Added a discussion on statistical tests, and the raw results files

.github/workflows/gh-pages-deploy.yml

@@ -34,3 +34,6 @@ jobs:
       with:
         github_token: ${{ secrets.GITHUB_TOKEN }}
         publish_dir: ./docs/build/html
+        # `keep_files` added to keep results and other `gh-pages` branch-only files intact 
+        keep_files: true    
+

docs/Doxyfile

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 # spaces. See also FILE_PATTERNS and EXTENSION_MAPPING
 # Note: If this tag is empty the current directory is searched.
 
-INPUT                  = ../README.md ./quickstart.md ../include ../examples
+INPUT                  = ../README.md ./quickstart.md ./statistical_results.md ../include ../examples
 
 # This tag can be used to specify the character encoding of the source files
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docs/statistical_results.md

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+# Statistical Quality Test Results
+
+OpenRAND generators has been tested with two statistical suites: TestU01 and PractRand. 
+
+There are two ways to test the generators:
++ Single stream: we use PractRand to test a single random number stream to their theoratical limit of 2^32 numbers (2^34 bytes). See the files [practrand_[tyche|squares|threefry|phillox].txt](https://github.com/msu-sparta/OpenRAND/tree/gh-pages/results) for details. 
+
++ Parallel Streams: This tests for correlation among parallel streams. This is not quite straightforward in those statistical frameworks- as they are designed to work on a single stream. To stitch parallel streams to a single one, we use the following approach: 
+  + Start with a set of `N` random number generators.
+  + Generate `C` random numbers from each generator.
+  + Combine them, now we have a buffer of length `N*C`.
+  + Repeat the above steps to extend the buffer by `N*C` at each iteration as long as necessary.
+
+In this document, our primary focus is on parallel streams, as single streams are already well-examined in existing literature. For the results discussed here, we've set N at 100 and C at 3.
+
+## Practrand
+Practrand can consume practically infinite amount of data. Given that each stream contains 2^32 numbers, and each number is 4 bytes (2^2), with 100 streams (or 2^6.64), the combined buffer is little over 2^40 bytes long. Put another way, we shouldn't expect to see any failures in Practrand before reaching the 2^40 bytes (1-terabyte) mark.
+
+None of the generators failed this test. For detailed output from the practrand suite for for each generator, see: [practrandm_[tyche|squares|threefry|phillox].txt](https://github.com/msu-sparta/OpenRAND/tree/gh-pages/results).
+
+
+## TestU01
+Testu01 has 3 batteries of tests: SmallCrush, Crush, and BigCrush. BigCrush is the most comprehensive one with a total of 106 tests, and is the one we use.
+
+Some generators do fail one or two BigCRush tests. For example, this is the output from Philox:
+```
+========= Summary results of BigCrush =========
+
+ Version:          TestU01 1.2.3
+ Generator:        philox
+ Number of statistics:  160
+ Total CPU time:   04:36:59.22
+ The following tests gave p-values outside [0.001, 0.9990]:
+ (eps  means a value < 1.0e-300):
+ (eps1 means a value < 1.0e-15):
+
+       Test                          p-value
+ ----------------------------------------------
+  9  CollisionOver, t = 14           0.9994 
+ ----------------------------------------------
+ All other tests were passed
+```
+
+This is not unusual on some runs, authors of curand also [observed](https://docs.nvidia.com/cuda/curand/testing.html) some failures on Bigcrush. It also depends on the configuration we used. 
+
+For detailed output from Bigcrush suite, see [testu01m_[tyche|squares|threefry|phillox].txt](https://github.com/msu-sparta/OpenRAND/tree/gh-pages/results).