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

<table class="fixed-width table-group" id="tab34"><tr><td><table class="table"><colgroup><col style="width:8.44em"/><col style="width:5.63em"/><col style="width:5.63em"/><col style="width:5.63em"/><col style="width:5.85em"/></colgroup><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="2"> </td><td class="align_center">20 points</td><td class="align_center">50 points</td><td class="align_center">100 points</td><td class="align_center">1000 points</td></tr><tr class="thead"><td class="align_center">Location <svg height="9.52079pt" id="M216" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.32745 7.57655 9.52079" width="7.57655pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="7.47692" x="0" y="-8.6252"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z"></path></g></svg></td><td class="align_center">Location <svg height="9.52079pt" id="M217" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.32745 7.57655 9.52079" width="7.57655pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="7.47692" x="0" y="-8.6252"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z"></path></g></svg></td><td class="align_center">Location <svg height="9.52079pt" id="M218" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.32745 7.57655 9.52079" width="7.57655pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="7.47692" x="0" y="-8.6252"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z"></path></g></svg></td><td class="align_center">Location <svg height="9.52079pt" id="M219" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.32745 7.57655 9.52079" width="7.57655pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="7.47692" x="0" y="-8.6252"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z"></path></g></svg></td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left">Mode</td><td class="align_center">1.06174</td><td class="align_center">0.71622</td><td class="align_center">0.56485</td><td class="align_center">0.24778</td></tr><tr><td class="align_left" colspan="5"><hr/></td></tr><tr><td class="align_left">Mean</td><td class="align_center">1.99158</td><td class="align_center">1.98960</td><td class="align_center">2.00813</td><td class="align_center">2.00242</td></tr><tr><td class="align_left"><span style="width: 40.6365ptpx;"><svg height="12.0621pt" id="M220" 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 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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">1.83644</td><td class="align_center">1.82000</td><td class="align_center">1.83880</td><td class="align_center">1.82967</td></tr><tr><td class="align_left" colspan="5"><hr/></td></tr><tr><td class="align_left">Trimmed Mean</td><td class="align_center">1.69601</td><td class="align_center">1.66348</td><td class="align_center">1.67930</td><td class="align_center">1.66705</td></tr><tr><td class="align_left"><span style="width: 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275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 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">1.52821</td><td class="align_center">1.49809</td><td class="align_center">1.51042</td><td class="align_center">1.50207</td></tr><tr><td class="align_left" colspan="5"><hr/></td></tr><tr><td class="align_left">Winsorized Mean</td><td class="align_center">1.75591</td><td class="align_center">1.73586</td><td class="align_center">1.75454</td><td class="align_center">1.74685</td></tr><tr><td class="align_left"><span style="width: 47.35ptpx;"><svg height="12.0621pt" id="M222" 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 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="M969 650H746V617C823 609 825 598 814 540C798 470 743 278 700 144H698C638 319 581 476 525 641H491C440 478 384 317 329 145H327C283 286 245 418 205 553C192 601 192 609 267 617V650H20V617C80 609 98 602 122 522C177 343 233 172 285 -9H321C372 152 428 309 485 485H487C542 330 604 151 660 -9H696C779 269 841 460 862 521C891 599 900 609 969 617V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,27.592,3.132)"><path d="M138 535C169 535 191 560 191 589C191 620 168 644 139 644S85 620 85 589C85 560 110 535 138 535ZM257 0V30C192 35 185 43 185 109V453C142 435 94 422 40 414V386C101 378 104 372 104 311V109C104 41 97 35 33 30V0H257Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.094,3.132)"><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><g transform="matrix(.0091,0,0,-0.0091,35.081,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,38.575,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,43.189,3.132)"><path 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">1.59822</td><td class="align_center">1.58391</td><td class="align_center">1.60236</td><td class="align_center">1.59234</td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Exponential distributions with 50% noise level.</div>

Complexity

tab34

Table 34

Table 34: Geodesic Distance on Gaussian Manifolds to Reduce the Statistical Errors in the Investigation of Complex Systems 