## 5 An Integrated Inductive-Deductive Framework for Data Mapping
### AMA Framework Details
#### 5.1.2 Scale-Based Local Distance Metric  
In this section, we modify the distance metric defined in Shepard interpolation to include the knowledge given by the domain model. The domain model helps to significantly reduce the size of the support nodes set to lower than that of the conventional Euclidean-distance-based interpolation methods. The increase in the size of the support set can lead to an increase in the computation and processing times of the interpolation algorithm. Therefore, the proposed metric attempts to balance the size of the support set with the interpolation algorithm complexity as well as interpolation accuracy.  
We define the term _scale_ for determining the weight of every given dimension with respect to \(P\) based on a combined Euclidean distance criteria as well as information already known about the application domain a priori to network deployment. While the term _weight_ is reserved for the relevance of a data site by calculating the Euclidean distance between \(P\) and a particular data point. A special case is when \(f_{i}\) is identical for all \(S_{i}\) which means that all sets have the same scale.  
We define a new scale-based weighting metric, \(m_{P}\), which uses the information given by the domain model to alter the distance weighting function to improve the interpolation results when applied to an arbitrary number of dimensions. All data sets are ordered pairs \(\left(x_{i},y_{i}\right)\) such that \(x_{i}\) is the sampling location and \(y_{i}\) is a measured scalar value associated with \(x_{i}\).  
Given a set of data points \(S_{i}\), standard spatial data interpolation methods define the interpolation function based on the distance from \(P\) to \(C_{i}\) where \(C_{i}\subseteq S_{i}\). In local interpolators, \(C_{i}\) is a small collection of the total set of data points. When \(C_{i}\) is extended to include a large proportion of the data points or the total set of points the interpolation function becomes global. In AMA, the set \(C_{i}\) for each dimension is determined using \(m_{P}\). Symbolically, \(C_{i}\) is calculated as  
\[C_{i}=L\left(d\left(P,E_{j}\right),\delta\left(S_{i}\right)\right)\ \ \forall E_{j}\in S_{i} \tag{11}\]  
where \(i\in\left[1,n\right]\), \(L\) is a local model that selects the support set for calculating \(P\), \(d\) is an Euclidean distance function, \(E_{j}\) is an observation point in the dimension \(S_{i}\), and \(\delta(S_{i})\) a set of parameters for dimension \(S_{i}\). Each dimension can have a different set of parameters. These parameters are usually a set of relationships between different dimensions or other application domain characteristics such as obstacles. When predicting the value of a point in dimension \(S_{i}\) we refer to that dimension as \(S_{P}\).  
In uni-dimensional distance weighting methods, the weight, \(\upomega\) can be calculated as follows  
\[\upomega=d\left(P,E_{j}\right),E_{j}\in S_{i} \tag{12}\]  
This function can be extended to multi-dimensional distance weighting systems as follows  
\[\upomega=K\left(P,S_{i}\right),\ \ i\in\left[0,n\right] \tag{13}\]  
where \(K\left(P,S_{i}\right)\) is the distance from \(P\) to data set \(S_{i}\) and \(n\) is the number of dimensions in the system. Equation (13) can now be extended to include the domain model parameters of arbitrary dimensional system. Then, the dimension-based scaling metric can be defined as  
\[m_{P}=\sum\limits_{i}L\left(K(P,S_{i}),\delta\left(S_{i}\right)\right)\ \ \ i\in\left[0,n\right]\text{and}\ S_{i}\
eq C_{P} \tag{14}\]  
where \(C_{P}\) is the dimension containing \(P\).