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The pattern frequency distribution theory: a mathematic establishment toward rational and reliable pattern mining

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Abstract

In big data science, the classic frequent pattern mining is fundamental to various pattern mining applications. Extensive research on this mining has been undertaken for nearly 30 years but left with no reliable mining approach. One of the main issues is the lack of study on the imperative pattern frequency distribution theory. With an emphasis on mining reliability and methodological change, this paper makes up the absent theory, which consists of a bundle of findings on the frequency distribution properties. The primary property is that the frequency distribution curves from different pattern generation modes are quasi-concave and ultimately resultant bell-shaped curves over large datasets. All the findings are well-formed with no exogenous input but rigorous mathematical proofs that every classic pattern mining approach should observe. This paper thus builds up a solid block of the theoretical foundation for rational and ultimately reliable pattern mining. Moreover, the findings inspire interesting rethinking and new conceptions not merely in pattern mining but also extended deeply to set theory and combinatorics. With this inspiration plus the pure mathematic nature of the explorations presented, the contributions of this study may not be restricted to pattern mining only but spring to data science in general or even broader.

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Notes

  1. Note: A real number sequence \((s_n)\) is a Cauchy sequence if \(\forall \epsilon > 0, \exists t \in N\), such that if \( m, n \ge t\), then \(|s_n - s_m|< \epsilon \). The Cauchy criterion says that any Cauchy sequence is convergent, and vice versa, any convergent series of real numbers is a Cauchy sequence [59,60,61].

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Acknowledgements

Special thanks are to Dr. Bipin C. Desai for his help and advice in the early stage of the writing of this paper.

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    Tongyuan Wang

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Appendix: The detailed empirical results from the “Retail” dataset

Appendix: The detailed empirical results from the “Retail” dataset

Total number of elements n = 16470. Total number of tuples u = 88162. Longest pattern length \(\alpha \) = 76.

The \(g_i\) distribution:

\(3016\,\,5516\,\,6919\,\,7210\,\,6814\,\,6163\,\,5746\,\,5143\,\,4660\,\,4086\,\,3751\,\,3285\,\,2866 2620\,\,2310\,\,2115\,\,1874\,\,1645\,\,1469\,\,1290\,\,1205 \,\,981\,\,887\,\,819\,\,684\,\,586\,\,582\,\,472 480\,\,355\,\,310\,\,303\,\,272\,\, 234\,\,194\,\,136\,\,153\,\,123\,\,115\,\,112\,\,76\,\,66\,\,71\,\,60\,\,50\,\,44\,\,37 37\,\,33\,\,22\,\,24\,\,21 \,\,21\,\,10\,\,11\,\,10\,\,9\,\,11\,\,4\,\,9\,\,7\,\,4\,\,5\,\,2\,\,2\,\,5\,\,3\,\,30\,\,0\,\,1\,\,0\,\,1\,\,1\,\,0\,\,1\)

The \(H_k\) Series: \((1)\,\,908576\,\,(2)\,\,7164335\,\,(3)\, 52502539\,\,(4)\,\,366817927\,\,(5)\,\,2447321444\,\,(6) \,15534598332 \,\,(7)\,\,93307736462\,\,(8)\,\,527550301625\,\,(9)\,\,2796416534241\,\,(10) \,13863139450195\,\,(11)\,\,64204046715896\,\,(12)\,\,277757200264229\,\,(13) \,1.12312584494064E\!+\!15\,\,(14)\,\,4.24904654295735E\!+\!15\,\,(15) \,1.5058885990449E\!+\!16 \,\,(16)\,\,5.00625023811958E\!\!+\!\!16\,\,(17) 1.56327472119759E\!\!+\!\!17\,\,(18)\,\, 4.59121121980175E\!+\!17\,(19)\,\, 1.26976301242088E\!\!+18\,\,(20)\,\,3.31067777753649E+18\,\,(21)\,\,8.14636975894685E+18\,\,(22)\,\,1.8935525717633E+19\,\,(23)\,\,4.16131440789675E+19\,\,(24)\,\,8.65288386954046E+19\,\,(25)\,\,1.70361679656958E+20\,\,(26)\,\,3.17787016348576E+20\,\,(27) \,5.61949830792223E +20\,\,(28)\,\,9.4248316680112E+20\,\,(29)\,\,1.49988185541216E+21\,\,(30)\,\,2.26577690430042E+21\,\,(31)\,\, 3.2501290738274E +21\,\,(32)\,\,4.42825143223911E+21\,\,(33)\,\,5.73212226482143E+21\,\,(34)\,\,7.05069716806149E+21\,\,(35)\,\,8.24219004470137E +21\,\,(36)\,\,9.15769085268638E+21\,\,(37)\,\,9.67116815427317E+21\,\,(38)\,\,9.70768639718059E+21\,(39) \,9.26120299243453E +\!21\,\,(40)\,\,8.39616377302037E+21\,\,(41)\,\,7.23232941850291E\!+\!21\,\,(42)\,\,5.91772992838759E\!+\!21\,\,(43) 4.59811839985694E\!+\!21\,\,(44)\,\,3.39150713519183E\!+\!21\,\,(45) \,2.37356853011541E\!+\!21\,\,(46)\,\,1.57537354482505E\!+21\,\,(47) \,9.91013257283473E\!+\!20\,\,(48)5.9046404820075E\!+\!20\,\,(49) \,3.32957641309953E\!+\!20\,\, (50)\,\,1.77535631861495E\!+\!20\,\,(51) \,8.94240979386786E+19\,\,(52)\,\,4.25025368313273E+19\,\,(53) \,1.90382440924376E+19\,\,(54)\,\,8.0257650807726E+18\,\,(55) \,3.17919252128806E+18\,\,(56)\,\,1.18129875755773E+18\,\,(57) \,4.10926627354641E+17\,\,(58)\,\,1.33528383786941E +17\,\,(59) \,4.04304582589712E+16\,\,(60)\,\,1.13749193542012E+16\,\,(61) \,2.96417842579272E+15\,\,(62)\,\,712836643924391 (63)\,\,157536096738717\,\,(64) \,31838940145963\,\,(65)\,\,5851270021055\,\,(66)\,\,971240578752\,\,(67) 144437457258\,\,(68) \,\,19056211853\,\,(69)\,\,2203389079\,\,(70)\,\,219831833\,\,(71) \,18542293\,(72)\,\,1285749\,\,(73)\,\,70375\,\,(74)\,\,2851\,\,(75)\,\,76\,\,(76)\,\,1\)

The \(H_k\) Quasi Concavity: + Increase, - decrease, = equality.

\(1<>\,\,\,\,2+\,\,3+\,\,4+\,\,5+\,\,6+\,\,7+\,\,8+\,\,9+\,\,10+\,\,11+\,\,12+\,\,13+\,\,14+\,\,15+\,\,16+\,\,17+\,\,18+ 19+\,\,20+\,\,21+\,\,22+\,\,23+\,\,24+\,\,25+\,\,26+\,\,27+\,\,28+\,\,29+\,\,30+\,\,31+\,\,32+\,\,33+\,\,34+\,\,35+ 36+\,\,37+\,\,38+\,\,39-\,\,40-\,\,41-\,\,42-\,\,43-\,\,44-\,\,45-\,\,46-\,\,47-\,\,48-\,\,49-\,\,50-\,\,51-\,\,52- 53-\,\,54-\,\,55-\,\,56-\,\,57-\,\,58-\,\,59-\,\,60-\,\,61-\,\,62-\,\,63-\,\,64-\,\,65-\,\,66-\,\,67-\,\,68-\,\,69- 70-\,\,71-\,\,72-\,\,73-\,\,74-\,\,75-\,\,76-\)

The \(H_k\) Genuine Concavity: (\(H_k - (H_{k-1} + H_{k+1})/2 \ge 0?\))

Theoretic concavity domain = [33, 43]; exact = [33, 43], detailed as below:

\(1<>\,\,2\,\, :(-19541222.5)\,\,3:(-134488592)\,\,4:(-883094064.5) \,\,5:(-5503386685.5)\,\,6:(-32342930621)\,\,7: (-178234713516.5) \,8:(-917311833726.5)\,\,9:(-4398928341669)\,\,10:(-19637092174873.5) \,11:(-81606123141316)12:(-315907745564039) \,13:(-1.14027602667015E\!+\!15)\,\,14:(-3.84195937473746E\!+\!15) 15:(-1.20968884716276E\!+\!16)\,\,16:(-3.56306766739083E+16) \,\,17:(-9.8264340060926E\!+\!16)\,\,18: (-2.53924120290146E\!+\!17) \,19:(-6.15136437337449E\!+\!17)\,\,20:(-1.39738860814738E\!+\!18) 21:(-2.97673198863789E\!+\!18)\,\,22:(-5.94423120132421E\!+\!18) \,23:(-1.11190381275513E\!+\!19)\,\,24: (-1.94585731725583E\!+\!19) \,25:(-3.1796247865032E\!+\!19)\,\,26:(-4.83687388760149E\!+\!19) 27:(-6.81852607826247E\!+\!19)\,\,28:(-8.84326763010704E\!+\!19)\,\, 29:(-1.04248180138614E\!+\!20)\,\,30: (-1.09228560319354E\!+\!20) \,31:(-9.68850944423698E\!+\!19)\,\,32:(-6.28742370853074E\!+\!19) 33:(-7.35203532886717E\!+\!18)\,\,34:(6.35410133000913E\!+\!19) \,35:(1.37996034327435E\!+\!20) \,\,36:(2.01011753199109E\!+\!20) \,37:(2.38479529339683E\!+\!20)\,\,38:(2.41500823826744E\!+\!20) 39:(2.09277907334049E\!+\!20)\,\,40:(1.49397567551648E\!+\!20) \,41:(7.53825677989307E\!+\!19) \,\,42:(2.50601920766935E\!+\!18) \,43:(-5.65001319327722E\!+\!19)\,44:(-9.43363297943486E\!+\!19) \,45: (-1.09871809893028E\!+\!20)\,\,46:(-1.06917348874388E\!+\!20) \,47:(-9.19055392294298E\!+\!19) \,48:(-7.1521401095963E\!+\!19) 49:(-5.10421987211691E\!+\!19)\,\,50:(-3.36552377628212E\!+\!19) \,51: (-2.05949864077324E\!+\!19)\,\,52:(-1.17286341842308E\!+\!19) \,53:(-6.22590686361234E\!+\!18) \,\,54:(-3.08295322609023E\!+\!18) \,55:(-1.4243393978771E\!+\!18)\,\,56:(-6.13760816763625E\!+\!17) \,57: (-2.46486943317692E\!+\!17)\,\,58:(-9.21501590198654E\!+\!16) \,59:(-3.20211933115998E\!+\!16) \,\,60:(-1.03223989881807E\!+\!16) \,61:(-3.07969957327008E\!+\!15)\,\,62:(-848020617341325) 63:(-214801695296460)\,\,64:(-49854743233923)\,\,65:(-10553820341302.5) \,66:(-2026613160404.5)\,\,67: (-350710938044.5)\,\,68:(-54264211315.5) \,69:(-7434632764)\,\,70:(-891133853)\,\,71:(-92016498)\,\,72: (-8020585) \,73:(-573925)\,\,74:(-32374.5)\,\,75:(-1350)\)

The \(R_k\) Series:

\(0.21027292525152.9\,\,0.297094051410328\,\,0.382831246282321 \,\,0.463316718039126\,\,0.536416236147605\) \(0.600644667263741 \,\,0.65552176787858\,\,0.701570924935985\,\,0.739920275015639 \,\,0.771879594163558 \,\,0.798676329287781\,\,0.82134661868907 \,\,0.840718401553246\,\,0.857434446016042\,\,0.871986691039956\, \,\,0.884749696089944\,\,0.896009184273812\,0.905984998255989\, \,\,0.914848923844976\,\,0.922738141516824 0.929765099200748 \,\,0.936024549137197\,\,0.941598411268332\,\,0.94655903067817 \,\,0.950971295213339 0.954893979352975\,\,0.958380588604126 \,\,0.961479900011198\,\,0.964236331011297\,\,0.96669022077285 \,\,0.968878073679284\,\,0.970832791317424\,\,0.972583904574083 \,\,0.974157808823637\,\,0.975578000710482 \,\,0.976865313167724 \,\,0.978038144975912\,\,0.979112681618661 \,\,0.980103104973037 \,\,0.981021790211479 0.981879489047814\,\,0.98268549907259 \,\,0.983447819379726\,\,0.984173293000019\,\,0.984867736850611 \,\,0.985536060009938\,\,0.98618237115967\,\,0.986810076020608 \,\,0.987421965565713\,0.988020295733908 \,\,0.988606859302998 \,\,0.989183050515657\,\,0.989749922993555\,\, 0.990308241423762 \,\,0.990858527459561 \,\,0.991401100244955\,\,0.991936111947395 \,\,0.992463578665651\,\,0.992983407067619\,\,0.993495417104725 \,\,0.993999361143785\,\,0.994494939852474\,\,0.994981815169571 \,\,0.995459620685058\,\,0.995927969747206 \,\,0.996386461603664 \,\,0.996834685870954\,\,0.997272225611996\,\,0.997698659284314 \,\,0.998113561803096\,\, 0.99851650494359\,\,0.998907057287231 \,\,0.999284783895796\,\,0.9996492458786391\)

The \(R_k\) Monotonic: + Increase, - decrease, = equality.

\(=\,\,1+\,\,2+\,\,3+\,\,4+\,\,5+\,\,6+\,\,7+\,\,8+\,\,9+\,\,10+\,\,11+\,\,12+\,\,13+\,\,14+\,\,15+\,\,16+\,\,17+\,\,18+\,\,19+ 20+\,\,21+\,\,22+\,\,23+\,\,24+\,\,25+\,\,26+\,\,27+\,\,28+\,\,29+\,\,30+\,\,31+\,\,32+\,\,33+\,\,34+\,\,35+\,\,36+ 37+\,\,38+\,\,39+\,\,40+\,\,41+\,\,42+\,\,43+\,\,44+\,\,45+\,\,46+\,\,47+\,\,48+\,\,49+\,\,50+\,\,51+\,\,52+\,\,53+ 54+\,\,55+\,\,56+\,\,57+\,\,58+\,\,59+\,\,60+\,\,61+\,\,62+\,\,63+\,\,64+\,\,65+\,\,66+\,\,67+\,\,68+\,\,69+\,\,70+ 71+\,\,72+\,\,73+\,\,74+\,\,75+\)

The accumulative frequency \(w_0\) = 1.08160582031538E+23

The sum of odd length pattern frequencies \(H_{ odd}\) = 5.4080291015769E+22

The sum of even length pattern frequencies \(H_{ even}\) = 5.4080291015769E+22

The \(h_k\) Series:

\((1)\,\,88162\,\,(2)\,\,820414\,\,(3)\,\,6343921\,\,(4)\,\,46158618\,\,(5)\, 320659309\,\,(6) \,2126662135\,\,(7)\,\,13407936197\,\,(8)\,\,79899800265\,\,(9)\,\,447650501360\,\,(10) \,2348766032881\,\,(11)\,\,11514373417314\,\,(12)\,\,52689673298582\,\,(13) \,225067526965647(14)\,\,898058317974992\,\,(15)\,\,3.35098822498236E\!+\!15\,\,(16) \,1.17078977654666E\!+\!16\,\,(17)\,\,3.83546046157292E\!+\!16\,\,(18) \,1.1797286750403E+17\,\,(19)\,\,3.41148254476145E\!+\!17\,\,(20) \,\,9.28614757944738E\!+\!17\,\,(21)\,\,2.38206301959175E\!+\!18\,\,(22) \,\,5.7643067393551E+18\,\,(23)\,\,1.31712189782779E+19\,\,(24) \,2.84419251006897E+19\,\,(25)\,\,5.8086913594715E+19\,\,(26) \,\,1.12274766062243E\!+\!20\,\,(27)\,\,2.05512250286333E\!+\!20\,(28) \,3.56437580505891E\!+\!20\,\,(29)\,\,5.86045586295229 E+20\,\,(30) \,9.13836269116928E+20\,\,(31)\,\,1.35194063518349E+21\,\,(32) \,1.8981884386439E+21\,\,(33)\,\,2.53006299359521 E+21\,\,(34) \,3.20205927122623E+21\,\,(35)\,\,3.84863789683527E+21\,\,(36) \,4.3935521478661E+21\,\,(37)\,\,4.76413870482027 E+21\,\,(38) \,4.90702944945289E+21\,\,(39)\,\,4.80065694772769E+21\,\,(40) \,4.46054604470683E+21\,\,(41)\, 3.93561772831 354E+21\,\,(42) \,3.29671169018937E+21\,(43)\,\,2.62101823819822E+21\,\,(44) \,1.97710016165872E+21\,\,(45)\,\,1.414406973 53311E+21\,\,(46) \,9.59161556582305E+20\,\,(47)\,\,6.1621198824275E+20\,\,(48) \,3.74801269040722E+20\,\,(49)\,\,2.156627791 60028E+20\,\,(50) \,1.17294862149926E+20\,\,(51)\,\,6.02407697115692E+19\,(52) \,2.91833282271095E+19\,\,(53)\,\,1.3319208 6042178E+19\,\,(54) \,5.71903548821977E+18\,\,(55)\,\,2.30672959255284E+18\,\,(56) \,8.72462928735225E+17\,\,(57)\,\,3.088358 28822501E+17\,\,(58) \,1.02090798532141E+17\,\,(59)\,\,3.14375852548001E+16\,\,(60) \,8.99287300417102E+15\,\,(61)\,\,2.38204 635003018E+15\,\,(62)\,\,582132075762540\, \,(63)\,\,130704568161851\,\,(64)\,\,26831528576866\,\,(65)\,\,5007411569097\,\,(66) \,84385 8451958\,\,(67)\,\,127382126794\,\,(68)\,\,17055330464\,\,(69)\,2000881389\,\,(70) \,202507690\,(71)\,\,17324143\,\,(72)\,\,1218\,150\,\,(73) \,\,67599\,\,(74)\,\,2776\,\,(75)\,\,75 \,(76)\,\,1\)

The \(h_k\) Quasi Concavity: + Increase, - decrease, = equality.

\(1<>\,\,\,\,2+\,\,3+\,\,4+\,\,5+\,\,6+\,\,7+\,\,8+\,\,9+\,\,10+\,\,11+\,\,12+\,\,13+\,\,14+\,\,15+\,\,16+\,\,17+\,\,18+ 19+\,\,20+\,\,21+\,\,22+\,\,23+\,\,24+\,\,25+\,\,26+\,\,27+\,\,28+\,\,29+\,\,30+\,\,31+\,\,32+\,\,33+\,\,34+\,\,35+ 36+\,\,37+\,\,38+\,\,39-\,\,40-\,\,41-\,\,42-\,\,43-\,\,44-\,\,45-\,\,46-\,\,47-\,\,48-\,\,49-\,\,50-\,\,51-\,\,52- 53-\,\,54-\,\,55-\,\,56-\,\,57-\,\,58-\,\,59-\,\,60-\,\,61-\,\,62-\,\,63-\,\,64-\,\,65-\,\,66-\,\,67-\,\,68-\,\,69- 70-\,\,71-\,\,72-\,\,73-\,\,74-\,\,75-\,\,76-\)

The \(h_k\) Genuine Concavity: (\(h_k - (h_{k-1} + h_{k+1})/2 \ge 0?\))

Concavity domain = [33, 43], detailed as below:

\(1<>\,\,\,\,2:(-2395627.5)\,\,3:(-17145595)\,\,4:(-117342997)\,\,5:(-765751067.5) \,6:(-4737635618)\,\,7: (-27605295\) \(003)\,\,8:(-150629418513.5) \,9:(-766682415213)\,\,10:(-3632245926456)\,\,11:(-16004846248417.5) \,12: (-65601276892898.5)\,\,13:(-250306468671140)\,\,14:(-889969557999009) \,15:(-2.95198981673845E\!+\!15)\,\,16: (-9.14489865488914E\!+\!15) \,17:(-2.64857780190192E\!+\!16)\,\,18:(-7.17785620419068E\!+\!16) \,\,19:(-1.82145558248 24E\!+\!17)\,\,20:(-4.3299087908921E\!+\!17) \,21:(-9.64397729058168E\!+\!17)\,\,22:(-2.01233425957972E\!+\!18) \,23:(-3.93189694174449E\!+\!18)\,\,24:(-7.18714118580676E\!+\!18) \,25:(-1.22714319867515E\!+\!19)\,\,26:(-1.95248158782 805E\!+\!19) \,27:(-2.88439229977343E\!+\!19)\,\,28:(-3.93413377848903E\!+\!19) \,29:(-4.90913385161801E\!+\!19)\,\,30: (-5.51568416224334E\!+\!19) \,31:(-5.407171869692E\!+\!19)\,\,32:(-4.28133757454501E\!+\!19) \,\,33:(-2.0060861339 8568E\!+\!19)\,\,34:(1.27088260109896E\!+\!19) \,\,35:(5.08321872891022E\!+\!19)\,\,36:(8.71638470383339E\!+\!19) \,37:(1.13847906160774E\!+\!20)\,\,38:(1.2463162317891E\!+\!20) \,39:(1.16869200647833E\!+\!20)\,\,40:(9.24087066862158E\!+\!19) \,\,41:(5.69888608654323E\!+\!19)\,\,42:(1.83937069334968E\!+\!19) \,43:(-1.5887687725827E\!+\!19)\,\,44:(-4.0612444 2069455E\!+\!19) \,\,45:(-5.37238855874031E\!+\!19)\,\,46:(-5.6147924305625E\!+\!19) \,47:(-5.07694245687632E\!+\!19)\,\,48:(-4.11361146606667E\!+\!19) \,\,49:(-3.03852864352963E\!+\!19)\,\,50:(-2.06569122858727E\!+\!19) \,51:(-1.29983254769 484E\!+\!19)\,\,52:(-7.59666093078404E\!+\!18) \,\,53:(-4.13197325344678E\!+\!18)\,\,54:(-2.0939336101 6557E\!\!+\!\!18) \,55:(-9.8901961592466E\!\!+\!\!17)\,\,56:(-4.35319781952443E\!\!+\!17) \,57:(-1.78441034811182E\!\!+\!17)\,\,58:(-6.80459085065098E\!+\!16) \,59:(-2.41042505133557E\!+\!16)\,\,60:(-7.91694279824414E\!+\!15) \,61:(-2.40545618993661E\!+\!15)\,\,62: (-674243383333473) \,63:(-173777234007852)\,\,64:(-41024461288608)\,\,65:(-8830281945315) \,66:(-1723538395987.5)\,\,67:(-303074764417)\,\,68:(-47636173627.5) \,69:(-6628037688)\,\,70:(-806595076)\,\,71:(-84538777)\,\,72:(-7477721) \,73:(-542864)\,\,74:(-31061)\,\,75:(-1313.5)\)

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Wang, T. The pattern frequency distribution theory: a mathematic establishment toward rational and reliable pattern mining. Int J Data Sci Anal 16, 43–83 (2023). https://doi.org/10.1007/s41060-022-00340-1

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