## 5 An Integrated Inductive-Deductive Framework for Data Mapping
### AMA Framework Details
#### 5.1.1 Multi-Modal Spatial Interpolation  
Most spatial data interpolation methods are based on the distance between \(P\), where the interpolation function has to be determined, and the given set of data points. The AMA defines a new metric for distance, suitable for higher dimensions, in which the concept of closeness is described in terms of relationships between sets rather than in terms of the Euclidean distance between points. Using this distance metric, a new generalised interpolation function \(f\), that is suitable for an arbitrary number of variables, is defined. Figure 3: The process of building the domain model by deductive and inductive steps. In multi-dimensional interpolation, every set \(S_{i}\) corresponds to an input variable, _i.e._, a sense modality, called \(i\), and referred to as a dimension. In AMA, the distance function does not need to satisfy the formal mathematical requirements for the Euclidean distance definition. The power of such a generalisation can be seen when we include the time variable as one dimension. The spatial data interpolation problem can be stated as follows:  
Given a set of randomly distributed data points  
\[x_{i}\in\Omega,\ i\in\left[1,N\right],\ \Omega\subset\mathbb{R}^{n} \tag{2}\]  
with function values \(y_{i}\in\mathbb{R}\), and \(i\in\left[1,N\right]\), we require a continuous function \(f:\Omega\longrightarrow\mathbb{R}\) to interpolate unknown intermediate points such that  
\[f\left(x_{i}\right)=y_{i}\ \text{where}\ i\in\left[1,N\right] \tag{3}\]  
We refer to \(x_{i}\) as the observation points. The integer \(n\) is the number of dimensions and \(\Omega\) is a suitable domain containing the observation points. When rewriting this definition in terms of relationships between sets we get the following:  
Given \(N\) ordered pairs of _separated_ sets \(S_{i}\subset\Omega\) with continuous functions  
\[f_{i}:S_{i}\longrightarrow\mathbb{R},\ i\in\left[1,n\right] \tag{4}\]  
we require a multi-dimensional continuous function \(f:\Omega\longrightarrow\mathbb{R}\), defined in the domain \(\Omega=S_{1}\cup S_{2}\cup...\cup S_{n-1}\cup S_{n}\) of the \(n\)-dimensional Euclidean space where  
\[f\left(x_{i}\right)=f_{i}\left(x_{i}\right)\forall x_{i}\in S_{i}\ \text{where}\ i\in\left[1,n\right] \tag{5}\]  
The existence of the global continuous function \(f\) can be verified as follows. First, the data set is defined as  
\[S=\left\{ \begin{array}{cccc}v_{1}^{(0)},&v_{1}^{(1)},&\cdots,&v_{1}^{(n^{ \prime})}\\ v_{2}^{(0)},&v_{2}^{(1)},&\cdots,&v_{2}^{(n^{\prime})}\\ \vdots&\vdots&&\vdots\\ v_{n}^{(0)},&v_{n}^{(1)},&\cdots,&v_{n}^{(n^{\prime})}\end{array} \right\} \tag{6}\]  
where \(n^{\prime}\leq N\) and \(v_{i}=\left(x_{i},r_{i}\right),i\in\left[1,N\right]\) and \(r_{i}\) is a reading value of some distinctive modality (e.g., temperature). Let \(\Phi\) be a topological space on \(S\) and there exists open subsets \(S_{i},i\in\left[1,n\right]\)  
\[ \begin{array}{l}S_{1}=\left\{v_{1}^{(0)},v_{1}^{(1)},\cdots,v_{1}^{(n^{ \prime})}\right\}\\ S_{2}=\left\{v_{2}^{(0)},v_{2}^{(1)},\cdots,v_{2}^{(n^{\prime})}\right\}\\ \vdots\\ S_{n}=\left\{v_{n}^{(0)},v_{n}^{(1)},\cdots,v_{n}^{(n^{\prime})}\right\}\end{array}  \tag{7}\]  
which are topological subspaces of \(\Phi\) such that  
\[\Phi_{S_{i}}=\left\{S_{i}\cap U|U\in\Phi\right\} \tag{8}\]  
Also define \(\Psi\) as a topological space on the co-domain \(\mathbb{R}\) of function \(f\). Then, there exists a function, \(f\), that has the following properties:1. Let \(f:S_{1}\cup S_{2}\cup...\cup S_{n-1}\cup S_{n}\) be a mapping defined on the union of subsets \(S_{i},i\in[1,N]\) such that the restriction mappings \(f_{\left|S_{i}\right. }\) are continuous. If subsets \(S_{i}\) are _open_ subspaces of \(S\) or _weakly_ separated, then there exist a function \(f\) that is continuous over \(S\) (proved by [PERSON] [27]). 2. If \(f:\Omega\rightarrow\Psi\) is continuous, then the restriction to \(S_{i},i\in[1,N]\) is continuous (property, see [28]).