CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation
Authors: YUSUKE TASHIRO
Jiaming Song
Yang Song
Stefano Ermon
Venue: NeurIPS 2021 Poster (2021)
Keywords: time series imputation
generative modeling
deep learning
score-based diffusion models

Abstract: The imputation of missing values in time series has many applications in healthcare and finance. While autoregressive models are natural candidates for time series imputation, score-based diffusion models have recently outperformed existing counterparts including autoregressive models in many tasks such as image generation and audio synthesis, and would be promising for time series imputation. In this paper, we propose Conditional Score-based Diffusion model (CSDI), a novel time series imputation method that utilizes score-based diffusion models conditioned on observed data. Unlike existing score-based approaches, the conditional diffusion model is explicitly trained for imputation and can exploit correlations between observed values. On healthcare and environmental data, CSDI improves by 40-65% over existing probabilistic imputation methods on popular performance metrics. In addition, deterministic imputation by CSDI reduces the error by 5-20% compared to the state-of-the-art deterministic imputation methods. Furthermore, CSDI can also be applied to time series interpolation and probabilistic forecasting, and is competitive with existing baselines. The code is available at https://github.com/ermongroup/CSDI.

CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation
Yusuke Tashiro 123* Jiaming Song', Yang Song', Stefano Ermon
Depaarment of Computer Science, Stanford University, Stanford, CA, USA Mitsubishi UFJ Trust Investment Technology Institute, Tokyo, Japan
*Japan Digital Design, Tokyo, Japan fytashiro, tsong, songyang, ermon) @ecooonneccccs stanford.
Abstract
The imputation of missing values in time series has many applications in healthcare and finance. While autoregressive models are natural candidates for time series imputation, score-based diffusion models have recently outperformed existing counterparts including autoregressive models in many tasks such as image genera- tion and audio synthesis, and would be promising for time series imputation. In this paper, we propose Conditional Score-based Diffusion models for Imputation (CSDI), novel time series imputation method that utilizes score based diffusion models conditioned on observed data. Unlike existing score-based approaches, the conditional diffusion model is explicitly trained for imputation and can exploit correlations between observed values. healthcare and environmental data, CSDI improves by 40-65% over existing probabilistic imputation methods popular performance metrics. In addition, deterministic imputation by CSDI reduces the eπror by 5-20% compared to the state-of-the the-hrt deterministic imputation methods. Furthermore, CSDI can also be applied to time series interpolation and probabilistic forecasting, and is competitive with existing baselines. The code is available at https://gil hhtb.......tt../ com/ www.ernoooo/CSDI......
1 Introduction
Multivariate time series are abundant in real world applications such as finance, meteorology and healthcare. These time series data often contain missing values due to various reasons, including device failures and human errors [1, 3]. Since missing values can hamper the interpretation of time series, many studies have addressed the task of imputing missing values using machine learning techniques [4, 5, 6]. the past few years, imputation methods based on deep neural networks have shown great success for both deterministic imputation [7, 8, 9] and probabilistic imputation [10]. These imputation methods typically utilize autoregressive models to deal with time series.
Score-based diffiusion models a class of deep generative models and generate samples by gradually converting noise into plausible data sample through denoising have recently achieved of the-art sample quality in many tasks such as image generation [11, 12] and audio synthesis [13, 14]. outperforming counterparts including autoregressive models. Diffusion models can also be used to impute missing values by approximating the scores of the posterior distribution obtained from the prior by conditioning on the observed values [12, 15, 16). While these approximations may work well in practice, they do not correspond to the exact conditional distribution.
In this paper, we propose CSDI, a novel probabilistic imputation method that directly learns the conditional distribution with conditional score-based diffusion models. Unlike existing score-based approaches, the conditional diffusion model is designed for imputation and can exploit useful information in observed values. We illustrate the procedure of time series imputation with CSDI in
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Conditional observations x6
Px [x xx))
x
x
Random noise x
Imputation targets x
The procedure of time series imputation with CSDI. The reverse process gradually converts random noise into plausible time series, conditioned on observed values x Dashed lines in each box represent observed values, which are plotted in order to show the relationship with generated imputation and not included each x!
Figure 1:
Figure 1. We start imputation from random noise the left of the figure and gradually convert the noise into plausible time series through the reverse process of the conditional diffusion model. At each step f, the reverse process removes noise from the output of the previous step (t + 1). Unlike existing score-based diffusion models, the reverse process can take observations (on the top left of the figure) as a conditional input, allowing the model exploit information in the observations for denoising. We utilize an attention mechanism to capture the temporal and feature dependencies of time series.
For training the conditional diffusion model, we need observed values (i.e., conditional information) and ground-truth missing values (i.e., imputation targets). However, in practice we do not know the ground-trum truth missing values, or training data may not contain missing values at all. Then, inspired by masked language modeling, develop self-supervised training method that separates observed values into conditional information and imputation targets. We note that CSDI is formulated for general imputation tasks, and is not restricted to time series imputation.
Our main contributions are as follows:
We propose conditional score-based diffusion models for probabilistic imputation (CSDI), and implement CSDI for time series imputation. To train the conditional diffusion model, we develop a self-supervised training method.
We empirically show that CSDI improves the continuous ranked probability score (CRPS) by 40-65% over existing probabilistic methods on healthcare and environmental data. Moreover, deterministic imputation with CSDI decreases the mean absolute error (MAE) by 5-20% compared to the state-of-the-ar methods developed for deterministic imputation.
We demonstrate that CSDI can also be applied to time series interpolations and probabilistic forecasting, and competitive with existing baselines designed for these tasks.
2
Related works
Time series imputations with deep learning Previous studies have shown deep learning models capture the temporal dependency of time series and give accurate imputation than statistical methods. A popular approach using deep learning is to use RNNs, including LSTMs and GRUs, for sequence modeling [17, 8, 7]. Subsequent studies combined RNNs with other methods to improve imputation performance, such as GANs [9, 18, 19] and self-training [20]. Among them, the combination of RNNs with attention mechanisms is particularly successful for imputation and interpolation of time series [21, 22). While these methods focused on deterministic imputation, GP-VAE [10] has been recently developed probabilistic imputation method.
Score-based generative models Score-based generative models, including score matching with Langevin dynamics [23] and denoising diffusion probabilistic models [11], have outperformed existing methods with other deep generative models in many domains, such images [23, 11],

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audio [13, 14], and graphs [24]. Most recently. TimeGrad [25] utilized diffusion probabilistic models for probabilistic time series forecasting. While the method has shown state-of-the-an performance, cannot be applied to time series imputation due to the use of RNNs to handle past time series.
3 Background
3.1 Multivariate time series imputation
We consider N multivariate time series with missing values. Let us denote the values of each time series as X (II. R where K is the number of features and L is the length of time series. While the length L can be different for each time series, we treat the length of all time series as the same for simplicity, unless otherwise stated. We also denote observation mask as M = {m.:K.1.L} E {0, _| where m = O if I&, is missing, and m if I I is observed. We assume time intervals between two consecutive data entries can be different, and define the timestamps of the time series as = { E R In summary, each time series is expressed as {X, M,
Probabilistic time series imputation is the task of estimating the distribution of the missing values of X by exploiting the observed values of X. We note that this definition of imputation includes other related tasks, such as interpolation, which imputes all features at target time points, and forecasting, which imputes all features at future time points.
3.2 Denoising diffusion probabilistic models
Let us consider learning model distribution (xx) that approximates data distribution ((xo)- Let X2 for t 1...., T be sequence of latent variables in the same sample space as Xo. which is denoted as X. Diffusion probabilistic models [26] are latent variable models that are composed of two processes: the forward process and the reverse process. The forward process is defined by the following Markov chain:
T
q(x ((X xo) = II (xx (x Xt-1) where g(x: == (VI-((VVVV -Ix--
(1)
and Be is a small positive constant that represents a noise level. Sampling of x, has the closed-form written as q(x; Xo) = =N(x;; V (1 -x.)I) where á, := 1 - 3, and or == II d;. Then, x, be expressed X; = Xa (1 or)e where NO, I). On the other hand, the reverse process denoises x2 to recover Xo. and is defined by the following Markov chain:
T
po(Xo:)) p(xT)
II px xx), XT ~ NO., I),
t=1
-0 x,) == (x t). t)...
Ho et al. [1]] has recently proposed denoising diffusion probabilistic models (DDPM), which considers the following specific parameterization of px xx):
1 p 1) -
[xx Bi ar
VI at E 10(xe.1)
00 (x 6) = 32 where B,
1-at-1
1-ai
t> 1 Br
B1
t = 1
(3)
where eg is trainable denoising function. We denote (xo () and (x( in Eq. (3) as DDPM (xx, L, ()) and DDDM (xe, (). respectively. The denoising function in Eq. (3) also corre- sponds to a rescaled score model for score-based generative models [23]. Under this parameterization. Ho et al. [11] have shown that the reverse process can be trained by solving the following optimization
problem:
min L(0) min ex
() where x, VorX0 + (1 oz)e. (4)
The denoising function eg estimates the noise vector e that was added to its noisy input x,, This training objective also be viewed as weighted combination of denoising score matching used for training score-bace basee generative models [23, 27, 12]. Once trained, we can sample Xg from Eq. (2).
We provide the details of DDPM in Appendix A.
3

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3.3 Imputation with diffusion models
Here, we focus on general imputation tasks that are not restricted time series imputation. Let us consider the following imputation problem: given a sample X, which contains missing values, we generate imputation targets xi E Xt by exploiting conditional observations X E A where X and Ao are part of the sample space X' and vary per sample. Then, the goal of probabilistic imputation is to estimate the true conditional data distribution g(x)* xx with model distribution (x5" x We typically impute all missing values using all observed values, and set all observed values as X) and all missing values as x respectively. Note that time series imputation in Section 3.1 can be considered as special case of this task.
Let us consider modeling pe po(x X with diffusion model. In the unconditional case, the reverse process po (Xo:r) is used to define the final data model pa (xu). Then, a natural approach is to extend the reverse process in Eq. (2) to a conditional one:
po x = p(x))) ) II po (x)=1 X ,60)
x7 ~ NO,,I)
(5)
po(x x x6 == M>{] 140 (x)"", , x6 oo IT
However, existing diffusion models generally designed for data generation and do not take conditional observations x6 as inputs. To utilize diffusion models for imputation, previous stud- ies [12, 15, 16] approximated the conditional reverse process pe (x1= x xx with the reverse process in Eq. (2). With this approximation, in the reverse process they add noise to both the target and the conditional observations x6 While this approach can impute missing values, the added noise can harm useful information in the observations. This suggests that modeling pe (x/=_ x x(") without approximations can improve the imputation quality. Hereafter, we call the model defined in Section 3.2 as the unconditional diffusion model.
4 Conditional score-based diffusion model for imputation (CSDI)
In this section, we propose CSDI, a novel imputation method based on conditional score-based diffusion model. The conditional diffusion model allows us to exploit useful information in observed values for accurate imputation. We provide the reverse process of the conditional diffusion model, and then develop self-supervised training method. We note that CSDI is not restricted to time series.
4.1 Imputation with CSDI
We focus on the conditional diffusion model with the reverse process in Eq. (5) and aim to model the conditional distribution p(x{__ xx xxx)) without approximations. Specifically, we extend the parameterization of DDPM in Eq. (3) the conditional case. We define a conditional denoising function co : (Ata x R 0)) - Ata which takes conditional observations xf as inputs. Then, we consider the following parameterization with Cg:
(6) where pt DDPM and DDPM are the functions defined in Section 3.2. Given the function Gø and data Xy. we can sample xi using the reverse process in Eq. (5) and Eq. (6). For the sampling, we set all observed values of xg as conditional observations x and all missing values as imputation targets x Note that the conditional model is reduced to the unconditional one under no conditional observations and can also be used for data generation.
x 1 DDDP Ex("", x))),
ao(x! / x = DDPM (x} ;))
4.2 Training of CSDI
Since Eq. (6) uses the same parameterization as Eq. (3) and the difference between Eq. (3) and Eq. (6) is only the form of eg. we can follow the training procedure for the unconditional model in Section 3.2. Namely, given conditional observations xi and imputation targets xi we sample noisy targets x! /aixh" + (1 a,)c. and train Eq by minimizing the following loss function:
min C(0) = min (0a - cx(" t xxx)) @
(7)
9

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Noïse
minimize ||e - Eg (xt xo")
Imputation targets x
Noisy targets x!"
Observed values
Eg (xxx xx")
function Eg
Conditional observations X
Figure 2: The self-supervised training procedure of CSDI. On the middle left rectangle, the green and white areas represent observed and missing values, respectively. The observed values are separated into red imputation targets x) and blue conditional observations x6 and used for training of eg. The colored areas in each rectangle mean the existence of values.
Table 1: Imputation targets x and conditional observations xij for CSDI at training and sampling.
imputation targets xij
conditional observations xi)
sampling (imputation)
all missing values
all observed values
subset of the observed values (sampled by a target choice strategy)
the remaining observed values
training
where the dimension of e corresponds to that the imputation targets xb
However, this training procedure has an issue. Since we do not know the ground-truth missing values in practice, is clear how to select x and x from a training sample Xg. To address this issue, we develop self-supervised learning method inspired by masked language modeling [28]. We illustrate the training procedure Figure 2. Given sample Xo, we separate observed values of xo into two parts, and set one of them as imputation targets x and the other conditional observations xh. We choose the targets xh through target choice strategy, which discussed in Section 4.3. Then, sample noisy targets xta and train ελ by solving Eq. (7). We summarize how set x and xta for training and sampling in Table 1. also provide the algorithm of training and sampling in Appendix B.1.
4.3 Choice of imputation targets in self-supervised learning
In the proposed self-supervised learning, the choice of imputation targets is important. We provide four target choice strategies depending on what is known about the missing patterns in the test dataset. We describe the algorithm for these strategies in Appendix B.2.
(1) Random strategy: : this strategy is used when we do not know about missing patterns, and randomly chooses certain percentage of observed values imputation targets. The percentage is sampled from [0%, 100%] to adapt to various missing ratios in the test dataset.
(2) Historical strategy: this strategy exploits missing patterns in the training dataset. Given training sample Xg, we randomly draw another sample X, from the training dataset. Then, we set the intersection of the observed indices of Xy and the missing indices of Xy as imputation targets. The motivation of this strategy comes from structured missing patterns example, missing values often appear consecutively in time series data. When missing patterns in the training and test dataset are highly correlated, this strategy helps the model learn a good conditional distribution.
the real world. For
5

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(3) Mix strategy: this strategy is the mix of the above two strategies. The historical strategy may lead to overfitting to missing patterss in the training dataset. The Mix strategy can benefit from generalization by the random strategy and structured missing patterns by the historical strategy.
(4) Test pattern strategy: when we know the missing patterns in the test dataset, we just set the patterns as imputation targets. For example, this strategy is used for time series forecasting, since the missing patterns in the test dataset are fixed given future time points.
5 Implementation of CSDI for time series imputation
Temporal Transformer layer
Feature Transformer layer
(K, Input Split
(1, C) XX
1K
II
(K,4,)
Concat /split
xL
K
Concat
Output
Figure 3: The architecture of 2D attention. Given a tensor with K features, L length, and C' channels, the temporal Transformer layer takes tensors with (1, L, L, C) shape as inputs and learns temporal dependency. The feature Transformer layer takes tensors with (K, 1, C*) shape as inputs and learns feature dependency. The output shape of each layer the same as the input shape.
In this section, we implement CSDI for time series imputation. For the implementation, we need the inputs and the architecture of ep.
First, we describe how process time series data as inputs for CSDI. As defined in Section 3.1, time series is denoted as (X, M. s), and the sample space X' of X is R We want to handle X in the sample space R x for learning dependencies in time series using a neural network, but the conditional denoising function Eş takes inputs x!" and x5" in varying sample spaces that are part of X' as shown in white areas of x and xi in Figure 2. To address this issue, we adjust the conditional denoising function co to inputs in the fixed sample space R RKxL, Concretely, we fix the shape of the inputs x! and x6" to (K L) by applying zero padding to xt" and x² In other words, we set zero values to white areas for x and x in Figure 2. To indicate which indices are padded, we introduce the conditional mask nd no E {0, 10,11Kx0 1 as an additional input to cp. which corresponds to x and takes value 1 for indices of conditional observations. For ease of handling, we also fix the output shape in the sample space IR. K by applying zero padding. Then, the conditional denoising function (x(x / x =)) can be written as 6g (R& x R R (0. 1) 1)XX)) -, K We discuss the effect this adjustment on training and sampling in Appendix D.
Under the adjustment, we set conditional observations x5" and imputation targets xij for time series imputation by following Table 1. At sampling time, since conditional observations xi are all observed values, we set m = M and x m @ X where @ represents element-wise products. For training, we sample x and x through target choice strategy, and set the indices of x5" as m Then, X1 is written as x m X and x is obtained as x (M - m²)) & X.
Next, describe the architecture of eg. We adopt the architecture in DiffWave [13] as the base, which is composed of multiple residual layers with residual channel C. We refine this architecture for time series imputation. We the diffusion step T 50. We discuss the main differences from DiffWave (see Appendix E. for the whole architecture and details).
Attention mechanism To capture temporal and feature dependencies of multivariate time series, we utilize a two dimensional attention mechanism in each residual layer instead of a convolution architecture. As shown in Figure 3, we introduce temporal Transformer layer and a feature Trans- former layer, which are 1-layer Transformer encoders. The temporal Transformer layer takes tensors for each feature inputs to leam temporal dependency, whereas the feature Transformer layer takes tensors for each time point as inputs to learn temporal dependency.
Note that while the length L can be different for each time series as mentioned in Section 3.1, the attention mechanism allows the model to handle various lengths. For batch training, we apply zero padding to each sequence so that the lengths of the sequences are the same.

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Side information In addition to the arguments of Eg. we provide some side information as ad- ditional inputs to the model. First, we use time embedding of s {s1.1. to learn the temporal dependency. Following previous studies [29, 30], we use 128-dimensions temporal embedding. Second, we exploit categorical feature embedding for K features, where the dimension 16.
6 Experimental results
In this section, we demonstrate the effectiveness of CSDI for time series imputation. Since CSDI can be applied to other related tasks such as interpolation and forecasting, we also evaluate CSDI for these tasks to show the flexibility of CSDI. Due the page limitation, we provide the detailed setup for experiments including train/ haai/vvaiiation/testt splits and hyperparameters in Appendix E.2.
6.1 Time series imputation
Dataset and experiment settings We run experiments for two datasets. The first one is the healthcare dataset in PhysioNet Challenge 2012 [1], which consists of 4000 clinical time series with 35 variables for 48 hours from intensive care unit (ICU). Following previous studies [7, 8], we process the dataset to hourly time series with 48 time steps. The processed dataset contains around 80% missing values. Since the dataset has no ground-truth, we randomly choose 10/50/90% of observed values as ground-truth on the test data.
The second one is the air quality dataset [2]. Following previous studies [7, 21]. use hourly sampled PM2.5 measurements from 36 stations in Beijing for 12 months and set 36 consecutive time steps one time series. There are around 13% missing values and the missing patterns not random. The dataset contains artificial ground-truth, whose missing patterns are also structured.
For both dataset, we run each experiment five times. As the target choice strategy for training, we adopt the random strategy for the healthcare dataset and the mix the random and historical strategy for the air quality dataset, based on the missing patterns of each dataset.
Results of probabilistic imputation CSDI is compared with three baselines. 1) Multitask GP [31]: the method learns the covariance between timepoints and features simulraneously. 2) GP-VAE [10]: the method showed the state-of-the arr results for probabilistic imputation. 3) V-RIN [32]: a deter- ministic imputation method that uses the uncertainty quantified by VAE to improve imputation. For
V-RIN, we regard the quantified uncertainty as probabilistic imputation. In addition. we compare CSDI with imputation using the unconditional diffusion model in order to show the effectiveness of the conditional one (see Appendix C for training and imputation with the unconditional diffusion model).
We first show quantitative results. We adopt the continuous ranked probability score (CRPS) [33] as the metric, which is freuquently used for evaluating probabilistic time series forecasting and measures the compatibility of estimated probability distribution with observation. We generate 100 samples to approximate the probability distribution over missing values and report the normalized average of CRPS for all missing values following previous studies [34] (see Appendix E.3 for details of the computation).
Table 2: Comparing CRPS for probabilistic imputation baselines and CSDI (lower better). We report the mean and the standard error of CRPS for five trials.
healthcare
air quality 90% missing
10% missing
50% missing
Multitask GP [31]
0.489(00000)
00.00().003)
0.942(0.010)
0.301(0.003)
GP-VAE [10]
0.574
0.774(0.004)
0.998(0.001)
0.397(0.009)
V-RIN [32]
0.808(0.008)
0.831 (0.005
0.922(0.003)
0.526(0.025)
unconditional
0.360(0.007)
(.458(0.008)
(0.00()
0.135(0.001)
CSDI (proposed)
0.238(0.001)
0.330(0.002)
0.522(0.002)
0.108(0.001)
7

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healthcare(50% missing)
air quality
2.5
GP-VAE
400
GP-VAE
CSDI
CSDI
300
1.5
value
1.0 ve
200
0.5
100
0.0
10
20
30
30
time
time
Figure 4: Examples of probabilistic time series imputation for the healthcare dataset with 50% missing (left) and the air quality dataset (right). The red crosses show the observed values and the blue circles show the ground-truth imputation targets. For each method, median values of imputations are shown as the line and 5% and 95% quantiles are shown as the shade.
Table 2 represents CRPS for each method. CSDI reduces CRPS by 40-65% compared to the existing baselines for both datasets. This indicates that CSDI generates more realistic distributions than other methods. We also observe that the imputation with CSDI outperforms that with the unconditional model. This suggests CSDI benefits from explicitly modeling the conditional distribution.
We provide imputation examples in Figure 4. For the air quality dataset, CSDI (green solid line) provides accurate imputations with high confidence, while those by GP-VAE (gray dashed line) are far from ground-truth. CSDI also gives reasonable imputations for the healthcare dataset. These results indicate that CSDI exploits temporal and feature dependencies to provide accurate imputations. We give more examples in Appendix G.
Table 3: Comparing MAE for deterministic imputation methods and CSDI. We report the mean and the standard error for five trials. The asterisks mean the results of the method are cited from the original paper.
healthcare
air quality
10% missing
50% missing
90% missing
V-RIN [32]
0.271(0.001)
0.365(0.0022
0.606(0.000)
25.4(0.62)
BRITS [7]
(.28-4(0.001)
(0.000)
0.517(0.002)
14.11(0.26)
BRITS [7] (*)
0.278
11.56
GLIMA (*))
0.265
10.54
RDIS [20]
(.319(0.002)
0.419(0.002)
0.631
(0.11(
unconditional
0.326(0.00g
0.417(0.010)
0.625(0.010)
12.13(0.07)
CSDI (proposed)
(..07(
(0.00().002)
0.481(
9.60(0.04)
Results of deterministic imputation We demonstrate that CSDI also provides accurate determin- istic imputations, which are obtained as the median of 100 generated samples. We compare CSDI with four baselines developed for deterministic imputation including GLIMA [21], which combined recurrent imputations with an attention mechanism to capture temporal and feature dependencies and showed the state-of-the-arr performance. These methods are based on autoregressive models. We use the original implementations except RDIS
We evaluate each method by the mean absolute error (MAE). In Table 3, CSDI improves MAE by 5-20% compared to the baselines. This suggests that the conditional diffusion model is effective to learn temporal and feature dependencies for imputation. For the healthcare dataset, the gap between the baselines and CSDI particularly significant when the missing ratio is small, because more observed values help CSDI capture dependencies.

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Table 4: Comparing the oftt--of-the-ant interpolation methods with CSDI for the healthcare dataset. We report the mean and the standard error of CRPS for five trials.
10% missing
50% missing
90% missing
Latent ODE [35]
0.700(0.002)
0.676(0.003) 0.567(0.003) 0.418(0.001)
0.761(0.010)
mTANN [22]
0.526(0.004)
0.689(0.015)
CSDI (proposed)
0.380 0.380(0.002)
0.556(0.003)
6.2 Interpolation of irregularly sampled time series
Dataset and experiment settings We use the same healthcare dataset as the previous section, but process the dataset irregularly sampled time series, following previous studies [22, 35J. Since the dataset has no ground-truth, we randomly choose 10/50/90% time points and use observed values at these time points ground-truth the test data. As the target choice strategy for training, adopt the random strategy. which is adjusted for interpolation so that some time points are sampled.
We compare CSDI with two baselines including mANNs [22], which utilized an attention mechanism and showed state-of ofttth-rrtt results for the interpolation of irregularly sampled time series. We generate 100 samples to approximate the probability distribution as with the previous section. The result is shown in Table 4. CSDI outperforms the baselines for all cases.
Results
Table 5: Comparing probabilistic forecasting methods with CSDI. We report the mean and the standard error of CRPS-sum for three trials. The baseline results are cited from the original paper. "TransMAF" the abbreviation for TTransformer MAF"
solar
electricity
traffic
taxi
wiki
GP-copula [34]
0.024(0.002) 0.337(0.024)
0.078(0.002)
0.208(0.183)
0.086(0.004)
TransMAF [36]
(..01()
0.056(0.001) (0.000)
0.179(0.002)
0.063(0.003)
TLAE [37]
0.124(0.033)
0.069(0.001) 0.040(0.002)
0.130(0.006)
0.241(0.001)
TimeGrad [25]
0.287(0.020)
0.021(0.001)
0.044(0.006)
0.114(0.020)
0.049(0.002)
CSDI (proposed)
(0290()
0.017(0.000)
0.020(0.001)
0..20(
0.047(0.003)
6.3 Time series Forecasting
Dataset and Experiment settings We use five datasets that are commonly used for evaluating probabilistic time series forecasting. Each dataset is composed of around 100 to 2000 features. We predict all features at future time steps using past time series. We use the same prediction steps as previous studies [34, 37]. For the target choice strategy, adopt the Test pattern strategy.
Results We compare CSDI with four baselines. Specifically, TimeGrad [25) combined the diffusion model with a RNN-based encoder. We evaluate each method for CRPS-sum, which is CRPS for the distribution of the sum of all time series across K features and accounts for joint effect (see Appendix E.3 for details).
In Table 5, CSDI outperforms the baselines for electricity and traffic datasets, and is competitive with the baselines as whole. The advantage of CSDI over baselines for forecasting is smaller than that for imputation in Section 1. We hypothesize is because the datasets for forecasting seldom contains missing values and are suitable for existing encoders including RNNs. For imputation, it is relatively difficult for RNNs to handle time series due to missing values.
7 Conclusion
In this paper, we have proposed CSDI, novel approach to impute multivariate time series with conditional diffusion models. We have shown that CSDI outperforms the existing probabilistic and deterministic imputation methods.

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There are some interesting directions for future work. One direction is to improve the computation efficiency. While diffusion models generate plausible samples, sampling is generally slower than other generative models. To mitigate the issue, several recent studies leverage an ODE solver to accelerate the sampling procedure [12, 38, 13]. Combining our method with these approaches would likely improve the sampling efficiency.
Another direction is to extend CSDI downstream tasks such classifications. Many previous studies have shown that accurate imputation improves the performance on downstream tasks [7, 18, 22]. Since conditional diffusion models can learn temporal and feature dependencies with uncertainty, joint training of imputations and downstream tasks using conditional diffusion models would be helpful improve the performance of the downstream tasks.
Finally, although our focus was on time series, it would be interesting to explore CSDI as imputation technique other modalities.
Acknowledgements and Disclosure of Funding
This research was supported by NSF(#16515665, #1522054, #1733686), ONR (N000141912145) AFOSR (FA95501910024), ARO (W911NF-21-1-0125)) and Sloan Fellowship.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] We described our proposed method and empirical results in the abstract and introduction.
(b) Did you describe the limitations of your work? [Yes] We have mentioned some limitations of our work in Section 7.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] Please see Appendix H.
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(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] We gave error bars in figures and the standard errors in tables.
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13


Reviews:
[{"review_id": "rwAcF8fRPfj", "reviewer": "Reviewer_8rwP", "summary": "The authors use diffusion probabilistic models for imputation / conditional generation and propose self-supervised training schemes for this task.\n\nThe authors demonstrate their method outperforms other state-of-art approaches for time-series data\n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "- Originality:\n  - This paper explains a novel albeit incremental application of existing techniques.\n  - Inpainting/ imputation/ infilling using score-matching has previously been explored for images and recently also for time-series, specifically for audio data [1]. Similarly [1] uses a mask for conditional generation, though this is not the focus of the paper.\n  - Class conditioned generative modeling has been explored previously [2], where the score network is augmented by a class-dependent embedding. This paper’s approach may be seen as an extension of this but instead based on an embedding with inputs of observed/conditional data rather than class.\n  - Modeling missing data as masked is standard for inpainting [1,3,4,5] and generating masks for self-supervised training has been performed before in other domains e.g. NLP. but not in combination with score-matching or time-series imputation as far as I am aware.\n\n- Quality\n  - The authors’ experimental section is excellent and they compare their approach to many state of the art methods in a variety of domains.\n\n- Clarity\n  - The paper is in general very well written. \n  - A few important details are left for the appendices but are helpful for clarity and understanding the paper, specifically:\n    - Definition of $\\mu^{DDPM}$, line 97\n    - The details on padding to ensure being same dimension inputs into the score network\n\n- Significance\n  - Although the individual techniques are not novel, the combination may be novel\n  - The method is general, shows state of art performance and may be used in practice\n\nSummary\n\nStrengths:\n - Excellent experimental section and comparisons, the proposed method demonstrates state of art performance\n - Well written and clear paper\n\nWeaknesses:\n - The work lacks significant novelty and is essentially an application of two existing methods 1) self-supervised training and 2) diffusion probabilistic models.  Score based models have previously been applied to time-series imputation and masked self-supervised training is fairly standard in training sequential models in NLP. I find it surprising that self-supervised techniques have not before been applied to train models for time-series imputation, as it would be an obvious extension of previous data imputation methods using pre-selected masks.\n\n- Missing reference [1]\n\nIn summary, although there is nothing incorrect with the paper and the paper demonstrates good empirical results, unfortunately I believe the work is too incremental and lacks sufficient novelty to justify acceptance. \n\nReferences\n[1] Mittal, Gautam, et al. \"Symbolic music generation with diffusion models.\" arXiv preprint arXiv:2103.16091 (2021). \\\n[2] Nichol, Alex, and Prafulla Dhariwal. \"Improved denoising diffusion probabilistic models.\" arXiv preprint arXiv:2102.09672 (2021). \\\n[3]  Jinsung Yoon, James Jordon, Mihaela Schaar “GAIN: Missing Data Imputation using Generative Adversarial Nets” PMLR 80:5689-5698, 2018. \\\n[4] Luo, Yonghong, et al. \"Multivariate time series imputation with generative adversarial networks.\" Proceedings of the 32nd International Conference on Neural Information Processing Systems. 2018. \\\n[5] Guo, Zijian, Yiming Wan, and Hao Ye. \"A data imputation method for multivariate time series based on generative adversarial network.\" Neurocomputing 360 (2019): 185-197.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors discuss societal / ethical impacts in Appendix H.", "review_text": "", "extra_scores": {"ethical_concerns": "The authors discuss societal / ethical impacts in Appendix H.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "muyXKT1rV-L", "reviewer": "Reviewer_2d7W", "summary": "This paper focuses on the problem of the probabilistic imputation of missing data in time series. This work proposes a novel approach for probabilistic imputation based on the score-based diffusion models. Experiments on time series imputation tasks show that the proposed method achieves better performance than the current SOTA methods in this space.  ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "### Novelty and Significance \n- The problem addressed is very important in many domains. The proposed approach is novel.\n\n### Quality\n- The paper is missing comparisons with some standard baselines for probabilistic imputation/interpolation such as Single-Task [1] and Multi-task Gaussian Processes [2] and SOTA methods based on neural ODEs [3]. ODE methods also perform particularly well on forecasting tasks so should be compared in Table 5 as well. Furthermore, the authors are also missing a comparison with mTANs in Tables 2 and 3 considering mTANs can produce probabilistic interpolation by sampling multiple latent states and passing through the decoder. \n- Negative log-likelihood is a standard metric for evaluation uncertainty computations. What is the reason behind omitting such a standard metric when evaluating probabilistic interpolations and imputations?\n- The authors should perform ablation to show the advantage of using temporal and feature transformer layers.\n\n### Clarity\n- The paper is well written and easy to follow.\n\n### Additional Comments\n- How are the interpolation and imputation tasks on the PhysioNet dataset different after performing discretization on the irregularly sampled dataset? Is the proposed approach applicable for handling irregularly sampled sequences directly since the discretization loses a lot of information about the time series?\n\n#### References\n1. C. E. Rasmussen and C. K. I. Williams. Gaussian processes for machine learning. Adaptive computation and machine learning. 2006.\n2. E. V. Bonilla, K. M. Chai, and C. Williams. Multi-task Gaussian process prediction. In Advances in Neural Information Processing Systems, pages 153–160, 2008.\n3. Y. Rubanova, R. T. Q. Chen, and D. K. Duvenaud. Latent ordinary differential equations for irregularly sampled time series. In Advances in Neural Information Processing Systems, pages 5320–5330. 2019.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "mj0ONFmS6xu", "reviewer": "Reviewer_yacf", "summary": "This work proposes a probabilistic data imputation model for multivariate time series. The proposed method uses a conditional diffusion model to estimate the (masked) conditional distribution of missing data given observed values. To utilise information from observed values, the authors employ a neural attention mechanism to capture spatiotemporal dependencies of time series. The authors use self-supervised learning for training the model.  In experiments, the author demonstrate competitive performance of their method against a number of comparable methodologies on time series imputation and forecasting tasks.\n", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The proposed model is an extension of the previously proposed denoising diffusion probabilistic model (DDPM), where the Markov chain reverse process of DDPM is conditioned on observed data. More specifically, the denoising function that parameterises the mean of the conditional Markov chain distributions in the DDPM reverse process is provided with the observed data as an additional argument. \n\nThe paper is well written and easy to follow. The proposed method builds on a methodology that has been shown to be competitive in a variety of domains including time series modelling. The task addressed by the authors is quite relevant. The utilisation of attention mechanism for capturing both spatial and temporal dependencies appears interesintg. In comparison to recent relevant methodologies, the method has been shown to be state-of-the-art on time series imputation and forecasting tasks involving a number of benchmark datasets.  \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "My main concern with the work is that methodologically, its contribution is rather limited. The main methodological contribution of this work is the modification of denoising function parameterising mu in eq (5), which in my view is a straightforward and minimal extension of DDPM. The architectural modification involving the use of neural attention layer appears to be interesting, but without any comparison it is difficult to say how crucial it is to the performance of the model as opposed to for instance using (bidrectional) RNNs.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper uses conditional score-based diffusion models for probabilistic time series imputation. The authors use an attention mechanism to capture the temporal and feature dependencies. A self-supervised training method is developed. The paper is overall well-written and the method appears to outperform alternative imputation techniques on some metrics.

The proposed method is novel but was also perceived as a rather straightforward combination of existing techniques (conditional score ideas have already been used in different domains and the self-supervised technique has been applied in related areas). That said, it is by no means trivial to put this combination in practice and the empirics are compelling.

After the rebuttal of the authors, some of the reviewers increase their score. With scores of 6, 6, 6, there remains a lack of enthusiasm for the paper. However, I recommend acceptance of the paper as it is a useful contribution to probabilistic time series imputation.

Decision: Accept (Poster)