Statistical Analysis of CF Digits of $\pi$

Large scale statistical analysis of the randomness of the continued fraction digits on $\pi$

INSTALL

1. Clone this repo

First we will clone this repo and set up a virtual environment nad download all the required packages.

git clone https://github.com/akhatua2/pi_cf_digits_statistics.git
cd pi_cf_digits_statistics
python3 -m venv igl_venv
source igl_venv/bin/activate
pip install --upgrade pip
pip install -r requirements.txt

2. Download the data

Since the raw and processed data is very large for github here is the google drive link where you can download the data from. Please down this and move the data and raw_data folders into the pi_cf_digits_statistics folder. For running the provided scripts you only need the data folder. Please see in Misc section for how to convert the raw_data into the processed format.

Google Drive link -

https://drive.google.com/drive/folders/1C7-0ixzZUMBNtBLvP6AvTjj2qstHGARq?usp=sharing

Chi-Square Test on CF digits of $\pi$

The results from the runs in the following table are in the results folder. To run this test yourself

python chi_square_test.py 

Use the --help flag to see the input parameters.

Number of digits	Number of blocks	Number of groups	KS-test on Chi-Square Vals
30B	30K	10	0.418
30B	30K	50	0.470
1B	10K	10	0.414
1B	10K	50	0.184
1B	1K	10	0.077
1B	1K	50	0.840
10M	10	10	0.8053
10M	10	50	0.6403
10M	100	10	0.8657
10M	100	50	0.1637
100M	1K	10	0.1165
100M	1K	50	0.3459
100M	100	10	0.1819
100M	100	50	0.5548

TODO:

Optimal values -> 10, 50 - groups, 10M, 1M, 100K - blocksize (6 total runs)

Chi-Square-Pairs Test on CF digits of $\pi$

The results from the runs in the following table are in the results folder. To run this test yourself

python chi_square_pairs_test.py 

Use the --help flag to see the input parameters.

Number of digits	Number of blocks	Number of pairs	KS-test on Chi-Square Vals
1B	1K	5	3.1474e-8
1B	10K	5	4.2490e-74
10M	10	5	0.8372
10M	10	5	0.0020
100M	1K	5	4.9684e-11
100M	100	5	0.1267

TODO:

Optimal values -> 10, 50 - groups, 10M, 1M, 100K - blocksize (6 total runs)

Random and Pi Walk Tests on CF digits of $\pi$

To run this test for random walks mod 4

python walk.py --use_rand True --mod 4

To run this test for random walks mod 6

python walk.py --use_rand True --mod 6

To run this test for pi walks mod 4

python walk.py --mod 4

To run this test for pi walks mod 6

python walk.py --mod 6

Number of digits	Number of blocks	Type or digits	Mod	MinMax	Mean	Variance
30M	1K	Random_1	4	(12971844.338, 103153685.155)	40769453.987	210973198162334.23
30M	1K	Random_2	4	(12876544.325, 108576945.287)	40586421.642	208297314523442.82
30M	1K	$\pi$	4	(12832847.898, 110378456.214)	40659703.784	208373904890053.84
100K	1K	Random	4	(0.525, 5.208)	1.771	0.537
100K	1K	$\pi$	4	(0.562, 4.682)	1.746	0.513
100K	1K	Random	6	(0.869, 7.718)	3.014	1.118
100K	1K	$\pi$	6	(0.878, 8.388)	2.988	1.206

Random and Pi Sites visited Tests on CF digits of $\pi$

To run this test for random walks mod 4

python sites_visited.py --use_rand True --mod 4

To run this test for random walks mod 6

python sites_visited.py --use_rand True --mod 6

To run this test for pi walks mod 4

python sites_visited.py --mod 4

To run this test for pi walks mod 6

python sites_visited.py --mod 6

Number of digits	Number of blocks	Type or digits	Mod	MinMax	Mean	Variance	Std Dev	Runtime
30M	1000	Random_1	4	(23045836, 25101953)	24318792.021	119744160752.9294	346040.692	16hrs
30M	1000	Random_2	4	(21384756, 24987201)	24128376.236	137956343984.6548	371424.748	16hrs
30M	1000	$\pi$	4	(22725453, 25109780)	24280608.064	148322090202.4152	385126.070	18hrs
30M	1000	Random_1	6	(28237148, 29019277)	29623879.345	350466.2355	592.001	18hrs
30M	1000	Random_2	6	(29465349, 29129376)	29821463.987	357655.1923	598.042	18hrs
30M	1000	$\pi$	6	(29744353, 29748462)	29746267.162	379760.6432	616.247	20hrs
100K	1K	Random	4	(90854, 176387)	147905.047	134260828.4412	11587.097	-
100K	1K	$\pi$	4	(97143, 172690)	146080.107	137343287.3529	11719.3552	-
100K	1K	Random	6	(17784, 21548)	20094.057	207318.9787	455.322	-
100K	1K	$\pi$	6	(17880, 21135)	19854.228	193608.8428	440.0100	-

Cut-off value on CF digits of $\pi$

Number of digits	Cut-of number(m)	Blocksize(n)	Theoretical value(Pr_T)	Experimental value(Pr_E)	Difference
10M	600	1000	0.9097688732593405	0.907	-0.0027688732593404985
10M	1000	1000	0.7637854600148878	0.7666	0.0028145399851121633
10M	2000	1000	0.513942043357302	0.5132	-0.0007420433573019913
10M	5000	1000	0.25065200098279206	0.2507	4.79990172079225e-05
10M	10000	1000	0.1343483455185307	0.136	0.0016516544814693113
10M	100000	10000	0.13434585722629822	0.128	-0.006345857226298213
10M	1000000	10000	0.01432338377020792	0.02	0.00567661622979208
10M	10000000	10000	0.0014416548888593894	0.001	-0.0004416548888593894
100M	10000	1000	0.1343483455185307	0.13569	0.0013416544814693065
100M	100000	1000	0.014323412100246347	0.01449	0.00016658789975365282
100M	1000000	1000	0.0014416551754701246	0.00169	0.00024834482452987553

Extreme value statistics of $\pi$ for Quartiles in Max_Digits

Number of digits	Blocksize(n)	Quartiles(1-Pr)	Experimental value(m_E)	Theoretical value(m_T)
10M	100	10%	63	62.6554495327263
10M	100	25%	104	104.06844905028039
10M	100	50%	208	208.13689810056078
10M	100	75%	501	501.48937978426767
10M	100	90%	1377	1369.2938306930143
10M	1000	10%	620	626.554495327263
10M	1000	25%	1046	1040.684490502804
10M	1000	50%	2072	2081.368981005608
10M	1000	75%	5016	5014.893797842677
10M	1000	90%	13460	13692.938306930142
10M	10000	10%	6090	6265.54495327263
10M	10000	25%	10479	10406.84490502804
10M	10000	50%	20311	20813.68981005608
10M	10000	75%	49379	50148.937978426766
10M	10000	90%	126163	136929.38306930143
100M	100	10%	63	62.6554495327263
100M	100	25%	104	104.0684490502803
100M	100	50%	208	208.13689810056078
100M	100	75%	502	501.48937978426767
100M	100	90%	1372	1369.2938306930143
100M	1000	10%	628	626.554495327263
100M	1000	25%	1044	1040.684490502804
100M	1000	50%	2081	2081.368981005608
100M	1000	75%	5037	5014.893797842677
100M	1000	90%	13818	13692.938306930142
100M	10000	10%	6444	6265.54495327263
100M	10000	25%	10654	10406.84490502804
100M	10000	50%	20970	20813.68981005608
100M	10000	75%	51469	50148.937978426766
100M	10000	90%	135329	136929.38306930143