Inferring causality in networks of WSS processes by pairwise estimation methods

Abstract

Abstract-Inferring causal dependences in a family of dynamic systems from a finite set of observations is a problem encountered in many applications that arise in a diverse variety of fields; ranging from economics and finance to climatology and neuroscience. Given a set of random processes, the objective is to determine whether one process is influenced by the others and to investigate the nature of this influence in case a dependence relation is identified. The notion of Granger-causality may be used in this context to measure and quantify causal structures. Ideally, in order to infer the complete interdependence structure of a complex system, one should simultaneously consider the dynamic behaviour of all the processes involved. However, for large networks, such a method becomes exceedingly complicated. In this paper, we consider an interdependent group of jointly wide sense stationary real-valued stochastic processes and investigate the problem of determining Granger-causality by identifying pairwise causal relations. It is seen that while such methods may not reveal all details of a system, they can nonetheless provide useful and reasonably accurate information.

Custom Extended Sobel Filters

“…filter with the mathematical defined extended Sobel (Sob.) filter from [Gupta and Mazumdar, 2013] and the extended Prewitt (Prew. ), Modified Prewitt (Prew*) and Scharr (Sch.)…”

“…Even though we showed previously that 7 × 7 extended filter obtained the best results, we have used for the comparison only the 5 × 5 extended Sobel in order to match with those predefined mathemathical filters, in [Gupta and Mazumdar, 2013] and [Levkine, 2012]. The results on each set are listed in Table4, where the 5 × 5 custom extended Sobel filter has obtained a better overall F1-score than all other filters.…”

“…Visually, the results of the custom extended Sobel filters seem to get more blurry as the size of the extension increases. By comparing the custom extended Sobel filter with other edge detection filters, which are built based on a mathematical explanation from [Gupta and Mazumdar, 2013] and [Levkine, 2012], we can notice in Table 4 that the custom extended Sobel 5 × 5 obtained a better overall F1-score than the other filters.…”

“…When we extend the kernels, we are filling the newly added kernel positions with 0s, see the Figures 1 or 2. With our approach we obtain in most of the test cases better edge detection results than the filters presented in [Gupta and Mazumdar, 2013] or [Levkine, 2012], even if it doesn't respect the geometrical gradient formulas. Another advantage of our filter is the reduced runtime complexity due to the simplicity of the extension.…”

“…In [Gupta and Mazumdar, 2013] they have built an extended 5 × 5 Sobel filter and they mathematically proved how to build it starting from classical 3 × 3 Sobel operator. They conclude that a bigger filter is useful in order to find more edge pixels.…”