Curriculum-Aware Interpolate-then-Refine: Learned Physiological Time-Series Imputation under Realistic Missingness

Yu-Chao Huang; Haochen Zhang; Nicholas Konz; Tianlong Chen

cs.LGAug 21, 2026

Curriculum-Aware Interpolate-then-Refine: Learned Physiological Time-Series Imputation under Realistic Missingness

Authors: Yu-Chao Huang, Haochen Zhang, Nicholas Konz, Tianlong Chen

Organizations: UNITES Lab, University of North Carolina at Chapel Hill

Abstract

Imputing physiological time series (arterial blood pressure, blood glucose, etc.) is essential for addressing the missingness that pervades clinical data. Yet modern imputation methods perform poorly in this domain: a recent benchmark found that simple linear interpolation outperformed every learned imputer on real-world clinical signals with realistic gaps. We show that this reflects two properties of physiological missingness that generic imputers ignore: gaps may occur when the signal is clinically extreme rather than typical, and gap lengths can easily span orders of magnitude. To this end, we introduce Curriculum-Aware Interpolate-then-Refine (CAIR), a two-stage framework for physiological time-series imputation. Our key motivation is to learn a coarse base curve and then repeatedly correct it toward physiological realism, rather than predict a gap in a single pass. Consequently, CAIR couples a bidirectional-GRU interpolator with a Transformer refiner that corrects its own estimate over three successive passes, trained jointly under a broad, signal-agnostic random-gap curriculum. We evaluate imputers stratified by gap length and missingness mechanism (MCAR, MAR, NMAR) rather than by a single average, and CAIR is the most accurate under every mechanism on continuous glucose monitoring (AI-READI) and arterial pressure in intensive care (MIMIC-III). Its margin over the strongest baseline grows with difficulty, from 9% under MCAR to 19% under value-dependent dropout, where generic learned imputers are weakest. We further show low reconstruction error alone does not recover the burden metrics clinicians act on: interpolants matching CAIR's error fail to preserve those metrics, imputers that recover them are far less accurate, and CAIR alone ranks among the best on both axes.

Figures & tables

Panel of Figure 1 — Figure 1: Method Overview. Notation: $x\in\mathbb{R}^{T}$ is the complete signal and $m\in\{0,1\}^{T}$ its observation mask ( $m_{t}=1$ where $x_{t}$ is recorded), so the elementwise product $x\odot m$ is the observed (gapped) trace; $c$ collects the co-recorded context channels (steps, sleep, stress, a second vital); $f$ is an imputer and $\widehat{x}=f(x\odot m,\,m)$ its reconstruction; and $y^{0}$ is the Stage-1 base curve. Shaded bands mark masked spans throughout. Every trace is recorded AI-READI CGM: (a) is one participant’s 24 h day, (b)–(d) zoom into one postprandial excursion from it, and the top row of (b) zooms further so that a $15$ min gap is visible. (a) The five masking regimes this paper evaluates, all on one signal: MCAR scatters isolated points; MAR deletes contiguous blocks triggered by co-recorded activity; NMAR deletes blocks triggered by the target’s own clinically extreme values (here every run above $150$ mg/dL); the gap-length protocol sweeps a single block from $15$ min to hours; and the physiological protocol masks event-aligned blocks. Dotted grey is the signal each mask removes. (b) Deterministic interpolants are the deployed default and are accurate on short gaps, but whatever they place inside a gap is a function of the two endpoints alone, so a long gap is filled by a chord that cannot express the excursion it spans. (c) Generic learned imputers are trained under MCAR, the one regime that does not occur, and damp the excursion they must recover. (d) CAIR learns that base curve instead of computing it: a jointly trained BiGRU interpolator whose base curve $y^{0}$ enters a Transformer refiner both as conditioning and as a residual base, unrolled for three passes and conditioned on whatever co-recorded context the deployment setting provides.

Method	MCAR $\downarrow$	MAR $\downarrow$	NMAR $\downarrow$
CAIR (Ours)	2.66	10.33	23.50
linear interp	2.91	12.34	28.94
MICE	3.26	19.83	49.35
missForest	3.33	16.54	40.90
hot-deck	5.04	16.06	43.44
$k$ NN	5.19	15.45	43.54

Table 1: Realistic missingness imputation on AI-READI CGM (all

352

test participants; RMSE mg/dL averaged over six missingness rates, lower is better). CAIR is best under all three mechanisms and its margin grows with difficulty. The two generic learned imputers fall below every non-constant baseline throughout. Bold = best, underline = second best, per column.

Method	15 min	30 min	45 min	60 min	mean $\downarrow$
CAIR (Ours)	2.84	5.03	6.66	8.23	5.69
akima	2.76	5.01	6.87	8.61	5.81
PCHIP	2.84	5.14	6.92	8.58	5.87
linear	3.20	5.64	7.45	9.09	6.34
SAITS	4.60	7.54	9.76	11.79	8.42
BRITS	6.25	9.42	11.67	13.33	10.17

Table 2: Imputation performance vs. gap length on AI-READI CGM (all

352

test participants; RMSE mg/dL). CAIR is best on the mean and at

45

–

60

min, and second by a small margin at

15

–

30

min. Bold = best, underline = second, per column.

Figure 2: Missingness mechanism examples. (AI-READI CGM at

30\%

missing; five representative methods, drawn at held-out positions only). Each panel is its mechanism’s median-difficulty window; insets give in-gap RMSE (mg/dL). MCAR gaps are near-trivial; NMAR puts contiguous gaps on the excursions crossing the dashed

150

mg/dL line.

Method	meal	sleep	asc	dip	comb	AVG $\downarrow$
CAIR (Ours)	15.25	24.94	9.01	7.49	16.37	14.61
PCHIP	16.56	29.26	7.86	7.36	18.65	15.94
linear	16.97	29.27	8.42	7.31	18.30	16.05
akima	17.82	36.73	8.02	7.24	19.64	17.89
Kalman	17.97	34.70	8.88	8.09	20.29	17.99
Savitzky–Golay	17.27	29.45	13.41	11.22	19.43	18.15

Table 3: Imputation of physiologically-significant missingness on AI-READI (all

352

test participants; RMSE mg/dL, lower is better). CAIR attains the best average over the full baseline suite, and the gain concentrates in the long sleep blocks. The four weakest classical fills are in the supplement. Bold = best, underline = second best, per column.

(a) Physiological vs. random-gap training masks
Training masks	ABP RMSE $\downarrow$
	MCAR	MAR	NMAR
linear	4.40	9.23	11.85
CAIR, CGM masks	5.47	8.74	12.10
CAIR, random-gap	4.30	8.60	11.42
(b) Clinical-burden recovery

Table 4: Two analyses behind the MIMIC-III results (ABP). (a) The identical architecture, trained on the same ABP data, is less accurate than linear under MCAR and NMAR when trained on masks carried over from glucose physiology, and more accurate under all three under the random-gap curriculum. (b) Linear is worse than mean-fill (

\mathrm{MRR}<0

) under MCAR/MAR; CAIR and

k

NN recover most of the burden. Bold = best, underline = second, per column. A third analysis, in which no imputer separates from any other when predicting mortality from arterial pressure alone, is reported in Sec. 4.7 .

	Heart rate (bpm) $\downarrow$				Resp. (br/min) $\downarrow$
Method	15	30	45	60	15	30	45	60
Akima	4.54	5.93	6.86	7.87	2.73	3.23	3.72	4.30
AR (bidir.)	4.71	5.62	6.09	6.72	2.77	3.09	3.33	3.57
PCHIP	4.45	5.62	6.27	7.05	2.63	3.07	3.43	3.76
Linear	4.40	5.52	6.12	6.86	2.61	3.00	3.34	3.65
CAIR (ours)

Table 5: Cross-modal conditioning is what transfers (AI-READI, gap-length protocol, RMSE in native units, lower is better). Column headings are gap lengths in minutes; “Resp.” is respiration and “AR (bidir.)” is bidirectional AR. Univariate CAIR is less accurate than linear at every gap length on both signals; adding the co-recorded context makes it best everywhere. On glucose the same context adds nothing (Sec. 4.5 ). Bold = best, underline = second, per column.

	ABP-mean (mmHg) $\downarrow$			HR (bpm) $\downarrow$
Method	MCAR	MAR	NMAR	MCAR	MAR	NMAR
CAIR (Ours)	4.30	8.60	11.42	2.26	5.28	6.48
linear interp	4.40	9.23	11.85	2.22	5.50	6.42
GP-VAE	11.03	12.82	17.94	9.17	14.01	18.00
M-RNN	13.45	13.19	18.35	15.28	14.53	18.60

Table 6: Cross-domain transfer to MIMIC-III ICU vitals (reconstruction RMSE, mean over six missingness rates, lower is better). CAIR is the most accurate method on arterial pressure under every mechanism; heart rate is a boundary case. The CAIR row is the multivariate variant; both variants, the remaining baselines and per-mechanism Wilcoxon tests are in the supplement. Bold = best, underline = second, per column.

Variant	MCAR	MAR	NMAR	ALL $\downarrow$	$\Delta$
CAIR (full)	2.66	10.33	23.50	12.16	n/a
w/o aux. loss	6.36	10.75	27.59	14.90	$+2.74$
w/o residual	4.45	10.34	27.38	14.06	$+1.90$
GRU $\to$ self-attn.	4.15	10.22	27.14	13.84	$+1.68$
linear interp	2.91	12.34	28.94	14.73	$+2.57$

Table 7: Component ablation on AI-READI CGM, on the protocol of Table 1 (all

352

test participants; RMSE mg/dL over six missingness rates). ALL is the mean of the three mechanisms,

\Delta

the change against the full model. Rows are single training runs under the published recipe, differing only in the ablated component; linear is repeated from Table 1 for scale. Bold = best, underline = second best, per metric column.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Pretrain, then fine-tune	meal	sleep	asc	dip	comb	AVG
control (AI-READI only)	15.60	27.52	5.96	5.14	14.74	13.79
$+$ HUPA-UCM (T1D)	15.99	26.32	6.21	5.36	15.66	13.91
$+$ OhioT1DM (T1D)	15.77	27.14	6.08	5.21	15.25	13.89
$+$ Shanghai (T2D)	15.71	27.58	6.13	5.20	15.18	13.96

Appendix

Table 8: External-cohort pretraining does not improve accuracy (physiological protocol,

352

participants, single seed; RMSE mg/dL). Every external cell is within

0.2

mg/dL of the control and none improves on it. Bold = best, underline = second best, per column.

Method	meal	sleep	asc	dip	comb	AVG $\downarrow$
CAIR (Ours)	15.25	24.94	9.01	7.49	16.37	14.61
Savitzky–Golay	17.27	29.45	13.41	11.22	19.43	18.15
EWMA	20.59	31.69	21.45	18.32	23.11	23.03
local mean	22.66	32.23	24.00	20.65	24.40	24.79
cubic spline	22.64	45.22	21.90	18.62	23.85	26.44
LOCF	25.91	37.33	34.19	29.36	31.81	31.72

Appendix

Table 9: The four baselines omitted from Table 3 of the main paper (AI-READI, physiological five-strategy protocol, all

352

test participants; RMSE mg/dL, lower is better). The lower block is the omitted group; CAIR and Savitzky–Golay are repeated from Table 3 as the best method and the weakest one carried there. Every omitted method is at least

4.9

mg/dL behind that floor on the average. Bold = best, underline = second best, per column.

Conditioning set	meal	sleep	asc	dip	AVG
target only (control)	14.94	25.17	6.07	4.93	13.11
$+$ heart rate	14.93	25.16	6.03	4.92	13.10
$+$ steps, calories	14.89	25.25	6.10	4.97	13.14
$+$ sleep state	14.89	25.27	5.98	4.92	13.09
$+$ respiration, stress	14.79	24.96	6.00	4.89	12.98
$+$ environment	14.82	24.83	6.24	5.06	13.10

Appendix

Table 10: Conditioning-modality ladder on AI-READI CGM (five-seed ensemble per rung; RMSE mg/dL). Adding modalities does not improve glucose imputation; the AVG spread (

12.98

–

13.14

) is within seed noise. AVG is over all five physiological strategies; the combined column is omitted for space. Absolute values are not comparable to Table 3 of the main paper. Bold / underline mark the best/second AVG.

Method	MCAR	MAR	NMAR
CAIR (univariate)	4.26	8.58	11.44
CAIR (multivariate)	4.30	8.60	11.42
linear interp	4.40	9.23	11.85
missForest	4.62	9.60	12.91
MICE	4.88	9.02	12.65
$k$ NN	5.27	8.91	12.69

Appendix

Table 11: Complete MIMIC-III arterial-pressure reconstruction RMSE (mmHg, mean over six missingness rates). Bold = best, underline = second best, per column.

Method	MCAR	MAR	NMAR
CAIR (univariate)	2.26	5.36	6.72
CAIR (multivariate)	2.26	5.28	6.48
linear interp	2.22	5.50	6.42
missForest	2.37	7.40	9.25
MICE	2.49	7.55	9.63
$k$ NN	2.69	7.09	9.21

Appendix

Table 12: Complete MIMIC-III heart-rate reconstruction RMSE (bpm, mean over six missingness rates). The CAIR (multivariate) and linear rows are those of Table 6 of the main paper. CAIR is the most accurate method under MAR and linear interpolation under MCAR and NMAR, the two separated by at most

0.22

bpm; the per-mechanism significance tests for those differences are in Table 13 . Bold = best, underline = second best, per column.

Signal	Mechanism	$\Delta$	$p$
ABP-mean (mmHg)	MCAR	$-0.105$	$7.9\times 10^{-9}$
	MAR	$-0.625$	$4.1\times 10^{-21}$
	NMAR	$-0.375$	$2.9\times 10^{-3}$
Heart rate (bpm)	MCAR	$+0.040$	$7.0\times 10^{-40}$
	MAR	$-0.220$	$0.83$
	NMAR	$+0.060$	$0.37$

Appendix

Table 13: Paired Wilcoxon signed-rank tests on MIMIC-III , CAIR (multivariate) versus linear interpolation, per (window, rate) pair.

\Delta

is the mean of the per-pair RMSE differences in native units, which need not equal the difference of the aggregate RMSEs in Table 6 of the main paper because RMSE does not aggregate linearly; negative favors CAIR. All three arterial-pressure tests favor CAIR; on heart rate, linear interpolation is significantly better under MCAR and the remaining two differences are not significant (

\alpha=0.05

Method	RMSE (mmHg) $\downarrow$	Burden MRR $\uparrow$
Low RMSE, burden not preserved
linear interp	11.85	$+0.14$
LOCF	12.79	$+0.07$
Fourier	13.56	$+0.05$
Recovers burden, high RMSE
MICE	12.65	$+0.63$

Appendix

Table 14: Only CAIR ranks among the best on both axes (MIMIC-III ABP, NMAR, mean over six rates). Reconstruction RMSE (mmHg, lower better) and clinical-burden recovery (MRR on time in/above/below range, higher better). Interpolants match the RMSE of CAIR but fail to preserve the burden; tabular and neural imputers recover the burden at much higher RMSE, with

k

NN and hot-deck matching its recovery at

{\sim}11\%

higher RMSE. Bold = best, underline = second best, per column.

Fill	MCAR AUROC	NMAR AUROC
oracle (recorded trace)	0.679	0.678
mean-fill	0.677	0.683
$k$ NN	0.680	0.685
CAIR (multivar.)	0.688	0.688
linear interp	0.692	0.688
PCHIP	0.692	0.689

Appendix

Table 15: Hard outcomes do not reward reconstruction fidelity (MIMIC-III, in-hospital mortality predicted from arterial pressure alone,

30\%

missingness, 1D-CNN readout;

n=1200

under MCAR and

1181

under NMAR). The oracle is the recorded trace and is the ceiling for this task, yet four fills exceed it under MCAR and all five do under NMAR, because interpolation denoises the vital. No column is ordered as reconstruction accuracy would predict, and no best value is marked.

	RMSE (mg/dL) $\downarrow$		TIR recovery $\uparrow$
Study group	CAIR	linear	CAIR	linear
healthy	18.29	21.49	0.165	0.097
pre-diabetes	15.90	21.15	0.467	0.382
oral medication	33.96	40.19	0.601	0.556
insulin-dependent	27.23	35.17	0.608	0.489

Appendix

Table 16: The CGM result holds in every diabetes study group (AI-READI, NMAR, all

352

test participants). CAIR attains lower reconstruction error and higher time-in-range recovery than linear interpolation in all four groups. Bold = better of the two, per group and metric.

Figure 3: AI-READI CGM, both missingness rates (columns: mechanism; rows: rate). The main paper shows the

30\%

row. Raising the rate from

15\%

30\%

lengthens and multiplies the blocks but does not change the ordering: CAIR and linear track the trace under MCAR, separate under MAR, and separate furthest under NMAR, where the dashed

70

150

mg/dL thresholds mark the excursions that trigger the deletion. GP-VAE and M-RNN revert to a near-constant fill under every mechanism, which is what puts them below every non-constant baseline in Table 1 of the main paper.

Figure 4: MIMIC-III arterial pressure, same protocol. Pale regions marked not recorded are positions the ICU monitor never sampled (

23\%

of ABP positions); they are neither observed nor scored, and no ground truth is drawn there. ABP is smoother and more locally linear than glucose, so the per-window gap between CAIR and linear is small and its sign varies from window to window (three of these six median-difficulty panels favour CAIR, two favour linear and one is a tie), while the six-rate aggregate in Table 6 of the main paper favours CAIR under all three mechanisms. The neural baselines fail here exactly as they do on CGM.

Figure 5: Easiest to hardest, at fixed mechanism and rate (AI-READI CGM, NMAR,

30\%

; panels at the

0

th,

33

rd,

67

th and

100

th percentile of CAIR’s in-gap RMSE). These panels support the claim in Sec. 4.8 of the main paper that CAIR degrades gracefully. As the blocks lengthen and swallow whole excursions, no method recovers the excursion, but the failures differ in kind: GP-VAE and M-RNN sit at a constant fixed by the training distribution, linear draws the chord its endpoints imply, and CAIR bends part of the way toward the excursion and stops. In the hardest panel the recorded in-gap peak is

354

mg/dL; CAIR reaches

215

and linear

176

. Under-shooting is the safe direction of error for a clinical burden metric, since an invented excursion would create a treatment signal that never occurred.

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