Forecasting Related Time Series

Singular spectrum analysis is a powerful non-parametric time series method that unfolds an observed time series into a special structured matrix, called a trajectory matrix. The trajectory matrix has a Hankel structure, whereby off-diagonal elements are non-unique. Singular value decomposition is then applied to the trajectory matrix to extract time series structures. The method can handle complex time series structures that include combinations of polynomials, sinusoids and exponentials. Time series structures that SSA can handle are very typical of seasonal series researched by economists and econometricians alike. The possible presence of multiple additive time series outliers can adversely affect time series forecasting and construction of bootstrap results. An additive time series outlier typically involves a single time series observation which either resulted due to some recording error or temporal shock (upwards/downwards) to the time series at a specific time. When more than a single additive outlier is present in a time series, we are faced with multiple additive outliers. An algorithm is proposed in this paper whereby Robust Principal Component Analysis (ROBPCA) methods are applied to the trajectory matrix, in order to identify multiple additive outliers and also estimate their size and appropriate imputations, based on the underlying time series structure. The procedure is iterative in the sense that, following identification of a single additive outlier, a signal processing procedure (Cadzow signal reconstruction) is used to impute identified outliers at each stage of the iterations. Monte Carlo simulations are used to illustrate the effectiveness of the iterative approach to handle identification of multiple additive time series outliers. The practical usefulness of the method is also illustrated by application to a real-life time series containing multiple additive outliers in a South African tourism time series.

A data mining framework for time series estimation

The presence of outliers in time series may have serious implications in the estimation of parameters or testing of hypothesis. The analysis of time series in the presence of outlier is not much explored using Bayesian framework. The present paper deals with the Bayesian analysis of an autoregressive model involving linear time trend and contaminated by additive outlier. The issue of unit root hypothesis is dealt in Bayesian set-up and posterior odds ratio for the unit root hypothesis has been derived under appropriate prior assumptions. The numerical illustration is carried out to observe the impact of presence of an additive outlier on posterior odds ratio for unit root hypothesis.

Based on the time series analysis method, this article develops a Bayesian method of detecting and repairing the cycle slips in the GNSS carrier-phase data. Firstly, this article analyses the characteristics of the cycle slips in the GNSS carrier-phase observations and establishes the relationships between the cycle slips and the additive outliers (AOs) in the stationary time series. When the ARMA (autoregressive moving-average) model is used to fit the stationary time series obtained by differencing the GNSS carrier-phase observations, the detection of cycle slips in the GNSS carrier-phase observations can be transformed to the detection of AOs in the ARMA model. Then, this article proposes a Bayesian method of detecting the AOs in the ARMA model, and the implementation of detecting the cycle slips in the GNSS carrier-phase observations is also developed. Finally, the new Bayesian method of detecting the cycle slips is used to the real GNSS carrier-phase data. From the comparison among the Bayesian method, the high-order differences method and ionospheric residual method, we can find that the Bayesian method has a better detection efficiency for several kinds of cycle slips in the GNSS carrier-phase observations than other methods.

This study investigated the impact of three prior distributions: matched, standard vague, and hierarchical in Bayesian estimation parameter recovery in two and one parameter models. Two Bayesian estimation methods were utilized: Markov chain Monte Carlo (MCMC) and the relatively new, Variational Bayesian (VB). Conditional (CML) and Marginal Maximum Likelihood (MML) estimates were used as baseline methods for comparison. Vague priors produced large errors or convergence issues and are not recommended. For both MCMC and VB, the hierarchical and matched priors showed the lowest root mean squared errors (RMSEs) for ability estimates; RMSEs of difficulty estimates were similar across estimation methods. For the standard errors (SEs), MCMC-hierarchical displayed the largest values across most conditions. SEs from the VB estimation were among the lowest in all but one case. Overall, VB-hierarchical, VB-matched, and MCMC-matched performed best. VB with hierarchical priors are recommended in terms of their accuracy, and cost and (subsequently) time effectiveness.

To obtain higher range resolution without incurring significant hardware costs, this paper proposes a novel method for high-resolution sparse subband imaging based on Bayesian learning. The signal model is derived and a probabilistic model is constructed. In particular, hierarchical sparse-promoting priors are imposed on the distribution of scattering centers, which is conjugate to the likelihood function. Then, a closed-form solution is derived based on the MAP-expectation–maximization framework. A multilevel dictionary which automatically adjusts the distance between adjacent atoms is adopted to achieve refined estimation with moderate computational burden. Finally, a coherent processing method is addressed. Experimental results have demonstrated the effectiveness of the proposed method in low signal-to-noise ratio and complex target scenarios.

Many applications naturally involve time series data, and the vector auto regression (VAR) and the structural VAR (SVAR) are dominant tools to investigate relations between variables in time series. In the first part of this work, we show that the SVAR method is incapable of identifying contemporaneous causal relations when data follow Gaussian distributions. In addition, least squares estimators become unreliable when the scales of the problems are large and observations are limited. In the remaining part, we propose an approach to apply Bayesian network learning algorithms to identify SVARs from time series data in order to capture both temporal and contemporaneous causal relations and avoid high-order statistical tests. The difficulty of applying Bayesian network learning algorithms to time series is that the sizes of the networks corresponding to time series tend to be large and high-order statistical tests are required by Bayesian network learning algorithms in this case. To overcome the difficulty, we show that the search space of conditioning sets d-separating two vertices should be subsets of Markov blankets. Based on this fact, we propose an algorithm learning Bayesian networks locally and making the largest order of statistical tests independent of the scales of the problems. Empirical results show that our algorithm outperforms existing methods in terms of both efficiency and accuracy.

Bayesian inference using hierarchical and spatial priors for intravoxel incoherent motion MR imaging in the brain: Analysis of cancer and acute stroke.

Change point models using hierarchical priors share in the information of each regime when estimating the parameter values of a regime.Because of this sharing, hierarchical priors have been very successful when estimating the parameter values of short-lived regimes and predicting the out-of-sample behavior of the regime parameters.However, the hierarchical priors have been parametric.Their parametric nature leads to global shrinkage that biases the estimates of the parameter coefficient of extraordinary regimes toward the value of the average regime.To overcome this shrinkage, we model the hierarchical prior nonparametrically by letting the hyperparameter's prior-in other words, the hyperprior-be unknown and modeling it with a Dirichlet processes prior.To apply a nonparametric hierarchical prior to the probability of a break occurring, we extend the change point model to a multiple-change-point panel model.The hierarchical prior then shares in the cross-sectional information of the break processes to estimate the transition probabilities.We apply our multiple-change-point panel model to a longitudinal data set of actively managed, U.S. equity, mutual fund returns to measure fund performance and investigate the chances of a skilled fund being skilled in the future.

We present an object matching system where Bayesian inference is used for estimating the probability distribution of object feature locations in a digital image. The representation of the object contains two parts: the likelihood part which defines the probability of perceiving a pixel image corresponding to the object detail, and a hierarchical prior part which defines the distributions of the feature locations relative to the other features. In this study the objects are human faces. The likelihood part is based on Gabor filters, which are a type of optimal bandpass digital filters. A distortion-tolerant measure of feature similarity is produced by performing statistical analysis on the filter response vectors, and the actual likelihood of observing an image detail given the feature location is computed by analyzing the distribution of the similarities. The prior part defines a priori distributions for the relative locations of the features, which correspond to their average locations on the object. The prior is hierarchical in nature, that is, a hyperprior is set for the width of the prior distribution, which allows the system to automatically determine the feature location variance appropriate for the image at hand. The object matching is carried out as Bayesian inference: the goal is to find the posterior probability distribution of the possible matches given the image and the hierarchical prior. Markov Chain Monte Carlo methods, especially Gibbs sampling, are used to draw samples from the posterior distributions of the feature locations, allowing simple estimation of the essential distribution statistics.

Can one model fit all? Evaluating foundation models for time series forecasting across clinical medicine.

Portfolio selection using hierarchical Bayesian analysis and MCMC methods

In this paper we propose a new procedure for detecting additive outliers in a univariate time series based on a bootstrap implementation of the test of Perron and Rodríguez (2003,Journal of Time Series Analysis24, 193‐220). This procedure is used to test the null hypothesis that a time series is uncontaminated by additive outliers against the alternative that one or more additive outliers are present. We demonstrate that the existing tests of,inter alia, Vogelsang (1999,Journal of Time Series Analysis20, 237–52) Perron and Rodríguez (2003) and Burridge and Taylor (2006,Journal of Time Series Analysis27, 685–701) are unable to strike a balance between size and power when the order of integration of a time series is unknown and the time series is driven by innovations drawn from an unknown distribution. We show that the proposed bootstrap testing procedure is able to control size to such an extent that its size properties are comparable with the robust test of Burridge and Taylor (2006) when the distribution of the innovations is not assumed known, whilst maintaining power in the Gaussian environment close to that of the test of Perron and Rodríguez (2003).

This article discusses the properties of the univariate Dickey–Fuller test and the Johansen test for the cointegrating rank when there exist additive outlying observations in the time series. We provide analytical as well as numerical evidence that additive outliers may produce spurious stationary. Hence the Dickey–Fuller test will reject a unit root too frequently and the Johansen test will indicate too many cointegrating vectors. The results easily generalize to models with “temporary change” outliers. Through an empirical example we discuss how additive and temporary change outliers can be detected in practice, and we show how dummy variables can be used to remove the influence of such extreme observations.

This article discusses the properties of the univariate Dickey-Fuller test and the Johansen test for the cointegrating rank when there exist additive outlying observations in the time series. We provide analytical as well as numerical evidence that additive outliers may produce spurious stationarity. Hence the Dickey-Fuller test will reject a unit root too frequently and the Johansen test will indicate too many cointegrating vectors. The results easily generalize to models with ''temporary change'' outliers. Through an empirical example we discuss how additive and temporary change outliers can be detected in practice, and we show how dummy variables can be used to remove the influence of such extreme observations.