**2. FCM background**

Well-developed modeling methodology for complex systems that allow to describe the behavior of a system in terms of concepts, Fuzzy Cognitive Maps (FCM) are powerful tools for modeling dynamic systems that was introduced by Kosko [32, 68, 69]. The resulting model describes expert knowledge (semantic concepts and/or computed values for example) of complex systems with high dimensions and a variety of factors. The scientific community is expressing a growing interest about the theory and application of FCMs in complex systems, and their validity and usefulness has been proved in various fields [22, 62–67, 70, 71]. FCMs are fuzzy causality

*Fuzzy Photogrammetric Algorithm for City Built Environment Capturing into Urban Augmented… DOI: http://dx.doi.org/10.5772/intechopen.110551*

backpropagation approach of modeling which combine fuzzy logic, nonlinear computing, semantic and neural networks.

#### **2.1 Theoretical foundation of FCM**

FCMs are fuzzy signed directed graphs with feedback. They are appropriate to encode knowledge thanks to concepts organized and causally linked to each other with weightings. Each concept is materialized by a network node. Different FCM networks have been used as a decision modeling tool under different approaches [63–67, 72]. FCMs are based on the theory of fuzzy logic and fuzzy subsets, thus improving the ability of cognitive maps to present and model qualitatively and quantitatively dynamic nonlinear systems. So, FCM is a soft computing modeling technique used for dynamic causal knowledge acquisition and process reasoning. Under its most general approach, each concept represents an entity, a state, a variable, or a feature of the system. An FCM embeds the topology of a fuzzy signed direct graph and a nonlinear neural networks feedback dynamic [26, 33, 61]. Concepts are equivalent to neurons which state value is not binary but belongs to a fuzzy subset. The value *wij* of the directed edge from causal concept *Ci* to concept *Cj* measures how much *Ci* causes *Cj*. Value *wij* belongs the fuzzy causal interval [1, +1], *wij = 0* indicates no causality; *wij > 0* indicates causal increase, this means that *Cj* increases as *Ci* increases and vice versa, *Cj* decreases as *Ci* decreases; *wij < 0* indicates causal decrease or negative causality. *Cj* decreases as *Ci* increases and *Cj* increases as *Ci* decreases.

Depending on the direction and size of this effect, and on the threshold levels of the dependent concepts, the affected concepts may subsequently change their state as well, thus activating further concepts within the network. Because FCMs allow feedback loops, newly activated concepts can influence concepts that have already been activated before. As a result, the activation spreads in a nonlinear fashion through the FCM net until the system reaches a stable limit cycle or fixed point.

#### **2.2 FCM representation**

To illustrate the description made above of FCMs, we will consider one, composed of 5 concepts and 10 causality links in total as shown in **Figure 1**. Concepts variables are represented by nodes, such as C1, C2, C3, C4 and C5.

In the relation C1 ➔ C2, C1 is said to impact C2. So, C1 is the causal variable, whereas C2 is the effect variable, and the intensity of the causality is expressed by the value of *w12*. Also, in the relation C2 ➔ C1, C2 is said to impact C1, and the intensity of the causality is expressed by the value of *w21*. Each concept is characterized by a

**Figure 1.** *Simple fuzzy cognitive map model illustration.*

number Ai that results from its computed value through the transformation of the real value of the hole system's variables.

There are 3 possible types of causal relationships between concepts:


The value of wij indicates how strongly concept Ci influence concept Cj . The sign of wij indicates whether the relationships between concept Ci and Cj is direct or inverse. The direction of causality indicates whether concept Ci causes concept Cj or vice versa. These parameters must be considered when a value is assigned to weight wij.

#### **2.3 Mathematical formalization of FCM**

The operation of FCMs is based on an inferential process whose dynamics can be formalized mathematically. A FCM model acts as a network of threshold or continuous concepts [64, 66, 68, 69]. At this level, they differ from a simple neural network because they are based on extracting knowledge from experts [33, 64, 73] and do not require a data input layer. The nonlinear structure of each concept is expressed during the dynamics of the whole system through backpropagation [74, 75]. Then, the value Ai t+1 for each concept Ci at each time step is calculated [22, 65, 74] by the following general rule:

$$A\_i^{t+1} = f\left(k\_1 \sum\_{\substack{j=1 \\ j\neq i}}^n \mathcal{W}\_{ji} \mathcal{A}\_j^t + k\_2 \mathcal{A}\_i^t\right) \tag{1}$$

The k1 expresses the influence of the interconnected concepts in the configuration of the new value of the concept Ai and k2 represents the proportion of the contribution of the previous value of concept in the computation of the new value. This formulation assumes that a concept links to itself with a weight wii <sup>¼</sup> *<sup>k</sup>*2. Namely, *At <sup>i</sup>* and *At*þ<sup>1</sup> *<sup>i</sup>* are respectively the values of concept Ci at times *t* respectively *t+1*. wji is the weight of the interconnection from concept Cj to concept Ci and *f* is a threshold function defined in Eq. (2). The unipolar sigmoid function is the most used threshold function [57, 65, 67] where λ > 0 determines the steepness of the continuous function *f*. For the purposes of this research, the value of λ is fixed at unity, i.e. 1. The sigmoid function ensures that the calculated value of each concept will belong to the interval [0,1].

$$f(\mathbf{x}) = \frac{1}{\mathbf{1} + e^{-\lambda \mathbf{x}}} \tag{2}$$
