lib.spectral_densities.model_free

1 ############################################################################### 2 # # 3 # Copyright (C) 2002-2005,2008,2024 Edward d'Auvergne # 4 # # 5 # This file is part of the program relax (http://www.nmr-relax.com). # 6 # # 7 # This program is free software: you can redistribute it and/or modify # 8 # it under the terms of the GNU General Public License as published by # 9 # the Free Software Foundation, either version 3 of the License, or # 10 # (at your option) any later version. # 11 # # 12 # This program is distributed in the hope that it will be useful, # 13 # but WITHOUT ANY WARRANTY; without even the implied warranty of # 14 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # 15 # GNU General Public License for more details. # 16 # # 17 # You should have received a copy of the GNU General Public License # 18 # along with this program. If not, see <http://www.gnu.org/licenses/>. # 19 # # 20 ############################################################################### 21 22 23 ############################ 24 # Spectral density values. # 25 ############################ 26 27 28 # Original {S2, te}. 29 #################### 30

31 -def calc_S2_te_jw_comps(data, params):

32 r"""Spectral density component function. 33 34 Calculate the components of the spectral density value for the original model-free formula with 35 the parameters {S2, te}. 36 37 The model-free formula is:: 38 39 _n_ 40 2 \ / S2 (1 - S2)(te + ti)te \ 41 J(w) = - > ci . ti | ------------ + ------------------------- |. 42 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / 43 i=m 44 45 Replicated calculations are:: 46 47 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 48 49 te_ti = te + ti, 50 te_ti_te = (te + ti).te. 51 52 53 Calculations which are replicated in the gradient equations are:: 54 55 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 56 57 one_s2 = 1 - S2, 58 59 te_ti_sqrd = (te + ti)^2, 60 w_te_ti_sqrd = (w.te.ti)^2, 61 62 (te + ti)te 63 fact_te = -------------------------. 64 (te + ti)^2 + (w.te.ti)^2 65 66 """ 67 68 # Order parameter factors. 69 data.one_s2 = 1.0 - params[data.s2_i] 70 71 # Internal correlation time, te, factors. 72 data.te_ti = params[data.te_i] + data.ti 73 data.te_ti_te = data.te_ti * params[data.te_i] 74 data.te_ti_sqrd = data.te_ti**2 75 76 data.w_te_ti_sqrd = data.w_ti_sqrd * params[data.te_i]**2 77 data.inv_te_denom = 1.0 / (data.te_ti_sqrd + data.w_te_ti_sqrd) 78 data.fact_te = data.te_ti_te * data.inv_te_denom

79 80 81 82 # Extended {S2f, S2, ts}. 83 ######################### 84

85 -def calc_S2f_S2_ts_jw_comps(data, params):

86 r"""Spectral density component function. 87 88 Calculate the components of the spectral density value for the extended model-free formula with 89 the parameters {S2f, S2, ts}. 90 91 The model-free formula is:: 92 93 _n_ 94 2 \ / S2 (S2f - S2)(ts + ti)ts \ 95 J(w) = - > ci . ti | ------------ + ------------------------- |. 96 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / 97 i=m 98 99 Replicated calculations are:: 100 101 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 102 103 ts_ti = ts + ti, 104 ts_ti_ts = (ts + ti).ts. 105 106 107 Calculations which are replicated in the gradient equations are:: 108 109 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 110 111 s2f_s2 = S2f - S2, 112 113 ts_ti_sqrd = (ts + ti)^2, 114 w_ts_ti_sqrd = (w.ts.ti)^2, 115 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 116 117 (ts + ti)ts 118 fact_ts = -------------------------. 119 (ts + ti)^2 + (w.ts.ti)^2 120 121 """ 122 123 # Order parameter factors. 124 data.s2f_s2 = params[data.s2f_i] - params[data.s2_i] 125 126 # Internal correlation time, ts, factors. 127 data.ts_ti = params[data.ts_i] + data.ti 128 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 129 data.ts_ti_sqrd = data.ts_ti**2 130 131 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 132 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 133 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

134 135 136 137 # Extended 2 {S2f, S2s, ts}. 138 ############################ 139

140 -def calc_S2f_S2s_ts_jw_comps(data, params):

141 r"""Spectral density component function. 142 143 Calculate the components of the spectral density value for the extended model-free formula with 144 the parameters {S2f, S2s, ts}. 145 146 The model-free formula is:: 147 148 _n_ 149 2 \ / S2s (1 - S2s)(ts + ti)ts \ 150 J(w) = - S2f > ci . ti | ------------ + ------------------------- |. 151 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / 152 i=m 153 154 Replicated calculations are:: 155 156 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 157 158 ts_ti = ts + ti, 159 ts_ti_ts = (ts + ti).ts. 160 161 162 Calculations which are replicated in the gradient equations are:: 163 164 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 165 166 one_s2s = 1 - S2s, 167 168 ts_ti_sqrd = (ts + ti)^2, 169 w_ts_ti_sqrd = (w.ts.ti)^2, 170 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 171 172 (ts + ti)ts 173 fact_ts = -------------------------. 174 (ts + ti)^2 + (w.ts.ti)^2 175 176 """ 177 178 # Order parameter factors. 179 data.one_s2s = 1.0 - params[data.s2s_i] 180 data.s2f_s2 = data.one_s2s * params[data.s2f_i] 181 182 # Internal correlation time, ts, factors. 183 data.ts_ti = params[data.ts_i] + data.ti 184 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 185 data.ts_ti_sqrd = data.ts_ti**2 186 187 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 188 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 189 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

190 191 192 193 # Extended {S2f, tf, S2, ts}. 194 ############################# 195

196 -def calc_S2f_tf_S2_ts_jw_comps(data, params):

197 r"""Spectral density component function. 198 199 Calculate the components of the spectral density value for the extended model-free formula with 200 the parameters {S2f, tf, S2, ts}. 201 202 The model-free formula is:: 203 204 _n_ 205 2 \ / S2 (1 - S2f)(tf + ti)tf 206 J(w) = - > ci . ti | ------------ + ------------------------- 207 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 208 i=m 209 210 (S2f - S2)(ts + ti)ts \ 211 + ------------------------- |. 212 (ts + ti)^2 + (w.ts.ti)^2 / 213 214 Replicated calculations are:: 215 216 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 217 218 tf_ti = tf + ti, 219 ts_ti = ts + ti, 220 tf_ti_tf = (tf + ti).tf, 221 ts_ti_ts = (ts + ti).ts. 222 223 224 Calculations which are replicated in the gradient equations are:: 225 226 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 227 228 one_s2f = 1 - S2f, 229 s2f_s2 = S2f - S2, 230 231 tf_ti_sqrd = (tf + ti)^2, 232 ts_ti_sqrd = (ts + ti)^2, 233 w_tf_ti_sqrd = (w.tf.ti)^2, 234 w_ts_ti_sqrd = (w.ts.ti)^2, 235 inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2), 236 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 237 238 (tf + ti)tf 239 fact_tf = -------------------------, 240 (tf + ti)^2 + (w.tf.ti)^2 241 242 243 (ts + ti)ts 244 fact_ts = -------------------------. 245 (ts + ti)^2 + (w.ts.ti)^2 246 247 """ 248 249 # Order parameter factors. 250 data.one_s2f = 1.0 - params[data.s2f_i] 251 data.s2f_s2 = params[data.s2f_i] - params[data.s2_i] 252 253 # Internal correlation time, tf, factors. 254 data.tf_ti = params[data.tf_i] + data.ti 255 data.tf_ti_tf = data.tf_ti * params[data.tf_i] 256 data.tf_ti_sqrd = data.tf_ti**2 257 258 data.w_tf_ti_sqrd = data.w_ti_sqrd * params[data.tf_i]**2 259 data.inv_tf_denom = 1.0 / (data.tf_ti_sqrd + data.w_tf_ti_sqrd) 260 data.fact_tf = data.tf_ti_tf * data.inv_tf_denom 261 262 # Internal correlation time, ts, factors. 263 data.ts_ti = params[data.ts_i] + data.ti 264 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 265 data.ts_ti_sqrd = data.ts_ti**2 266 267 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 268 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 269 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

270 271 272 273 # Extended 2 {S2f, tf, S2s, ts}. 274 ################################ 275

276 -def calc_S2f_tf_S2s_ts_jw_comps(data, params):

277 r"""Spectral density component function. 278 279 Calculate the components of the spectral density value for the extended model-free formula with 280 the parameters {S2f, tf, S2s, ts}. 281 282 The model-free formula is:: 283 284 _n_ 285 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf 286 J(w) = - > ci . ti | ------------ + ------------------------- 287 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 288 i=m 289 290 S2f(1 - S2s)(ts + ti)ts \ 291 + ------------------------- |. 292 (ts + ti)^2 + (w.ts.ti)^2 / 293 294 Replicated calculations are:: 295 296 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 297 298 tf_ti = tf + ti, 299 ts_ti = ts + ti, 300 tf_ti_tf = (tf + ti).tf, 301 ts_ti_ts = (ts + ti).ts. 302 303 304 Calculations which are replicated in the gradient equations are:: 305 306 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 307 308 one_s2s = 1 - S2s, 309 one_s2f = 1 - S2f, 310 s2f_s2 = S2f(1 - S2s) = S2f - S2, 311 312 tf_ti_sqrd = (tf + ti)^2, 313 ts_ti_sqrd = (ts + ti)^2, 314 w_tf_ti_sqrd = (w.tf.ti)^2, 315 w_ts_ti_sqrd = (w.ts.ti)^2, 316 inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2), 317 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 318 319 (tf + ti)tf 320 fact_tf = -------------------------, 321 (tf + ti)^2 + (w.tf.ti)^2 322 323 324 (ts + ti)ts 325 fact_ts = -------------------------. 326 (ts + ti)^2 + (w.ts.ti)^2 327 328 """ 329 330 # Order parameter factors. 331 data.one_s2s = 1.0 - params[data.s2s_i] 332 data.one_s2f = 1.0 - params[data.s2f_i] 333 data.s2f_s2 = params[data.s2f_i] * data.one_s2s 334 335 # Internal correlation time, tf, factors. 336 data.tf_ti = params[data.tf_i] + data.ti 337 data.tf_ti_tf = data.tf_ti * params[data.tf_i] 338 data.tf_ti_sqrd = data.tf_ti**2 339 340 data.w_tf_ti_sqrd = data.w_ti_sqrd * params[data.tf_i]**2 341 data.inv_tf_denom = 1.0 / (data.tf_ti_sqrd + data.w_tf_ti_sqrd) 342 data.fact_tf = data.tf_ti_tf * data.inv_tf_denom 343 344 # Internal correlation time, ts, factors. 345 data.ts_ti = params[data.ts_i] + data.ti 346 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 347 data.ts_ti_sqrd = data.ts_ti**2 348 349 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 350 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 351 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

352 353 354 355 356 ############################### 357 # Spectral density gradients. # 358 ############################### 359 360 361 # Original {} and {S2} with diffusion parameters. 362 ################################################# 363

364 -def calc_diff_djw_comps(data, params):

365 r"""Spectral density gradient component function. 366 367 Calculate the components of the spectral density gradient for the original model-free formula 368 with no parameters {} or the parameter {S2} together with diffusion tensor parameters. 369 370 Replicated calculations are:: 371 372 1 - (w.ti)^2 373 fact_ti_djw_dti = ----------------. 374 (1 + (w.ti)^2)^2 375 """ 376 377 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2

378 379 380 381 # Original {S2, te}. 382 #################### 383

384 -def calc_S2_te_djw_comps(data, params):

385 r"""Spectral density gradient component function. 386 387 Calculate the components of the spectral density gradient for the original model-free formula 388 with the parameters {S2, te}. 389 390 Replicated calculations are:: 391 392 (te + ti)^2 - (w.te.ti)^2 393 fact_djw_dte = ti^2 -----------------------------. 394 ((te + ti)^2 + (w.te.ti)^2)^2 395 """ 396 397 data.fact_djw_dte = data.ti**2 * (data.te_ti_sqrd - data.w_te_ti_sqrd) * data.inv_te_denom**2

398 399 400 401 # Original {S2, te} with diffusion parameters. 402 ############################################## 403

404 -def calc_diff_S2_te_djw_comps(data, params):

405 r"""Spectral density gradient component function. 406 407 Calculate the components of the spectral density gradient for the original model-free formula 408 with the parameters {S2, te} together with diffusion tensor parameters. 409 410 Replicated calculations are:: 411 412 1 - (w.ti)^2 413 fact_ti_djw_dti = ----------------, 414 (1 + (w.ti)^2)^2 415 416 417 (te + ti)^2 - (w.te.ti)^2 418 fact_djw = -----------------------------, 419 ((te + ti)^2 + (w.te.ti)^2)^2 420 421 422 (te + ti)^2 - (w.te.ti)^2 423 fact_te_djw_dti = te^2 -----------------------------, 424 ((te + ti)^2 + (w.te.ti)^2)^2 425 426 427 (te + ti)^2 - (w.te.ti)^2 428 fact_djw_dte = ti^2 -----------------------------. 429 ((te + ti)^2 + (w.te.ti)^2)^2 430 """ 431 432 fact_djw = (data.te_ti_sqrd - data.w_te_ti_sqrd) * data.inv_te_denom**2 433 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 434 data.fact_te_djw_dti = params[data.te_i]**2 * fact_djw 435 data.fact_djw_dte = data.ti**2 * fact_djw

436 437 438 439 # Extended {S2f, S2, ts}. 440 ######################### 441

442 -def calc_S2f_S2_ts_djw_comps(data, params):

443 r"""Spectral density gradient component function. 444 445 Calculate the components of the spectral density gradient for the extended model-free formula 446 with the parameters {S2f, S2, ts}. 447 448 Replicated calculations are:: 449 450 451 (ts + ti)^2 - (w.ts.ti)^2 452 fact_djw_dts = ti^2 -----------------------------. 453 ((ts + ti)^2 + (w.ts.ti)^2)^2 454 """ 455 456 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

457 458 459 460 # Extended {S2f, S2, ts} with diffusion parameters. 461 ################################################### 462

463 -def calc_diff_S2f_S2_ts_djw_comps(data, params):

464 r"""Spectral density gradient component function. 465 466 Calculate the components of the spectral density gradient for the extended model-free formula 467 with the parameters {S2f, S2, ts} together with diffusion tensor parameters. 468 469 Replicated calculations are:: 470 471 1 - (w.ti)^2 472 fact_ti_djw_dti = ----------------, 473 (1 + (w.ti)^2)^2 474 475 476 (ts + ti)^2 - (w.ts.ti)^2 477 fact_djw = -----------------------------, 478 ((ts + ti)^2 + (w.ts.ti)^2)^2 479 480 481 (ts + ti)^2 - (w.ts.ti)^2 482 fact_ts_djw_dti = ts^2 -----------------------------, 483 ((ts + ti)^2 + (w.ts.ti)^2)^2 484 485 486 (ts + ti)^2 - (w.ts.ti)^2 487 fact_djw_dts = ti^2 -----------------------------. 488 ((ts + ti)^2 + (w.ts.ti)^2)^2 489 """ 490 491 492 fact_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 493 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 494 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_djw 495 data.fact_djw_dts = data.ti**2 * fact_djw

496 497 498 499 # Extended {S2f, tf, S2, ts}. 500 ############################# 501

502 -def calc_S2f_tf_S2_ts_djw_comps(data, params):

503 r"""Spectral density gradient component function. 504 505 Calculate the components of the spectral density gradient for the extended model-free formula 506 with the parameters {S2f, tf, S2, ts}. 507 508 Replicated calculations are:: 509 510 (tf + ti)^2 - (w.tf.ti)^2 511 fact_djw_dtf = ti^2 -----------------------------, 512 ((tf + ti)^2 + (w.tf.ti)^2)^2 513 514 515 (ts + ti)^2 - (w.ts.ti)^2 516 fact_djw_dts = ti^2 -----------------------------. 517 ((ts + ti)^2 + (w.ts.ti)^2)^2 518 """ 519 520 data.fact_djw_dtf = data.ti**2 * (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 521 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

522 523 524 525 # Extended {S2f, tf, S2, ts} with diffusion parameters. 526 ####################################################### 527

528 -def calc_diff_S2f_tf_S2_ts_djw_comps(data, params):

529 r"""Spectral density gradient component function. 530 531 Calculate the components of the spectral density gradient for the extended model-free formula 532 with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters. 533 534 Replicated calculations are:: 535 536 1 - (w.ti)^2 537 fact_ti_djw_dti = ----------------, 538 (1 + (w.ti)^2)^2 539 540 541 (tf + ti)^2 - (w.tf.ti)^2 542 fact_tf_djw = -----------------------------, 543 ((tf + ti)^2 + (w.tf.ti)^2)^2 544 545 546 (ts + ti)^2 - (w.ts.ti)^2 547 fact_ts_djw = -----------------------------, 548 ((ts + ti)^2 + (w.ts.ti)^2)^2 549 550 551 (tf + ti)^2 - (w.tf.ti)^2 552 fact_tf_djw_dti = tf^2 -----------------------------, 553 ((tf + ti)^2 + (w.tf.ti)^2)^2 554 555 556 (ts + ti)^2 - (w.ts.ti)^2 557 fact_ts_djw_dti = ts^2 -----------------------------, 558 ((ts + ti)^2 + (w.ts.ti)^2)^2 559 560 561 (tf + ti)^2 - (w.tf.ti)^2 562 fact_djw_dtf = ti^2 -----------------------------, 563 ((tf + ti)^2 + (w.tf.ti)^2)^2 564 565 566 (ts + ti)^2 - (w.ts.ti)^2 567 fact_djw_dts = ti^2 -----------------------------. 568 ((ts + ti)^2 + (w.ts.ti)^2)^2 569 """ 570 571 # ti. 572 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 573 574 # tf. 575 fact_tf_djw = (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 576 data.fact_tf_djw_dti = params[data.tf_i]**2 * fact_tf_djw 577 data.fact_djw_dtf = data.ti**2 * fact_tf_djw 578 579 # ts. 580 fact_ts_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 581 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_ts_djw 582 data.fact_djw_dts = data.ti**2 * fact_ts_djw

583 584 585 586 # Extended 2 {S2f, S2s, ts}. 587 ############################ 588

589 -def calc_S2f_S2s_ts_djw_comps(data, params):

590 r"""Spectral density gradient component function. 591 592 Calculate the components of the spectral density gradient for the extended model-free formula 593 with the parameters {S2f, S2s, ts}. 594 595 Replicated calculations are:: 596 597 (ts + ti)^2 - (w.ts.ti)^2 598 fact_djw_dts = ti^2 -----------------------------. 599 ((ts + ti)^2 + (w.ts.ti)^2)^2 600 """ 601 602 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

603 604 605 606 # Extended 2 {S2f, S2s, ts} with diffusion parameters. 607 ###################################################### 608

609 -def calc_diff_S2f_S2s_ts_djw_comps(data, params):

610 r"""Spectral density gradient component function. 611 612 Calculate the components of the spectral density gradient for the extended model-free formula 613 with the parameters {S2f, S2s, ts} together with diffusion tensor parameters. 614 615 Replicated calculations are:: 616 617 1 - (w.ti)^2 618 fact_ti_djw_dti = ----------------, 619 (1 + (w.ti)^2)^2 620 621 622 (ts + ti)^2 - (w.ts.ti)^2 623 fact_djw = -----------------------------, 624 ((ts + ti)^2 + (w.ts.ti)^2)^2 625 626 627 (ts + ti)^2 - (w.ts.ti)^2 628 fact_ts_djw_dti = ts^2 -----------------------------, 629 ((ts + ti)^2 + (w.ts.ti)^2)^2 630 631 632 (ts + ti)^2 - (w.ts.ti)^2 633 fact_djw_dts = ti^2 -----------------------------. 634 ((ts + ti)^2 + (w.ts.ti)^2)^2 635 """ 636 637 fact_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 638 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 639 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_djw 640 data.fact_djw_dts = data.ti**2 * fact_djw

641 642 643 644 # Extended 2 {S2f, tf, S2s, ts}. 645 ################################ 646

647 -def calc_S2f_tf_S2s_ts_djw_comps(data, params):

648 r"""Spectral density gradient component function. 649 650 Calculate the components of the spectral density gradient for the extended model-free formula 651 with the parameters {S2f, tf, S2s, ts}. 652 653 Replicated calculations are:: 654 655 (tf + ti)^2 - (w.tf.ti)^2 656 fact_djw_dtf = ti^2 -----------------------------, 657 ((tf + ti)^2 + (w.tf.ti)^2)^2 658 659 660 (ts + ti)^2 - (w.ts.ti)^2 661 fact_djw_dts = ti^2 -----------------------------. 662 ((ts + ti)^2 + (w.ts.ti)^2)^2 663 """ 664 665 data.fact_djw_dtf = data.ti**2 * (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 666 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

667 668 669 670 # Extended 2 {S2f, tf, S2s, ts} with diffusion parameters. 671 ########################################################## 672

673 -def calc_diff_S2f_tf_S2s_ts_djw_comps(data, params):

674 r"""Spectral density gradient component function. 675 676 Calculate the components of the spectral density gradient for the extended model-free formula 677 with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters. 678 679 Replicated calculations are:: 680 681 1 - (w.ti)^2 682 fact_ti_djw_dti = ----------------, 683 (1 + (w.ti)^2)^2 684 685 686 (tf + ti)^2 - (w.tf.ti)^2 687 fact_tf_djw = -----------------------------, 688 ((tf + ti)^2 + (w.tf.ti)^2)^2 689 690 691 (ts + ti)^2 - (w.ts.ti)^2 692 fact_ts_djw = -----------------------------, 693 ((ts + ti)^2 + (w.ts.ti)^2)^2 694 695 696 (tf + ti)^2 - (w.tf.ti)^2 697 fact_tf_djw_dti = tf^2 -----------------------------, 698 ((tf + ti)^2 + (w.tf.ti)^2)^2 699 700 701 (ts + ti)^2 - (w.ts.ti)^2 702 fact_ts_djw_dti = ts^2 -----------------------------, 703 ((ts + ti)^2 + (w.ts.ti)^2)^2 704 705 706 (tf + ti)^2 - (w.tf.ti)^2 707 fact_djw_dtf = ti^2 -----------------------------, 708 ((tf + ti)^2 + (w.tf.ti)^2)^2 709 710 711 (ts + ti)^2 - (w.ts.ti)^2 712 fact_djw_dts = ti^2 -----------------------------. 713 ((ts + ti)^2 + (w.ts.ti)^2)^2 714 """ 715 716 # ti. 717 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 718 719 # tf. 720 fact_tf_djw = (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 721 data.fact_tf_djw_dti = params[data.tf_i]**2 * fact_tf_djw 722 data.fact_djw_dtf = data.ti**2 * fact_tf_djw 723 724 # ts. 725 fact_ts_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 726 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_ts_djw 727 data.fact_djw_dts = data.ti**2 * fact_ts_djw

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Source Code for Module lib.spectral_densities.model_free_components