139.1 3680.514 58645.189 | |
9 9347 3699.9 8619.270 67264.459 | |
10 8546 2666.3 12965.884 80230.343 | |
11 37542 1923.3 659632.159 739862.502 | |
12 523 1388.7 539.702 740402.204 | |
13 915 1003.7 7.841 740410.045 | |
14 2445 726.1 4068.740 744478.786 | |
15 3536 525.8 17231.780 761710.566 | |
16 6221 381.2 89476.023 851186.589 | |
17 1696 276.5 7286.006 858472.595 | |
18 859 200.8 2156.989 860629.584 | |
19 584 146.0 1314.234 861943.818 | |
20 259 106.2 219.773 862163.590 | |
21 10352 287.1 352828.246 1214991.837 | |
Chisq=1214991.84 for 20 degrees of freedom, p= 0.00000 | |
SUMMARY of craptest on out.bin | |
p-value for no. of wins: 0.000000 | |
p-value for throws/game: 0.000000 | |
_____________________________________________________________ |
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