What is Leverage Points

Feedforward neural networks (FFNs) are often considered as universal tools and find their applications in areas such as function approximation, pattern recognition, or signal and image processing. One of the main advantages of using FFNs is that they usually do not require, in the learning process, exact mathematical knowledge about input-output dependencies. In other words, they may be regarded as model-free approximators (Hornik, 1989). They learn by minimizing some kind of an error function to fit training data as close as possible. Such learning scheme doesn’t take into account a quality of the training data, so its performance depends strongly on the fact whether the assumption, that the data are reliable and trustable, is hold. This is why when the data are corrupted by the large noise, or when outliers and gross errors appear, the network builds a model that can be very inaccurate. In most real-world cases the assumption that errors are normal and iid, simply doesn’t hold. The data obtained from the environment are very often affected by noise of unknown form or outliers, suspected to be gross errors. The quantity of outliers in routine data ranges from 1 to 10% (Hampel, 1986). They usually appear in data sets during obtaining the information and pre-processing them when, for instance, measurement errors, long-tailed noise, or results of human mistakes may occur. Intuitively we can define an outlier as an observation that significantly deviates from the bulk of data. Nevertheless, this definition doesn’t help in classifying an outlier as a gross error or a meaningful and important observation. To deal with the problem of outliers a separate branch of statistics, called robust statistics (Hampel, 1986, Huber, 1981), was developed. Robust statistical methods are designed to act well when the true underlying model deviates from the assumed parametric model. Ideally, they should be efficient and reliable for the observations that are very close to the assumed model and simultaneously for the observations containing larger deviations and outliers. The other way is to detect and remove outliers before the beginning of the model building process. Such methods are more universal but they do not take into account the specific type of modeling philosophy (e.g. modeling by the FFNs). In this article we propose new robust FFNs learning algorithm based on the least trimmed squares estimator.