## 2 Preliminaries
### Multidimensional F-Transform  
Let \(f\): \(X\subseteq R^{n}\to Y\subseteq R\) be a continuous n-dimensional function defined in a closed interval \(X=[a_{1},b_{1}]\times[a_{2},b_{2}]\times\dots\times[a_{n},b_{n}]\subseteq R^{n}\) and known in a discrete set of N points \(P=\{(p_{11},p_{12},\dots,p_{1n}),(p_{21},p_{22},\dots,p_{2n}),\dots,(p_{N1},p_{ N2},\dots,p_{Nn})\}\). For each i = 1,... \(n\) let \(x_{i1}\), \(x_{i2}\),..., \(x_{imi}\) with \(m_{i}\geq 2\) be a set of \(m_{i}\) points of \([a_{i},b_{i}]\), called nodes, such that \(x_{i1}=a_{i}<x_{i2}<\dots<x_{imi}=b_{i}\). For each i = 1,... \(n\) let \(A_{i1}\), \(A_{i2}\),..., \(A_{imi}\): \([a_{i},b_{i}]\to[0,1]\) be a family of fuzzy sets forming a fuzzy partition of \([a_{i},b_{i}]\), where:  
1. \(A_{ih}(x_{ih})=1\) for every h =1,2,..., \(m_{i}\);
2. \(A_{ih}(x)=0\) if x is not in \((x_{ih-1},x_{ih+1})\), where we assume \(x_{i0}=x_{i1}=a_{i}\) and \(x_{in+1}=x_{in}=b_{i}\) by commodity of presentation;
3. \(A_{ih}(x)\) strictly increases on \([x_{ih-1},x_{ih}]\) for h =2,..., \(m_{i}\) and strictly decreases on \([x_{ih},x_{ih+1}]\) for h = 1,..., \(m_{i}-1\);
4. \(\sum_{h=1}^{m_{i}}A_{ih}(x)=1\) for every x \(\in[a_{i},b_{i}]\). The fuzzy sets \(A_{i1}\), \(A_{i2}\),..., \(A_{imi}\) are called basic functions. Let \(c_{i}=(p_{i}-a_{i})/(m_{i}-1)\). The basic functions \(A_{i1}\), \(A_{i2}\),..., \(A_{imi}\) form a uniform fuzzy partition of \([a_{i},b_{i}]\) if:  
1. \(m_{i}\geq 3\) and the nodes are equidistant, i.e., \(x_{ih}=a_{i}+d_{i}\cdot(h-1)\), where \(d_{i}=(b_{i}-a_{i})/(m_{i}-1)\) and h = 1, 2,..., \(m_{i}\). 2. \(A_{ih}(x_{ih}-x)=A_{ih}(x_{ih}+x)\)\(\forall\) x \(\in[0,h]\) and \(\forall\) h = 2,..., \(m_{i}-1\);
3. \(A_{ih}+1(x)=A_{ih}(x-d_{i})\)\(\forall\) x \(\in[x_{ih},x_{ih}+1]\) and \(\forall\) h = 1,2,..., \(m_{i}-1\). We say that the set \(P=\{(p_{11},p_{12},\dots,p_{1n}),(p_{21},p_{22},\dots,p_{2n}),\dots,(p_{N1},p_{ N2},\dots,p_{Nn})\}\) is sufficiently dense w.r.t. the set of fuzzy partitions \(\{A_{11}A_{12}\dots A_{1m_{1}}\},\dots,\{A_{i1}A_{i2}\dots A_{im_{i}}\}\),..., \(\{A_{n1}A_{n2}\dots A_{nm_{n}}\}\) if for each combination \(A_{1h_{1}}A_{2h_{2}}\dots A_{nn_{n}}\) exists at least a point \(p_{i}=(p_{1i},p_{22},\dots,p_{jn})\in P\), such that \(A_{1h_{1}}(p_{j1})\cdot A_{2h_{2}}(p_{j2})\cdot\dots\cdot A_{nn_{n}}(p_{jn})>0\). In this case, we can define the direct multidimensional F-transform of \(f\) with the \((h_{1},h_{2},\dots,h_{5})th\) component \(F_{h_{1}h_{2}\dots h_{n}}\) given by  
\[F_{h_{1}h_{2}\dots h_{n}}=\frac{\sum_{j=1}^{N}f(p_{j1},p_{j2},\dots p_{jn}) \cdot A_{1h_{1}}(p_{j1})\cdot A_{2h_{2}}(p_{j2})\cdot\dots\cdot A_{nn_{n}}(p_{ jn})}{\sum_{j=1}^{N}A_{1h_{1}}(p_{j1})\cdot A_{2h_{2}}(p_{j2})\cdot\dots\cdot A_{ nn_{n}}(p_{jn})} \tag{5}\]  
The multidimensional inverse F-transform, calculated in the point \(p_{j}\), is given by:  
\[f_{n_{1}n_{2}\dots n_{n}}^{F}(p_{j1},p_{j2},\dots,p_{jn})=\sum_{h_{1}=1}^{m_{1 }}\hskip-14.226378 pt\sum_{h_{2}=1}^{m_{2}}\dots\hskip-14.226378 pt\sum_{h_{n}=1}^{m_ {n}}F_{h_{1}h_{2}\dots h_{n}}\cdot A_{1h_{1}}(p_{j1})\cdot\dots\cdot A_{nh_{n}}(p _{jn}) \tag{6}\]  
It approximates the function \(f\) in the point \(p_{j}\). In [11; 12] the multidimensional inverse F-transform (6) is applied in regression analysis to find dependencies between attributes in datasets. To highlight the use of the multidimensional F-transform, consider, as an example, a dataset, given by two input features defined in the close intervals, respectively, [1.1, 4.9] and [0.1, 1.0]. Suppose we create for each of the two input variables a fuzzy partition consisting of three basic functions, setting m\({}_{1}\) = 3 and m\({}_{2}\) = 3. We obtain c\({}_{1}\) = 1.9 and c\({}_{2}\) = 0.45. Table 1 shows the values of the three nodes for the two input variables. Figure 1 shows the points in the input variable plane.