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23 from numpy import sum
24
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26
27
28
29
30 """
31 The spectral density equation
32 =============================
33
34 Data structure: data.jw
35 Dimension: 2D, (number of NMR frequencies, 5 spectral density frequencies)
36 Type: numpy matrix, float64
37 Dependencies: None
38 Required by: data.ri, data.dri, data.d2ri
39
40
41 Formulae
42 ========
43
44 Original::
45
46 _k_
47 2 \ 1
48 J(w) = - S2 > ci . ti ------------,
49 5 /__ 1 + (w.ti)^2
50 i=-k
51
52
53 _k_
54 2 \ / S2 (1 - S2)(te + ti)te \
55 J(w) = - > ci . ti | ------------ + ------------------------- |,
56 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
57 i=-k
58
59
60 Extended::
61
62 _k_
63 2 \ / S2 (S2f - S2)(ts + ti)ts \
64 J(w) = - > ci . ti | ------------ + ------------------------- |,
65 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
66 i=-k
67
68
69 _k_
70 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
71 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |,
72 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
73 i=-k
74
75
76 Extended 2::
77
78 _k_
79 2 \ / S2s (1 - S2s)(ts + ti)ts \
80 J(w) = - S2f > ci . ti | ------------ + ------------------------- |,
81 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
82 i=-k
83
84
85 _k_
86 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
87 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
88 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
89 i=-k
90 """
91
92
93
94
95
96
98 """Spectral density function.
99
100 Calculate the spectral density values for the original model-free formula with no parameters {}
101 with or without diffusion tensor parameters.
102
103 The formula is::
104
105 _k_
106 2 \ 1
107 J(w) = - > ci . ti ------------.
108 5 /__ 1 + (w.ti)^2
109 i=-k
110 """
111
112 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
113
114
115
116
117
118
120 """Spectral density function.
121
122 Calculate the spectral density values for the original model-free formula with the single
123 parameter {S2} with or without diffusion tensor parameters.
124
125 The formula is::
126
127 _k_
128 2 \ 1
129 J(w) = - S2 > ci . ti ------------.
130 5 /__ 1 + (w.ti)^2
131 i=-k
132 """
133
134 return 0.4 * params[data.s2_i] * sum(data.ci * data.ti * data.fact_ti, axis=2)
135
136
137
138
139
140
142 """Spectral density function.
143
144 Calculate the spectral density values for the original model-free formula with the parameters
145 {S2, te} with or without diffusion tensor parameters.
146
147 The model-free formula is::
148
149 _k_
150 2 \ / S2 (1 - S2)(te + ti)te \
151 J(w) = - > ci . ti | ------------ + ------------------------- |.
152 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
153 i=-k
154 """
155
156 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
157
158
159
160
161
162
164 """Spectral density function.
165
166 Calculate the spectral density values for the extended model-free formula with the parameters
167 {S2f, S2, ts} with or without diffusion tensor parameters.
168
169 The model-free formula is::
170
171 _k_
172 2 \ / S2 (S2f - S2)(ts + ti)ts \
173 J(w) = - > ci . ti | ------------ + ------------------------- |.
174 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
175 i=-k
176 """
177
178 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
179
180
181
182
183
184
186 """Spectral density function.
187
188 Calculate the spectral density values for the extended model-free formula with the parameters
189 {S2f, tf, S2, ts} with or without diffusion tensor parameters.
190
191 The model-free formula is::
192
193 _k_
194 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
195 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
196 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
197 i=-k
198 """
199
200 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
201
202
203
204
205
206
208 """Spectral density function.
209
210 Calculate the spectral density values for the extended model-free formula with the parameters
211 {S2f, S2s, ts} with or without diffusion tensor parameters.
212
213 The model-free formula is::
214
215 _k_
216 2 \ / S2s (1 - S2s)(ts + ti)ts \
217 J(w) = - S2f > ci . ti | ------------ + ------------------------- |.
218 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
219 i=-k
220 """
221
222 return 0.4 * params[data.s2f_i] * sum(data.ci * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
223
224
225
226
227
228
230 """Spectral density function.
231
232 Calculate the spectral density values for the extended model-free formula with the parameters
233 {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
234
235 The model-free formula is::
236
237 _k_
238 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
239 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
240 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
241 i=-k
242 """
243
244 return 0.4 * sum(data.ci * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
245
246
247
248
249
250
251
252
253 """
254 The spectral density gradients
255 ==============================
256
257 Data structure: data.djw
258 Dimension: 3D, (number of NMR frequencies, 5 spectral density frequencies,
259 model-free parameters)
260 Type: numpy 3D matrix, float64
261 Dependencies: None
262 Required by: data.dri, data.d2ri
263
264
265 Formulae
266 ========
267
268 Original::
269
270 _k_
271 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
272 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
273 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
274 i=-k
275
276 dci / S2 (1 - S2)(te + ti)te \ \
277 + --- . ti | ------------ + ------------------------- | |,
278 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
279
280
281 _k_
282 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
283 ----- = - > --- . ti | ------------ + ------------------------- |,
284 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
285 i=-k
286
287
288 _k_
289 dJ(w) 2 \ / 1 (te + ti)te \
290 ----- = - > ci . ti | ------------ - ------------------------- |,
291 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
292 i=-k
293
294
295 _k_
296 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
297 ----- = - (1 - S2) > ci . ti^2 -----------------------------,
298 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
299 i=-k
300
301
302 dJ(w)
303 ----- = 0,
304 dRex
305
306
307 dJ(w)
308 ----- = 0,
309 dcsa
310
311
312 dJ(w)
313 ----- = 0.
314 dr
315
316
317 Extended::
318
319 _k_
320 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
321 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
322 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
323 i=-k
324
325 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
326 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
327 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
328
329
330 _k_
331 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
332 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
333 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
334 i=-k
335
336
337 _k_
338 dJ(w) 2 \ / 1 (ts + ti).ts \
339 ----- = - > ci . ti | ------------ - ------------------------- |,
340 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
341 i=-k
342
343
344 _k_
345 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
346 ----- = - - > ci . ti | ------------------------- - ------------------------- |,
347 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
348 i=-k
349
350
351 _k_
352 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
353 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
354 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
355 i=-k
356
357
358 _k_
359 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
360 ----- = - (S2f - S2) > ci . ti^2 -----------------------------,
361 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
362 i=-k
363
364
365 dJ(w)
366 ----- = 0,
367 dRex
368
369
370 dJ(w)
371 ----- = 0,
372 dcsa
373
374
375 dJ(w)
376 ----- = 0.
377 dr
378
379
380 Extended 2::
381
382 _k_
383 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
384 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
385 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
386 i=-k
387
388 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
389 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
390 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
391
392
393 _k_
394 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
395 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
396 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
397 i=-k
398
399
400 _k_
401 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
402 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |,
403 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
404 i=-k
405
406
407 _k_
408 dJ(w) 2 \ / 1 (ts + ti).ts \
409 ----- = - S2f > ci . ti | ------------ - ------------------------- |,
410 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
411 i=-k
412
413
414 _k_
415 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
416 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
417 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
418 i=-k
419
420
421 _k_
422 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
423 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------,
424 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
425 i=-k
426
427
428 dJ(w)
429 ----- = 0,
430 dRex
431
432
433 dJ(w)
434 ----- = 0,
435 dcsa
436
437
438 dJ(w)
439 ----- = 0.
440 dr
441 """
442
443
444
445
446
447
448
449
451 """Spectral density gradient.
452
453 Calculate the spectral desity values for the Gj partial derivative of the original model-free
454 formula with no parameters {} together with diffusion tensor parameters.
455
456 The model-free gradient is::
457
458 _k_
459 dJ(w) 2 \ dti 1 - (w.ti)^2
460 ----- = - > ci . --- ----------------.
461 dGj 5 /__ dGj (1 + (w.ti)^2)^2
462 i=-k
463 """
464
465 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
466
467
469 """Spectral density gradient.
470
471 Calculate the spectral desity values for the Gj partial derivative of the original model-free
472 formula with no parameters {} together with diffusion tensor parameters.
473
474 The model-free gradient is::
475
476 _k_
477 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
478 ----- = - > | ci . --- ---------------- + --- . ti ------------ |.
479 dGj 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
480 i=-k
481 """
482
483 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti + data.dci[j] * data.ti * data.fact_ti, axis=2)
484
485
486
487
489 """Spectral density gradient.
490
491 Calculate the spectral desity values for the Gj partial derivative of the original model-free
492 formula with the parameter {S2} together with diffusion tensor parameters.
493
494 The model-free gradient is::
495
496 _k_
497 dJ(w) 2 \ dti 1 - (w.ti)^2
498 ----- = - S2 > ci . --- ----------------.
499 dGj 5 /__ dGj (1 + (w.ti)^2)^2
500 i=-k
501 """
502
503 return 0.4 * params[data.s2_i] * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
504
505
507 """Spectral density gradient.
508
509 Calculate the spectral desity values for the Gj partial derivative of the original model-free
510 formula with the parameter {S2} together with diffusion tensor parameters.
511
512 The model-free gradient is::
513
514 _k_
515 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
516 ----- = - S2 > | ci . --- ---------------- + --- . ti ------------ |.
517 dGj 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
518 i=-k
519 """
520
521 return 0.4 * params[data.s2_i] * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti + data.dci[j] * data.ti * data.fact_ti, axis=2)
522
523
524
525
527 """Spectral density gradient.
528
529 Calculate the spectral desity values for the Gj partial derivative of the original model-free
530 formula with the parameters {S2, te} together with diffusion tensor parameters.
531
532 The model-free gradient is::
533
534 _k_
535 dJ(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
536 ----- = - > ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
537 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
538 i=-k
539 """
540
541 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti), axis=2)
542
543
545 """Spectral density gradient.
546
547 Calculate the spectral desity values for the Gj partial derivative of the original model-free
548 formula with the parameters {S2, te} together with diffusion tensor parameters.
549
550 The model-free gradient is::
551
552 _k_
553 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
554 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
555 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
556 i=-k
557
558 dci / S2 (1 - S2)(te + ti)te \ \
559 + --- . ti | ------------ + ------------------------- | |.
560 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
561 """
562
563 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti) + data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
564
565
566
567
568
569
570
571
573 """Spectral density gradient.
574
575 Calculate the spectral desity values for the O partial derivative of the original model-free
576 formula with no parameters {} together with diffusion tensor parameters.
577
578 The model-free gradient is::
579
580 _k_
581 dJ(w) 2 \ dci 1
582 ----- = - > --- . ti ------------.
583 dOj 5 /__ dOj 1 + (w.ti)^2
584 i=-k
585 """
586
587 return 0.4 * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
588
589
590
591
593 """Spectral density gradient.
594
595 Calculate the spectral desity values for the O partial derivative of the original model-free
596 formula with the parameter {S2} together with diffusion tensor parameters.
597
598 The model-free gradient is::
599
600 _k_
601 dJ(w) 2 \ dci 1
602 ----- = - S2 > --- . ti ------------.
603 dOj 5 /__ dOj 1 + (w.ti)^2
604 i=-k
605 """
606
607 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
608
609
610
611
613 """Spectral density gradient.
614
615 Calculate the spectral desity values for the O partial derivative of the original model-free
616 formula with the parameters {S2, te} together with diffusion tensor parameters.
617
618 The model-free gradient is::
619
620 _k_
621 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
622 ----- = - > --- . ti | ------------ + ------------------------- |.
623 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
624 i=-k
625 """
626
627 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
628
629
630
631
632
633
634
635
637 """Spectral density gradient.
638
639 Calculate the spectral desity values for the S2 partial derivative of the original model-free
640 formula with the single parameter {S2} with or without diffusion tensor parameters.
641
642 The model-free gradient is::
643
644 _k_
645 dJ(w) 2 \ 1
646 ----- = - > ci . ti ------------.
647 dS2 5 /__ 1 + (w.ti)^2
648 i=-k
649 """
650
651 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
652
653
654
655
657 """Spectral density gradient.
658
659 Calculate the spectral desity values for the S2 partial derivative of the original model-free
660 formula with the parameters {S2, te} with or without diffusion tensor parameters.
661
662 The model-free gradient is::
663
664 _k_
665 dJ(w) 2 \ / 1 (te + ti)te \
666 ----- = - > ci . ti | ------------ - ------------------------- |.
667 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
668 i=-k
669 """
670
671 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_te), axis=2)
672
673
674
675
676
677
678
679
681 """Spectral density gradient.
682
683 Calculate the spectral desity values for the te partial derivative of the original model-free
684 formula with the parameters {S2, te} with or without diffusion tensor parameters.
685
686 The model-free gradient is::
687
688 _k_
689 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
690 ----- = - (1 - S2) > ci . ti^2 -----------------------------.
691 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
692 i=-k
693 """
694
695 return 0.4 * data.one_s2 * sum(data.ci * data.fact_djw_dte, axis=2)
696
697
698
699
700
701
702
703
705 """Spectral density gradient.
706
707 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
708 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
709
710 The formula is::
711
712 _k_
713 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
714 ----- = - > ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
715 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
716 i=-k
717 """
718
719 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
720
721
723 """Spectral density gradient.
724
725 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
726 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
727
728 The formula is::
729
730 _k_
731 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
732 ----- = - > | ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
733 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
734 i=-k
735
736 dci / S2 (S2f - S2)(ts + ti)ts \ \
737 + --- . ti | ------------ + ------------------------- | |.
738 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
739 """
740
741 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
742
743
744
745
747 """Spectral density gradient.
748
749 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
750 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
751
752 The formula is::
753
754 _k_
755 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
756 ----- = - > ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
757 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
758 i=-k
759 """
760
761 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
762
763
765 """Spectral density gradient.
766
767 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
768 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
769
770 The formula is::
771
772 _k_
773 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
774 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
775 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
776 i=-k
777
778 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
779 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
780 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
781 """
782
783 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
784
785
786
787
788
789
790
791
793 """Spectral density gradient.
794
795 Calculate the spectral desity values for the O partial derivative of the extended model-free
796 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
797
798 The formula is::
799
800 _k_
801 dJ(w) 2 \ dci / S2 (S2f - S2)(ts + ti)ts \
802 ----- = - > --- . ti | ------------ + ------------------------- |.
803 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
804 i=-k
805 """
806
807 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
808
809
810
811
813 """Spectral density gradient.
814
815 Calculate the spectral desity values for the O partial derivative of the extended model-free
816 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
817
818 The formula is::
819
820 _k_
821 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
822 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
823 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
824 i=-k
825 """
826
827 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
828
829
830
831
832
833
834
835
837 """Spectral density gradient.
838
839 Calculate the spectral desity values for the S2 partial derivative of the extended model-free
840 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
841 parameters.
842
843 The formula is::
844
845 _k_
846 dJ(w) 2 \ / 1 (ts + ti).ts \
847 ----- = - > ci . ti | ------------ - ------------------------- |.
848 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
849 i=-k
850 """
851
852 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
853
854
855
856
857
858
859
860
862 """Spectral density gradient.
863
864 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
865 formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.
866
867 The formula is::
868
869 _k_
870 dJ(w) 2 \ (ts + ti).ts
871 ----- = - > ci . ti -------------------------.
872 dS2f 5 /__ (ts + ti)^2 + (w.ts.ti)^2
873 i=-k
874 """
875
876 return 0.4 * sum(data.ci * data.ti * data.fact_ts, axis=2)
877
878
879
880
882 """Spectral density gradient.
883
884 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
885 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
886
887 The formula is::
888
889 _k_
890 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
891 ----- = - - > ci . ti | ------------------------- - ------------------------- |.
892 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
893 i=-k
894 """
895
896 return -0.4 * sum(data.ci * data.ti * (data.fact_tf - data.fact_ts), axis=2)
897
898
899
900
901
902
903
904
906 """Spectral density gradient.
907
908 Calculate the spectral desity values for the tf partial derivative of the extended model-free
909 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
910
911 The formula is::
912
913 _k_
914 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
915 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
916 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
917 i=-k
918 """
919
920 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
921
922
923
924
925
926
927
928
930 """Spectral density gradient.
931
932 Calculate the spectral desity values for the ts partial derivative of the extended model-free
933 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
934 parameters.
935
936 The formula is::
937
938 _k_
939 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
940 ----- = - (S2f - S2) > ci . ti^2 -----------------------------.
941 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
942 i=-k
943 """
944
945 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
946
947
948
949
950
951
952
953
955 """Spectral density gradient.
956
957 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
958 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
959
960 The formula is::
961
962 _k_
963 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
964 ----- = - S2f > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
965 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
966 i=-k
967 """
968
969 return 0.4 * params[data.s2f_i] * sum(data.ci * data.dti[j] * (params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2s * data.fact_ts_djw_dti), axis=2)
970
971
973 """Spectral density gradient.
974
975 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
976 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
977
978 The formula is::
979
980 _k_
981 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
982 ----- = - S2f > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
983 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
984 i=-k
985
986 dci / S2s (1 - S2s)(ts + ti)ts \ \
987 + --- . ti | ------------ + ------------------------- | |.
988 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
989 """
990
991 return 0.4 * params[data.s2f_i] * sum(data.ci * data.dti[j] * (params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2s * data.fact_ts_djw_dti) + data.dci[j] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
992
993
994
995
997 """Spectral density gradient.
998
999 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
1000 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1001
1002 The formula is::
1003
1004 _k_
1005 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1006 ----- = - > ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1007 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1008 i=-k
1009 """
1010
1011 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
1012
1013
1015 """Spectral density gradient.
1016
1017 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
1018 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1019
1020 The formula is::
1021
1022 _k_
1023 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1024 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f).tf^2 ----------------------------- + S2f(1 - S2s).ts^2 ----------------------------- |
1025 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1026 i=-k
1027
1028 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1029 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
1030 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1031 """
1032
1033 return 0.4 * sum(data.ci * data.dti[j] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.dci[j] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
1034
1035
1036
1037
1038
1039
1040
1041
1043 """Spectral density gradient.
1044
1045 Calculate the spectral desity values for the O partial derivative of the extended model-free
1046 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
1047
1048 The formula is::
1049
1050 _k_
1051 dJ(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
1052 ----- = - S2f > --- . ti | ------------ + ------------------------- |.
1053 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1054 i=-k
1055 """
1056
1057 return 0.4 * params[data.s2f_i] * sum(data.dci[j] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
1058
1059
1060
1061
1063 """Spectral density gradient.
1064
1065 Calculate the spectral desity values for the O partial derivative of the extended model-free
1066 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1067
1068 The formula is::
1069
1070 _k_
1071 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1072 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
1073 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1074 i=-k
1075 """
1076
1077 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
1078
1079
1080
1081
1082
1083
1084
1085
1087 """Spectral density gradient.
1088
1089 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1090 formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.
1091
1092 The formula is::
1093
1094 _k_
1095 dJ(w) 2 \ / S2s (1 - S2s)(ts + ti).ts \
1096 ----- = - > ci . ti | ------------ + ------------------------- |.
1097 dS2f 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1098 i=-k
1099 """
1100
1101 return 0.4 * sum(data.ci * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
1102
1103
1104
1105
1107 """Spectral density gradient.
1108
1109 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1110 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1111
1112 The formula is::
1113
1114 _k_
1115 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
1116 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |.
1117 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1118 i=-k
1119 """
1120
1121 return 0.4 * sum(data.ci * data.ti * (params[data.s2s_i] * data.fact_ti - data.fact_tf + data.one_s2s * data.fact_ts), axis=2)
1122
1123
1124
1125
1126
1127
1128
1129
1131 """Spectral density gradient.
1132
1133 Calculate the spectral desity values for the S2s partial derivative of the extended model-free
1134 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1135 tensor parameters.
1136
1137 The formula is::
1138
1139 _k_
1140 dJ(w) 2 \ / 1 (ts + ti).ts \
1141 ----- = - S2f > ci . ti | ------------ - ------------------------- |.
1142 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1143 i=-k
1144 """
1145
1146 return 0.4 * params[data.s2f_i] * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
1147
1148
1149
1150
1151
1152
1153
1154
1156 """Spectral density gradient.
1157
1158 Calculate the spectral desity values for the tf partial derivative of the extended model-free
1159 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1160
1161 The formula is::
1162
1163 _k_
1164 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1165 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
1166 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1167 i=-k
1168 """
1169
1170 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
1171
1172
1173
1174
1175
1176
1177
1178
1180 """Spectral density gradient.
1181
1182 Calculate the spectral desity values for the ts partial derivative of the extended model-free
1183 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1184 tensor parameters.
1185
1186 The formula is::
1187
1188 _k_
1189 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1190 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------.
1191 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1192 i=-k
1193 """
1194
1195 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
1196
1197
1198
1199
1200
1201
1202
1203
1204 """
1205 The spectral density Hessians
1206 =============================
1207
1208 Data structure: data.d2jw
1209 Dimension: 4D, (number of NMR frequencies, 5 spectral density frequencies, model-free
1210 parameters, model-free parameters)
1211 Type: numpy 4D matrix, float64
1212 Dependencies: None
1213 Required by: data.d2ri
1214
1215
1216 Formulae
1217 ========
1218
1219 Original: Model-free parameter - Model-free parameter::
1220
1221 _k_
1222 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \
1223 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1224 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1225 i=-k
1226
1227 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1228 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1229 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1230
1231
1232 d2ci / S2 (1 - S2)(te + ti)te \ \
1233 + ------- ti | ------------ + ------------------------- | |,
1234 dGj.dGk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1235
1236
1237 _k_
1238 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1239 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1240 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1241 i=-k
1242
1243 d2ci / S2 (1 - S2)(te + ti)te \ \
1244 + ------- ti | ------------ + ------------------------- | |,
1245 dGj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1246
1247
1248 _k_
1249 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1250 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
1251 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1252 i=-k
1253
1254 dci / 1 (te + ti)te \ \
1255 + --- . ti | ------------ - ------------------------- | |,
1256 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1257
1258
1259 _k_
1260 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
1261 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1262 dGj.dte 5 /__ \ dGj ((te + ti)^2 + (w.te.ti)^2)^3 dGj ((te + ti)^2 + (w.te.ti)^2)^2 /
1263 i=-k
1264
1265
1266 _k_
1267 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
1268 ------- = - > ------- . ti | ------------ + ------------------------- |,
1269 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1270 i=-k
1271
1272
1273 _k_
1274 d2J(w) 2 \ dci / 1 (te + ti)te \
1275 ------- = - > --- . ti | ------------ - ------------------------- |,
1276 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1277 i=-k
1278
1279
1280 _k_
1281 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
1282 ------- = - (1 - S2) > --- . ti^2 -----------------------------,
1283 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
1284 i=-k
1285
1286
1287 d2J(w)
1288 ------ = 0,
1289 dS2**2
1290
1291
1292 _k_
1293 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
1294 ------- = - - > ci . ti^2 -----------------------------,
1295 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
1296 i=-k
1297
1298
1299 _k_
1300 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
1301 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
1302 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
1303 i=-k
1304
1305
1306 Original: Other parameters::
1307
1308 d2J(w) d2J(w) d2J(w)
1309 -------- = 0, -------- = 0, ------ = 0,
1310 dS2.dRex dS2.dcsa dS2.dr
1311
1312
1313 d2J(w) d2J(w) d2J(w)
1314 -------- = 0, -------- = 0, ------ = 0,
1315 dte.dRex dte.dcsa dte.dr
1316
1317
1318 d2J(w) d2J(w) d2J(w)
1319 ------- = 0, --------- = 0, ------- = 0,
1320 dRex**2 dRex.dcsa dRex.dr
1321
1322
1323 d2J(w) d2J(w)
1324 ------- = 0, ------- = 0,
1325 dcsa**2 dcsa.dr
1326
1327
1328 d2J(w)
1329 ------ = 0.
1330 dr**2
1331
1332
1333 Extended: Model-free parameter - Model-free parameter::
1334
1335 _k_
1336 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
1337 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1338 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1339 i=-k
1340
1341 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1342 + (S2f - S2)ts^2 ---------------------------------------------------- |
1343 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1344
1345
1346 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1347 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
1348 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1349
1350
1351 (ts + ti)^2 - (w.ts.ti)^2 \
1352 + (S2f - S2)ts^2 ----------------------------- |
1353 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1354
1355
1356 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1357 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1358 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1359
1360
1361 _k_
1362 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1363 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
1364 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1365 i=-k
1366
1367 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1368 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1369 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1370
1371
1372 _k_
1373 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1374 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
1375 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1376 i=-k
1377
1378 dci / 1 (ts + ti)ts \ \
1379 + --- . ti | ------------ - ------------------------- | |,
1380 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1381
1382
1383 _k_
1384 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1385 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
1386 dGj.dS2f 5 /__ \ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1387 i=-k
1388
1389 dci / (tf + ti)tf (ts + ti)ts \ \
1390 + --- . ti | ------------------------- - ------------------------- | |,
1391 dGj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1392
1393
1394 _k_
1395 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1396 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1397 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1398 i=-k
1399
1400
1401 _k_
1402 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1403 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1404 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1405 i=-k
1406
1407
1408 _k_
1409 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
1410 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1411 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1412 i=-k
1413
1414
1415 _k_
1416 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1417 ------- = - > --- . ti | ------------ - ------------------------- |,
1418 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1419 i=-k
1420
1421
1422 _k_
1423 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
1424 -------- = - - > --- . ti | ------------------------- - ------------------------- |,
1425 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1426 i=-k
1427
1428
1429 _k_
1430 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1431 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1432 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1433 i=-k
1434
1435
1436 _k_
1437 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1438 ------- = - (S2f - S2) > --- . ti^2 -----------------------------,
1439 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1440 i=-k
1441
1442
1443 d2J(w) d2J(w) d2J(w)
1444 ------ = 0, -------- = 0, ------- = 0,
1445 dS2**2 dS2.dS2f dS2.dtf
1446
1447
1448 _k_
1449 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1450 ------- = - - > ci . ti^2 -----------------------------,
1451 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1452 i=-k
1453
1454
1455 d2J(w)
1456 ------- = 0,
1457 dS2f**2
1458
1459
1460 _k_
1461 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1462 -------- = - - > ci . ti^2 -----------------------------,
1463 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1464 i=-k
1465
1466
1467 _k_
1468 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1469 -------- = - > ci . ti^2 -----------------------------,
1470 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1471 i=-k
1472
1473
1474 _k_
1475 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1476 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1477 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1478 i=-k
1479
1480
1481 d2J(w)
1482 ------- = 0,
1483 dtf.dts
1484
1485
1486 _k_
1487 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1488 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------,
1489 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1490 i=-k
1491
1492
1493 Extended: Other parameters::
1494
1495 d2J(w) d2J(w) d2J(w)
1496 --------- = 0, --------- = 0, ------- = 0,
1497 dS2f.dRex dS2f.dcsa dS2f.dr
1498
1499
1500 d2J(w) d2J(w) d2J(w)
1501 -------- = 0, -------- = 0, ------ = 0,
1502 dS2.dRex dS2.dcsa dS2.dr
1503
1504
1505 d2J(w) d2J(w) d2J(w)
1506 -------- = 0, -------- = 0, ------ = 0,
1507 dtf.dRex dtf.dcsa dtf.dr
1508
1509
1510 d2J(w) d2J(w) d2J(w)
1511 -------- = 0, -------- = 0, ------ = 0,
1512 dts.dRex dts.dcsa dts.dr
1513
1514
1515 d2J(w) d2J(w) d2J(w)
1516 ------- = 0, --------- = 0, ------- = 0,
1517 dRex**2 dRex.dcsa dRex.dr
1518
1519
1520 d2J(w) d2J(w)
1521 ------- = 0, ------- = 0,
1522 dcsa**2 dcsa.dr
1523
1524
1525 d2J(w)
1526 ------ = 0.
1527 dr**2
1528
1529
1530 Extended 2: Model-free parameter - Model-free parameter::
1531
1532 _k_
1533 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
1534 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1535 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1536 i=-k
1537
1538 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1539 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
1540 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1541
1542
1543 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1544 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
1545 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1546
1547
1548 (ts + ti)^2 - (w.ts.ti)^2 \
1549 + S2f(1 - S2s)ts^2 ----------------------------- |
1550 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1551
1552
1553 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1554 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1555 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1556
1557
1558 _k_
1559 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1560 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1561 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1562 i=-k
1563
1564 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1565 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1566 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1567
1568
1569 _k_
1570 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1571 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
1572 dGj.dS2f 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1573 i=-k
1574
1575 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
1576 + --- . ti | ------------ - ------------------------- + ------------------------- | |,
1577 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1578
1579
1580 _k_
1581 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1582 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
1583 dGj.dS2s 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1584 i=-k
1585
1586 dci / 1 (ts + ti)ts \ \
1587 + --- . ti | ------------ - ------------------------- | |,
1588 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1589
1590
1591 _k_
1592 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1593 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1594 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1595 i=-k
1596
1597
1598 _k_
1599 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1600 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1601 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1602 i=-k
1603
1604
1605 _k_
1606 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1607 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1608 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1609 i=-k
1610
1611
1612 _k_
1613 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
1614 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |,
1615 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1616 i=-k
1617
1618
1619 _k_
1620 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1621 -------- = - S2f > --- . ti | ------------ - ------------------------- |,
1622 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1623 i=-k
1624
1625
1626 _k_
1627 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1628 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1629 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1630 i=-k
1631
1632
1633 _k_
1634 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1635 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------,
1636 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1637 i=-k
1638
1639
1640 d2J(w)
1641 ------- = 0,
1642 dS2f**2
1643
1644
1645 _k_
1646 d2J(w) 2 \ / 1 (ts + ti).ts \
1647 --------- = - > ci . ti | ------------ - ------------------------- |,
1648 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1649 i=-k
1650
1651
1652 _k_
1653 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1654 -------- = - - > ci . ti^2 -----------------------------,
1655 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1656 i=-k
1657
1658
1659 _k_
1660 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1661 -------- = - (1 - S2s) > ci . ti^2 -----------------------------,
1662 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1663 i=-k
1664
1665
1666 d2J(w) d2J(w)
1667 ------- = 0, -------- = 0,
1668 dS2s**2 dS2s.dtf
1669
1670
1671 _k_
1672 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1673 -------- = - - S2f > ci . ti^2 -----------------------------,
1674 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1675 i=-k
1676
1677
1678 _k_
1679 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1680 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1681 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1682 i=-k
1683
1684
1685 d2J(w)
1686 ------- = 0,
1687 dtf.dts
1688
1689
1690 _k_
1691 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1692 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
1693 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1694 i=-k
1695
1696
1697
1698 Extended 2: Other parameters::
1699
1700 d2J(w) d2J(w) d2J(w)
1701 --------- = 0, --------- = 0, ------- = 0,
1702 dS2f.dRex dS2f.dcsa dS2f.dr
1703
1704
1705 d2J(w) d2J(w) d2J(w)
1706 --------- = 0, --------- = 0, ------- = 0,
1707 dS2s.dRex dS2s.dcsa dS2s.dr
1708
1709
1710 d2J(w) d2J(w) d2J(w)
1711 -------- = 0, -------- = 0, ------ = 0,
1712 dtf.dRex dtf.dcsa dtf.dr
1713
1714
1715 d2J(w) d2J(w) d2J(w)
1716 -------- = 0, -------- = 0, ------ = 0,
1717 dts.dRex dts.dcsa dts.dr
1718
1719
1720 d2J(w) d2J(w) d2J(w)
1721 ------- = 0, --------- = 0, ------- = 0,
1722 dRex**2 dRex.dcsa dRex.dr
1723
1724
1725 d2J(w) d2J(w)
1726 ------- = 0, ------- = 0,
1727 dcsa**2 dcsa.dr
1728
1729
1730 d2J(w)
1731 ------ = 0.
1732 dr**2
1733 """
1734
1735
1736
1737
1738
1739
1740
1742 """Spectral density Hessian.
1743
1744 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1745 model-free formula with no parameters {} together with diffusion tensor parameters.
1746
1747 The model-free Hessian is::
1748
1749 _k_
1750 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1751 ------- = - > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1752 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 dGj.dGk (1 + (w.ti)^2)^2 /
1753 i=-k
1754 """
1755
1756 return 0.4 * sum(data.ci * (-2.0 * data.dti[j] * data.dti[k] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.d2ti[j, k] * data.fact_ti_djw_dti), axis=2)
1757
1758
1760 """Spectral density Hessian.
1761
1762 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1763 model-free formula with no parameters {} together with diffusion tensor parameters.
1764
1765 The model-free Hessian is::
1766
1767 _k_
1768 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1769 ------- = - > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1770 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 \ dGj dGk dGk dGj dGj.dGk / (1 + (w.ti)^2)^2 dGj.dGk 1 + (w.ti)^2 /
1771 i=-k
1772 """
1773
1774 return 0.4 * sum(-2.0 * data.ci * data.dti[j] * data.dti[k] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * data.fact_ti_djw_dti + data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
1775
1776
1777
1778
1780 """Spectral density Hessian.
1781
1782 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1783 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1784
1785 The model-free Hessian is::
1786
1787 _k_
1788 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1789 ------- = - S2 > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1790 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 dGj.dGk (1 + (w.ti)^2)^2 /
1791 i=-k
1792 """
1793
1794 return 0.4 * params[data.s2_i] * sum(data.ci * (-2.0 * data.dti[j] * data.dti[k] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.d2ti[j, k] * data.fact_ti_djw_dti), axis=2)
1795
1796
1798 """Spectral density Hessian.
1799
1800 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1801 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1802
1803 The model-free Hessian is::
1804
1805 _k_
1806 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1807 ------- = - S2 > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1808 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 \ dGj dGk dGk dGj dGj.dGk / (1 + (w.ti)^2)^2 dGj.dGk 1 + (w.ti)^2 /
1809 i=-k
1810 """
1811
1812 return 0.4 * params[data.s2_i] * sum(-2.0 * data.ci * data.dti[j] * data.dti[k] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * data.fact_ti_djw_dti + data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
1813
1814
1815
1816
1818 """Spectral density Hessian.
1819
1820 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1821 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1822
1823 The model-free Hessian is::
1824
1825 _k_
1826 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \
1827 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1828 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1829 i=-k
1830
1831 d2ti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \ \
1832 + ------- | S2 ---------------- + (1 - S2)te^2 ----------------------------- | |.
1833 dGj.dGk \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 / /
1834 """
1835
1836
1837 a = -2.0 * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2 * params[data.te_i]**2 * (data.te_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.te_i]**3 * data.ti * data.te_ti - (data.frq_list_ext * params[data.te_i])**4 * data.ti**3) * data.inv_te_denom**3)
1838
1839
1840 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti)
1841
1842 return 0.4 * sum(data.ci * (a + b), axis=2)
1843
1844
1846 """Spectral density Hessian.
1847
1848 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1849 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1850
1851 The model-free Hessian is::
1852
1853 _k_
1854 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \
1855 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1856 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1857 i=-k
1858
1859 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1860 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1861 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1862
1863
1864 d2ci / S2 (1 - S2)(te + ti)te \ \
1865 + ------- ti | ------------ + ------------------------- | |.
1866 dGj.dGk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1867 """
1868
1869
1870 a = -2.0 * data.ci * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2 * params[data.te_i]**2 * (data.te_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.te_i]**3 * data.ti * data.te_ti - (data.frq_list_ext * params[data.te_i])**4 * data.ti**3) * data.inv_te_denom**3)
1871
1872
1873 b = (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti)
1874
1875
1876 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te)
1877
1878 return 0.4 * sum(a + b + c, axis=2)
1879
1880
1881
1882
1883
1884
1885
1886
1887
1889 """Spectral density Hessian.
1890
1891 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1892 model-free formula with no parameters {} together with diffusion tensor parameters.
1893
1894 The model-free Hessian is::
1895
1896 _k_
1897 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1898 ------- = - > --- . --- . ----------------.
1899 dGj.dOj 5 /__ dOj dGj (1 + (w.ti)^2)^2
1900 i=-k
1901 """
1902
1903 return 0.4 * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1904
1905
1907 """Spectral density Hessian.
1908
1909 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1910 model-free formula with no parameters {} together with diffusion tensor parameters.
1911
1912 The model-free Hessian is::
1913
1914 _k_
1915 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1916 ------- = - > | --- . --- . ---------------- + ------- ti ------------ |.
1917 dGj.dOj 5 /__ \ dOj dGj (1 + (w.ti)^2)^2 dGj.dOj 1 + (w.ti)^2 /
1918 i=-k
1919 """
1920
1921 return 0.4 * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti + data.d2ci[k, j] * data.ti * data.fact_ti, axis=2)
1922
1923
1924
1925
1927 """Spectral density Hessian.
1928
1929 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1930 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1931
1932 The model-free Hessian is::
1933
1934 _k_
1935 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1936 ------- = - S2 > --- . --- . ----------------.
1937 dGj.dOj 5 /__ dOj dGj (1 + (w.ti)^2)^2
1938 i=-k
1939 """
1940
1941 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1942
1943
1945 """Spectral density Hessian.
1946
1947 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1948 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1949
1950 The model-free Hessian is::
1951
1952 _k_
1953 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1954 ------- = - S2 > | --- . --- . ---------------- + ------- ti ------------ |.
1955 dGj.dOj 5 /__ \ dOj dGj (1 + (w.ti)^2)^2 dGj.dOj 1 + (w.ti)^2 /
1956 i=-k
1957 """
1958
1959 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti + data.d2ci[k, j] * data.ti * data.fact_ti, axis=2)
1960
1961
1962
1963
1965 """Spectral density Hessian.
1966
1967 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1968 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1969
1970 The model-free Hessian is::
1971
1972 _k_
1973 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1974 ------- = - > --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
1975 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1976 i=-k
1977 """
1978
1979 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti), axis=2)
1980
1981
1983 """Spectral density Hessian.
1984
1985 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1986 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1987
1988 The model-free Hessian is::
1989
1990 _k_
1991 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1992 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1993 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1994 i=-k
1995
1996 d2ci / S2 (1 - S2)(te + ti)te \ \
1997 + ------- ti | ------------ + ------------------------- | |.
1998 dGj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1999 """
2000
2001 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti) + data.d2ci[k, j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
2002
2003
2004
2005
2006
2007
2008
2009
2011 """Spectral density Hessian.
2012
2013 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2014 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2015
2016 The model-free Hessian is::
2017
2018 _k_
2019 d2J(w) 2 \ dti 1 - (w.ti)^2
2020 ------- = - > ci . --- ----------------.
2021 dGj.dS2 5 /__ dGj (1 + (w.ti)^2)^2
2022 i=-k
2023 """
2024
2025 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti, axis=2)
2026
2027
2029 """Spectral density Hessian.
2030
2031 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2032 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2033
2034 The model-free Hessian is::
2035
2036 _k_
2037 d2J(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
2038 ------- = - > | ci . --- ---------------- + --- . ti ------------ |.
2039 dGj.dS2 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
2040 i=-k
2041 """
2042
2043 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti + data.dci[k] * data.ti * data.fact_ti, axis=2)
2044
2045
2046
2047
2049 """Spectral density Hessian.
2050
2051 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2052 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2053
2054 The model-free Hessian is::
2055
2056 _k_
2057 d2J(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2058 ------- = - > ci . --- | ---------------- - te^2 ----------------------------- |.
2059 dGj.dS2 5 /__ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2060 i=-k
2061 """
2062
2063 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_te_djw_dti), axis=2)
2064
2065
2067 """Spectral density Hessian.
2068
2069 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2070 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2071
2072 The model-free Hessian is::
2073
2074 _k_
2075 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2076 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
2077 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2078 i=-k
2079
2080 dci / 1 (te + ti)te \ \
2081 + --- . ti | ------------ - ------------------------- | |.
2082 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
2083 """
2084
2085 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_te_djw_dti) + data.dci[k] * data.ti * (data.fact_ti - data.fact_te), axis=2)
2086
2087
2088
2089
2090
2091
2092
2093
2095 """Spectral density Hessian.
2096
2097 Calculate the spectral desity values for the Gj - te double partial derivative of the original
2098 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2099
2100 The model-free Hessian is::
2101
2102 _k_
2103 d2J(w) 4 \ dti (te + ti)^2 - 3(w.te.ti)^2
2104 ------- = - (1 - S2) . te > ci . --- . ti . (te + ti) -----------------------------.
2105 dGj.dte 5 /__ dGj ((te + ti)^2 + (w.te.ti)^2)^3
2106 i=-k
2107 """
2108
2109 return 0.8 * data.one_s2 * params[data.te_i] * sum(data.ci * data.dti[k] * data.ti * data.te_ti * (data.te_ti_sqrd - 3.0 * data.w_te_ti_sqrd) * data.inv_te_denom**3, axis=2)
2110
2111
2113 """Spectral density Hessian.
2114
2115 Calculate the spectral desity values for the Gj - te double partial derivative of the original
2116 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2117
2118 The model-free Hessian is::
2119
2120 _k_
2121 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
2122 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2123 dGj.dte 5 /__ \ dGj ((te + ti)^2 + (w.te.ti)^2)^3 dGj ((te + ti)^2 + (w.te.ti)^2)^2 /
2124 i=-k
2125 """
2126
2127 return 0.4 * data.one_s2 * sum(2.0 * data.ci * data.dti[k] * params[data.te_i] * data.ti * data.te_ti * (data.te_ti_sqrd - 3.0 * data.w_te_ti_sqrd) * data.inv_te_denom**3 + data.dci[k] * data.fact_djw_dte, axis=2)
2128
2129
2130
2131
2132
2133
2134
2135
2137 """Spectral density Hessian.
2138
2139 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2140 original model-free formula with no parameters {} together with diffusion tensor parameters.
2141
2142 The model-free Hessian is::
2143
2144 _k_
2145 d2J(w) 2 \ d2ci ti
2146 ------- = - > ------- . ------------.
2147 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2148 i=-k
2149 """
2150
2151 return 0.4 * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2152
2153
2154
2155
2157 """Spectral density Hessian.
2158
2159 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2160 original model-free formula with the parameter {S2} together with diffusion tensor parameters.
2161
2162 The model-free Hessian is::
2163
2164 _k_
2165 d2J(w) 2 \ d2ci ti
2166 ------- = - S2 > ------- . ------------.
2167 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2168 i=-k
2169 """
2170
2171 return 0.4 * params[data.s2_i] * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2172
2173
2174
2175
2177 """Spectral density Hessian.
2178
2179 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2180 original model-free formula with the parameters {S2, te} together with diffusion tensor
2181 parameters.
2182
2183 The model-free Hessian is::
2184
2185 _k_
2186 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
2187 ------- = - > ------- . ti | ------------ + ------------------------- |.
2188 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2189 i=-k
2190 """
2191
2192 return 0.4 * sum(data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
2193
2194
2195
2196
2197
2198
2199
2200
2202 """Spectral density Hessian.
2203
2204 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2205 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2206
2207 The model-free Hessian is::
2208
2209 _k_
2210 d2J(w) 2 \ dci 1
2211 ------- = - > --- . ti ------------.
2212 dOj.dS2 5 /__ dOj 1 + (w.ti)^2
2213 i=-k
2214 """
2215
2216 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ti, axis=2)
2217
2218
2219
2220
2222 """Spectral density Hessian.
2223
2224 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2225 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2226
2227 The model-free Hessian is::
2228
2229 _k_
2230 d2J(w) 2 \ dci / 1 (te + ti)te \
2231 ------- = - > --- . ti | ------------ - ------------------------- |.
2232 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2233 i=-k
2234 """
2235
2236 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_te), axis=2)
2237
2238
2239
2240
2241
2242
2243
2244
2246 """Spectral density Hessian.
2247
2248 Calculate the spectral desity values for the Oj - te double partial derivative of the original
2249 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2250
2251 The model-free Hessian is::
2252
2253 _k_
2254 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
2255 ------- = - (1 - S2) > --- . ti^2 -----------------------------.
2256 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
2257 i=-k
2258 """
2259
2260 return 0.4 * data.one_s2 * sum(data.dci[k] * data.fact_djw_dte, axis=2)
2261
2262
2263
2264
2265
2266
2267
2268
2270 """Spectral density Hessian.
2271
2272 Calculate the spectral desity values for the S2 - te double partial derivative of the original
2273 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2274
2275 The model-free Hessian is::
2276
2277 _k_
2278 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
2279 ------- = - - > ci . ti^2 -----------------------------.
2280 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
2281 i=-k
2282 """
2283
2284 return -0.4 * sum(data.ci * data.fact_djw_dte, axis=2)
2285
2286
2287
2288
2289
2290
2291
2292
2294 """Spectral density Hessian.
2295
2296 Calculate the spectral desity values for the te - te double partial derivative of the original
2297 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2298
2299 The model-free Hessian is::
2300
2301 _k_
2302 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
2303 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
2304 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
2305 i=-k
2306 """
2307
2308 return -0.8 * data.one_s2 * sum(data.ci * data.ti**2 * (data.te_ti**3 + 3.0 * data.frq_sqrd_list_ext * data.ti**3 * params[data.te_i] * data.te_ti - data.w_ti_sqrd**2 * params[data.te_i]**3) * data.inv_te_denom**3, axis=2)
2309
2310
2311
2312
2313
2314
2315
2316
2318 """Spectral density Hessian.
2319
2320 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2321 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2322
2323 The model-free Hessian is::
2324
2325 _k_
2326 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2327 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2328 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2329 i=-k
2330
2331 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2332 + ------- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- | |.
2333 dGj.dGk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2334 """
2335
2336
2337 a = -2.0 * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
2338
2339
2340 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2341
2342 return 0.4 * sum(data.ci * (a + b), axis=2)
2343
2344
2346 """Spectral density Hessian.
2347
2348 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2349 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2350
2351 The model-free Hessian is::
2352
2353 _k_
2354 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2355 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2356 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2357 i=-k
2358
2359 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2360 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2361 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2362
2363
2364 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2365 + ------- . ti | ------------ + ------------------------- | |.
2366 dGj.dGk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2367 """
2368
2369
2370 a = -2.0 * data.ci * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
2371
2372
2373 b = (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2374
2375
2376 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
2377
2378 return 0.4 * sum(a + b + c, axis=2)
2379
2380
2381
2382
2383
2385 """Spectral density Hessian.
2386
2387 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2388 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2389 parameters.
2390
2391 The model-free Hessian is::
2392
2393 _k_
2394 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
2395 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2396 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2397 i=-k
2398
2399 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2400 + (S2f - S2)ts^2 ---------------------------------------------------- |
2401 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2402
2403
2404 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2405 + ------- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- | |.
2406 dGj.dGk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2407 """
2408
2409
2410 a = -2.0 * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2f * params[data.tf_i]**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.tf_i]**3 * data.ti * data.tf_ti - (data.frq_list_ext * params[data.tf_i])**4 * data.ti**3) * data.inv_tf_denom**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
2411
2412
2413 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2414
2415 return 0.4 * sum(data.ci * (a + b), axis=2)
2416
2417
2419 """Spectral density Hessian.
2420
2421 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2422 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2423 parameters.
2424
2425 The model-free Hessian is::
2426
2427 _k_
2428 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
2429 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2430 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2431 i=-k
2432
2433 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2434 + (S2f - S2)ts^2 ---------------------------------------------------- |
2435 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2436
2437
2438 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
2439 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
2440 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
2441
2442
2443 (ts + ti)^2 - (w.ts.ti)^2 \
2444 + (S2f - S2)ts^2 ----------------------------- |
2445 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2446
2447
2448 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2449 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2450 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2451 """
2452
2453
2454 a = -2.0 * data.ci * data.dti[j] * data.dti[k] * (params[data.s2_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2f * params[data.tf_i]**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.tf_i]**3 * data.ti * data.tf_ti - (data.frq_list_ext * params[data.tf_i])**4 * data.ti**3) * data.inv_tf_denom**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
2455
2456
2457 b = (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2458
2459
2460 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts)
2461
2462 return 0.4 * sum(a + b + c, axis=2)
2463
2464
2465
2466
2467
2468
2469
2470
2472 """Spectral density Hessian.
2473
2474 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2475 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2476
2477 The model-free Hessian is::
2478
2479 _k_
2480 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2481 ------- = - > --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
2482 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2483 i=-k
2484 """
2485
2486 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
2487
2488
2490 """Spectral density Hessian.
2491
2492 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2493 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2494
2495 The model-free Hessian is::
2496
2497 _k_
2498 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2499 ------- = - > | --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2500 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2501 i=-k
2502
2503 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2504 + ------- . ti | ------------ + ------------------------- | |.
2505 dGj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2506 """
2507
2508 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.d2ci[k, j] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
2509
2510
2511
2512
2514 """Spectral density Hessian.
2515
2516 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2517 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2518 parameters.
2519
2520 The model-free Hessian is::
2521
2522 _k_
2523 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2524 ------- = - > --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
2525 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2526 i=-k
2527 """
2528
2529 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
2530
2531
2533 """Spectral density Hessian.
2534
2535 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2536 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2537 parameters.
2538
2539 The model-free Hessian is::
2540
2541 _k_
2542 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2543 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
2544 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2545 i=-k
2546
2547 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2548 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2549 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2550 """
2551
2552 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.d2ci[k, j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
2553
2554
2555
2556
2557
2558
2559
2560
2562 """Spectral density Hessian.
2563
2564 Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended
2565 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2566 diffusion tensor parameters.
2567
2568 The model-free Hessian is::
2569
2570 _k_
2571 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2572 ------- = - > ci . --- | ---------------- - ts^2 ----------------------------- |.
2573 dGj.dS2 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2574 i=-k
2575 """
2576
2577 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti), axis=2)
2578
2579
2581 """Spectral density Hessian.
2582
2583 Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended
2584 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2585 diffusion tensor parameters.
2586
2587 The model-free Hessian is::
2588
2589 _k_
2590 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2591 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
2592 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2593 i=-k
2594
2595 dci / 1 (ts + ti)ts \ \
2596 + --- . ti | ------------ - ------------------------- | |.
2597 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2598 """
2599
2600 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti) + data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
2601
2602
2603
2604
2605
2606
2607
2608
2610 """Spectral density Hessian.
2611
2612 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2613 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2614
2615 The model-free Hessian is::
2616
2617 _k_
2618 d2J(w) 2 \ dti (ts + ti)^2 - (w.ts.ti)^2
2619 -------- = - > ci . --- ts^2 -----------------------------.
2620 dGj.dS2f 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^2
2621 i=-k
2622 """
2623
2624 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti, axis=2)
2625
2626
2628 """Spectral density Hessian.
2629
2630 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2631 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2632
2633 The model-free Hessian is::
2634
2635 _k_
2636 d2J(w) 2 \ / dti (ts + ti)^2 - (w.ts.ti)^2 dci (ts + ti)ts \
2637 -------- = - > | ci . --- ts^2 ----------------------------- + --- . ti ------------------------- |.
2638 dGj.dS2f 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 dGj (ts + ti)^2 + (w.ts.ti)^2 /
2639 i=-k
2640 """
2641
2642 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti + data.dci[k] * data.ti * data.fact_ts, axis=2)
2643
2644
2645
2646
2648 """Spectral density Hessian.
2649
2650 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2651 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2652 parameters.
2653
2654 The model-free Hessian is::
2655
2656 _k_
2657 d2J(w) 2 \ dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2658 -------- = - - > ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |.
2659 dGj.dS2f 5 /__ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2660 i=-k
2661 """
2662
2663 return -0.4 * sum(data.ci * data.dti[k] * (data.fact_tf_djw_dti - data.fact_ts_djw_dti), axis=2)
2664
2665
2667 """Spectral density Hessian.
2668
2669 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2670 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2671 parameters.
2672
2673 The model-free Hessian is::
2674
2675 _k_
2676 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2677 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
2678 dGj.dS2f 5 /__ \ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2679 i=-k
2680
2681 dci / (tf + ti)tf (ts + ti)ts \ \
2682 + --- . ti | ------------------------- - ------------------------- | |.
2683 dGj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2684 """
2685
2686 return -0.4 * sum(data.ci * data.dti[k] * (data.fact_tf_djw_dti - data.fact_ts_djw_dti) + data.dci[k] * data.ti * (data.fact_tf - data.fact_ts), axis=2)
2687
2688
2689
2690
2691
2692
2693
2694
2696 """Spectral density Hessian.
2697
2698 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
2699 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2700 parameters.
2701
2702 The model-free Hessian is::
2703
2704 _k_
2705 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
2706 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
2707 dGj.dtf 5 /__ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3
2708 i=-k
2709 """
2710
2711 return 0.8 * data.one_s2f * params[data.tf_i] * sum(data.ci * data.dti[k] * data.ti * data.tf_ti * (data.tf_ti_sqrd - 3.0 * data.w_tf_ti_sqrd) * data.inv_tf_denom**3, axis=2)
2712
2713
2715 """Spectral density Hessian.
2716
2717 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
2718 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2719 parameters.
2720
2721 The model-free Hessian is::
2722
2723 _k_
2724 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
2725 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2726 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
2727 i=-k
2728 """
2729
2730 return 0.4 * data.one_s2f * sum(2.0 * data.ci * data.dti[k] * params[data.tf_i] * data.ti * data.tf_ti * (data.tf_ti_sqrd - 3.0 * data.w_tf_ti_sqrd) * data.inv_tf_denom**3 + data.dci[k] * data.fact_djw_dtf, axis=2)
2731
2732
2733
2734
2735
2736
2737
2738
2740 """Spectral density Hessian.
2741
2742 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
2743 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2744 diffusion tensor parameters.
2745
2746 The model-free Hessian is::
2747
2748 _k_
2749 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
2750 ------- = - (S2f - S2) . ts > ci . --- . ti . (ts + ti) -----------------------------.
2751 dGj.dts 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3
2752 i=-k
2753 """
2754
2755 return 0.8 * data.s2f_s2 * params[data.ts_i] * sum(data.ci * data.dti[k] * data.ti * data.ts_ti * (data.ts_ti_sqrd - 3.0 * data.w_ts_ti_sqrd) * data.inv_ts_denom**3, axis=2)
2756
2757
2759 """Spectral density Hessian.
2760
2761 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
2762 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2763 diffusion tensor parameters.
2764
2765 The model-free Hessian is::
2766
2767 _k_
2768 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
2769 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2770 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2771 i=-k
2772 """
2773
2774 return 0.4 * data.s2f_s2 * sum(2.0 * data.ci * data.dti[k] * params[data.ts_i] * data.ti * data.ts_ti * (data.ts_ti_sqrd - 3.0 * data.w_ts_ti_sqrd) * data.inv_ts_denom**3 + data.dci[k] * data.fact_djw_dts, axis=2)
2775
2776
2777
2778
2779
2780
2781
2782
2784 """Spectral density Hessian.
2785
2786 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2787 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2788 parameters.
2789
2790 The model-free Hessian is::
2791
2792 _k_
2793 d2J(w) 2 \ d2ci / S2 (S2f - S2)(ts + ti)ts \
2794 ------- = - > ------- . ti | ------------ + ------------------------- |.
2795 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2796 i=-k
2797 """
2798
2799 return 0.4 * sum(data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
2800
2801
2802
2803
2805 """Spectral density Hessian.
2806
2807 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2808 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2809 parameters.
2810
2811 The model-free Hessian is::
2812
2813 _k_
2814 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
2815 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
2816 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2817 i=-k
2818 """
2819
2820 return 0.4 * sum(data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
2821
2822
2823
2824
2825
2826
2827
2828
2830 """Spectral density Hessian.
2831
2832 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
2833 model-free formula with the parameters {S2f, S2, ts} and {S2f, tf, S2, ts} together with
2834 diffusion tensor parameters.
2835
2836 The model-free Hessian is::
2837
2838 _k_
2839 d2J(w) 2 \ dci / 1 (ts + ti)ts \
2840 ------- = - > --- . ti | ------------ - ------------------------- |.
2841 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2842 i=-k
2843 """
2844
2845 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
2846
2847
2848
2849
2850
2851
2852
2853
2855 """Spectral density Hessian.
2856
2857 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2858 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2859 parameters.
2860
2861 The model-free Hessian is::
2862
2863 _k_
2864 d2J(w) 2 \ dci (ts + ti)ts
2865 -------- = - > --- . ti -------------------------.
2866 dOj.dS2f 5 /__ dOj (ts + ti)^2 + (w.ts.ti)^2
2867 i=-k
2868 """
2869
2870 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ts, axis=2)
2871
2872
2873
2874
2876 """Spectral density Hessian.
2877
2878 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2879 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2880 parameters.
2881
2882 The model-free Hessian is::
2883
2884 _k_
2885 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
2886 -------- = - - > --- . ti | ------------------------- - ------------------------- |.
2887 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2888 i=-k
2889 """
2890
2891 return -0.4 * sum(data.dci[k] * data.ti * (data.fact_tf - data.fact_ts), axis=2)
2892
2893
2894
2895
2896
2897
2898
2899
2901 """Spectral density Hessian.
2902
2903 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
2904 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2905 parameters.
2906
2907 The model-free Hessian is::
2908
2909 _k_
2910 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
2911 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
2912 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
2913 i=-k
2914 """
2915
2916 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
2917
2918
2919
2920
2921
2922
2923
2924
2926 """Spectral density Hessian.
2927
2928 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
2929 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2930 diffusion tensor parameters.
2931
2932 The model-free Hessian is::
2933
2934 _k_
2935 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
2936 ------- = - (S2f - S2) > --- . ti^2 -----------------------------.
2937 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
2938 i=-k
2939 """
2940
2941 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
2942
2943
2944
2945
2946
2947
2948
2949
2951 """Spectral density Hessian.
2952
2953 Calculate the spectral desity values for the S2 - ts double partial derivative of the extended
2954 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
2955 diffusion tensor parameters.
2956
2957 The model-free Hessian is::
2958
2959 _k_
2960 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
2961 ------- = - - > ci . ti^2 -----------------------------.
2962 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
2963 i=-k
2964 """
2965
2966 return -0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
2967
2968
2969
2970
2971
2972
2973
2974
2976 """Spectral density Hessian.
2977
2978 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
2979 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
2980 parameters.
2981
2982 The model-free Hessian is::
2983
2984 _k_
2985 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
2986 -------- = - - > ci . ti^2 -----------------------------.
2987 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
2988 i=-k
2989 """
2990
2991 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
2992
2993
2994
2995
2996
2997
2998
2999
3001 """Spectral density Hessian.
3002
3003 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3004 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3005 diffusion tensor parameters.
3006
3007 The model-free Hessian is::
3008
3009 _k_
3010 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3011 -------- = - > ci . ti^2 -----------------------------.
3012 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3013 i=-k
3014 """
3015
3016 return 0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
3017
3018
3019
3020
3021
3022
3023
3024
3026 """Spectral density Hessian.
3027
3028 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3029 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
3030 parameters.
3031
3032 The model-free Hessian is::
3033
3034 _k_
3035 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3036 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3037 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3038 i=-k
3039 """
3040
3041 return -0.8 * data.one_s2f * sum(data.ci * data.ti**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * data.ti**3 * params[data.tf_i] * data.tf_ti - data.w_ti_sqrd**2 * params[data.tf_i]**3) * data.inv_tf_denom**3, axis=2)
3042
3043
3044
3045
3046
3047
3048
3049
3051 """Spectral density Hessian.
3052
3053 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3054 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3055 diffusion tensor parameters.
3056
3057 The model-free Hessian is::
3058
3059 _k_
3060 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3061 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------.
3062 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3063 i=-k
3064 """
3065
3066 return -0.8 * data.s2f_s2 * sum(data.ci * data.ti**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * data.ti**3 * params[data.ts_i] * data.ts_ti - data.w_ti_sqrd**2 * params[data.ts_i]**3) * data.inv_ts_denom**3, axis=2)
3067
3068
3069
3070
3071
3072
3073
3074
3076 """Spectral density Hessian.
3077
3078 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3079 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3080
3081 The model-free Hessian is::
3082
3083 _k_
3084 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3085 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3086 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3087 i=-k
3088
3089 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
3090 + ------- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- | |.
3091 dGj.dGk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
3092 """
3093
3094
3095 a = -2.0 * data.dti[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
3096
3097
3098 b = data.d2ti[j, k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
3099
3100 return 0.4 * sum(data.ci * (a + b), axis=2)
3101
3102
3104 """Spectral density Hessian.
3105
3106 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3107 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3108
3109 The model-free Hessian is::
3110
3111 _k_
3112 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3113 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3114 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3115 i=-k
3116
3117 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3118 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3119 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3120
3121
3122 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3123 + ------- . ti | ------------ + ------------------------- | |.
3124 dGj.dGk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3125 """
3126
3127
3128 a = -2.0 * data.ci * data.dti[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
3129
3130
3131 b = (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
3132
3133
3134 c = data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
3135
3136 return 0.4 * sum(a + b + c, axis=2)
3137
3138
3139
3140
3142 """Spectral density Hessian.
3143
3144 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3145 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3146 parameters.
3147
3148 The model-free Hessian is::
3149
3150 _k_
3151 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
3152 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3153 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3154 i=-k
3155
3156 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3157 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3158 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3159
3160
3161 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3162 + ------- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3163 dGj.dGk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3164 """
3165
3166
3167 a = -2.0 * data.dti[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2f * params[data.tf_i]**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.tf_i]**3 * data.ti * data.tf_ti - (data.frq_list_ext * params[data.tf_i])**4 * data.ti**3) * data.inv_tf_denom**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
3168
3169
3170 b = data.d2ti[j, k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
3171
3172 return 0.4 * sum(data.ci * (a + b), axis=2)
3173
3174
3176 """Spectral density Hessian.
3177
3178 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3179 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3180 parameters.
3181
3182 The model-free Hessian is::
3183
3184 _k_
3185 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
3186 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3187 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3188 i=-k
3189
3190 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3191 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3192 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3193
3194
3195 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
3196 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
3197 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
3198
3199
3200 (ts + ti)^2 - (w.ts.ti)^2 \
3201 + S2f(1 - S2s)ts^2 ----------------------------- |
3202 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3203
3204
3205 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3206 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3207 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3208 """
3209
3210
3211 a = -2.0 * data.ci * data.dti[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.frq_sqrd_list_ext * data.ti * (3.0 - data.w_ti_sqrd) * data.fact_ti**3 + data.one_s2f * params[data.tf_i]**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.tf_i]**3 * data.ti * data.tf_ti - (data.frq_list_ext * params[data.tf_i])**4 * data.ti**3) * data.inv_tf_denom**3 + data.s2f_s2 * params[data.ts_i]**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * params[data.ts_i]**3 * data.ti * data.ts_ti - (data.frq_list_ext * params[data.ts_i])**4 * data.ti**3) * data.inv_ts_denom**3)
3212
3213
3214 b = (data.dti[j] * data.dci[k] + data.dti[k] * data.dci[j] + data.ci * data.d2ti[j, k]) * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
3215
3216
3217 c = data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts)
3218
3219 return 0.4 * sum(a + b + c, axis=2)
3220
3221
3222
3223
3224
3225
3226
3227
3229 """Spectral density Hessian.
3230
3231 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3232 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3233
3234 The model-free Hessian is::
3235
3236 _k_
3237 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3238 ------- = - > --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3239 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3240 i=-k
3241 """
3242
3243 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
3244
3245
3247 """Spectral density Hessian.
3248
3249 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3250 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3251
3252 The model-free Hessian is::
3253
3254 _k_
3255 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3256 ------- = - > | --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3257 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3258 i=-k
3259
3260 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3261 + ------- . ti | ------------ + ------------------------- | |.
3262 dGj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3263 """
3264
3265 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.d2ci[k, j] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
3266
3267
3268
3269
3271 """Spectral density Hessian.
3272
3273 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3274 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3275 parameters.
3276
3277 The model-free Hessian is::
3278
3279 _k_
3280 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3281 ------- = - > --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3282 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3283 i=-k
3284 """
3285
3286 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti), axis=2)
3287
3288
3290 """Spectral density Hessian.
3291
3292 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3293 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3294 parameters.
3295
3296 The model-free Hessian is::
3297
3298 _k_
3299 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3300 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3301 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3302 i=-k
3303
3304 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3305 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3306 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3307 """
3308
3309 return 0.4 * sum(data.dci[j] * data.dti[k] * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2f * data.fact_tf_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti) + data.d2ci[k, j] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
3310
3311
3312
3313
3314
3315
3316
3317
3319 """Spectral density Hessian.
3320
3321 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3322 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3323
3324 The model-free Hessian is::
3325
3326 _k_
3327 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3328 -------- = - > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
3329 dGj.dS2f 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3330 i=-k
3331 """
3332
3333 return 0.4 * sum(data.ci * data.dti[k] * (params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2s * data.fact_ts_djw_dti), axis=2)
3334
3335
3337 """Spectral density Hessian.
3338
3339 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3340 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3341
3342 The model-free Hessian is::
3343
3344 _k_
3345 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3346 -------- = - > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
3347 dGj.dS2f 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3348 i=-k
3349
3350 dci / S2s (1 - S2s)(ts + ti)ts \ \
3351 + --- . ti | ------------ + ------------------------- | |.
3352 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3353 """
3354
3355 return 0.4 * sum(data.ci * data.dti[k] * (params[data.s2s_i] * data.fact_ti_djw_dti + data.one_s2s * data.fact_ts_djw_dti) + data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
3356
3357
3358
3359
3361 """Spectral density Hessian.
3362
3363 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3364 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3365 parameters.
3366
3367 The model-free Hessian is::
3368
3369 _k_
3370 d2J(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3371 -------- = - > ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |.
3372 dGj.dS2f 5 /__ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3373 i=-k
3374 """
3375
3376 return 0.4 * sum(data.ci * data.dti[k] * (params[data.s2s_i] * data.fact_ti_djw_dti - data.fact_tf_djw_dti + data.one_s2s * data.fact_ts_djw_dti), axis=2)
3377
3378
3380 """Spectral density Hessian.
3381
3382 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3383 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3384 parameters.
3385
3386 The model-free Hessian is::
3387
3388 _k_
3389 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3390 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
3391 dGj.dS2f 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3392 i=-k
3393
3394 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
3395 + --- . ti | ------------ - ------------------------- + ------------------------- | |.
3396 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3397 """
3398
3399 return 0.4 * sum(data.ci * data.dti[k] * (params[data.s2s_i] * data.fact_ti_djw_dti - data.fact_tf_djw_dti + data.one_s2s * data.fact_ts_djw_dti) + data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti - data.tf_ti_tf * data.inv_ts_denom + data.one_s2s * data.fact_ts), axis=2)
3400
3401
3402
3403
3404
3405
3406
3407
3409 """Spectral density Hessian.
3410
3411 Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended
3412 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3413 diffusion tensor parameters.
3414
3415 The model-free Hessian is::
3416
3417 _k_
3418 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3419 -------- = - S2f > ci . --- | ---------------- - ts^2 ----------------------------- |.
3420 dGj.dS2s 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3421 i=-k
3422 """
3423
3424 return 0.4 * params[data.s2f_i] * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti), axis=2)
3425
3426
3428 """Spectral density Hessian.
3429
3430 Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended
3431 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3432 diffusion tensor parameters.
3433
3434 The model-free Hessian is::
3435
3436 _k_
3437 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3438 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
3439 dGj.dS2s 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3440 i=-k
3441
3442 dci / 1 (ts + ti)ts \ \
3443 + --- . ti | ------------ - ------------------------- | |.
3444 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3445 """
3446
3447 return 0.4 * params[data.s2f_i] * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti) + data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3448
3449
3450
3451
3452
3453
3454
3455
3457 """Spectral density Hessian.
3458
3459 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
3460 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3461 parameters.
3462
3463 The model-free Hessian is::
3464
3465 _k_
3466 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
3467 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
3468 dGj.dtf 5 /__ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3
3469 i=-k
3470 """
3471
3472 return 0.8 * data.one_s2f * params[data.tf_i] * sum(data.ci * data.dti[k] * data.ti * data.tf_ti * (data.tf_ti_sqrd - 3.0 * data.w_tf_ti_sqrd) * data.inv_tf_denom**3, axis=2)
3473
3474
3476 """Spectral density Hessian.
3477
3478 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
3479 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3480 parameters.
3481
3482 The model-free Hessian is::
3483
3484 _k_
3485 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
3486 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3487 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
3488 i=-k
3489 """
3490
3491 return 0.4 * data.one_s2f * sum(2.0 * data.ci * data.dti[k] * params[data.tf_i] * data.ti * data.tf_ti * (data.tf_ti_sqrd - 3.0 * data.w_tf_ti_sqrd) * data.inv_tf_denom**3 + data.dci[k] * data.fact_djw_dtf, axis=2)
3492
3493
3494
3495
3496
3497
3498
3499
3501 """Spectral density Hessian.
3502
3503 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
3504 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3505 diffusion tensor parameters.
3506
3507 The model-free Hessian is::
3508
3509 _k_
3510 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
3511 ------- = - S2f(1 - S2s) . ts > ci . --- . ti . (ts + ti) -----------------------------.
3512 dGj.dts 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3
3513 i=-k
3514 """
3515
3516 return 0.8 * data.s2f_s2 * params[data.ts_i] * sum(data.ci * data.dti[k] * data.ti * data.ts_ti * (data.ts_ti_sqrd - 3.0 * data.w_ts_ti_sqrd) * data.inv_ts_denom**3, axis=2)
3517
3518
3520 """Spectral density Hessian.
3521
3522 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
3523 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3524 diffusion tensor parameters.
3525
3526 The model-free Hessian is::
3527
3528 _k_
3529 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
3530 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3531 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3532 i=-k
3533 """
3534
3535 return 0.4 * data.s2f_s2 * sum(2.0 * data.ci * data.dti[k] * params[data.ts_i] * data.ti * data.ts_ti * (data.ts_ti_sqrd - 3.0 * data.w_ts_ti_sqrd) * data.inv_ts_denom**3 + data.dci[k] * data.fact_djw_dts, axis=2)
3536
3537
3538
3539
3540
3541
3542
3543
3545 """Spectral density Hessian.
3546
3547 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3548 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3549 parameters.
3550
3551 The model-free Hessian is::
3552
3553 _k_
3554 d2J(w) 2 \ d2ci / S2f . S2s S2f(1 - S2s)(ts + ti)ts \
3555 ------- = - > ------- . ti | ------------ + ------------------------- |.
3556 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3557 i=-k
3558 """
3559
3560 return 0.4 * sum(data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
3561
3562
3563
3564
3566 """Spectral density Hessian.
3567
3568 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3569 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3570 parameters.
3571
3572 The model-free Hessian is::
3573
3574 _k_
3575 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
3576 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
3577 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3578 i=-k
3579 """
3580
3581 return 0.4 * sum(data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.one_s2f * data.fact_tf + data.s2f_s2 * data.fact_ts), axis=2)
3582
3583
3584
3585
3586
3587
3588
3589
3591 """Spectral density Hessian.
3592
3593 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3594 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3595 parameters.
3596
3597 The model-free Hessian is::
3598
3599 _k_
3600 d2J(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
3601 -------- = - > --- . ti | ------------ + ------------------------- |.
3602 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3603 i=-k
3604 """
3605
3606 return 0.4 * sum(data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
3607
3608
3609
3610
3612 """Spectral density Hessian.
3613
3614 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3615 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3616 parameters.
3617
3618 The model-free Hessian is::
3619
3620 _k_
3621 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
3622 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |.
3623 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3624 i=-k
3625 """
3626
3627 return 0.4 * sum(data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti - data.fact_tf + data.one_s2s * data.fact_ts), axis=2)
3628
3629
3630
3631
3632
3633
3634
3635
3637 """Spectral density Hessian.
3638
3639 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
3640 model-free formula with the parameters {S2f, S2s, ts} and {S2f, tf, S2s, ts} together with
3641 diffusion tensor parameters.
3642
3643 The model-free Hessian is::
3644
3645 _k_
3646 d2J(w) 2 \ dci / 1 (ts + ti)ts \
3647 -------- = - S2f > --- . ti | ------------ - ------------------------- |.
3648 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3649 i=-k
3650 """
3651
3652 return 0.4 * params[data.s2f_i] * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3653
3654
3655
3656
3657
3658
3659
3660
3662 """Spectral density Hessian.
3663
3664 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
3665 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3666 parameters.
3667
3668 The model-free Hessian is::
3669
3670 _k_
3671 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
3672 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
3673 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
3674 i=-k
3675 """
3676
3677 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
3678
3679
3680
3681
3682
3683
3684
3685
3687 """Spectral density Hessian.
3688
3689 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
3690 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3691 diffusion tensor parameters.
3692
3693 The model-free Hessian is::
3694
3695 _k_
3696 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
3697 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------.
3698 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
3699 i=-k
3700 """
3701
3702 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
3703
3704
3705
3706
3707
3708
3709
3710
3712 """Spectral density Hessian.
3713
3714 Calculate the spectral desity values for the S2f - S2s double partial derivative of the extended
3715 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3716 diffusion tensor parameters.
3717
3718 The model-free Hessian is::
3719
3720 _k_
3721 d2J(w) 2 \ / 1 (ts + ti).ts \
3722 --------- = - > ci . ti | ------------ - ------------------------- |.
3723 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3724 i=-k
3725 """
3726
3727 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3728
3729
3730
3731
3732
3733
3734
3735
3737 """Spectral density Hessian.
3738
3739 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
3740 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3741 parameters.
3742
3743 The model-free Hessian is::
3744
3745 _k_
3746 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
3747 -------- = - - > ci . ti^2 -----------------------------.
3748 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
3749 i=-k
3750 """
3751
3752 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
3753
3754
3755
3756
3757
3758
3759
3760
3762 """Spectral density Hessian.
3763
3764 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3765 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3766 diffusion tensor parameters.
3767
3768 The model-free Hessian is::
3769
3770 _k_
3771 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3772 -------- = - (1 - S2s) > ci . ti^2 -----------------------------.
3773 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3774 i=-k
3775 """
3776
3777 return 0.4 * data.one_s2s * sum(data.ci * data.fact_djw_dts, axis=2)
3778
3779
3780
3781
3782
3783
3784
3785
3787 """Spectral density Hessian.
3788
3789 Calculate the spectral desity values for the S2s - ts double partial derivative of the extended
3790 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3791 diffusion tensor parameters.
3792
3793 The model-free Hessian is::
3794
3795 _k_
3796 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3797 -------- = - - S2f > ci . ti^2 -----------------------------.
3798 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3799 i=-k
3800 """
3801
3802 return -0.4 * params[data.s2f_i] * sum(data.ci * data.fact_djw_dts, axis=2)
3803
3804
3805
3806
3807
3808
3809
3810
3812 """Spectral density Hessian.
3813
3814 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3815 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3816 parameters.
3817
3818 The model-free Hessian is::
3819
3820 _k_
3821 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3822 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3823 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3824 i=-k
3825 """
3826
3827 return -0.8 * data.one_s2f * sum(data.ci * data.ti**2 * (data.tf_ti**3 + 3.0 * data.frq_sqrd_list_ext * data.ti**3 * params[data.tf_i] * data.tf_ti - data.w_ti_sqrd**2 * params[data.tf_i]**3) * data.inv_tf_denom**3, axis=2)
3828
3829
3830
3831
3832
3833
3834
3835
3837 """Spectral density Hessian.
3838
3839 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3840 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3841 diffusion tensor parameters.
3842
3843 The model-free Hessian is::
3844
3845 _k_
3846 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3847 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
3848 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3849 i=-k
3850 """
3851
3852 return -0.8 * data.s2f_s2 * sum(data.ci * data.ti**2 * (data.ts_ti**3 + 3.0 * data.frq_sqrd_list_ext * data.ti**3 * params[data.ts_i] * data.ts_ti - data.w_ti_sqrd**2 * params[data.ts_i]**3) * data.inv_ts_denom**3, axis=2)
3853