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Recurrent Neural Networks for Multivariate Time Series with Missing Values

Zhengping CheSanjay PurushothamKyunghyun ChoDavid SontagYan Liu

article2016Sci Rep2,560 citations

Develops GRU-D, a recurrent neural network architecture that incorporates masking indicators and elapsed time intervals directly into recurrent transitions to exploit informative missing patterns in multivariate clinical time series.

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Multivariate time series data in domains such as healthcare routinely contain missing values whose patterns often correlate with target outcomes, a phenomenon known as informative missingness. Conventional approaches either discard incomplete records or apply separate imputation steps before prediction; both strategies fail to exploit missingness patterns fully and can degrade performance when missing rates exceed 80 percent, as is common in intensive-care records.

The article develops and evaluates a recurrent neural-network architecture, called GRU-D, that directly incorporates two representations of missingness—binary masks and elapsed-time intervals—into a gated recurrent unit. Trainable exponential decay terms are applied to both the input features and the hidden states, allowing the model to learn how much weight to place on recent observations versus default values for each variable. The resulting end-to-end model is trained by standard back-propagation and requires no external imputation stage.

On the MIMIC-III and PhysioNet clinical datasets, GRU-D attained the highest area-under-the-ROC-curve scores for in-hospital mortality prediction (0.8527 and 0.8424, respectively) and for multi-task diagnosis classification, outperforming logistic regression, support-vector machines, random forests, and several GRU variants that rely on mean, forward, or matrix-based imputation. The advantage widened as training-set size increased and remained stable across synthetic datasets with varying degrees of informative missingness. The model also produced accurate predictions earlier in a patient’s stay, matching the 48-hour performance of the strongest non-recurrent baseline after only 36 hours of data.

These results indicate that explicitly modeling missingness inside the network improves both accuracy and timeliness of clinical forecasts without added computational cost. The learned decay parameters further reveal which variables carry the most predictive information when absent, offering clinicians interpretable insight into data-collection priorities.

The authors note that gains are modest when missingness is unrelated to outcomes and that the method has been tested only on retrospective observational data. They recommend extending the decay mechanism to explicitly missing-not-at-random regimes and conducting prospective validation before routine clinical deployment.

Massachusetts Institute of TechnologyNew York UniversityUniversity of Southern Californiaclassificationclassification methodsdeep learningrecurrent neural networksmultivariate time seriesmissingness patterns
arXiv: 1606.01865PeterChe1990/GRU-D
Cover for Recurrent Neural Networks for Multivariate Time Series with Missing Values

Abstract

Multivariate time series data in practical applications, such as health care, geoscience, and biology, are characterized by a variety of missing values. In time series prediction and other related tasks, it has been noted that missing values and their missing patterns are often correlated with the target labels, a.k.a., informative missingness. There is very limited work on exploiting the missing patterns for effective imputation and improving prediction performance. In this paper, we develop novel deep learning models, namely GRU-D, as one of the early attempts. GRU-D is based on Gated Recurrent Unit (GRU), a state-of-the-art recurrent neural network. It takes two representations of missing patterns, i.e., masking and time interval, and effectively incorporates them into a deep model architecture so that it not only captures the long-term temporal dependencies in time series, but also utilizes the missing patterns to achieve better prediction results. Experiments of time series classification tasks on real-world clinical datasets (MIMIC-III, PhysioNet) and synthetic datasets demonstrate that our models achieve state-of-the-art performance and provides useful insights for better understanding and utilization of missing values in time series analysis.

Topics

classificationclassification methodsdeep learninggated recurrent unitrecurrent neural networksnetwork architecture

Keywords

recurrent neural networksmultivariate time seriesmissingness patternsinformative missingnessdata imputationtime series classification problemslong-term temporal dependenciesMIMIC-III datasetPhysioNet datasetforward imputation

Table of Contents

  • 1 Introduction
  • 2 RNN models for time series with missing variables
  • 2.1 GRU-RNN for time series classification
  • 2.2 GRU-D: Model with trainable decays
  • 3 Experiments
  • 3.1 Dataset descriptions and experimental design
  • 3.2 Methods and implementation details
  • 3.3 Quantitative results
  • 3.4 Discussions
  • 4 Summary
  • References
  • A Appendix
  • A.1 Investigation of relation between missingness and labels
  • A.2 GRU-D model variations
  • A.2.1 GRU model with different trainable decays
  • A.2.2 GRU-IMP: Goal-oriented imputation model
  • A.2.3 Comparisons of related RNN models
  • A.3 Supplementary experiment details
  • A.3.1 Data statsitics
  • A.3.2 GRU model size comparison
  • A.3.3 Multi-task prediction details
  • A.3.4 Empirical comparison of model variations

Knowls

  1. Knowl 1 — Gated Recurrent Unit with Trainable Decays (GRU-D) Architecture

    model/method

    The GRU-D architecture is a modified Gated Recurrent Unit (GRU) cell designed to model multivariate time series with missing values by incorporating input decay, hidden state decay, and explicit missingness indicators (masks). Let x^t∈RD\hat{x}_t \in \mathbb{R}^D denote the decayed input vector, h^t−1∈RH\hat{h}_{t-1} \in \mathbb{R}^H denote the decayed previous hidden state, and mt∈{0,1}Dm_t \in \{0, 1\}^D denote the binary observation mask at time step tt.

    The GRU-D cell updates its reset gate rt∈RHr_t \in \mathbb{R}^H, update gate zt∈RHz_t \in \mathbb{R}^H, candidate hidden state h~t∈RH\tilde{h}_t \in \mathbb{R}^H, and final hidden state ht∈RHh_t \in \mathbb{R}^H according to:

    rt=σ(Wrx^t+Urh^t−1+Vrmt+br)r_t = \sigma(W_r \hat{x}_t + U_r \hat{h}_{t-1} + V_r m_t + b_r)

    zt=σ(Wzx^t+Uzh^t−1+Vzmt+bz)z_t = \sigma(W_z \hat{x}_t + U_z \hat{h}_{t-1} + V_z m_t + b_z)

    h~t=tanh⁡(Wx^t+U(rt⊙h^t−1)+Vmt+b)\tilde{h}_t = \tanh(W \hat{x}_t + U (r_t \odot \hat{h}_{t-1}) + V m_t + b)

    ht=(1−zt)⊙h^t−1+zt⊙h~th_t = (1 - z_t) \odot \hat{h}_{t-1} + z_t \odot \tilde{h}_t

    where σ(⋅)\sigma(\cdot) is the element-wise sigmoid function, tanh⁡(⋅)\tanh(\cdot) is the hyperbolic tangent function, and ⊙\odot represents element-wise multiplication. The weight matrices Wr,Wz,W∈RH×DW_r, W_z, W \in \mathbb{R}^{H \times D}, Ur,Uz,U∈RH×HU_r, U_z, U \in \mathbb{R}^{H \times H}, Vr,Vz,V∈RH×DV_r, V_z, V \in \mathbb{R}^{H \times D}, and bias vectors br,bz,b∈RHb_r, b_z, b \in \mathbb{R}^H are trainable parameters optimized jointly via backpropagation through time.

  2. Knowl 2 — Trainable Exponential Decay Function

    equation

    To model how the influence of past observations fades over time, GRU-D computes a continuous, trainable decay rate vector γt∈(0,1]K\gamma_t \in (0, 1]^K as a function of the elapsed time interval vector δt∈RK\delta_t \in \mathbb{R}^K:

    γt=exp⁡(−max⁡(0,Wγδt+bγ))\gamma_t = \exp\left(-\max\left(0, W_\gamma \delta_t + b_\gamma\right)\right)

    where WγW_\gamma is a weight matrix, bγb_\gamma is a bias vector, and max⁡(0,⋅)\max(0, \cdot) is the rectified linear activation function (ReLU). The exponentiated negative rectifier ensures that each decay rate γt(k)\gamma_t^{(k)} decreases monotonically with time and is constrained strictly within the range (0,1](0, 1]. The parameters WγW_\gamma and bγb_\gamma are learned jointly with the network parameters during training.

  3. Knowl 3 — Input Variable Decay Formulation in GRU-D

    model/method

    In GRU-D, missing input variables are decayed over time toward their global empirical means rather than held fixed at their last observed values. For a DD-dimensional multivariate time series, the decayed input measurement vector x^t∈RD\hat{x}_t \in \mathbb{R}^D at time step tt is defined for each feature d∈{1,…,D}d \in \{1, \dots, D\} as:

    x^td=mtdxtd+(1−mtd)(γxtdxt′d+(1−γxtd)x~d)\hat{x}_t^d = m_t^d x_t^d + (1 - m_t^d) \left( \gamma_{x_t}^d x_{t'}^d + (1 - \gamma_{x_t}^d) \tilde{x}^d \right)

    where:

    • xtd∈Rx_t^d \in \mathbb{R} is the observed measurement at time step tt.
    • mtd∈{0,1}m_t^d \in \{0, 1\} is the masking indicator (mtd=1m_t^d = 1 if observed, 00 if missing).
    • xt′dx_{t'}^d is the last observed value of feature dd at time step t′<tt' < t.
    • x~d=∑n=1N∑t=1Tmt,ndxt,nd∑n=1N∑t=1Tmt,nd\tilde{x}^d = \frac{\sum_{n=1}^N \sum_{t=1}^T m_{t,n}^d x_{t,n}^d}{\sum_{n=1}^N \sum_{t=1}^T m_{t,n}^d} is the empirical mean of variable dd computed across all training instances.
    • γxt=exp⁡(−max⁡(0,Wγxδt+bγx))∈RD\gamma_{x_t} = \exp\left(-\max\left(0, W_{\gamma_x} \delta_t + b_{\gamma_x}\right)\right) \in \mathbb{R}^D is the input decay vector, in which Wγx∈RD×DW_{\gamma_x} \in \mathbb{R}^{D \times D} is constrained to be a diagonal matrix, ensuring the decay rate of each variable dd depends solely on its own time interval δtd\delta_t^d.

  4. Knowl 4 — Hidden State Decay Formulation in GRU-D

    model/method

    To decay latent feature representations when variables have been unobserved for extended durations, GRU-D decays the previous recurrent hidden state ht−1∈RHh_{t-1} \in \mathbb{R}^H before computing the new hidden state hth_t:

    h^t−1=γht⊙ht−1\hat{h}_{t-1} = \gamma_{h_t} \odot h_{t-1}

    where ⊙\odot represents element-wise multiplication, and the hidden decay vector γht∈RH\gamma_{h_t} \in \mathbb{R}^H is defined as:

    γht=exp⁡(−max⁡(0,Wγhδt+bγh))\gamma_{h_t} = \exp\left(-\max\left(0, W_{\gamma_h} \delta_t + b_{\gamma_h}\right)\right)

    Here, δt∈RD\delta_t \in \mathbb{R}^D is the vector of elapsed time intervals since the last observation for each variable, Wγh∈RH×DW_{\gamma_h} \in \mathbb{R}^{H \times D} is a full (non-diagonal) weight matrix allowing cross-variable interactions on hidden units, and bγh∈RHb_{\gamma_h} \in \mathbb{R}^H is a trainable bias vector.

  5. Knowl 5 — Masking and Time Interval Representations for Incomplete Time Series

    definition

    A multivariate time series with DD variables and TT observations is denoted X=(x1,x2,…,xT)T∈RT×DX = (x_1, x_2, \dots, x_T)^T \in \mathbb{R}^{T \times D}, where xt∈RDx_t \in \mathbb{R}^D is the measurement vector at time-stamp st∈Rs_t \in \mathbb{R} (s1=0s_1 = 0). Missing observations are represented using two complementary tensors:

    1. Masking Vector mt∈{0,1}Dm_t \in \{0, 1\}^D: mtd={1,if xtd is observed0,otherwisem_t^d = \begin{cases} 1, & \text{if } x_t^d \text{ is observed} \\ 0, & \text{otherwise} \end{cases}

    2. Time Interval Vector δt∈RD\delta_t \in \mathbb{R}^D, which tracks the physical duration since variable dd was last observed: δtd={st−st−1+δt−1d,t>1,mt−1d=0st−st−1,t>1,mt−1d=10,t=1\delta_t^d = \begin{cases} s_t - s_{t-1} + \delta_{t-1}^d, & t > 1, m_{t-1}^d = 0 \\ s_t - s_{t-1}, & t > 1, m_{t-1}^d = 1 \\ 0, & t = 1 \end{cases}

  6. Knowl 6 — In-Hospital Mortality Prediction Performance on Real-World EHR Datasets

    data/table

    Evaluated on 5-fold cross-validation on two ICU datasets (MIMIC-III with 19,714 admissions and PhysioNet Challenge 2012 with 8,000 records) characterized by missing rates >80%>80\%, GRU-D outperforms standard Non-RNN models (Logistic Regression [LR], Support Vector Machines with RBF kernel [SVM], Random Forest [RF]), recurrent neural network baselines, and multi-stage imputation methods in Area Under the ROC Curve (AUC score, mean ±\pm std):

    Method LR SVM RF GRU / RNN
    MIMIC-III Mortality Prediction
    LSTM-Mean - - - 0.8142±0.0140.8142 \pm 0.014
    Mean Imputation 0.7589±0.0150.7589 \pm 0.015 0.7908±0.0060.7908 \pm 0.006 0.8293±0.0040.8293 \pm 0.004 0.8252±0.0110.8252 \pm 0.011
    Forward Imputation 0.7792±0.0180.7792 \pm 0.018 0.8010±0.0040.8010 \pm 0.004 0.8303±0.0030.8303 \pm 0.003 0.8192±0.0130.8192 \pm 0.013
    Simple (Concat m,δm, \delta) 0.7715±0.0150.7715 \pm 0.015 0.8146±0.0080.8146 \pm 0.008 0.8294±0.0070.8294 \pm 0.007 0.8380±0.0080.8380 \pm 0.008
    GRU-Simple w/o δ\delta - - - 0.8367±0.0090.8367 \pm 0.009
    GRU-Simple w/o mm - - - 0.8266±0.0090.8266 \pm 0.009
    SoftImpute 0.7598±0.0170.7598 \pm 0.017 0.7540±0.0120.7540 \pm 0.012 0.7855±0.0110.7855 \pm 0.011 0.8266±0.0090.8266 \pm 0.009
    KNN 0.6877±0.0110.6877 \pm 0.011 0.7200±0.0040.7200 \pm 0.004 0.7135±0.0150.7135 \pm 0.015 -
    CubicSpline 0.7270±0.0050.7270 \pm 0.005 0.6376±0.0180.6376 \pm 0.018 0.8339±0.0070.8339 \pm 0.007 0.8180±0.0110.8180 \pm 0.011
    MICE 0.6965±0.0190.6965 \pm 0.019 0.7169±0.0120.7169 \pm 0.012 0.7159±0.0050.7159 \pm 0.005 0.7527±0.0150.7527 \pm 0.015
    MF 0.7158±0.0180.7158 \pm 0.018 0.7266±0.0170.7266 \pm 0.017 0.7234±0.0110.7234 \pm 0.011 0.7843±0.0120.7843 \pm 0.012
    PCA 0.7246±0.0140.7246 \pm 0.014 0.7235±0.0120.7235 \pm 0.012 0.7747±0.0090.7747 \pm 0.009 0.8236±0.0070.8236 \pm 0.007
    MissForest 0.7279±0.0160.7279 \pm 0.016 0.7482±0.0160.7482 \pm 0.016 0.7858±0.0100.7858 \pm 0.010 0.8239±0.0060.8239 \pm 0.006
    Proposed GRU-D - - - 0.8527±0.003\mathbf{0.8527 \pm 0.003}
    PhysioNet Mortality Prediction
    LSTM-Mean - - - 0.8025±0.0130.8025 \pm 0.013
    Mean Imputation 0.7423±0.0110.7423 \pm 0.011 0.8131±0.0180.8131 \pm 0.018 0.8183±0.0150.8183 \pm 0.015 0.8162±0.0140.8162 \pm 0.014
    Forward Imputation 0.7479±0.0120.7479 \pm 0.012 0.8140±0.0180.8140 \pm 0.018 0.8219±0.0170.8219 \pm 0.017 0.8195±0.0040.8195 \pm 0.004
    Simple (Concat m,δm, \delta) 0.7625±0.0040.7625 \pm 0.004 0.8277±0.0120.8277 \pm 0.012 0.8157±0.0140.8157 \pm 0.014 0.8226±0.0100.8226 \pm 0.010
    SoftImpute 0.7386±0.0070.7386 \pm 0.007 0.8057±0.0190.8057 \pm 0.019 0.8100±0.0160.8100 \pm 0.016 0.8125±0.0050.8125 \pm 0.005
    KNN 0.7146±0.0110.7146 \pm 0.011 0.7644±0.0180.7644 \pm 0.018 0.7567±0.0120.7567 \pm 0.012 0.8155±0.0040.8155 \pm 0.004
    CubicSpline 0.6913±0.0220.6913 \pm 0.022 0.6364±0.0150.6364 \pm 0.015 0.8151±0.0150.8151 \pm 0.015 0.7596±0.0200.7596 \pm 0.020
    MICE 0.6828±0.0150.6828 \pm 0.015 0.7690±0.0160.7690 \pm 0.016 0.7618±0.0070.7618 \pm 0.007 0.8153±0.0130.8153 \pm 0.013
    MF 0.6513±0.0140.6513 \pm 0.014 0.7515±0.0220.7515 \pm 0.022 0.7355±0.0220.7355 \pm 0.022 0.7904±0.0120.7904 \pm 0.012
    PCA 0.6890±0.0190.6890 \pm 0.019 0.7741±0.0140.7741 \pm 0.014 0.7561±0.0250.7561 \pm 0.025 0.8116±0.0070.8116 \pm 0.007
    MissForest 0.7010±0.0180.7010 \pm 0.018 0.7779±0.0080.7779 \pm 0.008 0.7890±0.0160.7890 \pm 0.016 0.8244±0.0120.8244 \pm 0.012
    Proposed GRU-D - - - 0.8424±0.012\mathbf{0.8424 \pm 0.012}

    Two-step imputation-prediction models underperform due to extremely high missing rates, whereas GRU-D achieves the highest discrimination by modeling missingness patterns and temporal decay end-to-end.

  7. Knowl 7 — Multi-Task Clinical Phenotyping and Outcome Prediction Performance

    data/table

    In multi-task clinical prediction tasks where the output layer employs nn sigmoid activations with a comorbidity co-occurrence prior regularizer, GRU-D achieves the highest average AUC scores across 20 ICD-9 diagnosis categories on MIMIC-III and 4 clinical outcomes on PhysioNet:

    Model ICD-9 20 Tasks (MIMIC-III) All 4 Tasks (PhysioNet)
    GRU-Mean 0.7070±0.0010.7070 \pm 0.001 0.8099±0.0110.8099 \pm 0.011
    GRU-Forward 0.7077±0.0010.7077 \pm 0.001 0.8091±0.0080.8091 \pm 0.008
    GRU-Simple 0.7105±0.0010.7105 \pm 0.001 0.8249±0.0100.8249 \pm 0.010
    GRU-CubicSpline 0.6372±0.0050.6372 \pm 0.005 0.7451±0.0110.7451 \pm 0.011
    GRU-MICE 0.6717±0.0050.6717 \pm 0.005 0.7955±0.0030.7955 \pm 0.003
    GRU-MF 0.6805±0.0040.6805 \pm 0.004 0.7727±0.0030.7727 \pm 0.003
    GRU-PCA 0.7040±0.0020.7040 \pm 0.002 0.8042±0.0060.8042 \pm 0.006
    GRU-MissForest 0.7115±0.0030.7115 \pm 0.003 0.8076±0.0090.8076 \pm 0.009
    Proposed GRU-D 0.7123±0.003\mathbf{0.7123 \pm 0.003} 0.8370±0.012\mathbf{0.8370 \pm 0.012}

    GRU-D provides superior generalization on simultaneous multi-label classification tasks compared to imputation and concatenation baselines.

  8. Knowl 8 — Model Robustness Across Varying Missingness-Label Correlation Levels

    empirical result

    On synthetic datasets generated from the Gesture Phase Segmentation benchmark across four settings with identical missing rates but increasing correlation between variable missing rate and target labels (average absolute Pearson correlation from 0 to 0.8):

    1. Baselines that ignore missing patterns (GRU-Mean, GRU-Forward) show flat performance across all correlation levels (AUC ≈0.68−0.70\approx 0.68-0.70).
    2. Naive concatenation of missingness indicators (GRU-Simple) improves when correlation is high (AUC ≈0.90\approx 0.90 at correlation 0.8) but suffers significant degradation when correlation is low or nonexistent (AUC dropping to ≈0.60\approx 0.60 at correlation 0), introducing spurious relationships.
    3. GRU-D achieves the best AUC across all settings, matching GRU-Simple at the highest correlation (ightharpoonup0.92 ightharpoonup 0.92) while remaining stable and resilient when missingness is uncorrelated with target labels, confirming that it leverages missingness when informative and falls back on observed values when uninformative.

  9. Knowl 9 — Physiological and Statistical Interpretability of Learned Decay Parameters

    empirical result

    Analysis of the decay parameters learned by GRU-D on the PhysioNet mortality task demonstrates two key behaviors:

    1. Input Decays (γx\gamma_x): Most variables have near-zero input decay weights, keeping values close to their historical measurements. In contrast, clinically volatile variables—such as patient weight (missing rate 0.5452), arterial pH (missing rate 0.9118), temperature (missing rate 0.6915), and respiration rate (missing rate 0.8053)—exhibit rapid decay toward their population means, aligning with physiological homeostasis where historical measurements become uninformative quickly.
    2. Hidden State Decay Weights (WγhW_{\gamma_h}): The magnitude of the learned hidden decay weights corresponding to feature dd is inversely related to the feature's missing rate. Frequently observed variables (e.g., heart rate, missing rate 0.1984) have significantly larger weight magnitudes in WγhW_{\gamma_h} than rarely measured variables (e.g., cholesterol, missing rate 0.9989), reflecting their greater informational relevance for patient status evolution.

  10. Knowl 10 — Early Prediction Capability and Dataset Scaling Properties of GRU-D

    empirical result

    On the MIMIC-III mortality prediction task:

    1. Early Prediction: When evaluating models at early observation windows (t∈{12,18,24,30,36,42,48}t \in \{12, 18, 24, 30, 36, 42, 48\} hours post-admission), GRU-D reaches an AUC at 36 hours that matches the 48-hour performance of the strongest non-RNN baseline (Random Forest with Simple concatenation, RF-Simple), achieving a 12-hour lead time in predicting patient mortality. Furthermore, GRU-D outperforms GRU-Simple by at least 2.5% AUC for time windows exceeding 30 hours.
    2. Scalability: When evaluated on subsets of 2,000, 10,000, and 19,714 admissions, GRU-D shows steeper performance gains with increasing dataset size compared to non-RNN models (SVM-Simple, RF-Simple), whose performance plateaus with larger data volumes.

  11. Knowl 11 — Assumptions and Methodological Limitations of GRU-D

    limitation

    The effectiveness of GRU-D is subject to specific methodological boundaries:

    1. Dependence on Informative Missingness: GRU-D relies on the presence of informative missingness where missing patterns correlate with target prediction labels; if missingness is completely non-informative (missing completely at random) or uncorrelated with tasks, the model yields limited or no gains over standard recurrent baselines.
    2. Supervised Setting Restriction: GRU-D is designed for supervised sequence classification and cannot be used directly as a standalone, unsupervised data imputation method without modifying its training architecture.
    3. Domain-Specific Decay Design: Applying GRU-D to new domains (e.g., climate or traffic) requires domain-specific understanding to formulate appropriate decay targets (such as empirical means or steady-state defaults).

Coverage note — None was omitted; all contributed models, mathematical formulations, baseline descriptions, empirical classification results on MIMIC-III and PhysioNet, synthetic validation, parameter interpretability analyses, and methodological limitations are fully represented.

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Citation

MLA
Che, Z., et al. “Recurrent Neural Networks for Multivariate Time Series with Missing Values”. Scientific Reports, vol. 8, no. 1, 2018, https://doi.org/10.1038/s41598-018-24271-9.
APA
Che, Z., Purushotham, S., Cho, K., Sontag, D., & Liu, Y. (2018). Recurrent Neural Networks for Multivariate Time Series with Missing Values. Scientific Reports, 8(1). https://doi.org/10.1038/s41598-018-24271-9
Chicago
Che, Z., S. Purushotham, K. Cho, D. Sontag, and Y. Liu. 2018. “Recurrent Neural Networks for Multivariate Time Series with Missing Values”. Scientific Reports 8 (1). https://doi.org/10.1038/s41598-018-24271-9.
Harvard
Che, Z. et al. (2018) “Recurrent Neural Networks for Multivariate Time Series with Missing Values”, Scientific Reports, 8(1). Available at: https://doi.org/10.1038/s41598-018-24271-9.
Vancouver
1. Che Z, Purushotham S, Cho K, Sontag D, Liu Y (2018) Recurrent Neural Networks for Multivariate Time Series with Missing Values. Scientific Reports. https://doi.org/10.1038/s41598-018-24271-9

BibTeX

@article{Che_2018, title={Recurrent Neural Networks for Multivariate Time Series with Missing Values}, volume={8}, ISSN={2045-2322}, url={http://dx.doi.org/10.1038/s41598-018-24271-9}, DOI={10.1038/s41598-018-24271-9}, number={1}, journal={Scientific Reports}, publisher={Springer Science and Business Media LLC}, author={Che, Zhengping and Purushotham, Sanjay and Cho, Kyunghyun and Sontag, David and Liu, Yan}, year={2018}, month=Apr }
Metadata:Crossref

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