###### Abstract  
Air quality modelling that relates meteorological, car traffic, and pollution data is a fundamental problem, approached in several different ways in the recent literature. In particular, a set of such data sampled at a specific location and during a specific period of time can be seen as a multivariate time series, and modelling the values of the pollutant concentrations can be seen as a multivariate temporal regression problem. In this paper, we propose a new method for symbolic multivariate temporal regression, and we apply it to several data sets that contain real air quality data from the city of Wroclaw (Poland). Our experiments show that our approach is superior to classical, especially symbolic, ones, both in statistical performances and the interpretability of the results. temporal regression; symbolic regression; air quality modelling +
Footnote †: journal:  
0000-0001-937X/0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-0000-0000-00000-0000-00000-00000-00000-00000-0  
regression models [12]. While, in general, functional regression systems tend to perform statistically well, their models tend to lack interpretability, defined not only as the possibility of understanding the process that is behind a prediction, but also explaining it. Attempts at amending this problem include optimizing the amount of lag per each independent variable have been done, for example, in [12]; by pinpointing exactly the amount of delay after which an independent variable has its maximal effect on the dependent one, it is possible to derive more reliable physical theories to explain the underlying phenomenon. However, the resulting model is still functional, and therefore not completely explicit. Symbolic regression is less common and, at least in problems of air quality prediction, usually limited to non-interpretable symbolic systems, such as random forests [13]. There are two typical, explicit approaches to symbolic regression, that is decision trees and rule based regression systems. Decision trees are part of the more general set of techniques often referred to as classification and regression trees, originally introduced in [14], but then improved by several authors and implemented in different versions and learning suites. Rule based regression is an alternative to decision tree regression based on the possibility of extracting independent rules instead of a tree, and it was introduced in [15], but as in the case of trees, improved and extended in different ways later on. In [16], a prototype interval temporal symbolic classification tree extraction algorithm, called Temporal J48, was presented. While originally designed for temporal classification (i.e., classification of time series), as shown in [17], it can be used for temporal regression. In this formulation, Temporal J48 features, on its own, many of the relevant characteristics for modern prediction systems, for example for air quality modelling: it is symbolic; therefore, its predictions are interpretable and explainable; and it allows the use of past values of the independent variables; therefore, it is comparable with lag regression systems. Interval temporal regression is based on interval temporal logic and, in particular, on [PERSON] and [PERSON]'s modal logic for time intervals [18]. In short, the extracted model is based on decisions made on the past values of the independent variables over intervals of time, and their temporal relations; for example, Temporal J48 may infer that if the amount of traffic in a certain interval of time is, on average, very high, while there are no gusts of wind during the same interval, then at the end of that interval, the concentration of NO\({}_{2}\) is high. The interaction between intervals is modelled via the so-called Allen's relations, which are, in a linear understanding of time, thirteen [19].