## 3 Theory and Calculation  
### General Formulation of the Problem  
Data that characterize the observed phenomenon are needed for mathematical modeling by using artificial intelligence methods. They can be gathered by observing the relevant phenomena, after which the data are measured (for objective values), assessed using subject matter expertise (for subjective values), or both (for combined values). In this paper, the proposed procedure is limited to the measured values, which are the result of actual bridges built by the incremental launching method. The described approach is otherwise general and independent of the type of data. Thus, the phenomenon under consideration has been discretized, which means that \(N\) observations, each of which is mathematically characterized by \(N\) model vectors (the name _model vector_ in the first part (model) refers to the _mathematical model_, and in the second part (vector) to \((D+E)\)_components of the vector_ in the \((D+E)\) dimensional hyper-space in which the considered phenomenon is described), are used to explain the complete phenomenon under consideration. The entire observed phenomenon can be expressed as:  
\[[\mathbf{X}_{1},\dots,\mathbf{X}_{i},\dots,\mathbf{X}_{N}]. \tag{1}\]  
The above Equation (1) formally represents an ordered, matrix description of the phenomenon under consideration and is more commonly referred to as a database.  
Different parameters (variables) can be used to describe each observation (for instance, the \(i\)-th model vector \(\mathbf{X}_{i}\)), where the parameters (variables) are viewed as components of the (\(i\)-th) model vector, which can be expressed as:  
\[\mathbf{X}_{i}=\left\{q_{i1},\dots,\ q_{ij},\dots,q_{iD};r_{i1},\dots,r_{ik},\dots,r_{iE}\right\}^{T} \tag{2}\]  
where \(q_{ij}\) and \(r_{ik}\) in the above Equation (2) represent the \(j\)-th in \(k\)-th component of model vector \(\mathbf{X}_{i}\). Vector \(\mathbf{X}_{i}\) is further divided into two vectors \(\mathbf{Q}_{i}\) and \(\mathbf{R}_{i}\), which can be written as:  
\[\mathbf{Q}_{i}=\left\{q_{i1},q_{i2},\dots,q_{iD}\right\}^{T}\text{ and }\mathbf{R}_{i}= \left\{r_{i1},\dots,r_{ik},\dots,r_{iE}\right\}^{T} \tag{3}\]  
where in Equation (3) vector \(\mathbf{Q}_{i}\) is complementary to vector \(\mathbf{R}_{i}\), implying that their concatenation yields the complete data model vector \(\mathbf{X}_{i}\).  
Now, one needs to define the predictive vector, which too, is composed of two truncated vectors, i.e., the given truncated vector \(\mathbf{Q}\) and the unknown complementary vector \(\hat{\mathbf{R}}\). Only the first part of the vector (\(\mathbf{Q}=\mathbf{Q}_{N+1}\)) includes known components (input parameters), while the components in the second part of the predictive vector (\(\mathbf{R}=\mathbf{R}_{N+1}\)) are unknown (output parameters). This can be written as:\[\mathbf{Q}=\{q_{1},\ldots,\,q_{2},\ldots,q_{D}\}^{T}\ \text{and}\ \hat{\mathbf{R}}=\{\hat{r}_{1}, \ldots,\hat{r}_{k},\ldots,\hat{r}_{E}\}^{T} \tag{4}\]  
where \(\hat{\ }\) in Equation (4) denotes estimation. The problem that needs to be solved is how the unknown complementary vector \(\hat{\mathbf{R}}\) can be estimated from a given truncated vector \(\mathbf{Q}\) and the model vectors \([\mathbf{X}_{1},\ldots,\mathbf{X}_{i},\ldots,\mathbf{X}_{N}]\). In presented case, the problem is formulated as how the k-th output parameters of the bridge \((\hat{r}_{k}=\hat{r}_{N+1,k})\) can be estimated from the known input parameters (\(q_{j}\); i.e., truncated vector \(\mathbf{Q}\)) and the data in the database.