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# Correlations about random_walks

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The correlation coefficient between random walk and time has a characteristic shape of histogram or density function. Some findings has been revealed and it is desirable to be investigated more.

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• ランダムウォークの長さを変えて、分布が収束する様子を観察すると、長さに対して2次の収束を長さが128以下だと示すのに、それ以上だと長さ2048までのところで、1次の収束をするように見えました。その理由がまだよく分からないので、将来の課題の１つとしたいです。

理論的な考察で無くて、さらに数値計算をする場合は、さらに1000倍程度の計算をすることになります。

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### Correlations about random_walks

1. 1. Correlations About Random Walks A (single) corr. coef. > .8 or .9 unnecessary means high degree of relationship in time series 2014-06-06 Toshiyuki Shimono The correlation coefficient between two independt random walks has a characteristic shape of histogram or density function as shown above. A random walk is defined as its each increment at each time step is +1 or -1 with each probability 50%.
2. 2. Random walks are various. And their correlation coefficients have a broad distribution even if they are just chosen `randomly’.
3. 3. Remarks for time series analysis • Very often, random walks have very high correlation coefficients ρ such as > 0.8 or 0.9. • Thus, unless you know well about how numerals moves (such as economic indexes), do not take ρ as some relation degree. • High correlations in multiple may mean significant deviation from just a random chance, but some rigid methodology to detect that significance is desirable.
4. 4. § Numerical Calculations To investigate the distributions of correlation coefficients (1) between random walks and time, and also (2) between two independent random walks, Monte Carlo approach was employed and trillions of random variables are yielded from Mersenne Twister.
5. 5. The density functions for (1) corr. coef. of random walk and time (2) corr. coef. of two independent random walks • Mersenne Twister produced trillions of random variables to calculate billions of random walks here. • Each of vertical segments inside indicates 1.25% percentiles. 0.3099.. 1.018..1.018.. 0.6054..
6. 6. Main stats. • The two distributions seems to have the modes at ±0.910.. and 0, respectively. • The two distributions have the variances of 0.4341..=0.659..2 and 0.2405.. =0.490..2, respectively. • When only their positive parts are considered, the two distributions have the medians of 0.6613.. and 0.4136.., respectively. • For 0 < ε << 1, the density functions seems to be approximated as exp(negative const. ÷ ε2).
7. 7. Probabilities, percentiles  For the correlation coefficients btw. r.w. and time :  The probability being over 0.6 is 0.2823.. ≈ sin(0.6)/2  The probability being over 0.8 is 0.1582.. ≈ 1/401/2  The probability being over 0.81 is 0.1501..  The probability being over 0.8669.. ( ≈ π -1/8 ) is 0.1  For the correlation coefficients btw. two r.w. :  The prob. being over 0.4529 or below -0.4529 is 0.4529..  The 99.5 percentile is 0.9051..  The 99.95 percentile is 0.9481..  The 99.995 percentile is 0.9670..  The 99.9995 percentile is 0.9771..  The density function at 0 is 0.6055.. ≈ π -1/6 /2
8. 8. Conclusions and summary • The correlation coefficients above have wide distributions and their shapes are characteristic, which may have grave concerns about time series analysis. • The convergence behavior depends also on the length L of random walks, and L=512, 1024, 2048 are used to analyze the convergence, which is not well presented in this document. • Quadrillions (1015) of random numbers are desirable to determine more digits. Then we can easily compare discovered numerals and some meaningful mathematical numbers. • Mathematical analysis is desirable to determine the density functions of the distributions — as a matter of course! — and for establishing some new hypothesis tests.
9. 9. Appendix. • Raw data from calculations are presented here for the further analysis in the future.
10. 10. Percentile Tables Percentile tables of the absolute value of the correlation coefficients (1) between random walk and time (2) between two independent random walks +0% +1% +2% +3% +4% +5% +6% +7% +8% +9% 0% 0.00000 0.01613 0.03224 0.04834 0.06441 0.08044 0.09643 0.11234 0.12820 0.14400 10% 0.15971 0.17533 0.19085 0.20626 0.22156 0.23674 0.25178 0.26670 0.28148 0.29611 20% 0.31059 0.32491 0.33907 0.35307 0.36686 0.38053 0.39401 0.40731 0.42043 0.43335 30% 0.44610 0.45865 0.47101 0.48319 0.49518 0.50698 0.51858 0.53000 0.54123 0.55220 40% 0.56307 0.57371 0.58418 0.59445 0.60455 0.61446 0.62419 0.63374 0.64312 0.65232 50% 0.66135 0.67021 0.67891 0.68742 0.69580 0.70400 0.71209 0.71997 0.72772 0.73533 60% 0.74279 0.75011 0.75729 0.76434 0.77126 0.77805 0.78471 0.79126 0.79769 0.80400 70% 0.81020 0.81629 0.82228 0.82817 0.83396 0.83966 0.84528 0.85080 0.85625 0.86161 80% 0.86691 0.87213 0.87727 0.88239 0.88745 0.89245 0.89742 0.90236 0.90729 0.91222 90% 0.91713 0.92207 0.92707 0.93211 0.93732 0.94268 0.94834 0.95437 0.96114 0.96960 +0% +1% +2% +3% +4% +5% +6% +7% +8% +9% 0% 0.00000 0.00826 0.01652 0.02477 0.03303 0.04129 0.04955 0.05781 0.06607 0.07432 10% 0.08258 0.09084 0.09910 0.10736 0.11562 0.12387 0.13213 0.14039 0.14865 0.15691 20% 0.16516 0.17342 0.18168 0.18994 0.19820 0.20647 0.21475 0.22297 0.23125 0.23951 30% 0.24778 0.25605 0.26431 0.27258 0.28085 0.28913 0.29740 0.30568 0.31396 0.32225 40% 0.33053 0.33882 0.34712 0.35541 0.36372 0.37202 0.38034 0.38865 0.39699 0.40530 50% 0.41364 0.42199 0.43034 0.43870 0.44707 0.45545 0.46384 0.47225 0.48067 0.48910 60% 0.49755 0.50596 0.51444 0.52294 0.53147 0.54001 0.54858 0.55717 0.56578 0.57443 70% 0.58310 0.59181 0.60057 0.60936 0.61820 0.62708 0.63602 0.64502 0.65408 0.66321 80% 0.67243 0.68171 0.69113 0.70064 0.71025 0.72001 0.72991 0.73999 0.75026 0.76080 90% 0.77159 0.78270 0.79419 0.80615 0.81872 0.83208 0.84649 0.86252 0.88111 0.90511
11. 11. (1) Frequency distribution table of the absolute value of the correlation coefficient between random walks with the length 2048 and time. 1019620 1018456 1019738 1019155 1019477 1018350 1017541 1018397 1019744 1019610 1020347 1019912 1019809 1019366 1020079 1018360 1018681 1019116 1019079 1019001 1019518 1019547 1020546 1021081 1019455 1019470 1020458 1021345 1020650 1019621 1019112 1019295 1020733 1019688 1022002 1021391 1020548 1021067 1021078 1022718 1021095 1020390 1019218 1021225 1021311 1020060 1021508 1020667 1020830 1021836 1021070 1021211 1022527 1021645 1023239 1023166 1021930 1022523 1023502 1024473 1024305 1022056 1023056 1022534 1023513 1023531 1024793 1026272 1026001 1022309 1026272 1023789 1025114 1025317 1024976 1026933 1025946 1026098 1025811 1026435 1025826 1026637 1029633 1028261 1028735 1027057 1027797 1026614 1029649 1028241 1027701 1029665 1030240 1027753 1028782 1030491 1030782 1030562 1028729 1031417 1033709 1030921 1031445 1033754 1032794 1031375 1032197 1032499 1032132 1031421 1032555 1033543 1033550 1034419 1034122 1032093 1034654 1035163 1033929 1037131 1036494 1034059 1036326 1037575 1038153 1037274 1036146 1036807 1037303 1038107 1039338 1040173 1040556 1040382 1039625 1040280 1040100 1041068 1041691 1043562 1042532 1042069 1042436 1042148 1043099 1042086 1044459 1046082 1044850 1044888 1047343 1046047 1048240 1047091 1045741 1048560 1047772 1047058 1048822 1050309 1048703 1050823 1051253 1049533 1052499 1052546 1053606 1053034 1052275 1051661 1056236 1052759 1053434 1054605 1054334 1054020 1056833 1057423 1058973 1057032 1058056 1058963 1059421 1058578 1060704 1059608 1059539 1061166 1061685 1061159 1062106 1063769 1063745 1065263 1064796 1065495 1066731 1067584 1065760 1066769 1068070 1068484 1069055 1068024 1070818 1070352 1070109 1070013 1072269 1071028 1073649 1072594 1072980 1074294 1075129 1075806 1075019 1076992 1078148 1077343 1078951 1078516 1080848 1079582 1080749 1081447 1081551 1082712 1083700 1083152 1084803 1085095 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3056418 3058399 3071327 3075200 3083495 3094143 3104300 3110429 3117596 3124921 3132622 3140672 3149001 3159129 3163757 3171908 3178414 3183536 3192464 3202692 3204999 3211302 3220440 3224716 3234436 3240862 3248559 3253107 3256734 3266232 3269553 3276791 3281574 3284820 3291998 3299230 3301777 3306552 3311355 3316281 3319520 3324962 3327553 3330382 3331092 3338454 3336112 3343407 3341872 3345542 3344519 3344062 3344898 3345962 3344181 3346576 3345864 3345186 3340937 3337262 3334591 3329499 3325985 3318835 3317818 3309915 3303608 3296835 3288824 3276803 3271487 3257238 3248364 3233763 3223480 3206270 3191316 3179456 3158888 3141222 3124000 3103848 3080106 3057122 3034750 3009903 2981071 2949014 2925134 2889476 2859045 2823399 2787136 2750366 2710319 2668410 2623400 2580513 2534892 2483823 2434780 2377697 2323529 2266577 2206739 2144049 2076274 2013411 1943951 1873649 1796018 1723241 1642958 1567465 1487130 1401459 1317890 1233507 1146685 1056530 970443 883803 794925 710598 623934 543759 464832 390442 320485 256043 198816 147822 104250 69854 42988 23419 11265 4186 1136 190 12 0 0 Each of the integer part of the value times 1000 which ranges from 0 to 999 are counted. The counted number are arranged from left to right then top to bottom in 25x40 table form. The whole number is 1,644,000,000.
12. 12. (2) Frequency distribution table of the absolute value of the correlation coefficient between two independent random walks with the lengths 2048. 2670506 2667800 2669087 2670212 2671105 2671604 2669255 2667542 2668964 2669612 2671668 2674404 2670966 2669501 2667456 2669375 2668526 2668789 2671883 2667989 2664272 2668323 2665721 2667733 2668242 2669600 2670119 2668185 2666356 2668084 2670759 2669235 2663748 2666588 2667300 2667838 2672344 2669755 2672154 2667782 2667574 2672602 2669717 2672871 2667977 2667807 2670580 2668612 2667639 2667654 2665703 2666537 2668696 2666316 2668744 2669663 2667864 2667257 2669367 2665722 2665675 2672237 2668861 2666813 2668482 2669814 2667583 2667192 2667804 2669969 2667400 2671099 2668821 2669170 2668805 2668996 2668855 2669799 2671140 2668060 2669786 2670402 2665861 2666843 2669024 2667830 2669509 2667474 2668825 2669659 2666548 2670933 2667597 2670690 2666825 2666655 2669251 2668134 2667791 2665317 2671228 2667345 2667821 2671365 2667693 2669223 2670530 2667892 2669356 2669464 2668098 2667187 2669406 2669545 2671074 2671960 2668665 2667095 2667287 2668990 2669127 2668833 2666344 2669438 2665977 2668023 2670216 2667733 2669556 2670605 2670257 2670989 2668403 2666887 2667441 2669254 2667197 2670650 2668912 2669237 2669784 2666284 2665103 2668031 2668143 2669750 2670035 2667108 2667273 2665861 2665997 2669987 2668834 2669384 2667305 2667018 2669692 2670671 2669546 2670981 2668321 2668669 2667932 2668568 2667106 2666871 2668206 2669172 2666022 2668120 2669285 2668254 2667619 2667437 2665593 2668940 2666796 2671563 2667115 2671315 2668899 2666276 2665629 2672164 2671898 2669476 2668780 2662913 2670192 2668711 2667894 2667855 2665698 2667488 2670859 2666162 2668303 2668693 2665854 2667268 2663158 2668152 2668592 2667789 2665316 2668965 2665940 2665478 2666167 2667792 2668793 2668771 2666986 2665089 2666246 2665129 2666909 2666149 2668703 2664621 2666500 2667837 2665853 2668269 2665902 2666131 2669083 2669591 2665613 2665281 2664040 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2504434 2503835 2500077 2501634 2500968 2498910 2496831 2494260 2492218 2491705 2491318 2487914 2487604 2484993 2482436 2480616 2480837 2478771 2478692 2475445 2476513 2472403 2472216 2472583 2467095 2465696 2462663 2461993 2459273 2457074 2455621 2454629 2455062 2451887 2450259 2452017 2445645 2443228 2443120 2438633 2437276 2437667 2436242 2431242 2430466 2428301 2425498 2421714 2423616 2421030 2417315 2413585 2413045 2408720 2407958 2408040 2405240 2401138 2399493 2399540 2394467 2391988 2390715 2387817 2386763 2386239 2381935 2376248 2377612 2374968 2368455 2370800 2367525 2367235 2360439 2359677 2357213 2351608 2349480 2349545 2344973 2343766 2338880 2334861 2336912 2334397 2330775 2326693 2323688 2319830 2317820 2313950 2312664 2310808 2307511 2304535 2299462 2294820 2297540 2291189 2286959 2283922 2281104 2280729 2273435 2269812 2268737 2262064 2260671 2259422 2254243 2251741 2247436 2245851 2240314 2234477 2232739 2231139 2224386 2223080 2216480 2217824 2211294 2207481 2201639 2199879 2195212 2193448 2187206 2181879 2180163 2174582 2172585 2168280 2163794 2158451 2154792 2150967 2147494 2146296 2137218 2131981 2128891 2123638 2120560 2117445 2109866 2104700 2100048 2097783 2092168 2085302 2081641 2080296 2073302 2067951 2061723 2057345 2053581 2048858 2042222 2036133 2030608 2025894 2020826 2014195 2011389 2004646 2000201 1995002 1989127 1979622 1977689 1971478 1967473 1961388 1954379 1949811 1941506 1932891 1930389 1922986 1921722 1910083 1907424 1901199 1893551 1886629 1882157 1873622 1869000 1861504 1852338 1847080 1839329 1833010 1823988 1818271 1809887 1808857 1797928 1790935 1783114 1775976 1768085 1762200 1752206 1746471 1739495 1732281 1723723 1716785 1707760 1699628 1691554 1683811 1673125 1668560 1658173 1649934 1643920 1634828 1625016 1617211 1609113 1601067 1590880 1581066 1571606 1565245 1553902 1542654 1536457 1525198 1517525 1509891 1496975 1487894 1479854 1467829 1457218 1450267 1436824 1428719 1419234 1408649 1399939 1389480 1378329 1367001 1359596 1348065 1336936 1324817 1314191 1304867 1291096 1282243 1272258 1260605 1248442 1235685 1224077 1214842 1200217 1190741 1176515 1167320 1155080 1140278 1129850 1118126 1107184 1093695 1080206 1066121 1055857 1044963 1030714 1016494 1005667 993463 980620 967607 952577 940039 926759 914242 902386 889650 874066 860195 846871 833427 818575 807486 793371 778810 765201 751795 736937 723201 708986 696152 681654 669400 654521 639953 626091 612201 595431 583741 570819 556296 542086 528511 515773 500883 485506 472182 459136 444646 431718 418309 405069 390986 377260 363800 351054 337609 325743 313019 300552 288760 275771 263457 251740 240493 227610 216254 205722 195900 184421 172445 163942 152964 143630 134076 125921 116736 107618 99371 91782 83988 76500 69782 63494 56948 50720 45805 40651 35715 31110 26602 23525 20039 17030 14092 11767 9544 7672 6095 4724 3607 2678 1950 1390 939 606 421 206 101 61 27 13 2 1 0 0 0 0 0 0 0 Each of the integer part of the value times 1000 which ranges from 0 to 999 are counted. The counted number are arranged from left to right then top to bottom in 25x40 table form. The whole number is 2,204,000,000.