--- kind: concept slug: time-series-imputation title: Time-series imputation summary: >- Filling missing values in a time series. Two methods are on the wiki. GRIL, a bidirectional graph-recurrent imputer, suits multivariate series whose channels sit on a fixed graph and beats BRITS on the paper's real data. A denoising autoencoder on Gramian angular field images fills gaps in single series, with gains only on the missing points. A 2023 survey sets out the task and its settings. tags: [time-series, imputation, graph-neural-network, autoencoder] papers: ["1506.00327", "2108.00298", "2307.03759"] related: [graph-neural-network, recurrent-neural-network, denoising-autoencoder, time-series-forecasting, time-series-classification] status: draft updated_at: 2026-10-24 --- ## Overview Sensors drop out, so some entries of a multivariate series are missing. The task is to reconstruct them, using the series' own past and future and, where available, the relations between channels [arxiv:2108.00298 §1]. The paper assumes that the missingness is stationary and mostly missing at random, and it puts no limit on the number or length of missing blocks [arxiv:2108.00298 §3]. It distinguishes two evaluation settings. In-sample imputation trains and tests on the same sequence. Out-of-sample imputation trains and tests on disjoint sequences [arxiv:2108.00298 §3]. The survey uses the same two settings and describes imputation as a task head on a spatial-temporal encoder, trained without clean values for the gaps, so it needs a surrogate objective such as synthetic missing values [arxiv:2307.03759 §3.1]. The invariant GRIL keeps is that observed entries are not changed. Only the gaps receive predictions. GRIL enforces this with a fill operator that keeps the observed values and uses the model output only where the mask is zero [arxiv:2108.00298 §4, eq. 8]. The GASF method does not state this rule. Its reported errors cover the missing points only, and its full-series error is higher in every row [arxiv:1506.00327 tab. 2]. If you use it, apply the same fill step so observed values stay as they are. That is a recommendation of this page, not a claim from the paper. The survey gives no masking scheme for evaluation [arxiv:2307.03759 §3.1]. GRIL's masks are the paper's own choices: random block drops and 25% point masking [arxiv:2108.00298 §5]. The GASF paper uses another protocol. Missing points are simulated by setting them to zero, and 20% of training points are zeroed [arxiv:1506.00327 §4.1]. Compare methods only under the same masking protocol, and do not read the two papers' numbers against each other. ## Variants ### Graph recurrent imputation (GRIL) A bidirectional recurrent network runs once forward and once backward in time. Its GRU gates are message-passing layers over a sensor graph. Each step imputes twice, first from the hidden state and then from the one-hop neighbours, and a final MLP combines the two directions [arxiv:2108.00298 §4, eq. 3-11]. The paper's message operator is diffusion convolution [arxiv:2108.00298 §5] [app. A]. Card: [Filling the Gaps: Multivariate Time Series Imputation by Graph Neural Networks](/papers/2108.00298). The survey lists the same model, under the name GRIN, in its imputation section [arxiv:2307.03759 §7.1]. Reported results (author-reported, mean of 5 runs): - Out-of-sample on AQI-36: MAE 12.08 against 14.50 for BRITS [arxiv:2108.00298 tab. 1]. - Point-missing PEMS-BAY: MAE 0.67 against 1.47 for BRITS [arxiv:2108.00298 tab. 2]. - Block-missing METR-LA: MAE 2.03 against 2.34 for BRITS [arxiv:2108.00298 tab. 2]. - Synthetic particle data with a fixed charge: MAE 0.0500 against 0.1203 [arxiv:2108.00298 tab. 6]. BRITS is the paper's main baseline. It has no card in this wiki, so it is named here only, and no content is taken from it. Notes for implementers, from the card's "Paper vs code" section: - The paper calls the model GRIL. The released code calls it GRIN. The decoder in the code has projections the paper's equations do not write down, and it adds node embeddings the paper's eq. 11 omits [arxiv:2108.00298 §4] [code: lib/nn/layers/gril.py#L15]. - The paper cuts edges beyond about 40 km. The code keeps edges whose kernel weight is at least 0.1 and sets no distance cutoff [arxiv:2108.00298 app. B.1] [code: lib/utils/utils.py#L209-L211]. - The code uses dense N×N supports at every step. The paper gives O(E) for a sparse implementation and O(N²) for a dense one, and it reports no runtime figures [arxiv:2108.00298 app. C.3] [code: lib/nn/layers/spatial_conv.py#L58]. - The graph is assumed fixed over time. The paper gives no adaptation for dynamic graphs [arxiv:2108.00298 §3]. ### Gramian angular field denoising autoencoder (GASF) The series is rescaled to $[0,1]$ and turned into a Gramian angular summation field (GASF). This is an $n \times n$ image whose cell $(i, j)$ is the cosine of the sum of two angles, one per time step [arxiv:1506.00327 §2.1]. A single-hidden-layer denoising autoencoder with 500 sigmoid units learns to rebuild clean GASF images from corrupted ones [arxiv:1506.00327 §4.1]. At imputation time, the gaps are set to zero, the autoencoder repairs the image, and the values are read back from its diagonal. The inverse used there, $\cos\phi = \sqrt{(\cos 2\phi + 1)/2}$, holds only for $[0,1]$ scaling [arxiv:1506.00327 §4]. Card: [Imaging Time-Series to Improve Classification and Imputation](/papers/1506.00327). The paper's main subject is classification. Imputation is one section of it. Reported results (author-reported, MSE averaged over 10 runs, five UCR series): - Interpolation MSE, which counts only the missing points: ECG 0.01196 for GASF against 0.02301 for the raw-series autoencoder [arxiv:1506.00327 tab. 2]. - The paper reports reductions of 12.18% to 48.02% across the five series [arxiv:1506.00327 §4.2]. - Full-series MSE is higher with GASF in all five rows, for example ECG 0.01148 against 0.01001 [arxiv:1506.00327 tab. 2]. Notes for implementers: - The corruption is zeros, so a real zero in the data cannot be told apart from a removed point unless a mask is kept. The test-time corruption rate is not stated [arxiv:1506.00327 §4.1]. - Which minimum and maximum set the rescaling range is not stated. Store them per series to undo the scaling [arxiv:1506.00327 §4]. - The image has one cell per pair of time steps, so memory grows with the square of the series length [arxiv:1506.00327 §2.1]. - Four of the five series in tab. 2 are also in the autoencoder's training list. The held-out series is "7 Misc" [arxiv:1506.00327 §4.1]. - No official code was found for this paper, so the sketch below follows the paper's equations. ## When to use which | Situation | Use | Why | |---|---|---| | Channels have a known or computable similarity graph that stays fixed over time | GRIL | Its spatial decoder uses neighbour information. On METR-LA block missing, the learned graph gives MAE 2.03, against 2.34 for BRITS [arxiv:2108.00298 tab. 2] [arxiv:2108.00298 tab. 9] | | One univariate series, no graph needed | GASF autoencoder, with the notes above | It reports lower interpolation MSE than raw-series autoencoding on five UCR series, by 12.18% to 48.02% [arxiv:1506.00327 §4.2] [arxiv:1506.00327 tab. 2]. It is compared with raw-series autoencoding only, not with GRIL or BRITS | | Multivariate channels with no useful graph | Not established by this wiki | The graph ablation on METR-LA block missing gives MAE 3.42 with no edges, worse than BRITS at 2.34 [arxiv:2108.00298 tab. 9]. The GASF method is for single series | | Full-series reconstruction matters, not only the gaps | GRIL, with its fill step | GASF raises full-series MSE in every row of its table [arxiv:1506.00327 tab. 2]. GRIL keeps observed entries [arxiv:2108.00298 §4, eq. 8] | | Graph changes over time | Not supported as written | The method assumes a fixed W [arxiv:2108.00298 §3]. The paper gives no dynamic variant | | Non-stationary missingness | Not supported as written | The paper lists this as future work [arxiv:2108.00298 §6] | | Many channels, tight compute budget | Check before committing | Dense N×N messages dominate the released code's cost [arxiv:2108.00298 app. C.3]. The paper reports no timings or hardware | | Point-missing data (scattered gaps) | GRIL, with the caveat on dataset size | Largest reported gains are in point-missing settings: 54% lower MAE than BRITS on PEMS-BAY and 55% on CER-E [arxiv:2108.00298 tab. 2]. Five runs on a few benchmarks [arxiv:2108.00298 §5] | | Natural missing rate of your data is unknown or very different from the benchmarks | Measure it first, then test on your own masks | The survey's benchmark table gives natural missing ratios of 0.003% for PeMS-BAY, 8.109% for METR-LA and 25.7% for AQI [arxiv:2307.03759 app. F]. The reported gains of both methods come from injected masks [arxiv:2108.00298 §5] [arxiv:1506.00327 §4.1] | ## Implementation sketch The GRIL core is the fill step and the masked loss. The GASF core is a repair of an image followed by a read-back of the diagonal. Both sketches follow the papers' equations, not their code. The papers' own pseudocode is not reproduced. ```pseudocode # Keep observed entries, fill gaps with the model's prediction [§4, eq. 8] # Inputs: X (values), M (1 = observed), Y (prediction), all (T, N, d) def fill(Y, X, M): return M * X + (1 - M) * Y # Loss: MAE on originally observed entries only [app. A] # Hide an extra 5% of observed inputs during training [app. A] def training_loss(model, X, M, W): M_in = M * bernoulli(0.95) # (T, N, d) out = model(X, M_in, W) # prediction for every entry return mean(abs(M * (X - out))) # computed on M, not on M_in ``` For GRIL, `model` is the bidirectional network described above, and `W` is the fixed adjacency built from sensor distances. The paper's kernel is `exp(-dist^2 / gamma)`, cut at a threshold [arxiv:2108.00298 app. B.1, eq. 12]. ```pseudocode # Gramian angular summation field of one series, values in [0, 1] [§2.1] def gasf(z): phi = arccos(z) # angle per time step return cos(phi[:, None] + phi[None, :]) # cell (i, j) = cos(phi_i + phi_j) # Impute gaps with a GASF denoising autoencoder [§4, §4.1] # Inputs: x (n,), observed mask m (n,) with 1 = observed, trained dae # Output: x with gaps filled, observed points unchanged def impute_gasf(x, m, dae): # Scale to [0, 1]. The paper does not say which values set the range; # this sketch uses the observed points [§4] lo, hi = min(x[m]), max(x[m]) z = (x - lo) / (hi - lo) # (n,) in [0, 1] # Gaps are zeroed, as in the training corruption [§4.1] z = where(m, z, 0) # (n,) # Repair the GASF image; diagonal cell (i, i) holds cos(2 phi_i) [§4] img = dae(gasf(z)) # (n, n) # Invert the diagonal: cos(phi) = sqrt((cos(2 phi) + 1) / 2) [§4] z_hat = sqrt((diagonal(img) + 1) / 2) # (n,) # Undo the scaling and keep observed values [§4] [arxiv:2108.00298 §4] x_hat = z_hat * (hi - lo) + lo # (n,) return where(m, x, x_hat) ``` ## Contradictions - **Synthetic varying-charge setting, text against table.** The text says BRITS is "more accurate" in the varying-charge setting [arxiv:2108.00298 app. C.1]. The table shows GRIL ahead in both settings: MAE 0.0530 against 0.1089 for BRITS with varying charge [arxiv:2108.00298 tab. 6]. The card reads the sentence as "BRITS does better with varying than with fixed charge". Trust the table. This page does not resolve it. - **No shared baseline.** GRIL is compared with BRITS on multivariate data. The GASF paper compares only with raw-series autoencoding, on univariate UCR series [arxiv:1506.00327 §4.1, tab. 2]. Their numbers measure different things and cannot be ranked against each other. The survey reports no results of its own, so it cannot conflict with GRIL's numbers [arxiv:2307.03759 §1]. It calls the model GRIN, where the paper calls it GRIL [arxiv:2307.03759 §7.1] [arxiv:2108.00298 §4]. The name is the same model. ## Papers - [Imaging Time-Series to Improve Classification and Imputation](/papers/1506.00327): encodes series as GASF, GADF and MTF images. Its imputation section trains a denoising autoencoder on GASF images and reports lower interpolation MSE than raw-series autoencoding on five UCR series. - [Filling the Gaps: Multivariate Time Series Imputation by Graph Neural Networks](/papers/2108.00298): introduces GRIL, a bidirectional graph-recurrent imputer. It reports lower MAE than BRITS on AQI, traffic, smart-meter and synthetic data. - [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](/papers/2307.03759): a survey with no experiments. It defines imputation as a task with a head and a surrogate objective, names the in-sample and out-of-sample settings, and lists benchmark missing ratios.