Suppose we have a series of integer numbers which are not truly random (For example, the numbers at a given position in decimal pi representation).

Is there a machine learning algorithm that can capture this?

Thanks for your time.

  • $\begingroup$ You can use the Runs test. There is an implementation in stats models that you can use directly. If you find this useful let me know and I can extend a full answer with code and implementation. (Long ago I ran this with Excel VBA before using python) $\endgroup$
    – Multivac
    Commented Jun 27, 2023 at 19:32
  • $\begingroup$ Thanks very much for your response $\endgroup$
    – dddr rddd
    Commented Jun 27, 2023 at 22:06
  • $\begingroup$ Yes I think an implementation of this would be helpfull $\endgroup$
    – dddr rddd
    Commented Jun 27, 2023 at 22:07

1 Answer 1


Here is an example, it converts the series into binary. It needs a threshold that by default is the average.

I ran some tests with both random and not random and It gives the expected results.

from scipy.stats import norm
import numpy as np
import random

class WaldWolfowitzRunsTest:
    def __init__(self, series, threshold=None):
        self.series = series
        self.threshold = threshold

    def _calculate_p_value(self, z_value):
        Calculates the p value given a z value.
        Here we assume using the normal distribution.
        return 2 * norm.sf(abs(z_value))

    def _get_runs_list(self):
        Given the series, create the runs list.
        i.e. label the series and sort the concatenated lists, and then return the list of labels.
        labels = []
        if self.threshold is None:
            threshold = np.mean(self.series)
            threshold = self.threshold

        for value in self.series:
            if value > threshold:

        return labels

    def _calculate_z_score(self, nruns, mean, variance):
        Calculates the z-score.
        return (nruns - mean) / np.sqrt(variance)

    def _calculate_mean(self, n_plus, n_min):
        Calculates the mean, defined as:
        (2 * n_plus * n_min)/N + 1
        numerator = 2 * n_plus * n_min
        denominator = n_plus + n_min
        return numerator / float(denominator) + 1

    def _calculate_variance(self, mean, n):
        Calculates the variance.
        assert n > 1, "The number of elements should be higher than 1"
        return ((mean - 1) * (mean - 2)) / float(n - 1)

    def _calculate_nruns(self, labels):
        Calculate the number of runs.
        A run is defined as the follow-up of the same signs.
        nruns = 0
        last_seen = None

        for label in labels:
            if label != last_seen:
                nruns += 1
            last_seen = label

        return nruns

    def runs_test(self, alpha=0.05):
        Executes the whole Wald-Wolfowitz runs test.
        labels = self._get_runs_list()
        nruns = self._calculate_nruns(labels)
        mean = self._calculate_mean(labels.count("X"), labels.count("Y"))
        variance = self._calculate_variance(mean, len(labels))
        z_score = self._calculate_z_score(nruns, mean, variance)
        p_value = self._calculate_p_value(z_score)

        return p_value >= alpha, p_value

# Example usage
# series = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
series = [random.randint(1, 100) for _ in range(10)]
test = WaldWolfowitzRunsTest(series)
result = test.runs_test()
print("Is random:", result[0])
print("p-value:", result[1])

Hope it helps!


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