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

<div>Left: case of log normal distribution sampled 20 times and 30 % of Gaussian noise. Right: case of log normal distribution sampled 100 times and 30 % of Gaussian noise. The level of noise is expressed as percentage of the mean of the distributions. The parameters of the log normal are <svg height="9.52079pt" id="M69" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -9.32745 29.0243 9.52079" width="29.0243pt" 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><g transform="matrix(.013,0,0,-0.013,11.109,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.372,0)"><path d="M440 175C403 110 395 108 335 108H173L293 222C391 316 432 379 432 457C432 568 349 635 250 635C187 635 137 610 100 573L36 479L65 456C97 502 138 544 199 544C268 544 303 495 303 425C303 349 264 274 194 191C145 133 86 78 33 26V0H434C445 42 457 105 473 164L440 175Z"></path></g></svg> and <span class="nowrap"><svg height="8.47691pt" id="M70" style="vertical-align:-0.1933393pt" version="1.1" viewbox="-0.0498162 -8.28357 26.8582 8.47691" width="26.8582pt" 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="M386 382C386 422 341 454 281 454C239 454 191 429 152 402C103 368 86 331 86 294C86 233 150 188 200 160C257 128 270 106 270 89C270 68 251 52 222 52C184 52 134 88 92 134C82 145 75 144 58 135C39 125 21 100 21 76C21 27 89 -12 154 -12C239 -12 362 60 362 151C362 209 318 247 257 280C220 300 179 327 179 355C179 375 192 390 219 390C263 390 295 362 318 327C325 317 336 317 347 324C370 339 386 361 386 382Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.943,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.206,0)"><path d="M409 0V34C329 39 322 56 322 125V635C234 611 160 593 94 583V554L145 551C186 547 190 541 190 488V125C190 54 181 39 102 34V0H409Z"></path></g></svg>.</span></div>

Complexity

tab4

Table 4

Table 4: Geodesic Distance on Gaussian Manifolds to Reduce the Statistical Errors in the Investigation of Complex Systems 