Birisi zaman serisi benzerliğini belirlemek için dinamik zaman eğrilmesini açıklayabilir mi?

Zaman serilerini karşılaştırmak için dinamik zaman çözgüsü ölçüsünü kavramaya çalışıyorum. Bunun gibi üç zaman serisi veri setim var:

T1 <- structure(c(0.000213652387565, 0.000535045478866, 0, 0, 0.000219346347883, 
0.000359669104424, 0.000269469145783, 0.00016051364366, 0.000181950509461, 
0.000385579332948, 0.00078170803205, 0.000747244535774, 0, 0.000622858922454, 
0.000689084895259, 0.000487983408564, 0.000224744353298, 0.000416449765747, 
0.000308388157895, 0.000198906016907, 0.000179549331179, 9.06289650172e-05, 
0.000253506844685, 0.000582896161212, 0.000386473429952, 0.000179839942451, 
0, 0.000275608635737, 0.000622665006227, 0.00036075036075, 0.00029057097196, 
0.000353232073472, 0.000394710874285, 0.000207555002076, 0.000402738622634, 
0, 0.000309693403531, 0.000506521463847, 0.000226988991034, 0.000414164423276, 
9.6590360282e-05, 0.000476689865573, 0.000377572210685, 0.000378967314069, 
9.25240562546e-05, 0.000172309813044, 0.000447627573859, 0, 0.000589333071408, 
0.000191699415317, 0.000362943471554, 0.000287549122975, 0.000311688311688, 
0.000724112961622, 0.000434656621269, 0.00122292103424, 0.00177549812586, 
0.00308008213552, 0.00164338537387, 0.00176056338028, 0.00180072028812, 
0.00258939580764, 0.00217548948513, 0.00493015612161, 0.00336344416683, 
0.00422716412424, 0.00313360554553, 0.00540144648906, 0.00425728829246, 
0.0046828437633, 0.00397219463754, 0.00501656412683, 0.00492700729927, 
0.00224424911165, 0.000634696755994, 0.00120550276557, 0.00125313283208, 
0.00164551010813, 0.00143575017947, 0.00237006940918, 0.00236686390533, 
0.00420336269015, 0.00329840900272, 0.00242005185825, 0.00326554846371, 
0.006217237596, 0.0037103784586, 0.0038714672861, 0.00455830066551, 
0.00361747518783, 0.00304147465438, 0.00476801760499, 0.00569875504121, 
0.00583855136233, 0.0050566695728, 0.0042220072126, 0.00408237321963, 
0.00255222610833, 0.00123507616303, 0.00178136133508, 0.00147434637311, 
0.00126742712294, 0.00186590371937, 0.00177226406735, 0.00249154653853, 
0.00549127279859, 0.00349072202829, 0.00348027842227, 0.00229555236729, 
0.00336862367661, 0.00383477593952, 0.00273999412858, 0.00349618180145, 
0.00376108175875, 0.00383351588171, 0.00368928059028, 0.00480028982882, 
0.00388823582602, 0.00745054380406, 0.0103754506287, 0.00822677278011, 
0.00778350981989, 0.0041831792162, 0.00537228238059, 0.00723645609231, 
0.0144428396845, 0.00893333333333, 0.0106231171714, 0.0158367059652, 
0.01811729548, 0.0207095263821, 0.0211700064641, 0.017604180993, 
0.0165804327375, 0.0188679245283, 0.0191859923629, 0.0269251008595, 
0.0351239669421, 0.0283510318573, 0.0346557651212, 0.0270022042616, 
0.0260845175767, 0.0349758630112, 0.0207069247809, 0.0106362024818, 
0.00981093510475, 0.00916507201128, 0.00887198986058, 0.0073929115025, 
0.00659077291791, 0.00716191546131, 0.00942304513143, 0.0106886280007, 
0.0123527175979, 0.0171022290546, 0.0142909490656, 0.0157642220699, 
0.0265140538974, 0.0194395354708, 0.0241685144124, 0.0229897123662, 
0.017921889568, 0.0155115839714, 0.0145263157895, 0.017609281127, 
0.0157671315949, 0.0190258751903, 0.0138453217956, 0.00958058335108, 
0.0122924304507, 0.00929741151611, 0.00885235535884, 0.00509319462505, 
0.0061314863177, 0.0063104189044, 0.00729117134253, 0.010843373494, 
0.0217755443886, 0.0181687353841, 0.0155402963498, 0.017310022503, 
0.0214746959003, 0.026357827476, 0.0194751217195, 0.0196820590462, 
0.0184317400812, 0.0130208333333, 0.0128666035951, 0.0120045731707, 
0.0122374253228, 0.00874940561103, 0.0114368092263, 0.00922893718369, 
0.00479041916168, 0.00644107774653, 0.00775830595108, 0.00829578041786, 
0.00681348095875, 0.00573782551125, 0.00772002058672, 0.0112488083889, 
0.00908907291456, 0.0157722638969, 0.00994270306707, 0.0134179772039, 
0.0126050420168, 0.0113648781554, 0.0153894803415, 0.0126959699913, 
0.0116655865198, 0.0112065745237, 0.0122006737686, 0.010251878038, 
0.010891174691, 0.0148273273273, 0.0138516532618, 0.0136552722011, 
0.00986993819758, 0.0097852677358, 0.00889011089726, 0.00816723383568, 
0.00917641660931, 0.00884466556108, 0.0182179529646, 0.0183156760639, 
0.0217806648835, 0.0171099125907, 0.0186579938377, 0.019360390076, 
0.0144603654529, 0.0177730696798, 0.0153226598566, 0.0134016909516, 
0.0126480805202, 0.0115501519757, 0.0127156322248, 0.0124326204138, 
0.0240245215806, 0.0130234933606, 0.0144222706691, 0.00854005693371, 
0.0053560967445, 0.00504132231405, 0.00288778877888, 0.00593526847816, 
0.00455653279644, 0.00433014040152, 0.00535770564135, 0.0131095962244, 
0.0126319758673, 0.0154982879798, 0.0125940464508, 0.0169948745616, 
0.0257535512184, 0.0256175663312, 0.0265191262043, 0.0228974403622, 
0.0193122555411, 0.0165794768612, 0.015658837248, 0.0168208578638, 
0.0129912843282, 0.0119498443154, 0.0112663755459, 0.00838112042347, 
0.00925767186696, 0.0113408269771, 0.0210861519924, 0.0156036134684, 
0.0121687119728, 0.011006497812, 0.0107891491985, 0.0134615384615, 
0.0147229755909, 0.015756893641, 0.0176257128046, 0.016776075857, 
0.0169553999263, 0.0179193118984, 0.0190055672874, 0.0183088625509, 
0.0155489923558, 0.0152507401094, 0.0160748342567, 0.0161532350605, 
0.0139190952588, 0.0161469457497, 0.0118186629035, 0.0109259765092, 
0.00950587391265, 0.00928986154533, 0.00815520645549, 0.00702576112412, 
0.00709539362541, 0.00827287768869, 0.0104688211197, 0.0130375888927, 
0.0160891089109, 0.0188415910677, 0.0203265044814, 0.0183175033921, 
0.0139940353292, 0.0124648170487, 0.0131685758095, 0.00957428620277, 
0.0119647893342, 0.00835800104475, 0.0101892285298, 0.00904207699194, 
0.00772134522992, 0.00740740740741, 0.00776823249863, 0.00642254601227, 
0.00484237572883, 0.00361539964823, 0.00414811817078, 0.00358072916667, 
0.00433306007729, 0.00485008818342, 0.00905280804694, 0.00931847250137, 
0.00779271381259, 0.00779912497622, 0.00908230842006, 0.0058152538582, 
0.0102777777778, 0.00807537012113, 0.00648535564854, 0.0145492582731, 
0.00694127317563, 0.00759878419453, 0.00789242911429, 0.00635050701629, 
0.00785233530492, 0.00607964332759, 0.00531968282646, 0.00361944157187, 
0.00305157155935, 0.00276327909119, 0.00318820364651, 0.00184464029514, 
0.00412550211703, 0.00516567972786, 0.00463655399342, 0.00702897308418, 
0.0100714154917, 0.00791168353266, 0.00959190791768, 0.00736, 
0.00738007380074, 0.012573964497, 0.0117919562013, 0.00842919476398, 
0.00778887565289, 0.00623967700496, 0.0062232955601, 0.00447815755803, 
0.00511135450894, 0.00502557659517, 0.00330328263712), .Tsp = c(1, 
15.9583333333333, 24), class = "ts")

T2 <- structure(c(0, 0, 0, 0, 0.000109673173942, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9.66183574879e-05, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9.43930526713e-05, 
0, 0, 0, 8.95255147717e-05, 0, 0, 0, 0, 0.000191699415317, 0.000207792207792, 
0, 0, 0, 0.00019727756954, 0.000205338809035, 0.000205423171734, 
0.000704225352113, 0.000450180072029, 0.000493218249075, 0.000120860526952, 
0.000410846343468, 0.000384393619066, 0.000643264105863, 0.000189915487608, 
0.000915499404925, 0.000185099490976, 0.000936568752661, 0.000451385754266, 
0.000757217226692, 0.000273722627737, 0.000187020759304, 0.000211565585331, 
0.000141823854772, 9.63948332369e-05, 0.000117536436295, 0.000287150035894, 
0, 0, 0.000400320256205, 0.000388048117967, 0.000345721694036, 
0.000296868042155, 0.000609533097647, 0.000424043252412, 0.000290360046458, 
0.000546996079861, 0.000556534644282, 0.00036866359447, 0.000275077938749, 
0.000964404699281, 0.00152310035539, 0.00113339145597, 0.00061570938517, 
0.000362877619523, 0.000472634464505, 0.000102923013586, 0.000187511719482, 
0.000294869274622, 0.00011522064754, 0.000248787162582, 0, 0.00035593521979, 
0.000392233771328, 0.000551166636046, 0.000165727543918, 0.000143472022956, 
0.00012030798845, 0.000438260107374, 0.000195713866327, 0.000184009568498, 
0.000537297394108, 0.000365096750639, 0.000102480016397, 0.000452857531021, 
0.000180848177955, 0.000770745910765, 0.00219818869252, 0.000357685773048, 
0.000362023712553, 0.000660501981506, 0.000419709560984, 0.000488949735967, 
0.00177758026886, 4e-04, 0.000475661962898, 0.000879816998064, 
0.0014942099365, 0.00378173960022, 0.00274725274725, 0.00192545729611, 
0.0016462841016, 0.00176238855484, 0.00260780478718, 0.00447289949132, 
0.0034435261708, 0.00290522941294, 0.002694416055, 0.0041329904482, 
0.00729244577412, 0.0296930503689, 0.00982375036117, 0.00453023439039, 
0.00327031170158, 0.00221573169503, 0.00211237853823, 0.00108719286801, 
0.00131815458358, 0.000983008004494, 0.00132253265002, 0.00227790432802, 
0.00247054351957, 0.00307455803228, 0.0029314767314, 0.00222755311857, 
0.00492610837438, 0.00454430699318, 0.00753880266075, 0.00671845475541, 
0.00590490003108, 0.00288356368698, 0.00294736842105, 0.00248601615911, 
0.00197089144936, 0.00326157860404, 0.00302866414278, 0.00202256759634, 
0.00258788009489, 0.00169043845747, 0.00137000737696, 0.000433463372345, 
0.000908368343363, 0.000805585392052, 0.00142653352354, 0.00189328743546, 
0.00558347292016, 0.00161899622234, 0.00162631008312, 0.00276960360048, 
0.00585673524553, 0.00519169329073, 0.0045125282033, 0.00562344544176, 
0.00322815786733, 0.00330528846154, 0.00255439924314, 0.00285823170732, 
0.00240894199268, 0.00218735140276, 0.00201826045171, 0.00168701002282, 
0.000460617227084, 0.00127007166833, 0.00109529025192, 0.000819336337567, 
0.00158170093685, 0.000588494924231, 0.00120089209127, 0.00305052430887, 
0.00161583518481, 0.00211579149837, 0.0010111223458, 0.00346270379455, 
0.00228091236495, 0.00207627581685, 0.00295140718878, 0.0022121765894, 
0.00240718451995, 0.00224131490474, 0.0031867431485, 0.00176756517897, 
0.00233382314807, 0.00178303303303, 0.00169794459339, 0.00162778079219, 
0.000737939304492, 0.00135906496331, 0.000733205022454, 0.000875060768109, 
0.00114705207616, 0.000967385295744, 0.00182179529646, 0.00359130903214, 
0.00420328620558, 0.00446345545843, 0.00376583361862, 0.00659687365553, 
0.00433810963586, 0.00353107344633, 0.00333955407131, 0.00341788091383, 
0.0024939877082, 0.00538428137212, 0.00906989151698, 0.00773778473309, 
0.0210421671775, 0.00859720803541, 0.00511487506289, 0.00406669377796, 
0.00117164616286, 0.00206611570248, 0.00107260726073, 0.00148381711954, 
0.000741761152909, 0.00104973100643, 0.00110305704381, 0.00209753539591, 
0.00452488687783, 0.00486574157506, 0.00850507033039, 0.0101159967629, 
0.0163991223005, 0.0150452373691, 0.0156443766097, 0.0112310639039, 
0.00635593220339, 0.00627766599598, 0.00583041812427, 0.00622371740959, 
0.00624897220852, 0.00420769166036, 0.00305676855895, 0.00291133656815, 
0.00120006857535, 0.00501806503412, 0.00490575781048, 0.00593119810202, 
0.00226874291018, 0.00304999336958, 0.00339087546239, 0.00541958041958, 
0.00445563734986, 0.00431438754455, 0.0038016243304, 0.0037928519329, 
0.00491460867428, 0.00460782305959, 0.00508734881935, 0.00300725278613, 
0.00390896455872, 0.00367811967345, 0.00953591862683, 0.00529614264278, 
0.00243584167029, 0.00427167876976, 0.00291056623743, 0.00227624510607, 
0.00439422473321, 0.00232246538633, 0.00317623830372, 0.00263466042155, 
0.00180200473026, 0.00190912562047, 0.0034896070399, 0.00338638672536, 
0.00548090523338, 0.00697836706211, 0.00720230473752, 0.00746268656716, 
0.00367056664373, 0.0032167269803, 0.00523135203391, 0.00299196443837, 
0.00299119733356, 0.00287306285913, 0.00154657933042, 0.00214861235452, 
0.00163006177076, 0.00157407407407, 0.00137086455858, 0.00124616564417, 
0.000790591955727, 0.00107484854407, 0.00121408336706, 0.00108506944444, 
0.00105398758637, 0.000881834215168, 0.00184409052808, 0.00237529691211, 
0.0013637249172, 0.00190222560396, 0.00264900662252, 0.00156564526951, 
0.00263888888889, 0.00183531139117, 0.00303347280335, 0.0120768352986, 
0.00365330167139, 0.00351443768997, 0.00263080970476, 0.0029703984431, 
0.00265143789517, 0.0014185834431, 0.00150557061126, 0.00144777662875, 
0.00111890957176, 0.000716405690308, 0.000797050911627, 0.000512400081984, 
0.000868526761481, 0.00113392969636, 0.00134609632067, 0.00240013715069, 
0.00128181651712, 0.00110395584177, 0.00156958493198, 0.00208, 
0.00184501845018, 0.00110946745562, 0.000736997262582, 0.00208250694169, 
0.00229084578026, 0.00137639933933, 0.00111462010032, 0.000822518735149, 
0.00200803212851, 0.000987166831194, 0.00041291032964), .Tsp = c(1, 
15.9583333333333, 24), class = "ts")

T3 <- structure(c(0.00192287148809, 0.00149812734082, 0.00192410475681, 
0.00151122625216, 0.00120640491336, 0.00167845582065, 0.00121261115602, 
0.000802568218299, 0.00109170305677, 0.00250626566416, 0.00273597811218, 
0.00242854474127, 0.00160915430002, 0.00124571784491, 0.00192943770673, 
0.00329388800781, 0.00191032700303, 0.00156168662155, 0.00174753289474, 
0.0014917951268, 0.00143639464943, 0.000543773790103, 0.000929525097178, 
0.00141560496294, 0.000966183574879, 0.000719359769805, 0.00190740419629, 
0.00137804317869, 0.00197177251972, 0.001443001443, 0.00203399680372, 
0.00158954433063, 0.00256562068285, 0.00228310502283, 0.00302053966975, 
0.00227352221056, 0.00263239393001, 0.00202608585539, 0.00272386789241, 
0.00269206875129, 0.0027045300879, 0.00276480122033, 0.00405890126487, 
0.00341070582662, 0.00351591413768, 0.00336004135436, 0.00358102059087, 
0.00257289879931, 0.00235733228563, 0.00239624269146, 0.00136103801833, 
0.000862647368926, 0.00145454545455, 0.00168959691045, 0.00246305418719, 
0.0020964360587, 0.00335371868219, 0.00390143737166, 0.00349219391947, 
0.00334507042254, 0.00255102040816, 0.00332922318126, 0.00386753686246, 
0.00246507806081, 0.00432442821449, 0.00312442565705, 0.00408318298357, 
0.00375354756019, 0.00416473854697, 0.00263942103023, 0.0028888688273, 
0.00321817321344, 0.00310218978102, 0.002150738732, 0.00296191819464, 
0.00134732662034, 0.00221708116445, 0.00152797367184, 0.00157932519742, 
0.00220077873709, 0.00207100591716, 0.00260208166533, 0.00310438494373, 
0.00311149524633, 0.00385928454802, 0.00292575886871, 0.00222622707516, 
0.00329074719319, 0.00282614641262, 0.00287542899545, 0.00221198156682, 
0.00311754997249, 0.00315623356128, 0.00287696733796, 0.00296425457716, 
0.00263875450787, 0.00208654631226, 0.00179601096512, 0.00164676821737, 
0.00206262891431, 0.00235895419697, 0.00241963359834, 0.0028610523697, 
0.00516910352976, 0.00160170848905, 0.00254951951363, 0.00275583318023, 
0.00298309579052, 0.00286944045911, 0.00288739172281, 0.00394434096636, 
0.00254428026226, 0.00285214831171, 0.0034924330617, 0.00246440306681, 
0.00266448042632, 0.00389457476678, 0.00253187449136, 0.00171276869059, 
0.00184647850171, 0.00134132164893, 0.00153860077835, 0.000990752972259, 
0.00117518677075, 0.00312927831019, 0.00188867903566, 0.0024, 
0.00269541778976, 0.00263945099419, 0.00242809114681, 0.00378173960022, 
0.00274725274725, 0.00165039196809, 0.00211665098777, 0.00290275761974, 
0.00149017416411, 0.00105244693913, 0.00309917355372, 0.00240432779002, 
0.00297314875035, 0.0015613519471, 0.00196335078534, 0.00227707441479, 
0.00279302706347, 0.00295450068938, 0.00316811446091, 0.00211501661799, 
0.00168990283059, 0.00195694716243, 0.00131815458358, 0.00112343771942, 
0.00214911555629, 0.00157701068863, 0.00171037628278, 0.00230591852421, 
0.00183217295713, 0.00102810143934, 0.00130396986381, 0.00151476899773, 
0.00188470066519, 0.00220449296662, 0.00238267895991, 0.00238639753406, 
0.00147368421053, 0.00113942407292, 0.0018192844148, 0.00152207001522, 
0.00151433207139, 0.00117096018735, 0.000862626698296, 0.00095087163233, 
0.00137000737696, 0.00119202427395, 0.00170319064381, 0.000805585392052, 
0.0012680297987, 0.00189328743546, 0.00186115764005, 0.000719553876597, 
0.000903505601735, 0.000865501125151, 0.00210241778045, 0.00146432374867, 
0.00130625816411, 0.0011895749973, 0.00135374362178, 0.00120192307692, 
0.00160832544939, 0.0015243902439, 0.00240894199268, 0.00218735140276, 
0.00230658337338, 0.00188548179022, 0.0016582220175, 0.00263086274154, 
0.00155166119022, 0.00204834084392, 0.00194670884536, 0.00308959835221, 
0.00154400411734, 0.00152526215443, 0.00343364976772, 0.00269282554337, 
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0.00167193015047, 0.00201060135259, 0.00219058050383, 0.00233330341919, 
0.000963457435827), .Tsp = c(1, 15.9583333333333, 24), class = "ts")

T1 ve T2'nin birbiriyle ilişkili olduğunu biliyorum ve bunları gerçek gerçeği olarak görüyorum, böylece herhangi bir mesafe ölçüsü (T1, T2) 'nin (T2, T3) ve (T1, T3)' den daha yakın olduğunu söylemelidir. Ancak, dtwR'de kullanırken , aşağıdakileri alıyorum:

> dtw(T1, T2, k = TRUE)$distance; dtw(T1, T3, k = TRUE)$distance; dtw(T3, T2, k = TRUE)$distance
[1] 1.107791
[1] 1.568011
[1] 0.4102962

Birisi en yakın komşu sorgular için Dinamik Zaman Çözümü'nü nasıl kullanacağını açıklayabilir mi?

Dinamik zaman atlama , veri kümenizde belirli bir varsayım yapar: bir vektör, diğerinin doğrusal olmayan zamana bağlı serisidir. Ancak gerçek değerlerin aynı ölçekte olduğunu varsayar.

Diyelim ki: , , , .a ( x ) = 1 ⋅ günah ( 0.01 ∗ x ) b ( x ) = 1 ⋅ günah ( 0.01234 ∗ x ) c ( x ) = 1000 ⋅ günah ( 0.01 ∗ x $x=1..10000$$a(x)=1\cdot\sin(0.01*x)$$b(x)=1\cdot\sin(0.01234*x)$$c(x)=1000\cdot\sin(0.01*x)$

Daha sonra DTW için ve son derece benzer olurken, ve neredeyse Manhattan mesafesiyle aynıdır. Bununla birlikte, bir frekans analizi yaparsanız, ve frekansları ile aynı olur ve sadece büyüklük bakımından farklılık gösterirken, ve açıkça farklı bir frekansa sahiptir.b a c a c a $a$$b$$a$$c$$a$$c$$a$$b$

DTW, tüm zaman serisi eşleştirme ihtiyaçlarınızı çözmek için sihirli silahınız değildir. İlgilendiğiniz benzerlik konusunda özel varsayımlar yapar . Bu, verilerinizle eşleşmezse iyi çalışmaz. Paylaştığınız veri serisinden yola çıkarak, zamansal hizalamaya (DTW'nin yaptığı) değil, aslında bazı uygun normalizasyona ve belki de dört katlı dönüşümlere ihtiyacınız yoktur. Çaydanlık geçiş mesafeleri de sizin için iyi sonuç verebilir, örneğin bakınız:

• Eşik Sorgularına Dayalı Zaman Serisinde Benzerlik Araması
  Johannes Aßfalg, Hans-Peter Kriegel, Peer Kröger, Peter Kunath, Alexey Pryakhin ve Matthias Renz, EDBT 2006

1980'lerde dinamik zaman çarpıtma, konuşma tanımada şablon eşleştirme için kullanılan yöntemdi. Amaç, analiz edilen konuşmanın zaman serilerini genellikle tam kelimelerin yer aldığı kayıtlı şablonlarla eşleştirmeye çalışmaktı. Zorluk, insanların farklı oranlarda konuşmasıdır. DTW, bilinmeyen kalıbı şablona kaydetmek için kullanıldı. Buna "lastik levha" eşleşmesi deniyordu. Temel olarak, küresel uyumu optimize etmek için zaman serilerinin yerel olarak nasıl genişletilebileceğine dair bazı kısıtlı olasılıkları araştırırsınız. Bu yaklaşımın, gizli Markov modelleri ile hemen hemen aynı şey olduğu gösterilmiştir.

İlk olarak, "dinamik zaman bükme metriği" diyorsunuz, ancak DTW bir mesafe ölçüsüdür, ancak bir metrik değildir (üçgen eşitsizliğine uymaz).

Kağıt [a] DTW'yi 43 veri kümesindeki 12 alternatifle karşılaştırır, DTW çoğu sorun için gerçekten iyi çalışır.

DTW hakkında daha fazla bilgi edinmek istiyorsanız, http://www.cs.ucr.edu/~eamonn/Keogh_Time_Series_CDrom.zip Keoghs öğreticisine göz atabilirsiniz (500 meg uyarısı)

Geçiş peggy.

Ayrıca SAX hakkında bir eğitim var http://www.cs.ucr.edu/~eamonn/SIGKDD_2007.ppt

[a] Xiaoyue Wang, Hui Ding, Goce Trajcevski, Peter Scheuermann, Eamonn J. Keogh: Zaman Serisi Verileri için Temsil Yöntemleri ve Mesafe Ölçümlerinin Deneysel Karşılaştırması CoRR abs / 1012.2789: (2010)