*5.4. Multidimensional Case*

The functions considered in DC and DC4F are unary, i.e., they are of the form *f* (*t*), where *t* is the time parameter. In this subsection, we consider integrals over multidimensional sets and spaces. Such cases may occur when, for instance, images of an area are considered and measurements are performed. The development of values over time may concern specific areas or multidimensional spaces. In this case, the integrated function has a number of variables, apart from the time variable. We show that in fact such a multidimensional case can be reduced to the case of unary functions. In fact, we utilize such integrals in the examples presented in the following section.

An image may be represented by a number of pixels with two coordinates *x*, *y*. Thus, the area of an image may be considered as a set *A* ⊆ R2. In the multidimensional case, it has the form *A* ⊆ R*<sup>n</sup>*. If the images are used to measure certain values, then the measurement can be modeled by a function. The measurements may be time-dependent. Thus, we consider time-dependent integrable functions of the form *f* (*t*, *x*1, ... , *xn*) and multidimensional sets of the form R+ × *A*. For an interval *I*, the corresponding integral has the form: ∫*<sup>I</sup>*×*<sup>A</sup> f d*(*<sup>t</sup>*, *x*1, ... *xn*), where *d*(*<sup>t</sup>*, *x*1, ... , *xn*) is a product measure on the space R × R*<sup>n</sup>*. Equivalently, we can present the integral using the characteristic function 1*A* corresponding to set *A*, the function has value 1 for the elements of *A* and 0 for all other arguments: ∫ 1*A f* (*t*, *x*1,..., *xn*) *d*(*<sup>t</sup>*, *x*1,..., *xn*).

This integral can be presented also in the form ∫*I* ∫*A f d*(*<sup>x</sup>*1, ... , *xn*)*dt*. Thus, the integral ∫*<sup>I</sup>*×*<sup>A</sup> f d*(*<sup>t</sup>*, *x*1, ... *xn*) has the form ∫*I g*(*t*) *dt*, where *g*(*t*) = ∫*A f* (*t*, *x*1, ... , *xn*) *d*(*<sup>x</sup>*1, ... , *xn*).Consequently, it can be presented as an integral of unary function *g*(*t*) depending on the time parameter *t* only.

#### **6. Applications and Validation**

In the case of multisensor systems, the data generated may be of various types, such as temperature, pressure, or density measurements, videos, or sound, among others. It is hard to bring them into a uniform and consistent model. Thus, quite often, textual specifications and informal validations are used, which is inherently imprecise. Specification, interpretation, comparison, and reasoning about such heterogeneous data pose a nontrivial problem. In this section, we apply the DC4F calculus to express measurements and complex behaviors. The goal is to illustrate the way DC4F can be applied and to demonstrate its capabilities for specifying temporal series. We use examples from the area of volcanic activity monitoring by multisensor systems because of their complexity and, thus, the challenge that they pose. We aim to show that the extension of DC that we propose in this paper provides a uniform language, models, and a reasoning system to integrate, describe, and reason about such data. The examples are used to show the practical applicability of DC4F and to demonstrate how it can be used. However, it is not our goal to adequately describe those phenomena per se. It should be also pointed out that the units of measurement do not play an essential role for the illustration. Our concepts concern mathematical properties and, thus, are independent of the units used.
