Development of the nearest neighbour multivariate localized regression modelling technique for steady state engine calibration and comparison with neural networks and global regression

Abstract A new modelling technique has been developed to aid steady state diesel engine calibration by accurately predicting engine system response and emissions at steady state operating conditions. This new modelling technique, referred to as the nearest neighbour multivariate localized regression (NNMLR) in this work, is built upon the particular localized regression technique for multiple independent variables developed by M. C. Sharp et al. at Cummins Inc., Columbus and referred to as the multivariate localized regression (MLR) technique in this work. Among other advantages, MLR has been demonstrated to generalize better than other similar data-driven empirical modelling techniques such as global regression. Although MLR has been proven and tested, it is computationally expensive which makes it unwieldy with current optimization schemes, particularly random search methods; NNMLR is significantly faster than MLR. Additionally, the primary localization parameter which remains fixed in MLR over the entire dataset is allowed to vary over the dataset in NNMLR. Therefore, in NNMLR it is the dataset that decides the degree of localization. This variable degree of localization is unique to each response being modelled and can change with data density and with the complexity of the relationship between the dependent and independent variables. This greater degree of localization makes NNMLR slightly more accurate than the current MLR scheme. The motivation for developing NNMLR was to reduce optimization routine runtimes while maintaining accuracy at MLR levels or even improving on these levels. The other modelling approach that promises similar or better accuracy levels and model runtimes is neural networks. Neural networks are increasingly being applied over a wide range of applications, hence they were examined for their suitability for modelling engine responses. Their robustness to noise, training data requirements, and ability to extrapolate were compared with localized and global regression. In the following sections, after background that describes the MLR technique, the algorithm for NNMLR is described. Some heuristics are unavoidable and these have been discussed and justified. Results have been presented for many datasets of different sizes over different sets of dependent variables, comparing MLR, NNMLR, global regression, and neural networks. In addition to performance over standard engine responses, test functions have been used to create pseudo-responses, so that noise can be added in a controlled manner. Accuracy and robustness of MLR, NNMLR, global regression, and neural networks over different levels of noisy data have been studied. The study shows how neural networks work better than other approaches for steady state calibration in terms of mean accuracy, maximum error, robustness to noise, training data size, ability to handle non-linearities as well as ability to extrapolate. However, localized regression (MLR and NNMLR) produces comparable performance and the reasons why a regression-based approach is favoured by the authors are outlined. The reduction of training data afforded by localized regression over response surface methods is evident. The study concludes that the relative performance of global regression, localized regression, and neural networks will change depending on the application at hand and there is no single answer for choice of modelling approach. While choosing a modelling approach, a number of considerations based on the training dataset and model application need to be taken into account.