Runs Test for Randomness Testing in Python

Hello there,

This is my first post on this site. I've been running a Turkish/English blog on Medium platform for 28 months now and have just discovered this site. I absolutely fell in love with its retroesque design, so here we are. As you can see, I am looking forward for job opportunities abroad. Feel free to contact me for any vacancies.

Runs Test Implementation in Python | by Sıddık Açıl | Medium

Sıddık Açıl ・ Oct 28, 2019 ・ 2 min read

Medium

Methodology

Runs test is a hypothesis testing based methodology that is widely used in statistical analysis to test if a set of values are generated randomly or not. It is a hypothesis test so we have a pair of a null hypothesis and an alternative one.

Null hypothesis: The values are randomly generated.

Alternative hypothesis: The values are NOT randomly generated.

A Z score for hypothesis can be acquired by simply following the general formula:

(Observed-Excepted) / Standard Deviation

The score is then tested against the confidence interval(two-tailed) we specify. If the value is higher, we conclude that our alternative hypothesis holds. Otherwise, if the value is lower, we cannot say anything about the randomness of data at this significance level. We will be using %95 confidence interval(alpha = 0.05) through the rest of this article.

Definitions and Formulas

• A run: A series of positive or negative values:

Data: [1, -2, -3, 4, 5, 6, –7]

Runs: [[1], [-2, -3], [4, 5, 6], [-7]]

• Score Formula

(Number of runs - Excepted Value for Number of Runs) / Standard Deviation

• Expected Value Formula for Runs Test

n_p = Number of positive values
n_n = Number of negative values
Excepted value of runs = (2 * n_p * n_n) / (n_p + n_n) + 1

• Standard Deviation Formula for Runs Test

n_p = Number of positive values
n_n = Number of negative values
Excepted value of runs = (2 * n_p * n_n) / (n_p + n_n) + 1

Test Data

I will using the data provided by NIST.

Octave/Matlab Implementation and Results

Using online Octave:

1. Upload text file

2. Load ‘statistics’ package
   

   pkg load statistics
   

3. Import data to array
   

   x = importdata("test.txt")
   

4. Run runstest
   

   [h, v, stats] = runstest(x, median(x))
   

Python Implementation

Implement the test using Python.

	# H0: the sequence was produced in a random manner
	# Ha: the sequence was not produced in a random manner
	# Read test data
	import numpy as np
	import scipy.stats as st
	# Assuming number of runs greater than 10
	def runs_test(d, v, alpha = 0.05):
	# Get positive and negative values
	mask = d > v
	# get runs mask
	p = mask == True
	n = mask == False
	xor = np.logical_xor(p[:-1], p[1:])
	# A run can be identified by positive
	# to negative (or vice versa) changes
	d = sum(xor) + 1 # Get number of runs
	n_p = sum(p) # Number of positives
	n_n = sum(n)
	# Temporary intermediate values
	tmp = 2 * n_p * n_n
	tmps = n_p + n_n
	# Expected value
	r_hat = np.float64(tmp) / tmps + 1
	# Variance
	s_r_squared = (tmp*(tmp - tmps)) / (tmps*tmps*(tmps-1))
	# Standard deviation
	s_r = np.sqrt(s_r_squared)
	# Test score
	z = (d - r_hat) / s_r
	# Get normal table
	z_alpha = st.norm.ppf(1-alpha)
	# Check hypothesis
	return z, z_alpha
	filename = "test.txt"
	# Load array
	d = np.array(np.loadtxt(filename))
	print(runs_test(d, np.median(d)))

view raw runs_test.py hosted with ❤ by GitHub

Running the code produces the results below (Calculated Z Score, Z Score at %95 confidence):

(2.8355606218883844, 1.6448536269514722)

Since our test score is higher, alternative hypothesis holds. This means that our values are genuinely random.

Thanks for reading. Any corrections are welcome.