-
Notifications
You must be signed in to change notification settings - Fork 14
/
dfa.stan
681 lines (635 loc) · 27.1 KB
/
dfa.stan
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
functions {
// this function subsets a matrix by dropping the row/column labeled 'drop'. P represents dimensions
matrix subset(matrix x, int drop, int P) {
// count number of rows in result
// assign rows in result
{
matrix[P-1,P-1] result;
int rowindx;
int colindx;
rowindx = 0;
for (i in 1:P) {
if (i != drop) {
rowindx = rowindx + 1;
colindx = 0;
for (j in 1:P) {
if (j != drop) {
colindx = colindx + 1;
result[rowindx, colindx] = x[i, j];
}
} // end j loop
} // end i!= drop
} // end i loop
return result;
}
}
matrix subsetvec(matrix x, int drop, int P) {
// assign rows in result
{
matrix[P-1,1] result;
int rowindx;
rowindx = 0;
for (i in 1:P) {
if (i != drop) {
rowindx = rowindx + 1;
result[rowindx,1] = x[i, drop];
} // end i!= drop
} // end i loop
return result;
}
}
matrix subsetvec2(vector x, int drop, int P) {
// assign rows in result
{
matrix[P-1,1] result;
int rowindx;
rowindx = 0;
for (i in 1:P) {
if (i != drop) {
rowindx = rowindx + 1;
result[rowindx,1] = x[i];
} // end i!= drop
} // end i loop
return result;
}
}
}
data {
int<lower=0> N; // number of data points
int<lower=0> P; // number of time series of data
int<lower=0> K; // number of trends
int<lower=0> nZ; // number of unique z elements
array[nZ] int<lower=0> row_indx;
array[nZ] int<lower=0> col_indx;
int<lower=0> nVariances;
array[P] int<lower=0> varIndx;
int<lower=0> nZero;
array[nZero] int<lower=0> row_indx_z;
array[nZero] int<lower=0> col_indx_z;
int<lower=0> n_pos; // number of non-missing observations
array[n_pos] int<lower=0> row_indx_pos; // row indices of non-missing obs
array[n_pos] int<lower=0> col_indx_pos; // col indices of non-missing obs
array[n_pos] real y; // vectorized matrix of observations
array[n_pos] int<lower=0> y_int; // vectorized matrix of observations
array[n_pos] real input_offset; // vector of offset (no estimated coefficient)
int<lower=0> n_na; // number of missing observations
array[n_na] int<lower=0> row_indx_na; // row indices of missing obs
array[n_na] int<lower=0> col_indx_na; // col indices of missing obs
real<lower=1> nu_fixed; // df on student-t
int estimate_nu; // Estimate degrees of freedom?
int use_normal; // flag, for large values of nu > 100, use normal instead
int est_cor; // whether to estimate correlation in obs error (=1) or not (=0)
int est_phi; // whether to estimate autocorrelation in trends (=1) or not (= 0)
int est_theta; // whether to estimate moving-average in trends (=1) or not (= 0
int<lower=0> num_obs_covar; // number of unique observation covariates, dimension of matrix
int<lower=0> n_obs_covar; // number of unique covariates included
array[num_obs_covar,3] int obs_covar_index;// indexed by time, trend, covariate #, covariate value. +1 because of indexing issues
array[num_obs_covar] real obs_covar_value;
array[num_obs_covar] int match_obs_covar;
int<lower=0> num_pro_covar; // number of unique process covariates, dimension of matrix
int<lower=0> n_pro_covar; // number of unique process covariates included
array[num_pro_covar,3] int pro_covar_index;// indexed by time, trend, covariate #, covariate value. +1 because of indexing issues
array[num_pro_covar] real pro_covar_value;
array[2] real z_bound;
int<lower=0> long_format; // data shape, 0 == wide (default), 1 = long with potential for multiple observations
int<lower=0> proportional_model;
int<lower=0> est_sigma_process; // optional, 0 == not estimate sigma_pro (default), 1 == estimate
int<lower=0> n_sigma_process; // single value, or equal number of tre
int<lower=0> est_rw; // // single value, 0 if false 1 if true [model trends as latend AR process = default]
int<lower=0> est_spline; // single value, 0 if false 1 if true to model trends with b-splines
int<lower=0> est_gp; // single value, 0 if false 1 if true to model trends with predictive gaussian process
int<lower=0> n_knots; // single value representing knots for b-spline or gp process
matrix[N, n_knots] X_spline;
array[n_knots] real knot_locs; // inputs of knot locations for GP model
//array[N] real data_locs; // locations of data
//matrix[n_knots, n_knots] distKnots;
//matrix[N, n_knots] distKnots21;
matrix[1, n_knots] distKnots21_pred;
int obs_model; // 1 = normal, 2 = gamma, 3 = bernoulli, 4 = poisson, 5 = neg bin, 6 = lognormal
int<lower=0, upper=1> est_sigma_params;
int<lower=0, upper=1> est_gamma_params;
int<lower=0, upper=1> est_nb2_params;
int<lower=0, upper=1> use_expansion_prior;
array[2] real gp_theta_prior;
array[n_pos] real inv_var_weights_vec;
array[n_pos] real weights_vec;
}
transformed data {
int n_pcor; // dimension for cov matrix
int n_loglik; // dimension for loglik calculation
vector[K] zeros;
array[N] real data_locs; // for gp model
vector[K*proportional_model] alpha_vec;
vector[n_knots] muZeros;
real gp_delta = 1e-9; // stabilizing value for GP model
real lower_bound_z;
for(i in 1:N) {
data_locs[i] = i;
}
for(k in 1:K) {
zeros[k] = 0; // used in MVT / MVN below
}
for(k in 1:n_knots) {
muZeros[k] = 0; // used for GP
}
// this is number of points of log-likelihood, depends if model is MVN or not and data is in wide/long format
n_loglik = n_pos;
if(long_format==0) {
if(est_cor == 0) {
n_loglik = P * N;
} else {
n_loglik = N;
}
}
if(est_cor == 0) {
// MVN correlation matrix not estimated
n_pcor = P;
if(nVariances < 2) {
// minimum bound, just for Stan types
n_pcor = 2;
}
} else {
// MVN correlation matrix is estimated
n_pcor = P;
}
// for compositional model
if(proportional_model==1) {
for(k in 1:K) alpha_vec[k] = 1;
}
// for zpos
lower_bound_z = -100;
if(use_expansion_prior==1) lower_bound_z = 0;
}
parameters {
matrix[K * est_rw,(N-1) * est_rw] devs; // random deviations of trends
vector[K] x0; // initial state
vector<lower=0>[K*(1-proportional_model)*use_expansion_prior] psi; // expansion parameters
vector<lower=z_bound[1],upper=z_bound[2]>[nZ*(1-proportional_model)] z; // estimated loadings in vec form
vector<lower=lower_bound_z>[K*(1-proportional_model)] zpos; // constrained positive values
array[P*proportional_model] simplex[K] p_z; // alternative for proportional Z
matrix[K * est_spline, n_knots * est_spline] spline_a; // weights for b-splines
matrix[n_obs_covar, P] b_obs; // coefficients on observation model
matrix[n_pro_covar, K] b_pro; // coefficients on process model
array[nVariances*est_sigma_params] real<lower=0> sigma;
array[nVariances*est_gamma_params] real<lower=0> gamma_a;
array[nVariances*est_nb2_params] real<lower=0> nb2_phi;
array[estimate_nu] real<lower=2> nu; // df on student-t
array[n_na] real ymiss;
array[est_phi*K] real<lower=-1,upper=1> phi; // AR(1) coefficients specific to each trend
array[est_theta*K] real<lower=-1,upper=1> theta;// MA(1) coefficients specific to each trend
array[est_gp*K] real<lower=0> gp_theta;// gp_theta coefficients specific to each trend
cholesky_factor_corr[n_pcor] Lcorr; // matrix for correlated errros
array[est_sigma_process * n_sigma_process] real<lower=0> sigma_process; // process variances, potentially unique
array[K * est_gp] vector[n_knots* est_gp] effectsKnots; // gaussian predictive process
}
transformed parameters {
matrix[P,N] pred;
matrix[P,K] Z;
matrix[P,N] yall;
vector<lower=0>[P*est_sigma_params] sigma_vec;
vector<lower=0>[P*est_gamma_params] gamma_a_vec;
vector<lower=0>[P*est_nb2_params] nb_phi_vec;
vector[K] phi_vec; // for AR(1) part
vector[K] theta_vec; // for MA(1) part
matrix[K,N] x; // random walk-trends
vector[K] indicator; // indicates whether diagonal is neg or pos
vector<lower=0>[K*use_expansion_prior] psi_root; // derived sqrt(expansion parameter psi)
matrix[n_pcor*long_format*est_cor, n_pcor*long_format*est_cor] Sigma_derived;// temporary for calculations for MVN model
matrix[(n_pcor-1)*long_format*est_cor, (n_pcor-1)*long_format*est_cor] Sigma_temp;// temporary for calculations for MVN model
matrix[n_pcor-1,1] sigma12_vec;// temporary for calculations for MVN model
matrix[P*long_format*est_cor, N*long_format*est_cor] temp_sums;// temporary for calculations for MVN model
matrix[P*long_format*est_cor, N*long_format*est_cor] temp_counts;// temporary for calculations for MVN model
vector[P*long_format*est_cor] cond_sigma_vec;// temporary for calculations for MVN model
vector[P*long_format*est_cor] cond_mean_vec;// temporary for calculations for MVN model
real sigma11;// temporary for calculations for MVN model
vector[K] sigma_pro;
matrix[K * est_spline, n_knots * est_spline] spline_a_trans; // weights for b-splines
array[K] matrix[n_knots, n_knots] SigmaKnots; // matrix for GP model, unique for each trend K
//matrix[N, n_knots] SigmaOffDiag;// matrix for GP model
//matrix[N, n_knots] SigmaOffDiagTemp;// matrix for GP model
vector[n_pos] obs_cov_offset;
// block for process errors - can be estimated or not, and shared or not
for(k in 1:K) {
sigma_pro[k] = 1; // default constraint of all DFAs
if(est_sigma_process==1) {
if(n_sigma_process==1) {
sigma_pro[k] = sigma_process[1];
} else {
sigma_pro[k] = sigma_process[k];
}
}
}
// phi is the ar(1) parameter, fixed or estimated
if(est_phi == 1) {
for(k in 1:K) {phi_vec[k] = phi[k];}
//phi_vec = to_vector(phi);
} else {
for(k in 1:K) {phi_vec[k] = 1;}
//phi_vec = rep_vector(1.0, K);
}
// theta is the ma(1) parameter, fixed or estimated
if(est_theta == 1) {
for(k in 1:K) {theta_vec[k] = theta[k];}
//theta_vec = to_vector(theta);
} else {
for(k in 1:K) {theta_vec[k] = 0;}
//theta_vec = rep_vector(1.0, K);
}
if(est_sigma_params == 1) {
for(p in 1:P) {sigma_vec[p] = sigma[varIndx[p]];} // convert estimated sigmas to vec form
}
if(est_gamma_params == 1) {
for(p in 1:P) {gamma_a_vec[p] = gamma_a[varIndx[p]];} // convert estimated sigmas to vec form
}
if(est_nb2_params == 1) {
for(p in 1:P) {nb_phi_vec[p] = nb2_phi[varIndx[p]];} // convert estimated sigmas to vec form
}
if(long_format==0) {
// Fill yall with non-missing values
for(i in 1:n_pos) {
yall[row_indx_pos[i], col_indx_pos[i]] = y[i];
}
// Include missing observations
if(n_na > 0) {
for(i in 1:n_na) {
yall[row_indx_na[i], col_indx_na[i]] = ymiss[i];
}
}
}
if(proportional_model == 0) {
for(i in 1:nZ) {
Z[row_indx[i],col_indx[i]] = z[i]; // convert z to from vec to matrix
}
// fill in zero elements in upper diagonal
if(nZero > 2) {
for(i in 1:(nZero-2)) {
Z[row_indx_z[i],col_indx_z[i]] = 0;
}
}
for(k in 1:K) {
Z[k,k] = zpos[k];// add constraint for Z diagonal
}
// this block is for the expansion prior
if(use_expansion_prior==1) {
for(k in 1:K) {
if(zpos[k] < 0) {
indicator[k] = -1;
} else {
indicator[k] = 1;
}
// psi_root here should really be named inv_psi_root, because it's the inv
psi_root[k] = sqrt(psi[k]);
for(p in 1:P) {
// see Ghosh & Dunson 2009 eq 3
Z[p,k] = Z[p,k] * indicator[k] * (1/psi_root[k]);
}
}
}
// initial state for each trend
if(est_rw == 1) {
for(k in 1:K) {
x[k,1] = x0[k];
// trend is modeled as random walk, with optional
// AR(1) component = phi, and optional MA(1) component
// theta. Theta is included in the model block below.
for(t in 2:N) {
x[k,t] = phi_vec[k]*x[k,t-1] + devs[k,t-1];
}
}
}
if(est_spline==1) {
// P-spline adapted from Crainiceanu et al. 2005
// B-spline modified from Milad Kharratzadeh's example on B-splines/stan
for(k in 1:K) spline_a_trans[k] = spline_a[k] * sigma_pro[k];
x = spline_a_trans * X_spline';
for(k in 1:K) {x[k] = x0[k] + x[k];}
}
if(est_gp == 1) {
// for the GP model, we use Stan's built in gp_exp_quad_cov for the distance between knots
for (k in 1:K) {
SigmaKnots[k] = gp_exp_quad_cov(knot_locs, sigma_pro[k], gp_theta[k]);
//SigmaKnots = SigmaKnots+diag_matrix(rep_vector(gp_delta, n_knots));
for(i in 1:n_knots) {
SigmaKnots[k][i,i] += gp_delta; // stabilizing
}
// cov matrix between knots and projected locs
//SigmaOffDiagTemp = square(sigma_pro[k]) * exp(-distKnots21 / (2.0*pow(gp_theta[k],2.0)));
//SigmaOffDiagTemp = gp_exp_quad_cov(data_locs, knot_locs, sigma_pro[k], gp_theta[k]);
// multiply and invert once, used below:
//SigmaOffDiag = SigmaOffDiagTemp * inverse_spd(SigmaKnots[k]);
// cholesky_decompose(SigmaKnots[k]) * effectsKnots[k] is equivalent to drawn MVN deviations
//x[k] = to_row_vector(SigmaOffDiagTemp * inverse_spd(SigmaKnots[k]) * cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
if(n_knots == N) {
// full rank
x[k] = to_row_vector(cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
} else {
x[k] = to_row_vector(gp_exp_quad_cov(data_locs, knot_locs, sigma_pro[k], gp_theta[k]) * inverse_spd(SigmaKnots[k]) * cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
}
}
}
// this block also for the expansion prior, used to convert trends
if(use_expansion_prior==1) {
for(k in 1:K) {
//x[k,1:N] = x[k,1:N] * indicator[k] * psi_root[k];
// see Ghosh and Dunson 2009 eq 3. psi_root[k] here is really psi^(-1/2)
x[k] = x[k] * indicator[k] * psi_root[k];
}
}
}
if(proportional_model == 1) {
// initial state for each trend
if(est_rw == 1) {
for(k in 1:K) {
x[k,1] = x0[k];
// trend is modeled as random walk, with optional
// AR(1) component = phi, and optional MA(1) component
// theta. Theta is included in the model block below.
for(t in 2:N) {
x[k,t] = phi_vec[k]*x[k,t-1] + devs[k,t-1];
}
}
}
if(est_spline==1) {
for(k in 1:K) spline_a_trans[k] = spline_a[k] * sigma_pro[k];
x = spline_a_trans * X_spline';
for(k in 1:K) {x[k] = x0[k] + x[k];}
}
if(est_gp == 1) {
for (k in 1:K) {
SigmaKnots[k] = gp_exp_quad_cov(knot_locs, sigma_pro[k], gp_theta[k]);
//SigmaKnots = SigmaKnots+diag_matrix(rep_vector(gp_delta, n_knots));
for(i in 1:n_knots) {
SigmaKnots[k][i,i] += gp_delta; // stabilizing
}
// cov matrix between knots and projected locs
//SigmaOffDiagTemp = square(sigma_pro[k]) * exp(-distKnots21 / (2.0*pow(gp_theta[k],2.0)));
//SigmaOffDiagTemp = gp_exp_quad_cov(data_locs, knot_locs, sigma_pro[k], gp_theta[k]);
// multiply and invert once, used below:
//SigmaOffDiag = SigmaOffDiagTemp * inverse_spd(SigmaKnots[k]);
//cholesky_decompose(SigmaKnots[k]) * effectsKnots[k] is equivalent to drawn MVN deviations
//x[k] = to_row_vector(SigmaOffDiagTemp * inverse_spd(SigmaKnots[k]) * cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
if(n_knots == N) {
// full rank
x[k] = to_row_vector(cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
} else {
x[k] = to_row_vector(gp_exp_quad_cov(data_locs, knot_locs, sigma_pro[k], gp_theta[k]) * inverse_spd(SigmaKnots[k]) * cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]);
}
}
}
// proportional model
for(p in 1:P) {
//for(k in 1:K) {
// Z[p,k] = p_z[p,k]; // compositions sum to 1 for a time series
//}
Z[p] = to_row_vector(p_z[p]);
}
}
// adjust predictions if process covariates exist
if(num_pro_covar > 0) {
for(i in 1:num_pro_covar) {
// indexed by time, trend, covariate #, covariate value
x[pro_covar_index[i,2],pro_covar_index[i,1]] += b_pro[pro_covar_index[i,3], pro_covar_index[i,2]] * pro_covar_value[i];
}
}
// N is sample size, P = time series, K = number trends
// [PxN] = [PxK] * [KxN]
pred = Z * x;
// obs_cov_offset is specific to each non-NA observation
for(i in 1:n_pos) {
obs_cov_offset[i] = 0;
}
// adjust predictions if observation covariates exist
if(num_obs_covar > 0) {
if(long_format==0) {
for(i in 1:num_obs_covar) {
// if data are in wide format, only 1 obs exists per prediction + pred matrix can just be adjusted
// indexed by time, trend, covariate #, covariate value
pred[obs_covar_index[i,2],obs_covar_index[i,1]] += b_obs[obs_covar_index[i,3], obs_covar_index[i,2]] * obs_covar_value[i];
}
} else {
// if data are in long format, multiple obs might exist per time point, and need to use ugly loops
// loop over dataframe of obs error covariates -- may be > observations if multiple covariates exist
for(i in 1:num_obs_covar) {
obs_cov_offset[match_obs_covar[i]] += b_obs[obs_covar_index[i,3], obs_covar_index[i,2]] * obs_covar_value[i];
}
}
}
if(long_format==1 && est_cor==1) {
// this is a pain, but we need to calculate the deviations (basically mean y - E[y] for each time point/time series)
for(n in 1:N) {
for(p in 1:P) {
temp_sums[p,n] = 0.0;
temp_counts[p,n] = 0.0;
}
}
for(i in 1:n_pos) {
temp_sums[row_indx_pos[i],col_indx_pos[i]] += (y[i] - pred[row_indx_pos[i],col_indx_pos[i]]);//PxN
temp_counts[row_indx_pos[i],col_indx_pos[i]] += 1;
}
for(n in 1:N) {
for(p in 1:P) {
// Now temp_sums will hold the mean residual for each time - timeseries combination
temp_sums[p,n] = temp_sums[p,n]/temp_counts[p,n];
}
}
//Omega_derived = multiply_lower_tri_self_transpose(Lcorr);
Sigma_derived = quad_form_diag(multiply_lower_tri_self_transpose(Lcorr), sigma_vec);
for(p in 1:P) {
sigma11 = Sigma_derived[p,p]; //
Sigma_temp = inverse(subset(Sigma_derived, p, P)); // this is sigma22^-1
sigma12_vec = subsetvec(Sigma_derived, p, P); // P-1 x 1 matrix
// conditional mean for multivariate normal, e.g. https://en.wikipedia.org/wiki/Multivariate_normal_distribution
cond_mean_vec[p] = to_row_vector(sigma12_vec) * Sigma_temp * to_vector(subsetvec2(col(temp_sums,p), p, P));
// conditional variance of multivariate normal, e.g. https://en.wikipedia.org/wiki/Multivariate_normal_distribution
cond_sigma_vec[p] = sqrt(sigma11 - to_row_vector(sigma12_vec) * Sigma_temp * to_vector(sigma12_vec));
}
}
}
model {
// initial state for each trend
x0 ~ normal(0, 1); // initial state estimate at t=1
if(use_expansion_prior==1) {
psi ~ gamma(2, 1); // expansion parameter for par-expanded priors
}
// prior for df parameter for t-distribution
if (estimate_nu == 1) {
nu[1] ~ gamma(2, 0.1);
}
// prior on AR(1) component if included
if(est_phi == 1) {
phi ~ normal(0,1); // K elements
}
// prior on MA(1) component if included
if(est_theta == 1) {
theta ~ normal(0,1); // K elements
}
// prior on process variance if included
if(est_sigma_process) {
sigma_process ~ normal(0,1);
}
// observation variance, which depend on family
if(est_sigma_params==1) sigma ~ student_t(3, 0, 1);
if(est_gamma_params==1) gamma_a ~ student_t(3, 0, 1);
if(est_nb2_params==1) nb2_phi ~ student_t(3, 0, 1);
// MVN model for observation error variance
if(est_cor == 1) {
Lcorr ~ lkj_corr_cholesky(1);
}
if(est_gp==1) {
// Use Betancort prior, https://betanalpha.github.io/assets/case_studies/gp_part3/part3.html
// P[gp_theta < 2.0] = 0.01
// P[gp_theta > 10] = 0.01
//gp_theta ~ inv_gamma(8.91924, 34.5805);
gp_theta ~ inv_gamma(gp_theta_prior[1], gp_theta_prior[2]);
// random effects estimated for each trend
for(k in 1:K) {
effectsKnots[k] ~ std_normal();
}
}
// This is deviations - either normal or Student t, and
// if Student-t, df parameter nu can be estimated or fixed.
// Tried putting most of this in the transformed param block, with devs ~ std_normal(), but slowed down
if(est_rw == 1) {
for(k in 1:K) {
if(use_normal == 0) {
for(t in 1:1) {
if (estimate_nu == 1) {
devs[k,t] ~ student_t(nu[1], 0, sigma_pro[k]); // random walk
} else {
devs[k,t] ~ student_t(nu_fixed, 0, sigma_pro[k]); // random walk
}
}
for(t in 2:(N-1)) {
// if MA is not included, theta_vec = 0
if (estimate_nu == 1) {
devs[k,t] ~ student_t(nu[1], theta_vec[k]*devs[k,t-1], sigma_pro[k]); // random walk
} else {
devs[k,t] ~ student_t(nu_fixed, theta_vec[k]*devs[k,t-1], sigma_pro[k]); // random walk
}
}
} else {
devs[k,1] ~ normal(0, 1);
for(t in 2:(N-1)) {
// if MA is not included, theta_vec = 0
devs[k,t] ~ normal(theta_vec[k]*devs[k,t-1], sigma_pro[k]);
}
}
}
}
if(est_spline==1) {
for(k in 1:K) {
spline_a[k] ~ std_normal();
}
}
if(proportional_model == 0) {
// prior on loadings
z ~ std_normal(); // off-diagonal
zpos ~ std_normal();// diagonal
} else {
for(p in 1:P) {
p_z[p] ~ dirichlet(alpha_vec);
}
}
// likelihood for independent
if(est_cor == 0) {
if(long_format==0) {
if(obs_model == 1) {for(i in 1:P) target += normal_lpdf(yall[i] | pred[i], sigma_vec[i]);} // gaussian
} else {
if(obs_model == 1) {for(i in 1:n_pos) target += weights_vec[i] * normal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i], sigma_vec[row_indx_pos[i]] * inv_var_weights_vec[i]);}
if(obs_model == 2) {for(i in 1:n_pos) target += weights_vec[i] * gamma_lpdf(y[i] | gamma_a_vec[row_indx_pos[i]], gamma_a_vec[row_indx_pos[i]] / exp(input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]));} // gamma
if(obs_model == 3) {for(i in 1:n_pos) target += weights_vec[i] * poisson_log_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]);} // poisson
if(obs_model == 4) {for(i in 1:n_pos) target += weights_vec[i] * neg_binomial_2_log_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i], nb_phi_vec[row_indx_pos[i]]);} // negbin
if(obs_model == 5) {for(i in 1:n_pos) target += weights_vec[i] * bernoulli_logit_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]);} // binomial
if(obs_model == 6) {for(i in 1:n_pos) target += weights_vec[i] * lognormal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i], sigma_vec[row_indx_pos[i]]);} // lognormal
}
} else {
// need to loop over time slices / columns - each ~ MVN
if(long_format==0) {
if(obs_model == 1) {for(i in 1:N) target += multi_normal_cholesky_lpdf(col(yall,i) | col(pred,i), diag_pre_multiply(sigma_vec, Lcorr));}
} else {
if(obs_model == 1) {for(i in 1:n_pos) target += weights_vec[i] * normal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i] + cond_mean_vec[row_indx_pos[i]], cond_sigma_vec[row_indx_pos[i]] * inv_var_weights_vec[i]);}
}
}
}
generated quantities {
vector[n_loglik] log_lik;
matrix[n_pcor, n_pcor] Omega;
matrix[n_pcor, n_pcor] Sigma;
matrix[K,1] xstar; //random walk-trends in future
vector[K] future_devs; // deviations in future
matrix[n_knots, n_knots] SigmaKnots_pred; // matrix for GP model
row_vector[n_knots] SigmaOffDiag_pred;// matrix for GP model
int<lower=0> j;
j = 0;
if(est_cor == 1) {
Omega = multiply_lower_tri_self_transpose(Lcorr);
Sigma = quad_form_diag(Omega, sigma_vec);
}
// calculate pointwise log_lik for loo package:
if(est_cor == 0) {
if(long_format==0) {
j = 0;
for(n in 1:N) {
for(p in 1:P) {
j = j + 1;
log_lik[j] = normal_lpdf(yall[p,n] | pred[p,n], sigma_vec[p]);
}
}
} else {
if(obs_model == 1) {for(i in 1:n_pos) log_lik[i] = normal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i], sigma_vec[row_indx_pos[i]] * inv_var_weights_vec[i]);}
if(obs_model == 2) {for(i in 1:n_pos) log_lik[i] = gamma_lpdf(y[i] | gamma_a_vec[row_indx_pos[i]], gamma_a_vec[row_indx_pos[i]] / exp(input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]));} // gamma
if(obs_model == 3) {for(i in 1:n_pos) log_lik[i] = poisson_log_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]);} // poisson
if(obs_model == 4) {for(i in 1:n_pos) log_lik[i] = neg_binomial_2_log_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i], nb_phi_vec[row_indx_pos[i]]);} // negbin
if(obs_model == 5) {for(i in 1:n_pos) log_lik[i] = bernoulli_logit_lpmf(y_int[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i]);} // binomial
if(obs_model == 6) {for(i in 1:n_pos) log_lik[i] = lognormal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]], sigma_vec[row_indx_pos[i]] + obs_cov_offset[i]);} // lognormal
}
} else {
if(long_format==0) {
for(i in 1:N) {
log_lik[i] = multi_normal_cholesky_lpdf(col(yall,i) | col(pred,i), diag_pre_multiply(sigma_vec, Lcorr));
}
} else {
for(i in 1:n_pos) {
// //row_indx_pos[i] is the time series, col_index_pos is the time
log_lik[i] = normal_lpdf(y[i] | input_offset[i] + pred[row_indx_pos[i],col_indx_pos[i]] + obs_cov_offset[i] + cond_mean_vec[row_indx_pos[i]], cond_sigma_vec[row_indx_pos[i]] * inv_var_weights_vec[i]);
}
}
}
for(k in 1:K) {
future_devs[k] = 0;
}
// future deviations
if(est_rw==1) {
for(k in 1:K) {
if(use_normal == 0) {
// if MA is not included, theta_vec = 0
if (estimate_nu == 1) {
future_devs[k] = student_t_rng(nu[1], theta_vec[k]*devs[k,N-1], sigma_pro[k]); // random walk
} else {
future_devs[k] = student_t_rng(nu_fixed, theta_vec[k]*devs[k,N-1], sigma_pro[k]); // random walk
}
} else {
// if MA is not included, theta_vec = 0
future_devs[k] = normal_rng(theta_vec[k]*devs[k,N-1], sigma_pro[k]);
}
xstar[k,1] = x[k,N] + future_devs[k]; // future value of trend at t+1
}
}
if(est_spline == 1) {
// b-spline, only affected by endpoint where weight -> 1
for(k in 1:K) {
xstar[k,1] = spline_a_trans[k,n_knots] * X_spline[N,n_knots];
}
}
if(est_gp == 1) {
for (k in 1:K) {
SigmaKnots_pred = gp_exp_quad_cov(knot_locs, sigma_pro[k], gp_theta[k]);
for(i in 1:n_knots) {
SigmaKnots_pred[i,i] += gp_delta; // stabilizing
}
// cov matrix between knots and projected locs
SigmaOffDiag_pred = to_row_vector(square(sigma_pro[k]) * exp(-distKnots21_pred / (2.0*pow(gp_theta[k],2.0)))) * inverse_spd(SigmaKnots_pred);
xstar[k,1] = SigmaOffDiag_pred * cholesky_decompose(SigmaKnots[k]) * effectsKnots[k]; // RHS is real
}
}
}