Title: Self-Attention-based Imputation for Time Series Published in the journal Expert Systems with Applications. Its DOI link: https://doi.org/10.1016/j.eswa.2023.119619
URL Source: https://arxiv.org/html/2202.08516
Markdown Content: Wenjie Du
Concordia University
Montréal, Canada
&David Cote
Ciena Corporation
Ottawa, Canada
&Yan Liu
Concordia University
Montréal, Canada
Abstract
Missing data in time series is a pervasive problem that puts obstacles in the way of advanced analysis. A popular solution is imputation, where the fundamental challenge is to determine what values should be filled in. This paper proposes SAITS, a novel method based on the self-attention mechanism for missing value imputation in multivariate time series. Trained by a joint-optimization approach, SAITS learns missing values from a weighted combination of two diagonally-masked self-attention (DMSA) blocks. DMSA explicitly captures both the temporal dependencies and feature correlations between time steps, which improves imputation accuracy and training speed. Meanwhile, the weighted-combination design enables SAITS to dynamically assign weights to the learned representations from two DMSA blocks according to the attention map and the missingness information. Extensive experiments quantitatively and qualitatively demonstrate that SAITS outperforms the state-of-the-art methods on the time-series imputation task efficiently and reveal SAITS’ potential to improve the learning performance of pattern recognition models on incomplete time-series data from the real world. The code is open source on GitHub at https://github.com/WenjieDu/SAITS.
K eywords Time Series ⋅⋅\cdot⋅ Missing Values ⋅⋅\cdot⋅ Imputation Model ⋅⋅\cdot⋅ Self-Attention ⋅⋅\cdot⋅ Neural Network
1 Introduction
Multivariate time-series data is ubiquitous in many application domains, for instance, transportation, economics, healthcare, and meteorology. State-of-the-art (SOTA) pattern recognition methods can well handle various time-series analysis tasks on complete datasets. However, due to all kinds of reasons, including failure of collection sensors, communication error, and unexpected malfunction, missing values are common to see in time series from the real-world environment, for example, patient health measurement[1], air-quality monitoring[2], telecommunication networking[3], and educational systems[4]. They impair the interpretability of data and pose challenges for advanced analysis and pattern recognition tasks, for instance, classification and clustering. To learn with such incomplete time series, thinking ahead of how to deal with missing parts before modeling is an inevitable step.
Traditional missing value processing methods fall into two categories. One is deletion, which removes samples or features that are partially observed. However, deletion makes data incomplete and can yield biased parameter estimates[5]. The other one is data imputation that estimates missing data from observed values[6]. There are two main advantages to imputation over deletion[7]: (1). Deletion introduces bias, while correctly specified imputation estimates are unbiased[8]; (2). Partially observed data may still be informative and helpful to the analysis. Nevertheless, the problem of imputation is what values should be filled in. Amounts of prior work are proposed to solve this problem with statistics and machine learning methods[9, 10, 11, 12, 13, 14, 15]. However, most of them require strong assumptions on missing data[16], for example, linear regression[10], mean/median averaging and k-nearest neighbors[11]. Such assumptions may introduce a strong bias that influences the outcome of the analysis. Recently, much literature utilizes deep learning to solve this imputation problem and achieves SOTA results. Non-time series imputation models, which are not in the scope of this paper, can be referred to[17, 18, 19, 20, 21, 22, 23, 24]. A more detailed description of time-series imputation models is presented in Section2, including[16, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
Missing values can be characterized into three types: (1). Missing completely at random (MCAR), missing values are independent of any other values; (2). Missing at random (MAR), missing values depend only on observed values; (3). Missing not at random (MNAR), missing values depend on both observed and unobserved values[35, 36]. This work focuses on the MCAR case, as is standard in most of literature in the field of imputation. We uniformly sample values to be missing independently and introduce them as artificial missingness to evaluate all imputation methods used in this paper.
The self-attention mechanism is now widely applied, whereas its application on time-series imputation is still limited. Previous SOTA time-series imputation models are mostly based on recurrent neural networks (RNN), such as[16, 25, 27, 28, 26]. Among them, the methods[16, 25, 27, 28] are autoregressive models that are highly susceptible to compounding errors[37, 26]. Although the work[26] is not autoregressive, the multi-resolution imputation algorithm it proposed is made up of a loop, which can greatly slow the imputation speed. The self-attention mechanism, which is non-autoregressive and can overcome RNNs’ drawbacks of slow speed and memory constraints, can avoid compounding error and be helpful to achieve better imputation quality and faster speed. This paper proposes a novel model called SAITS (S elf-A ttention-based I mputation for T ime S eries) to learn missing values by a joint-optimization training approach of imputation and reconstruction. Particularly, our contributions in this work are summarized as the following:
I. We design a joint-optimization training approach of imputation and reconstruction for self-attention models to perform missing value imputation for multivariate time series. Transformer trained by this approach outperforms SOTA methods.
II. We design a model called SAITS that consists of a weighted combination of two diagonally-masked self-attention (DMSA) blocks, which emancipates SAITS from RNN and enables it to capture temporal dependencies and feature correlations between time steps explicitly.
III. We conduct adequate experiments and ablation studies on four real-world public datasets to quantitatively and qualitatively evaluate our methodology and justify its design. The experimental results do not only prove that SAITS achieves the new SOTA position for the imputation accuracy, but also show SAITS’ potential of facilitating pattern recognition models to learn with partially-observed time series from the real world.
The rest of this work is organized as follows: Section2 reviews the related literature. Section3 introduces our joint-optimization training approach and the SAITS model. Experiments and conclusions are presented in Section4 and5, respectively.
2 Related Work
We review the prior related work of time-series imputation in the following four categories:
RNN-based Che et al.[38] propose GRU-D, a gated recurrent unit (GRU) variant, to handle missing data in time series classification problems. The concept of time decay on the last observation is firstly proposed by[38] and continues to be used in[16, 25, 27, 28]. M-RNN[25] and BRITS[16] impute missing values according to hidden states from bidirectional RNN. However, M-RNN treats missing values as constants, while BRITS treats missing values as variables of the RNN graph. Furthermore, BRITS takes correlations among features into consideration while M-RNN does not.
GAN-based Models in [27, 28, 26] are also RNN-based. However, due to they adopt the generative adversarial network (GAN) structure, they are listed separately as GAN-based. Luo et al.[27] propose GRUI (GRU for Imputation) to model temporal information of incomplete time series. Both the generator and the discriminator in their GAN model are based on GRUI. Moreover, based on[27], Luo et al.[28] propose E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN, which is an end-to-end method, comparing to the method in[27] having two stages. E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN adopts an auto-encoder based on GRUI to form its generator to ease the difficulty of model training and improve imputation performance. Liu et al.[26] propose a non-autoregressive model called NAOMI for spatiotemporal sequence imputation, which consists of a bidirectional encoder and a multiresolution decoder. NAOMI is further enhanced by adversarial training.
All the above RNN-based imputation models are susceptible to the recurrent network structure. They are time-consuming and have memory constraints, which make it hard for them to handle long-term dependency in time series, especially when the number of time steps in data samples is big. Apart from these disadvantages, most of these models are autoregressive that leads to the problem which is called compounding error in the field of time-series analysis[37]. Although NAOMI alleviates this problem, its imputation algorithm consists of a loop, which can greatly slow the imputation speed together with its RNN structure. In addition, RNN models are usually unidirectional[38, 27, 28]. Even the models in[16, 25, 26] leverage bidirectional RNN to further improve the performance. Their final imputation results come from the average of two RNN models working on two directions (forward and backward) separately. As a result, these models are not deeply bidirectional[39].
VAE-based Inspired by GPPVAE[40] and the non-time-series imputation model HI-VAE[22], Fortuin et al.[29] propose GP-VAE, a variational auto-encoder (VAE) architecture for time series imputation with a Gaussian process (GP) prior in the latent space. The GP-prior is used to help embed the data into a smoother and more explainable representation. L-VAE[30] uses an additive multi-output GP-prior to accommodate auxiliary covariate information other than time. To support sparse GP approximations based on inducing points and handle missing values in spatiotemporal datasets, Ashman et al.[31] propose SGP-VAE.
The GAN-based and VAE-based models are all generative ones, and they are difficult to train[23]. Particularly, GAN models suffer from the problems of non-convergence and mode collapse due to their loss formulation[41]. VAE models tend to involve latent variables used in the sampling and imputation, while they often do not correspond to concrete structures or distributions of the data, and this can make it difficult to interpret the imputation process for further understanding[23].
Self-Attention-based Ma et al.[32] apply cross-dimensional self-attention (CDSA) jointly from three dimensions (time, location, and measurement) to impute missing values in geo-tagged data, namely spatiotemporal datasets. Bansal et al.[33] propose DeepMVI for missing value imputation in multidimensional time-series data. Their model includes a Transformer with a convolutional window feature and a kernel regression. Shan et al.[34] propose NRTSI, a time-series imputation approach treating time series as a set of (time,data)𝑡 𝑖 𝑚 𝑒 𝑑 𝑎 𝑡 𝑎(time,data)( italic_t italic_i italic_m italic_e , italic_d italic_a italic_t italic_a ) tuples. Such a design makes NRTSI applicable to irregularly-sampled time series. The method directly uses a Transformer encoder for modeling and achieves SOTA performance in their work. Such above prior literature explores applying self-attention in the field of time-series imputation. However, CDSA[32] is specifically designed for spatiotemporal data rather than general time series, and both CDSA and DeepMVI[33] are not open-source, which makes it hard for other researchers to reproduce their methods and results. Regarding NRTSI[34], its algorithm design consists of two nested loops, which weaken the advantage of self-attention that is parallelly computational. Even worse, such loops can lead NRTSI to slow processing. Related work of self-attention-based models for time-series imputation is not much. There are some non-time series imputation models based on self-attention, such as AimNet[23] and MAIN[24].
3 Methodology
Our methodology is made up of two parts: (1). the joint-optimization training approach of imputation and reconstruction; (2). the SAITS model, a weighted combination of two DMSA blocks.
3.1 Joint-optimization Training Approach
A general illustration of the joint-optimization approach is shown in Figure1. We are going to first give the definition of multivariate time series bearing missing data, then introduce the two learning objectives in detail.
Figure 1: A graphical overview of the joint-optimization training approach, illustrating the implementation details of the two training tasks: masked imputation task (MIT) and observed reconstruction task (ORT).
3.1.1 Definition of Multivariate Time-Series with Missing Values
Given a collection of multivariate time series with T 𝑇 T italic_T time steps and D 𝐷 D italic_D dimensions, it is denoted as X={x 1,x 2,…,x t,…,x T}∈ℝ T×D 𝑋 subscript 𝑥 1 subscript 𝑥 2…subscript 𝑥 𝑡…subscript 𝑥 𝑇 superscript ℝ 𝑇 𝐷 X={x_{1},x_{2},...,x_{t},...,x_{T}}\in\mathbb{R}^{T\times D}italic_X = { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT } ∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_D end_POSTSUPERSCRIPT, where the t 𝑡 t italic_t-th step x t={x t 1,x t 2,…,x t d,…,x t D}∈ℝ 1×D subscript 𝑥 𝑡 superscript subscript 𝑥 𝑡 1 superscript subscript 𝑥 𝑡 2…superscript subscript 𝑥 𝑡 𝑑…superscript subscript 𝑥 𝑡 𝐷 superscript ℝ 1 𝐷 x_{t}={x_{t}^{1},x_{t}^{2},...,x_{t}^{d},...,x_{t}^{D}}\in\mathbb{R}^{1% \times D}italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = { italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT } ∈ blackboard_R start_POSTSUPERSCRIPT 1 × italic_D end_POSTSUPERSCRIPT and each value in it could be missing. Accordingly, X t d superscript subscript 𝑋 𝑡 𝑑 X_{t}^{d}italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT represents the d 𝑑 d italic_d-th dimension variable of the t 𝑡 t italic_t-th step in X 𝑋 X italic_X. To represent the missing variables in X 𝑋 X italic_X, the missing mask vector M∈ℝ T×D 𝑀 superscript ℝ 𝑇 𝐷 M\in\mathbb{R}^{T\times D}italic_M ∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_D end_POSTSUPERSCRIPT is introduced, where
M t d={1 ifX t dis observed 0 ifX t dis missing superscript subscript 𝑀 𝑡 𝑑 cases 1 if superscript subscript 𝑋 𝑡 𝑑 is observed 0 if superscript subscript 𝑋 𝑡 𝑑 is missing M_{t}^{d}=\left{\begin{array}[]{ll}1&\text{if }X_{t}^{d}\text{ is observed}\ 0&\text{if }X_{t}^{d}\text{ is missing}\end{array}\right.italic_M start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT = { start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL if italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is observed end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL if italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is missing end_CELL end_ROW end_ARRAY
3.1.2 Two Learning Tasks
(a) Imputation MAE
(b) Reconstruction MAE
Figure 2: Imputation MAE and reconstruction MAE of the models in the validation stage. All models are trained on the same data. BRITS is trained with ORT, namely in the same way as its original paper. Transformer-ORT is trained with only ORT as well, i.e. without MIT. Transformer-MIT is trained with only MIT. Transformer-ORT+MIT is trained with the joint-optimization approach, namely with both ORT and MIT.
To well train self-attention-based imputation models on the above defined multivariate time series with missing values, a joint-optimization training approach of imputation and reconstruction is designed. Here "imputation" is defined as the process in which the model fills the missing part in the given samples from null, and "reconstruction" means the model restores the observed values as exactly as possible after its processing. This joint-optimization approach consists of two learning tasks: Masked Imputation Task (MIT) and Observed Reconstruction Task (ORT). Correspondingly, the training loss is accumulated from two losses: the imputation loss of MIT and the reconstruction loss of ORT.
Before illustrating the joint-optimization approach in detail, it is necessary to discuss why we need a new training approach for self-attention-based imputation models. Here, for the straightforward visualization, BRITS[16] and Transformer[42] are taken as examples for explanation. The former stands for mainstream RNN-based imputation methods. The latter is a standard self-attention-based model. Figure2 is plotted to make comparisons and to further illustrate the effects.
A normal way to train an RNN for imputation consists of three main steps: (1). Input time-series feature vectors X 𝑋 X italic_X together with missing mask M 𝑀 M italic_M to alert the model that input data has observations and missing values; (2). Let the model reconstruct the observed part of the input time series and calculate the reconstruction error in each time step as the loss; (3). Finally, utilize the reconstruction loss to update the model. This training method is ORT. ORT works well with RNN-based models, for instance, BRITS. However, different from RNN which is autoregressive, self-attention itself is non-autoregressive and processes all input data parallelly and globally. Thus, if only trained on ORT, Transformer can distinguish the observed part from X 𝑋 X italic_X according to M 𝑀 M italic_M, and as a result, it will focus only on minimizing the reconstruction error on the observed values. Taking a look at Figure2(b), Transformer-ORT’s reconstruction MAE is much smaller than BRITS’. However, in Figure2(a), the imputation MAE of Transformer-ORT goes up from the beginning and is much larger than BRITS’. Transformer-ORT ignores the missing values because there is no penalty posed to it no matter what values it fills in. Hence, ORT can only ensure that Transformer gets well trained on observed values. In other words, there is no guarantee that Transformer-ORT can predict missing values accurately. To solve this optimization problem, we make another learning task MIT to become this guarantee and bind it together with ORT. This is how the joint-optimization training approach comes. The details of the two tasks are described as follows, and their concrete implementation is illustrated in Figure2.
Task #1: Masked Imputation Task (MIT) MIT is a prediction task on artificially-masked values, which explicitly forces the model to predict missing values accurately. In MIT, for every batch input into the model, some percentage (such as 20% in our work) of observed values gets artificially masked at random. These values are not visible to the model, namely missing to the model. After artificially masking, the actual input time series is denoted as X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG, and its corresponding missing mask vector is M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG. The output estimated time series bearing reconstructions and imputations is denoted as X~~𝑋\tilde{X}over~ start_ARG italic_X end_ARG. To distinguish artificially-missing values and originally-missing values, the indicating mask vector I 𝐼 I italic_I is introduced. Math definitions of M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG and I 𝐼 I italic_I are:
M^t d={1 ifX^t dis observed 0 ifX^t dis missing,I t d={1 ifX^t dis artificially masked 0 otherwise formulae-sequence superscript subscript^𝑀 𝑡 𝑑 cases 1 if superscript subscript^𝑋 𝑡 𝑑 is observed 0 if superscript subscript^𝑋 𝑡 𝑑 is missing superscript subscript 𝐼 𝑡 𝑑 cases 1 if superscript subscript^𝑋 𝑡 𝑑 is artificially masked 0 otherwise\hat{M}{t}^{d}=\left{\begin{array}[]{ll}1&\text{if }\hat{X}{t}^{d}\text{ is% observed}\ 0&\text{if }\hat{X}{t}^{d}\text{ is missing}\end{array}\right.,\hskip 10.0000% 2ptI{t}^{d}=\left{\begin{array}[]{ll}1&\text{if }\hat{X}_{t}^{d}\text{ is % artificially masked}\ 0&\text{otherwise}\end{array}\right.over^ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT = { start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL if over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is observed end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL if over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is missing end_CELL end_ROW end_ARRAY , italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT = { start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL if over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is artificially masked end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL otherwise end_CELL end_ROW end_ARRAY
After the model imputes all missing values, the imputation loss is the mean absolute error (MAE) calculated between the artificially missing values and their respective imputations. The calculation of MAE and MIT loss are defined in Equation1 and Equation2 below.
ℓ MAE(estimation,target,mask)=∑d=1 D∑t=1 T|(estimation−target)⊙mask|t d∑d=1 D∑t=1 T mask t d subscript ℓ MAE 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 superscript subscript direct-product 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 𝑡 𝑑 superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 𝑚 𝑎 𝑠 superscript subscript 𝑘 𝑡 𝑑\ell_{\text{MAE}}\left(estimation,target,mask\right)=\frac{\sum_{d=1}^{D}\sum_% {t=1}^{T}\lvert\left(estimation-target\right)\odot mask\rvert_{t}^{d}}{\sum_{d% =1}^{D}\sum_{t=1}^{T}mask_{t}^{d}}roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n , italic_t italic_a italic_r italic_g italic_e italic_t , italic_m italic_a italic_s italic_k ) = divide start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT | ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n - italic_t italic_a italic_r italic_g italic_e italic_t ) ⊙ italic_m italic_a italic_s italic_k | start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_m italic_a italic_s italic_k start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG(1)
ℒ MIT=ℓ MAE(X,X,I)subscript ℒ MIT subscript ℓ MAE𝑋 𝑋 𝐼\mathcal{L}{\text{MIT}}=\ell{\text{MAE}}\left(\tilde{X},X,I\right)caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT = roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG , italic_X , italic_I )(2)
Note that learning tasks similar to MIT, which mask some objects and then predict them, are commonly used to train models in NLP (Natural Language Processing) field, for example, the Cloze task[43], and the Masked Language Modeling (MLM) used to pre-train BERT[39]. MIT is inspired by MLM, but the differences are: (1). MLM predicts missing tokens (time steps), while MIT predicts missing values in time steps; (2). One disadvantage of MLM is that it causes pretrain-finetune discrepancy because masking symbols used during pretraining are absent from real data in finetuning[44]. However, the original objective of imputation is to predict missing or masked values. Therefore, MIT does not cause such discrepancies.
Task #2: Observed Reconstruction Task (ORT) ORT is a reconstruction task on the observed values. It is widely applied in the training of imputation models for both time-series and non-time series [16, 27, 28, 29, 17, 18]. After model processing, observed values in the output are different from their original values, and they are called reconstructions. In our work, the reconstruction loss is MAE calculated between the observed values and their respective reconstructions, defined in Equation3 as below.
ℒ ORT=ℓ MAE(X,X,M^)subscript ℒ ORT subscript ℓ MAE𝑋 𝑋^𝑀\mathcal{L}{\text{ORT}}=\ell{\text{MAE}}\left(\tilde{X},X,\hat{M}\right)caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT = roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG , italic_X , over^ start_ARG italic_M end_ARG )(3)
In our training approach, MIT and ORT are integral. MIT is utilized to force the model to predict missing values as accurately as possible, and ORT is leveraged to ensure that the model converges to the distribution of observed data. As shown in Figure2, both the imputation MAE and reconstruction MAE of Transformer-ORT+MIT drop steadily. A comparison between Transformer-MIT and Transformer-ORT+MIT tells that MIT makes the main contribution to decreasing the imputation MAE. Compared with Transformer-ORT+MIT, Transformer-MIT has a slightly higher imputation MAE. This proves that ORT can help models further optimize performance on the imputation task. On the reconstruction MAE, Transformer-MIT climes up because it is not required to converge on the observed data. Furthermore, in the aspect of the reconstruction MAE in Figure2(b), Transformer-ORT+MIT is slightly higher than Transformer-ORT because the gradient of the reconstruction loss gets influenced by the imputation loss. This is another piece of evidence proving that our joint-optimization approach works.
It is worth mentioning that our training approach is designed not only for time-series self-attention models but can be applied to train other imputation models. As shown in Figure1, the imputation model in the blue box is not specified and can be replaced with other models for training. Moreover, the data can be non-time-series. We also discuss applying the joint-optimization approach to train BRITS in AppendixB.
3.2 SAITS
Figure 3: The SAITS model architecture.
As illustrated in Figure3, SAITS is composed of two diagonally-masked self-attention (DMSA) blocks and a weighted combination. Firstly, some fundamental components of SAITS get introduced in Subsection3.2.1 and3.2.2. Then SAITS’ three-part structure is illustrated in Subsection3.2.3,3.2.4, and3.2.5, respectively. Finally, the loss functions of learning tasks are discussed in Subsection3.2.6.
3.2.1 Diagonally-Masked Self-Attention
The conventional self-attention mechanism is proposed by Vaswani et al.[42] to solve the language translation task. Now it is widely applied in sequence modeling. A given sequence is mapped into a query vector Q 𝑄 Q italic_Q of dimension d k subscript 𝑑 𝑘 d_{k}italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, a key vector K 𝐾 K italic_K of dimension d k subscript 𝑑 𝑘 d_{k}italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and a value vector V 𝑉 V italic_V of dimension d v subscript 𝑑 𝑣 d_{v}italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT. The scaled dot-product can effectively calculate attention scores (or the attention map) between Q 𝑄 Q italic_Q and K 𝐾 K italic_K. After that, a softmax function is applied to obtain attention weights. The final output is attention-weighted V 𝑉 V italic_V. The whole process is as shown in Equation4 below:
SelfAttention(Q,K,V)=Softmax(QK 𝖳 d k)V SelfAttention 𝑄 𝐾 𝑉 Softmax 𝑄 superscript 𝐾 𝖳 subscript 𝑑 𝑘 𝑉\operatorname{SelfAttention}\left(Q,K,V\right)=\operatorname{Softmax}\left(% \frac{QK^{\mathsf{T}}}{\sqrt{d_{k}}}\right)V roman_SelfAttention ( italic_Q , italic_K , italic_V ) = roman_Softmax ( divide start_ARG italic_Q italic_K start_POSTSUPERSCRIPT sansserif_T end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_ARG ) italic_V(4)
To enhance SAITS’ imputation ability, the diagonal masks are applied inside the self-attention. As formulated in Equation5 and6, the diagonal entries of the attention map (∈ℝ T×T absent superscript ℝ 𝑇 𝑇\in\mathbb{R}^{T\times T}∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_T end_POSTSUPERSCRIPT) are set as −∞-\infty- ∞ (set as −1×10 9 1 superscript 10 9-1\times 10^{9}- 1 × 10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT in practice) so the diagonal attention weights approach 0 after the softmax function. Figure4 gives a vivid illustration of the DMSA mechanism.
[DiagMask(x)](i,j)={−∞i=j x(i,j)i≠j delimited-[]DiagMask 𝑥 𝑖 𝑗 cases 𝑖 𝑗 𝑥 𝑖 𝑗 𝑖 𝑗\left[\operatorname{DiagMask}\left(x\right)\right]\left(i,j\right)=\left{% \begin{array}[]{ll}-\infty&i=j\ x\left(i,j\right)&i\neq j\end{array}\right.[ roman_DiagMask ( italic_x ) ] ( italic_i , italic_j ) = { start_ARRAY start_ROW start_CELL - ∞ end_CELL start_CELL italic_i = italic_j end_CELL end_ROW start_ROW start_CELL italic_x ( italic_i , italic_j ) end_CELL start_CELL italic_i ≠ italic_j end_CELL end_ROW end_ARRAY(5)
DiagMaskedSelfAttention(Q,K,V)DiagMaskedSelfAttention 𝑄 𝐾 𝑉\displaystyle\operatorname{DiagMaskedSelfAttention}\left(Q,K,V\right)roman_DiagMaskedSelfAttention ( italic_Q , italic_K , italic_V )=Softmax(DiagMask(QK 𝖳 d k))V absent Softmax DiagMask 𝑄 superscript 𝐾 𝖳 subscript 𝑑 𝑘 𝑉\displaystyle=\operatorname{Softmax}\left(\operatorname{DiagMask}\left(\frac{% QK^{\mathsf{T}}}{\sqrt{d_{k}}}\right)\right)V= roman_Softmax ( roman_DiagMask ( divide start_ARG italic_Q italic_K start_POSTSUPERSCRIPT sansserif_T end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_ARG ) ) italic_V(6) =AV,whereAis attention weights absent 𝐴 𝑉 where 𝐴 is attention weights\displaystyle=AV,\ \ \text{where }A\text{ is attention weights}= italic_A italic_V , where italic_A is attention weights
Figure 4: Illustration of Equation6. Diagonally-masked self-attention on a time-series sample with five time steps.
With these diagonal masks, input values at the t 𝑡 t italic_t-th step can not see themselves and are prohibited from contributing to their own estimations. Consequently, their estimations depend only on the input values from other (T−1)𝑇 1(T-1)( italic_T - 1 ) time steps. Such a mechanism makes DMSA able to capture the temporal dependencies and feature correlations between time steps in the high dimensional space with only one attention operation. Subsequently, the diagonally-masked multi-head attention (DiagMaskedMHA DiagMaskedMHA\operatorname{DiagMaskedMHA}roman_DiagMaskedMHA) is formulated as:
DiagMaskedMHA(x)=Concat DiagMaskedMHA 𝑥 Concat\displaystyle\operatorname{DiagMaskedMHA}\left(x\right)=\operatorname{Concat}roman_DiagMaskedMHA ( italic_x ) = roman_Concat(head 1,head 2,…,head i,…,head h)W O ℎ 𝑒 𝑎 subscript 𝑑 1 ℎ 𝑒 𝑎 subscript 𝑑 2…ℎ 𝑒 𝑎 subscript 𝑑 𝑖…ℎ 𝑒 𝑎 subscript 𝑑 ℎ superscript 𝑊 𝑂\displaystyle\left(head_{1},head_{2},...,head_{i},...,head_{h}\right)W^{O}( italic_h italic_e italic_a italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_h italic_e italic_a italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_h italic_e italic_a italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , … , italic_h italic_e italic_a italic_d start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) italic_W start_POSTSUPERSCRIPT italic_O end_POSTSUPERSCRIPT(7) wherehead i=DiagMaskedSelfAttention where ℎ 𝑒 𝑎 subscript 𝑑 𝑖 DiagMaskedSelfAttention\displaystyle\text{where }head_{i}=\operatorname{DiagMaskedSelfAttention}where italic_h italic_e italic_a italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_DiagMaskedSelfAttention(xW i Q,xW i K,xW i V),his the number of heads 𝑥 superscript subscript 𝑊 𝑖 𝑄 𝑥 superscript subscript 𝑊 𝑖 𝐾 𝑥 superscript subscript 𝑊 𝑖 𝑉 ℎ is the number of heads\displaystyle\left(xW_{i}^{Q},xW_{i}^{K},xW_{i}^{V}\right),h\text{ is the % number of heads}( italic_x italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT , italic_x italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT , italic_x italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT ) , italic_h is the number of heads
In the above Equation7, W i Q∈ℝ d model×d k superscript subscript 𝑊 𝑖 𝑄 superscript ℝ subscript 𝑑 model subscript 𝑑 𝑘 W_{i}^{Q}\in\mathbb{R}^{d_{\text{model }}\times d_{k}}italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, W i K∈ℝ d model×d k superscript subscript 𝑊 𝑖 𝐾 superscript ℝ subscript 𝑑 model subscript 𝑑 𝑘 W_{i}^{K}\in\mathbb{R}^{d_{\text{model }}\times d_{k}}italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, and W i V∈ℝ d model×d v superscript subscript 𝑊 𝑖 𝑉 superscript ℝ subscript 𝑑 model subscript 𝑑 𝑣 W_{i}^{V}\in\mathbb{R}^{d_{\text{model}}\times d_{v}}italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT are parameters of the linear layers projecting input x 𝑥 x italic_x to Q 𝑄 Q italic_Q, K 𝐾 K italic_K, and V 𝑉 V italic_V separately. W O∈ℝ hd v×d model superscript 𝑊 𝑂 superscript ℝ ℎ subscript 𝑑 𝑣 subscript 𝑑 model W^{O}\in\mathbb{R}^{hd_{v}\times d_{\text{model}}}italic_W start_POSTSUPERSCRIPT italic_O end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_h italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is the parameter of the output layer in DiagMaskedMHA DiagMaskedMHA\operatorname{DiagMaskedMHA}roman_DiagMaskedMHA.
To prove the effectiveness of DMSA, an ablation study is performed in Section4.5.1. Note that attention masks are widely applied in self-attention modeling, especially in NLP field, including the diagonal masks used here, for example[45, 44, 46].
3.2.2 Positional Encoding and Feed-Forward Network
In Transformer, Vaswani et al.[42] apply the positional encoding to make use of the sequence order because there is no notion of sequence order in the original Transformer architecture. Additionally, there is a fully-connected feed-forward network applied behind each attention layer. In SAITS, both the positional encoding and the feed-forward network are kept.
The positional encoding consists of sine and cosine functions, which is formulated as Equation8 below. Note that p 𝑝 p italic_p is used to refer to the positional encoding in the following equations for brevity.
PosEnc(pos,2i)=sin(pos 10000 2i d model),PosEnc 𝑝 𝑜 𝑠 2 𝑖 𝑝 𝑜 𝑠 superscript 10000 2 𝑖 subscript 𝑑 model\displaystyle\operatorname{PosEnc}(pos,2i)=\sin\left(\frac{pos}{10000^{\frac{2% i}{d_{\text{model}}}}}\right),roman_PosEnc ( italic_p italic_o italic_s , 2 italic_i ) = roman_sin ( divide start_ARG italic_p italic_o italic_s end_ARG start_ARG 10000 start_POSTSUPERSCRIPT divide start_ARG 2 italic_i end_ARG start_ARG italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT end_ARG ) ,PosEnc(pos,2i+1)=cos(pos 10000 2i d model)PosEnc 𝑝 𝑜 𝑠 2 𝑖 1 𝑝 𝑜 𝑠 superscript 10000 2 𝑖 subscript 𝑑 model\displaystyle\hskip 10.00002pt\operatorname{PosEnc}(pos,2i+1)=\cos\left(\frac{% pos}{10000^{\frac{2i}{d_{\text{model}}}}}\right)roman_PosEnc ( italic_p italic_o italic_s , 2 italic_i + 1 ) = roman_cos ( divide start_ARG italic_p italic_o italic_s end_ARG start_ARG 10000 start_POSTSUPERSCRIPT divide start_ARG 2 italic_i end_ARG start_ARG italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT end_ARG )(8) whereposis the time-step where 𝑝 𝑜 𝑠 is the time-step\displaystyle\text{where }pos\text{ is the time-step }where italic_p italic_o italic_s is the time-step position,iis the dimension position,𝑖 is the dimension\displaystyle\text{position, }i\text{ is the dimension}position, italic_i is the dimension
The feed-forward network has two linear transformations with a ReLU activation function between them, as shown in Equation9:
FFN(x)FFN 𝑥\displaystyle\operatorname{FFN}\left(x\right)roman_FFN ( italic_x )=ReLU(xW 1+b 1)W 2+b 2 absent ReLU 𝑥 subscript 𝑊 1 subscript 𝑏 1 subscript 𝑊 2 subscript 𝑏 2\displaystyle=\operatorname{ReLU}\left(xW_{1}+b_{1}\right)W_{2}+b_{2}= roman_ReLU ( italic_x italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT(9) whereW 1∈ℝ d model×d ffn where subscript 𝑊 1 superscript ℝ subscript 𝑑 model subscript 𝑑 ffn\displaystyle\text{where }W_{1}\in\mathbb{R}^{d_{\text{model}}\times d_{\text{% ffn}}}where italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT ffn end_POSTSUBSCRIPT end_POSTSUPERSCRIPT,b 1∈ℝ d ffn,W 2∈ℝ d ffn×d model,b 2∈ℝ d model\displaystyle,b_{1}\in\mathbb{R}^{d_{\text{ffn}}},W_{2}\in\mathbb{R}^{d_{\text% {ffn}}\times d_{\text{model}}},b_{2}\in\mathbb{R}^{d_{\text{model}}}, italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT ffn end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT ffn end_POSTSUBSCRIPT × italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT
3.2.3 The First DMSA Block
e=[Concat(X^,M^)W e+b e]+p 𝑒 delimited-[]Concat^𝑋^𝑀 subscript 𝑊 𝑒 subscript 𝑏 𝑒 𝑝 e=\left[\operatorname{Concat}\left(\hat{X},\hat{M}\right)W_{e}+b_{e}\right]+p italic_e = [ roman_Concat ( over^ start_ARG italic_X end_ARG , over^ start_ARG italic_M end_ARG ) italic_W start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ] + italic_p(10)
z={FFN(DiagMaskedMHA(e))}N 𝑧 superscript FFN DiagMaskedMHA 𝑒 𝑁 z={\operatorname{FFN}(\operatorname{DiagMaskedMHA}\left(e\right))}^{N}italic_z = { roman_FFN ( roman_DiagMaskedMHA ( italic_e ) ) } start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT(11)
X1=zW z+b z subscript𝑋 1 𝑧 subscript 𝑊 𝑧 subscript 𝑏 𝑧\tilde{X}{1}=zW{z}+b_{z}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_z italic_W start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT(12)
X^′=M^⊙X^+(1−M^)⊙X1 superscript^𝑋′direct-product^𝑀^𝑋 direct-product 1^𝑀 subscript𝑋 1\hat{X}^{\prime}=\hat{M}\odot\hat{X}+\left(1-\hat{M}\right)\odot\tilde{X}_{1}over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = over^ start_ARG italic_M end_ARG ⊙ over^ start_ARG italic_X end_ARG + ( 1 - over^ start_ARG italic_M end_ARG ) ⊙ over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT(13)
In the first DMSA block, the actual input feature vector X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG and its missing mask vector M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG are concatenated as the input. Equation10 projects the input to d model subscript 𝑑 model d_{\text{model}}italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT dimensions and adds up with the positional encoding p 𝑝 p italic_p to produce e 𝑒 e italic_e. W e subscript 𝑊 𝑒 W_{e}italic_W start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT and b e subscript 𝑏 𝑒 b_{e}italic_b start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT are parameters (W e∈ℝ 2D×d model subscript 𝑊 𝑒 superscript ℝ 2 𝐷 subscript 𝑑 model W_{e}\in\mathbb{R}^{2D\times d_{\text{model}}}italic_W start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_D × italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, b e∈ℝ d model subscript 𝑏 𝑒 superscript ℝ subscript 𝑑 model b_{e}\in\mathbb{R}^{d_{\text{model}}}italic_b start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT). Operation {}N superscript 𝑁{}^{N}{ } start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT in Equation11 means stacking N 𝑁 N italic_N layers. Equation11 transfers e 𝑒 e italic_e to z 𝑧 z italic_z with N 𝑁 N italic_N stacked layers of the diagonally-masked multi-head attention and the feed-forward network 1 1 1 Note that the layer normalization[47] and residual connection[48] are applied after each attention layer and feed-forward layer in the same way as[42]. Figure3 shows these details. They are suppressed here for simplicity.. Equation12 reduces z 𝑧 z italic_z from d model subscript 𝑑 model d_{\text{model}}italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT dimensions to D 𝐷 D italic_D dimensions and produces X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (Learned Representation 1). Parameter W z∈ℝ d model×D subscript 𝑊 𝑧 superscript ℝ subscript 𝑑 model 𝐷 W{z}\in\mathbb{R}^{d_{\text{model}}\times D}italic_W start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_D end_POSTSUPERSCRIPT and b z∈ℝ D subscript 𝑏 𝑧 superscript ℝ 𝐷 b_{z}\in\mathbb{R}^{D}italic_b start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT. In Equation13, missing values in X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG are replaced with corresponding values in X1 subscript𝑋 1\tilde{X}_{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to obtain the completed feature vector X^′superscript^𝑋′\hat{X}^{\prime}over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT with the observed part in X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG kept intact. Here, ⊙direct-product\odot⊙ is the Hadamard product, also known as the element-wise product.
3.2.4 The Second DMSA Block
α=[Concat(X^′,M^)W α+b α]+p 𝛼 delimited-[]Concat superscript^𝑋′^𝑀 subscript 𝑊 𝛼 subscript 𝑏 𝛼 𝑝\alpha=\left[\operatorname{Concat}\left(\hat{X}^{\prime},\hat{M}\right)W_{% \alpha}+b_{\alpha}\right]+p italic_α = [ roman_Concat ( over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , over^ start_ARG italic_M end_ARG ) italic_W start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ] + italic_p(14)
β={FFN(DiagMaskedMHA(α))}N 𝛽 superscript FFN DiagMaskedMHA 𝛼 𝑁\beta={\operatorname{FFN}\left(\operatorname{DiagMaskedMHA}\left(\alpha\right% )\right)}^{N}italic_β = { roman_FFN ( roman_DiagMaskedMHA ( italic_α ) ) } start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT(15)
X2=ReLU(βW β+b β)W γ+b γ subscript𝑋 2 ReLU 𝛽 subscript 𝑊 𝛽 subscript 𝑏 𝛽 subscript 𝑊 𝛾 subscript 𝑏 𝛾\tilde{X}{2}=\operatorname{ReLU}\left(\beta W{\beta}+b_{\beta}\right)W_{% \gamma}+b_{\gamma}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = roman_ReLU ( italic_β italic_W start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ) italic_W start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT(16)
The second DMSA block takes the output X^′superscript^𝑋′\hat{X}^{\prime}over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT of the first DMSA block and continues learning. Similar to Equation10, Equation14 projects the concatenation of X^′superscript^𝑋′\hat{X}^{\prime}over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG from D 𝐷 D italic_D dimensions to d model subscript 𝑑 model d_{\text{model}}italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT dimensions and then adds the result together with p 𝑝 p italic_p to generate α 𝛼\alpha italic_α. Parameter W α∈ℝ 2D×d model subscript 𝑊 𝛼 superscript ℝ 2 𝐷 subscript 𝑑 model W_{\alpha}\in\mathbb{R}^{2D\times d_{\text{model}}}italic_W start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_D × italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, b α∈ℝ d model subscript 𝑏 𝛼 superscript ℝ subscript 𝑑 model b_{\alpha}\in\mathbb{R}^{d_{\text{model}}}italic_b start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. Equation15 performs N 𝑁 N italic_N times of nested attention functions and feed-forward networks on α 𝛼\alpha italic_α and outputs β 𝛽\beta italic_β. In Equation16, to obtain X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (Learned Representation 2), two linear projections are applied on β 𝛽\beta italic_β with a ReLU activation in between, where parameter W β∈ℝ d model×D subscript 𝑊 𝛽 superscript ℝ subscript 𝑑 model 𝐷 W{\beta}\in\mathbb{R}^{d_{\text{model}}\times D}italic_W start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT × italic_D end_POSTSUPERSCRIPT, b β∈ℝ D subscript 𝑏 𝛽 superscript ℝ 𝐷 b_{\beta}\in\mathbb{R}^{D}italic_b start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT, W γ∈ℝ D×D subscript 𝑊 𝛾 superscript ℝ 𝐷 𝐷 W_{\gamma}\in\mathbb{R}^{D\times D}italic_W start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_D × italic_D end_POSTSUPERSCRIPT, b γ∈ℝ D subscript 𝑏 𝛾 superscript ℝ 𝐷 b_{\gamma}\in\mathbb{R}^{D}italic_b start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT. Empirically, a deeper structure can learn better a representation to capture more complicated correlations in time series. Here, in Equation16, we apply one more non-linear layer than Equation12 to build a deeper block. In practice, such an operation does help achieve a better imputation performance than applying a single linear projection. The same transformation is not applied to obtain X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in the first DMSA block because the learnable parameters in the following weighted combination can dynamically adjust the weights for X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT to form better X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT (Learned Representation 3). Moreover, we find that even applying the same transformation here to obtain X1 subscript𝑋 1\tilde{X}_{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT does not help achieve better results than the current design. It validates the effectiveness of our weighted combination, described as below.
3.2.5 The Weighted Combination Block
A^=1 h∑i=1 h A i^𝐴 1 ℎ superscript subscript 𝑖 1 ℎ subscript 𝐴 𝑖\hat{A}=\frac{1}{h}\sum_{i=1}^{h}A_{i}over^ start_ARG italic_A end_ARG = divide start_ARG 1 end_ARG start_ARG italic_h end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT(17)
η=Sigmoid(Concat(A^,M^)W η+b η)𝜂 Sigmoid Concat^𝐴^𝑀 subscript 𝑊 𝜂 subscript 𝑏 𝜂\eta=\operatorname{Sigmoid}\left(\operatorname{Concat}\left(\hat{A},\hat{M}% \right)W_{\eta}+b_{\eta}\right)italic_η = roman_Sigmoid ( roman_Concat ( over^ start_ARG italic_A end_ARG , over^ start_ARG italic_M end_ARG ) italic_W start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT )(18)
X3=(1−η)⊙X1+η⊙X2 subscript𝑋 3 direct-product 1 𝜂 subscript𝑋 1 direct-product 𝜂 subscript𝑋 2\tilde{X}{3}=\left(1-\eta\right)\odot\tilde{X}{1}+\eta\odot\tilde{X}_{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = ( 1 - italic_η ) ⊙ over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_η ⊙ over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT(19)
X^c=M^⊙X^+(1−M^)⊙X3 subscript^𝑋 𝑐 direct-product^𝑀^𝑋 direct-product 1^𝑀 subscript𝑋 3\hat{X}{c}=\hat{M}\odot\hat{X}+\left(1-\hat{M}\right)\odot\tilde{X}{3}over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = over^ start_ARG italic_M end_ARG ⊙ over^ start_ARG italic_X end_ARG + ( 1 - over^ start_ARG italic_M end_ARG ) ⊙ over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT(20)
To obtain a better learned representation X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, the weighted combination block is designed to dynamically weigh X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT according to temporal dependencies and missingness information. A^^𝐴\hat{A}over^ start_ARG italic_A end_ARG (∈ℝ T×T absent superscript ℝ 𝑇 𝑇\in\mathbb{R}^{T\times T}∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_T end_POSTSUPERSCRIPT) in Equation17 is averaged from attention weights A 𝐴 A italic_A output by multi heads in the last layer of the second DMSA block. Equation18 takes averaged attention weights A^^𝐴\hat{A}over^ start_ARG italic_A end_ARG and missing masks M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG as references to produce the combining weights η 𝜂\eta italic_η (∈(0,1)T×D absent superscript 0 1 𝑇 𝐷\in(0,1)^{T\times D}∈ ( 0 , 1 ) start_POSTSUPERSCRIPT italic_T × italic_D end_POSTSUPERSCRIPT) with the learnable parameters W η subscript 𝑊 𝜂 W{\eta}italic_W start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT (∈ℝ(T+D)×D absent superscript ℝ 𝑇 𝐷 𝐷\in\mathbb{R}^{(T+D)\times D}∈ blackboard_R start_POSTSUPERSCRIPT ( italic_T + italic_D ) × italic_D end_POSTSUPERSCRIPT) and b η subscript 𝑏 𝜂 b_{\eta}italic_b start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT (∈ℝ D absent superscript ℝ 𝐷\in\mathbb{R}^{D}∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT). Equation19 combines X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT by weights η 𝜂\eta italic_η to form X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Finally, in Equation20, missing values in X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG are replaced with corresponding values in X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT to produce the complement vector X^c subscript^𝑋 𝑐\hat{X}_{c}over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, i.e. the imputed data. To further discuss the rationality of the weighted combination, an ablation experiment is performed in Section4.5.2.
Moreover, the second DMSA block and the weighted combination block are added to extend the learning process of our model and to obtain better performance. We do not apply more than two DMSA blocks because the benefit brought is marginal. Experiments and analysis are conducted to prove our points here in Section4.5.3.
3.2.6 Loss Functions of Learning Objectives
ℒ ORT=1 3(ℓ MAE(X1,X,M^)+ℓ MAE(X2,X,M^)+ℓ MAE(X3,X,M^))subscript ℒ ORT 1 3 subscript ℓ MAE subscript𝑋 1 𝑋^𝑀 subscript ℓ MAE subscript𝑋 2 𝑋^𝑀 subscript ℓ MAE subscript𝑋 3 𝑋^𝑀\mathcal{L}{\text{ORT}}=\frac{1}{3}\left(\ell{\text{MAE}}\left(\tilde{X}{1}% ,X,\hat{M}\right)+\ell{\text{MAE}}\left(\tilde{X}{2},X,\hat{M}\right)+\ell{% \text{MAE}}\left(\tilde{X}_{3},X,\hat{M}\right)\right)caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 3 end_ARG ( roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_X , over^ start_ARG italic_M end_ARG ) + roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_X , over^ start_ARG italic_M end_ARG ) + roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_X , over^ start_ARG italic_M end_ARG ) )(21)
ℒ MIT=ℓ MAE(X^c,X,I)subscript ℒ MIT subscript ℓ MAE subscript^𝑋 𝑐 𝑋 𝐼\mathcal{L}{\text{MIT}}=\ell{\text{MAE}}\left(\hat{X}_{c},X,I\right)caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT = roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT ( over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , italic_X , italic_I )(22)
ℒ=ℒ ORT+λℒ MIT ℒ subscript ℒ ORT 𝜆 subscript ℒ MIT\mathcal{L}=\mathcal{L}{\text{ORT}}+\lambda,\mathcal{L}{\text{MIT}}caligraphic_L = caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT + italic_λ caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT(23)
There are two learning tasks in the model training: MIT and ORT. The imputation loss of MIT (ℒ MIT subscript ℒ MIT\mathcal{L}{\text{MIT}}caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT) and the reconstruction loss of ORT (ℒ ORT subscript ℒ ORT\mathcal{L}{\text{ORT}}caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT) are both calculated by the MAE loss function (ℓ MAE subscript ℓ MAE\ell_{\text{MAE}}roman_ℓ start_POSTSUBSCRIPT MAE end_POSTSUBSCRIPT) defined in Equation1, which takes three inputs: estimation 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 estimation italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n, target 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 target italic_t italic_a italic_r italic_g italic_e italic_t, and mask 𝑚 𝑎 𝑠 𝑘 mask italic_m italic_a italic_s italic_k (all of them ∈ℝ T×D absent superscript ℝ 𝑇 𝐷\in\mathbb{R}^{T\times D}∈ blackboard_R start_POSTSUPERSCRIPT italic_T × italic_D end_POSTSUPERSCRIPT). It calculates MAE between values indicated by mask 𝑚 𝑎 𝑠 𝑘 mask italic_m italic_a italic_s italic_k in estimation 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 estimation italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n and target 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 target italic_t italic_a italic_r italic_g italic_e italic_t. target 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 target italic_t italic_a italic_r italic_g italic_e italic_t and mask 𝑚 𝑎 𝑠 𝑘 mask italic_m italic_a italic_s italic_k of ℒ ORT subscript ℒ ORT\mathcal{L}{\text{ORT}}caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT in Equation21 are the input feature vector X^^𝑋\hat{X}over^ start_ARG italic_X end_ARG and its missing mask vector M^^𝑀\hat{M}over^ start_ARG italic_M end_ARG. We let X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT directly participate in the composition of X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Accordingly, here ℒ ORT subscript ℒ ORT\mathcal{L}{\text{ORT}}caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT is accumulated from three learned representations: X1 subscript𝑋 1\tilde{X}{1}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, X2 subscript𝑋 2\tilde{X}{2}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and X3 subscript𝑋 3\tilde{X}{3}over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Such an accumulated loss can lead to faster convergence speed. To ensure ℒ ORT subscript ℒ ORT\mathcal{L}{\text{ORT}}caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT not too large to dominate the direction of the gradient, it is reduced by a factor of three, i.e. averaged. Inputs estimation 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 estimation italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n, target 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 target italic_t italic_a italic_r italic_g italic_e italic_t and mask 𝑚 𝑎 𝑠 𝑘 mask italic_m italic_a italic_s italic_k of ℒ MIT subscript ℒ MIT\mathcal{L}{\text{MIT}}caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT in Equation22 are the complement feature vector X^c subscript^𝑋 𝑐\hat{X}{c}over^ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, the original feature vector X 𝑋 X italic_X without artificially-masked values, and the indicating mask vector I 𝐼 I italic_I, respectively. At last, Equation23 adds ℒ ORT subscript ℒ ORT\mathcal{L}{\text{ORT}}caligraphic_L start_POSTSUBSCRIPT ORT end_POSTSUBSCRIPT and ℒ MIT subscript ℒ MIT\mathcal{L}_{\text{MIT}}caligraphic_L start_POSTSUBSCRIPT MIT end_POSTSUBSCRIPT together by a weighted sum, where λ 𝜆\lambda italic_λ is the weighting coefficient that can be tuned. λ 𝜆\lambda italic_λ is fixed as 1 in our experiments. Our SAITS model is updated by minimizing the final loss ℒ ℒ\mathcal{L}caligraphic_L.
4 Experiments
For the sake of the reproducibility of our results, we make our work publicly available to the community. Our data preprocessing scripts, model implementations, as well as hyper-parameter search configurations, are all available in the GitHub repositoryhttps://github.com/WenjieDu/SAITS.
4.1 Datasets
The descriptions of four datasets used in this work and their preprocessing details are elaborated as below. General information of all datasets is listed in Table1. Note that standardization is applied in the preprocessing of all datasets.
Table 1: General information of four datasets used in this work.
PhysioNet-2012 Air-Quality Electricity ETT Number of total samples 11,988 1,461 1,400 5,803 Number of features 37 132 370 7 Sequence length 48 24 100 24 Original missing rate 80.0%1.6%0%0%
PhysioNet 2012 Mortality Prediction Challenge (PhysioNet-2012) The PhysioNet 2012 challenge dataset[51] contains 12,000 multivariate clinical time-series samples collected from patients in ICU (Intensive Care Unit). Each sample is recorded during the first 48 hours after admission to the ICU. Depending on the status of patients, there are up to 37 time-series variables measured, for instance, temperature, heart rate, blood pressure. Measurements might be collected at regular intervals (hourly or daily), and also may be recorded at irregular intervals (only collected as required). Not all variables are available in all samples. Note that this dataset is very sparse and has 80% missing values in total. The dataset is firstly split into the training set and the test set according to 80% and 20%. Then 20% of samples are split from the training set as the validation set. We randomly eliminate 10% of observed values in the validation set and the test set and use these values as ground truth to evaluate the imputation performance of models. Following 12 samples are dropped because of containing no time-series information at all: 147514, 142731, 145611, 140501, 155655, 143656, 156254, 150309, 140936, 141264, 150649, 142998.
Beijing Multi-Site Air-Quality (Air-Quality) This air-quality dataset[49] includes hourly air pollutants data from 12 monitoring sites in Beijing. Data is collected from 2013/03/01 to 2017/02/28 (48 months in total). For each monitoring site, there are 11 continuous time series variables measured (e.g. PM2.5, PM10, SO2). We aggregate variables from 12 sites together so this dataset has 132 features. There are a total of 1.6% missing values in this dataset. The test set takes data from the first 10 months (2013/03 - 2013/12). The validation set contains data from the following 10 months (2014/01 - 2014/10). The training set takes the left 28 months (2014/11 - 2017/02). To generate time series data samples, we select every 24 hours data, i.e. every 24 consecutive steps, as one sample. Similar to dataset PhysioNet-2012, 10% observed values in the validation set and test set are eliminated and held out as ground-truth for evaluation.
Electricity Load Diagrams (Electricity) This is another widely-used public dataset from UCI[50]. It contains electricity consumption data (in kWh) collected from 370 clients every 15 minutes and has no missing data. The period of this dataset is from 2011/01/01 to 2014/12/31 (48 months in total). Similar to processing Air-Quality, we use the first 10 months of data (2011/01 - 2011/10) as the test set, the following 10 months of data (2011/11 - 2012/08) as the validation set and the left (2012/09 - 2014/12) as the training set. Every 100 consecutive steps are selected as a sample to generate time-series data for model training. Due to this dataset having no missing values, we vary artificial missing rate from 10% ∼similar-to\sim∼ 90% to eliminate observed values in the training set, validation set, and test set. This can make the comparison between our method and other SOTA models more comprehensive. Artificial missing values in the validation and test set are held out for model evaluation. Experiment results of 10% missing values are displayed in Table2. Results of 20% ∼similar-to\sim∼ 90% missing values are shown in Table4.
Electricity Transformer Temperature (ETT) This dataset[52] is collected from electricity transformers in two years from 2016/07/01 to 2018/06/26. Here we select the 15-minute-level version dataset that contains data collected every 15 minutes and has a total of 69680 sample points without missing values. Each sample has seven features, including oil temperature and six different types of external power load features. The data in the first four months (2016/07 - 2016/10) is taken as the test set. The following four-month data (2016/11 - 2017/02) is held out as the validation set. The left sixteen months (2017/03 - 2018/06) are for training use. The sliding-window method is applied for time-series sample generation. The window size is set as 6 hours (i.e. 24 consecutive steps) and the sliding size is 3 hours (i.e. 12 steps). Still, 10% of the observed values in the validation set and test set are randomly masked for model evaluation.
4.2 Baseline Methods
4.3 Experimental Setup
Three metrics are utilized to evaluate the imputation performance of methods: MAE (Mean Absolute Error), RMSE (Root Mean Square Error), and MRE (Mean Relative Error). The math definitions of three evaluation metrics are presented below. Note that errors are only computed on the values indicated by mask 𝑚 𝑎 𝑠 𝑘 mask italic_m italic_a italic_s italic_k in the input of the equations.
MAE(estimation,target,mask)MAE 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘\displaystyle\text{MAE}\left(estimation,target,mask\right)MAE ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n , italic_t italic_a italic_r italic_g italic_e italic_t , italic_m italic_a italic_s italic_k )=∑d=1 D∑t=1 T|(estimation−target)⊙mask|t d∑d=1 D∑t=1 T mask t d absent superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 superscript subscript direct-product 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 𝑡 𝑑 superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 𝑚 𝑎 𝑠 superscript subscript 𝑘 𝑡 𝑑\displaystyle=\frac{\sum_{d=1}^{D}\sum_{t=1}^{T}\lvert\left(estimation-target% \right)\odot mask\rvert_{t}^{d}}{\sum_{d=1}^{D}\sum_{t=1}^{T}mask_{t}^{d}}= divide start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT | ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n - italic_t italic_a italic_r italic_g italic_e italic_t ) ⊙ italic_m italic_a italic_s italic_k | start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_m italic_a italic_s italic_k start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG RMSE(estimation,target,mask)RMSE 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘\displaystyle\text{RMSE}\left(estimation,target,mask\right)RMSE ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n , italic_t italic_a italic_r italic_g italic_e italic_t , italic_m italic_a italic_s italic_k )=∑d=1 D∑t=1 T(((estimation−target)⊙mask)2)t d∑d=1 D∑t=1 T mask t d absent superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 superscript subscript superscript direct-product 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 2 𝑡 𝑑 superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 𝑚 𝑎 𝑠 superscript subscript 𝑘 𝑡 𝑑\displaystyle=\sqrt{\frac{\sum_{d=1}^{D}\sum_{t=1}^{T}\left(\left(\left(% estimation-target\right)\odot mask\right)^{2}\right){t}^{d}}{\sum{d=1}^{D}% \sum_{t=1}^{T}mask_{t}^{d}}}= square-root start_ARG divide start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( ( ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n - italic_t italic_a italic_r italic_g italic_e italic_t ) ⊙ italic_m italic_a italic_s italic_k ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_m italic_a italic_s italic_k start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG end_ARG MRE(estimation,target,mask)MRE 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘\displaystyle\text{MRE}\left(estimation,target,mask\right)MRE ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n , italic_t italic_a italic_r italic_g italic_e italic_t , italic_m italic_a italic_s italic_k )=∑d=1 D∑t=1 T|(estimation−target)⊙mask|t d∑d=1 D∑t=1 T|target⊙mask|t d absent superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 superscript subscript direct-product 𝑒 𝑠 𝑡 𝑖 𝑚 𝑎 𝑡 𝑖 𝑜 𝑛 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 𝑡 𝑑 superscript subscript 𝑑 1 𝐷 superscript subscript 𝑡 1 𝑇 superscript subscript direct-product 𝑡 𝑎 𝑟 𝑔 𝑒 𝑡 𝑚 𝑎 𝑠 𝑘 𝑡 𝑑\displaystyle=\frac{\sum_{d=1}^{D}\sum_{t=1}^{T}\lvert\left(estimation-target% \right)\odot mask\rvert_{t}^{d}}{\sum_{d=1}^{D}\sum_{t=1}^{T}|target\odot mask% |_{t}^{d}}= divide start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT | ( italic_e italic_s italic_t italic_i italic_m italic_a italic_t italic_i italic_o italic_n - italic_t italic_a italic_r italic_g italic_e italic_t ) ⊙ italic_m italic_a italic_s italic_k | start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT | italic_t italic_a italic_r italic_g italic_e italic_t ⊙ italic_m italic_a italic_s italic_k | start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG
The batch size is fixed as 128, and the early stopping strategy is applied in the model training. Training of models is stopped after 30 epochs without any decrease of MAE. To permit a fair comparison between the models, the hyper-parameter searches are executed for every model on each dataset, except SAITS-base. For SAITS-base, we fix its hyper-parameters to form a base model and observe its performance. SAITS-base is also applied in ablation experiments in Section4.5 as a baseline to make the comparisons more straightforward. Please consult AppendixA for further details of models’ hyper-parameters. Transformer used in this paper only includes the encoder part because the imputation problem is not treated as a generative task in this work, therefore, the decoder part is in no need. All models are trained with the Adam optimizer[53] on Nvidia Quadro RTX 5000 GPUs. Our models are implemented with PyTorch[54].
4.4 Experimental Results
The adequate experiments performed to benchmark the performance of SAITS are made up of three parts in this section. In Subsection4.4.1, the baseline methods and the self-attention models are impartially compared on four datasets. To further discuss the influence of imputation quality on pattern recognition tasks, an experiment in Subsection4.4.2 is performed on a downstream classification task. In addition, we experiment to compare SAITS with NRTSI, another SOTA self-attention-based imputation model for time series. Due to bugs in the official implementation of NRTSI, it is not appropriate to put it together with other baseline methods. Hence, we run SAITS on preprocessed datasets provided by authors of NRTSI to make a fair comparison in Subsection4.4.3.
4.4.1 Imputation Performance Comparison
Table 2: Performance comparison between methods on four datasets. 10% observations in the test set are held out for evaluation. Metrics are reported in the order of MAE / RMSE / MRE. The lower, the better. Bold font indicates the best performance. GRUI-GAN and E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN have no results on Electricity because they fail in the training due to loss explosion.
Method PhysioNet-2012 Air-Quality Electricity ETT Median 0.726 / 0.988 / 103.5%0.763 / 1.175 / 107.4%2.056 / 2.732 / 110.1%1.145 / 1.847 / 139.1% Last 0.862 / 1.207 / 123.0%0.967 / 1.408 / 136.3%1.006 / 1.533 / 53.9%1.007 / 1.365 / 96.4% GRUI-GAN 0.765 / 1.040 / 109.1%0.788 / 1.179 / 111.0%/0.612 / 0.729 / 95.1% E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN 0.702 / 0.964 / 100.1%0.750 / 1.126 / 105.6%/0.584 / 0.703 / 89.0% M-RNN 0.533 / 0.776 / 76.0%0.294 / 0.643 / 41.4%1.244 / 1.867 / 66.6%0.376 / 0.428 / 31.6% GP-VAE 0.398 / 0.630 / 56.7%0.268 / 0.614 / 37.7%1.094 / 1.565 / 58.6%0.274 / 0.307 / 15.5% BRITS 0.256 / 0.767 / 36.5%0.153 / 0.525 / 21.6%0.847 / 1.322 / 45.3%0.130 / 0.259 / 12.5% Transformer 0.190 / 0.445 / 26.9%0.158 / 0.521 / 22.3%0.823 / 1.301 / 44.0%0.114 / 0.173 / 10.9% SAITS-base 0.192 / 0.439 / 27.3%0.146 / 0.521 / 20.6%0.822 / 1.221 / 44.0%0.121 / 0.197 / 11.6% SAITS 0.186 / 0.431 / 26.6%0.137 / 0.518 / 19.3%0.735 / 1.162 / 39.4%0.092 / 0.139 / 8.8%
Table 3: Models’ parameter number (in million) and training time of each epoch (in seconds) on datasets PhysioNet-2012, Air-Quality, Electricity, and ETT are listed from left to right. GRUI-GAN and E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN have no results for the Electricity dataset because they fail on this dataset due to loss explosion.
PhysioNet-2012 Air-Quality Electricity ETT Model# of param s / epoch# of param s / epoch# of param s / epoch# of param s / epoch GRUI-GAN 0.16M 14.4 2.32M 2.0//1.52M 9.7 E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN 0.08M 22.8 1.13M 2.2//0.29M 11.9 M-RNN 0.07M 6.8 1.09M 1.3 18.63M 3.9 0.15M 7.6 GP-VAE 0.15M 40.1 0.36M 8.7 13.45M 106.0 0.27M 22.1 BRITS 0.73M 12.8 11.25M 1.9 7.00M 5.2 0.57M 10.2 Transformer 4.36M 3.1 5.13M 0.9 14.78M 2.6 4.67M 3.7 SAITS-base 1.38M 2.7 1.56M 1.1 2.20M 2.1 1.33M 2.5 SAITS 5.32M 5.0 3.07M 0.9 11.51M 2.6 4.27M 4.6
Table 4: Performance comparison between methods on the Electricity dataset across different missing rates from 20% ∼similar-to\sim∼ 90%. Metrics are reported in the order of MAE / RMSE / MRE. The lower, the better. Values in bold are the best.
Method 20%30%40%50% Median 2.053 / 2.726 / 109.9%2.055 / 2.732 / 110.0%2.058 / 2.734 / 110.2%2.053 / 2.728 / 109.9% Last 1.012 / 1.547 / 54.2%1.018 / 1.559 / 54.5%1.025 / 1.578 / 54.9%1.032 / 1.595 / 55.2% M-RNN 1.242 / 1.854 / 66.5%1.258 / 1.876 / 67.3%1.269 / 1.884 / 68.0%1.283 / 1.902 / 68.7% GP-VAE 1.124 / 1.502 / 60.2%1.057 / 1.571 / 56.6%1.090 / 1.578 / 58.4%1.097 / 1.572 / 58.8% BRITS 0.928 / 1.395 / 49.7%0.943 / 1.435 / 50.4%0.996 / 1.504 / 53.4%1.037 / 1.538 / 55.5% Transformer 0.843 / 1.318 / 45.1%0.846 / 1.321 / 45.3%0.876 / 1.387 / 46.9%0.895 / 1.410 / 47.9% SAITS-base 0.838 / 1.264 / 44.9%0.845 / 1.247 / 45.2%0.873 / 1.325 / 46.7%0.939 / 1.537 / 50.3% SAITS 0.763 / 1.187 / 40.8%0.790 / 1.223 / 42.3%0.869 / 1.314 / 46.7%0.876 / 1.377 / 46.9%
Method 60%70%80%90% Median 2.057 / 2.734 / 110.2%2.050 / 2.726 / 109.8%2.059 / 2.734 / 110.2%2.056 / 2.723 / 110.1% Last 1.040 / 1.615 / 55.7%1.049 / 1.640 / 56.2%1.059 / 1.663 / 56.7%1.070 / 1.690 / 57.3% M-RNN 1.298 / 1.912 / 69.4%1.305 / 1.928 / 69.9%1.318 / 1.951 / 70.5%1.331 / 1.961 / 71.3% GP-VAE 1.101 / 1.616 / 59.0%1.037 / 1.598 / 55.6%1.062 / 1.621 / 56.8%1.004 / 1.622 / 53.7% BRITS 1.101 / 1.602 / 59.0%1.090 / 1.617 / 58.4%1.138 / 1.665 / 61.0%1.163 / 1.702 / 62.3% Transformer 0.891 / 1.404 / 47.7%0.920 / 1.437 / 49.3%0.924 / 1.472 / 49.5%0.934 / 1.491 / 49.8% SAITS-base 0.969 / 1.565 / 51.9%0.972 / 1.601 / 52.0%1.012 / 1.608 / 54.2%1.001 / 1.630 / 53.6% SAITS 0.892 / 1.328 / 47.9%0.898 / 1.273 / 48.1%0.908 / 1.327 / 48.6%0.933 / 1.354 / 49.9%
Table2 reports the imputation performance of models on four datasets in three evaluation metrics (MAE / RMSE / MRE). On PhysioNet-2012 and Air-Quality, GRUI-GAN achieves better results than Last but is slightly inferior to Median. E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN performs better than these three methods. On Electricity, GRUI-GAN and E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN both fail because of loss explosion. We find this is caused by the long sequence length. Dataset Electricity’s sequence length is 100. If the sequence length of the Air-Quality dataset is increased from 24 to 100, both GAN models will be confronted with loss explosion and fail again. M-RNN outperforms both naive imputation methods a lot on PhysioNet-2012 and Air-Quality but gets worse results than Last on Electricity. On ETT, all the deep-learning methods performs obviously better than the naive imputation methods. GP-VAE and BRITS both perform much better than the methods mentioned above on all the four datasets. BRITS is the best one among baseline methods. When it comes to the self-attention-based models, Transformer surpasses BRITS obviously on datasets PhysioNet-2012, Electricity, and ETT, and obtains comparable results to BRITS on Air-Quality. SAITS-base achieves similar results to Transformer on all datasets. SAITS exceeds all baseline methods significantly on all metrics and all datasets, and it outperforms Transformer and SAITS-base as well.
To show further details of the models in Table2, the parameter number and the training speed of them are listed in the following Table3. We can see that GP-VAE is the slowest model and consumes the most seconds for each epoch training. The RNN-based models are all slower than the self-attention-based models. Compared to BRITS that yields the best results in the baseline methods, SAITS takes half the training time or even less as BRITS on each epoch. Compared to Transformer, SAITS-base has only 15% ∼similar-to\sim∼ 30% parameters of Transformer’s, but it still obtains comparable performance to Transformer. It confirms that SAITS’ model structure is more efficient than Transformer on the time-series imputation task.
To further compare the performance of the imputation methods on different missing rates, we also experiment to introduce missing values into the Electricity dataset at different rates between 20% ∼similar-to\sim∼ 90%. The results of this experiment are elaborated in Table4. The results of 10% missing rate have been displayed in Table2. Dataset Electricity is selected because it has no missing data, and it is the most complex among the four datasets because each sample has 370 features and 100 time steps (please refer to Table1). In Table2, the models achieve the highest error on the Electricity dataset, which also proves that it is the most difficult one to impute among four datasets. Therefore, Electricity is the most suitable dataset to experiment with different missing rates. GRUI-GAN and E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN are omitted because they fail on the Electricity dataset due to loss explosion as is discussed above. The baseline methods are all inferior to self-attention-based models in all cases. SAITS-base performs better than Transformer in cases of 20%, 30%, and 40%. However, its performance becomes worse than Transformer in the left cases where missing rates become higher. This is because the hyper-parameters of SAITS-base are fixed, and its number of parameters is limited to a low level, only 15% of Transformer (refer to Table3). Such a situation makes the capacity of SAITS-base not enough to well handle the imputation problem with the higher missing rate. And given enough model capacity (with 78% parameters of Transformer), SAITS achieves the best performance in eight out of nine cases, demonstrating its distinct advantage.
4.4.2 Downstream Classification Task
In the PhysioNet-2012 dataset, each sample has a label indicating if the patient is deceased that makes PhysioNet-2012 a binary-classification dataset and there are 1,707 (14.2%) samples with the positive mortality label. Consequently, mortality prediction is one of the main tasks on this dataset. However, 80% missing values make this task challenging. Similar to PhysioNet-2012, in real-world datasets, missingness often makes tasks of pattern recognition tricky. To further discuss the benefits that SAITS can bring to pattern recognition, the experiment here is conducted on a downstream classification task on the PhysioNet-2012 dataset to qualitatively compare the imputation quality of each method. Note that this experiment is inspired by prior work[16, 27, 28, 29, 30]. The idea behind this experimental design is that, if one method’s imputation is better in terms of overall data quality, datasets imputed by the method should achieve better performance on downstream pattern recognition tasks, such as classification here.
Table 5: Results of the downstream classification task on the PhysioNet-2012 dataset. Performance metrics of methods are calculated by five independent runs. The reported values are means ±plus-or-minus\pm± standard deviations. The higher, the better. Values in bold are the best.
Method ROC-AUC PR-AUC F1-score Median 83.4% ±plus-or-minus\pm± 0.4%46.0% ±plus-or-minus\pm± 0.6%38.5% ±plus-or-minus\pm± 3.1% Last 82.8% ±plus-or-minus\pm± 0.3%46.9% ±plus-or-minus\pm± 0.4%39.5% ±plus-or-minus\pm± 2.4% GRUI-GAN 83.0% ±plus-or-minus\pm± 0.2%45.1% ±plus-or-minus\pm± 0.7%38.8% ±plus-or-minus\pm± 2.0% E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN 83.0% ±plus-or-minus\pm± 0.2%45.5% ±plus-or-minus\pm± 0.5%35.6% ±plus-or-minus\pm± 2.0% M-RNN 82.2% ±plus-or-minus\pm± 0.2%45.4% ±plus-or-minus\pm± 0.6%38.8% ±plus-or-minus\pm± 3.5% GP-VAE 83.4% ±plus-or-minus\pm± 0.2%48.1% ±plus-or-minus\pm± 0.7%40.9% ±plus-or-minus\pm± 3.3% BRITS 83.5% ±plus-or-minus\pm± 0.1%49.1% ±plus-or-minus\pm± 0.4%41.3% ±plus-or-minus\pm± 1.8% Transformer 84.3% ±plus-or-minus\pm± 0.5%49.2% ±plus-or-minus\pm± 1.4%41.2% ±plus-or-minus\pm± 1.9% SAITS-base 84.6% ±plus-or-minus\pm± 0.2%49.8% ±plus-or-minus\pm± 0.4%41.5% ±plus-or-minus\pm± 2.0% SAITS 84.8% ±plus-or-minus\pm± 0.2%51.0% ±plus-or-minus\pm± 0.5%42.7% ±plus-or-minus\pm± 2.8%
We first let each method impute the dataset and then train a classifier on each imputed dataset to obtain the classification results. Since this is a time-series dataset, a simple RNN classification model is employed as the classifier. This RNN classifier consists of a GRU layer followed by a fully connected layer. All hyper-parameters are fixed as follows: the learning rate (1×10−3 1 superscript 10 3 1\times 10^{-3}1 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT), the batch size (128), the RNN hidden size (128), the patience of the early stopping strategy (20). The classifier and the training procedure are kept exactly the same for each imputed dataset to obtain the equitably comparable results. Considering that classes in this dataset are imbalanced, metrics ROC-AUC (Area Under ROC Curve), PR-AUC (Area Under Precision-Recall Curve), and F1-score are used to measure performance. The experiment results are reported in Table5. The method names in the table annotate that the dataset is imputed by which method.
As displayed in Table5, the classifier trained on the dataset imputed by SAITS achieves the greatest results on all evaluation metrics and obtains obvious improvements than RNN-based models (1.3%, 1.9%, and 1.4% better than BRITS in ROC-AUC, PR-AUC, and F1-score). Even though comparing SAITS with Transformer, the improvements on PR-AUC and F1-score are noteworthy (increased 1.8% and 1.5% respectively), considering this is on an imbalanced and sparse dataset and the classifier is trivial. Such an indirect comparison reaches the same conclusion that the imputation quality of SAITS is the best among all methods. In the meanwhile, it demonstrates that SAITS does not only have higher accuracy in the imputation metrics but also can help improve the performance of trivial models on pattern recognition tasks on time-series datasets with missing values.
4.4.3 Comparison with NRTSI
To make a fair comparison with NRTSI[34], another SOTA self-attention-based imputation model for time series, we tried to incorporate it into our framework of imputation models. However, its official open-source implementation in the GitHub repository https://github.com/lupalab/NRTSI has fatal bugs that stop us from reproducing their results or adding their model into our baselines. Fortunately, the authors provide their preprocessed datasets Air and Gas in their code repository. This makes us able to run SAITS on their datasets for an impartial comparison with NRTSI. Regarding the experiments in this section, we have uploaded the related dataset generating script and SAITS configuration files to our code repository to ensure our results are reproducible.
Both datasets Air[49] and Gas[55] are from UCI machine learning repository[50]. The raw data of the Air dataset is the same as the Air-Quality dataset used in our work, but the preprocessing methods are different that results in different time steps and feature numbers. Our Air-Quality dataset is 132-dimensional and has 24 time steps, while the Air dataset is 11-dimensional and has 48 time steps. The experiment results on the datasets Air and Gas are listed in Table6 and7 respectively. The results of NRTSI are from Table 5 in the original paper[34], where the evaluation metric is mean squared error (MSE), so we keep using it here.
Table 6: The performance comparison between NRTSI and SAITS on dataset Air across different missing rates from 10% ∼similar-to\sim∼ 80%. The evaluation metric is MSE. The lower, the better. The best results are in bold.
Method 10%20%30%40%50%60%70%80% NRTSI 0.1230 0.1155 0.1189 0.1250 0.1297 0.1378 0.1542 0.1790 SAITS 0.0980 0.0911 0.0916 0.1021 0.1109 0.1030 0.1179 0.1409
Table 7: The performance comparison between NRTSI and SAITS on dataset Gas across different missing rates from 10% ∼similar-to\sim∼ 80%. The evaluation metric used here is MSE. The lower, the better. The best results are in bold.
Method 10%20%30%40%50%60%70%80% NRTSI 0.0165 0.0195 0.0196 0.0229 0.0286 0.0311 0.0362 0.0445 SAITS 0.0100 0.0123 0.0145 0.0192 0.0239 0.0289 0.0337 0.0326
(a) Comparison on dataset Air
(b) Comparison on dataset Gas
Figure 5: The visualized comparison with NRTSI on datasets Air and Gas. The percentage numbers above the bars indicate, compared with NRTSI, the amount of imputation MSE reduced by SAITS.
With the above results, it is obvious that SAITS outperforms NRTSI in all cases on both datasets. In particular, SAITS achieves 7% ∼similar-to\sim∼ 39% smaller MSE (above 20% in nine out of sixteen cases) than NRTSI. To make the comparison more straightforward, the bar graphs in Figure5 are plotted to visualize the results in Table6 and7.
Table 8: The models’ parameter number (in million) and training time of each epoch (in seconds) on datasets Air and Gas are listed below. NRTSI have no results of the training speed because the original paper does not include them.
Air Gas Model# of param s / epoch# of param s / epoch NRTSI 84.00M/84.00M/ SAITS 10.00M 2.6 2.78M 18.4
Besides model performance, the model parameter numbers and training speed are recorded in Table8. NRTSI’s number of parameters is from Appendix B in[34]. The results in Table8 tell us that SAITS needs much fewer parameters than NRTSI on both datasets (only 12% and 3% of NRTSI’s parameters on datasets Air and Gas respectively). Considering the hyper-parameters of NRTSI share across the datasets and do not get adjusted accordingly in the original paper[34], NRTSI may not need so many parameters to obtain such performance. Nevertheless, NRTSI directly takes a Transformer encoder as the backbone, and it has been proven in the experiments and discussion in Section4.4.1 that SAITS architecture is more efficient than Transformer on the imputation task. Concerning the training speed, we can not run NRTSI, so NRTSI’s speed does not have records. However, according to the algorithms of NRTSI listed in Appendix A in[34], both the training procedure and imputation procedure have two nested loops, which can make NRTSI much slower than SAITS because SAITS has no loop in neither training nor imputation stage.
4.5 Ablation Studies
In this section, three ablation experiments are leveraged to discuss the rationality of SAITS architecture design. The first one4.5.1 is to validate the improvement brought by the diagonally-masked self-attention (DMSA). The second one4.5.2 is to discuss the necessity of the weighted combination. The third one4.5.3 is to explain why we do not apply more than two DMSA blocks.
4.5.1 Ablation Study of the Diagonal Masks in Self-Attention
Table 9: Ablation experiment results of the diagonal masks in self-attention. SAITS-base-w/o is the exact same as SAITS-base, except it is without the diagonal masks in self-attention layers.
Model PhysioNet-2012 Air-Quality Electricity ETT SAITS-base-w/o 0.200 / 0.446 / 28.5%0.148 / 0.528 / 21.3%0.898 / 1.504 / 48.1%0.147 / 0.211 / 13.8% SAITS-base 0.192 / 0.439 / 27.3%0.146 / 0.521 / 20.6%0.822 / 1.221 / 44.0%0.121 / 0.197 / 11.6%
To prove that DMSA has better imputation performance than the conventional self-attention, a comparison is made between SAITS-base and SAITS-base-w/o in Table9. SAITS-base-w/o is without the diagonal masks. SAITS-base outperforms SAITS-base-w/o on all datasets, and this demonstrates DMSA does improve SAITS’ imputation ability.
4.5.2 Ablation Study of the Weighted Combination
Table 10: Ablation experiment results of the weighted combination. SAITS-base-1block does not have the second DMSA block nor the weighted-combination block, and its final representation is directly from the only DMSA block. SAITS-base-R2 directly takes Learned Representation 2 as the final representation. In other words, it has no combination of representations. SAITS-base-residual applies a residual connection to combine Learned Representation 1 and 2.
Model PhysioNet-2012 Air-Quality Electricity ETT SAITS-base-1block 0.204 / 0.496 / 29.2%0.178 / 0.544 / 25.1%0.876 / 1.381 / 46.9%0.149 / 0.221 / 14.1% SAITS-base-R2 0.199 / 0.451 / 28.4%0.149 / 0.522 / 21.0%0.906 / 1.456 / 48.5%0.141 / 0.203 / 13.5% SAITS-base-residual 0.200 / 0.477 / 28.5%0.160 / 0.527 / 22.6%0.819 / 1.223 / 43.7%0.143 / 0.207 / 13.4% SAITS-base 0.192 / 0.439 / 27.3%0.146 / 0.521 / 20.6%0.822 / 1.221 / 44.0%0.121 / 0.197 / 11.6%
During the model design process, after applying the diagonal masks to self-attention, we further think about how to enhance the imputation ability. As a result, the second DMSA block is added to increase our model’s depth and extend the learning process. Rather than simply raising the layer number of the first DMSA block that can also increase the network depth, the second DMSA block is employed as a learner to play a role of verification. Different from the first DMSA block that can only make imputation from scratch, the second DMSA block has its input containing the imputed data from the first DMSA block. Accordingly, its learning target is to verify these imputation values. However, there is no guarantee that the second DMSA block can perform better than the first one. In other words, the imputations from the second DMSA block are not necessarily better than those from the first block. For example, SAITS-base-R2 achieves better performance than SAITS-base-1block on datasets PhysioNet-2012, Air-Quality, and ETT, but performs worse on the Electricity dataset. Hence, taking imputation values from either block is not wise. Therefore, we let representations from both blocks form the final imputation together, namely in the way of the weighted combination discussed in Section3.2.5.
We compare the weighted combination with the other two designs to discuss its rationality. As shown in Table10, one is no combination, directly taking Learned Representation 2 as the final representation, referring to SAITS-base-R2. The other is the residual combination, which combines Learned Representation 1 and 2 by a residual connection, referring to SAITS-base-residual. Compared with the residual connection, the weighted combination design parameterizes the connection process and makes it actively assign weights for the learned representations rather than being a simple addition.
As shown in Table10, SAITS-base obtains the best results on datasets PhysioNet-2012, Air-Quality, and ETT. On these three datasets, SAITS-base-residual is even inferior to SAITS-base-R2. That is to say, the residual combination makes results worse. On dataset Electricity, SAITS-base and SAITS-base-residual achieve comparable results, and both are better than SAITS-base-R2. In summary, our weighted combination is the most practical design in all of the three.
4.5.3 Ablation Study of the Third DMSA Block
Similar to applying the second DMSA block to obtain better performance, theoretically, we can apply more than two DMSA blocks. However, the benefit is marginal. Taking three DMSA blocks as an example, the experiments are conducted and the results are listed in Table11 above.
Figure 6: Structure illustration of SAITS with three residual-connected DMSA blocks. The second DMSA block takes in data imputed by the first DMSA block, and the third DMSA block takes in data from the second one. The final output is from a residual connection of the representations produced by three DMSA blocks.
Figure 7: Structure illustration of SAITS with three cascade-weighted DMSA blocks. The representations from the first two DMSA blocks are merged by the first weighted combination block to produce Learned Representation 3, which is used to impute data for the input of the third DMSA block. The final output is a weighted combination of the representation from the third DMSA block and Learned Representation 3.
Table 11: Ablation experiment results of the third DMSA block. Results of SAITS here are from Table2 in our paper. Both SAITS-3residual and SAITS-3cascade apply the same hyper-parameters with SAITS.
Model PhysioNet-2012 Air-Quality Electricity ETT SAITS-3residual 0.189 / 0.620 / 27.0%0.158 / 0.509 / 22.2%0.740 / 1.020 / 39.6%0.103 / 0.145 / 9.6% SAITS-3cascade 0.185 / 0.418 / 26.4%0.146 / 0.512 / 20.5%0.800 / 1.147 / 42.8%0.096 / 0.141 / 8.8% SAITS 0.186 / 0.431 / 26.6%0.137 / 0.518 / 19.3%0.735 / 1.162 / 39.4%0.092 / 0.139 / 8.8%
Regarding how to combine representations from three DMSA blocks, there are still two options: residual connection and weighted combination. Residual connection is easy to implement, and SAITS-3residual takes this way. The weighted combination can only combine two blocks’ representation at a time, so the cascade-weighted combination is used here to implement SAITS-3cascade. The graphs in Figure6 and Figure7 are plotted to clearly illustrate both models’ structure.
With the results in Table11, we can see, in general, SAITS-3residual and SAITS-3cascade do not obtain better results than SAITS, which means that adding one more block brings nothing but more parameters and computation resource waste.
5 Conclusion
This paper proposes SAITS, a novel self-attention-based model to impute missing values in multivariate time series. Specifically, a joint-optimization training approach is designed for self-attention-based models to perform the imputation task. Compared to BRITS, a SOTA RNN-based imputation model, SAITS reduces mean absolute error (MAE) by 12% ∼similar-to\sim∼ 38% and achieves 2.0 ∼similar-to\sim∼ 2.6 times faster training speed. In the comparison with another SOTA model NRTSI, which takes a Transformer as the backbone, SAITS achieves 7% ∼similar-to\sim∼ 39% better imputation accuracy. Moreover, when Transformer is trained by our joint-optimization approach, SAITS still obtains MAE 2% ∼similar-to\sim∼ 19% smaller than it, with comparable training speed. Especially on dataset Electricity, the most complex dataset among all the four, the improvement is still obvious (11%), which means SAITS has an obvious advantage over Transformer when datasets become complex. Furthermore, the experiments also tell us that SATIS has a more efficient model structure than Transformer on the imputation task. To obtain comparable performance, SAITS needs only 15% ∼similar-to\sim∼ 30% parameters of Transformer. Additionally, to justify the design of SAITS architecture, a series of ablation experiments are performed to further discuss the reasons for our design and prove its effectiveness. All of the experimental results lead to the same conclusion that SAITS efficiently achieves the new state-of-the-art accuracy on the time-series imputation task. In addition to imputation accuracy that evaluates SAITS quantitatively, our empirical results in the downstream classification experiment qualitatively show that classification performance can directly get improved by letting SAITS impute the missing part, which reveals SAITS’ potential of becoming a bridge for pattern recognition models to learn with incomplete time-series data.
Our future work will investigate the imputation performance of SAITS on partially-observed time series with other missing patterns. Note that we add completely-random artificial missingness in MIT because the missing pattern is assumed to be MCAR in the settings of this work. If one already knows the missing pattern of the dataset to be imputed, one can apply the specific pattern to introduce artificially-missing values. This is intuitive and still keeps the functionality of MIT, though whether it can help improve imputation accuracy compared to applying MCAR missingness is open to discussion. Additionally, we will investigate the performance of SAITS on other real-world large datasets to further validate the model’s generality in other domains.
Acknowledgments and Disclosure of Funding
We sincerely appreciate Dr. Wei Cao at Microsoft Research Asia, Prof. Christian Desrosiers at ÉTS Montréal, Dr. Ziyu Jia at Beijing Jiaotong University, and our anonymous reviewers for their constructive comments. We thank Ciena for providing computing resources.
Wenjie Du is supported by a Mitacs accelerate program (# FR61813) cooperating with Ciena. This research is partially supported by the Natural Sciences and Engineering Research Council of Canada (# RGPIN-2020-06797).
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Appendix A Details of Hyper-parameter Searching
We expose the details about hyper-parameter searches in this section.
General For all models, the learning rate is log-uniformly sampled between 1×10−4 1 superscript 10 4 1\times 10^{-4}1 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT and 1×10−2 1 superscript 10 2 1\times 10^{-2}1 × 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT. If applicable, the dropout rate is sampled from the values (0.0, 0.1, 0.2, 0.3, 0.4, 0.5).
RNN-based models For all RNN-based models (GRUI-GAN, E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN, M-RNN, and BRITS), the RNN hidden size is sampled from the values (32, 64, 128, 256, 512, 1024). For GRUI-GAN and E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN, the dimension of z 𝑧 z italic_z is sampled from (32, 64, 128, 256, 512, 1024), the number of pretrain epochs is sampled from (5, 10, 15, 20). For GRUI-GAN, hyper-parameter λ 𝜆\lambda italic_λ, which controls the proportion between the masked reconstruction loss and the discriminative loss, is sampled from (0, 0.15, 0.3, 0.45). For E 2 2{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT GAN, hyper-parameter λ 𝜆\lambda italic_λ, which controls the weight of the discriminative loss and the squared error loss, is sampled from (2, 4, 8, 16, 32, 64).
GP-VAE The encoder size and decoder size are sampled from values (64, 128, 256, 512, 1024). The length scale is sampled from (4, 8, 12, 16) and the window size is sampled from (4, 8, 16, 32, 64). β 𝛽\beta italic_β is sampled from (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8). σ 𝜎\sigma italic_σ is set as 1.005.
Self-attention-based models For self-attention models (Transformer and SAITS), we sample the number of layers N 𝑁 N italic_N from (1, 2, 3, 4, 5, 6, 7, 8), d model subscript 𝑑 model d_{\text{model}}italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT from (64, 128, 256, 512, 1024), d ffn subscript 𝑑 ffn d_{\text{ffn}}italic_d start_POSTSUBSCRIPT ffn end_POSTSUBSCRIPT from (128, 256, 512, 1024, 2048, 4096), d v subscript 𝑑 𝑣 d_{v}italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT from (32, 64, 128, 256, 512), the number of heads h ℎ h italic_h from (2, 4, 8). d k subscript 𝑑 𝑘 d_{k}italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is set as the value of d model subscript 𝑑 model d_{\text{model}}italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT divided by h ℎ h italic_h.
For model SAITS-base, we fix the learning rate = 0.001, the dropout rate = 0.1, N=2,d model=256,d ffn=128,h=4,d v=d k=64 formulae-sequence 𝑁 2 formulae-sequence subscript 𝑑 model 256 formulae-sequence subscript 𝑑 ffn 128 formulae-sequence ℎ 4 subscript 𝑑 𝑣 subscript 𝑑 𝑘 64 N=2,d_{\text{model}}=256,d_{\text{ffn}}=128,h=4,d_{v}=d_{k}=64 italic_N = 2 , italic_d start_POSTSUBSCRIPT model end_POSTSUBSCRIPT = 256 , italic_d start_POSTSUBSCRIPT ffn end_POSTSUBSCRIPT = 128 , italic_h = 4 , italic_d start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 64.
Appendix B BRITS Trained by the joint-optimization approach
Table 12: Performance comparison between BRITS trained without MIT and with MIT.
Model PhysioNet-2012 Air-Quality Electricity ETT BRITS-w/oMIT 0.256 / 0.767 / 36.5%0.153 / 0.525 / 21.6%0.847 / 1.322 / 45.3%0.130 / 0.259 / 12.5% BRITS-wMIT 0.251 / 0.691 / 35.8%0.144 / 0.521 / 20.3%0.910 / 1.363 / 48.7%0.123 / 0.237 / 12.6%
To discuss how our joint-optimization approach can influence the performance of RNN-based models, we apply it in the training of model BRITS and show experimental results in this section.
BRITS models from Table2 are used here. That is to say, hyper-parameters are kept exactly the same. The difference between the training way in the original paper[16] and our joint-optimization approach is whether to apply MIT. Consequently, we use suffix "w/oMIT" to represent the original training way and suffix "wMIT" to represent our joint-optimization training approach.
As displayed in Table12, BRITS-wMIT outperforms BRITS-w/oMIT on datasets PhysioNet-2012, Air-Quality, and ETT, but achieves worse performance on the Electricity dataset. Therefore, applying MIT in the training of BRITS can bring further improvement on some datasets, but this is not necessary, and it depends on the dataset. Note that despite BRITS-wMIT obtains better results on datasets PhysioNet-2012, Air-Quality, and ETT, its performance is still inferior to Transformer and SAITS.