Moved. See All new content goes to the new site. This is a legacy, and will likely be dropped five years after the last post in Jan 2023.

Tuesday, January 25, 2022

No one wins at Code Golf vs. This is more noise than signal

Looking at code. Came to a 20-line block of code that did exactly this.

sorted(Path.cwd().glob("some_pattern[1-9]*.*"), reverse=True)

Twenty lines. Seriously. 

To be fair, 8 of the 20 lines were comments. 3 were blank. Which leaves 9 lines of code to perform the task of a one-liner.

I often say "no one wins at code golf" as a way to talk people out of trying to minimize Python code into vanishingly small black holes where no information about the code's design escapes.

However. Blowing a line of code into 9 lines seems to be just as bad. 

I'll spare you the 9 lines. I will say this, though, the author was blissfully ignorant that Path objects are comparable. So. There were needless conversions. And. Even after commenting on this, they seemed to somehow feel (without evidence of any kind) that Path objects were incomparable.

This is not the first time I've seen folks who like assembler-style code. There is at most one state-change or attribute reference on each line of code. The code has a very voluble verticality (VVV™).

This seems as wrong as code golf.  Neither style provides meaningful code. 

How can we measure "meaningful"?

Of the 8 lines of comments, the English summary, the "reverse alphabetic order" phrase is only a few words. Therefore, the matching code can be an equally terse few symbols. I think code can parallel natural language.

Tuesday, January 18, 2022

How to Test a Random Number Generator

Nowadays, we don't have the same compelling reasons to test a random number generator. The intervening decades have seen a lot of fruitful research. Good algorithms.

Looking back to my 1968 self, however, I still feel a need to work out the solution to an old problem. See The Old Days -- ca. 1968 for some background on this.

What could I have done on that ancient NCE Fortran -- with four digit integers -- to create random numbers? Step 1 was to stop using the middle-squared generator. It doesn't work.

Step 2 is to find a Linear Congruential Generator that works. LCG's have a (relatively) simple form:

\[X_{n+1} = (X_n \times a + c) \bmod m\]

In this case, the modulo value, m, is 10,000. What's left is step 3: find a and c parameters.

To find suitable parameters, we need battery of empirical tests. Most of them are extensions to the following class:

from collections import Counter
from typing import Hashable
from functools import cache

class Chi2Test:
    """The base class for empirical PRNG tests based on the Chi-2 testing."""
    #: The actual distribution, created by ``test()``.
    actual_fq : dict[Hashable, int]
    #: The expected distribution, created by ``__init__()``.
    expected_fq: dict[Hashable, int]
    #: The lower and upper bound on acceptable chi-squared values.
    expected_chi_2_range: tuple[float, float]
    def __init__(self):
        A subclass will override this to call ``super().__init__()`` and then
        create the expected distribution.
        self._chi2 = None
    def test(self):
        A subclass will override this to call ``super().test()`` and then
        create an actual distribution, usually with a distinct seed value.
        self._chi2 = None
    def chi2(self) -> float:
        """Return chi-squared metric between actual and expected observations."""
        if self._chi2 is None:
            a_e = (
                (self.actual_fq[k], self.expected_fq[k]) 
                for k in self.expected_fq 
                if self.expected_fq[k] > 0
            v = sum((a-e)**2/e for a, e in a_e)
            self._chi2 = v
        return self._chi2

    def pass_test(self) -> bool:
        return self.expected_chi_2_range[0] <= self.chi2 <= self.expected_chi_2_range[1]

This defines the essence of a chi-squared test. There's another test that isn't based on chi-squared. The serial correlation where a correlation coefficient is computed between adjacent pairs of samples. We'll ignore this special case for now. Instead, we'll focus on the battery of chi-squared tests. 

Linear Congruential Pseudo-Random Number Generator

We'll also need an LC PRNG that's constrained to 4 decimal digits.

It looks like this:

class LCM4:
    """Constrained by the NCE Fortran 4-digit integer type."""
    def __init__(self, a: int, c: int) -> None:
        self.a = a
        self.c = c
    def seed(self, v: int) -> None:
        self.v = v
    def random(self) -> int:
        self.v = (self.a*self.v % 10_000 + self.c) % 10_000
        return self.v

This mirrors the old NCE Fortran on the IBM 1620 computer. 4 decimal digits. No more. 

We can use this to generate a pile of samples that can be evaluated. I'm a fan of using generators because they're so efficient. The use of a set to create a list seems weird, but it's very fast.

def lcg_samples(rng: LCM4, seed: int, n_samples: int = N_SAMPLES) -> list[int]:
    Generate a bunch of sample values. A repeat implies a cycle, and we'll stop early.

    >>> lcg_samples(LCM4(1621, 3), 1234)[:12]
    [317, 3860, 7063, 9126, 3249, 6632, 475, 9978, 4341, 6764, 4447, 8590]

    def until_dup(f: Callable[..., Hashable], n_samples: int) -> Iterator[Hashable]:
        seen: set[Hashable] = set()
        while (v := f()) not in seen and len(seen) < n_samples:
            yield v
    return list(until_dup(rng.random, n_samples))

This function builds a list of values for us. We can then subject the set of samples to a battery of tests. We'll look at one test as an example for the others. They're each devilishy clever, and require a little bit of coding smarts to get them to work correctly and quickly.

Frequency Test

Here's one of the tests in the battery of chi-squared tests. This is the frequency test that examines values to see if they have the right number of occurrences. We pick a domain, d, and parcel numbers out into this domain. We use \(\frac{d \times X_{n}}{10,000}\) because this tends to leverage the left-most digits which are somewhat more random than the right-most digits.

class FQTest(Chi2Test):
    expected_chi_2_range = (7.261, 25.00)

    def __init__(self, d: int = 16, size_samples: int = 6_400) -> None:
        #: Size of the domain
        self.d = d
        #: Number of samples expected
        self.size_samples = size_samples
        #: Frequency for Chi-squared comparison
        self.expected_fq = {e: int(self.size_samples/self.d) for e in range(self.d)}
    def test(self, sequence: list[int]) -> None:
        self.actual_fq = Counter(int(self.d*s/10_000) for s in sequence)

We can apply this test to some samples, compare with the expectation, and save the chi-squared value. This lets us look at LCM parameters to see if the generator creates suitably random values.

The essential test protocol is this:

samples = lcg_samples(LCM4(1621, 3), seed=1234)
fqt = FQTest()

The test creates some samples, applies the frequency test. The next step is to examine the chi-squared value to see if it's in the allowable range, \(7.261 \leq \chi^2 < 25\).

The search space

Superficially, it seems like there could be 10,000 choices of a and 10,000 choices of c parameter values for this PRNG. That's 100 million combinations. It takes a bit of processing to look at all of those. 

Looking more deeply, the values of c are often small prime numbers. 1 or 11 or some such. That really cuts down on the search. The values of a have a number of other constraints with respect to the modulo value. Because 10,000 has factors of 4 and 5, this suggests values like \(20k + 1\) will work. Sensible combinations are defined by the following domain:

combinations = [
    (a, c)
    for c in (1, 3, 7, 11,)
    for a in range(21, 10_000, 20)

This is 2,000 distinct combinations, something we can compute on our laptop. 

The problem we have trying to evaluate these is each combination's testing is compute-intensive. This means we want to use as many cores of our machine as we have available. We don't want this to process each combination serially on a single core. A thread pool isn't going to help much because the OS doesn't scatter threads among all the cores. 

Because the OS likes to scatter processes among all the cores, we need a process pool.

Here's how to spread the work among the cores:

    from concurrent.futures import ProcessPoolExecutor, as_completed

    combinations = [
        (a, c)
        for c in (1, 3, 7, 11)
        for a in range(21, 10_000, 20)

    with Progress() as progress:
        setup_task = progress.add_task("setup ...", total=len(combinations))
        finish_task = progress.add_task("finish...", total=len(combinations))

        with ProcessPoolExecutor(max_workers=8) as pool:
            futures = [
                pool.submit(evaluate, (a, c))
                for a, c in progress.track(combinations, task_id=setup_task, total=len(combinations))
            results = [
                for f in progress.track(as_completed(futures), task_id=finish_task, total=len(combinations))

This will occupy *all* the cores of the computer executing the `evaluate()` function. This function applies the battery of tests to each combination of a and c. We can then check the results for combinations where the chi-squared results for each test are in the acceptable ranges for the test.

It's fun.


Use a=1621 and c=3 can generate acceptable random numbers using 4 decimal digits.

Here's some output using only a subset of the tests.

(rngtest2) % python
setup ... ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 100% 0:00:00
finish... ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 100% 0:00:00
2361  1  11.46  14.22  46.64  63.76   2.30  11.33   2.16   2.16 
 981  3  10.28  15.24  52.56  66.32   2.28  11.08  10.47  10.47 
1221  3  10.19  14.12  48.72  62.08   3.03  10.08   2.59   2.59 
1621  3  11.70  14.91  47.12  69.52   2.23   9.69   0.86   0.86 

The output shows the a and c values followed by the minimum and maximum chi-squared values for each test. The chi-squared values are in pairs for the frequency test, serial pairs test, gap test, and poker test. 

Each test uses about two dozen seed values to generate piles of 3,200 samples and subject each pile of samples to a battery of tests. The seed values, BTW, are range(1, 256, 11); kind of arbitrary. Once I find the short list of candidates, I can test with more seeds. There are only 10,000 seed values, so, this can be done in finite time.

For example, a=1621, c=3, had chi-squared values between 11.70 and 14.91 for the frequency test. Well within the 7.261 to 25.0 range required. The remaining numbers show that it passed the other tests, also.

For completeness, I intend to implement the remaining half-dozen or so tests. Then I need to make sure the sphinx-produced documentation looks good. I've done this before. It's kind of an obsession, I think.

Looking back to my 1968 self, this would have been better than the middle-squared nonsense that caused me to struggle with bad games that behaved badly.

Tuesday, January 11, 2022

The Old Days -- ca. 2000 -- Empirical Tests of Random Numbers (Python and Chi-Square Testing)

See The Old Days -- ca. 1974 Random Numbers Before Python for some background.

We'll get to Python after reminiscing about the olden days. I want to provide some back story on why sympy has had a huge impact on ordinary hacks like myself.

What we're talking about is how we struggled with randomness before

  1. /dev/random
  2. The Mersenne Twister Pseudo-Random Number Generator (PRNG)

Pre-1997, we performed empirical tests of PRNG's to find one that was random enough for our application. Maybe we were doing random samples of data to compare statistical measures. Maybe we were writing a game. What was important was a way to create a sequence of values that passed a battery of statistical tests.

See for the kind of material we salivated over. 

While there are an infinite number of bad algorithms, some math reveals that the Linear Congruential Generator (LCG) is simple and effective. Each new number is based on the previous number: \(X_{n+1} = (X_n \times a + c) \bmod m\). There's a multiply and an add, modulo some big number. The actual samples are often a subset of the bits in \(X_{n}\). 

After the Mersenne Twister became widely used, we essentially stopped looking at alternative random number algorithms. Before then -- well -- things weren't so good.

Here are some classics that I tested.

  • The ACM Collected Algorithms (CALGO) number 294 is a random-number generator. This is so obsolete, I have trouble finding links to it. It was a 28-bit generator.
  • The ACM Collected Algorithms (CALGO) number 266 has code still available. See toms/266
  • The Cheney-Kincaid generator is available. See random.f plus dependencies.

These formed a kind of benchmark I used when looking at Python's built-in Mersenne Twister.

Nowadays, you can find a great list of LCM PRNG's at

Python Empirical Testing

One of the early questions I had was whether or not the random module in Python stacked up against these older RNG's that I was a little more familiar with.

So, I wrote a big, fancy random number testing tool in Python. 

When? Around 2000. I started this in the Python 1.6 and 2.1 era. I have files showing results from Python 2.3 (#2, Jul 30 2003). This is about when I stopped fooling around with this and moved on to trusting that Python really did work and was -- perhaps -- the best approach to working with randomly-sampled data for statistical work. 

The OO design for the test classes was Lavish Over The Top (LOTT™) OO:

  • Too Many Methods
  • Too Many Superclasses
  • No Duck Typing

We won't look at that code. It's regrettable and stems from trying to make Python into C++.

What I do want to look at is the essential Chi-Squared test methodology. This is some cool stuff.

Comparing Expected and Actual

The chi-squared metric is a way to compare actual and expected distributions. You can read about it on your own time. It's a way to establish if data is random or there's something else going on that's not random. i.e., a trend or a bias. 

The empirical tests for PRNG's that Knuth defines all come with chi-squared values that bracket acceptable levels of randomness. For the purposes of writing a working set of tests the magic chi-squared values supplied by Knuth are fine. Magical. But fine. Really. Trust them.

If you make modifications, you'd use your statistics text-book. You'd open to the back where it had a Chi-Squared table. That table gave you chi-squared values for a given degree of freedom and a given probability of being random.

Or, You could look for the NIST handbook online. It has a section on chi-squared testing. See Same drill. Degrees of freedom and probability map to a chi-squared threshold.


Were do these magical Chi-Squared values come from? This gets interesting in a useless-sidebar kind of way.

Chi-Squared Values

There's a really, really terse summary of chi-squared numbers here: This is all you need to know. It may be too terse to help you learn about it, but it's a handy reference.

We need to evaluate two functions: partial gamma and gamma. These are defined as integrals. And they're nasty levels of complexity. Nasty.

This kind of nasty:

\[\gamma (s,z)=\int _{0}^{z}t^{s-1} e^{-t} dt\].

\[\Gamma (z)=\int _{0}^{\infty} t^{z-1} e^{-t} dt\].

These are not easy things to evaluate. Back to the ACM Collected Algorithms (CALGO) to find ways to evaluate these integrals. There are algorithms in CALGO 435 and 654 that are expressed as Fortran for evaluating these. This ain't all, of course, we need Stirling Numbers and Bernoulli Numbers. So there's a lot going on here.

A lot of this can be transliterated from Fortran. The resulting code is frankly quite ugly, and requires extensive test cases. Fortran with GOTO's requires some cleverness to unwind the conceptual for/while/if constructs.


Enter Sympy

In the 20+ years since I implemented my empirical PRNG tests "the hard way," sympy has come of age.

Check this out

from sympy import Sum, rf
from import k, s, z
from sympy.functions import exp
from sympy import oo
Sum(z**s * exp(-z) * z**k / rf(s, k+1), (k, 0, oo)).simplify()

I could use this in Jupyter Lab to display a computation for the partial gamma function.

\[z^{s}e^{-z}\sum _{k=0}^{\infty }{\dfrac {z^{k}}{s^{\overline {k+1}}}}\]

This requires a fancy Rising Factorial computation, the \(s^{\overline {k+1}}\) term. This is available in sympy as the rf(s, k+1) expression.

It turns out that sympy offers lowergamma() and gamm() as first-class functions. I don't even need to work through the closed-form simplifications.

I could do this...

def gammap(s: float, z: float) -> float:
    return (z**s * exp(-z) * Sum(z**k / rf(s, k+1), (k, 0, oo))).evalf()

def gamma(z: float) -> float:
    return integrate(t**(z-1) * exp(-t), (t, 0, oo)).doit()

It works well. And it provides elegant documentation. But I don't need to. I can write this, instead,

def chi2P(chi2: float, degF: int) -> float:
   return lowergamma(degF/2, chi2/2) / gamma(degF/2)

This is used to compute the probability of seeing a chi-squared value. 

For the frequency test, as an example. We partition the random numbers into 16 bins. These gives us 15 degrees of freedom. We want chi-squared values between 7.2578125 and 25.0.


Given a chi-squared value of 6.0, we can say the probability of 0.02 is suspiciously low, less than 0.05 level that we've decided signifies mostly random. The data is "too random"; that is to say it's too close to the ideal distribution to be trusted.

The established practice was to lookup a chi-squared value because you couldn't easily compute the probability of that value. With sympy, we can compute the probability. It's slow, so we have to optimize this carefully and not compute probabilities more frequently than necessary.

We can, for example, compute chi-squared values for a number of seeds, take the max and min of these and compute the probability of those two boundary values. This will bracket the probability that the pseudo random number generator is producing suitably random numbers.

This also applies to any process we're measuring with results that might vary randomly or might indicate a consistent problem that requires evaluation.

Using sympy eliminates the complexity of understanding these beautifully hand-crafted antique algorithms. It acts as a kind of super-compiler. From Math to an intermediate AST to a concrete implementation.

Tuesday, January 4, 2022

The Old Days -- ca. 1974 -- Random Numbers before Python

See "The Old Old Days -- ca. 1968" for my first exposure to an actual computer. Nothing about Python there. But. It's what motivated me to get started learning to code -- I was fascinated by games that involved randomization. Games with cards or dice.

After filling in a little background, I will get to the Python part of this. First, however, I want to compare the olden days with what we have now.

From 1969 to 1974 I had access to the high school's IBM 1620. This means programming in IBM's SPS assembler, or using the NCE Load-and-Go Fortran compiler. See for a scathing review of the problems with this machine.

See Page 36 has this:

That's a quick overview of my earliest programming language. What's essential here is the NCE Fortran used 4-digit integers.

I'll repeat that for those skimming, and wondering what the Python connection is.

Four. Digit. Integers.

That's four decimal digits. Decimal digits required at least 4 hardware bits. IBM 1620 digits also had flags and signs, so, there were maybe 6 bits per digit. 24 bits of hardware used to represent just under 14 bits of useful information.

My interest is in simulation and randomness. So. I have this question of how to create random sequences of numbers limited to 4-digit integers.

PRNG Algorithms

There are a number of classic Pseudo-Random Number Generator (PRNG) algorithms from the early days before Mersenne Twister took over in 1997.

We used to be super-careful to emphasize the letter P in PRNG because the numbers we're really random. They just behaved randomish. This is contrasted with real randomness, also known as entropy. For example, /dev/random device driver has a fair amount of entropy. I think it's comparable to a person throwing dice across a table. I think it's as random as a noise-generating diode with a sample-and-hold circuit to pluck out random values from the noise.

Pre-Mersenne-Twister -- pre-1997 -- we worried a lot about random number generation. See Knuth, Donald E. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Addison-Wesley, 1969. Section 3.3.2. covers empirical testing of random number generators. Section 3.3.1. covers the Chi-squared test for fit between actual and expected frequency distributions.

Back in the olden days, it was stylish to perform an empirical test (or ten) to confirm we really had "good" random numbers. The built-in libraries that came with your compiler or OS could not be trusted without evidence.

One of the classic (bad) PRNG's is the "Middle-Squared" method. See I learned about this in the 70's. And used it in the old NCE Fortran. 

With Four. Digit. Integers.

Did I mention that the Fortran compiler used four decimal digits for integers? That means plucking the middle two digits out of a four-digit number. How random can that be?

Not very. The longest possible sequence is 100 numbers. If, by some miracle, you found a seed number with the right properties and only two digits.

Nowadays I can, in Python, do a quick middle-squared analysis for all 100 seed values.

This kind of thing.

def csqr4(value: int) -> list[int]:
    """The 4 decimal digit center-squared PRNG."""
    sequence = []
    while value not in sequence:
        value = (((value**2) // 10) % 100)
    return sequence

Which you can run and see that all of my early attempts at games and simulations were doomed. The seed values of 76, 42, and 69 provided kind of long sequences of almost random-seeming numbers. Otherwise, pfft, this was junk computer science. 50% of the seeds provide 5 or fewer numbers before repeating. 

For blackjack, a few random numbers for shuffling might be enough. For other games, the lack of randomness made the outcomes trivially predictable. 

What's funny is how far the state of the art has moved since then.

  1. Hardware now has more than 20,000 decimal digits (about 10K bytes) of storage.
  2. Software with algorithms that are really, really clever.

It's hard to understate these two advances, particularly, the second one. I'll return to the algorithm thing a lot in the next few posts.

My focus was on games and randomization. Ideally, simple stuff. But... under the hood, it's not simple. I've spent some time (not much, and not in much depth) looking at the tip of this iceberg. 

It served me as an incentive to dive just a little more deeply into a topic, like math or a programming language or a statistical tool.