## 2 Materials and Methods
### Fuzzy Vector Spaces and the Object-Oriented Model (FVSOOM)  
This section commences with a brief summary of fuzzy vector space (FVS) literature and then describes the fuzzy vector space model FVSOOM, within FSVOOM, the model and method that was introduced in Section 1. Many papers concerning fuzzy vector spaces rely on extensions of fuzzy set theory and fuzzy logic as proposed by [PERSON] [4; 14; 15; 16; 17]. [PERSON] and [PERSON] defined the notion of fuzzy vector space from [PERSON]'s fuzzy set theory [18]. [PERSON] proved the properties of the dimensions of these fuzzy vector spaces [19]. Fuzzy set theory allows combination of the values of several attributes with \(\in,AND\), and \(OR\), fuzzy operators. Fuzzy set theory has been used to implement analogical problem solving, for example, to assess the results of students in a classroom solving mathematical problems [20]. One interesting application of the use of fuzzy rules was to implement fuzzy cognitive maps to residential mobility in urban spaces, describing the relationship between attributes, causality relationships and linguistic variables [21]. However, most of these works describe domains that have only a few attributes and do not rely on semantic models, such as the entity-relationship model containing entity sets, entities, attributes, and relationships, or the object-oriented model, containing classes, objects, attributes, and associations. FSVOOM, the model of FVSOOM, relies on a fuzzy function that represents an attribute as a vector, with a scalar value in its definition set, \([min,max]\) at each time instant t. This vector represents a force and can be combined with other vectors in a vector space using the addition of vectors and the scalar product and, more generally, all vector space properties. These properties simplify the combination of attributes and object composition that are considered vector addition operators in object-oriented models. Fuzzy vector space (FVS) is based on the following function that depicts two opposite vectors (Figure 1). The function is defined in Equation (4) and shown on Figure 2, where \(\frac{2}{1+e^{-2x}}-1=tanh(x)\), which is uniformly continuous and defined on \([-1,1]\). The limits \(\forall x\in\mathbb{R}\), \(\lim_{x\rightarrow-\infty}tanh(x)=-1\)_and_\(\lim_{x\rightarrow+\infty}tanh(x)=1\). Thus, according to the continuous extension theorem, \(tanh(x)\) is continuous on \([-1,1]\) in Equation (4). Equations (1)-(3), respectively, describe the positive vector \(f(x)\), the negative vector \(f^{\prime}(x)\) and the hyperbolic tangent \(F(x)\). Each attribute \(x_{i}\) of an object is described by a function \(F(x_{i}\) and its corresponding vector (Figure 1). If the attribute \(i\) is changing at each time \(t\), the fuzzy value is calculated by the function \(F_{i,t}(x_{i,t})\), where \(i\) is the order number of the attribute in the object, \(x_{i,t}\) is its value, and the scalar \(a_{i,t}\) is the weight of the attribute in the object at each instant \(t\). The weight in the object-\(\alpha_{i,t}\) or the function \(F_{i,t}\)--could change over the time (for example, from linear to exponential, or to logarithmic) but it is rarely needed. \[f:[0,1]\rightarrow[0,1],f(x)=\frac{2}{1+e^{-k(x-s)}}-1 \tag{1}\]  
\[f^{\prime}:[-1,0]\rightarrow[-1,0],f^{\prime}(x)=\frac{2}{1+e^{-k(x-s)}}-1 \tag{2}\]  
\[F:[-1,1]\rightarrow[-1,1],F(x)=tanh(\frac{kx-s}{2}) \tag{3}\]  
\[F:[-1,1]\rightarrow[-1,1]/F(x_{i},t)= \begin{cases}x=-1&F(x_{i},t)=-1,\\ x\in[-1,1]&F(x_{i},t)=tanh(\frac{kx_{i}-s}{2}),\\ x=1&F(x_{i},t)=1.\end{cases}  \tag{4}\]  
Figure 2: A sigmoid membership function on \([-1,1]\longrightarrow[-1,1]\). It follows that \(\forall x\in[-1,1],f(x)=-f^{\prime}(-x)\).