17708138
High-dimensional Time Series Prediction with Missing Values
High-dimensional time series prediction is needed in applications as diverse as demand forecasting and climatology. Often, such applications require methods that are both highly scalable, and deal with noisy data in terms of corruptions or missing values. Classical time series methods usually fall short of handling both these issues. In this paper, we propose to adapt matrix matrix completion approaches that have previously been successfully applied to large scale noisy data, but which fail to adequately model high-dimensional time series due to temporal dependencies. We present a novel temporal regularized matrix factorization (TRMF) framework which supports data-driven temporal dependency learning and enables forecasting ability to our new matrix factorization approach. TRMF is highly general, and subsumes many existing matrix factorization approaches for time series data. We make interesting connections to graph regularized matrix factorization methods in the context of learning the dependencies. Experiments on both real and synthetic data show that TRMF outperforms several existing approaches for common time series tasks.
121876157
Bayesian forecasting and dynamic models
to the DLM: The First-Order Polynomial Model.- to the DLM: The Dynamic Regression Model.- The Dynamic Linear Model.- Univariate Time Series DLM Theory.- Model Specification and Design.- Polynomial Trend Models.- Seasonal Models.- Regression, Autoregression, and Related Models.- Illustrations and Extensions of Standard DLMs.- Intervention and Monitoring.- Multi-Process Models.- Non-Linear Dynamic Models: Analytic and Numerical Approximations.- Exponential Family Dynamic Models.- Simulation-Based Methods in Dynamic Models.- Multivariate Modelling and Forecasting.- Distribution Theory and Linear Algebra.
14448486
Online Time Series Prediction with Missing Data
We consider the problem of time series prediction in the presence of missing data. We cast the problem as an online learning problem in which the goal of the learner is to minimize prediction error. We then devise an efficient algorithm for the problem, which is based on autoregressive model, and does not assume any structure on the missing data nor on the mechanism that generates the time series. We show that our algorithm's performance asymptotically approaches the performance of the best AR predictor in hindsight, and corroborate the theoretic results with an empirical study on synthetic and real-world data.
117555136
Dynamic Linear Models with R
Introduction: basic notions about Bayesian inference.- Dynamic linear models.- Model specification.- Models with unknown parameters.- Sequential Monte Carlo methods.
62731894
An R Package for Dynamic Linear Models
We describe an R package focused on Bayesian analysis of dynamic linear models. The main features of the package are its flexibility to deal with a variety of constant or time-varying, univariate or multivariate models, and the numerically stable singular value decomposition-based algorithms used for filtering and smoothing. In addition to the examples of "out-of-the-box" use, we illustrate how the package can be used in advanced applications to implement a Gibbs sampler for a user-specified model.
15320727
Parallel matrix factorization for recommender systems
Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two popular approaches to compute matrix factorization, and there has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been thoroughly explored. In this paper, we show that coordinate descent-based methods have a more efficient update rule compared to ALS and have faster and more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updates rank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, with a synthetic dataset containing 14.6 billion ratings, on a distributed memory cluster with 64 processors, to deliver the desired test RMSE, CCD++ is 49 times faster than SGD and 20 times faster than ALS. When the number of processors is increased to 256, CCD++ takes only 16 s and is still 40 times faster than SGD and 20 times faster than ALS.
7012164
Large-scale Multi-label Learning with Missing Labels
The multi-label classification problem has generated significant interest in recent years. However, existing approaches do not adequately address two key challenges: (a) scaling up to problems with a large number (say millions) of labels, and (b) handling data with missing labels. In this paper, we directly address both these problems by studying the multi-label problem in a generic empirical risk minimization (ERM) framework. Our framework, despite being simple, is surprisingly able to encompass several recent label-compression based methods which can be derived as special cases of our method. To optimize the ERM problem, we develop techniques that exploit the structure of specific loss functions - such as the squared loss function - to obtain efficient algorithms. We further show that our learning framework admits excess risk bounds even in the presence of missing labels. Our bounds are tight and demonstrate better generalization performance for low-rank promoting trace-norm regularization when compared to (rank insensitive) Frobenius norm regularization. Finally, we present extensive empirical results on a variety of benchmark datasets and show that our methods perform significantly better than existing label compression based methods and can scale up to very large datasets such as a Wikipedia dataset that has more than 200,000 labels.
1469055
Exploiting temporal stability and low-rank structure for localization in mobile networks
Localization is a fundamental operation for many wireless networks. While GPS is widely used for location determination, it is unavailable in many environments either due to its high cost or the lack of line of sight to the satellites (e.g., indoors, under the ground, or in a downtown canyon). The limitations of GPS have motivated researchers to develop many localization schemes to infer locations based on measured wireless signals. However, most of these existing schemes focus on localization in static wireless networks. As many wireless networks are mobile (e.g., mobile sensor networks, disaster recovery networks, and vehicular networks), we focus on localization in mobile networks in this paper. We analyze real mobility traces and find that they exhibit temporal stability and low-rank structure. Motivated by this observation, we develop three novel localization schemes to accurately determine locations in mobile networks: (i) Low Rank based Localization (LRL), which exploits the low-rank structure in mobility, (ii) Temporal Stability based Localization (TSL), which leverages the temporal stability, and (iii) Temporal Stability and Low Rank based Localization (TSLRL), which incorporates both the temporal stability and the low-rank structure. These localization schemes are general and can leverage either mere connectivity (i.e., range-free localization) or distance estimation between neighbors (i.e., range-based localization). Using extensive simulations and testbed experiments, we show that our new schemes significantly outperform state-of-the-art localization schemes under a wide range of scenarios and are robust to measurement errors.
1502954
Spatio-Temporal Compressive Sensing and Internet Traffic Matrices (Extended Version)
Despite advances in measurement technology, it is still challenging to reliably compile large-scale network datasets. For example, because of flaws in the measurement systems or difficulties posed by the measurement problem itself, missing, ambiguous, or indirect data are common. In the case where such data have spatio-temporal structure, it is natural to try to leverage this structure to deal with the challenges posed by the problematic nature of the data. Our work involving network datasets draws on ideas from the area of compressive sensing and matrix completion, where sparsity is exploited in estimating quantities of interest. However, the standard results on compressive sensing are: 1) reliant on conditions that generally do not hold for network datasets; and 2) do not allow us to exploit all we know about their spatio-temporal structure. In this paper, we overcome these limitations with an algorithm that has at its heart the same ideas espoused in compressive sensing, but adapted to the problem of network datasets. We show how this algorithm can be used in a variety of ways, in particular on traffic data, to solve problems such as simple interpolation of missing values, traffic matrix inference from link data, prediction, and anomaly detection. The elegance of the approach lies in the fact that it unifies all of these tasks and allows them to be performed even when as much as 98% of the data is missing.
1173916
Temporal Collaborative Filtering with Bayesian Probabilistic Tensor Factorization
Real-world relational data are seldom stationary, yet traditional collaborative filtering algorithms generally rely on this assumption. Motivated by our sales prediction problem, we propose a factor-based algorithm that is able to take time into account. By introducing additional factors for time, we formalize this problem as a tensor factorization with a special constraint on the time dimension. Further, we provide a fully Bayesian treatment to avoid tuning parameters and achieve automatic model complexity control. To learn the model we develop an efficient sampling procedure that is capable of analyzing large-scale data sets. This new algorithm, called Bayesian Probabilistic Tensor Factorization (BPTF), is evaluated on several real-world problems including sales prediction and movie recommendation. Empirical results demonstrate the superiority of our temporal model.
3411705
Spatio-temporal compressive sensing and internet traffic matrices
Many basic network engineering tasks (e.g., traffic engineering, capacity planning, anomaly detection) rely heavily on the availability and accuracy of traffic matrices. However, in practice it is challenging to reliably measure traffic matrices. Missing values are common. This observation brings us into the realm of compressive sensing, a generic technique for dealing with missing values that exploits the presence of structure and redundancy in many real-world systems. Despite much recent progress made in compressive sensing, existing compressive-sensing solutions often perform poorly for traffic matrix interpolation, because real traffic matrices rarely satisfy the technical conditions required for these solutions.
 To address this problem, we develop a novel spatio-temporal compressive sensing framework with two key components: (i) a new technique called Sparsity Regularized Matrix Factorization (SRMF) that leverages the sparse or low-rank nature of real-world traffic matrices and their spatio-temporal properties, and (ii) a mechanism for combining low-rank approximations with local interpolation procedures. We illustrate our new framework and demonstrate its superior performance in problems involving interpolation with real traffic matrices where we can successfully replace up to 98% of the values. Evaluation in applications such as network tomography, traffic prediction, and anomaly detection confirms the flexibility and effectiveness of our approach.
7326173
Kernels and Regularization on Graphs
We introduce a family of kernels on graphs based on the notion of regularization operators. This generalizes in a natural way the notion of regularization and Greens functions, as commonly used for real valued functions, to graphs. It turns out that diffusion kernels can be found as a special case of our reasoning. We show that the class of positive, monotonically decreasing functions on the unit interval leads to kernels and corresponding regularization operators.
13154909
DynaMMo: mining and summarization of coevolving sequences with missing values
Given multiple time sequences with missing values, we propose DynaMMo which summarizes, compresses, and finds latent variables. The idea is to discover hidden variables and learn their dynamics, making our algorithm able to function even when there are missing values.
 We performed experiments on both real and synthetic datasets spanning several megabytes, including motion capture sequences and chlorine levels in drinking water. We show that our proposed DynaMMo method (a) can successfully learn the latent variables and their evolution; (b) can provide high compression for little loss of reconstruction accuracy; (c) can extract compact but powerful features for segmentation, interpretation, and forecasting; (d) has complexity linear on the duration of sequences.
8997605
Time Series Clustering: Complex is Simpler!
Given a motion capture sequence, how to identify the category of the motion? Classifying human motions is a critical task in motion editing and synthesizing, for which manual labeling is clearly inefficient for large databases. Here we study the general problem of time series clustering. We propose a novel method of clustering time series that can (a) learn joint temporal dynamics in the data; (b) handle time lags; and (c) produce interpretable features. We achieve this by developing complex-valued linear dynamical systems (CLDS), which include real-valued Kalman filters as a special case; our advantage is that the transition matrix is simpler (just diagonal), and the transmission one easier to interpret. We then present Complex-Fit, a novel EM algorithm to learn the parameters for the general model and its special case for clustering. Our approach produces significant improvement in clustering quality, 1.5 to 5 times better than well-known competitors on real motion capture sequences.
6050694
Collaborative Filtering with Graph Information: Consistency and Scalable Methods
Low rank matrix completion plays a fundamental role in collaborative filtering applications, the key idea being that the variables lie in a smaller subspace than the ambient space. Often, additional information about the variables is known, and it is reasonable to assume that incorporating this information will lead to better predictions. We tackle the problem of matrix completion when pairwise relationships among variables are known, via a graph. We formulate and derive a highly efficient, conjugate gradient based alternating minimization scheme that solves optimizations with over 55 million observations up to 2 orders of magnitude faster than state-of-the-art (stochastic) gradient-descent based methods. On the theoretical front, we show that such methods generalize weighted nuclear norm formulations, and derive statistical consistency guarantees. We validate our results on both real and synthetic datasets.
2002298
Transition Matrix Estimation in High Dimensional Time Series
In this paper, we propose a new method in estimating transition matrices of high dimensional vector autoregressive (VAR) models. Here the data are assumed to come from a stationary Gaussian VAR time series. By formulating the problem as a linear program, we provide a new approach to conduct inference on such models. In theory, under a doubly asymptotic framework in which both the sample size T and dimensionality d of the time series can increase (with possibly d ≫ T), we provide explicit rates of convergence between the estimator and the population transition matrix under different matrix norms. Our results show that the spectral norm of the transition matrix plays a pivotal role in determining the final rates of convergence. This is the first work analyzing the estimation of transition matrices under a high dimensional doubly asymptotic framework. Experiments are conducted on both synthetic and real-world stock data to demonstrate the effectiveness of the proposed method compared with the existing methods. The results of this paper have broad impact on different applications, including finance, genomics, and brain imaging.
38469068
Structured Regularization for Large Vector Autoregression
The vector autoregression (VAR), has long proven to be an effective method for modeling the joint dynamics of macroeconomic time series as well as forecasting. One of the major disadvantages of the VAR that has hindered its applicability is its heavy parameterization; the parameter space grows quadratically with the number of series included, quickly exhausting the available degrees of freedom. Consequently, forecasting using VARs is intractable for low-frequency, high dimensional macroeconomic data. However, empirical evidence suggests that VARs which incorporate more component series tend to result in more accurate forecasts than their smaller counterparts. Existing methods which allow for the estimation of large VARs either tend to require ad-hoc specifications or are computationally intractable. We adapt several prominent scalar regression regularization techniques to a vector time series context to greatly reduce the parameter space of VARs. We formulate convex optimization procedures that are amenable to efficient solutions for the time ordered high-dimensional problems we aim to solve. Through this framework, we propose a structured family of models and provide implementations which allow for both the efficient estimation and accurate forecasting of high-dimensional VARs. We demonstrate their efficacy in simulated data examples as well as an application to a large set of macroeconomic indicators. ∗PhD Student, Department of Statistical Science, Cornell University, 301 Malott Hall, Ithaca, NY 14853 (E-mail: wbn8@cornell.edu; Webpage: http://www.wbnicholson.com) †Assistant Professor, Department of Statistical Science and Department of Social Statistics, Cornell University, 1196 Comstock Hall, Ithaca, NY 14853, (E-mail: matteson@cornell.edu; Webpage: https://courses.cit.cornell. edu/~dm484/) ‡Assistant Professor, Department of Biological Statistics and Computational Biology and Department of Statistical Science, Cornell University, 1178 Comstock Hall, Ithaca, NY 14853 (E-mail: jbien@cornell.edu; Webpage: http://faculty.bscb.cornell.edu/~bien/)
12534912
Parameter estimation for linear dynamical systems
Linear systems have been used extensively in engineering to model and control the behavior of dynamical systems. In this note, we present the Expectation Maximization (EM) algorithm for estimating the parameters of linear systems (Shumway and Sto er, 1982). We also point out the relationship between linear dynamical systems, factor analysis, and hidden Markov models. Introduction The goal of this note is to introduce the EM algorithm for estimating the parameters of linear dynamical systems (LDS). Such linear systems can be used both for supervised and unsupervised modeling of time series. We rst describe the model and then brie y point out its relation to factor analysis and other data modeling techniques. The Model Linear time-invariant dynamical systems, also known as linear Gaussian state-space models, can be described by the following two equations: xt+1 = Axt +wt (1) yt = Cxt+ vt: (2) Time is indexed by the discrete index t. The output yt is a linear function of the state, xt, and the state at one time step depends linearly on the previous state. Both state and output noise, wt and vt, are zero-mean normally distributed random variables with covariance matrices Q and R, respectively. Only the output of the system is observed, the state and all the noise variables are hidden. Rather than regarding the state as a deterministic value corrupted by random noise, we combine the state variable and the state noise variable into a single Gaussian random
2070579
Dynamic matrix factorization: A state space approach
Matrix factorization from a small number of observed entries has recently garnered much attention as the key ingredient of successful recommendation systems. One unresolved problem in this area is how to adapt current methods to handle changing user preferences over time. Recent proposals to address this issue are heuristic in nature and do not fully exploit the time-dependent structure of the problem. As a principled and general temporal formulation, we propose a dynamical state space model of matrix factorization. Our proposal builds upon probabilistic matrix factorization, a Bayesian model with Gaussian priors. We utilize results in state tracking, i.e. the Kalman filter, to provide accurate recommendations in the presence of both process and measurement noise. We show how system parameters can be learned via expectation-maximization and provide comparisons to current published techniques.
12373442
Provable Inductive Matrix Completion
Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimation using rank-1 measurements. We study this generic problem and provide conditions that the set of measurements should satisfy so that the alternating minimization method (which otherwise is a non-convex method with no convergence guarantees) is able to recover back the {\em exact} underlying low-rank matrix. 
In addition to inductive matrix completion, we show that two other low-rank estimation problems can be studied in our framework: a) general low-rank matrix sensing using rank-1 measurements, and b) multi-label regression with missing labels. For both the problems, we provide novel and interesting bounds on the number of measurements required by alternating minimization to provably converges to the {\em exact} low-rank matrix. In particular, our analysis for the general low rank matrix sensing problem significantly improves the required storage and computational cost than that required by the RIP-based matrix sensing methods \cite{RechtFP2007}. Finally, we provide empirical validation of our approach and demonstrate that alternating minimization is able to recover the true matrix for the above mentioned problems using a small number of measurements.
8999678
Speedup Matrix Completion with Side Information: Application to Multi-Label Learning
In standard matrix completion theory, it is required to have at least O(n ln2 n) observed entries to perfectly recover a low-rank matrix M of size n x n, leading to a large number of observations when n is large. In many real tasks, side information in addition to the observed entries is often available. In this work, we develop a novel theory of matrix completion that explicitly explore the side information to reduce the requirement on the number of observed entries. We show that, under appropriate conditions, with the assistance of side information matrices, the number of observed entries needed for a perfect recovery of matrix M can be dramatically reduced to O(ln n). We demonstrate the effectiveness of the proposed approach for matrix completion in transductive incomplete multi-label learning.
706026
Kernelized Probabilistic Matrix Factorization: Exploiting Graphs and Side Information
We propose a new matrix completion algorithm— Kernelized Probabilistic Matrix Factorization (KPMF), which effectively incorporates external side information into the matrix factorization process. Unlike Probabilistic Matrix Factorization (PMF) [14], which assumes an independent latent vector for each row (and each column) with Gaussian priors, KMPF works with latent vectors spanning all rows (and columns) with Gaussian Process (GP) priors. Hence, KPMF explicitly captures the underlying (nonlinear) covariance structures across rows and columns. This crucial difference greatly boosts the performance of KPMF when appropriate side information, e.g., users’ social network in recommender systems, is incorporated. Furthermore, GP priors allow the KPMF model to fill in a row that is entirely missing in the original matrix based on the side information alone, which is not feasible for standard PMF formulation. In our paper, we mainly work on the matrix completion problem with a graph among the rows and/or columns as side information, but the proposed framework can be easily used with other types of side information as well. Finally, we demonstrate the efficacy of KPMF through two different applications: 1) recommender systems and 2) image restoration.
36795331
Regression coefficient and autoregressive order shrinkage and selection via the lasso
Summary. The least absolute shrinkage and selection operator (‘lasso’) has been widely used in regression shrinkage and selection. We extend its application to the regression model with autoregressive errors. Two types of lasso estimators are carefully studied. The first is similar to the traditional lasso estimator with only two tuning parameters (one for regression coefficients and the other for autoregression coefficients). These tuning parameters can be easily calculated via a data‐driven method, but the resulting lasso estimator may not be fully efficient. To overcome this limitation, we propose a second lasso estimator which uses different tuning parameters for each coefficient. We show that this modified lasso can produce the estimator as efficiently as the oracle. Moreover, we propose an algorithm for tuning parameter estimates to obtain the modified lasso estimator. Simulation studies demonstrate that the modified estimator is superior to the traditional estimator. One empirical example is also presented to illustrate the usefulness of lasso estimators. The extension of the lasso to the autoregression with exogenous variables model is briefly discussed.