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24 from numpy import sum
25
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29
30
31 """
32 The spectral density equation
33 =============================
34
35 Data structure: data.jw
36 Dimension: 2D, (number of NMR frequencies, 5 spectral density frequencies)
37 Type: numpy matrix, float64
38 Dependencies: None
39 Required by: data.ri, data.dri, data.d2ri
40
41
42 Formulae
43 ========
44
45 Original::
46
47 _k_
48 2 \ 1
49 J(w) = - S2 > ci . ti ------------,
50 5 /__ 1 + (w.ti)^2
51 i=-k
52
53
54 _k_
55 2 \ / S2 (1 - S2)(te + ti)te \
56 J(w) = - > ci . ti | ------------ + ------------------------- |,
57 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
58 i=-k
59
60
61 Extended::
62
63 _k_
64 2 \ / S2 (S2f - S2)(ts + ti)ts \
65 J(w) = - > ci . ti | ------------ + ------------------------- |,
66 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
67 i=-k
68
69
70 _k_
71 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
72 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |,
73 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
74 i=-k
75
76
77 Extended 2::
78
79 _k_
80 2 \ / S2s (1 - S2s)(ts + ti)ts \
81 J(w) = - S2f > ci . ti | ------------ + ------------------------- |,
82 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
83 i=-k
84
85
86 _k_
87 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
88 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
89 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
90 i=-k
91 """
92
93
94
95
96
97
99 """Spectral density function.
100
101 Calculate the spectral density values for the original model-free formula with no parameters {}
102 with or without diffusion tensor parameters.
103
104 The formula is::
105
106 _k_
107 2 \ 1
108 J(w) = - > ci . ti ------------.
109 5 /__ 1 + (w.ti)^2
110 i=-k
111 """
112
113 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
114
115
116
117
118
119
121 """Spectral density function.
122
123 Calculate the spectral density values for the original model-free formula with the single
124 parameter {S2} with or without diffusion tensor parameters.
125
126 The formula is::
127
128 _k_
129 2 \ 1
130 J(w) = - S2 > ci . ti ------------.
131 5 /__ 1 + (w.ti)^2
132 i=-k
133 """
134
135 return 0.4 * params[data.s2_i] * sum(data.ci * data.ti * data.fact_ti, axis=2)
136
137
138
139
140
141
143 """Spectral density function.
144
145 Calculate the spectral density values for the original model-free formula with the parameters
146 {S2, te} with or without diffusion tensor parameters.
147
148 The model-free formula is::
149
150 _k_
151 2 \ / S2 (1 - S2)(te + ti)te \
152 J(w) = - > ci . ti | ------------ + ------------------------- |.
153 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
154 i=-k
155 """
156
157 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
158
159
160
161
162
163
165 """Spectral density function.
166
167 Calculate the spectral density values for the extended model-free formula with the parameters
168 {S2f, S2, ts} with or without diffusion tensor parameters.
169
170 The model-free formula is::
171
172 _k_
173 2 \ / S2 (S2f - S2)(ts + ti)ts \
174 J(w) = - > ci . ti | ------------ + ------------------------- |.
175 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
176 i=-k
177 """
178
179 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
180
181
182
183
184
185
187 """Spectral density function.
188
189 Calculate the spectral density values for the extended model-free formula with the parameters
190 {S2f, tf, S2, ts} with or without diffusion tensor parameters.
191
192 The model-free formula is::
193
194 _k_
195 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
196 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
197 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
198 i=-k
199 """
200
201 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)
202
203
204
205
206
207
209 """Spectral density function.
210
211 Calculate the spectral density values for the extended model-free formula with the parameters
212 {S2f, S2s, ts} with or without diffusion tensor parameters.
213
214 The model-free formula is::
215
216 _k_
217 2 \ / S2s (1 - S2s)(ts + ti)ts \
218 J(w) = - S2f > ci . ti | ------------ + ------------------------- |.
219 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
220 i=-k
221 """
222
223 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)
224
225
226
227
228
229
231 """Spectral density function.
232
233 Calculate the spectral density values for the extended model-free formula with the parameters
234 {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
235
236 The model-free formula is::
237
238 _k_
239 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
240 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
241 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
242 i=-k
243 """
244
245 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)
246
247
248
249
250
251
252
253
254 """
255 The spectral density gradients
256 ==============================
257
258 Data structure: data.djw
259 Dimension: 3D, (number of NMR frequencies, 5 spectral density frequencies,
260 model-free parameters)
261 Type: numpy 3D matrix, float64
262 Dependencies: None
263 Required by: data.dri, data.d2ri
264
265
266 Formulae
267 ========
268
269 Original::
270
271 _k_
272 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
273 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
274 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
275 i=-k
276
277 dci / S2 (1 - S2)(te + ti)te \ \
278 + --- . ti | ------------ + ------------------------- | |,
279 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
280
281
282 _k_
283 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
284 ----- = - > --- . ti | ------------ + ------------------------- |,
285 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
286 i=-k
287
288
289 _k_
290 dJ(w) 2 \ / 1 (te + ti)te \
291 ----- = - > ci . ti | ------------ - ------------------------- |,
292 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
293 i=-k
294
295
296 _k_
297 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
298 ----- = - (1 - S2) > ci . ti^2 -----------------------------,
299 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
300 i=-k
301
302
303 dJ(w)
304 ----- = 0,
305 dRex
306
307
308 dJ(w)
309 ----- = 0,
310 dcsa
311
312
313 dJ(w)
314 ----- = 0.
315 dr
316
317
318 Extended::
319
320 _k_
321 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
322 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
323 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
324 i=-k
325
326 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
327 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
328 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
329
330
331 _k_
332 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
333 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
334 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
335 i=-k
336
337
338 _k_
339 dJ(w) 2 \ / 1 (ts + ti).ts \
340 ----- = - > ci . ti | ------------ - ------------------------- |,
341 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
342 i=-k
343
344
345 _k_
346 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
347 ----- = - - > ci . ti | ------------------------- - ------------------------- |,
348 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
349 i=-k
350
351
352 _k_
353 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
354 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
355 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
356 i=-k
357
358
359 _k_
360 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
361 ----- = - (S2f - S2) > ci . ti^2 -----------------------------,
362 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
363 i=-k
364
365
366 dJ(w)
367 ----- = 0,
368 dRex
369
370
371 dJ(w)
372 ----- = 0,
373 dcsa
374
375
376 dJ(w)
377 ----- = 0.
378 dr
379
380
381 Extended 2::
382
383 _k_
384 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
385 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
386 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
387 i=-k
388
389 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
390 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
391 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
392
393
394 _k_
395 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
396 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
397 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
398 i=-k
399
400
401 _k_
402 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
403 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |,
404 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
405 i=-k
406
407
408 _k_
409 dJ(w) 2 \ / 1 (ts + ti).ts \
410 ----- = - S2f > ci . ti | ------------ - ------------------------- |,
411 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
412 i=-k
413
414
415 _k_
416 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
417 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
418 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
419 i=-k
420
421
422 _k_
423 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
424 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------,
425 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
426 i=-k
427
428
429 dJ(w)
430 ----- = 0,
431 dRex
432
433
434 dJ(w)
435 ----- = 0,
436 dcsa
437
438
439 dJ(w)
440 ----- = 0.
441 dr
442 """
443
444
445
446
447
448
449
450
452 """Spectral density gradient.
453
454 Calculate the spectral desity values for the Gj partial derivative of the original model-free
455 formula with no parameters {} together with diffusion tensor parameters.
456
457 The model-free gradient is::
458
459 _k_
460 dJ(w) 2 \ dti 1 - (w.ti)^2
461 ----- = - > ci . --- ----------------.
462 dGj 5 /__ dGj (1 + (w.ti)^2)^2
463 i=-k
464 """
465
466 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
467
468
470 """Spectral density gradient.
471
472 Calculate the spectral desity values for the Gj partial derivative of the original model-free
473 formula with no parameters {} together with diffusion tensor parameters.
474
475 The model-free gradient is::
476
477 _k_
478 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
479 ----- = - > | ci . --- ---------------- + --- . ti ------------ |.
480 dGj 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
481 i=-k
482 """
483
484 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti + data.dci[j] * data.ti * data.fact_ti, axis=2)
485
486
487
488
490 """Spectral density gradient.
491
492 Calculate the spectral desity values for the Gj partial derivative of the original model-free
493 formula with the parameter {S2} together with diffusion tensor parameters.
494
495 The model-free gradient is::
496
497 _k_
498 dJ(w) 2 \ dti 1 - (w.ti)^2
499 ----- = - S2 > ci . --- ----------------.
500 dGj 5 /__ dGj (1 + (w.ti)^2)^2
501 i=-k
502 """
503
504 return 0.4 * params[data.s2_i] * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
505
506
508 """Spectral density gradient.
509
510 Calculate the spectral desity values for the Gj partial derivative of the original model-free
511 formula with the parameter {S2} together with diffusion tensor parameters.
512
513 The model-free gradient is::
514
515 _k_
516 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
517 ----- = - S2 > | ci . --- ---------------- + --- . ti ------------ |.
518 dGj 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
519 i=-k
520 """
521
522 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)
523
524
525
526
528 """Spectral density gradient.
529
530 Calculate the spectral desity values for the Gj partial derivative of the original model-free
531 formula with the parameters {S2, te} together with diffusion tensor parameters.
532
533 The model-free gradient is::
534
535 _k_
536 dJ(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
537 ----- = - > ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
538 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
539 i=-k
540 """
541
542 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)
543
544
546 """Spectral density gradient.
547
548 Calculate the spectral desity values for the Gj partial derivative of the original model-free
549 formula with the parameters {S2, te} together with diffusion tensor parameters.
550
551 The model-free gradient is::
552
553 _k_
554 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
555 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
556 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
557 i=-k
558
559 dci / S2 (1 - S2)(te + ti)te \ \
560 + --- . ti | ------------ + ------------------------- | |.
561 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
562 """
563
564 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)
565
566
567
568
569
570
571
572
574 """Spectral density gradient.
575
576 Calculate the spectral desity values for the O partial derivative of the original model-free
577 formula with no parameters {} together with diffusion tensor parameters.
578
579 The model-free gradient is::
580
581 _k_
582 dJ(w) 2 \ dci 1
583 ----- = - > --- . ti ------------.
584 dOj 5 /__ dOj 1 + (w.ti)^2
585 i=-k
586 """
587
588 return 0.4 * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
589
590
591
592
594 """Spectral density gradient.
595
596 Calculate the spectral desity values for the O partial derivative of the original model-free
597 formula with the parameter {S2} together with diffusion tensor parameters.
598
599 The model-free gradient is::
600
601 _k_
602 dJ(w) 2 \ dci 1
603 ----- = - S2 > --- . ti ------------.
604 dOj 5 /__ dOj 1 + (w.ti)^2
605 i=-k
606 """
607
608 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
609
610
611
612
614 """Spectral density gradient.
615
616 Calculate the spectral desity values for the O partial derivative of the original model-free
617 formula with the parameters {S2, te} together with diffusion tensor parameters.
618
619 The model-free gradient is::
620
621 _k_
622 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
623 ----- = - > --- . ti | ------------ + ------------------------- |.
624 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
625 i=-k
626 """
627
628 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
629
630
631
632
633
634
635
636
638 """Spectral density gradient.
639
640 Calculate the spectral desity values for the S2 partial derivative of the original model-free
641 formula with the single parameter {S2} with or without diffusion tensor parameters.
642
643 The model-free gradient is::
644
645 _k_
646 dJ(w) 2 \ 1
647 ----- = - > ci . ti ------------.
648 dS2 5 /__ 1 + (w.ti)^2
649 i=-k
650 """
651
652 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
653
654
655
656
658 """Spectral density gradient.
659
660 Calculate the spectral desity values for the S2 partial derivative of the original model-free
661 formula with the parameters {S2, te} with or without diffusion tensor parameters.
662
663 The model-free gradient is::
664
665 _k_
666 dJ(w) 2 \ / 1 (te + ti)te \
667 ----- = - > ci . ti | ------------ - ------------------------- |.
668 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
669 i=-k
670 """
671
672 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_te), axis=2)
673
674
675
676
677
678
679
680
682 """Spectral density gradient.
683
684 Calculate the spectral desity values for the te partial derivative of the original model-free
685 formula with the parameters {S2, te} with or without diffusion tensor parameters.
686
687 The model-free gradient is::
688
689 _k_
690 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
691 ----- = - (1 - S2) > ci . ti^2 -----------------------------.
692 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
693 i=-k
694 """
695
696 return 0.4 * data.one_s2 * sum(data.ci * data.fact_djw_dte, axis=2)
697
698
699
700
701
702
703
704
706 """Spectral density gradient.
707
708 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
709 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
710
711 The formula is::
712
713 _k_
714 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
715 ----- = - > ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
716 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
717 i=-k
718 """
719
720 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)
721
722
724 """Spectral density gradient.
725
726 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
727 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
728
729 The formula is::
730
731 _k_
732 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
733 ----- = - > | ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
734 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
735 i=-k
736
737 dci / S2 (S2f - S2)(ts + ti)ts \ \
738 + --- . ti | ------------ + ------------------------- | |.
739 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
740 """
741
742 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)
743
744
745
746
748 """Spectral density gradient.
749
750 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
751 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
752
753 The formula is::
754
755 _k_
756 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
757 ----- = - > ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
758 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
759 i=-k
760 """
761
762 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)
763
764
766 """Spectral density gradient.
767
768 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
769 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
770
771 The formula is::
772
773 _k_
774 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
775 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
776 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
777 i=-k
778
779 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
780 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
781 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
782 """
783
784 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)
785
786
787
788
789
790
791
792
794 """Spectral density gradient.
795
796 Calculate the spectral desity values for the O partial derivative of the extended model-free
797 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
798
799 The formula is::
800
801 _k_
802 dJ(w) 2 \ dci / S2 (S2f - S2)(ts + ti)ts \
803 ----- = - > --- . ti | ------------ + ------------------------- |.
804 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
805 i=-k
806 """
807
808 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
809
810
811
812
814 """Spectral density gradient.
815
816 Calculate the spectral desity values for the O partial derivative of the extended model-free
817 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
818
819 The formula is::
820
821 _k_
822 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
823 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
824 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
825 i=-k
826 """
827
828 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)
829
830
831
832
833
834
835
836
838 """Spectral density gradient.
839
840 Calculate the spectral desity values for the S2 partial derivative of the extended model-free
841 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
842 parameters.
843
844 The formula is::
845
846 _k_
847 dJ(w) 2 \ / 1 (ts + ti).ts \
848 ----- = - > ci . ti | ------------ - ------------------------- |.
849 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
850 i=-k
851 """
852
853 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
854
855
856
857
858
859
860
861
863 """Spectral density gradient.
864
865 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
866 formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.
867
868 The formula is::
869
870 _k_
871 dJ(w) 2 \ (ts + ti).ts
872 ----- = - > ci . ti -------------------------.
873 dS2f 5 /__ (ts + ti)^2 + (w.ts.ti)^2
874 i=-k
875 """
876
877 return 0.4 * sum(data.ci * data.ti * data.fact_ts, axis=2)
878
879
880
881
883 """Spectral density gradient.
884
885 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
886 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
887
888 The formula is::
889
890 _k_
891 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
892 ----- = - - > ci . ti | ------------------------- - ------------------------- |.
893 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
894 i=-k
895 """
896
897 return -0.4 * sum(data.ci * data.ti * (data.fact_tf - data.fact_ts), axis=2)
898
899
900
901
902
903
904
905
907 """Spectral density gradient.
908
909 Calculate the spectral desity values for the tf partial derivative of the extended model-free
910 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
911
912 The formula is::
913
914 _k_
915 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
916 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
917 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
918 i=-k
919 """
920
921 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
922
923
924
925
926
927
928
929
931 """Spectral density gradient.
932
933 Calculate the spectral desity values for the ts partial derivative of the extended model-free
934 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
935 parameters.
936
937 The formula is::
938
939 _k_
940 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
941 ----- = - (S2f - S2) > ci . ti^2 -----------------------------.
942 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
943 i=-k
944 """
945
946 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
947
948
949
950
951
952
953
954
956 """Spectral density gradient.
957
958 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
959 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
960
961 The formula is::
962
963 _k_
964 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
965 ----- = - S2f > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
966 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
967 i=-k
968 """
969
970 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)
971
972
974 """Spectral density gradient.
975
976 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
977 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
978
979 The formula is::
980
981 _k_
982 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
983 ----- = - S2f > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
984 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
985 i=-k
986
987 dci / S2s (1 - S2s)(ts + ti)ts \ \
988 + --- . ti | ------------ + ------------------------- | |.
989 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
990 """
991
992 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)
993
994
995
996
998 """Spectral density gradient.
999
1000 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
1001 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1002
1003 The formula is::
1004
1005 _k_
1006 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1007 ----- = - > ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1008 dGj 5 /__ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1009 i=-k
1010 """
1011
1012 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)
1013
1014
1016 """Spectral density gradient.
1017
1018 Calculate the spectral desity values for the Gj partial derivative of the extended model-free
1019 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1020
1021 The formula is::
1022
1023 _k_
1024 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1025 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f).tf^2 ----------------------------- + S2f(1 - S2s).ts^2 ----------------------------- |
1026 dGj 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1027 i=-k
1028
1029 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1030 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
1031 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1032 """
1033
1034 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)
1035
1036
1037
1038
1039
1040
1041
1042
1044 """Spectral density gradient.
1045
1046 Calculate the spectral desity values for the O partial derivative of the extended model-free
1047 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
1048
1049 The formula is::
1050
1051 _k_
1052 dJ(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
1053 ----- = - S2f > --- . ti | ------------ + ------------------------- |.
1054 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1055 i=-k
1056 """
1057
1058 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)
1059
1060
1061
1062
1064 """Spectral density gradient.
1065
1066 Calculate the spectral desity values for the O partial derivative of the extended model-free
1067 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1068
1069 The formula is::
1070
1071 _k_
1072 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1073 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
1074 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1075 i=-k
1076 """
1077
1078 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)
1079
1080
1081
1082
1083
1084
1085
1086
1088 """Spectral density gradient.
1089
1090 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1091 formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.
1092
1093 The formula is::
1094
1095 _k_
1096 dJ(w) 2 \ / S2s (1 - S2s)(ts + ti).ts \
1097 ----- = - > ci . ti | ------------ + ------------------------- |.
1098 dS2f 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1099 i=-k
1100 """
1101
1102 return 0.4 * sum(data.ci * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
1103
1104
1105
1106
1108 """Spectral density gradient.
1109
1110 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1111 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1112
1113 The formula is::
1114
1115 _k_
1116 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
1117 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |.
1118 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1119 i=-k
1120 """
1121
1122 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)
1123
1124
1125
1126
1127
1128
1129
1130
1132 """Spectral density gradient.
1133
1134 Calculate the spectral desity values for the S2s partial derivative of the extended model-free
1135 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1136 tensor parameters.
1137
1138 The formula is::
1139
1140 _k_
1141 dJ(w) 2 \ / 1 (ts + ti).ts \
1142 ----- = - S2f > ci . ti | ------------ - ------------------------- |.
1143 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1144 i=-k
1145 """
1146
1147 return 0.4 * params[data.s2f_i] * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
1148
1149
1150
1151
1152
1153
1154
1155
1157 """Spectral density gradient.
1158
1159 Calculate the spectral desity values for the tf partial derivative of the extended model-free
1160 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1161
1162 The formula is::
1163
1164 _k_
1165 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1166 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
1167 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1168 i=-k
1169 """
1170
1171 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
1172
1173
1174
1175
1176
1177
1178
1179
1181 """Spectral density gradient.
1182
1183 Calculate the spectral desity values for the ts partial derivative of the extended model-free
1184 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1185 tensor parameters.
1186
1187 The formula is::
1188
1189 _k_
1190 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1191 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------.
1192 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1193 i=-k
1194 """
1195
1196 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
1197
1198
1199
1200
1201
1202
1203
1204
1205 """
1206 The spectral density Hessians
1207 =============================
1208
1209 Data structure: data.d2jw
1210 Dimension: 4D, (number of NMR frequencies, 5 spectral density frequencies, model-free
1211 parameters, model-free parameters)
1212 Type: numpy 4D matrix, float64
1213 Dependencies: None
1214 Required by: data.d2ri
1215
1216
1217 Formulae
1218 ========
1219
1220 Original: Model-free parameter - Model-free parameter::
1221
1222 _k_
1223 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 \
1224 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1225 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1226 i=-k
1227
1228 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1229 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1230 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1231
1232
1233 d2ci / S2 (1 - S2)(te + ti)te \ \
1234 + ------- ti | ------------ + ------------------------- | |,
1235 dGj.dGk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1236
1237
1238 _k_
1239 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1240 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1241 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1242 i=-k
1243
1244 d2ci / S2 (1 - S2)(te + ti)te \ \
1245 + ------- ti | ------------ + ------------------------- | |,
1246 dGj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1247
1248
1249 _k_
1250 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1251 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
1252 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1253 i=-k
1254
1255 dci / 1 (te + ti)te \ \
1256 + --- . ti | ------------ - ------------------------- | |,
1257 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1258
1259
1260 _k_
1261 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
1262 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1263 dGj.dte 5 /__ \ dGj ((te + ti)^2 + (w.te.ti)^2)^3 dGj ((te + ti)^2 + (w.te.ti)^2)^2 /
1264 i=-k
1265
1266
1267 _k_
1268 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
1269 ------- = - > ------- . ti | ------------ + ------------------------- |,
1270 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1271 i=-k
1272
1273
1274 _k_
1275 d2J(w) 2 \ dci / 1 (te + ti)te \
1276 ------- = - > --- . ti | ------------ - ------------------------- |,
1277 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1278 i=-k
1279
1280
1281 _k_
1282 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
1283 ------- = - (1 - S2) > --- . ti^2 -----------------------------,
1284 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
1285 i=-k
1286
1287
1288 d2J(w)
1289 ------ = 0,
1290 dS2**2
1291
1292
1293 _k_
1294 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
1295 ------- = - - > ci . ti^2 -----------------------------,
1296 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
1297 i=-k
1298
1299
1300 _k_
1301 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
1302 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
1303 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
1304 i=-k
1305
1306
1307 Original: Other parameters::
1308
1309 d2J(w) d2J(w) d2J(w)
1310 -------- = 0, -------- = 0, ------ = 0,
1311 dS2.dRex dS2.dcsa dS2.dr
1312
1313
1314 d2J(w) d2J(w) d2J(w)
1315 -------- = 0, -------- = 0, ------ = 0,
1316 dte.dRex dte.dcsa dte.dr
1317
1318
1319 d2J(w) d2J(w) d2J(w)
1320 ------- = 0, --------- = 0, ------- = 0,
1321 dRex**2 dRex.dcsa dRex.dr
1322
1323
1324 d2J(w) d2J(w)
1325 ------- = 0, ------- = 0,
1326 dcsa**2 dcsa.dr
1327
1328
1329 d2J(w)
1330 ------ = 0.
1331 dr**2
1332
1333
1334 Extended: Model-free parameter - Model-free parameter::
1335
1336 _k_
1337 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
1338 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1339 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1340 i=-k
1341
1342 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1343 + (S2f - S2)ts^2 ---------------------------------------------------- |
1344 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1345
1346
1347 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1348 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
1349 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1350
1351
1352 (ts + ti)^2 - (w.ts.ti)^2 \
1353 + (S2f - S2)ts^2 ----------------------------- |
1354 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1355
1356
1357 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1358 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1359 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1360
1361
1362 _k_
1363 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1364 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
1365 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 /
1366 i=-k
1367
1368 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1369 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1370 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1371
1372
1373 _k_
1374 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1375 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
1376 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1377 i=-k
1378
1379 dci / 1 (ts + ti)ts \ \
1380 + --- . ti | ------------ - ------------------------- | |,
1381 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1382
1383
1384 _k_
1385 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1386 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
1387 dGj.dS2f 5 /__ \ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1388 i=-k
1389
1390 dci / (tf + ti)tf (ts + ti)ts \ \
1391 + --- . ti | ------------------------- - ------------------------- | |,
1392 dGj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1393
1394
1395 _k_
1396 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1397 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1398 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1399 i=-k
1400
1401
1402 _k_
1403 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1404 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1405 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1406 i=-k
1407
1408
1409 _k_
1410 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
1411 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1412 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1413 i=-k
1414
1415
1416 _k_
1417 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1418 ------- = - > --- . ti | ------------ - ------------------------- |,
1419 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1420 i=-k
1421
1422
1423 _k_
1424 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
1425 -------- = - - > --- . ti | ------------------------- - ------------------------- |,
1426 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1427 i=-k
1428
1429
1430 _k_
1431 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1432 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1433 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1434 i=-k
1435
1436
1437 _k_
1438 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1439 ------- = - (S2f - S2) > --- . ti^2 -----------------------------,
1440 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1441 i=-k
1442
1443
1444 d2J(w) d2J(w) d2J(w)
1445 ------ = 0, -------- = 0, ------- = 0,
1446 dS2**2 dS2.dS2f dS2.dtf
1447
1448
1449 _k_
1450 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1451 ------- = - - > ci . ti^2 -----------------------------,
1452 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1453 i=-k
1454
1455
1456 d2J(w)
1457 ------- = 0,
1458 dS2f**2
1459
1460
1461 _k_
1462 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1463 -------- = - - > ci . ti^2 -----------------------------,
1464 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1465 i=-k
1466
1467
1468 _k_
1469 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1470 -------- = - > ci . ti^2 -----------------------------,
1471 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1472 i=-k
1473
1474
1475 _k_
1476 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1477 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1478 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1479 i=-k
1480
1481
1482 d2J(w)
1483 ------- = 0,
1484 dtf.dts
1485
1486
1487 _k_
1488 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1489 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------,
1490 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1491 i=-k
1492
1493
1494 Extended: Other parameters::
1495
1496 d2J(w) d2J(w) d2J(w)
1497 --------- = 0, --------- = 0, ------- = 0,
1498 dS2f.dRex dS2f.dcsa dS2f.dr
1499
1500
1501 d2J(w) d2J(w) d2J(w)
1502 -------- = 0, -------- = 0, ------ = 0,
1503 dS2.dRex dS2.dcsa dS2.dr
1504
1505
1506 d2J(w) d2J(w) d2J(w)
1507 -------- = 0, -------- = 0, ------ = 0,
1508 dtf.dRex dtf.dcsa dtf.dr
1509
1510
1511 d2J(w) d2J(w) d2J(w)
1512 -------- = 0, -------- = 0, ------ = 0,
1513 dts.dRex dts.dcsa dts.dr
1514
1515
1516 d2J(w) d2J(w) d2J(w)
1517 ------- = 0, --------- = 0, ------- = 0,
1518 dRex**2 dRex.dcsa dRex.dr
1519
1520
1521 d2J(w) d2J(w)
1522 ------- = 0, ------- = 0,
1523 dcsa**2 dcsa.dr
1524
1525
1526 d2J(w)
1527 ------ = 0.
1528 dr**2
1529
1530
1531 Extended 2: Model-free parameter - Model-free parameter::
1532
1533 _k_
1534 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
1535 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1536 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1537 i=-k
1538
1539 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1540 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
1541 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1542
1543
1544 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1545 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
1546 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1547
1548
1549 (ts + ti)^2 - (w.ts.ti)^2 \
1550 + S2f(1 - S2s)ts^2 ----------------------------- |
1551 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1552
1553
1554 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1555 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1556 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1557
1558
1559 _k_
1560 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1561 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1562 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 /
1563 i=-k
1564
1565 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1566 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1567 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1568
1569
1570 _k_
1571 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1572 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
1573 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 /
1574 i=-k
1575
1576 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
1577 + --- . ti | ------------ - ------------------------- + ------------------------- | |,
1578 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1579
1580
1581 _k_
1582 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1583 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
1584 dGj.dS2s 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1585 i=-k
1586
1587 dci / 1 (ts + ti)ts \ \
1588 + --- . ti | ------------ - ------------------------- | |,
1589 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1590
1591
1592 _k_
1593 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1594 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1595 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1596 i=-k
1597
1598
1599 _k_
1600 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1601 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1602 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1603 i=-k
1604
1605
1606 _k_
1607 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1608 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1609 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1610 i=-k
1611
1612
1613 _k_
1614 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
1615 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |,
1616 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1617 i=-k
1618
1619
1620 _k_
1621 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1622 -------- = - S2f > --- . ti | ------------ - ------------------------- |,
1623 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1624 i=-k
1625
1626
1627 _k_
1628 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1629 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1630 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1631 i=-k
1632
1633
1634 _k_
1635 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1636 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------,
1637 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1638 i=-k
1639
1640
1641 d2J(w)
1642 ------- = 0,
1643 dS2f**2
1644
1645
1646 _k_
1647 d2J(w) 2 \ / 1 (ts + ti).ts \
1648 --------- = - > ci . ti | ------------ - ------------------------- |,
1649 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1650 i=-k
1651
1652
1653 _k_
1654 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1655 -------- = - - > ci . ti^2 -----------------------------,
1656 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1657 i=-k
1658
1659
1660 _k_
1661 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1662 -------- = - (1 - S2s) > ci . ti^2 -----------------------------,
1663 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1664 i=-k
1665
1666
1667 d2J(w) d2J(w)
1668 ------- = 0, -------- = 0,
1669 dS2s**2 dS2s.dtf
1670
1671
1672 _k_
1673 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1674 -------- = - - S2f > ci . ti^2 -----------------------------,
1675 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1676 i=-k
1677
1678
1679 _k_
1680 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1681 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1682 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1683 i=-k
1684
1685
1686 d2J(w)
1687 ------- = 0,
1688 dtf.dts
1689
1690
1691 _k_
1692 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1693 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
1694 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1695 i=-k
1696
1697
1698
1699 Extended 2: Other parameters::
1700
1701 d2J(w) d2J(w) d2J(w)
1702 --------- = 0, --------- = 0, ------- = 0,
1703 dS2f.dRex dS2f.dcsa dS2f.dr
1704
1705
1706 d2J(w) d2J(w) d2J(w)
1707 --------- = 0, --------- = 0, ------- = 0,
1708 dS2s.dRex dS2s.dcsa dS2s.dr
1709
1710
1711 d2J(w) d2J(w) d2J(w)
1712 -------- = 0, -------- = 0, ------ = 0,
1713 dtf.dRex dtf.dcsa dtf.dr
1714
1715
1716 d2J(w) d2J(w) d2J(w)
1717 -------- = 0, -------- = 0, ------ = 0,
1718 dts.dRex dts.dcsa dts.dr
1719
1720
1721 d2J(w) d2J(w) d2J(w)
1722 ------- = 0, --------- = 0, ------- = 0,
1723 dRex**2 dRex.dcsa dRex.dr
1724
1725
1726 d2J(w) d2J(w)
1727 ------- = 0, ------- = 0,
1728 dcsa**2 dcsa.dr
1729
1730
1731 d2J(w)
1732 ------ = 0.
1733 dr**2
1734 """
1735
1736
1737
1738
1739
1740
1741
1743 """Spectral density Hessian.
1744
1745 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1746 model-free formula with no parameters {} together with diffusion tensor parameters.
1747
1748 The model-free Hessian is::
1749
1750 _k_
1751 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1752 ------- = - > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1753 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 dGj.dGk (1 + (w.ti)^2)^2 /
1754 i=-k
1755 """
1756
1757 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)
1758
1759
1761 """Spectral density Hessian.
1762
1763 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1764 model-free formula with no parameters {} together with diffusion tensor parameters.
1765
1766 The model-free Hessian is::
1767
1768 _k_
1769 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1770 ------- = - > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1771 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 /
1772 i=-k
1773 """
1774
1775 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)
1776
1777
1778
1779
1781 """Spectral density Hessian.
1782
1783 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1784 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1785
1786 The model-free Hessian is::
1787
1788 _k_
1789 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1790 ------- = - S2 > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1791 dGj.dGk 5 /__ \ dGj dGk (1 + (w.ti)^2)^3 dGj.dGk (1 + (w.ti)^2)^2 /
1792 i=-k
1793 """
1794
1795 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)
1796
1797
1799 """Spectral density Hessian.
1800
1801 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1802 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1803
1804 The model-free Hessian is::
1805
1806 _k_
1807 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1808 ------- = - S2 > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1809 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 /
1810 i=-k
1811 """
1812
1813 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)
1814
1815
1816
1817
1819 """Spectral density Hessian.
1820
1821 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1822 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1823
1824 The model-free Hessian is::
1825
1826 _k_
1827 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 \
1828 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1829 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1830 i=-k
1831
1832 d2ti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \ \
1833 + ------- | S2 ---------------- + (1 - S2)te^2 ----------------------------- | |.
1834 dGj.dGk \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 / /
1835 """
1836
1837
1838 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)
1839
1840
1841 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti)
1842
1843 return 0.4 * sum(data.ci * (a + b), axis=2)
1844
1845
1847 """Spectral density Hessian.
1848
1849 Calculate the spectral desity values for the Gj - Gk double partial derivative of the original
1850 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1851
1852 The model-free Hessian is::
1853
1854 _k_
1855 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 \
1856 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1857 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1858 i=-k
1859
1860 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1861 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1862 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1863
1864
1865 d2ci / S2 (1 - S2)(te + ti)te \ \
1866 + ------- ti | ------------ + ------------------------- | |.
1867 dGj.dGk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1868 """
1869
1870
1871 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)
1872
1873
1874 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)
1875
1876
1877 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te)
1878
1879 return 0.4 * sum(a + b + c, axis=2)
1880
1881
1882
1883
1884
1885
1886
1887
1888
1890 """Spectral density Hessian.
1891
1892 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1893 model-free formula with no parameters {} together with diffusion tensor parameters.
1894
1895 The model-free Hessian is::
1896
1897 _k_
1898 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1899 ------- = - > --- . --- . ----------------.
1900 dGj.dOj 5 /__ dOj dGj (1 + (w.ti)^2)^2
1901 i=-k
1902 """
1903
1904 return 0.4 * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1905
1906
1908 """Spectral density Hessian.
1909
1910 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1911 model-free formula with no parameters {} together with diffusion tensor parameters.
1912
1913 The model-free Hessian is::
1914
1915 _k_
1916 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1917 ------- = - > | --- . --- . ---------------- + ------- ti ------------ |.
1918 dGj.dOj 5 /__ \ dOj dGj (1 + (w.ti)^2)^2 dGj.dOj 1 + (w.ti)^2 /
1919 i=-k
1920 """
1921
1922 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)
1923
1924
1925
1926
1928 """Spectral density Hessian.
1929
1930 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1931 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1932
1933 The model-free Hessian is::
1934
1935 _k_
1936 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1937 ------- = - S2 > --- . --- . ----------------.
1938 dGj.dOj 5 /__ dOj dGj (1 + (w.ti)^2)^2
1939 i=-k
1940 """
1941
1942 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1943
1944
1946 """Spectral density Hessian.
1947
1948 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1949 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1950
1951 The model-free Hessian is::
1952
1953 _k_
1954 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1955 ------- = - S2 > | --- . --- . ---------------- + ------- ti ------------ |.
1956 dGj.dOj 5 /__ \ dOj dGj (1 + (w.ti)^2)^2 dGj.dOj 1 + (w.ti)^2 /
1957 i=-k
1958 """
1959
1960 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)
1961
1962
1963
1964
1966 """Spectral density Hessian.
1967
1968 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1969 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1970
1971 The model-free Hessian is::
1972
1973 _k_
1974 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1975 ------- = - > --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
1976 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1977 i=-k
1978 """
1979
1980 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)
1981
1982
1984 """Spectral density Hessian.
1985
1986 Calculate the spectral desity values for the Gj - Oj double partial derivative of the original
1987 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1988
1989 The model-free Hessian is::
1990
1991 _k_
1992 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1993 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1994 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1995 i=-k
1996
1997 d2ci / S2 (1 - S2)(te + ti)te \ \
1998 + ------- ti | ------------ + ------------------------- | |.
1999 dGj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
2000 """
2001
2002 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)
2003
2004
2005
2006
2007
2008
2009
2010
2012 """Spectral density Hessian.
2013
2014 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2015 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2016
2017 The model-free Hessian is::
2018
2019 _k_
2020 d2J(w) 2 \ dti 1 - (w.ti)^2
2021 ------- = - > ci . --- ----------------.
2022 dGj.dS2 5 /__ dGj (1 + (w.ti)^2)^2
2023 i=-k
2024 """
2025
2026 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti, axis=2)
2027
2028
2030 """Spectral density Hessian.
2031
2032 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2033 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2034
2035 The model-free Hessian is::
2036
2037 _k_
2038 d2J(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
2039 ------- = - > | ci . --- ---------------- + --- . ti ------------ |.
2040 dGj.dS2 5 /__ \ dGj (1 + (w.ti)^2)^2 dGj 1 + (w.ti)^2 /
2041 i=-k
2042 """
2043
2044 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti + data.dci[k] * data.ti * data.fact_ti, axis=2)
2045
2046
2047
2048
2050 """Spectral density Hessian.
2051
2052 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2053 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2054
2055 The model-free Hessian is::
2056
2057 _k_
2058 d2J(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2059 ------- = - > ci . --- | ---------------- - te^2 ----------------------------- |.
2060 dGj.dS2 5 /__ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2061 i=-k
2062 """
2063
2064 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_te_djw_dti), axis=2)
2065
2066
2068 """Spectral density Hessian.
2069
2070 Calculate the spectral desity values for the Gj - S2 double partial derivative of the original
2071 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2072
2073 The model-free Hessian is::
2074
2075 _k_
2076 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2077 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
2078 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2079 i=-k
2080
2081 dci / 1 (te + ti)te \ \
2082 + --- . ti | ------------ - ------------------------- | |.
2083 dGj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
2084 """
2085
2086 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)
2087
2088
2089
2090
2091
2092
2093
2094
2096 """Spectral density Hessian.
2097
2098 Calculate the spectral desity values for the Gj - te double partial derivative of the original
2099 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2100
2101 The model-free Hessian is::
2102
2103 _k_
2104 d2J(w) 4 \ dti (te + ti)^2 - 3(w.te.ti)^2
2105 ------- = - (1 - S2) . te > ci . --- . ti . (te + ti) -----------------------------.
2106 dGj.dte 5 /__ dGj ((te + ti)^2 + (w.te.ti)^2)^3
2107 i=-k
2108 """
2109
2110 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)
2111
2112
2114 """Spectral density Hessian.
2115
2116 Calculate the spectral desity values for the Gj - te double partial derivative of the original
2117 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2118
2119 The model-free Hessian is::
2120
2121 _k_
2122 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
2123 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2124 dGj.dte 5 /__ \ dGj ((te + ti)^2 + (w.te.ti)^2)^3 dGj ((te + ti)^2 + (w.te.ti)^2)^2 /
2125 i=-k
2126 """
2127
2128 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)
2129
2130
2131
2132
2133
2134
2135
2136
2138 """Spectral density Hessian.
2139
2140 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2141 original model-free formula with no parameters {} together with diffusion tensor parameters.
2142
2143 The model-free Hessian is::
2144
2145 _k_
2146 d2J(w) 2 \ d2ci ti
2147 ------- = - > ------- . ------------.
2148 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2149 i=-k
2150 """
2151
2152 return 0.4 * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2153
2154
2155
2156
2158 """Spectral density Hessian.
2159
2160 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2161 original model-free formula with the parameter {S2} together with diffusion tensor parameters.
2162
2163 The model-free Hessian is::
2164
2165 _k_
2166 d2J(w) 2 \ d2ci ti
2167 ------- = - S2 > ------- . ------------.
2168 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2169 i=-k
2170 """
2171
2172 return 0.4 * params[data.s2_i] * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2173
2174
2175
2176
2178 """Spectral density Hessian.
2179
2180 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2181 original model-free formula with the parameters {S2, te} together with diffusion tensor
2182 parameters.
2183
2184 The model-free Hessian is::
2185
2186 _k_
2187 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
2188 ------- = - > ------- . ti | ------------ + ------------------------- |.
2189 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2190 i=-k
2191 """
2192
2193 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)
2194
2195
2196
2197
2198
2199
2200
2201
2203 """Spectral density Hessian.
2204
2205 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2206 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2207
2208 The model-free Hessian is::
2209
2210 _k_
2211 d2J(w) 2 \ dci 1
2212 ------- = - > --- . ti ------------.
2213 dOj.dS2 5 /__ dOj 1 + (w.ti)^2
2214 i=-k
2215 """
2216
2217 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ti, axis=2)
2218
2219
2220
2221
2223 """Spectral density Hessian.
2224
2225 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2226 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2227
2228 The model-free Hessian is::
2229
2230 _k_
2231 d2J(w) 2 \ dci / 1 (te + ti)te \
2232 ------- = - > --- . ti | ------------ - ------------------------- |.
2233 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2234 i=-k
2235 """
2236
2237 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_te), axis=2)
2238
2239
2240
2241
2242
2243
2244
2245
2247 """Spectral density Hessian.
2248
2249 Calculate the spectral desity values for the Oj - te double partial derivative of the original
2250 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2251
2252 The model-free Hessian is::
2253
2254 _k_
2255 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
2256 ------- = - (1 - S2) > --- . ti^2 -----------------------------.
2257 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
2258 i=-k
2259 """
2260
2261 return 0.4 * data.one_s2 * sum(data.dci[k] * data.fact_djw_dte, axis=2)
2262
2263
2264
2265
2266
2267
2268
2269
2271 """Spectral density Hessian.
2272
2273 Calculate the spectral desity values for the S2 - te double partial derivative of the original
2274 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2275
2276 The model-free Hessian is::
2277
2278 _k_
2279 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
2280 ------- = - - > ci . ti^2 -----------------------------.
2281 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
2282 i=-k
2283 """
2284
2285 return -0.4 * sum(data.ci * data.fact_djw_dte, axis=2)
2286
2287
2288
2289
2290
2291
2292
2293
2295 """Spectral density Hessian.
2296
2297 Calculate the spectral desity values for the te - te double partial derivative of the original
2298 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2299
2300 The model-free Hessian is::
2301
2302 _k_
2303 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
2304 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
2305 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
2306 i=-k
2307 """
2308
2309 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)
2310
2311
2312
2313
2314
2315
2316
2317
2319 """Spectral density Hessian.
2320
2321 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2322 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2323
2324 The model-free Hessian is::
2325
2326 _k_
2327 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 \
2328 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2329 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2330 i=-k
2331
2332 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2333 + ------- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- | |.
2334 dGj.dGk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2335 """
2336
2337
2338 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)
2339
2340
2341 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2342
2343 return 0.4 * sum(data.ci * (a + b), axis=2)
2344
2345
2347 """Spectral density Hessian.
2348
2349 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2350 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2351
2352 The model-free Hessian is::
2353
2354 _k_
2355 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 \
2356 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2357 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2358 i=-k
2359
2360 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2361 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2362 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2363
2364
2365 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2366 + ------- . ti | ------------ + ------------------------- | |.
2367 dGj.dGk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2368 """
2369
2370
2371 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)
2372
2373
2374 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)
2375
2376
2377 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
2378
2379 return 0.4 * sum(a + b + c, axis=2)
2380
2381
2382
2383
2384
2386 """Spectral density Hessian.
2387
2388 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2389 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2390 parameters.
2391
2392 The model-free Hessian is::
2393
2394 _k_
2395 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
2396 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2397 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2398 i=-k
2399
2400 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2401 + (S2f - S2)ts^2 ---------------------------------------------------- |
2402 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2403
2404
2405 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2406 + ------- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- | |.
2407 dGj.dGk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2408 """
2409
2410
2411 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)
2412
2413
2414 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)
2415
2416 return 0.4 * sum(data.ci * (a + b), axis=2)
2417
2418
2420 """Spectral density Hessian.
2421
2422 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
2423 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2424 parameters.
2425
2426 The model-free Hessian is::
2427
2428 _k_
2429 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
2430 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2431 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2432 i=-k
2433
2434 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2435 + (S2f - S2)ts^2 ---------------------------------------------------- |
2436 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2437
2438
2439 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
2440 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
2441 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
2442
2443
2444 (ts + ti)^2 - (w.ts.ti)^2 \
2445 + (S2f - S2)ts^2 ----------------------------- |
2446 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2447
2448
2449 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2450 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2451 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2452 """
2453
2454
2455 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)
2456
2457
2458 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)
2459
2460
2461 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)
2462
2463 return 0.4 * sum(a + b + c, axis=2)
2464
2465
2466
2467
2468
2469
2470
2471
2473 """Spectral density Hessian.
2474
2475 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2476 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2477
2478 The model-free Hessian is::
2479
2480 _k_
2481 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2482 ------- = - > --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
2483 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2484 i=-k
2485 """
2486
2487 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)
2488
2489
2491 """Spectral density Hessian.
2492
2493 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2494 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2495
2496 The model-free Hessian is::
2497
2498 _k_
2499 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2500 ------- = - > | --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2501 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2502 i=-k
2503
2504 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2505 + ------- . ti | ------------ + ------------------------- | |.
2506 dGj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2507 """
2508
2509 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)
2510
2511
2512
2513
2515 """Spectral density Hessian.
2516
2517 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2518 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2519 parameters.
2520
2521 The model-free Hessian is::
2522
2523 _k_
2524 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2525 ------- = - > --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
2526 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 /
2527 i=-k
2528 """
2529
2530 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)
2531
2532
2534 """Spectral density Hessian.
2535
2536 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
2537 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2538 parameters.
2539
2540 The model-free Hessian is::
2541
2542 _k_
2543 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2544 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
2545 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 /
2546 i=-k
2547
2548 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2549 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2550 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2551 """
2552
2553 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)
2554
2555
2556
2557
2558
2559
2560
2561
2563 """Spectral density Hessian.
2564
2565 Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended
2566 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2567 diffusion tensor parameters.
2568
2569 The model-free Hessian is::
2570
2571 _k_
2572 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2573 ------- = - > ci . --- | ---------------- - ts^2 ----------------------------- |.
2574 dGj.dS2 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2575 i=-k
2576 """
2577
2578 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti), axis=2)
2579
2580
2582 """Spectral density Hessian.
2583
2584 Calculate the spectral desity values for the Gj - S2 double partial derivative of the extended
2585 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2586 diffusion tensor parameters.
2587
2588 The model-free Hessian is::
2589
2590 _k_
2591 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2592 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
2593 dGj.dS2 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2594 i=-k
2595
2596 dci / 1 (ts + ti)ts \ \
2597 + --- . ti | ------------ - ------------------------- | |.
2598 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2599 """
2600
2601 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)
2602
2603
2604
2605
2606
2607
2608
2609
2611 """Spectral density Hessian.
2612
2613 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2614 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2615
2616 The model-free Hessian is::
2617
2618 _k_
2619 d2J(w) 2 \ dti (ts + ti)^2 - (w.ts.ti)^2
2620 -------- = - > ci . --- ts^2 -----------------------------.
2621 dGj.dS2f 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^2
2622 i=-k
2623 """
2624
2625 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti, axis=2)
2626
2627
2629 """Spectral density Hessian.
2630
2631 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2632 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2633
2634 The model-free Hessian is::
2635
2636 _k_
2637 d2J(w) 2 \ / dti (ts + ti)^2 - (w.ts.ti)^2 dci (ts + ti)ts \
2638 -------- = - > | ci . --- ts^2 ----------------------------- + --- . ti ------------------------- |.
2639 dGj.dS2f 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 dGj (ts + ti)^2 + (w.ts.ti)^2 /
2640 i=-k
2641 """
2642
2643 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti + data.dci[k] * data.ti * data.fact_ts, axis=2)
2644
2645
2646
2647
2649 """Spectral density Hessian.
2650
2651 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2652 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2653 parameters.
2654
2655 The model-free Hessian is::
2656
2657 _k_
2658 d2J(w) 2 \ dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2659 -------- = - - > ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |.
2660 dGj.dS2f 5 /__ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2661 i=-k
2662 """
2663
2664 return -0.4 * sum(data.ci * data.dti[k] * (data.fact_tf_djw_dti - data.fact_ts_djw_dti), axis=2)
2665
2666
2668 """Spectral density Hessian.
2669
2670 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
2671 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2672 parameters.
2673
2674 The model-free Hessian is::
2675
2676 _k_
2677 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2678 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
2679 dGj.dS2f 5 /__ \ dGj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2680 i=-k
2681
2682 dci / (tf + ti)tf (ts + ti)ts \ \
2683 + --- . ti | ------------------------- - ------------------------- | |.
2684 dGj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2685 """
2686
2687 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)
2688
2689
2690
2691
2692
2693
2694
2695
2697 """Spectral density Hessian.
2698
2699 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
2700 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2701 parameters.
2702
2703 The model-free Hessian is::
2704
2705 _k_
2706 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
2707 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
2708 dGj.dtf 5 /__ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3
2709 i=-k
2710 """
2711
2712 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)
2713
2714
2716 """Spectral density Hessian.
2717
2718 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
2719 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2720 parameters.
2721
2722 The model-free Hessian is::
2723
2724 _k_
2725 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
2726 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2727 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
2728 i=-k
2729 """
2730
2731 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)
2732
2733
2734
2735
2736
2737
2738
2739
2741 """Spectral density Hessian.
2742
2743 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
2744 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2745 diffusion tensor parameters.
2746
2747 The model-free Hessian is::
2748
2749 _k_
2750 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
2751 ------- = - (S2f - S2) . ts > ci . --- . ti . (ts + ti) -----------------------------.
2752 dGj.dts 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3
2753 i=-k
2754 """
2755
2756 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)
2757
2758
2760 """Spectral density Hessian.
2761
2762 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
2763 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2764 diffusion tensor parameters.
2765
2766 The model-free Hessian is::
2767
2768 _k_
2769 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
2770 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2771 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2772 i=-k
2773 """
2774
2775 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)
2776
2777
2778
2779
2780
2781
2782
2783
2785 """Spectral density Hessian.
2786
2787 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2788 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2789 parameters.
2790
2791 The model-free Hessian is::
2792
2793 _k_
2794 d2J(w) 2 \ d2ci / S2 (S2f - S2)(ts + ti)ts \
2795 ------- = - > ------- . ti | ------------ + ------------------------- |.
2796 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2797 i=-k
2798 """
2799
2800 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)
2801
2802
2803
2804
2806 """Spectral density Hessian.
2807
2808 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2809 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2810 parameters.
2811
2812 The model-free Hessian is::
2813
2814 _k_
2815 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
2816 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
2817 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2818 i=-k
2819 """
2820
2821 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)
2822
2823
2824
2825
2826
2827
2828
2829
2831 """Spectral density Hessian.
2832
2833 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
2834 model-free formula with the parameters {S2f, S2, ts} and {S2f, tf, S2, ts} together with
2835 diffusion tensor parameters.
2836
2837 The model-free Hessian is::
2838
2839 _k_
2840 d2J(w) 2 \ dci / 1 (ts + ti)ts \
2841 ------- = - > --- . ti | ------------ - ------------------------- |.
2842 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2843 i=-k
2844 """
2845
2846 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
2847
2848
2849
2850
2851
2852
2853
2854
2856 """Spectral density Hessian.
2857
2858 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2859 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2860 parameters.
2861
2862 The model-free Hessian is::
2863
2864 _k_
2865 d2J(w) 2 \ dci (ts + ti)ts
2866 -------- = - > --- . ti -------------------------.
2867 dOj.dS2f 5 /__ dOj (ts + ti)^2 + (w.ts.ti)^2
2868 i=-k
2869 """
2870
2871 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ts, axis=2)
2872
2873
2874
2875
2877 """Spectral density Hessian.
2878
2879 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2880 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2881 parameters.
2882
2883 The model-free Hessian is::
2884
2885 _k_
2886 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
2887 -------- = - - > --- . ti | ------------------------- - ------------------------- |.
2888 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2889 i=-k
2890 """
2891
2892 return -0.4 * sum(data.dci[k] * data.ti * (data.fact_tf - data.fact_ts), axis=2)
2893
2894
2895
2896
2897
2898
2899
2900
2902 """Spectral density Hessian.
2903
2904 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
2905 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2906 parameters.
2907
2908 The model-free Hessian is::
2909
2910 _k_
2911 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
2912 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
2913 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
2914 i=-k
2915 """
2916
2917 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
2918
2919
2920
2921
2922
2923
2924
2925
2927 """Spectral density Hessian.
2928
2929 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
2930 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2931 diffusion tensor parameters.
2932
2933 The model-free Hessian is::
2934
2935 _k_
2936 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
2937 ------- = - (S2f - S2) > --- . ti^2 -----------------------------.
2938 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
2939 i=-k
2940 """
2941
2942 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
2943
2944
2945
2946
2947
2948
2949
2950
2952 """Spectral density Hessian.
2953
2954 Calculate the spectral desity values for the S2 - ts double partial derivative of the extended
2955 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
2956 diffusion tensor parameters.
2957
2958 The model-free Hessian is::
2959
2960 _k_
2961 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
2962 ------- = - - > ci . ti^2 -----------------------------.
2963 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
2964 i=-k
2965 """
2966
2967 return -0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
2968
2969
2970
2971
2972
2973
2974
2975
2977 """Spectral density Hessian.
2978
2979 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
2980 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
2981 parameters.
2982
2983 The model-free Hessian is::
2984
2985 _k_
2986 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
2987 -------- = - - > ci . ti^2 -----------------------------.
2988 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
2989 i=-k
2990 """
2991
2992 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
2993
2994
2995
2996
2997
2998
2999
3000
3002 """Spectral density Hessian.
3003
3004 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3005 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3006 diffusion tensor parameters.
3007
3008 The model-free Hessian is::
3009
3010 _k_
3011 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3012 -------- = - > ci . ti^2 -----------------------------.
3013 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3014 i=-k
3015 """
3016
3017 return 0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
3018
3019
3020
3021
3022
3023
3024
3025
3027 """Spectral density Hessian.
3028
3029 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3030 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
3031 parameters.
3032
3033 The model-free Hessian is::
3034
3035 _k_
3036 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3037 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3038 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3039 i=-k
3040 """
3041
3042 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)
3043
3044
3045
3046
3047
3048
3049
3050
3052 """Spectral density Hessian.
3053
3054 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3055 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3056 diffusion tensor parameters.
3057
3058 The model-free Hessian is::
3059
3060 _k_
3061 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3062 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------.
3063 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3064 i=-k
3065 """
3066
3067 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)
3068
3069
3070
3071
3072
3073
3074
3075
3077 """Spectral density Hessian.
3078
3079 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3080 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3081
3082 The model-free Hessian is::
3083
3084 _k_
3085 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 \
3086 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3087 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3088 i=-k
3089
3090 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
3091 + ------- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- | |.
3092 dGj.dGk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
3093 """
3094
3095
3096 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)
3097
3098
3099 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)
3100
3101 return 0.4 * sum(data.ci * (a + b), axis=2)
3102
3103
3105 """Spectral density Hessian.
3106
3107 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3108 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3109
3110 The model-free Hessian is::
3111
3112 _k_
3113 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 \
3114 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3115 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3116 i=-k
3117
3118 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3119 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3120 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3121
3122
3123 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3124 + ------- . ti | ------------ + ------------------------- | |.
3125 dGj.dGk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3126 """
3127
3128
3129 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)
3130
3131
3132 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)
3133
3134
3135 c = data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
3136
3137 return 0.4 * sum(a + b + c, axis=2)
3138
3139
3140
3141
3143 """Spectral density Hessian.
3144
3145 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3146 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3147 parameters.
3148
3149 The model-free Hessian is::
3150
3151 _k_
3152 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
3153 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3154 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3155 i=-k
3156
3157 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3158 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3159 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3160
3161
3162 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3163 + ------- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3164 dGj.dGk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3165 """
3166
3167
3168 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)
3169
3170
3171 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)
3172
3173 return 0.4 * sum(data.ci * (a + b), axis=2)
3174
3175
3177 """Spectral density Hessian.
3178
3179 Calculate the spectral desity values for the Gj - Gk double partial derivative of the extended
3180 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3181 parameters.
3182
3183 The model-free Hessian is::
3184
3185 _k_
3186 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
3187 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3188 dGj.dGk 5 /__ \ dGj dGk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3189 i=-k
3190
3191 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3192 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3193 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3194
3195
3196 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
3197 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
3198 \ dGj dGk dGk dGj dGj.dGk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
3199
3200
3201 (ts + ti)^2 - (w.ts.ti)^2 \
3202 + S2f(1 - S2s)ts^2 ----------------------------- |
3203 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3204
3205
3206 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3207 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3208 dGj.dGk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3209 """
3210
3211
3212 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)
3213
3214
3215 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)
3216
3217
3218 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)
3219
3220 return 0.4 * sum(a + b + c, axis=2)
3221
3222
3223
3224
3225
3226
3227
3228
3230 """Spectral density Hessian.
3231
3232 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3233 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3234
3235 The model-free Hessian is::
3236
3237 _k_
3238 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3239 ------- = - > --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3240 dGj.dOj 5 /__ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3241 i=-k
3242 """
3243
3244 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)
3245
3246
3248 """Spectral density Hessian.
3249
3250 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3251 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3252
3253 The model-free Hessian is::
3254
3255 _k_
3256 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3257 ------- = - > | --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3258 dGj.dOj 5 /__ \ dOj dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3259 i=-k
3260
3261 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3262 + ------- . ti | ------------ + ------------------------- | |.
3263 dGj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3264 """
3265
3266 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)
3267
3268
3269
3270
3272 """Spectral density Hessian.
3273
3274 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3275 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3276 parameters.
3277
3278 The model-free Hessian is::
3279
3280 _k_
3281 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3282 ------- = - > --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3283 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 /
3284 i=-k
3285 """
3286
3287 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)
3288
3289
3291 """Spectral density Hessian.
3292
3293 Calculate the spectral desity values for the Gj - Oj double partial derivative of the extended
3294 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3295 parameters.
3296
3297 The model-free Hessian is::
3298
3299 _k_
3300 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3301 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3302 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 /
3303 i=-k
3304
3305 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3306 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3307 dGj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3308 """
3309
3310 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)
3311
3312
3313
3314
3315
3316
3317
3318
3320 """Spectral density Hessian.
3321
3322 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3323 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3324
3325 The model-free Hessian is::
3326
3327 _k_
3328 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3329 -------- = - > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
3330 dGj.dS2f 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3331 i=-k
3332 """
3333
3334 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)
3335
3336
3338 """Spectral density Hessian.
3339
3340 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3341 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3342
3343 The model-free Hessian is::
3344
3345 _k_
3346 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3347 -------- = - > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
3348 dGj.dS2f 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3349 i=-k
3350
3351 dci / S2s (1 - S2s)(ts + ti)ts \ \
3352 + --- . ti | ------------ + ------------------------- | |.
3353 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3354 """
3355
3356 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)
3357
3358
3359
3360
3362 """Spectral density Hessian.
3363
3364 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3365 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3366 parameters.
3367
3368 The model-free Hessian is::
3369
3370 _k_
3371 d2J(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3372 -------- = - > ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |.
3373 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 /
3374 i=-k
3375 """
3376
3377 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)
3378
3379
3381 """Spectral density Hessian.
3382
3383 Calculate the spectral desity values for the Gj - S2f double partial derivative of the extended
3384 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3385 parameters.
3386
3387 The model-free Hessian is::
3388
3389 _k_
3390 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3391 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
3392 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 /
3393 i=-k
3394
3395 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
3396 + --- . ti | ------------ - ------------------------- + ------------------------- | |.
3397 dGj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3398 """
3399
3400 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)
3401
3402
3403
3404
3405
3406
3407
3408
3410 """Spectral density Hessian.
3411
3412 Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended
3413 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3414 diffusion tensor parameters.
3415
3416 The model-free Hessian is::
3417
3418 _k_
3419 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3420 -------- = - S2f > ci . --- | ---------------- - ts^2 ----------------------------- |.
3421 dGj.dS2s 5 /__ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3422 i=-k
3423 """
3424
3425 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)
3426
3427
3429 """Spectral density Hessian.
3430
3431 Calculate the spectral desity values for the Gj - S2s double partial derivative of the extended
3432 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3433 diffusion tensor parameters.
3434
3435 The model-free Hessian is::
3436
3437 _k_
3438 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3439 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
3440 dGj.dS2s 5 /__ \ dGj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3441 i=-k
3442
3443 dci / 1 (ts + ti)ts \ \
3444 + --- . ti | ------------ - ------------------------- | |.
3445 dGj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3446 """
3447
3448 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)
3449
3450
3451
3452
3453
3454
3455
3456
3458 """Spectral density Hessian.
3459
3460 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
3461 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3462 parameters.
3463
3464 The model-free Hessian is::
3465
3466 _k_
3467 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
3468 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
3469 dGj.dtf 5 /__ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3
3470 i=-k
3471 """
3472
3473 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)
3474
3475
3477 """Spectral density Hessian.
3478
3479 Calculate the spectral desity values for the Gj - tf double partial derivative of the extended
3480 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3481 parameters.
3482
3483 The model-free Hessian is::
3484
3485 _k_
3486 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
3487 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3488 dGj.dtf 5 /__ \ dGj ((tf + ti)^2 + (w.tf.ti)^2)^3 dGj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
3489 i=-k
3490 """
3491
3492 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)
3493
3494
3495
3496
3497
3498
3499
3500
3502 """Spectral density Hessian.
3503
3504 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
3505 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3506 diffusion tensor parameters.
3507
3508 The model-free Hessian is::
3509
3510 _k_
3511 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
3512 ------- = - S2f(1 - S2s) . ts > ci . --- . ti . (ts + ti) -----------------------------.
3513 dGj.dts 5 /__ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3
3514 i=-k
3515 """
3516
3517 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)
3518
3519
3521 """Spectral density Hessian.
3522
3523 Calculate the spectral desity values for the Gj - ts double partial derivative of the extended
3524 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3525 diffusion tensor parameters.
3526
3527 The model-free Hessian is::
3528
3529 _k_
3530 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
3531 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3532 dGj.dts 5 /__ \ dGj ((ts + ti)^2 + (w.ts.ti)^2)^3 dGj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3533 i=-k
3534 """
3535
3536 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)
3537
3538
3539
3540
3541
3542
3543
3544
3546 """Spectral density Hessian.
3547
3548 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3549 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3550 parameters.
3551
3552 The model-free Hessian is::
3553
3554 _k_
3555 d2J(w) 2 \ d2ci / S2f . S2s S2f(1 - S2s)(ts + ti)ts \
3556 ------- = - > ------- . ti | ------------ + ------------------------- |.
3557 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3558 i=-k
3559 """
3560
3561 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)
3562
3563
3564
3565
3567 """Spectral density Hessian.
3568
3569 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3570 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3571 parameters.
3572
3573 The model-free Hessian is::
3574
3575 _k_
3576 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
3577 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
3578 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3579 i=-k
3580 """
3581
3582 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)
3583
3584
3585
3586
3587
3588
3589
3590
3592 """Spectral density Hessian.
3593
3594 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3595 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3596 parameters.
3597
3598 The model-free Hessian is::
3599
3600 _k_
3601 d2J(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
3602 -------- = - > --- . ti | ------------ + ------------------------- |.
3603 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3604 i=-k
3605 """
3606
3607 return 0.4 * sum(data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
3608
3609
3610
3611
3613 """Spectral density Hessian.
3614
3615 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3616 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3617 parameters.
3618
3619 The model-free Hessian is::
3620
3621 _k_
3622 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
3623 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |.
3624 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3625 i=-k
3626 """
3627
3628 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)
3629
3630
3631
3632
3633
3634
3635
3636
3638 """Spectral density Hessian.
3639
3640 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
3641 model-free formula with the parameters {S2f, S2s, ts} and {S2f, tf, S2s, ts} together with
3642 diffusion tensor parameters.
3643
3644 The model-free Hessian is::
3645
3646 _k_
3647 d2J(w) 2 \ dci / 1 (ts + ti)ts \
3648 -------- = - S2f > --- . ti | ------------ - ------------------------- |.
3649 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3650 i=-k
3651 """
3652
3653 return 0.4 * params[data.s2f_i] * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3654
3655
3656
3657
3658
3659
3660
3661
3663 """Spectral density Hessian.
3664
3665 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
3666 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3667 parameters.
3668
3669 The model-free Hessian is::
3670
3671 _k_
3672 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
3673 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
3674 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
3675 i=-k
3676 """
3677
3678 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
3679
3680
3681
3682
3683
3684
3685
3686
3688 """Spectral density Hessian.
3689
3690 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
3691 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3692 diffusion tensor parameters.
3693
3694 The model-free Hessian is::
3695
3696 _k_
3697 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
3698 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------.
3699 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
3700 i=-k
3701 """
3702
3703 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
3704
3705
3706
3707
3708
3709
3710
3711
3713 """Spectral density Hessian.
3714
3715 Calculate the spectral desity values for the S2f - S2s double partial derivative of the extended
3716 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3717 diffusion tensor parameters.
3718
3719 The model-free Hessian is::
3720
3721 _k_
3722 d2J(w) 2 \ / 1 (ts + ti).ts \
3723 --------- = - > ci . ti | ------------ - ------------------------- |.
3724 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3725 i=-k
3726 """
3727
3728 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3729
3730
3731
3732
3733
3734
3735
3736
3738 """Spectral density Hessian.
3739
3740 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
3741 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3742 parameters.
3743
3744 The model-free Hessian is::
3745
3746 _k_
3747 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
3748 -------- = - - > ci . ti^2 -----------------------------.
3749 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
3750 i=-k
3751 """
3752
3753 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
3754
3755
3756
3757
3758
3759
3760
3761
3763 """Spectral density Hessian.
3764
3765 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3766 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3767 diffusion tensor parameters.
3768
3769 The model-free Hessian is::
3770
3771 _k_
3772 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3773 -------- = - (1 - S2s) > ci . ti^2 -----------------------------.
3774 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3775 i=-k
3776 """
3777
3778 return 0.4 * data.one_s2s * sum(data.ci * data.fact_djw_dts, axis=2)
3779
3780
3781
3782
3783
3784
3785
3786
3788 """Spectral density Hessian.
3789
3790 Calculate the spectral desity values for the S2s - ts double partial derivative of the extended
3791 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3792 diffusion tensor parameters.
3793
3794 The model-free Hessian is::
3795
3796 _k_
3797 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3798 -------- = - - S2f > ci . ti^2 -----------------------------.
3799 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3800 i=-k
3801 """
3802
3803 return -0.4 * params[data.s2f_i] * sum(data.ci * data.fact_djw_dts, axis=2)
3804
3805
3806
3807
3808
3809
3810
3811
3813 """Spectral density Hessian.
3814
3815 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3816 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3817 parameters.
3818
3819 The model-free Hessian is::
3820
3821 _k_
3822 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3823 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3824 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3825 i=-k
3826 """
3827
3828 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)
3829
3830
3831
3832
3833
3834
3835
3836
3838 """Spectral density Hessian.
3839
3840 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3841 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3842 diffusion tensor parameters.
3843
3844 The model-free Hessian is::
3845
3846 _k_
3847 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3848 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
3849 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3850 i=-k
3851 """
3852
3853 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)
3854