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"""Compute statistics on the digits of pi.
This uses precomputed digits of pi from the website
of Professor Yasumasa Kanada at the University of
Tokoyo: https://www.super-computing.org/
Currently, there are only functions to read the
.txt (non-compressed, non-binary) files, but adding
support for compression and binary files would be
straightforward.
This focuses on computing the number of times that
all 1, 2, n digits sequences occur in the digits of pi.
If the digits of pi are truly random, these frequencies
should be equal.
"""
import os
from urllib.request import urlretrieve
import numpy as np
from matplotlib import pyplot as plt
# Top-level functions
def fetch_pi_file(filename):
"""This will download a segment of pi from super-computing.org
if the file is not already present.
"""
ftpdir = "ftp://pi.super-computing.org/.2/pi200m/"
if os.path.exists(filename):
# we already have it
return
else:
# download it
urlretrieve(ftpdir + filename, filename)
def compute_one_digit_freqs(filename):
"""
Read digits of pi from a file and compute the 1 digit frequencies.
"""
d = txt_file_to_digits(filename)
freqs = one_digit_freqs(d)
return freqs
def compute_two_digit_freqs(filename):
"""
Read digits of pi from a file and compute the 2 digit frequencies.
"""
d = txt_file_to_digits(filename)
freqs = two_digit_freqs(d)
return freqs
def reduce_freqs(freqlist):
"""
Add up a list of freq counts to get the total counts.
"""
allfreqs = np.zeros_like(freqlist[0])
for f in freqlist:
allfreqs += f
return allfreqs
def compute_n_digit_freqs(filename, n):
"""
Read digits of pi from a file and compute the n digit frequencies.
"""
d = txt_file_to_digits(filename)
freqs = n_digit_freqs(d, n)
return freqs
# Read digits from a txt file
def txt_file_to_digits(filename, the_type=str):
"""
Yield the digits of pi read from a .txt file.
"""
with open(filename) as f:
for line in f.readlines():
for c in line:
if c != '\n' and c != ' ':
yield the_type(c)
# Actual counting functions
def one_digit_freqs(digits, normalize=False):
"""
Consume digits of pi and compute 1 digit freq. counts.
"""
freqs = np.zeros(10, dtype='i4')
for d in digits:
freqs[int(d)] += 1
if normalize:
freqs = freqs / freqs.sum()
return freqs
def two_digit_freqs(digits, normalize=False):
"""
Consume digits of pi and compute 2 digits freq. counts.
"""
freqs = np.zeros(100, dtype='i4')
last = next(digits)
this = next(digits)
for d in digits:
index = int(last + this)
freqs[index] += 1
last = this
this = d
if normalize:
freqs = freqs / freqs.sum()
return freqs
def n_digit_freqs(digits, n, normalize=False):
"""
Consume digits of pi and compute n digits freq. counts.
This should only be used for 1-6 digits.
"""
freqs = np.zeros(pow(10, n), dtype='i4')
current = np.zeros(n, dtype=int)
for i in range(n):
current[i] = next(digits)
for d in digits:
index = int(''.join(map(str, current)))
freqs[index] += 1
current[0:-1] = current[1:]
current[-1] = d
if normalize:
freqs = freqs / freqs.sum()
return freqs
# Plotting functions
def plot_two_digit_freqs(f2):
"""
Plot two digits frequency counts using matplotlib.
"""
f2_copy = f2.copy()
f2_copy.shape = (10, 10)
ax = plt.matshow(f2_copy)
plt.colorbar()
for i in range(10):
for j in range(10):
plt.text(i - 0.2, j + 0.2, str(j) + str(i))
plt.ylabel('First digit')
plt.xlabel('Second digit')
return ax
def plot_one_digit_freqs(f1):
"""
Plot one digit frequency counts using matplotlib.
"""
ax = plt.plot(f1, 'bo-')
plt.title('Single digit counts in pi')
plt.xlabel('Digit')
plt.ylabel('Count')
return ax
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