Scaling and Warping in Time Series Querying Dear Reader: This file contains larger, full color versions of the images in “Scaling and Warping in Time Series

Scaling and Warping in Time Series Querying

Dear Reader: This file contains larger, full color versions of the images in “Scaling and Warping in Time Series Querying”. In addition, there are some extra experiments which we could not fit into the paper.

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Euclidean

DTW

Uniform Scaling

SWM

If we attempt simple Euclidean matching (after truncating the longer sequence) we get a large error because we are mapping part of the flight of one sequence to the takeoff drive in the other.

If we simply use DTW to match the entire sequences we get a large error because we are trying to explain part of the sequence in one attempt (the bounce from the mat) that simply does not exist in the other sequence.

If we attempt just uniform scaling, we get the best match when we stretch the shorter sequence by 112%. However the local alignment, particularly of the takeoff drive and up-flight is quite poor.

Finally, when we match the two sequences with SWM, we get an intuitive alignment between the two sequences. The global stretching (once again at 112%) allows DTW to align the small local differences. In this case, the fact that DTW needed to map a single point on time series onto 4 points in the other time series suggests an important local difference in one of these sequences. Inspection of the original videos suggest that the athlete misjudged his approach and attempted a clumsy correction just before his takeoff drive.

Indexing video: There is increasing interest in indexing sports data, both from sports fans who may wish to find particular types of shots or moves, and from coaches who are interested in analyzing their athletes performance over time. As a concrete example, we consider the high jump. We can automatically collect the athlete’s center of mass information from video and convert a time series (It is possible to correct for the cameras pan and tilt). We found that when we issued queries to a database of high jumps, we only got intuitive answers when doing SWM. It is easy to see why if we look at two particular examples from the same athlete, and consider all possible matching options, as shown in the Figure on the left. From top to bottom: Video plays in presentation mode

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C = candidatematch

Q = query

0 20 40 60 80 100 120

C

Q (rescaled 1.54 )

0 20 40 60 80 100 120 140

happ

y

happ

y

happ

y

happ

y

birt

h

birt

h

birt

hbirt

h

-day

-day

-day

-day

to

to to you

you

you

dear

----

- C

Q (rescaled 1.40)

140

Query by Humming: The need for both local and global alignment when working with music has been extensively demonstrated. For completeness we will briefly review it here. In the Figure on the left we demonstrate the problems with universally familiar piece of music, Happy Birthday to You. For clarity of illustration, the music was produced by the second author on a keyboard and converted into a pitch contour.

Two performances of Happy Birthday to You aligned with different metrics. Both performances were performed in the same key, but are shifted in the Y-axis for visual clarity.

(top) Because the query sequence was performed at a much faster tempo, direct application of DTW fails to produce an intuitive alignment.

(center) Rescaling the shorter performance by a scaling factor of 1.54 seems to improve the alignment, but note for example that the higher pitched note produced on the third “birth..” of the candidate is forced to align with the lower note of the third “happy..” in the query.

(bottom) Only the application of both uniform scaling and DTW produces the correct alignment.

Click to Play sound files

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0

2

4

6Time series in gray and query in blue

0 50 100 150 200 250-2

-1

0

1

2

3Query, and scaling bounds in green

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-1

0

1

2

3Time series, and SWM bounds in red

0 50 100 150 200 250-2

0

2

4

6The lower bound!!

Pruning Power

• The following slide shows how the pruning power of the proposed lower bounding measure varies as the lengths of data change on different datasets.– For a majority of datasets, the pruning power

increased with the length of data, suggesting that the proposed algorithm is likely to perform well in real-life environment, in which long sequences of data are collected for a long period of time.

– More than 60% of the datasets obtained a pruning power above 90%. All but two of the datasets exhibited a pruning power of over 60% at length 1024. Even at length 16, over 60% pruning power was achieved in three-fourths of the datasets.

Pruning Power

00.10.20.30.40.50.60.70.80.9

1

16 32 64 128 256 512 1024

Length of Data

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Average Pruning Power

• The following slide shows the pruning power averaged over all datasets; 87% of data sequences of length 1024 and 65% of data sequences of length 16 did not require computation of the actual time warping distances.

Average Pruning Power

0

0.2

0.4

0.6

0.8

1

16 32 64 128 256 512 1024

Length of Data

Ave

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Pruning Power – Raw NumbersPruning Power vs. Length of Data

Dimension

EEG ERP Data

Reality Check

ATTAS Ballbeam

Buoy Sensor

Burst Burstin Chaotic CSTR Darwin Earthquake EegEvaporat

or

160.3226

20

0.764575

0.841603

0.701505

0.883734

0.504575

0.876227

0.212102

0.912278

0.898905

0.700938

0.693372

0.506784

0.706061

320.3660

09

0.835115

0.877450

0.771041

0.931881

0.616600

0.912379

0.273170

0.928676

0.939355

0.758134

0.729497

0.686507

0.725354

640.3781

88

0.923458

0.929353

0.844811

0.960093

0.648186

0.959023

0.305665

0.934549

0.957856

0.771355

0.749861

0.727912

0.728513

1280.4052

50

0.935473

0.970443

0.889625

0.979563

0.728380

0.978911

0.394770

0.943150

0.967728

0.764390

0.760549

0.746065

0.753290

2560.4215

53

0.943048

0.971053

0.906387

0.981056

0.735730

0.984876

0.445970

0.952091

0.984185

0.771298

0.759630

0.822354

0.723838

5120.4225

01

0.952062

0.985270

0.960686

0.979073

0.766258

0.988171

0.501640

0.962880

0.989307

0.771883

0.746005

0.788726

0.789709

10240.4237

89

0.909400

0.976175

0.959490

0.982871

0.881104

0.990863

0.514037

0.966186

0.986775

0.769779

0.749507

0.804118

0.778116

Dimension

Foetal ECG

Glass Furnace

Great Lakes

Koski ECG

Leleccum Memory Network Ocean Ocean Shear Packet

PGT50 Alpha

PGT50 CDC15

Power Data

Power Plant

160.5823

66

0.667292

0.763617

0.788211

0.738788

0.767077

0.361056

0.838729

0.838729

0.454409

0.252942

0.201408

0.673431

0.763441

320.7544

60

0.805286

0.817176

0.868442

0.803527

0.873913

0.441143

0.905589

0.905589

0.485061

0.327846

0.309233

0.737997

0.801387

640.8250

94

0.827233

0.855985

0.914999

0.877705

0.946237

0.478237

0.947321

0.947321

0.533411

0.429885

0.378305

0.820879

0.878095

1280.8779

12

0.870167

0.907650

0.944805

0.924549

0.972812

0.546761

0.970419

0.970419

0.568309

0.519509

0.459351

0.864432

0.929772

2560.8881

85

0.871079

0.916292

0.972940

0.946995

0.983239

0.587142

0.983679

0.983679

0.606108

0.617305

0.561448

0.911212

0.941613

5120.8885

88

0.858204

0.924727

0.981762

0.964394

0.990447

0.612243

0.985778

0.985778

0.657920

0.682724

0.594279

0.937497

0.953994

10240.9372

12

0.850404

0.982488

0.988746

0.980213

0.993886

0.613137

0.986514

0.986514

0.692022

0.721404

0.678522

0.952692

0.969604

Dimension

Random Walk

Robot Arm Shuttle Soil Temp SpeechSpot Exrat

es

Standard & Poo

rSteamgen

Synthetic Control

Tide Tongue Winding Wool Average

160.6930

60

0.638651

0.848837

0.466393

0.802835

0.667351

0.708446

0.640215

0.197165

0.684809

0.702385

0.745298

0.782698

0.653535

320.7627

26

0.672360

0.887582

0.583468

0.813972

0.802003

0.815125

0.753073

0.242413

0.775340

0.821866

0.833679

0.897753

0.728029

640.8522

34

0.703463

0.927370

0.603629

0.865021

0.845587

0.898330

0.834774

0.357602

0.866745

0.860557

0.881309

0.934227

0.778302

1280.9092

91

0.657382

0.969701

0.635959

0.907711

0.917622

0.922461

0.939568

0.525450

0.902781

0.876713

0.905980

0.964483

0.819014

2560.9513

56

0.659780

0.975370

0.663489

0.914221

0.957046

0.950578

0.952791

0.647056

0.918573

0.883266

0.912312

0.980002

0.842435

5120.9717

88

0.636301

0.989539

0.667249

0.917585

0.965179

0.968773

0.963212

0.673337

0.932474

0.910305

0.929074

0.990091

0.857010

10240.9813

49

0.677865

0.996204

0.654225

0.916146

0.968208

0.978736

0.961485

0.714639

0.943143

0.909705

0.957939

0.996061

0.870275

Varying Scaling Factor

• The following slide shows the effect of varying the range of allowed scaling factors on pruning power.– Note the x-axis indicates the upper bound range of allowed

scaling factor.• The lower bound range of allowed scaling factor is the reciprocal of

the upper bound.• For instance, the label 2.0 indicates that the range of allowed

scaling factor is between 1 / 2.0 = 0.5 and 2.0.• In particular, the label 1.0 indicates that the time warping distance

was calculated without scaling.• It also implied that the size of the range was not increasing linearly.

– However, the important observation is that for all sizes of ranges, a pruning power of over 90% was achieved in nearly three-fourths of the datasets.

– For almost all datasets, the pruning powers never dropped below 60%.

Pruning Power vs. Scaling Factor

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1

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0Scaling Factor

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Varying Scaling Factor

• We note that vigorously fluctuating datasets are far less common than smooth datasets.– The following slide illustrates this claim by

showing the pruning power averaged over all the datasets, as the range of allowed scaling factor changes.

• For most ranges of scaling factors, the pruning powers achieved are above 90%.

Average Pruning Power vs. Scaling Factor

0

0.2

0.4

0.6

0.8

1

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Scaling Factor

Ave

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Pruning Power vs. Scaling Factor

SF EEG ERP DataReality Check

ATTASBallbea

mBuoy

SensorBurst Burstin Chaotic CSTR Darwin Earthquake Eeg Evaporat

or

1.0

0.995155 0.988530 0.999500 0.99165

2 0.9994

83 0.971044 0.999601

0.980451

0.999443 0.9996

19 0.998517 0.995555

0.997595

0.984193

1.1

0.569788 0.947583 0.998150 0.98273

8 0.9953

90 0.945588 0.998441

0.690906

0.992562 0.9982

02 0.871137 0.833163

0.892153

0.896910

1.2

0.510637 0.936655 0.996421 0.96994

8 0.9936

98 0.917838 0.997110

0.628075

0.984201 0.9954

61 0.841203 0.821058

0.880753

0.852248

1.3

0.487403 0.925715 0.989586 0.97042

0 0.9908

93 0.925152 0.996125

0.592877

0.981369 0.9922

92 0.809179 0.770709

0.808965

0.860910

1.4

0.374507 0.911637 0.979298 0.95933

6 0.9825

31 0.881410 0.993280

0.503659

0.967504 0.9899

82 0.747959 0.709324

0.771187

0.817780

1.5

0.423789 0.909400 0.976175 0.95949

0 0.9828

71 0.881104 0.990863

0.514037

0.966186 0.9867

75 0.769779 0.749507

0.804118

0.778116

1.6

0.599908 0.954994 0.990557 0.96355

7 0.9917

32 0.947908 0.997250

0.706783

0.988641 0.9937

66 0.868481 0.813228

0.837758

0.948679

1.7

0.599908 0.955004 0.990564 0.96355

7 0.9918

79 0.947928 0.997251

0.706783

0.988643 0.9939

09 0.868481 0.813228

0.837761

0.948679

1.8

0.595027 0.954620 0.987676 0.94951

3 0.9902

63 0.943670 0.996806

0.738644

0.988885 0.9934

69 0.893790 0.825144

0.834176

0.950930

1.9

0.595027 0.954620 0.987681 0.94951

3 0.9902

63 0.943671 0.996809

0.738644

0.988886 0.9935

04 0.893790 0.825146

0.834176

0.950930

2.0

0.571919 0.946946 0.986103 0.94873

0 0.9880

43 0.933296 0.995620

0.714954

0.986619 0.9895

41 0.882468 0.805741

0.816587

0.939334

SF Foetal ECG Glass Furnace

Great Lakes

Koski ECG


PGT50 Alpha

PGT50 CDC15

Power Data

Power Plant

1.0

0.997624 0.994921 0.999267 0.99923

9 0.9991

22 0.999352 0.989543

0.995874

0.995874 0.9499

36 0.953711 0.933391

0.997446

0.999292

1.1

0.976355 0.938664 0.991092 0.99737

2 0.9946

90 0.998441 0.788691

0.995218

0.995218 0.8119

66 0.866775 0.817116

0.979990

0.991370

1.2

0.973517 0.895859 0.990663 0.99486

0 0.9902

09 0.997347 0.715996

0.993081

0.993081 0.7702

96 0.790230 0.705462

0.973394

0.989214

1.3

0.963991 0.912847 0.991636 0.99496

1 0.9900

56 0.996592 0.661433

0.992768

0.992768 0.7549

45 0.817864 0.731899

0.963406

0.982271

1.4

0.944534 0.871009 0.970673 0.99189

3 0.9811

69 0.995002 0.593253

0.987532

0.987532 0.7517

45 0.728678 0.650973

0.956345

0.970720

1.5

0.937212 0.850404 0.982488 0.98874

6 0.9802

13 0.993886 0.613137

0.986514

0.986514 0.6920

22 0.721404 0.678522

0.952692

0.969604

1.6

0.966199 0.989953 0.991907 0.99519

8 0.9870

51 0.997280 0.761301

0.987839

0.987839 0.8314

17 0.874820 0.873153

0.975440

0.985869

1.7

0.966199 0.990256 0.991907 0.99519

9 0.9870

54 0.997281 0.761301

0.987839

0.987839 0.8314

17 0.874820 0.873153

0.975541

0.986111

1.8

0.966943 0.989489 0.974032 0.99534

8 0.9870

65 0.996516 0.738819

0.984802

0.984802 0.8632

54 0.898295 0.889821

0.958461

0.982541

1.9

0.966943 0.989489 0.974032 0.99534

9 0.9870

66 0.996526 0.738820

0.985037

0.985037 0.8632

54 0.898295 0.889821

0.958627

0.982550

2.0

0.960183 0.987919 0.970952 0.99207

6 0.9845

52 0.995183 0.712417

0.983227

0.983227 0.8452

65 0.883358 0.880298

0.951742

0.968619

SFRandom

WalkRobot Arm Shuttle

Soil Temp

SpeechSpot Exrat

esStandard &

PoorSteamg

enSynthetic Control

Tide Tongue Winding WoolAverag

e

1.0

0.999316 0.991823 0.999472 0.99717

8 0.9995

25 0.996780 0.998003

0.998463

0.977751 0.9988

42 0.999415 0.998395

0.999156

0.991684

1.1

0.995331 0.853549 0.998895 0.78669

1 0.9702

43 0.987656 0.992618

0.988999

0.785640 0.9937

85 0.985925 0.987407

0.998610

0.928805

1.2

0.991490 0.775236 0.998385 0.69240

3 0.9612

92 0.980151 0.986087

0.982789

0.752406 0.9845

17 0.957332 0.982430

0.997621

0.905870

1.3

0.989852 0.738884 0.998083 0.68941

7 0.9612

06 0.977120 0.985869

0.984748

0.711955 0.9832

90 0.960734 0.986088

0.997206

0.897890

1.4

0.980321 0.683294 0.997118 0.63031

8 0.9189

63 0.967930 0.977566

0.973481

0.677345 0.9619

34 0.914782 0.962338

0.997277

0.868613

1.5

0.981349 0.677865 0.996204 0.65422

5 0.9161

46 0.968208 0.978736

0.961485

0.714639 0.9431

43 0.909705 0.957939

0.996061

0.870275

1.6

0.986326 0.867726 0.998239 0.67684

7 0.9692

22 0.981636 0.988090

0.980783

0.819298 0.9851

29 0.985496 0.986275

0.997754

0.927593

1.7

0.986328 0.867726 0.998242 0.67684

7 0.9692

23 0.981644 0.988091

0.980825

0.819298 0.9851

37 0.985496 0.986278

0.997758

0.927619

1.8

0.981463 0.870426 0.997439 0.68777

9 0.9661

13 0.979830 0.987544

0.972622

0.859444 0.9790

27 0.981215 0.981085

0.997219

0.929122

1.9

0.981465 0.870426 0.997440 0.68777

9 0.9661

13 0.979830 0.987545

0.972625

0.859444 0.9790

27 0.981215 0.981085

0.997339

0.929142

2.0

0.977163 0.849937 0.996917 0.66735

6 0.9563

03 0.976385 0.982182

0.965784

0.841658 0.9718

08 0.976245 0.975663

0.996876

0.920468

Relative Page Access

• The following slide shows the relative page access of the datasets versus the length of data.– The relative page access varied significantly from approximately

less than 0.2 and up to less than 1.8.– For short data of length 16, the relative page access is almost

always larger than 1, suggesting that it is generally not a wise idea to use index for short data.

• However, as the length of data increases, the relative page access decreases in general, as evident in the slide following next, which shows that the relative page access decreases as the length of data increases.– This is an important result for the proposed index, signaling that

the index is likely to perform progressively better as the length of data increases.

Relative Page Access

00.20.40.60.8

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Length of Data

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EEGERP DataReality CheckATTASBallbeamBuoy SensorBurstBurstinChaoticCSTRDarwinEarthquakeEegEvaporatorFoetal ECGGlass FurnaceGreat LakesKoski ECGLeleccumMemoryNetworkOceanOcean ShearPacketPGT50 AlphaPGT50 CDC15Power DataPower PlantRandom WalkRobot ArmShuttleSoil TempSpeechSpot ExratesStandard and PoorSteamgenSynthetic ControlTideTongueWindingWool

Average Relative Page Access

• The following slide also shows that the relative page access approaches 1 when the length of data is between 64 and 128.– This suggests that the use of index should be

considered if the length of data is greater than 64.– The fact that about half of the datasets achieved a

relative page access below 1 at length 64 and that over 70% of the datasets achieved less-than-one relative page access at length 1024 backed up the above claim.

Average Relative Page Access

0

0.2

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1.2

1.4

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Relative Page Access vs. Length of Data

Dimension

EEG ERP Data

Reality Check

ATTAS Ballbeam

Buoy Sensor


or

161.5182

28

1.338657

1.241675

1.760202

1.197839

1.349615

1.256644

1.458875

1.189462

1.303555

1.542068

1.452640

1.480270

1.199762

321.5289

40

1.334180

1.186908

1.483841

1.174878

1.242334

1.083856

1.383041

1.267108

1.262357

1.435336

1.318991

1.331406

1.120413

641.5666

09

1.108589

0.830415

0.906578

0.852347

1.215321

0.763797

1.381750

1.155789

0.983249

1.213138

1.407445

1.186633

1.051831

1281.6171

53

0.875603

0.487917

0.602250

0.936104

0.985261

0.526650

1.185658

1.364667

0.590014

1.229549

1.549319

1.126215

0.761407

2561.5535

15

0.956367

0.370727

0.473387

0.690759

0.937086

0.428182

1.435439

1.114336

0.433531

1.291175

1.428954

0.936818

0.780737

5121.4665

99

0.708304

0.191088

0.257655

0.667033

0.894466

0.395736

1.218574

0.868454

0.309955

1.234848

1.292325

0.849211

0.725856

10241.4160

34

0.933733

0.319118

0.268805

0.578775

0.753478

0.431923

1.573359

0.829318

0.413916

0.989940

1.347502

1.094145

0.905848

Dimension

Foetal ECG

Glass Furnace

Great Lakes

Koski ECG


PGT50 Alpha

PGT50 CDC15

Power Data

Power Plant

161.3406

89

1.287433

1.435583

1.237885

1.224261

1.215665

1.383371

1.153702

1.153702

0.992752

1.481012

1.634052

1.327203

1.136920

321.2472

50

1.311823

1.207590

1.125165

1.138678

1.099045

1.163370

1.024475

1.024475

1.132718

1.260214

1.316821

1.150763

1.233051

641.0383

04

0.896539

0.988632

0.840303

1.022127

0.715251

1.195175

0.631686

0.631686

0.986160

1.221378

1.450676

0.858115

0.930927

1281.0486

38

0.581836

0.955289

0.619018

0.717476

0.479320

1.044484

0.440641

0.440641

1.058402

1.250114

1.376594

0.791496

0.730234

2560.9607

51

0.829880

0.831002

0.403638

0.327351

0.308918

1.206453

0.283952

0.283952

0.960267

1.471842

1.263139

0.618712

0.500736

5120.9305

44

0.667888

0.630370

0.440675

0.202979

0.157417

1.127305

0.205982

0.205982

1.019801

1.374783

1.382482

0.471118

0.462361

10240.9791

22

0.655376

0.231967

0.665916

0.154578

0.180978

1.139579

0.169438

0.169438

1.254587

1.405600

1.425001

0.465772

0.444837

Dimension

Random Walk


es

Standard & Poo

rSteamgen

Synthetic Control


161.3766

42

1.682913

1.234891

1.573792

1.398669

1.293338

1.345880

1.360767

1.365710

1.390704

1.308829

1.201163

1.128378

1.340376

321.3494

51

1.532538

1.104730

1.493701

1.496310

1.271063

1.329950

1.159717

1.355713

1.385898

1.370407

1.350632

1.140216

1.266570

641.0050

43

1.588033

0.744667

1.212753

1.230510

0.853253

0.998204

0.961043

1.321875

1.139378

1.311534

1.131427

0.938562

1.060164

1280.7883

86

1.460337

0.496912

1.067514

1.347617

0.633163

0.744228

0.519667

1.186771

1.056253

1.353303

1.047924

0.800327

0.923765

2560.4215

66

1.299988

0.195134

1.063398

1.259823

0.361241

0.352205

0.407806

0.864809

0.821942

1.402710

1.043574

0.498838

0.806699

5120.4577

25

1.585264

0.214273

1.167539

1.267393

0.306695

0.358125

0.540054

0.927356

0.748646

1.223565

0.923770

0.306352

0.741135

10240.2580

44

1.528005

0.088301

1.097235

1.168286

0.228104

0.169561

0.331320

0.568785

0.769215

1.220670

0.881667

0.132935

0.722932

Relative Page Access– Raw Numbers

Query Time

• The following slide shows the actual running time of the range queries as calculated by the difference between two calls to gettimeofday before and after the queries.– The time is averaged over all 50 queries performed fo

r each length of data of each dataset.– It shows that the query generally runs very fast.

• All queries completed within a fraction of a second for all datasets of length 16 and 32, and all queries completed in the magnitude of minutes.

– Note the logarithmic scale in both axes.

Query Time

0.001

0.01

0.1

1

10

100

1000

10000

16 32 64 128 256 512 1024

Length of Data

Que

ry T

ime

(sec

ond)

EEGERP DataReality CheckATTASBallbeamBuoy SensorBurstBurstinChaoticCSTRDarwinEarthquakeEegEvaporatorFoetal ECGGlass FurnaceGreat LakesKoski ECGLeleccumMemoryNetworkOceanOcean ShearPacketPGT50 AlphaPGT50 CDC15Power DataPower PlantRandom WalkRobot ArmShuttleSoil TempSpeechSpot ExratesStandard and PoorSteamgenSynthetic ControlTideTongueWindingWool

Average Query Time

• The following slide shows the running time averaged over all datasets.– It suggested that most queries actually completed well

within a minute, even for the larger length of data.• And even for the largest length of data, queries completed in

560 seconds on average.– Recall from previous slides that linear scan perform

better for datasets of shorter lengths; however, as the following slide shows, the query time for those datasets is actually not significant anyway.

– Moreover, the proposed index can significantly reduce the query time for datasets of longer lengths.

• Combining both advantages, our proposed index is capable as an all-round solution suitable for datasets of all lengths.

Average Query Time

0.001

0.01

0.1

1

10

100

1000

16 32 64 128 256 512 1024

Length of Data

Que

ry T

ime

(sec

ond)

Query Time vs. Length of Data

Dimension

EEG ERP Data

Reality Check

ATTAS Ballbeam

Buoy Sensor


or

16 0.008906 0.008991 0.007852 0.027820 0.008524 0.007687 0.010244 0.008537 0.007947 0.011091 0.010337 0.008509 0.008130 0.006077

32 0.034602 0.027548 0.050998 0.140272 0.020844 0.025829 0.025415 0.023906 0.023303 0.022733 0.021901 0.042887 0.032926 0.017459

64 2.857623 1.205225 0.721554 1.693815 0.564420 1.774876 0.783452 1.384006 1.049300 0.731008 0.928825 2.345460 1.825744 0.926490

128 18.497708 6.200565 1.999890 5.316119 3.602136 5.536449 1.899189 9.189511 10.817802 2.157083 5.650658 16.737971 8.200356 5.450292

256 72.653503 20.072262 6.090931 32.63387

0

14.102981

46.932799

9.189990 45.07391

3 30.044976 7.941388

26.603104

75.275616 42.68375

8

27.296181

512317.31845

2 62.417018 17.067263

40.477818

52.518124

154.425892

26.894878 204.2943

96

78.948243 21.00884

3

117.051126

552.629948

106.354172

138.263561

10241245.4144

22

604.961684

91.006146 106.8290

25

239.951857

509.289414

161.670165

1251.927631

279.658997

130.976992

448.449666

2228.000279

1231.362539

825.873218

Dimension

Foetal ECG

Glass Furnace

Great Lakes

Koski ECG Leleccum Memory Network Ocean Ocean Shear Packet

PGT50 Alpha

PGT50 CDC15

Power Data

Power Plant

16 0.012231 0.007972 0.007545 0.009204 0.007899 0.013536 0.008237 0.009436 0.010056 0.005425 0.008388 0.011404 0.008328 0.007762

32 0.022082 0.018874 0.020769 0.029771 0.059950 0.044397 0.028854 0.048008 0.040405 0.032746 0.028760 0.020838 0.027546 0.026521

64 1.666670 0.809296 0.979123 0.620342 1.871419 0.693061 2.680475 0.500041 0.650238 1.553650 1.484277 1.819446 0.986859 0.821533

128 10.400214 3.201489 8.553828 2.801950 8.966195 1.749249 12.950348 1.781316 1.789617 8.619517 9.966430 11.932752 4.126129 4.021875

256 23.348046 40.490286 32.690045 7.028240 10.14701

3

5.163696 84.270622 6.275355 5.075757 27.07833

6

92.258568

54.404754 14.70836

5

10.431865

512 98.134765 124.50753

8 162.57611

8 43.74687

6

16.635796

15.147429

254.149856 14.82258

0 14.385844

113.622047

287.367275

304.586963

39.681939

32.909189

1024352.86413

2 467.26862

0 149.53252

4

234.984439

50.042713

49.646797

1211.571944

64.286687

60.260655

917.933985

2013.349248

1313.071738

207.557484

148.655349

Dimension

Random Walk


es

Standard & Poo

rSteamgen

Synthetic Control


16 0.007986 0.009621 0.013935 0.009682 0.007806 0.010297 0.008018 0.009280 0.009646 0.009957 0.007609 0.009757 0.007734 0.009498

32 0.025615 0.045487 0.022771 0.034344 0.025284 0.021546 0.036322 0.026498 0.025737 0.028008 0.028174 0.025720 0.020987 0.032357

64 1.547834 2.595087 0.643038 1.672957 1.498341 0.847308 0.829322 1.747261 2.403264 0.942431 0.939633 0.855632 0.999961 1.303666

128 3.479501 17.817225 1.853674 11.71076

2 7.519265 3.157955 3.303026 2.632136 13.362863 4.667584 7.556816 4.631206 3.444505 6.762272

256 8.901375 68.145256 3.093776 59.23178

6

37.131263

6.517546 6.343443 17.15533

1 39.370829

21.753843

48.243180

25.453888 8.870395 29.76044

2

512 36.814731 375.71248

9 14.908316

325.899240

148.185988

32.366527

25.864971 56.19133

4 308.370053

93.215671

145.993875

90.096584 23.38742

8

124.120760

1024 85.938851 2033.8050

61

24.773419

1516.404957

662.354303

101.094878

64.856889 149.8172

56

408.384121

312.716380

704.605891

287.207593

38.630299

560.658250

Query Time – Raw Numbers

Documents

Scaling and Warping in Time Series Querying Dear Reader: This file contains larger, full color versions of the images in “Scaling and Warping in Time Series