This paper considers the problem of learning, from samples, the dependency structure of a system of linear stochastic differential equations, when some of the variables are latent. In particular, we observe the time evolution of some variables, and never observe other variables; from this, we would like to find the dependency structure between the observed variables - separating out the spurious interactions caused by the (marginalizing out of the) latent variables' time series. We develop a new method, based on convex optimization, to do so in the case when the number of latent variables is smaller than the number of observed ones. For the case when the dependency structure between the observed variables is sparse, we theoretically establish a high-dimensional scaling result for structure recovery. We verify our theoretical result with both synthetic and real data (from the stock market).
Time-series Forecasting: Motivated by finance applications, time-series forecasting has got a lot of attention during the past three decades @cite_4 . In the model based approaches, it is assumed that the time-series evolves according to some statistical model such as linear regression model @cite_44 , transfer function model @cite_15 , vector autoregressive model @cite_22 , etc. In each case, researchers have developed different methods to learn the parameters of the model for the purpose of forecasting. In this paper, we focus on linear stochastic dynamical systems that are an instance of vector autoregressive models. Previous work toward estimating this model parameters include ad-hoc use of neural network @cite_38 or support vector machine method @cite_19 , all without providing theoretical guarantees on the performance of the algorithm. Our work is different from these results because although our method provides better prediction comparing to similar algorithm, our main focus is sparse model selection not prediction. Perhaps, once a sparse model is selected, one can study the prediction quality as a separate subject.
@cite_38 @cite_4 @cite_22 @cite_44 @cite_19 @cite_15
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1. Overview. 2. Fundamental Concepts. 3. Stationary Time Series Models. 4. Non-Stationary Time Series Models. 5. Forecasting. 6. Model Identification. 7. Parameter Estimation, Diagnostic Checking, and Model Selection. 8. Seasonal Time Series Models. 9. Intervention Analysis and Outlier Detection. 10. Fourier Analysis. 11. Spectral Theory of Stationary Processes. 12. Estimation of the Spectrum. 13. Transfer Function Models. 14. Vector Time Series Models. 15. State Space Models and the Kalman Filter. 16. Aggregation and Systematic Sampling in Time Series. 17. References. 18. Appendix. An introduction to forecasting Basic statistical concepts Forecasting by using regression analysis Simple linear regression Multiple regression Topics in regression analysis Forecasting by using time series regression, decomposition methods and exponential smoothing Time series regression Decomposition methods Exponential smoothing Forecasting by using basic techniques of the box Jenkins methodology Nonseasonal, box-Jenkins models and their tentative identification Estimation, diagnostic checking and forecasting for nonseasonal box-Jenkins models An introduction to box-Jenkins seasonal modelling Forecasting by using advanced technology of the box-Jenkins methodology General box-Jenkins seasonal modelling Using the box-Jenkins methodology to improve time series regression models and to implement exponential smoothing Transfer functions and intervention models. Abstract Support vector machines (SVMs) are promising methods for the prediction of financial time-series because they use a risk function consisting of the empirical error and a regularized term which is derived from the structural risk minimization principle. This study applies SVM to predicting the stock price index. In addition, this study examines the feasibility of applying SVM in financial forecasting by comparing it with back-propagation neural networks and case-based reasoning. The experimental results show that SVM provides a promising alternative to stock market prediction. From the Publisher: This is a complete revision of a classic, seminal, and authoritative book that has been the model for most books on the topic written since 1970. It focuses on practical techniques throughout, rather than a rigorous mathematical treatment of the subject. It explores the building of stochastic (statistical) models for time series and their use in important areas of application forecasting, model specification, estimation, and checking, transfer function modeling of dynamic relationships, modeling the effects of intervention events, and process control. Features sections on: recently developed methods for model specification, such as canonical correlation analysis and the use of model selection criteria; results on testing for unit root nonstationarity in ARIMA processes; the state space representation of ARMA models and its use for likelihood estimation and forecasting; score test for model checking; and deterministic components and structural components in time series models and their estimation based on regression-time series model methods.