# Diffusion-based Time Series Data Imputation for Microsoft 365

Fangkai Yang∗, Wenjie Yin†, Lu Wang∗, Tianci Li∗, Pu Zhao∗, Bo Liu§, Paul
Wang§, Bo Qiao∗ and  Yudong Liu∗, Mårten Björkman†, Saravan Rajmohan§, Qingwei
Lin∗, Dongmei Zhang∗ Microsoft Research∗ Microsoft 365§ KTH Royal Institute of
Technology†

###### Abstract.

Reliability is extremely important for large-scale cloud systems like
Microsoft 365. Cloud failures such as disk failure, node failure, etc.
threaten service reliability, resulting in online service interruptions and
economic loss. Existing works focus on predicting cloud failures and
proactively taking action before failures happen. However, they suffer from
poor data quality like data missing in model training and prediction, which
limits the performance. In this paper, we focus on enhancing data quality
through data imputation by the proposed Diffusion $+$ , a sample-efficient
diffusion model, to impute the missing data efficiently based on the observed
data. Our experiments and application practice show that our model contributes
to improving the performance of the downstream failure prediction task.

Diffusion model, missing data imputation, cloud failure prediction

††ccs: Computer systems organization Cloud computing††ccs: Hardware Failure
prediction

##  1\. Introduction

Microsoft 365 cloud platform is a large-scale online service system and serves
millions of customer workloads on a 24/7 basis. It is extremely critical to
ensure high reliability as any cloud failure will result in financial loss and
degradation of user experience (Jayathilaka et al., 2017; Chen et al., 2019;
Levy et al., 2020). However, cloud failure, including hardware failure and
software failure, is inevitable in large-scale systems (Botezatu et al., 2016;
Meza et al., 2015; Gao et al., 2020). Recent research and works (Chen et al.,
2019; Luo et al., 2021; Liu et al., 2022; Ma et al., 2022) have proposed
approaches to predict cloud failures before they actually happen and take
actions proactively to mitigate potential failures, thus minimizing the
negative impact of cloud failure. Although significant progress has achieved
good results in practice, these failure prediction methods still suffer from
the issues of data missing (Deb et al., 2017; Chai et al., 2020; Luo et al.,
2021). Data missing is a practical and ubiquitous problem in large cloud
systems caused by data delay (Lu et al., 2020), monitoring error (Ward and
Barker, 2014), etc. In this paper, rather than designing better failure
prediction models, we focus on a new perspective of enhancing the data quality
by imputing missing data to improve the performance of downstream cloud
failure prediction.

There exist a large number of studies of data imputation concerning images and
time series data (Pratama et al., 2016; Cao et al., 2018; Fortuin et al.,
2020; Tashiro et al., 2021). However, very few focus on time series data
imputation in the domain of cloud systems, and rules-based and statistical
approaches are commonly used in industry (Ma et al., 2022). Most importantly,
there lacks an end-to-end evaluation of data imputation with the downstream
tasks, and the effect of different data imputation methods is still unexplored
connecting to the downstream tasks that utilize the data. In this paper, we
leverage the success of the diffusion models (Sohl-Dickstein et al., 2015; Ho
et al., 2020), which have outperformed state-of-the-art generative models with
higher sample quality, and we propose a new diffusion model, _i.e._ ,
Diffusion $+$ , to impute missing data with high efficiency. Figure 1 shows
the overview of the process. We use the diffusion model to do data imputation,
and the imputed data is fed into the downstream failure prediction task for
model training and prediction. We select disk failure prediction as the
downstream task since disk failure is one of the most frequent failures in
cloud systems (Sankar et al., 2013; Botezatu et al., 2016), and our model can
be easily adapted to other downstream tasks in cloud scenarios. Moreover, the
slow sampling issue of the diffusion model restricts its application in
industry. Inspired by the most recent work (Lu et al., 2022), we improve the
diffusion sampling efficiency with at least 4$\times$ speed up without
degrading the downstream prediction task.

Our main contributions are summarized as follows:

  * •

We propose a new perspective of improving cloud failure prediction by imputing
missing data.

  * •

Inspired by the diffusion model, we propose a new diffusion model with better
imputation performance and higher sampling efficiency in the cloud scenario.

  * •

We conduct extensive experiments on industrial data and demonstrate that our
model improves the performance on the downstream task.

Figure 1. The overview of data imputation with downstream failure prediction
tasks.

##  2\. Methodology

###  2.1. Problem Formulation

In practice, a disk’s status vector is recorded at each timestamp (_e.g._ ,
hourly), which is a multivariate time series data $x\in\mathbb{R}^{K\times L}$
where $K$ is the number of features and $L$ is the length of time series. At
each timestamp $l$, some data features are missing in $x_{l}$, then we can
partition $x_{l}$ into the missing part
$x^{ms}_{l}=\\{x_{l}^{k}|x_{l}^{k}\;\text{is missing}\\}^{1:K}$ and the
observed part $x^{ob}_{l}=\\{x_{l}^{k}|x_{l}^{k}\;\text{is
observed}\\}^{1:K}$, _i.e._ , $x_{l}=x^{ms}_{l}\cup x^{ob}_{l}$. Our goal is
to do data imputation for $x^{ms}=\\{x^{ms}_{l}\\}_{1:L}$ given
$x^{ob}=\\{x^{ob}_{l}\\}_{1:L}$ for all status feature vectors $x$, and the
imputed status feature vectors are then fed toward downstream prediction
tasks, which is trained to predict whether a disk will fail or not.

###  2.2. Overview of Diffusion $+$ Model

Denoising diffusion probabilistic models (DDPM) (Sohl-Dickstein et al., 2015;
Ho et al., 2020), known as diffusion models (DM) for brevity, are a class of
generative models inspired by non-equilibrium thermodynamics. DMs consist of a
forward process and a reverse process. In the forward process, DMs define a
fixed Markov chain of $T$ diffusion steps to slowly add noise to the data
$x_{0}\in\mathbb{R}^{K\times L}$ until the data distribution is close to a
standard Gaussian distribution $x_{T}\in\mathbb{R}^{K\times L}$. Note that the
subscripts in $x_{0}$ and $x_{T}$ represent the diffusion step, _i.e._ ,
$x_{0}=\\{x_{0,l}\\}_{1:L}$, and we omit $l$ for simplicity. The forward
process is defined as:

(1) $\displaystyle q(x_{1:T}|x_{0})$
$\displaystyle=\prod_{t=1}^{T}q(x_{t}|x_{t-1}),$ $\displaystyle
q(x_{t}|x_{t-1})$
$\displaystyle=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}\mathbf{I}),$

where $\beta_{1},\beta_{2},\cdots,\beta_{T}$ are the fixed noise schedulers
for controlling the noise scale (Ho et al., 2020).

On the other hand, in the reverse process, DMs learn to reverse the forward
process by denoising to get the desired data distribution from the noise
distribution, _i.e._ , sampling from $q(x_{t-1}|x_{t})$ will be able to create
the true sample $x_{0}$ from a Gaussian noise $x_{T}$. However, it is non-
trivial to estimate $q(x_{t-1}|x_{t})$, and we learn to model
$p_{\theta}(\cdot)$ as the approximate estimation. We adopt the conditional
diffusion model (Tashiro et al., 2021) which uses the observation $x_{0}^{ob}$
as the condition to generate imputation targets $x_{0}^{ms}$. More
specifically, the goal of data imputation is to estimate the true conditional
data distribution $q(x_{0}^{ms}|x_{0}^{ob})$ with a model distribution
$p_{\theta}(x_{0}^{ms}|x_{0}^{ob})$, and the missing data $x_{0}^{ms}$ can be
sampled from $p_{\theta}(\cdot)$ as shown in Figure 2. We model
$p_{\theta}(x_{0}^{ms}|x_{0}^{ob})$ with the diffusion model in the reverse
process:

(2) $\displaystyle
p_{\theta}(x_{0:T}^{ms}|x_{0}^{ob})=p(x_{T}^{ms})\prod_{t=1}^{T}p_{\theta}(x_{t-1}^{ms}|x_{t}^{ms},x_{0}^{ob}),\
x_{T}^{ms}\sim\mathcal{N}({\bm{0,I}}),$ $\displaystyle
p_{\theta}(x_{t-1}^{ms}|x_{t}^{ms},x_{0}^{ob})=\mathcal{N}(x_{t-1}^{ms};\bm{\mu}_{\theta}(x^{ms}_{t},t|x_{0}^{ob}),\sigma_{\theta}(x^{ms}_{t},t|x_{0}^{ob})\bm{I})$

Figure 2. Data imputation with the reverse process of the diffusion model.

We define a conditional denoising function $\epsilon_{\theta}$ in the reverse
process to estimate $\bm{\mu}_{\theta}(\cdot)$ and $\sigma_{\theta}(\cdot)$ of
the distribution $p_{\theta}(x_{t-1}^{ms}|x_{t}^{ms},x_{0}^{ob})$. In
particular,
$\bm{\mu}_{\theta}(x^{ms}_{t},t|x_{0}^{ob})=\bm{\mu}^{DDPM}(x^{ms}_{t},t,\bm{\epsilon}_{\theta}(x^{ms}_{t},t|x_{0}^{ob}))$
and $\sigma_{\theta}(x^{ms}_{t},t|x_{0}^{ob})=\sigma^{DDPM}(x^{ms}_{t},t)$,
where $\bm{\mu}^{DDPM}(\cdot)$ and $\sigma^{DDPM}(\cdot)$ are the
parameterization functions in denoising diffusion probabilistic models (DDPM)
(Ho et al., 2020). Then, given $\bm{\epsilon}_{\theta}$ and $x_{0}^{ob}$, we
can sample $x_{0}^{ms}$ in the reverse process in Equation 2, where
$\bm{\epsilon}_{\theta}$ is trainable.

Training. Since we do not have the ground-truth missing values, we first do
zero imputation for the missing data, and then we randomly partition the
observation $x_{0}^{ob}$ into two parts: the conditional observation
$\hat{x}_{0}^{ob}$, and the masked target that needs imputation
$\hat{x}_{0}^{ms}$. Our model is then trained in a self-supervised learning
manner (Devlin et al., 2018) to do data imputation for $\hat{x}_{0}^{ms}$
given $\hat{x}_{0}^{ob}$, and the imputation performance is evaluated on
$\hat{x}_{0}^{ms}$. With the formulated forward process and the reverse
process, the training process optimizes the log-likelihood in the reverse
process by maximizing the variational lower bound. The training is performed
for all diffusion steps. and it is trained by minimizing the simplified
objective function:

(3)
$\min_{\theta}\mathcal{L}=\mathbb{E}_{\bm{\epsilon}\sim\mathcal{N}(\bm{0},\bm{I})}||\bm{\epsilon}-\bm{\epsilon}_{\theta}(\hat{x}^{ms}_{t},t|\hat{x}_{0}^{ob})||_{2}^{2}$

.

Inference. When the training is done, we have good modeling of
$p_{\theta}(\cdot)$. Given the real observation $x_{0}^{ob}$ as the
conditional observation, we could impute the missing data $x_{0}^{ms}$ with
the reverse generation process $x_{t-1}^{ms}\sim
p_{\theta}(x_{t}^{ms}|x_{0}^{ob})$ according to Equation 2.

For each sample with missing data, we generate 100 data imputations and take
their median as the final imputed results. The data imputation is conducted
over the whole dataset before training the downstream failure prediction
models.

Speed up. As shown in Figure 2, DMs suffer from slow sampling as they require
a large number of diffusion steps $T$ of running large neural networks to draw
one sample (Lu et al., 2022), which makes it inefficient and impractical for
data imputation in industry and becomes a bottleneck for the downstream tasks.
Inspired by recent work (Lu et al., 2022), we speed up the data imputation by
reducing the diffusion steps in the reverse process without any further
training. The sampling of DMs in the reverse process can be viewed
alternatively as solving corresponding ordinary differential equations (ODEs)
(Jolicoeur-Martineau et al., 2021; Song et al., 2020), and the sampling
process is done by ODE solvers (Atkinson et al., 2011; Lu et al., 2022) which
results in high-quality and few-step sampling. Specifically, the noise
scheduler in $\bm{\mu}_{\theta}(\cdot)$ and $\sigma_{\theta}(\cdot)$ is
updated by the ODE solver, and we adopt a uniform step size schedule to
determine $M$ ($\ll T$) steps. Then the diffusion steps $T$ in the reverse
process is reduced to $M$ steps.

##  3\. Experiments

###  3.1. Experimental Settings

The data we used for experiments were collected from the Microsoft 365 online
service system in recent 6 months. The data is in the SMART format (Self-
Monitoring, Analysis and Reporting Technology) (Allen, 2004), which records
the disk status and provides important indicators during the lifetime of
disks. We predict the disk failure based on 72-hour data. All experiments are
performed on a workstation equipped with AMD EPYC 7V12 64-Core CPUs, NVIDIA
Tesla T4 GPU with CUDA 10.1, and running Linux (16.04.5) OS.

###  3.2. Baselines

Following the previous work (Fortuin et al., 2020; Tashiro et al., 2021; Ma et
al., 2022), we use imputation baselines as follows:

Zero imputation (Z): Zero imputation replaces the missing data with zero,
which is the most intuitive way.

Forward imputation (F): Forward imputation (Little and Rubin, 2002) is a
single imputation method that replaces the missing data with the previously
observed value.

Linear interpolation (L): Linear interpolation (Read, 1999) interpolates the
missing data by linear curve fitting.

BRITS: BRITS (Cao et al., 2018) is an RNN-based approach that utilizes a bi-
directional recurrent neural network that handles the missing data considering
the forward and backward temporal dependency.

Variational Autoencoders (VAE): VAE (Kingma and Welling, 2013; Fortuin et al.,
2020) is a generative model that learns a probability distribution
representing the data, and the missing data is sampled from the estimated
distribution.

Following the previous work on disk failure prediction (Luo et al., 2021; Liu
et al., 2022), we use the downstream prediction baselines: long short-term
memory (LSTM) (Zhang et al., 2018), Transformer (Trans) (Luo et al., 2021),
and temporal convolutional neural network (TCNN) (Sun et al., 2019).

###  3.3. Experimental Results

In this section, we aim to address three research questions:

  * •

RQ1: Do the diffusion and Diffusion $+$ models impute missing data
effectively?

As mentioned in Section 2.2, we randomly mask parts of the observation
$x_{0}^{ob}$ as the imputation target $\hat{x}_{0}^{ms}$, and we train and
evaluate data imputation models with 10%, 50%, 90% masked missing ratio
following previous work (Tashiro et al., 2021). Note that Z, F, and L are
rule-based methods without training, and we list them for reference.

We first present the quantitative results. We adopt two metrics following
previous work (Tashiro et al., 2021) to evaluate the performance of data
imputation, _i.e._ , MAE (mean absolute error) and CRPS (continuous ranked
probability score), where CRPS (Matheson and Winkler, 1976) is usually used to
measure the compatibility of an estimated probability distribution with an
observation. For the deterministic imputation methods, _i.e._ , Z, F, L, and
BRITS, we only use MAE since they are not probabilistic imputation methods. As
for probabilistic imputation methods (VAE, Diffusion, and Diffusion $+$ ), we
generate 100 samples for each missing data sample to estimate the probability
distribution of the missing data with the metric CRPS. The MAE of the
probabilistic imputation methods is computed using the median of 100 generated
samples. Note that the data are normalized within each feature dimension in
the evaluation. As shown in Table 1, the diffusion model has the lowest MAE,
49%-95% less compared with baselines. It suggests that the diffusion model is
more effective in capturing the feature and temporal dependency. The diffusion
model also shows the lowest CRPS metric compared with VAE, which indicates its
capability of generating more realistic distributions. Diffusion $+$ model has
a very close performance as the diffusion model, _i.e._ , second best, in
general (excluding models trained under 90% missing ratio). Most imputation
approaches have better performance with smaller missing ratios since more
observations are available. Thus, we use the models trained under the 10%
missing ratio for imputation, _i.e._ , models with the best performance
trained with three missing ratios.

Table 1. Data imputation performance evaluated with MAE and CRPS (lower is
better). CRPS is only available for probabilistic imputation methods.

Approach | Mssing Ratio (%)
---|---
10 | 50 | 90
MAE | CRPS | MAE | CRPS | MAE | CRPS
Z | 0.429 | — | 0.428 | — | 0.429 | —
F | 0.047 | — | 0.057 | — | 0.111 | —
L | 0.063 | — | 0.064 | — | 0.068 | —
BRITS | 0.052 | — | 0.054 | — | 0.081 | —
VAE | 0.039 | 0.613 | 0.045 | 0.616 | 0.075 | 0.648
Diffusion | 0.020 | 0.049 | 0.021 | 0.040 | 0.053 | 0.131
Diffusion $+$ | 0.025 | 0.046 | 0.034 | 0.068 | 0.099 | 0.253

Figure 3. Two data imputation examples of VAE, Diffusion, and Diffusion $+$ .
Each example is a time series sample of one feature. The red crosses represent
observed values and the blue dots represent the masked observation data for
imputation targets. The shaded areas are 5% and 95% quantiles and the line is
the median value of imputation.

We also provide imputation examples (shown in Figure 3). The diffusion and
Diffusion $+$ models generate imputations with high confidence and the
imputation distributions tightly cover masked missing targets (blue dots). VAE
imputations have larger variations and cannot cover the missing targets.

  * •

RQ2: Does data imputation contribute to improving the downstream disk failure
prediction task?

We impute the ground-truth missing data in the entire dataset and feed the
imputed data to downstream failure prediction tasks. As shown in Table 2, with
all prediction methods, the diffusion imputation model achieves the best
performance in precision and F1-score, and also in recall for most cases. In
the domain of cloud failure prediction, F1-score is the most important metric
(Ma et al., 2022; Liu et al., 2022). Diffusion $+$ shows a very close
performance as the diffusion model in F1-score, and it demonstrates the second
best of all the other approaches. Compared with different prediction methods,
Trans achieves the best performance in F1-score. Note that we use advanced
failure prediction models in practice (Liu et al., 2022; Luo et al., 2021),
which have better prediction performance than the prediction baselines.

Table 2. Failure prediction performance with different data imputation methods on three metrics, _i.e._ , precision, recall, and F1-score. Approach | Precision | Recall | F1-score
---|---|---|---
Z+LSTM | 60.00 | 50.45 | 54.81
F+LSTM | 64.69 | 46.41 | 54.05
L+LSTM | 59.13 | 44.74 | 50.94
BRITS+LSTM | 61.20 | 50.22 | 55.17
VAE+LSTM | 62.07 | 52.84 | 57.08
Diffusion+LSTM | 66.75 | 55.49 | 60.60
Diffusion $+$ +LSTM | 65.96 | 54.23 | 59.52
Z+Trans | 62.84 | 51.57 | 56.65
F+Trans | 68.15 | 47.98 | 56.32
L+Trans | 62.87 | 48.21 | 54.57
BRITS+Trans | 64.81 | 52.92 | 58.26
VAE+Trans | 66.85 | 52.45 | 58.78
Diffusion+Trans | 74.05 | 53.59 | 62.18
Diffusion $+$ +Trans | 72.01 | 52.34 | 60.62
Z+TCNN | 60.60 | 50.00 | 54.79
F+TCNN | 61.05 | 50.66 | 55.37
L+TCNN | 59.44 | 47.55 | 52.83
BRITS+TCNN | 60.61 | 50.32 | 54.99
VAE+TCNN | 60.50 | 54.64 | 57.42
Diffusion+TCNN | 79.24 | 48.31 | 60.03
Diffusion $+$ +TCNN | 72.93 | 49.78 | 59.17

  * •

RQ3: Does our Diffusion $+$ model speed up the sampling process in diffusion
models?

Diffusion models suffer from slow sampling issues since generating one sample
requires a large number of diffusion steps. Diffusion $+$ model aims to speed
up the sampling process with only a few sampling steps without degrading the
performance too much. As discussed in RQ1 and RQ2, Diffusion $+$ model
achieves similar performance as the diffusion model. Then we conduct the
analysis on time cost for imputing each sample. Figure 4 shows the averaged
time cost of data imputation for each data sample. As the diffusion step $T$
grows, the time cost for the diffusion model increases accordingly, while
Diffusion $+$ has a stable time cost that needs far fewer diffusion steps and
it has at least 4$\times$ speed up, and the speed up is more obvious with the
increase of diffusion steps.

Figure 4. The time cost (ms) for each diffusion imputation.

##  4\. Application in practice

We have run our Diffusion $+$ model for one month on Microsoft 365, which
contains millions of disks. In particular, our model takes effect in the data
process phase of the current disk failure prediction pipeline (Liu et al.,
2022). The SMART data is first collected by a data collection service,
transferred by a distributed streaming tool, and stored in Azure. Then, our
model imputes missing data before sending it to the feature engineering of the
downstream prediction tasks. We conduct A/B testing to measure the
effectiveness of our model and its contribution to service reliability. We
monitor the reduction of virtual machine (VM) interruptions by taking
proactive failure mitigation based on the prediction. Compared with the
original data process phase without Diffusion $+$ , the interruptions have
reduced the VM interruptions and enhanced the service reliability to avoid
potential financial loss.

##  5\. Related work

Cloud failure prediction. There exist many studies on cloud failure prediction
(Botezatu et al., 2016; Meza et al., 2015; Gao et al., 2020), and they are
commonly treated as binary classification problems (Liu et al., 2022). They
use collected monitoring metrics from services in a time window to predict
whether there will be a failure in the near future. They can capture temporal
dependency to make a good prediction. However, missing data is a critical
issue for these approaches since it requires the prediction models to infer
missing information, and it usually results in poor prediction performance
(Fletcher Mercaldo and Blume, 2020; Alcaraz and Strodthoff, 2022). Our paper
is orthogonal to these failure prediction methods, and it offers a new
perspective to improve cloud failure prediction by enhancing the data quality.

Time series data imputation. Time series data imputation is a rich topic
(Little et al., 2014). In particular, deep learning models including RNN-based
approaches (Cao et al., 2018; Che et al., 2018; Liu et al., 2019) and
generative models (Fortuin et al., 2020; Tashiro et al., 2021; Luo et al.,
2018) can capture the temporal dependency of the time series and generate
better data imputation than rule-based and statistical methods. Different from
these methods, our paper not only evaluates the imputation quality but also
focuses on the end-to-end performance of data imputation with practical
industrial problems, _i.e._ , disk failure prediction, and we speed up the
diffusion-model-based data imputation to make it applicable in industry.

##  6\. Conclusion

In this paper, we focus on enhancing the data quality for disk failure
prediction by imputing missing data. We propose our Diffusion $+$ model based
on diffusion models which imputes missing data effectively and efficiently.
Our experiments on industrial datasets collected in Microsoft 365 and A/B
testing show that our model outperforms baselines with fast sampling speed and
contributes to enhancing the failure prediction tasks, and then improving the
reliability of the Microsoft 365 cloud platform.

## References

  * (1)
  * Alcaraz and Strodthoff (2022) Juan Miguel Lopez Alcaraz and Nils Strodthoff. 2022.  Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models.  _arXiv preprint arXiv:2208.09399_ (2022).
  * Allen (2004) Bruce Allen. 2004\.  Monitoring hard disks with SMART.  _Linux Journal_ 2004, 117 (2004), 9.
  * Atkinson et al. (2011) Kendall Atkinson, Weimin Han, and David E Stewart. 2011\.  _Numerical solution of ordinary differential equations_.  John Wiley & Sons.
  * Botezatu et al. (2016) Mirela Madalina Botezatu, Ioana Giurgiu, Jasmina Bogojeska, and Dorothea Wiesmann. 2016.  Predicting disk replacement towards reliable data centers. In _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_. 39–48.
  * Cao et al. (2018) Wei Cao, Dong Wang, Jian Li, Hao Zhou, Lei Li, and Yitan Li. 2018.  Brits: Bidirectional recurrent imputation for time series.  _Advances in neural information processing systems_ 31 (2018).
  * Chai et al. (2020) Chengliang Chai, Lei Cao, Guoliang Li, Jian Li, Yuyu Luo, and Samuel Madden. 2020\.  Human-in-the-loop outlier detection. In _Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data_. 19–33.
  * Che et al. (2018) Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. 2018.  Recurrent neural networks for multivariate time series with missing values.  _Scientific reports_ 8, 1 (2018), 1–12.
  * Chen et al. (2019) Yujun Chen, Xian Yang, Qingwei Lin, Hongyu Zhang, Feng Gao, Zhangwei Xu, Yingnong Dang, Dongmei Zhang, Hang Dong, Yong Xu, et al. 2019\.  Outage prediction and diagnosis for cloud service systems. In _The World Wide Web Conference_. 2659–2665.
  * Deb et al. (2017) Supratim Deb, Zihui Ge, Sastry Isukapalli, Sarat Puthenpura, Shobha Venkataraman, He Yan, and Jennifer Yates. 2017.  Aesop: Automatic policy learning for predicting and mitigating network service impairments. In _Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_. 1783–1792.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_ (2018).
  * Fletcher Mercaldo and Blume (2020) Sarah Fletcher Mercaldo and Jeffrey D Blume. 2020.  Missing data and prediction: the pattern submodel.  _Biostatistics_ 21, 2 (2020), 236–252.
  * Fortuin et al. (2020) Vincent Fortuin, Dmitry Baranchuk, Gunnar Rätsch, and Stephan Mandt. 2020.  Gp-vae: Deep probabilistic time series imputation. In _International conference on artificial intelligence and statistics_. PMLR, 1651–1661.
  * Gao et al. (2020) Jiechao Gao, Haoyu Wang, and Haiying Shen. 2020.  Task failure prediction in cloud data centers using deep learning.  _IEEE transactions on services computing_ (2020).
  * Ho et al. (2020) Jonathan Ho, Ajay Jain, and Pieter Abbeel. 2020.  Denoising diffusion probabilistic models.  _Advances in Neural Information Processing Systems_ 33 (2020), 6840–6851.
  * Jayathilaka et al. (2017) Hiranya Jayathilaka, Chandra Krintz, and Rich Wolski. 2017\.  Performance monitoring and root cause analysis for cloud-hosted web applications. In _Proceedings of the 26th International Conference on World Wide Web_. 469–478.
  * Jolicoeur-Martineau et al. (2021) Alexia Jolicoeur-Martineau, Ke Li, Rémi Piché-Taillefer, Tal Kachman, and Ioannis Mitliagkas. 2021\.  Gotta go fast when generating data with score-based models.  _arXiv preprint arXiv:2105.14080_ (2021).
  * Kingma and Welling (2013) Diederik P Kingma and Max Welling. 2013.  Auto-encoding variational bayes.  _arXiv preprint arXiv:1312.6114_ (2013).
  * Levy et al. (2020) Sebastien Levy, Randolph Yao, Youjiang Wu, Yingnong Dang, Peng Huang, Zheng Mu, Pu Zhao, Tarun Ramani, Naga Govindaraju, Xukun Li, et al. 2020\.  Predictive and Adaptive Failure Mitigation to Avert Production Cloud $\\{$VM$\\}$ Interruptions. In _14th USENIX Symposium on Operating Systems Design and Implementation (OSDI 20)_. 1155–1170.
  * Little and Rubin (2002) Roderick JA Little and Donald B Rubin. 2002.  Single imputation methods.  _Statistical analysis with missing data_ (2002), 59–74.
  * Little et al. (2014) Todd D Little, Terrence D Jorgensen, Kyle M Lang, and E Whitney G Moore. 2014.  On the joys of missing data.  _Journal of pediatric psychology_ 39, 2 (2014), 151–162.
  * Liu et al. (2022) Yudong Liu, Hailan Yang, Pu Zhao, Minghua Ma, Chengwu Wen, Hongyu Zhang, Chuan Luo, Qingwei Lin, Chang Yi, Jiaojian Wang, et al. 2022\.  Multi-task Hierarchical Classification for Disk Failure Prediction in Online Service Systems. In _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. 3438–3446.
  * Liu et al. (2019) Yukai Liu, Rose Yu, Stephan Zheng, Eric Zhan, and Yisong Yue. 2019.  Naomi: Non-autoregressive multiresolution sequence imputation.  _Advances in neural information processing systems_ 32 (2019).
  * Lu et al. (2022) Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. 2022\.  DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps.  _arXiv preprint arXiv:2206.00927_ (2022).
  * Lu et al. (2020) Sidi Lu, Bing Luo, Tirthak Patel, Yongtao Yao, Devesh Tiwari, and Weisong Shi. 2020\.  Making Disk Failure Predictions $\\{$SMARTer$\\}$!. In _18th USENIX Conference on File and Storage Technologies (FAST 20)_. 151–167.
  * Luo et al. (2021) Chuan Luo, Pu Zhao, Bo Qiao, Youjiang Wu, Hongyu Zhang, Wei Wu, Weihai Lu, Yingnong Dang, Saravanakumar Rajmohan, Qingwei Lin, et al. 2021\.  NTAM: neighborhood-temporal attention model for disk failure prediction in cloud platforms. In _Proceedings of the Web Conference 2021_. 1181–1191.
  * Luo et al. (2018) Yonghong Luo, Xiangrui Cai, Ying Zhang, Jun Xu, et al. 2018\.  Multivariate time series imputation with generative adversarial networks.  _Advances in neural information processing systems_ 31 (2018).
  * Ma et al. (2022) Minghua Ma, Yudong Liu, Yuang Tong, Haozhe Li, Pu Zhao, Yong Xu, Hongyu Zhang, Shilin He, Lu Wang, Yingnong Dang, Saravanakumar Rajmohan, and Qingwei Lin. 2022.  An Empirical Investigation of Missing Data Handling in Cloud Node Failure Prediction. In _Proceedings of the European Software Engineering Conference and Symposium on the Foundations of Software Engineering (ESEC/FSE)_. 1453 – 1464\.
  * Matheson and Winkler (1976) James E Matheson and Robert L Winkler. 1976.  Scoring rules for continuous probability distributions.  _Management science_ 22, 10 (1976), 1087–1096.
  * Meza et al. (2015) Justin Meza, Qiang Wu, Sanjev Kumar, and Onur Mutlu. 2015\.  A large-scale study of flash memory failures in the field.  _ACM SIGMETRICS Performance Evaluation Review_ 43, 1 (2015), 177–190.
  * Pratama et al. (2016) Irfan Pratama, Adhistya Erna Permanasari, Igi Ardiyanto, and Rini Indrayani. 2016.  A review of missing values handling methods on time-series data. In _2016 international conference on information technology systems and innovation (ICITSI)_. IEEE, 1–6.
  * Read (1999) ALEXANDER L Read. 1999\.  Linear interpolation of histograms.  _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 425, 1-2 (1999), 357–360.
  * Sankar et al. (2013) Sriram Sankar, Mark Shaw, Kushagra Vaid, and Sudhanva Gurumurthi. 2013.  Datacenter scale evaluation of the impact of temperature on hard disk drive failures.  _ACM Transactions on Storage (TOS)_ 9, 2 (2013), 1–24.
  * Sohl-Dickstein et al. (2015) Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. 2015.  Deep unsupervised learning using nonequilibrium thermodynamics. In _International Conference on Machine Learning_. PMLR, 2256–2265.
  * Song et al. (2020) Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. 2020.  Score-based generative modeling through stochastic differential equations.  _arXiv preprint arXiv:2011.13456_ (2020).
  * Sun et al. (2019) Xiaoyi Sun, Krishnendu Chakrabarty, Ruirui Huang, Yiquan Chen, Bing Zhao, Hai Cao, Yinhe Han, Xiaoyao Liang, and Li Jiang. 2019.  System-level hardware failure prediction using deep learning. In _2019 56th ACM/IEEE design automation conference (DAC)_. IEEE, 1–6.
  * Tashiro et al. (2021) Yusuke Tashiro, Jiaming Song, Yang Song, and Stefano Ermon. 2021\.  CSDI: Conditional score-based diffusion models for probabilistic time series imputation.  _Advances in Neural Information Processing Systems_ 34 (2021), 24804–24816.
  * Ward and Barker (2014) Jonathan Stuart Ward and Adam Barker. 2014.  Observing the clouds: a survey and taxonomy of cloud monitoring.  _Journal of Cloud Computing_ 3, 1 (2014), 1–30.
  * Zhang et al. (2018) Jianguo Zhang, Ji Wang, Lifang He, Zhao Li, and S Yu Philip. 2018.  Layerwise perturbation-based adversarial training for hard drive health degree prediction. In _2018 IEEE International Conference on Data Mining (ICDM)_. IEEE, 1428–1433.