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21
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23 from Numeric import sum
24
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27
28
29
30 """
31 The spectral density equation
32 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
33
34 Data structure: data.jw
35 Dimension: 2D, (number of NMR frequencies, 5 spectral density frequencies)
36 Type: Numeric matrix, Float64
37 Dependencies: None
38 Required by: data.ri, data.dri, data.d2ri
39
40
41 Formulae
42 ~~~~~~~~
43 _n_
44 2 \ 1
45 J(w) = - S2 > ci . ti ------------,
46 5 /__ 1 + (w.ti)^2
47 i=m
48
49
50 _n_
51 2 \ / S2 (1 - S2)(te + ti)te \
52 J(w) = - > ci . ti | ------------ + ------------------------- |,
53 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
54 i=m
55
56
57 _n_
58 2 \ / S2 (S2f - S2)(ts + ti)ts \
59 J(w) = - > ci . ti | ------------ + ------------------------- |,
60 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
61 i=m
62
63
64 _n_
65 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
66 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |,
67 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
68 i=m
69
70
71 Extended 2
72 ~~~~~~~~~~
73
74 _n_
75 2 \ / S2s (1 - S2s)(ts + ti)ts \
76 J(w) = - S2f > ci . ti | ------------ + ------------------------- |,
77 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
78 i=m
79
80
81 _n_
82 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
83 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
84 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
85 i=m
86 """
87
88
89
90
91
92
94 """Spectral density function.
95
96 Calculate the spectral density values for the original model-free formula with no parameters {}
97 with or without diffusion tensor parameters.
98
99 The formula is
100
101 _n_
102 2 \ 1
103 J(w) = - > ci . ti ------------.
104 5 /__ 1 + (w.ti)^2
105 i=m
106 """
107
108 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
109
110
111
112
113
114
116 """Spectral density function.
117
118 Calculate the spectral density values for the original model-free formula with the single
119 parameter {S2} with or without diffusion tensor parameters.
120
121 The formula is
122
123 _n_
124 2 \ 1
125 J(w) = - S2 > ci . ti ------------.
126 5 /__ 1 + (w.ti)^2
127 i=m
128 """
129
130 return 0.4 * params[data.s2_i] * sum(data.ci * data.ti * data.fact_ti, axis=2)
131
132
133
134
135
136
138 """Spectral density function.
139
140 Calculate the spectral density values for the original model-free formula with the parameters
141 {S2, te} with or without diffusion tensor parameters.
142
143 The model-free formula is
144
145 _n_
146 2 \ / S2 (1 - S2)(te + ti)te \
147 J(w) = - > ci . ti | ------------ + ------------------------- |.
148 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
149 i=m
150 """
151
152 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
153
154
155
156
157
158
160 """Spectral density function.
161
162 Calculate the spectral density values for the extended model-free formula with the parameters
163 {S2f, S2, ts} with or without diffusion tensor parameters.
164
165 The model-free formula is
166
167 _n_
168 2 \ / S2 (S2f - S2)(ts + ti)ts \
169 J(w) = - > ci . ti | ------------ + ------------------------- |.
170 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
171 i=m
172 """
173
174 return 0.4 * sum(data.ci * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
175
176
177
178
179
180
182 """Spectral density function.
183
184 Calculate the spectral density values for the extended model-free formula with the parameters
185 {S2f, tf, S2, ts} with or without diffusion tensor parameters.
186
187 The model-free formula is
188
189 _n_
190 2 \ / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
191 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
192 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
193 i=m
194 """
195
196 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)
197
198
199
200
201
202
204 """Spectral density function.
205
206 Calculate the spectral density values for the extended model-free formula with the parameters
207 {S2f, S2s, ts} with or without diffusion tensor parameters.
208
209 The model-free formula is
210
211 _n_
212 2 \ / S2s (1 - S2s)(ts + ti)ts \
213 J(w) = - S2f > ci . ti | ------------ + ------------------------- |.
214 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
215 i=m
216 """
217
218 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)
219
220
221
222
223
224
226 """Spectral density function.
227
228 Calculate the spectral density values for the extended model-free formula with the parameters
229 {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
230
231 The model-free formula is
232
233 _n_
234 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
235 J(w) = - > ci . ti | ------------ + ------------------------- + ------------------------- |.
236 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
237 i=m
238 """
239
240 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)
241
242
243
244
245
246
247
248
249 """
250 The spectral density gradients
251 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
252
253 Data structure: data.djw
254 Dimension: 3D, (number of NMR frequencies, 5 spectral density frequencies,
255 model-free parameters)
256 Type: Numeric 3D matrix, Float64
257 Dependencies: None
258 Required by: data.dri, data.d2ri
259
260
261 Formulae
262 ~~~~~~~~
263
264 Original
265 ~~~~~~~~
266
267 _n_
268 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
269 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
270 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
271 i=m
272
273 dci / S2 (1 - S2)(te + ti)te \ \
274 + --- . ti | ------------ + ------------------------- | |,
275 dDj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
276
277
278 _n_
279 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
280 ----- = - > --- . ti | ------------ + ------------------------- |,
281 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
282 i=m
283
284
285 _n_
286 dJ(w) 2 \ / 1 (te + ti)te \
287 ----- = - > ci . ti | ------------ - ------------------------- |,
288 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
289 i=m
290
291
292 _n_
293 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
294 ----- = - (1 - S2) > ci . ti^2 -----------------------------,
295 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
296 i=m
297
298
299 dJ(w)
300 ----- = 0,
301 dRex
302
303
304 dJ(w)
305 ----- = 0,
306 dcsa
307
308
309 dJ(w)
310 ----- = 0.
311 dr
312
313
314 Extended
315 ~~~~~~~~
316
317 _n_
318 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
319 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
320 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
321 i=m
322
323 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
324 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
325 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
326
327
328 _n_
329 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
330 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
331 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
332 i=m
333
334
335 _n_
336 dJ(w) 2 \ / 1 (ts + ti).ts \
337 ----- = - > ci . ti | ------------ - ------------------------- |,
338 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
339 i=m
340
341
342 _n_
343 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
344 ----- = - - > ci . ti | ------------------------- - ------------------------- |,
345 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
346 i=m
347
348
349 _n_
350 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
351 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
352 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
353 i=m
354
355
356 _n_
357 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
358 ----- = - (S2f - S2) > ci . ti^2 -----------------------------,
359 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
360 i=m
361
362
363 dJ(w)
364 ----- = 0,
365 dRex
366
367
368 dJ(w)
369 ----- = 0,
370 dcsa
371
372
373 dJ(w)
374 ----- = 0.
375 dr
376
377
378 Extended 2
379 ~~~~~~~~~~
380
381 _n_
382 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
383 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
384 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
385 i=m
386
387 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
388 + --- . ti | ------------ + ------------------------- + ------------------------- | |,
389 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
390
391
392 _n_
393 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
394 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |,
395 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
396 i=m
397
398
399 _n_
400 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
401 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |,
402 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
403 i=m
404
405
406 _n_
407 dJ(w) 2 \ / 1 (ts + ti).ts \
408 ----- = - S2f > ci . ti | ------------ - ------------------------- |,
409 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
410 i=m
411
412
413 _n_
414 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
415 ----- = - (1 - S2f) > ci . ti^2 -----------------------------,
416 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
417 i=m
418
419
420 _n_
421 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
422 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------,
423 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
424 i=m
425
426
427 dJ(w)
428 ----- = 0,
429 dRex
430
431
432 dJ(w)
433 ----- = 0,
434 dcsa
435
436
437 dJ(w)
438 ----- = 0.
439 dr
440 """
441
442
443
444
445
446
447
448
450 """Spectral density gradient.
451
452 Calculate the spectral desity values for the Dj partial derivative of the original model-free
453 formula with no parameters {} together with diffusion tensor parameters.
454
455 The model-free gradient is
456
457 _n_
458 dJ(w) 2 \ dti 1 - (w.ti)^2
459 ----- = - > ci . --- ----------------.
460 dDj 5 /__ dDj (1 + (w.ti)^2)^2
461 i=m
462 """
463
464 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
465
466
468 """Spectral density gradient.
469
470 Calculate the spectral desity values for the Dj partial derivative of the original model-free
471 formula with no parameters {} together with diffusion tensor parameters.
472
473 The model-free gradient is
474
475 _n_
476 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
477 ----- = - > | ci . --- ---------------- + --- . ti ------------ |.
478 dDj 5 /__ \ dDj (1 + (w.ti)^2)^2 dDj 1 + (w.ti)^2 /
479 i=m
480 """
481
482 return 0.4 * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti + data.dci[j] * data.ti * data.fact_ti, axis=2)
483
484
485
486
488 """Spectral density gradient.
489
490 Calculate the spectral desity values for the Dj partial derivative of the original model-free
491 formula with the parameter {S2} together with diffusion tensor parameters.
492
493 The model-free gradient is
494
495 _n_
496 dJ(w) 2 \ dti 1 - (w.ti)^2
497 ----- = - S2 > ci . --- ----------------.
498 dDj 5 /__ dDj (1 + (w.ti)^2)^2
499 i=m
500 """
501
502 return 0.4 * params[data.s2_i] * sum(data.ci * data.dti[j] * data.fact_ti_djw_dti, axis=2)
503
504
506 """Spectral density gradient.
507
508 Calculate the spectral desity values for the Dj partial derivative of the original model-free
509 formula with the parameter {S2} together with diffusion tensor parameters.
510
511 The model-free gradient is
512
513 _n_
514 dJ(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
515 ----- = - S2 > | ci . --- ---------------- + --- . ti ------------ |.
516 dDj 5 /__ \ dDj (1 + (w.ti)^2)^2 dDj 1 + (w.ti)^2 /
517 i=m
518 """
519
520 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)
521
522
523
524
526 """Spectral density gradient.
527
528 Calculate the spectral desity values for the Dj partial derivative of the original model-free
529 formula with the parameters {S2, te} together with diffusion tensor parameters.
530
531 The model-free gradient is
532
533 _n_
534 dJ(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
535 ----- = - > ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
536 dDj 5 /__ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
537 i=m
538 """
539
540 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)
541
542
544 """Spectral density gradient.
545
546 Calculate the spectral desity values for the Dj partial derivative of the original model-free
547 formula with the parameters {S2, te} together with diffusion tensor parameters.
548
549 The model-free gradient is
550
551 _n_
552 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
553 ----- = - > | ci . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
554 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
555 i=m
556
557 dci / S2 (1 - S2)(te + ti)te \ \
558 + --- . ti | ------------ + ------------------------- | |.
559 dDj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
560 """
561
562 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)
563
564
565
566
567
568
569
570
572 """Spectral density gradient.
573
574 Calculate the spectral desity values for the O partial derivative of the original model-free
575 formula with no parameters {} together with diffusion tensor parameters.
576
577 The model-free gradient is
578
579 _n_
580 dJ(w) 2 \ dci 1
581 ----- = - > --- . ti ------------.
582 dOj 5 /__ dOj 1 + (w.ti)^2
583 i=m
584 """
585
586 return 0.4 * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
587
588
589
590
592 """Spectral density gradient.
593
594 Calculate the spectral desity values for the O partial derivative of the original model-free
595 formula with the parameter {S2} together with diffusion tensor parameters.
596
597 The model-free gradient is
598
599 _n_
600 dJ(w) 2 \ dci 1
601 ----- = - S2 > --- . ti ------------.
602 dOj 5 /__ dOj 1 + (w.ti)^2
603 i=m
604 """
605
606 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.ti * data.fact_ti, axis=2)
607
608
609
610
612 """Spectral density gradient.
613
614 Calculate the spectral desity values for the O partial derivative of the original model-free
615 formula with the parameters {S2, te} together with diffusion tensor parameters.
616
617 The model-free gradient is
618
619 _n_
620 dJ(w) 2 \ dci / S2 (1 - S2)(te + ti)te \
621 ----- = - > --- . ti | ------------ + ------------------------- |.
622 dOj 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
623 i=m
624 """
625
626 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te), axis=2)
627
628
629
630
631
632
633
634
636 """Spectral density gradient.
637
638 Calculate the spectral desity values for the S2 partial derivative of the original model-free
639 formula with the single parameter {S2} with or without diffusion tensor parameters.
640
641 The model-free gradient is
642
643 _n_
644 dJ(w) 2 \ 1
645 ----- = - > ci . ti ------------.
646 dS2 5 /__ 1 + (w.ti)^2
647 i=m
648 """
649
650 return 0.4 * sum(data.ci * data.ti * data.fact_ti, axis=2)
651
652
653
654
656 """Spectral density gradient.
657
658 Calculate the spectral desity values for the S2 partial derivative of the original model-free
659 formula with the parameters {S2, te} with or without diffusion tensor parameters.
660
661 The model-free gradient is
662
663 _n_
664 dJ(w) 2 \ / 1 (te + ti)te \
665 ----- = - > ci . ti | ------------ - ------------------------- |.
666 dS2 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
667 i=m
668 """
669
670 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_te), axis=2)
671
672
673
674
675
676
677
678
680 """Spectral density gradient.
681
682 Calculate the spectral desity values for the te partial derivative of the original model-free
683 formula with the parameters {S2, te} with or without diffusion tensor parameters.
684
685 The model-free gradient is
686
687 _n_
688 dJ(w) 2 \ (te + ti)^2 - (w.te.ti)^2
689 ----- = - (1 - S2) > ci . ti^2 -----------------------------.
690 dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
691 i=m
692 """
693
694 return 0.4 * data.one_s2 * sum(data.ci * data.fact_djw_dte, axis=2)
695
696
697
698
699
700
701
702
704 """Spectral density gradient.
705
706 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
707 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
708
709 The formula is
710
711 _n_
712 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
713 ----- = - > ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
714 dDj 5 /__ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
715 i=m
716 """
717
718 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)
719
720
722 """Spectral density gradient.
723
724 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
725 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
726
727 The formula is
728
729 _n_
730 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
731 ----- = - > | ci . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
732 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
733 i=m
734
735 dci / S2 (S2f - S2)(ts + ti)ts \ \
736 + --- . ti | ------------ + ------------------------- | |.
737 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
738 """
739
740 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)
741
742
743
744
746 """Spectral density gradient.
747
748 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
749 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
750
751 The formula is
752
753 _n_
754 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
755 ----- = - > ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
756 dDj 5 /__ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
757 i=m
758 """
759
760 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)
761
762
764 """Spectral density gradient.
765
766 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
767 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
768
769 The formula is
770
771 _n_
772 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
773 ----- = - > | ci . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
774 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
775 i=m
776
777 dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
778 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
779 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
780 """
781
782 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)
783
784
785
786
787
788
789
790
792 """Spectral density gradient.
793
794 Calculate the spectral desity values for the O partial derivative of the extended model-free
795 formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
796
797 The formula is
798
799 _n_
800 dJ(w) 2 \ dci / S2 (S2f - S2)(ts + ti)ts \
801 ----- = - > --- . ti | ------------ + ------------------------- |.
802 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
803 i=m
804 """
805
806 return 0.4 * sum(data.dci[j] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts), axis=2)
807
808
809
810
812 """Spectral density gradient.
813
814 Calculate the spectral desity values for the O partial derivative of the extended model-free
815 formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.
816
817 The formula is
818
819 _n_
820 dJ(w) 2 \ dci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
821 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
822 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
823 i=m
824 """
825
826 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)
827
828
829
830
831
832
833
834
836 """Spectral density gradient.
837
838 Calculate the spectral desity values for the S2 partial derivative of the extended model-free
839 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
840 parameters.
841
842 The formula is
843
844 _n_
845 dJ(w) 2 \ / 1 (ts + ti).ts \
846 ----- = - > ci . ti | ------------ - ------------------------- |.
847 dS2 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
848 i=m
849 """
850
851 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
852
853
854
855
856
857
858
859
861 """Spectral density gradient.
862
863 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
864 formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.
865
866 The formula is
867
868 _n_
869 dJ(w) 2 \ (ts + ti).ts
870 ----- = - > ci . ti -------------------------.
871 dS2f 5 /__ (ts + ti)^2 + (w.ts.ti)^2
872 i=m
873 """
874
875 return 0.4 * sum(data.ci * data.ti * data.fact_ts, axis=2)
876
877
878
879
881 """Spectral density gradient.
882
883 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
884 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
885
886 The formula is
887
888 _n_
889 dJ(w) 2 \ / (tf + ti).tf (ts + ti).ts \
890 ----- = - - > ci . ti | ------------------------- - ------------------------- |.
891 dS2f 5 /__ \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
892 i=m
893 """
894
895 return -0.4 * sum(data.ci * data.ti * (data.fact_tf - data.fact_ts), axis=2)
896
897
898
899
900
901
902
903
905 """Spectral density gradient.
906
907 Calculate the spectral desity values for the tf partial derivative of the extended model-free
908 formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.
909
910 The formula is
911
912 _n_
913 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
914 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
915 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
916 i=m
917 """
918
919 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
920
921
922
923
924
925
926
927
929 """Spectral density gradient.
930
931 Calculate the spectral desity values for the ts partial derivative of the extended model-free
932 formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
933 parameters.
934
935 The formula is
936
937 _n_
938 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
939 ----- = - (S2f - S2) > ci . ti^2 -----------------------------.
940 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
941 i=m
942 """
943
944 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
945
946
947
948
949
950
951
952
954 """Spectral density gradient.
955
956 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
957 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
958
959 The formula is
960
961 _n_
962 dJ(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
963 ----- = - S2f > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
964 dDj 5 /__ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
965 i=m
966 """
967
968 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)
969
970
972 """Spectral density gradient.
973
974 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
975 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
976
977 The formula is
978
979 _n_
980 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
981 ----- = - S2f > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
982 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
983 i=m
984
985 dci / S2s (1 - S2s)(ts + ti)ts \ \
986 + --- . ti | ------------ + ------------------------- | |.
987 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
988 """
989
990 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)
991
992
993
994
996 """Spectral density gradient.
997
998 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
999 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1000
1001 The formula is
1002
1003 _n_
1004 dJ(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1005 ----- = - > ci . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1006 dDj 5 /__ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1007 i=m
1008 """
1009
1010 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)
1011
1012
1014 """Spectral density gradient.
1015
1016 Calculate the spectral desity values for the Dj partial derivative of the extended model-free
1017 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1018
1019 The formula is
1020
1021 _n_
1022 dJ(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1023 ----- = - > | ci . --- | S2f.S2s ---------------- + (1 - S2f).tf^2 ----------------------------- + S2f(1 - S2s).ts^2 ----------------------------- |
1024 dDj 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1025 i=m
1026
1027 dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1028 + --- . ti | ------------ + ------------------------- + ------------------------- | |.
1029 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1030 """
1031
1032 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)
1033
1034
1035
1036
1037
1038
1039
1040
1042 """Spectral density gradient.
1043
1044 Calculate the spectral desity values for the O partial derivative of the extended model-free
1045 formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
1046
1047 The formula is
1048
1049 _n_
1050 dJ(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
1051 ----- = - S2f > --- . ti | ------------ + ------------------------- |.
1052 dOj 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1053 i=m
1054 """
1055
1056 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)
1057
1058
1059
1060
1062 """Spectral density gradient.
1063
1064 Calculate the spectral desity values for the O partial derivative of the extended model-free
1065 formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.
1066
1067 The formula is
1068
1069 _n_
1070 dJ(w) 2 \ dci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1071 ----- = - > --- . ti | ------------ + ------------------------- + ------------------------- |.
1072 dOj 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1073 i=m
1074 """
1075
1076 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)
1077
1078
1079
1080
1081
1082
1083
1084
1086 """Spectral density gradient.
1087
1088 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1089 formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.
1090
1091 The formula is
1092
1093 _n_
1094 dJ(w) 2 \ / S2s (1 - S2s)(ts + ti).ts \
1095 ----- = - > ci . ti | ------------ + ------------------------- |.
1096 dS2f 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1097 i=m
1098 """
1099
1100 return 0.4 * sum(data.ci * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
1101
1102
1103
1104
1106 """Spectral density gradient.
1107
1108 Calculate the spectral desity values for the S2f partial derivative of the extended model-free
1109 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1110
1111 The formula is
1112
1113 _n_
1114 dJ(w) 2 \ / S2s (tf + ti).tf (1 - S2s)(ts + ti).ts \
1115 ----- = - > ci . ti | ------------ - ------------------------- + ------------------------- |.
1116 dS2f 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1117 i=m
1118 """
1119
1120 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)
1121
1122
1123
1124
1125
1126
1127
1128
1130 """Spectral density gradient.
1131
1132 Calculate the spectral desity values for the S2s partial derivative of the extended model-free
1133 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1134 tensor parameters.
1135
1136 The formula is
1137
1138 _n_
1139 dJ(w) 2 \ / 1 (ts + ti).ts \
1140 ----- = - S2f > ci . ti | ------------ - ------------------------- |.
1141 dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1142 i=m
1143 """
1144
1145 return 0.4 * params[data.s2f_i] * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
1146
1147
1148
1149
1150
1151
1152
1153
1155 """Spectral density gradient.
1156
1157 Calculate the spectral desity values for the tf partial derivative of the extended model-free
1158 formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.
1159
1160 The formula is
1161
1162 _n_
1163 dJ(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1164 ----- = - (1 - S2f) > ci . ti^2 -----------------------------.
1165 dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1166 i=m
1167 """
1168
1169 return 0.4 * data.one_s2f * sum(data.ci * data.fact_djw_dtf, axis=2)
1170
1171
1172
1173
1174
1175
1176
1177
1179 """Spectral density gradient.
1180
1181 Calculate the spectral desity values for the ts partial derivative of the extended model-free
1182 formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
1183 tensor parameters.
1184
1185 The formula is
1186
1187 _n_
1188 dJ(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1189 ----- = - S2f(1 - S2s) > ci . ti^2 -----------------------------.
1190 dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1191 i=m
1192 """
1193
1194 return 0.4 * data.s2f_s2 * sum(data.ci * data.fact_djw_dts, axis=2)
1195
1196
1197
1198
1199
1200
1201
1202
1203 """
1204 The spectral density Hessians
1205 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1206
1207 Data structure: data.d2jw
1208 Dimension: 4D, (number of NMR frequencies, 5 spectral density frequencies, model-free
1209 parameters, model-free parameters)
1210 Type: Numeric 4D matrix, Float64
1211 Dependencies: None
1212 Required by: data.d2ri
1213
1214
1215 Formulae
1216 ~~~~~~~~
1217
1218 Original: Model-free parameter - Model-free parameter
1219 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1220
1221 _n_
1222 d2J(w) 2 \ / dti dti / 3 - (w.ti)^2 (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \
1223 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1224 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1225 i=m
1226
1227 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1228 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1229 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1230
1231
1232 d2ci / S2 (1 - S2)(te + ti)te \ \
1233 + ------- ti | ------------ + ------------------------- | |,
1234 dDj.dDk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1235
1236
1237 _n_
1238 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1239 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1240 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1241 i=m
1242
1243 d2ci / S2 (1 - S2)(te + ti)te \ \
1244 + ------- ti | ------------ + ------------------------- | |,
1245 dDj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1246
1247
1248 _n_
1249 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1250 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
1251 dDj.dS2 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1252 i=m
1253
1254 dci / 1 (te + ti)te \ \
1255 + --- . ti | ------------ - ------------------------- | |,
1256 dDj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1257
1258
1259 _n_
1260 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
1261 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1262 dDj.dte 5 /__ \ dDj ((te + ti)^2 + (w.te.ti)^2)^3 dDj ((te + ti)^2 + (w.te.ti)^2)^2 /
1263 i=m
1264
1265
1266 _n_
1267 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
1268 ------- = - > ------- . ti | ------------ + ------------------------- |,
1269 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1270 i=m
1271
1272
1273 _n_
1274 d2J(w) 2 \ dci / 1 (te + ti)te \
1275 ------- = - > --- . ti | ------------ - ------------------------- |,
1276 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
1277 i=m
1278
1279
1280 _n_
1281 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
1282 ------- = - (1 - S2) > --- . ti^2 -----------------------------,
1283 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
1284 i=m
1285
1286
1287 d2J(w)
1288 ------ = 0,
1289 dS2**2
1290
1291
1292 _n_
1293 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
1294 ------- = - - > ci . ti^2 -----------------------------,
1295 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
1296 i=m
1297
1298
1299 _n_
1300 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
1301 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
1302 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
1303 i=m
1304
1305
1306 Original: Other parameters
1307 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
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
1337 _n_
1338 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
1339 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1340 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1341 i=m
1342
1343 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1344 + (S2f - S2)ts^2 ---------------------------------------------------- |
1345 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1346
1347
1348 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1349 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
1350 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1351
1352
1353 (ts + ti)^2 - (w.ts.ti)^2 \
1354 + (S2f - S2)ts^2 ----------------------------- |
1355 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1356
1357
1358 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1359 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1360 dDj.dDk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1361
1362
1363 _n_
1364 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1365 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
1366 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1367 i=m
1368
1369 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
1370 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1371 dDj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1372
1373
1374 _n_
1375 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1376 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
1377 dDj.dS2 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1378 i=m
1379
1380 dci / 1 (ts + ti)ts \ \
1381 + --- . ti | ------------ - ------------------------- | |,
1382 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1383
1384
1385 _n_
1386 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1387 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
1388 dDj.dS2f 5 /__ \ dDj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1389 i=m
1390
1391 dci / (tf + ti)tf (ts + ti)ts \ \
1392 + --- . ti | ------------------------- - ------------------------- | |,
1393 dDj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1394
1395
1396 _n_
1397 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1398 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1399 dDj.dtf 5 /__ \ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3 dDj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1400 i=m
1401
1402
1403 _n_
1404 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1405 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1406 dDj.dts 5 /__ \ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3 dDj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1407 i=m
1408
1409
1410 _n_
1411 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
1412 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1413 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1414 i=m
1415
1416
1417 _n_
1418 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1419 ------- = - > --- . ti | ------------ - ------------------------- |,
1420 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1421 i=m
1422
1423
1424 _n_
1425 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
1426 -------- = - - > --- . ti | ------------------------- - ------------------------- |,
1427 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1428 i=m
1429
1430
1431 _n_
1432 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1433 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1434 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1435 i=m
1436
1437
1438 _n_
1439 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1440 ------- = - (S2f - S2) > --- . ti^2 -----------------------------,
1441 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1442 i=m
1443
1444
1445 d2J(w) d2J(w) d2J(w)
1446 ------ = 0, -------- = 0, ------- = 0,
1447 dS2**2 dS2.dS2f dS2.dtf
1448
1449
1450 _n_
1451 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1452 ------- = - - > ci . ti^2 -----------------------------,
1453 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1454 i=m
1455
1456
1457 d2J(w)
1458 ------- = 0,
1459 dS2f**2
1460
1461
1462 _n_
1463 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1464 -------- = - - > ci . ti^2 -----------------------------,
1465 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1466 i=m
1467
1468
1469 _n_
1470 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1471 -------- = - > ci . ti^2 -----------------------------,
1472 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1473 i=m
1474
1475
1476 _n_
1477 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1478 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1479 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1480 i=m
1481
1482
1483 d2J(w)
1484 ------- = 0,
1485 dtf.dts
1486
1487
1488 _n_
1489 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1490 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------,
1491 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1492 i=m
1493
1494
1495 Extended: Other parameters
1496 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
1497
1498 d2J(w) d2J(w) d2J(w)
1499 --------- = 0, --------- = 0, ------- = 0,
1500 dS2f.dRex dS2f.dcsa dS2f.dr
1501
1502
1503 d2J(w) d2J(w) d2J(w)
1504 -------- = 0, -------- = 0, ------ = 0,
1505 dS2.dRex dS2.dcsa dS2.dr
1506
1507
1508 d2J(w) d2J(w) d2J(w)
1509 -------- = 0, -------- = 0, ------ = 0,
1510 dtf.dRex dtf.dcsa dtf.dr
1511
1512
1513 d2J(w) d2J(w) d2J(w)
1514 -------- = 0, -------- = 0, ------ = 0,
1515 dts.dRex dts.dcsa dts.dr
1516
1517
1518 d2J(w) d2J(w) d2J(w)
1519 ------- = 0, --------- = 0, ------- = 0,
1520 dRex**2 dRex.dcsa dRex.dr
1521
1522
1523 d2J(w) d2J(w)
1524 ------- = 0, ------- = 0,
1525 dcsa**2 dcsa.dr
1526
1527
1528 d2J(w)
1529 ------ = 0.
1530 dr**2
1531
1532
1533 Extended 2: Model-free parameter - Model-free parameter
1534 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1535
1536 _n_
1537 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
1538 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
1539 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
1540 i=m
1541
1542 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
1543 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
1544 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
1545
1546
1547 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
1548 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
1549 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
1550
1551
1552 (ts + ti)^2 - (w.ts.ti)^2 \
1553 + S2f(1 - S2s)ts^2 ----------------------------- |
1554 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1555
1556
1557 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1558 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1559 dDj.dDk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1560
1561
1562 _n_
1563 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1564 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
1565 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1566 i=m
1567
1568 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
1569 + ------- . ti | ------------ + ------------------------- + ------------------------- | |,
1570 dDj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1571
1572
1573 _n_
1574 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1575 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
1576 dDj.dS2f 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1577 i=m
1578
1579 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
1580 + --- . ti | ------------ - ------------------------- + ------------------------- | |,
1581 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1582
1583
1584 _n_
1585 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
1586 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
1587 dDj.dS2s 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1588 i=m
1589
1590 dci / 1 (ts + ti)ts \ \
1591 + --- . ti | ------------ - ------------------------- | |,
1592 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
1593
1594
1595 _n_
1596 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
1597 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1598 dDj.dtf 5 /__ \ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3 dDj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
1599 i=m
1600
1601
1602 _n_
1603 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
1604 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |,
1605 dDj.dts 5 /__ \ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3 dDj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
1606 i=m
1607
1608
1609 _n_
1610 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
1611 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |,
1612 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1613 i=m
1614
1615
1616 _n_
1617 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
1618 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |,
1619 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1620 i=m
1621
1622
1623 _n_
1624 d2J(w) 2 \ dci / 1 (ts + ti)ts \
1625 -------- = - S2f > --- . ti | ------------ - ------------------------- |,
1626 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1627 i=m
1628
1629
1630 _n_
1631 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
1632 ------- = - (1 - S2f) > --- . ti^2 -----------------------------,
1633 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
1634 i=m
1635
1636
1637 _n_
1638 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
1639 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------,
1640 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
1641 i=m
1642
1643
1644 d2J(w)
1645 ------- = 0,
1646 dS2f**2
1647
1648
1649 _n_
1650 d2J(w) 2 \ / 1 (ts + ti).ts \
1651 --------- = - > ci . ti | ------------ - ------------------------- |,
1652 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
1653 i=m
1654
1655
1656 _n_
1657 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
1658 -------- = - - > ci . ti^2 -----------------------------,
1659 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
1660 i=m
1661
1662
1663 _n_
1664 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1665 -------- = - (1 - S2s) > ci . ti^2 -----------------------------,
1666 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1667 i=m
1668
1669
1670 d2J(w) d2J(w)
1671 ------- = 0, -------- = 0,
1672 dS2s**2 dS2s.dtf
1673
1674
1675 _n_
1676 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
1677 -------- = - - S2f > ci . ti^2 -----------------------------,
1678 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
1679 i=m
1680
1681
1682 _n_
1683 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
1684 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------,
1685 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
1686 i=m
1687
1688
1689 d2J(w)
1690 ------- = 0,
1691 dtf.dts
1692
1693
1694 _n_
1695 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
1696 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
1697 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
1698 i=m
1699
1700
1701
1702 Extended 2: Other parameters
1703 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1704
1705 d2J(w) d2J(w) d2J(w)
1706 --------- = 0, --------- = 0, ------- = 0,
1707 dS2f.dRex dS2f.dcsa dS2f.dr
1708
1709
1710 d2J(w) d2J(w) d2J(w)
1711 --------- = 0, --------- = 0, ------- = 0,
1712 dS2s.dRex dS2s.dcsa dS2s.dr
1713
1714
1715 d2J(w) d2J(w) d2J(w)
1716 -------- = 0, -------- = 0, ------ = 0,
1717 dtf.dRex dtf.dcsa dtf.dr
1718
1719
1720 d2J(w) d2J(w) d2J(w)
1721 -------- = 0, -------- = 0, ------ = 0,
1722 dts.dRex dts.dcsa dts.dr
1723
1724
1725 d2J(w) d2J(w) d2J(w)
1726 ------- = 0, --------- = 0, ------- = 0,
1727 dRex**2 dRex.dcsa dRex.dr
1728
1729
1730 d2J(w) d2J(w)
1731 ------- = 0, ------- = 0,
1732 dcsa**2 dcsa.dr
1733
1734
1735 d2J(w)
1736 ------ = 0.
1737 dr**2
1738 """
1739
1740
1741
1742
1743
1744
1745
1747 """Spectral density Hessian.
1748
1749 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1750 model-free formula with no parameters {} together with diffusion tensor parameters.
1751
1752 The model-free Hessian is
1753
1754 _n_
1755 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1756 ------- = - > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1757 dDj.dDk 5 /__ \ dDj dDk (1 + (w.ti)^2)^3 dDj.dDk (1 + (w.ti)^2)^2 /
1758 i=m
1759 """
1760
1761 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)
1762
1763
1765 """Spectral density Hessian.
1766
1767 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1768 model-free formula with no parameters {} together with diffusion tensor parameters.
1769
1770 The model-free Hessian is
1771
1772 _n_
1773 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1774 ------- = - > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1775 dDj.dDk 5 /__ \ dDj dDk (1 + (w.ti)^2)^3 \ dDj dDk dDk dDj dDj.dDk / (1 + (w.ti)^2)^2 dDj.dDk 1 + (w.ti)^2 /
1776 i=m
1777 """
1778
1779 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)
1780
1781
1782
1783
1785 """Spectral density Hessian.
1786
1787 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1788 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1789
1790 The model-free Hessian is
1791
1792 _n_
1793 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 d2ti 1 - (w.ti)^2 \
1794 ------- = - S2 > ci | -2 --- . --- w^2.ti ---------------- + ------- ---------------- |.
1795 dDj.dDk 5 /__ \ dDj dDk (1 + (w.ti)^2)^3 dDj.dDk (1 + (w.ti)^2)^2 /
1796 i=m
1797 """
1798
1799 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)
1800
1801
1803 """Spectral density Hessian.
1804
1805 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1806 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1807
1808 The model-free Hessian is
1809
1810 _n_
1811 d2J(w) 2 \ / dti dti 3 - (w.ti)^2 / dti dci dti dci d2ti \ 1 - (w.ti)^2 d2ci 1 \
1812 ------- = - S2 > | -2ci --- . --- w^2.ti ---------------- + | --- . --- + --- . --- + ci ------- | ---------------- + ------- ti ------------ |.
1813 dDj.dDk 5 /__ \ dDj dDk (1 + (w.ti)^2)^3 \ dDj dDk dDk dDj dDj.dDk / (1 + (w.ti)^2)^2 dDj.dDk 1 + (w.ti)^2 /
1814 i=m
1815 """
1816
1817 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)
1818
1819
1820
1821
1823 """Spectral density Hessian.
1824
1825 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1826 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1827
1828 The model-free Hessian is
1829
1830 _n_
1831 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 \
1832 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1833 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1834 i=m
1835
1836 d2ti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \ \
1837 + ------- | S2 ---------------- + (1 - S2)te^2 ----------------------------- | |.
1838 dDj.dDk \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 / /
1839 """
1840
1841
1842 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)
1843
1844
1845 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.one_s2 * data.fact_te_djw_dti)
1846
1847 return 0.4 * sum(data.ci * (a + b), axis=2)
1848
1849
1851 """Spectral density Hessian.
1852
1853 Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
1854 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1855
1856 The model-free Hessian is
1857
1858 _n_
1859 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 \
1860 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2)te^2 ---------------------------------------------------- |
1861 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((te + ti)^2 + (w.te.ti)^2)^3 /
1862 i=m
1863
1864 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1865 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1866 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1867
1868
1869 d2ci / S2 (1 - S2)(te + ti)te \ \
1870 + ------- ti | ------------ + ------------------------- | |.
1871 dDj.dDk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
1872 """
1873
1874
1875 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)
1876
1877
1878 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)
1879
1880
1881 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.one_s2 * data.fact_te)
1882
1883 return 0.4 * sum(a + b + c, axis=2)
1884
1885
1886
1887
1888
1889
1890
1891
1892
1894 """Spectral density Hessian.
1895
1896 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1897 model-free formula with no parameters {} together with diffusion tensor parameters.
1898
1899 The model-free Hessian is
1900
1901 _n_
1902 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1903 ------- = - > --- . --- . ----------------.
1904 dDj.dOj 5 /__ dOj dDj (1 + (w.ti)^2)^2
1905 i=m
1906 """
1907
1908 return 0.4 * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1909
1910
1912 """Spectral density Hessian.
1913
1914 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1915 model-free formula with no parameters {} together with diffusion tensor parameters.
1916
1917 The model-free Hessian is
1918
1919 _n_
1920 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1921 ------- = - > | --- . --- . ---------------- + ------- ti ------------ |.
1922 dDj.dOj 5 /__ \ dOj dDj (1 + (w.ti)^2)^2 dDj.dOj 1 + (w.ti)^2 /
1923 i=m
1924 """
1925
1926 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)
1927
1928
1929
1930
1932 """Spectral density Hessian.
1933
1934 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1935 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1936
1937 The model-free Hessian is
1938
1939 _n_
1940 d2J(w) 2 \ dci dti 1 - (w.ti)^2
1941 ------- = - S2 > --- . --- . ----------------.
1942 dDj.dOj 5 /__ dOj dDj (1 + (w.ti)^2)^2
1943 i=m
1944 """
1945
1946 return 0.4 * params[data.s2_i] * sum(data.dci[j] * data.dti[k] * data.fact_ti_djw_dti, axis=2)
1947
1948
1950 """Spectral density Hessian.
1951
1952 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1953 model-free formula with the parameter {S2} together with diffusion tensor parameters.
1954
1955 The model-free Hessian is
1956
1957 _n_
1958 d2J(w) 2 \ / dci dti 1 - (w.ti)^2 d2ci 1 \
1959 ------- = - S2 > | --- . --- . ---------------- + ------- ti ------------ |.
1960 dDj.dOj 5 /__ \ dOj dDj (1 + (w.ti)^2)^2 dDj.dOj 1 + (w.ti)^2 /
1961 i=m
1962 """
1963
1964 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)
1965
1966
1967
1968
1970 """Spectral density Hessian.
1971
1972 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1973 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1974
1975 The model-free Hessian is
1976
1977 _n_
1978 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1979 ------- = - > --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |.
1980 dDj.dOj 5 /__ dOj dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1981 i=m
1982 """
1983
1984 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)
1985
1986
1988 """Spectral density Hessian.
1989
1990 Calculate the spectral desity values for the Dj - Oj double partial derivative of the original
1991 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
1992
1993 The model-free Hessian is
1994
1995 _n_
1996 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
1997 ------- = - > | --- . --- | S2 ---------------- + (1 - S2)te^2 ----------------------------- |
1998 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
1999 i=m
2000
2001 d2ci / S2 (1 - S2)(te + ti)te \ \
2002 + ------- ti | ------------ + ------------------------- | |.
2003 dDj.dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
2004 """
2005
2006 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)
2007
2008
2009
2010
2011
2012
2013
2014
2016 """Spectral density Hessian.
2017
2018 Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
2019 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2020
2021 The model-free Hessian is
2022
2023 _n_
2024 d2J(w) 2 \ dti 1 - (w.ti)^2
2025 ------- = - > ci . --- ----------------.
2026 dDj.dS2 5 /__ dDj (1 + (w.ti)^2)^2
2027 i=m
2028 """
2029
2030 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti, axis=2)
2031
2032
2034 """Spectral density Hessian.
2035
2036 Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
2037 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2038
2039 The model-free Hessian is
2040
2041 _n_
2042 d2J(w) 2 \ / dti 1 - (w.ti)^2 dci 1 \
2043 ------- = - > | ci . --- ---------------- + --- . ti ------------ |.
2044 dDj.dS2 5 /__ \ dDj (1 + (w.ti)^2)^2 dDj 1 + (w.ti)^2 /
2045 i=m
2046 """
2047
2048 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ti_djw_dti + data.dci[k] * data.ti * data.fact_ti, axis=2)
2049
2050
2051
2052
2054 """Spectral density Hessian.
2055
2056 Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
2057 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2058
2059 The model-free Hessian is
2060
2061 _n_
2062 d2J(w) 2 \ dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2063 ------- = - > ci . --- | ---------------- - te^2 ----------------------------- |.
2064 dDj.dS2 5 /__ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2065 i=m
2066 """
2067
2068 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_te_djw_dti), axis=2)
2069
2070
2072 """Spectral density Hessian.
2073
2074 Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
2075 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2076
2077 The model-free Hessian is
2078
2079 _n_
2080 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (te + ti)^2 - (w.te.ti)^2 \
2081 ------- = - > | ci . --- | ---------------- - te^2 ----------------------------- |
2082 dDj.dS2 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((te + ti)^2 + (w.te.ti)^2)^2 /
2083 i=m
2084
2085 dci / 1 (te + ti)te \ \
2086 + --- . ti | ------------ - ------------------------- | |.
2087 dDj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / /
2088 """
2089
2090 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)
2091
2092
2093
2094
2095
2096
2097
2098
2100 """Spectral density Hessian.
2101
2102 Calculate the spectral desity values for the Dj - te double partial derivative of the original
2103 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2104
2105 The model-free Hessian is
2106
2107 _n_
2108 d2J(w) 4 \ dti (te + ti)^2 - 3(w.te.ti)^2
2109 ------- = - (1 - S2) . te > ci . --- . ti . (te + ti) -----------------------------.
2110 dDj.dte 5 /__ dDj ((te + ti)^2 + (w.te.ti)^2)^3
2111 i=m
2112 """
2113
2114 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)
2115
2116
2118 """Spectral density Hessian.
2119
2120 Calculate the spectral desity values for the Dj - te double partial derivative of the original
2121 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2122
2123 The model-free Hessian is
2124
2125 _n_
2126 d2J(w) 2 \ / dti (te + ti)^2 - 3(w.te.ti)^2 dci (te + ti)^2 - (w.te.ti)^2 \
2127 ------- = - (1 - S2) > | 2ci . --- . te . ti . (te + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2128 dDj.dte 5 /__ \ dDj ((te + ti)^2 + (w.te.ti)^2)^3 dDj ((te + ti)^2 + (w.te.ti)^2)^2 /
2129 i=m
2130 """
2131
2132 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)
2133
2134
2135
2136
2137
2138
2139
2140
2142 """Spectral density Hessian.
2143
2144 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2145 original model-free formula with no parameters {} together with diffusion tensor parameters.
2146
2147 The model-free Hessian is
2148
2149 _n_
2150 d2J(w) 2 \ d2ci ti
2151 ------- = - > ------- . ------------.
2152 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2153 i=m
2154 """
2155
2156 return 0.4 * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2157
2158
2159
2160
2162 """Spectral density Hessian.
2163
2164 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2165 original model-free formula with the parameter {S2} together with diffusion tensor parameters.
2166
2167 The model-free Hessian is
2168
2169 _n_
2170 d2J(w) 2 \ d2ci ti
2171 ------- = - S2 > ------- . ------------.
2172 dOj.dOk 5 /__ dOj.dOk 1 + (w.ti)^2
2173 i=m
2174 """
2175
2176 return 0.4 * params[data.s2_i] * sum(data.d2ci[j, k] * data.ti * data.fact_ti, axis=2)
2177
2178
2179
2180
2182 """Spectral density Hessian.
2183
2184 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2185 original model-free formula with the parameters {S2, te} together with diffusion tensor
2186 parameters.
2187
2188 The model-free Hessian is
2189
2190 _n_
2191 d2J(w) 2 \ d2ci / S2 (1 - S2)(te + ti)te \
2192 ------- = - > ------- . ti | ------------ + ------------------------- |.
2193 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2194 i=m
2195 """
2196
2197 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)
2198
2199
2200
2201
2202
2203
2204
2205
2207 """Spectral density Hessian.
2208
2209 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2210 model-free formula with the parameter {S2} together with diffusion tensor parameters.
2211
2212 The model-free Hessian is
2213
2214 _n_
2215 d2J(w) 2 \ dci 1
2216 ------- = - > --- . ti ------------.
2217 dOj.dS2 5 /__ dOj 1 + (w.ti)^2
2218 i=m
2219 """
2220
2221 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ti, axis=2)
2222
2223
2224
2225
2227 """Spectral density Hessian.
2228
2229 Calculate the spectral desity values for the Oj - S2 double partial derivative of the original
2230 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2231
2232 The model-free Hessian is
2233
2234 _n_
2235 d2J(w) 2 \ dci / 1 (te + ti)te \
2236 ------- = - > --- . ti | ------------ - ------------------------- |.
2237 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 /
2238 i=m
2239 """
2240
2241 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_te), axis=2)
2242
2243
2244
2245
2246
2247
2248
2249
2251 """Spectral density Hessian.
2252
2253 Calculate the spectral desity values for the Oj - te double partial derivative of the original
2254 model-free formula with the parameters {S2, te} together with diffusion tensor parameters.
2255
2256 The model-free Hessian is
2257
2258 _n_
2259 d2J(w) 2 \ dci (te + ti)^2 - (w.te.ti)^2
2260 ------- = - (1 - S2) > --- . ti^2 -----------------------------.
2261 dOj.dte 5 /__ dOj ((te + ti)^2 + (w.te.ti)^2)^2
2262 i=m
2263 """
2264
2265 return 0.4 * data.one_s2 * sum(data.dci[k] * data.fact_djw_dte, axis=2)
2266
2267
2268
2269
2270
2271
2272
2273
2275 """Spectral density Hessian.
2276
2277 Calculate the spectral desity values for the S2 - te double partial derivative of the original
2278 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2279
2280 The model-free Hessian is
2281
2282 _n_
2283 d2J(w) 2 \ (te + ti)^2 - (w.te.ti)^2
2284 ------- = - - > ci . ti^2 -----------------------------.
2285 dS2.dte 5 /__ ((te + ti)^2 + (w.te.ti)^2)^2
2286 i=m
2287 """
2288
2289 return -0.4 * sum(data.ci * data.fact_djw_dte, axis=2)
2290
2291
2292
2293
2294
2295
2296
2297
2299 """Spectral density Hessian.
2300
2301 Calculate the spectral desity values for the te - te double partial derivative of the original
2302 model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.
2303
2304 The model-free Hessian is
2305
2306 _n_
2307 d2J(w) 4 \ (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
2308 ------ = - - (1 - S2) > ci . ti^2 -----------------------------------------------------.
2309 dte**2 5 /__ ((te + ti)^2 + (w.te.ti)^2)^3
2310 i=m
2311 """
2312
2313 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)
2314
2315
2316
2317
2318
2319
2320
2321
2323 """Spectral density Hessian.
2324
2325 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
2326 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2327
2328 The model-free Hessian is
2329
2330 _n_
2331 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 \
2332 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2333 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2334 i=m
2335
2336 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2337 + ------- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- | |.
2338 dDj.dDk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2339 """
2340
2341
2342 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)
2343
2344
2345 b = data.d2ti[j, k] * (params[data.s2_i] * data.fact_ti_djw_dti + data.s2f_s2 * data.fact_ts_djw_dti)
2346
2347 return 0.4 * sum(data.ci * (a + b), axis=2)
2348
2349
2351 """Spectral density Hessian.
2352
2353 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
2354 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2355
2356 The model-free Hessian is
2357
2358 _n_
2359 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 \
2360 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (S2f - S2)ts^2 ---------------------------------------------------- |
2361 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2362 i=m
2363
2364 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2365 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2366 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2367
2368
2369 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2370 + ------- . ti | ------------ + ------------------------- | |.
2371 dDj.dDk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2372 """
2373
2374
2375 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)
2376
2377
2378 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)
2379
2380
2381 c = data.d2ci[j, k] * data.ti * (params[data.s2_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
2382
2383 return 0.4 * sum(a + b + c, axis=2)
2384
2385
2386
2387
2388
2390 """Spectral density Hessian.
2391
2392 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
2393 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2394 parameters.
2395
2396 The model-free Hessian is
2397
2398 _n_
2399 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
2400 ------- = - > ci | -2 --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2401 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2402 i=m
2403
2404 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2405 + (S2f - S2)ts^2 ---------------------------------------------------- |
2406 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2407
2408
2409 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
2410 + ------- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- | |.
2411 dDj.dDk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
2412 """
2413
2414
2415 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)
2416
2417
2418 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)
2419
2420 return 0.4 * sum(data.ci * (a + b), axis=2)
2421
2422
2424 """Spectral density Hessian.
2425
2426 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
2427 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2428 parameters.
2429
2430 The model-free Hessian is
2431
2432 _n_
2433 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
2434 ------- = - > | -2ci --- . --- | S2.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
2435 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
2436 i=m
2437
2438 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
2439 + (S2f - S2)ts^2 ---------------------------------------------------- |
2440 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
2441
2442
2443 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
2444 + | --- . --- + --- . --- + ci ------- | | S2 ---------------- + (1 - S2f)tf^2 -----------------------------
2445 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
2446
2447
2448 (ts + ti)^2 - (w.ts.ti)^2 \
2449 + (S2f - S2)ts^2 ----------------------------- |
2450 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2451
2452
2453 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2454 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2455 dDj.dDk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2456 """
2457
2458
2459 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)
2460
2461
2462 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)
2463
2464
2465 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)
2466
2467 return 0.4 * sum(a + b + c, axis=2)
2468
2469
2470
2471
2472
2473
2474
2475
2477 """Spectral density Hessian.
2478
2479 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
2480 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2481
2482 The model-free Hessian is
2483
2484 _n_
2485 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2486 ------- = - > --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |.
2487 dDj.dOj 5 /__ dOj dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2488 i=m
2489 """
2490
2491 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)
2492
2493
2495 """Spectral density Hessian.
2496
2497 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
2498 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2499
2500 The model-free Hessian is
2501
2502 _n_
2503 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2504 ------- = - > | --- . --- | S2 ---------------- + (S2f - S2)ts^2 ----------------------------- |
2505 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2506 i=m
2507
2508 d2ci / S2 (S2f - S2)(ts + ti)ts \ \
2509 + ------- . ti | ------------ + ------------------------- | |.
2510 dDj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2511 """
2512
2513 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)
2514
2515
2516
2517
2519 """Spectral density Hessian.
2520
2521 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
2522 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2523 parameters.
2524
2525 The model-free Hessian is
2526
2527 _n_
2528 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2529 ------- = - > --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |.
2530 dDj.dOj 5 /__ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2531 i=m
2532 """
2533
2534 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)
2535
2536
2538 """Spectral density Hessian.
2539
2540 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
2541 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2542 parameters.
2543
2544 The model-free Hessian is
2545
2546 _n_
2547 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2548 ------- = - > | --- . --- | S2 ---------------- + (1 - S2f)tf^2 ----------------------------- + (S2f - S2)ts^2 ----------------------------- |
2549 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2550 i=m
2551
2552 d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \ \
2553 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
2554 dDj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2555 """
2556
2557 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)
2558
2559
2560
2561
2562
2563
2564
2565
2567 """Spectral density Hessian.
2568
2569 Calculate the spectral desity values for the Dj - S2 double partial derivative of the extended
2570 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2571 diffusion tensor parameters.
2572
2573 The model-free Hessian is
2574
2575 _n_
2576 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2577 ------- = - > ci . --- | ---------------- - ts^2 ----------------------------- |.
2578 dDj.dS2 5 /__ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2579 i=m
2580 """
2581
2582 return 0.4 * sum(data.ci * data.dti[k] * (data.fact_ti_djw_dti - data.fact_ts_djw_dti), axis=2)
2583
2584
2586 """Spectral density Hessian.
2587
2588 Calculate the spectral desity values for the Dj - S2 double partial derivative of the extended
2589 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2590 diffusion tensor parameters.
2591
2592 The model-free Hessian is
2593
2594 _n_
2595 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2596 ------- = - > | ci . --- | ---------------- - ts^2 ----------------------------- |
2597 dDj.dS2 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2598 i=m
2599
2600 dci / 1 (ts + ti)ts \ \
2601 + --- . ti | ------------ - ------------------------- | |.
2602 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2603 """
2604
2605 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)
2606
2607
2608
2609
2610
2611
2612
2613
2615 """Spectral density Hessian.
2616
2617 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
2618 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2619
2620 The model-free Hessian is
2621
2622 _n_
2623 d2J(w) 2 \ dti (ts + ti)^2 - (w.ts.ti)^2
2624 -------- = - > ci . --- ts^2 -----------------------------.
2625 dDj.dS2f 5 /__ dDj ((ts + ti)^2 + (w.ts.ti)^2)^2
2626 i=m
2627 """
2628
2629 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti, axis=2)
2630
2631
2633 """Spectral density Hessian.
2634
2635 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
2636 model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.
2637
2638 The model-free Hessian is
2639
2640 _n_
2641 d2J(w) 2 \ / dti (ts + ti)^2 - (w.ts.ti)^2 dci (ts + ti)ts \
2642 -------- = - > | ci . --- ts^2 ----------------------------- + --- . ti ------------------------- |.
2643 dDj.dS2f 5 /__ \ dDj ((ts + ti)^2 + (w.ts.ti)^2)^2 dDj (ts + ti)^2 + (w.ts.ti)^2 /
2644 i=m
2645 """
2646
2647 return 0.4 * sum(data.ci * data.dti[k] * data.fact_ts_djw_dti + data.dci[k] * data.ti * data.fact_ts, axis=2)
2648
2649
2650
2651
2653 """Spectral density Hessian.
2654
2655 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
2656 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2657 parameters.
2658
2659 The model-free Hessian is
2660
2661 _n_
2662 d2J(w) 2 \ dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2663 -------- = - - > ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |.
2664 dDj.dS2f 5 /__ dDj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2665 i=m
2666 """
2667
2668 return -0.4 * sum(data.ci * data.dti[k] * (data.fact_tf_djw_dti - data.fact_ts_djw_dti), axis=2)
2669
2670
2672 """Spectral density Hessian.
2673
2674 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
2675 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2676 parameters.
2677
2678 The model-free Hessian is
2679
2680 _n_
2681 d2J(w) 2 \ / dti / (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
2682 -------- = - - > | ci . --- | tf^2 ----------------------------- - ts^2 ----------------------------- |
2683 dDj.dS2f 5 /__ \ dDj \ ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2684 i=m
2685
2686 dci / (tf + ti)tf (ts + ti)ts \ \
2687 + --- . ti | ------------------------- - ------------------------- | |.
2688 dDj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
2689 """
2690
2691 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)
2692
2693
2694
2695
2696
2697
2698
2699
2701 """Spectral density Hessian.
2702
2703 Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
2704 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2705 parameters.
2706
2707 The model-free Hessian is
2708
2709 _n_
2710 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
2711 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
2712 dDj.dtf 5 /__ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3
2713 i=m
2714 """
2715
2716 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)
2717
2718
2720 """Spectral density Hessian.
2721
2722 Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
2723 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2724 parameters.
2725
2726 The model-free Hessian is
2727
2728 _n_
2729 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
2730 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2731 dDj.dtf 5 /__ \ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3 dDj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
2732 i=m
2733 """
2734
2735 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)
2736
2737
2738
2739
2740
2741
2742
2743
2745 """Spectral density Hessian.
2746
2747 Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
2748 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2749 diffusion tensor parameters.
2750
2751 The model-free Hessian is
2752
2753 _n_
2754 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
2755 ------- = - (S2f - S2) . ts > ci . --- . ti . (ts + ti) -----------------------------.
2756 dDj.dts 5 /__ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3
2757 i=m
2758 """
2759
2760 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)
2761
2762
2764 """Spectral density Hessian.
2765
2766 Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
2767 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2768 diffusion tensor parameters.
2769
2770 The model-free Hessian is
2771
2772 _n_
2773 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
2774 ------- = - (S2f - S2) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
2775 dDj.dts 5 /__ \ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3 dDj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
2776 i=m
2777 """
2778
2779 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)
2780
2781
2782
2783
2784
2785
2786
2787
2789 """Spectral density Hessian.
2790
2791 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2792 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2793 parameters.
2794
2795 The model-free Hessian is
2796
2797 _n_
2798 d2J(w) 2 \ d2ci / S2 (S2f - S2)(ts + ti)ts \
2799 ------- = - > ------- . ti | ------------ + ------------------------- |.
2800 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2801 i=m
2802 """
2803
2804 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)
2805
2806
2807
2808
2810 """Spectral density Hessian.
2811
2812 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
2813 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2814 parameters.
2815
2816 The model-free Hessian is
2817
2818 _n_
2819 d2J(w) 2 \ d2ci / S2 (1 - S2f)(tf + ti)tf (S2f - S2)(ts + ti)ts \
2820 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
2821 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2822 i=m
2823 """
2824
2825 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)
2826
2827
2828
2829
2830
2831
2832
2833
2835 """Spectral density Hessian.
2836
2837 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
2838 model-free formula with the parameters {S2f, S2, ts} and {S2f, tf, S2, ts} together with
2839 diffusion tensor parameters.
2840
2841 The model-free Hessian is
2842
2843 _n_
2844 d2J(w) 2 \ dci / 1 (ts + ti)ts \
2845 ------- = - > --- . ti | ------------ - ------------------------- |.
2846 dOj.dS2 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2847 i=m
2848 """
2849
2850 return 0.4 * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
2851
2852
2853
2854
2855
2856
2857
2858
2860 """Spectral density Hessian.
2861
2862 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2863 extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
2864 parameters.
2865
2866 The model-free Hessian is
2867
2868 _n_
2869 d2J(w) 2 \ dci (ts + ti)ts
2870 -------- = - > --- . ti -------------------------.
2871 dOj.dS2f 5 /__ dOj (ts + ti)^2 + (w.ts.ti)^2
2872 i=m
2873 """
2874
2875 return 0.4 * sum(data.dci[k] * data.ti * data.fact_ts, axis=2)
2876
2877
2878
2879
2881 """Spectral density Hessian.
2882
2883 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
2884 extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2885 parameters.
2886
2887 The model-free Hessian is
2888
2889 _n_
2890 d2J(w) 2 \ dci / (tf + ti)tf (ts + ti)ts \
2891 -------- = - - > --- . ti | ------------------------- - ------------------------- |.
2892 dOj.dS2f 5 /__ dOj \ (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
2893 i=m
2894 """
2895
2896 return -0.4 * sum(data.dci[k] * data.ti * (data.fact_tf - data.fact_ts), axis=2)
2897
2898
2899
2900
2901
2902
2903
2904
2906 """Spectral density Hessian.
2907
2908 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
2909 model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
2910 parameters.
2911
2912 The model-free Hessian is
2913
2914 _n_
2915 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
2916 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
2917 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
2918 i=m
2919 """
2920
2921 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
2922
2923
2924
2925
2926
2927
2928
2929
2931 """Spectral density Hessian.
2932
2933 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
2934 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
2935 diffusion tensor parameters.
2936
2937 The model-free Hessian is
2938
2939 _n_
2940 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
2941 ------- = - (S2f - S2) > --- . ti^2 -----------------------------.
2942 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
2943 i=m
2944 """
2945
2946 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
2947
2948
2949
2950
2951
2952
2953
2954
2956 """Spectral density Hessian.
2957
2958 Calculate the spectral desity values for the S2 - ts double partial derivative of the extended
2959 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
2960 diffusion tensor parameters.
2961
2962 The model-free Hessian is
2963
2964 _n_
2965 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
2966 ------- = - - > ci . ti^2 -----------------------------.
2967 dS2.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
2968 i=m
2969 """
2970
2971 return -0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
2972
2973
2974
2975
2976
2977
2978
2979
2981 """Spectral density Hessian.
2982
2983 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
2984 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
2985 parameters.
2986
2987 The model-free Hessian is
2988
2989 _n_
2990 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
2991 -------- = - - > ci . ti^2 -----------------------------.
2992 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
2993 i=m
2994 """
2995
2996 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
2997
2998
2999
3000
3001
3002
3003
3004
3006 """Spectral density Hessian.
3007
3008 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3009 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3010 diffusion tensor parameters.
3011
3012 The model-free Hessian is
3013
3014 _n_
3015 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3016 -------- = - > ci . ti^2 -----------------------------.
3017 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3018 i=m
3019 """
3020
3021 return 0.4 * sum(data.ci * data.fact_djw_dts, axis=2)
3022
3023
3024
3025
3026
3027
3028
3029
3031 """Spectral density Hessian.
3032
3033 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3034 model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
3035 parameters.
3036
3037 The model-free Hessian is
3038
3039 _n_
3040 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3041 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3042 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3043 i=m
3044 """
3045
3046 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)
3047
3048
3049
3050
3051
3052
3053
3054
3056 """Spectral density Hessian.
3057
3058 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3059 model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
3060 diffusion tensor parameters.
3061
3062 The model-free Hessian is
3063
3064 _n_
3065 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3066 ------ = - - (S2f - S2) > ci . ti^2 -----------------------------------------------------.
3067 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3068 i=m
3069 """
3070
3071 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)
3072
3073
3074
3075
3076
3077
3078
3079
3081 """Spectral density Hessian.
3082
3083 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
3084 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3085
3086 The model-free Hessian is
3087
3088 _n_
3089 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 \
3090 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3091 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3092 i=m
3093
3094 d2ti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \ \
3095 + ------- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- | |.
3096 dDj.dDk \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 / /
3097 """
3098
3099
3100 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)
3101
3102
3103 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)
3104
3105 return 0.4 * sum(data.ci * (a + b), axis=2)
3106
3107
3109 """Spectral density Hessian.
3110
3111 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
3112 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3113
3114 The model-free Hessian is
3115
3116 _n_
3117 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 \
3118 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3119 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3120 i=m
3121
3122 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3123 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3124 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3125
3126
3127 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3128 + ------- . ti | ------------ + ------------------------- | |.
3129 dDj.dDk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3130 """
3131
3132
3133 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)
3134
3135
3136 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)
3137
3138
3139 c = data.d2ci[j, k] * data.ti * (params[data.s2f_i] * params[data.s2s_i] * data.fact_ti + data.s2f_s2 * data.fact_ts)
3140
3141 return 0.4 * sum(a + b + c, axis=2)
3142
3143
3144
3145
3147 """Spectral density Hessian.
3148
3149 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
3150 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3151 parameters.
3152
3153 The model-free Hessian is
3154
3155 _n_
3156 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
3157 ------- = - > ci | -2 --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3158 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3159 i=m
3160
3161 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3162 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3163 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3164
3165
3166 d2ti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3167 + ------- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3168 dDj.dDk \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3169 """
3170
3171
3172 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)
3173
3174
3175 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)
3176
3177 return 0.4 * sum(data.ci * (a + b), axis=2)
3178
3179
3181 """Spectral density Hessian.
3182
3183 Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
3184 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3185 parameters.
3186
3187 The model-free Hessian is
3188
3189 _n_
3190 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
3191 ------- = - > | -2ci --- . --- | S2f.S2s.w^2.ti ---------------- + (1 - S2f)tf^2 ----------------------------------------------------
3192 dDj.dDk 5 /__ \ dDj dDk \ (1 + (w.ti)^2)^3 ((tf + ti)^2 + (w.tf.ti)^2)^3
3193 i=m
3194
3195 (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \
3196 + S2f(1 - S2s)ts^2 ---------------------------------------------------- |
3197 ((ts + ti)^2 + (w.ts.ti)^2)^3 /
3198
3199
3200 / dti dci dti dci d2ti \ / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2
3201 + | --- . --- + --- . --- + ci ------- | | S2f.S2s ---------------- + (1 - S2f)tf^2 -----------------------------
3202 \ dDj dDk dDk dDj dDj.dDk / \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2
3203
3204
3205 (ts + ti)^2 - (w.ts.ti)^2 \
3206 + S2f(1 - S2s)ts^2 ----------------------------- |
3207 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3208
3209
3210 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3211 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3212 dDj.dDk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3213 """
3214
3215
3216 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)
3217
3218
3219 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)
3220
3221
3222 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)
3223
3224 return 0.4 * sum(a + b + c, axis=2)
3225
3226
3227
3228
3229
3230
3231
3232
3234 """Spectral density Hessian.
3235
3236 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
3237 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3238
3239 The model-free Hessian is
3240
3241 _n_
3242 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3243 ------- = - > --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3244 dDj.dOj 5 /__ dOj dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3245 i=m
3246 """
3247
3248 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)
3249
3250
3252 """Spectral density Hessian.
3253
3254 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
3255 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3256
3257 The model-free Hessian is
3258
3259 _n_
3260 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3261 ------- = - > | --- . --- | S2f.S2s ---------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3262 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3263 i=m
3264
3265 d2ci / S2f.S2s S2f(1 - S2s)(ts + ti)ts \ \
3266 + ------- . ti | ------------ + ------------------------- | |.
3267 dDj.dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3268 """
3269
3270 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)
3271
3272
3273
3274
3276 """Spectral density Hessian.
3277
3278 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
3279 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3280 parameters.
3281
3282 The model-free Hessian is
3283
3284 _n_
3285 d2J(w) 2 \ dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3286 ------- = - > --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |.
3287 dDj.dOj 5 /__ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3288 i=m
3289 """
3290
3291 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)
3292
3293
3295 """Spectral density Hessian.
3296
3297 Calculate the spectral desity values for the Dj - Oj double partial derivative of the extended
3298 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3299 parameters.
3300
3301 The model-free Hessian is
3302
3303 _n_
3304 d2J(w) 2 \ / dci dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3305 ------- = - > | --- . --- | S2f.S2s ---------------- + (1 - S2f)tf^2 ----------------------------- + S2f(1 - S2s)ts^2 ----------------------------- |
3306 dDj.dOj 5 /__ \ dOj dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3307 i=m
3308
3309 d2ci / S2f.S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \ \
3310 + ------- . ti | ------------ + ------------------------- + ------------------------- | |.
3311 dDj.dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3312 """
3313
3314 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)
3315
3316
3317
3318
3319
3320
3321
3322
3324 """Spectral density Hessian.
3325
3326 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
3327 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3328
3329 The model-free Hessian is
3330
3331 _n_
3332 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3333 -------- = - > ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |.
3334 dDj.dS2f 5 /__ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3335 i=m
3336 """
3337
3338 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)
3339
3340
3342 """Spectral density Hessian.
3343
3344 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
3345 model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.
3346
3347 The model-free Hessian is
3348
3349 _n_
3350 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3351 -------- = - > | ci . --- | S2s ---------------- + (1 - S2s)ts^2 ----------------------------- |
3352 dDj.dS2f 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3353 i=m
3354
3355 dci / S2s (1 - S2s)(ts + ti)ts \ \
3356 + --- . ti | ------------ + ------------------------- | |.
3357 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3358 """
3359
3360 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)
3361
3362
3363
3364
3366 """Spectral density Hessian.
3367
3368 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
3369 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3370 parameters.
3371
3372 The model-free Hessian is
3373
3374 _n_
3375 d2J(w) 2 \ dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3376 -------- = - > ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |.
3377 dDj.dS2f 5 /__ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3378 i=m
3379 """
3380
3381 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)
3382
3383
3385 """Spectral density Hessian.
3386
3387 Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
3388 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3389 parameters.
3390
3391 The model-free Hessian is
3392
3393 _n_
3394 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (tf + ti)^2 - (w.tf.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3395 -------- = - > | ci . --- | S2s ---------------- - tf^2 ----------------------------- + (1 - S2s)ts^2 ----------------------------- |
3396 dDj.dS2f 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((tf + ti)^2 + (w.tf.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3397 i=m
3398
3399 dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \ \
3400 + --- . ti | ------------ - ------------------------- + ------------------------- | |.
3401 dDj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3402 """
3403
3404 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)
3405
3406
3407
3408
3409
3410
3411
3412
3414 """Spectral density Hessian.
3415
3416 Calculate the spectral desity values for the Dj - S2s double partial derivative of the extended
3417 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3418 diffusion tensor parameters.
3419
3420 The model-free Hessian is
3421
3422 _n_
3423 d2J(w) 2 \ dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3424 -------- = - S2f > ci . --- | ---------------- - ts^2 ----------------------------- |.
3425 dDj.dS2s 5 /__ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3426 i=m
3427 """
3428
3429 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)
3430
3431
3433 """Spectral density Hessian.
3434
3435 Calculate the spectral desity values for the Dj - S2s double partial derivative of the extended
3436 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3437 diffusion tensor parameters.
3438
3439 The model-free Hessian is
3440
3441 _n_
3442 d2J(w) 2 \ / dti / 1 - (w.ti)^2 (ts + ti)^2 - (w.ts.ti)^2 \
3443 -------- = - S2f > | ci . --- | ---------------- - ts^2 ----------------------------- |
3444 dDj.dS2s 5 /__ \ dDj \ (1 + (w.ti)^2)^2 ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3445 i=m
3446
3447 dci / 1 (ts + ti)ts \ \
3448 + --- . ti | ------------ - ------------------------- | |.
3449 dDj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / /
3450 """
3451
3452 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)
3453
3454
3455
3456
3457
3458
3459
3460
3462 """Spectral density Hessian.
3463
3464 Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
3465 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3466 parameters.
3467
3468 The model-free Hessian is
3469
3470 _n_
3471 d2J(w) 4 \ dti (tf + ti)^2 - 3(w.tf.ti)^2
3472 ------- = - (1 - S2f) . tf > ci . --- . ti . (tf + ti) -----------------------------.
3473 dDj.dtf 5 /__ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3
3474 i=m
3475 """
3476
3477 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)
3478
3479
3481 """Spectral density Hessian.
3482
3483 Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
3484 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3485 parameters.
3486
3487 The model-free Hessian is
3488
3489 _n_
3490 d2J(w) 2 \ / dti (tf + ti)^2 - 3(w.tf.ti)^2 dci (tf + ti)^2 - (w.tf.ti)^2 \
3491 ------- = - (1 - S2f) > | 2ci . --- . tf . ti . (tf + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3492 dDj.dtf 5 /__ \ dDj ((tf + ti)^2 + (w.tf.ti)^2)^3 dDj ((tf + ti)^2 + (w.tf.ti)^2)^2 /
3493 i=m
3494 """
3495
3496 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)
3497
3498
3499
3500
3501
3502
3503
3504
3506 """Spectral density Hessian.
3507
3508 Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
3509 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3510 diffusion tensor parameters.
3511
3512 The model-free Hessian is
3513
3514 _n_
3515 d2J(w) 4 \ dti (ts + ti)^2 - 3(w.ts.ti)^2
3516 ------- = - S2f(1 - S2s) . ts > ci . --- . ti . (ts + ti) -----------------------------.
3517 dDj.dts 5 /__ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3
3518 i=m
3519 """
3520
3521 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)
3522
3523
3525 """Spectral density Hessian.
3526
3527 Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
3528 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3529 diffusion tensor parameters.
3530
3531 The model-free Hessian is
3532
3533 _n_
3534 d2J(w) 2 \ / dti (ts + ti)^2 - 3(w.ts.ti)^2 dci (ts + ti)^2 - (w.ts.ti)^2 \
3535 ------- = - S2f(1 - S2s) > | 2ci . --- . ts . ti . (ts + ti) ----------------------------- + --- . ti^2 ----------------------------- |.
3536 dDj.dts 5 /__ \ dDj ((ts + ti)^2 + (w.ts.ti)^2)^3 dDj ((ts + ti)^2 + (w.ts.ti)^2)^2 /
3537 i=m
3538 """
3539
3540 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)
3541
3542
3543
3544
3545
3546
3547
3548
3550 """Spectral density Hessian.
3551
3552 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3553 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3554 parameters.
3555
3556 The model-free Hessian is
3557
3558 _n_
3559 d2J(w) 2 \ d2ci / S2f . S2s S2f(1 - S2s)(ts + ti)ts \
3560 ------- = - > ------- . ti | ------------ + ------------------------- |.
3561 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3562 i=m
3563 """
3564
3565 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)
3566
3567
3568
3569
3571 """Spectral density Hessian.
3572
3573 Calculate the spectral desity values for the Oj - Ok double partial derivative of the
3574 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3575 parameters.
3576
3577 The model-free Hessian is
3578
3579 _n_
3580 d2J(w) 2 \ d2ci / S2f . S2s (1 - S2f)(tf + ti)tf S2f(1 - S2s)(ts + ti)ts \
3581 ------- = - > ------- . ti | ------------ + ------------------------- + ------------------------- |.
3582 dOj.dOk 5 /__ dOj.dOk \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3583 i=m
3584 """
3585
3586 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)
3587
3588
3589
3590
3591
3592
3593
3594
3596 """Spectral density Hessian.
3597
3598 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3599 extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
3600 parameters.
3601
3602 The model-free Hessian is
3603
3604 _n_
3605 d2J(w) 2 \ dci / S2s (1 - S2s)(ts + ti)ts \
3606 -------- = - > --- . ti | ------------ + ------------------------- |.
3607 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3608 i=m
3609 """
3610
3611 return 0.4 * sum(data.dci[k] * data.ti * (params[data.s2s_i] * data.fact_ti + data.one_s2s * data.fact_ts), axis=2)
3612
3613
3614
3615
3617 """Spectral density Hessian.
3618
3619 Calculate the spectral desity values for the Oj - S2f double partial derivative of the
3620 extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3621 parameters.
3622
3623 The model-free Hessian is
3624
3625 _n_
3626 d2J(w) 2 \ dci / S2s (tf + ti)tf (1 - S2s)(ts + ti)ts \
3627 -------- = - > --- . ti | ------------ - ------------------------- + ------------------------- |.
3628 dOj.dS2f 5 /__ dOj \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3629 i=m
3630 """
3631
3632 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)
3633
3634
3635
3636
3637
3638
3639
3640
3642 """Spectral density Hessian.
3643
3644 Calculate the spectral desity values for the Oj - S2 double partial derivative of the extended
3645 model-free formula with the parameters {S2f, S2s, ts} and {S2f, tf, S2s, ts} together with
3646 diffusion tensor parameters.
3647
3648 The model-free Hessian is
3649
3650 _n_
3651 d2J(w) 2 \ dci / 1 (ts + ti)ts \
3652 -------- = - S2f > --- . ti | ------------ - ------------------------- |.
3653 dOj.dS2s 5 /__ dOj \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3654 i=m
3655 """
3656
3657 return 0.4 * params[data.s2f_i] * sum(data.dci[k] * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3658
3659
3660
3661
3662
3663
3664
3665
3667 """Spectral density Hessian.
3668
3669 Calculate the spectral desity values for the Oj - tf double partial derivative of the extended
3670 model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
3671 parameters.
3672
3673 The model-free Hessian is
3674
3675 _n_
3676 d2J(w) 2 \ dci (tf + ti)^2 - (w.tf.ti)^2
3677 ------- = - (1 - S2f) > --- . ti^2 -----------------------------.
3678 dOj.dtf 5 /__ dOj ((tf + ti)^2 + (w.tf.ti)^2)^2
3679 i=m
3680 """
3681
3682 return 0.4 * data.one_s2f * sum(data.dci[k] * data.fact_djw_dtf, axis=2)
3683
3684
3685
3686
3687
3688
3689
3690
3692 """Spectral density Hessian.
3693
3694 Calculate the spectral desity values for the Oj - ts double partial derivative of the extended
3695 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
3696 diffusion tensor parameters.
3697
3698 The model-free Hessian is
3699
3700 _n_
3701 d2J(w) 2 \ dci (ts + ti)^2 - (w.ts.ti)^2
3702 ------- = - S2f(1 - S2s) > --- . ti^2 -----------------------------.
3703 dOj.dts 5 /__ dOj ((ts + ti)^2 + (w.ts.ti)^2)^2
3704 i=m
3705 """
3706
3707 return 0.4 * data.s2f_s2 * sum(data.dci[k] * data.fact_djw_dts, axis=2)
3708
3709
3710
3711
3712
3713
3714
3715
3717 """Spectral density Hessian.
3718
3719 Calculate the spectral desity values for the S2f - S2s double partial derivative of the extended
3720 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3721 diffusion tensor parameters.
3722
3723 The model-free Hessian is
3724
3725 _n_
3726 d2J(w) 2 \ / 1 (ts + ti).ts \
3727 --------- = - > ci . ti | ------------ - ------------------------- |.
3728 dS2f.dS2s 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 /
3729 i=m
3730 """
3731
3732 return 0.4 * sum(data.ci * data.ti * (data.fact_ti - data.fact_ts), axis=2)
3733
3734
3735
3736
3737
3738
3739
3740
3742 """Spectral density Hessian.
3743
3744 Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
3745 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3746 parameters.
3747
3748 The model-free Hessian is
3749
3750 _n_
3751 d2J(w) 2 \ (tf + ti)^2 - (w.tf.ti)^2
3752 -------- = - - > ci . ti^2 -----------------------------.
3753 dS2f.dtf 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^2
3754 i=m
3755 """
3756
3757 return -0.4 * sum(data.ci * data.fact_djw_dtf, axis=2)
3758
3759
3760
3761
3762
3763
3764
3765
3767 """Spectral density Hessian.
3768
3769 Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
3770 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3771 diffusion tensor parameters.
3772
3773 The model-free Hessian is
3774
3775 _n_
3776 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3777 -------- = - (1 - S2s) > ci . ti^2 -----------------------------.
3778 dS2f.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3779 i=m
3780 """
3781
3782 return 0.4 * data.one_s2s * sum(data.ci * data.fact_djw_dts, axis=2)
3783
3784
3785
3786
3787
3788
3789
3790
3792 """Spectral density Hessian.
3793
3794 Calculate the spectral desity values for the S2s - ts double partial derivative of the extended
3795 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3796 diffusion tensor parameters.
3797
3798 The model-free Hessian is
3799
3800 _n_
3801 d2J(w) 2 \ (ts + ti)^2 - (w.ts.ti)^2
3802 -------- = - - S2f > ci . ti^2 -----------------------------.
3803 dS2s.dts 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^2
3804 i=m
3805 """
3806
3807 return -0.4 * params[data.s2f_i] * sum(data.ci * data.fact_djw_dts, axis=2)
3808
3809
3810
3811
3812
3813
3814
3815
3817 """Spectral density Hessian.
3818
3819 Calculate the spectral desity values for the tf - tf double partial derivative of the extended
3820 model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
3821 parameters.
3822
3823 The model-free Hessian is
3824
3825 _n_
3826 d2J(w) 4 \ (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
3827 ------ = - - (1 - S2f) > ci . ti^2 -----------------------------------------------------.
3828 dtf**2 5 /__ ((tf + ti)^2 + (w.tf.ti)^2)^3
3829 i=m
3830 """
3831
3832 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)
3833
3834
3835
3836
3837
3838
3839
3840
3842 """Spectral density Hessian.
3843
3844 Calculate the spectral desity values for the ts - ts double partial derivative of the extended
3845 model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
3846 diffusion tensor parameters.
3847
3848 The model-free Hessian is
3849
3850 _n_
3851 d2J(w) 4 \ (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
3852 ------ = - - S2f(1 - S2s) > ci . ti^2 -----------------------------------------------------.
3853 dts**2 5 /__ ((ts + ti)^2 + (w.ts.ti)^2)^3
3854 i=m
3855 """
3856
3857 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)
3858