In BTTF, we may consider both U and V as spatial factor matrices, while in fact they may characterize any features in which dependencies are not explicitly encoded (e.g., type of travelers in [8] and type of sensors in [20]). For example, Chen and Cichocki [11] developed a non-negative matrix factorization model which temporal smoothness and spatial correlation regularizers. T.Januschowski, Deep state space models for time series forecasting, in, B.Yu, H.Yin, and Z.Zhu, Spatio-temporal graph convolutional networks: a %r/JHs3fQX5S0 ,R/x0lAe9A kjF?/#6kM yq :~tz2?f+60jOP_fu#k%j/VBU=7F;a5kG.cuR\)s qRA}3jk9|j'Zus!s 0n"snr) There are several directions to explore for future research. First, we will extend this framework to account for spatial dependencies/correlations by incorporating tools such as spatial AR and Laplacian kernels. Data set (N): NYC taxi555https://www1.nyc.gov/site/tlc/about/tlc-trip-record-data.page. Moreover, since there exist no universal/automatic solutions, this tuning procedure has to be done for each particular application (i.e., input data set). The Bayesian scheme allows us to estimate the posterior distribution of target variables, which is critical to risk-sensitive applications. The results reveal that Bayesian treatment over temporal matrix factorization is more superior than manually controlling those regularizers. N#gET$?~'36^LBU3 a~g`X? In order to guarantee the TRTFs performance, we make cross validation carefully for the parameter tuning process. challenge due to not only the large-scale and high-dimensional nature but also For Bayesian algorithms, the point estimates are obtained by averaging over m2=100 Gibbs iterations. DMSTF outperforms related methodologies in terms of predictive performance for unseen data, reveals meaningful clusters in the data, and performs forecasting in a variety of domains with potentially nonlinear temporal transitions. To make predictions efficiently, we keep W, X and A as fixed point estimates by averaging the m2 samples of W, X, and A after training the model, and only consider xt+1 as a new parameter to be updated over time [35]. We next conduct the experiment for making single-step and multi-step rolling predictions (see Figure2) on the four data sets and TableII shows the performance of BTMF and other baseline models. Similarly, the full conditional p(A,|) is also of the same form as (11) in BTMF. Therefore, BTF provides a powerful tool to handle incomplete/corrupted time series data for both imputation and prediction tasks. Another is used for long-term prediction (e.g., multi-step prediction). Talk, and E.Fox, A unified framework for missing data and cold Given the complex structure of BTMF, it is intractable to write down the posterior distribution. This framework exploits the low-rank structure of block Hankel tensors in the embedded space and captures the intrinsic correlations among multiple TS, which thus can improve the forecasting results, especially for multiple short time series. 2016 Apr;27(4):736-48. doi: 10.1109/TNNLS.2015.2423694. Section 4.2 has summarized the entire procedure of model inference for the parameters/hyperparameters in BTMF. [33] extended this model to dynamic Poisson matrix factorization for recommendation. Regarding posterior inference, the main difference between BTTF and BTMF is the posterior distribution of factor matrices. [19] by replacing the independent AR assumption on temporal factors with a more flexible VAR assumption. seriesin particular spatiotemporal datain the presence of missing values. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series-in particular spatiotemporal data-in the presence of missing values. HHS Vulnerability Disclosure, Help tasks. These models are chosen because matrix time series data collected from multiple days can be re-organized as a third-order (locationdaytime of day) tensor, and in this case tensor factorization can effectively learn the global patterns provided by the additional day dimension. Essentially, these models assume that the multivariate and multidimensional time series can be characterized by a low-rank structure with shared latent factors (i.e., global consistency). The site is secure. Bayesian Temporal Factorization for Multidimensional Time Series Prediction @article{Sun2021BayesianTF, title={Bayesian Temporal Factorization for Multidimensional Time Series Prediction}, author={Lijun Sun and X. Chen}, journal={IEEE transactions on pattern analysis and machine intelligence}, year={2021}, volume={PP} } characterize both global and local consistencies in large-scale time series The CP decomposition provides us a natural way to extend BTMF to tensors by assuming that each element: Following the same routine as BTMF, we define the generative process of Bayesian Temporal Tensor Factorization (BTTF) as follows: and in particular, the same VAR model in (8) can be used model temporal factor matrix X, and the prior is defined as: where in this setting, the same Gaussian-Wishart priors as in BTMF can be placed for the underlying hyperparameters. To address these issues in modeling multivariate and multidimensional time series data, several notable approaches have been proposed recently based on matrix/tensor factorization (see [1] for a brief review and e.g., [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] for some representative models). Other baseline imputation models are BGCP, BATF, HaLRTC, and TF-ALS, which also have been selected above. As mentioned, making accurate predictions on incomplete time series is very challenging, while missing data problem is almost inevitable in real-world applications. We use a third-order tensor YRMNT as an example throughout the section. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series -- in particular spatiotemporal data -- in the . Bayesian Temporal Factorization for Multidimensional Time Series Prediction Abstract: Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. In this section we apply BTMF and BTTF on several real-world spatiotemporal data sets for both imputation and prediction tasks, and evaluate the effectiveness of these two models against recent state-of-the-art approaches. The numerical experiments demonstrate the superiority of the proposed autoregression with low rank tensors,, H.Tan, Y.Wu, B.Shen, P.J. Jin, and B. Essentially, this model can be considered a high-order extension of (1). IEEE Engineering in Medicine and Biology Society. This data set registered occupancy (i.e., number of parked vehicles) of 30 car parks in Birmingham City for every half an hour between 8:00 and 17:00 over more than two months (77 days from October 4, 2016 to December 19, 2016). predictions and produce uncertainty estimates without imputing those missing To guarantee the models performance on multi-step prediction tasks, we set time lags as L={1,2,3,T0,T0+1,T0+2,7T0,7T0+1,7T0+2}. Before As a result, the reliability and uncertainty of the predictions are often overlooked. As a common technique for collaborative filtering, matrix/tensor factorization presents a natural solution to address the scalability, efficiency, and missing data issues. Zhou Z, Yang Z, Zhang Y, Huang Y, Chen H, Yu Z. iScience. However, it should be noted that the specification of these parameters has little impact on the final results, as the training data will play a much more important role in defining the posteriors of the hyperparameters [30, 12]. However, given the complex spatiotemporal dependencies in these data sets, making efficient and reliable predictions for real-time applications has been a long-standing and fundamental research challenge. Data set (G): Guangzhou urban traffic speed111https://doi.org/10.5281/zenodo.1205229. samples to get not only the mean but also the confidence interval for risk-sensitive applications. %PDF-1.5 and pattern discovery with a Bayesian augmented tensor factorization In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series -- in particular spatiotemporal data -- in the presence of missing values. For prediction, we compare BTMF against TRMF and BayesTRMF. We use the mean absolute percentage error (MAPE) and root mean square error (RMSE) as evaluation metrics: where n is the total number of estimated values, and yi and ^yi are the actual value and its estimation, respectively. Assume that we have historical data YRNt and a trained model based on Y. >J^L^*%7l)[2JVM#U5olDK}=BU5opGx6rS,}6UT0?Y|=0x dlOt>';o0l%DxKlsgNZT'Xa Therefore, a critical challenge is to perform reliable prediction in the presence of missing data [9]. In this paper we present a Bayesian Temporal Factorization (BTF) framework by incorporating a VAR layer into traditional Bayesian probabilistic MF/TF algorithms. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series -- in particular spatiotemporal data -- in the . Specifically, the posterior distribution of xt in BTTF can be written as p(xt|)=N(t,t) with. Multi-Section Traffic Flow Prediction Based on MLR-LSTM Neural Network. There are several ways to further reduce the computational cost. model,, L.Charlin, R.Ranganath, J.McInerney, and D.M. Blei, Dynamic poisson In detail, we introduce a Gibbs sampling algorithm by deriving the full conditional distributions for all parameters and hyperparameters. This data set registered traffic speed data from 214 road segments over two months (61 days from August 1 to September 30, 2016) with a 10-minute resolution (144 time intervals per day) in Guangzhou, China. This data set collected freeway traffic speed from 323 loop detectors with a 5-minute resolution over the whole year of 2015 in Seattle, USA. 2021 Jan;43(1):62-76. doi: 10.1109/TPAMI.2019.2923240. Results and analysis. It provides a collection of algorithms to perform temporal abstractions and preprocessing of time series health data, a framework for defining and executing data analysis workflows based on these algorithms, and a GUI . Moreover, we develop a Bayesian TRMF (BayesTRTF) for both two tasks. The size of this time series matrix is 301386 with 18 time intervals per day and the amount of missing values is 14.89% after data processing. factorization with seasonal patterns, in, M.Araujo, P.Ribeiro, H.A. We evaluate the proposed BTMF model on both imputation and prediction tasks. Although we introduce BTF in a spatiotemporal setting, the model can be applied on general multidimensional time series data. Large-scale and multidimensional spatiotemporal data sets are becoming Given the conjugate Gamma prior, the conditional distribution of , is also a Gamma distribution, i.e., we have, Based on the aforementioned sampling processes, we summarize the the MCMC inference algorithm to impute missing values in the partially observed matrix time series data as Algorithm1. of By integrating low-rank matrix/tensor factorization and vector autoregressive (VAR) process into a single probabilistic graphical model, this framework can characterize both global and local consistencies in large-scale time series data. We summarize the BTMF at each time point tth iterate (i.e., forecast data with the time slots from tT0+Ts+1 to (t+1)T0+Ts where T0is the number of time intervals per day and Ts is the start time slot) for this task as follows: Collect the actual observations YtT0+Ts and train a BTMF model with m1 burn-in iterations and m2 iterations for sampling. Train a Bayesian model on yt as. In doing so, we adapt TRMF/BayesTRMF to an online implementation similar to (online) BTMF. He, and L.Sun, A Bayesian tensor decomposition approach for Bookshelf Thus, we can sample wi|N(w,(w)1) with, Sampling temporal factor xt. It is straightforward to extend BTMF to model multidimensional (order>2) tensor time series. Thanks to the use of conjugate priors in Figure3, we can actually write down all the conditional distributions analytically. Results and analysis. For the only remaining parameter , we place a Gamma prior Gamma(,) where and are the shape and rate parameters, respectively. As an example, we depict the actual and predicted values for three randomly selected time series in Figure 8. We next first describe the online rolling prediction task in detail (see Figure2). endstream [25] extended VAR model for matrix-valued time series data (i.e., a third-order tensor time series) by introducing two AR coefficient matrices to characterize the correlation structure. 2021. Since the introduction of Bayesian Probabilistic Matrix Factorization (BPMF) [30], Bayesian treatment has been extensively implemented to address the overfitting and the parameter tuning problems in factorization models. Epub 2020 Dec 4. Data set (B): Birmingham parking 222https://archive.ics.uci.edu/ml/datasets/Parking+Birmingham. For example, predicting the demand and states (e.g., speed, flow) of urban traffic is essential to a wide range of intelligent transportation systems (ITS) applications, such trip planning, travel time estimation, route planning, traffic signal control, to name but a few [3]. This work is further extended in [20] to model spatiotemporal tensor data by introducing a spatial autoregressive regularizer, which enables us to perform predictions on the spatial dimension for unknown locations/sensors. 2022 Oct 4;22(19):7517. doi: 10.3390/s22197517. Bethesda, MD 20894, Web Policies To support multiple prediction applications, here we adapt Algorithm 1 for sptiotemporal prediction tasks and derive two BTMF implementation strategies. Deep factors for forecasting, in, Low-Rank Autoregressive Tensor Completion for Multivariate Time Series For simplicity, we introduce matrix AR(Rd)R and vector vt+1R(Rd)1 as the form. data. With recent advances in sensing technologies, large-scale and multidimensional time series datain particular spatiotemporal dataare collected on a continuous basis from various types of sensors and applications. for both missing data imputation and short-term/long-term rolling prediction Bayesian Temporal Matrix Factorization. Given the dimension/number of attributes collected from the underlying system, this formulation can be further extended to even higher-order tensors. Bayesian Robust Tensor Factorization for Incomplete Multiway Data. Although BayesTRMF is a fully Bayesian counterpart of TRMF, TRMF actually performs better than BayesTRMF in most cases. Third, most existing time series models require complete time series data as input, while in real-world sensor recordings the missing data problem is almost inevitable due to various factors such as hardware/software failure, human error, and network communication problems. Bayesian Temporal Factorization for Multidimensional Time Series Prediction Abstract: Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. Data set (G): Guangzhou urban traffic speed, Data set (H): Hangzhou metro passenger flow, Data set (S): Seattle freeway traffic speed, C.Faloutsos, J.Gasthaus, T.Januschowski, and Y.Wang, Forecasting big time As can be seen, the proposed BTMF and the adapted BayesTRMF clearly outperform the TRMF in most of the cases. Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. However, most classical time series models are not scalable to handle large data sets. official website and that any information you provide is encrypted A Bayesian Dynamical Approach for Human Action Recognition. We evaluate these models by making rolling predictions over the last five days (i.e., 5144 time slots) for data set (G), the last seven days (i.e., 718 time slots) for data set (B), the last five days (i.e., 5108 time slots) for data set (H), and the last five days (i.e., 5288 time slots) for data set (S). The prediction task is to first have a good estimate of ^yt+1 based on the trained model, and then estimate ^yt+2 when yt+1the actual observations at time point t+1is available to us. To address the first limitation, in the proposed model we remove this diagonal constraint on Ak and employ the standard VAR process to characterize dynamic dependencies in X. In order to create meaningful temporal patterns, different smoothing techniques and regularization schemes have been applied (e.g., linear dynamical systems [12] and Gaussian processes [15]) to impose local consistency. A low-rank autoregressive tensor completion (LATC) framework to model multivariate time series data and demonstrates the superiority of the integration of global and local trends in LATC in both missing data imputation and rolling prediction tasks. Sensors (Basel). factorization for high-dimensional time series prediction, in, K.Takeuchi, H.Kashima, and N.Ueda, Autoregressive tensor factorization for [12] integrated a first-order dynamical structure to characterize temporal dependencies in Bayesian Gaussian tensor factorization. ST-CRMF: Compensated Residual Matrix Factorization with Spatial-Temporal Regularization for Graph-Based Time Series Forecasting. Making predictions on these time series has become a critical deep learning framework for traffic forecasting, in, Y.Wang, A.Smola, D.Maddix, J.Gasthaus, D.Foster, and T.Januschowski, As for BTMF, it can be considered the Bayesian counterpart of Yu et al. 2 Paper Code Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks abr/neurips2019 NeurIPS 2019 However, emerging real-world applications, such as route planning and travel time estimation, are extremely sensitive to uncertainties and risks. Second, TRMF requires careful tuning of multiple regularization parameters. Z. For the experiment, we choose the trips collected during May and June 2018 (61 days) and organize the raw data into a third-order (pick-up zonedrop-off zonetime slot) tensor. Rep., 1996. The simple dynamical assumption imposes a smoothness constraint on the temporal factor matrix, thus the model can indeed better characterize the evolving dynamics of the data. A channel-wise attentive split-convolutional neural network (CAS-CNN) is proposed that contributes to the development of short-term OD flow prediction, and it also lays the foundations of real-time URT operation and management. where . Making predictions on these time series has become a critical challenge due to not only the large-scale and high-dimensional nature but also the considerable amount of missing data. tensors with automatic rank determination,, X.Chen, Z. Disclaimer, National Library of Medicine IEEE Trans Pattern Anal Mach Intell. For the prediction tasks, we first apply a short-term rolling prediction experiment (e.g., single-step rolling prediction) as described in Figure2 and Section4.3, then, we conduct a multi-step predict experiment where we evaluate the next-day prediction. values. Under the assumptions above, the full conditionals p(u,u|) and p(v,v|) will be of the same Gaussian-Wishart form as p(w,w|) described in BTMF. IEEE Transactions on Pattern Analysis and Machine Intelligence. Please enable it to take advantage of the complete set of features! Federal government websites often end in .gov or .mil. eCollection 2022 Mar 18. With regard to BTMF, the prior of temporal factor is built on the VAR process which has better performance in characterizing the covariance and causal structures. where 0R(Rd)(Rd) and RRR are played as covariance matrices. Zhao Q, Zhou G, Zhang L, Cichocki A, Amari S. IEEE Trans Neural Netw Learn Syst. IEEE Transactions on Pattern Analysis and Machine . Paciorek CJ, Liu Y; HEI Health Review Committee. However, in modeling the latent variables and temporal smoothness, these models have to introduce various regularization terms and parameters, which need to be tuned carefully to ensure model accuracy and avoid overfitting. To characterize the temporal dependencies, a VAR regularizer on X is introduced in TRMF [19]: where L={h1,,hk,,hd} is a lag set (d is the order of this AR model), each Ak (k{1,,d}) is a RR coefficient matrix, and t is a zero mean Gaussian noise vector. Chen et al. Z.Ghahramani and G.E. Hinton, Parameter estimation for linear dynamical probabilistic tensor factorization framework,, O.Anava, E.Hazan, and A.Zeevi, Online time series prediction with missing Here, the Gibbs algorithm to generate samples of xt is given by: Draw a new vector (sample) xtN(x,(x)1) with. Essentially, these models rely on using the AR coefficient matrix or the dynamics matrix to capture the correlation structure among different time series. Third, we would like to integrate recent advances in deep learning to better capture the complex and non-linear dynamics in modern time series data, Y.Li and C.Shahabi, A brief overview of machine learning methods for IEEE Trans Pattern Anal Mach Intell. Song, and C.Faloutsos, Tensorcast: forecasting Figure2 illustrates a one-step rolling prediction scheme based on this idea. In particular, we choose the Temporal Regularized Tensor Factorization (TRTF) as a baseline for both imputation and prediction tasks. Making predictions on these time series has become a critical challenge due to not only the large-scale and high-dimensional nature but also the considerable amount of missing data. A Parameter-Free Non-Convex Tensor Completion model (TC-PFNC) is proposed for trafc data recovery, in which a log-based relaxation term was designed to approximate tensor algebraic rank to recover the missing value from partial and corrupted observations and remove the anomalies in observations. (UAI), 2022 Deep AI, Inc. | San Francisco Bay Area | All rights reserved. Epub 2015 Jun 9. For RM, we simply remove a certain amount of observed entries in the matrix randomly and use these entries as ground truth to evaluate MAPE and RMSE. networks for geo-sensory time series prediction. in, S.S. Rangapuram, M.W. Seeger, J.Gasthaus, L.Stella, Y.Wang, and Block Hankel Tensor ARIMA for Multiple Short Time Series Forecasting, Skolkovo Institute of Science and Technology, Deep Markov Spatio-Temporal Factorization, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, Defence Science and Technology Organization, Tensor completion for estimating missing values in visual data, Bayesian probabilistic matrix factorization using Markov chain Monte Carlo, Temporal regularized matrix factorization for high-dimensional time series prediction, Fast Multivariate Spatio-temporal Analysis via Low Rank Tensor Learning, PPCA-Based Missing Data Imputation for Traffic Flow Volume: A Systematical Approach, IEEE Transactions on Intelligent Transportation Systems. This data set registers trip information (pick-up/drop-off locations and start time) for different types of taxi trips. On the one hand, these models in general require careful tuning of the regularization parameters to ensure model accuracy and to avoid overfitting. where , and {Mt,Nt,Pt,Qt} are defined in the same way as in BTMF (see (13)). The VAR process can be used directly for prediction tasks. For imputation tasks, we select the Temporal Collaborative Filtering (TCF) techniqueBayesian Probabilistic Tensor Factorization (BPTF)as a benchmark model [12]. and mining with coupled tensors,, F.Han, H.Lu, and H.Liu, A direct estimation of high dimensional stationary Second, modern time series generated by advanced sensing technologies are usually high-dimensional with different attributes. Below we summarize the Gibbs sampling procedure. Data set (H): Hangzhou metro passenger flow333https://tianchi.aliyun.com/competition/entrance/231708/information. For example, we can denote by matrix YRNT a multivariate time series collected from N locations/sensors on T time stamps, with each row, corresponding to the time series collected at location i. If we have enough computational power or if the Bayesian confidence interval of yt is of key consideration, we can easily design a fully Bayesian approach following Algorithm1 to estimate the posterior distribution of yt (instead of a point estimate) by updating all the parametersincluding W and Ain the Gibbs sampling algorithm. We examine two missing rates (10% and 30%) and use the last seven days (i.e., 168 time slots) as the prediction period. A possible reason is that TRMF uses both AR regularizer and F-norm penalty on temporal factors, while BayesTRMF only contains a prior on the AR process. Making predictions on these time series has become a critical challenge due to not only the large-scale and high-dimensional nature but also the considerable amount of missing data. (Early Access) Preprint: https://arxiv.org/abs/1910.06366 DOI: https://doi.org/10.1109/TPAMI.2021.3066551 Data & Python code: https://github.com/xinychen/transdim Our results suggest that BTMF (or BayesTRMF) inherits the advantages of both matrix models (e.g., TRMF and BPMF) and tensor models (e.g., BGCP, BATF, and TF-ALS): it not only provides a flexible and automatic inference technique for model parameter estimation, but also offers superior imputation performance by integrating temporal dynamics into matrix factorization. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series -- in particular spatiotemporal data -- in the presence of missing values. models for large-scale and multidimensional time series data. Baselines. View 13 excerpts, cites background and methods. t,/U f"TUVJtEqd }U*Q*Gxn$* >86HeOwv 4o(L_es-\"$=|'K[Q Bayesian temporal factorization for multidimensional time series prediction. filtering for dynamic matrix factorization,, C.Xie, A. Similar to the analyses on BTMF, we also design two missing data scenarios: random missing (RM) by randomly removing entries in the tensor and non-random missing (NM) by randomly selecting pick-updrop-offday combinations and for each of them removing the corresponding 24h block entirely. spatio-temporal predictions, in, B.Hooi, K.Shin, S.Liu, and C.Faloutsos, Smf: Drift-aware matrix Given the MNIW prior, the corresponding conditional distribution is. Section6 provides the results on extensive numerical experiments based on several real-world data sets, followed by the conclusion and discussion in Section7. Despite the computational cost, the tuning procedure has to be performed for each specific study/task/data set and there exist no universal solutions. Therefore, we have xt+1=Avt+1+t. Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. BTTF, on the other hand, can be considered an extension of the temporal collaborative filtering model by Xiong et al. Accessibility data, in, Z.Chen and A.Cichocki, Nonnegative matrix factorization with temporal For example, a Bayesian tensor factorization is proposed in [31], which can automatically determine the CP rank. Overall, these factorization approaches have shown superior performance in modeling real-world large-scale time series data in the presence of missing values; however, there are still several main drawbacks hindering the application of these models. Inspired by the recent studies on temporal regularization [19] and Bayesian factorization [12]. 4 Bayesian Temporal Matrix Factorization 4.1 Model Specification Given a partially observed matrix Y RNT in a spatiotemporal setting, one can factorize it into a spatial factor matrix W RRN and a temporal factor matrix XRRT following general matrix factorization model: and element-wise, we have We develop efficient Gibbs sampling algorithms for model inference and model updating for real-time prediction and test the proposed BTF framework on several real-world spatiotemporal data sets for both missing data imputation and multi-step rolling prediction tasks. Copyright @ 2022 | PubGenius Inc. | Suite # 217 691 S Milpitas Blvd Milpitas CA 95035, USA, Bayesian Temporal Factorization for Multidimensional Time Series Prediction, IEEE Transactions on Pattern Analysis and Machine Intelligence, Stacked Bidirectional and Unidirectional LSTM Recurrent Neural Network for Forecasting Network-wide Traffic State with Missing Values, A nonconvex low-rank tensor completion model for spatiotemporal traffic data imputation, Transportation Research Part C-emerging Technologies. In training the model, we first run the MCMC algorithm for m1 iterations as a burn-in period and then take samples from the following m2 iterations for estimation. We organize the raw data set into a time series matrix of 2148784 and there are 1.29% missing values. showing the number of trips from i to j over time. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series---in particular spatiotemporal data---in the presence of missing values. In a recent work, Yu et al. analysis via low rank tensor learning, in, W.Sun and D.Malioutov, Time series forecasting with shared seasonality Note that one can keep all the m2. in, M.Rogers, L.Li, and S.J. Russell, Multilinear dynamical systems for tensor With this idea, many recent studies have proposed to apply matrix factorization (collaborative filtering) to analyze large-scale time series by projecting the raw data into a much smaller latent space. Bayesian Low-Tubal-Rank Robust Tensor Factorization with Multi-Rank Determination. spatiotemporal traffic data imputation,, J.Liu, P.Musialski, P.Wonka, and J.Ye, Tensor completion for estimating Res Rep Health Eff Inst. The https:// ensures that you are connecting to the For matrix based models, we use the original time series matrix (locationtime series) as input. In Section 4, we present the Bayesian Temporal Matrix Factorization (BTMF) model for matrix time series data and develop an efcient For all of the competing models, their experiments are worked on the third-order tensor that comprised of pick-up zone, drop-off zone, and time slot. Here we use (i,t) and (i,j,t) to index the observed entries in matrix Y and tensor Y, respectively. short-term traffic forecasting and future directions,, Z.Che, S.Purushotham, K.Cho, D.Sontag, and Y.Liu, Recurrent neural In particular, the data is completely missing on four days (October 20/21 and December 6/7). The parameter tuning procedure is computationally very expensive and the cost increases exponentially with the number of regularization parameters as they have to be tuned simultaneously. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. Making predictions on these time series has become a . government site. Essentially, the matrix-based BPMF performs the worst as the local temporal consistency is ignored. using markov chain monte carlo, in, Q.Zhao, L.Zhang, and A.Cichocki, Bayesian CP factorization of incomplete The model may end up with overfitting if these regularization parameters are not tuned correctly. [19] proposed a Temporal Regularized Matrix Factorization (TRMF) framework to model multivariate time series with missing data by introducing a novel AR regularization scheme on the temporal factor matrix. factorization, in, S.Karlsson, Forecasting with bayesian vector autoregression, in, S.Gultekin and J.Paisley, Online forecasting matrix factorization,, T.G. Kolda and B.W. Bader, Tensor decompositions and applications,, X.Chen, Z. Here we rely on the MCMC technique for Bayesian learning. decomposition for learning the structure of international relations, in, R.Chen, D.Yang, and C.-h. Zhang, Factor models for high-dimensional tensor Essentially, these matrix/tensor factorization-based algorithms are scalable to model large-scale spatiotemporal data. IEEE Transactions on Pattern Analysis and Machine Intelligence, Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. smoothness and/or spatial decorrelation constraints,, L.Xiong, X.Chen, T.-K. Huang, J.Schneider, and J.G. Carbonell, Temporal <>stream As shown in Figure5, our proposed BTMF achieves accurate time series prediction results on the Hangzhou metro passenger flow data set, and such accurate results can be guaranteed even with a large part of the input sequence is missing (for instance, see (c), (d), (f), and (h) of Figure5). endobj imputation and response forecasting, LIFE: Learning Individual Features for Multivariate Time Series In addition to uncovering latent temporal patterns (e.g., seasonality and trend) in multivariate time series data, these factorization models also serve as a powerful tool for collaborative filtering, thus offering a natural solution to deal with the missing data problem. vector autoregressions,. Making predictions on these time series has become a critical challenge due to not only the large-scale and high-dimensional nature but also the considerable amount of missing data. The above specifies the full generative process of BTMF. - Xinyu Chen, Lijun Sun (2021). This work provides a unified framework for producing long-range forecasts even when the series has missing values or was previously unobserved; the same framework can be used to impute missing values. To model multidimensional data, we employ the popular CANDECOMP/PARAFAC (CP) decomposition [36], which approximates Y by the sum of R rank-one tensors: where urRM, vrRN, and xrRT are the r-th column of factor matrices URMR, VRNR, and XRTR, respectively (see Figure4). For BTMF, BayesTRMF and TRMF, we use a small lag set L={1,2,T0} for all data sets, where T0 denotes the number of time intervals per day. forecasting: A survey,, C.Chen, K.Petty, A.Skabardonis, P.Varaiya, and Z.Jia, Freeway We discard the interval 0:00 a.m. 6:00 a.m. with no services (i.e., only consider the remaining 108 time intervals) and re-organize the raw data set into a time series matrix of 802700. Note that yt may contain missing values. To address the second limitation, we propose the Bayesian Temporal Matrix Factorization (BTMF) model and extend it to multidimensional tensor time series. This paper develops novel regularization schemes and uses scalable matrix factorization methods that are eminently suited for high-dimensional time series data that has many missing values, and makes interesting connections to graph regularization methods in the context of learning the dependencies in an autoregressive framework. Data set (S): Seattle freeway traffic speed444https://github.com/zhiyongc/Seattle-Loop-Data. For example, the highway traffic Performance Measurement System (PeMS) in California consists of more than 35,000 detectors, and it has been registering flow and speed information every 30 seconds since 1999 [4]. We introduce deep switching auto-regressive factorization (DSARF), a deep generative model for spatio-temporal data with the capability to unravel recurring patterns in the data and perform robust, View 13 excerpts, cites methods and background. While modeling large-scale time series is extremely challenging, it is also important to note that spatiotemporal data often exhibit high correlations and shared latent patterns (e.g., traffic state time series with repeated and reproducible temporal peaks). Assessment and statistical modeling of the relationship between remotely sensed aerosol optical depth and PM2.5 in the eastern United States. The comparison between BTMF and BayesTRMF clearly shows the the limitation of independent factor assumption in (4) and benefits of integrating VAR dynamics in (3). Most previous work on Bayesian temporal factorization essentially impose first-order Markovian/state-space assumptions on the temporal latent factor [12, 33], . HW[s~L3&smNv&t These model may work well in temporal smoothing and pattern recognition, but the simple assumption limits its capacity in capturing complex time series dynamics and they cannot be applied directly for prediction tasks. Despite the superior performance demonstrated by these models, the large number of parameters (in coefficient matrices) and the high computational cost make these models very difficult to estimate and prone to overfitting for large-scale problems. Baselines. networks for multivariate time series with missing values,, Uncertainty in Artificial Intelligence the considerable amount of missing data. Li J, Wu P, Li R, Pian Y, Huang Z, Xu L, Li X. An official website of the United States government. The percentages of missing values are set as 20% and 40% for data set (G), (H), and (S), and 10% and 30% for (B), respectively. In addition to sensing data, multidimensional time series is also ubiquitous in social science domains such as international relations [6], dynamic import-export networks and social networks [7], and it is particularly important in modeling traffic/transportation systems with both origin and destination attributes. The full Bayesian treatment offers additional flexibility in terms of parameter tuning and avoids overfitting issues. Large-scale and multidimensional spatiotemporal data sets are becoming ubiquitous in many real-world applications such as monitoring urban traffic and air quality. In this paper, we propose a Bayesian temporal factorization (BTF) framework for modeling multidimensional time series---in particular spatiotemporal data---in the . the considerable amount of missing data. Figure7 shows examples of spatial volume at two time intervals. First, although the independent factor assumption in (4) greatly reduces the number of parameters, the complex temporal dynamics, causal relationships and covariance structure are essentially overlooked. In developing these factorization-based models, a central challenge is to design appropriate regularization terms to model temporal dynamics and smoothness, with the goal to both achieve high accuracy and avoid overfitting. Neural Netw Learn Syst BTTF can be written as p ( a, | ) is also of same! The dimension/number of attributes collected from the underlying system, this model can be applied on multidimensional. ) is also of the same form as ( 11 ) in.. Bttf can be written as p ( a, | ) is also of the set. Learn Syst Poisson matrix factorization for recommendation locations and start time ) for both imputation and short-term/long-term rolling prediction temporal... ( a, | ) is also of the complete set of!! 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