Locally Adaptive Bayesian Multivariate Time Series
In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If such locally adaptive smoothness is not accounted for, one can obtain misleading inferences and predictions, with over-smoothing across erratic time intervals and under-smoothing across times exhibiting slow variation. This can lead to miscalibration of predictive intervals, which can be substantially too narrow or wide depending on the time. We propose a continuous multivariate stochastic process for time series having locally varying smoothness in both the mean and covariance matrix. This process is constructed utilizing latent dictionary functions in time, which are given nested Gaussian process priors and linearly related to the observed data through a sparse mapping. Using a differential equation representation, we bypass usual computational bottlenecks in obtaining MCMC and online algorithms for approximate Bayesian inference. The performance is assessed in simulations and illustrated in a financial application.
Summary: The paper proposes a multivariate stochastic 
      process for modeling time series which incorporates locally varying 
      smoothness in the mean and in the covariance matrix. The process uses 
      latent dictionary functions with nested Gaussian process priors; the 
      dictionary functions are linearly related to the observations through a 
      sparse mapping. The authors outline MCMC and online algorithms for 
      approximate Bayesian inference and assess performances using simulation 
      and processing of financial data. Quality: The paper extends the 
      application of the nested Gaussian process priors in [23] to the 
      multivariate case and employs them for both the mean and covariance. This 
      constitutes a sensible extension, and the authors develop an effective 
      inference algorithm. The authors outline an online algorithm and 
      suggest that it could prove beneficial for high-frequency data. While this 
      is certainly attractive, the paper lacks any clear characterization of the 
      computational overhead (or complexity). Nor is there any indication of the 
      reduction in accuracy that is induced by executing the online method as 
      opposed to rerunning the full posterior computation. As such it is 
      difficult for the reader to gain a sense of the settings in which the 
      online approach might prove useful and appropriate. The authors do 
      not provide compelling evidence that the extended model proposed in this 
      paper is important in a practical setting. The simulation examples are 
      very toy cases tailored to the technique, so that one would be 
      disappointed if the proposed strategy did not provide improved results. 
      The studies certainly hint at a setting where the local smoothness in the 
      model could prove beneficial, but they do not model a practical 
      measurement setting. The paper provides an analysis of correlation 
      estimation for national stock indices, with a qualitative analysis of the 
      derived correlation results. Unfortunately, there is limited discussion of 
      the choice of hyperparameters for the inference and as a result it is 
      difficult to determine whether the comparison between LBCR and BCR in 
      Figure 3 is meaningful. The BCR graph has the appearance of a correlation 
      estimate that is constructed using much greater smoothing than the LBCR 
      method. It is not clear that LBCR is using less smoothing where necessary 
      and similar levels of smoothing where appropriate; it just appears to be a 
      much less smooth estimate of the correlation throughout the entire 
      time-series. I wonder whether BCR might achieve a more similar result with 
      different parameter choices. Since there is no ground truth, it is 
      impossible to know whether the generally increased level of correlation is 
      real or an artifact of the analysis. One cannot know if the oscillations 
      in first portion of the data (2004-2006) are a true reflection of 
      significant variation in the correlations over this period or if they 
      simply indicate that there has been inadequate smoothing in the formation 
      of the estimate. Although the authors provide plausible explanations for 
      the changes in the LBCR estimates, it is not at all clear that these were 
      hypothesized before the results were obtained. If not, then one cannot 
      place much value in the explanations, since it is almost always possible 
      to find some plausible explanation for an increase or decrease in 
      correlation of financial time series at the national level. The 
      final section of the paper suggests that the model could be employed to 
      obtain better predictions of the log-returns of the national stock 
      indices. The results here add very little of value to the paper. The 
      authors suggest that the boxplots in Figure 4 indicate improved prediction 
      performance, but this figure provides no real evidence of any meaningful 
      improvement. It is extremely unlikely that any analysis conducted on only 
      7 weeks of log-returns (at the weekly level) would be able to provides any 
      evidence of any improvement in estimation quality. Clarity: The 
      paper is well-written and easy to follow. At times it is not completely 
      clear what choices have been made in algorithmic simulation and data 
      analysis comparisons. Originality: The model in the paper is a 
      variation of the model in [22], replacing Gaussian process priors with 
      nested Gaussian process priors. The use of nested Gaussian process priors 
      is suggested in [23], where it is applied to the univariate case, for the 
      mean only. The paper is not significantly original. Significance: 
      The introduced model is a relatively minor adaptation of an existing model 
      and the inference techniques are fairly straightforward adaptations of 
      existing methods. Although there is some novelty and the model and method 
      are of some interest, the paper is unlikely to have a significant impact.