Anatomy: how does u attach to y?

Spec: {'id': 'row_id', 'model': 'iid'} with 10 distinct row_id values

One latent value per observation. Classic use: absorbing overdispersion. n_levels = n_obs = 10.

Spec: {'id': 'school_id', 'model': 'iid'} with 4 schools, 10 students

Many observations share the same latent value (students in schools). n_levels (4) is smaller than n_obs (10), so the same u appears on several rows.

Spec: {'id': 'level', 'model': 'iid', 'replicate': rep, 'nrep': 2}

The whole iid block is repeated nrep times, independently. Each observation carries two integer indices: its level (1..5) and which replicate it belongs to (1..2). Together they pick out one of 5 × 2 = 10 latent values. The two replicate blocks share one precision τ (estimated jointly), but the realized u-draws across reps are uncorrelated: same level in different reps gives a different u.

Why colors split into two families? Cool colors (indigo → teal) mark rep = 1; warm colors (pink → amber) mark rep = 2. Observations 1 and 6 both have level = 1 but their badges have different colors, because they pick up different latent values (u_1,1 vs u_1,2).

Spec: {'id': 'region', 'model': 'iid', 'group': week, 'ngroup': 2, 'control.group': {'model': 'ar1'}}

Same 2D layout as case 3 (5 regions × 2 weeks = 10 latent values), but now the second axis carries correlation, not independent copies. Within each week the 5 regions are still iid. Between the two weeks, the matching latent values are tied together: u_k,1 and u_k,2 share a one-step AR1 correlation ρ. So obs 1 (region 1, week 1) and obs 6 (region 1, week 2) are not the same value, but they are correlated: knowing one tells you something about the other.

Why colors come in light/dark pairs? Each region gets one hue. The saturated shade marks week 1, the lighter shade marks week 2. Same hue says "same region, AR1-correlated"; different shade says "different week so different draw." Contrast with case 3, where rep 1 and rep 2 use unrelated cool / warm families because they are independent.

Spec: {'id': 'time', 'model': 'rw1'} with 10 ordered time points

One latent value per ordered position, like IID, but neighbors are tied together by an increment penalty. Colors shade smoothly to remind you of the neighbor coupling.

Spec: {'id': 'region', 'model': 'iid', 'group': week, 'ngroup': 2}   (no control.group set)

group introduces a second indexing axis on top of id. By itself, with no control.group dict, the across-group axis defaults to iid: all 5 × 2 = 10 latent values are independent draws. Mathematically, this case is identical to case 3 (replicate). The point of group is not what it does on its own; it is what it lets you do next: add a control.group dict to make the across-group axis correlated (case 4).

Why colors split into two families (cool / warm)? Same scheme as case 3: cool for g = 1, warm for g = 2. Independent groups get unrelated hue families. Compare with case 4, where the same hue (saturated vs light) marks the AR1 tie between matched regions.

Spec: {'id': 'hour', 'model': 'rw1', 'cyclic': True}

Same setup as case 5 (linear RW1): one latent value per ordered position, smooth-with-neighbors prior. The only difference is the graph topology: setting cyclic: True adds one extra increment u₁ − u_n to the prior, so the start and end of the index are tied together. Use it when the index wraps naturally (hour-of-day, day-of-week, month-of-year, any angular variable).

Precision matrix Q = τR: linear vs cyclic

The structure matrix R for cyclic RW1 is a circulant tridiagonal: 2 on every diagonal entry, −1 on every nearest-neighbor off-diagonal, plus −1 in the two corners for the wrap edge between positions 1 and n. Compared with linear RW1, the only differences are the four highlighted cells: the boundary diagonals jump from 1 to 2 (positions 1 and n now have two neighbors each), and the two corners pick up −1 entries (the wrap edge).

Why the colors form a wheel? The 10 badges trace the hue circle in steps of 36°. Position 10 (purple-red) sits visually next to position 1 (red) just like its prior neighbor relationship. Compare case 5 where colors run from teal to violet along a line: in linear RW1 position 10 has no special tie to position 1.

Spec: {'id': 'group_id', 'model': 'iid', 'weights': w}   (w is a 1-D numpy array of length n_obs)

weights scales the random-effect contribution per observation: w_i multiplies u_g(i) in the linear predictor. The latent vector is the same as case 2 (one u per group). What changes is the design matrix: instead of putting a 1 wherever observation i picks up group g, it puts w_i. So obs sharing a group keep the same colour (same u), but the badge label carries the weight factor.

Reading the badges. Colour = group (indigo / green / amber). The "w · u_g" label inside the badge is the literal contribution to that row's η_i. Rows 1, 2, 3 all share u₁ but multiply it by 1.0, 2.0, and 0.5 respectively. Same u, different pull.

Spec: {'id': 'school_id', 'model': 'iid', 'values': [1, 2, 3, 4, 5]}   (10 students across 4 schools; school 3 has no data yet)

values declares the complete set of allowed levels for the id column. Without it, pyINLA infers the level set from sort(unique(id)) in the data, so the latent vector has one entry per observed level. With values, the latent vector has one entry per declared level, even those that never appear in the data. Those extra slots are governed purely by the prior; there are no y_i rows that would update them.

Spec: {'id': 'time', 'model': 'seasonal', 'season.length': 5}   (10 ordered time points = 2 full cycles)

The seasonal model puts the prior on sliding sums of length m, not on neighbor differences. Every consecutive window of m latent values is constrained to look like a small Gaussian. Patterns that repeat with period m (so each cycle of length m roughly averages to zero) sit comfortably under this prior; non-periodic drift accumulates inside the windows and is penalized. The model is intrinsic: it has rank deficiency m − 1, identifiable once you have an intercept or other fixed effects.