Noise simple linear regression

Models of this type allow solving the simplest predictive tasks or making the basic description of simple linear systems. Usually, however, they are developed further in the direction of extending their application possibilities. In practice, the ARMA type model, due to the emergence of the measurement noise, inaccuracies, etc., is usually extended to a simple stochastic model of the ARX type (Auto Regressive with auXUiary input) it is extended by an additional signal, i.e. white noise. In this model, as input, we can also use the values of other (known) time series, data in the current time moments or historical data. 

Although ANNs can, in theory, model any relation between predictors and predictands, it has been found that common regression methods such as PLS can outperform ANN solutions when linear or slightly non-linear problems are considered. In fact, although ANNs can model linear relationships, they require a long training time because a non-linear technique is applied to linear data. Although, ideally, for a perfectly linear and noise-free dataset, the ANN performance tends asymptotically towards the linear model one, in practical situations ANNs can reach a performance qualitatively similar to that of linear methods. Therefore, it is not advised to apply them before more simple alternatives have been considered. 

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