Geodesic Distance on Gaussian Manifolds to Reduce the Statistical Errors in the Investigation of Complex Systems

<table class="fixed-width table-group" id="tab79"><tr><td><table class="table"><colgroup><col style="width:8.44em"/><col style="width:5.02em"/><col style="width:3.82em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left"> </td><td class="align_center">No Noise</td><td class="align_center">Noise</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">Mode</td><td class="align_center">2.590</td><td class="align_center">-0.465</td></tr><tr><td class="align_left" colspan="3"><hr/></td></tr><tr><td class="align_left">Mean</td><td class="align_center">3.249</td><td class="align_center">0.335</td></tr><tr><td class="align_left"><span style="width: 40.6365ptpx;"><svg height="12.0621pt" id="M400" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.7664 40.6365 12.0621" width="40.6365pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.295,0)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.921,3.132)"><path d="M871 0V33C783 41 780 45 777 152L768 506C765 610 771 609 860 617V650H683L453 157L228 650H59V617C150 609 154 606 146 481L132 275C124 170 117 125 112 93C105 50 84 39 27 34V0H267V34C200 39 176 48 173 93C170 130 171 172 176 256L190 540H192L414 8H438L682 554H685L689 152C689 45 686 41 604 33V0H871Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,26.9,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.886,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,34.89,3.132)"><path d="M532 0V30C473 38 467 38 467 107V298C467 395 416 451 336 451C307 451 279 439 251 420C226 404 205 389 185 374V453C142 434 94 422 43 413V386C98 378 104 373 104 309V107C104 41 100 38 31 30V0H251V30C191 35 185 40 185 107V337C216 364 255 389 295 389C360 389 386 347 386 276V110C386 44 380 35 321 30V0H532Z"></path></g></svg></span></td><td class="align_center">3.120</td><td class="align_center">0.245</td></tr><tr><td class="align_left" colspan="3"><hr/></td></tr><tr><td class="align_left">Trimmed Mean</td><td class="align_center">3.051</td><td class="align_center">0.148</td></tr><tr><td class="align_left"><span style="width: 54.7029ptpx;"><svg height="12.0621pt" id="M401" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.7664 54.7029 12.0621" width="54.7029pt" 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30V0H810Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,45.08,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,49.076,3.132)"><path d="M526 56L493 58C457 61 448 68 448 119V710C411 697 344 682 288 675V646C362 641 367 639 367 573V439C344 448 315 451 300 451C164 451 40 342 40 202C40 61 147 -12 227 -12C239 -12 266 -7 305 16L367 53V-12C427 10 505 21 526 26V56ZM367 88C342 69 306 54 271 54C207 54 133 111 133 229C133 373 219 409 264 409C301 409 342 394 367 359V88Z"></path></g></svg></span></td><td class="align_center">2.539</td><td class="align_center">0.126</td></tr><tr><td class="align_left" colspan="3"><hr/></td></tr><tr><td class="align_left">Winsorized Mean</td><td class="align_center">3.099</td><td class="align_center">0.199</td></tr><tr><td class="align_left"><span style="width: 47.35ptpx;"><svg height="12.0621pt" id="M402" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.7664 47.35 12.0621" width="47.35pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.295,0)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 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d="M185 344V453C137 433 92 421 41 413V386C100 378 104 375 104 310V106C104 41 97 35 33 30V0H268V30C190 35 185 41 185 107V283C206 340 240 371 267 371C282 371 296 364 310 351C316 344 325 343 336 348C357 357 369 378 369 399C369 424 345 451 310 451C262 451 219 397 187 344H185Z"></path></g></svg></span></td><td class="align_center">2.737</td><td class="align_center">0.164</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Log-normal distributions with step noise (Type 3). The uniform noise is applied in the range from the minus maximum value to the minimum value of the distribution. The frequency of the samples decreases linearly, going to create a noise-shaped step. The noise is composed of 1000 points.</div>

Complexity

tab79

Table 79

Table 79: Geodesic Distance on Gaussian Manifolds to Reduce the Statistical Errors in the Investigation of Complex Systems 