Sparse modeling using orthogonal forward regression with PRESS statistic and regularization

The paper introduces an efficient construction algorithm for obtaining sparse linear-in-the-weights regression models based on an approach of directly optimizing model generalization capability. This is achieved by utilizing the delete-1 cross validation concept and the associated leave-one-out test error also known as the predicted residual sums of squares (PRESS) statistic, without resorting to any other validation data set for model evaluation in the model construction process. Computational efficiency is ensured using an orthogonal forward regression, but the algorithm incrementally minimizes the PRESS statistic instead of the usual sum of the squared training errors. A local regularization method can naturally be incorporated into the model selection procedure to further enforce model sparsity. The proposed algorithm is fully automatic, and the user is not required to specify any criterion to terminate the model construction procedure. Comparisons with some of the existing state-of-art modeling methods are given, and several examples are included to demonstrate the ability of the proposed algorithm to effectively construct sparse models that generalize well.

A robust nonlinear identification algorithm using PRESS statistic and forward regression

This letter introduces a new robust nonlinear identification algorithm using the Predicted REsidual Sums of Squares (PRESS) statistic and for-ward regression. The major contribution is to compute the PRESS statistic within a framework of a forward orthogonalization process and hence construct a model with a good generalization property. Based on the properties of the PRESS statistic the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation.

Automatic nonlinear predictive model-construction algorithm using forward regression and the PRESS statistic

An automatic nonlinear predictive model-construction algorithm is introduced based on forward regression and the predicted-residual-sums-of-squares (PRESS) statistic. The proposed algorithm is based on the fundamental concept of evaluating a model's generalisation capability through crossvalidation. This is achieved by using the PRESS statistic as a cost function to optimise model structure. In particular, the proposed algorithm is developed with the aim of achieving computational efficiency, such that the computational effort, which would usually be extensive in the computation of the PRESS statistic, is reduced or minimised. The computation of PRESS is simplified by avoiding a matrix inversion through the use of the orthogonalisation procedure inherent in forward regression, and is further reduced significantly by the introduction of a forward-recursive formula. Based on the properties of the PRESS statistic, the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation. Numerical examples are used to demonstrate the efficacy of the algorithm.

Significance tests in climate science

A large fraction of papers in the climate literature includes erroneous uses of significance tests. A Bayesian analysis is presented to highlight the meaning of significance tests and why typical misuse occurs. The significance statistic is not a quantitative measure of how confident we can be of the ‘reality’ of a given result. It is concluded that a significance test very rarely provides useful quantitative information.

Bayesian retrieval of complete posterior PDFs of oceanic rain rate from microwave observations

A new Bayesian algorithm for retrieving surface rain rate from Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) over the ocean is presented, along with validations against estimates from the TRMM Precipitation Radar (PR). The Bayesian approach offers a rigorous basis for optimally combining multichannel observations with prior knowledge. While other rain-rate algorithms have been published that are based at least partly on Bayesian reasoning, this is believed to be the first self-contained algorithm that fully exploits Bayes’s theorem to yield not just a single rain rate, but rather a continuous posterior probability distribution of rain rate. To advance the understanding of theoretical benefits of the Bayesian approach, sensitivity analyses have been conducted based on two synthetic datasets for which the “true” conditional and prior distribution are known. Results demonstrate that even when the prior and conditional likelihoods are specified perfectly, biased retrievals may occur at high rain rates. This bias is not the result of a defect of the Bayesian formalism, but rather represents the expected outcome when the physical constraint imposed by the radiometric observations is weak owing to saturation effects. It is also suggested that both the choice of the estimators and the prior information are crucial to the retrieval. In addition, the performance of the Bayesian algorithm herein is found to be comparable to that of other benchmark algorithms in real-world applications, while having the additional advantage of providing a complete continuous posterior probability distribution of surface rain rate.

An analysis of the impact of R&D on productivity using Bayesian model averaging with a reversible jump algorithm.

A Bayesian Model Averaging approach to the estimation of lag structures is introduced, and applied to assess the impact of R&D on agricultural productivity in the US from 1889 to 1990. Lag and structural break coefficients are estimated using a reversible jump algorithm that traverses the model space. In addition to producing estimates and standard deviations for the coe¢ cients, the probability that a given lag (or break) enters the model is estimated. The approach is extended to select models populated with Gamma distributed lags of di¤erent frequencies. Results are consistent with the hypothesis that R&D positively drives productivity. Gamma lags are found to retain their usefulness in imposing a plausible structure on lag coe¢ cients, and their role is enhanced through the use of model averaging.

Bayesian model averaging and identification of structural breaks in time series

Bayesian Model Averaging (BMA) is used for testing for multiple break points in univariate series using conjugate normal-gamma priors. This approach can test for the number of structural breaks and produce posterior probabilities for a break at each point in time. Results are averaged over specifications including: stationary; stationary around trend and unit root models, each containing different types and number of breaks and different lag lengths. The procedures are used to test for structural breaks on 14 annual macroeconomic series and 11 natural resource price series. The results indicate that there are structural breaks in all of the natural resource series and most of the macroeconomic series. Many of the series had multiple breaks. Our findings regarding the existence of unit roots, having allowed for structural breaks in the data, are largely consistent with previous work.