Prediction intervals for non-linear projection to latent structures regression models

Linear and non-linear projection to latent structures (PLS) have been shown to be flexible and reliable methodologies for developing empirical regression models when dealing with noisy, correlated and/or collinear data. However, as for every regression technique, a measure for assessing the reliability of the predicted values is required. A common approach is through the use of prediction intervals. These are the upper and lower confidence limits of the predicted values. The larger the magnitude of these intervals, the less precise is the prediction. Several approaches have been proposed in the literature for calculating prediction intervals for linear and non-linear regression techniques including multiple linear regression (MLR), linear projection to latent structures and feed-forward neural networks (FNN). In this work, a methodology proposed to evaluate prediction intervals for neural network models is extended to linear and non-linear projection to latent structures algorithms. The prediction intervals are computed using a first-order Taylor series expansion and the Jacobian matrix of the functional mapping provided by the projection to latent structures algorithms. From previous work on feed-forward neural networks, this approach is known to give approximate but reliable results. Furthermore, it is less computationally expensive than alternative but mathematically more precise approaches. The methodology is illustrated on data generated from a non-linear pH neutralisation system.