# Condition Rating Prediction Using an Interactive Deterioration Model Development Package
## 3 Dmdp
### Weighted LASSO  
[PERSON] et al. (2017) [21] proposed a framework for the selection of explanatory variables using penalized regression, known as LASSO [51]. By controlling tuning parameter in LASSO, the particular contribution for candidate variables is determined. Under the given condition ratings, which possibly contain innate human error, the explanatory variables can be determined without bias. To establish an efficient bridge management system with a nearly stationary set of explanatory variables, we adopted the weighted LASSO in the DMDP. Figure 1: Flowchart for Deterioration Model Development Package (DMDP). Suppose that the condition ratings and inspection data collected from \(m\) bridges in the \(i\)th year are \(\mathbf{Y}\) and \(\mathbf{X}\), respectively, for which the linear regression model can be defined as follows:  
\[\mathbf{Y}=\mathbf{X}\beta+\varepsilon \tag{1}\]  
In Equation (1), \(\mathbf{X}\in\Re^{m\times n}\) is a matrix form of the inspection data for \(n\) explanatory variables, and \(\beta\) is the regression coefficient vector that minimizes the sum of the squared error, \(\varepsilon\). The traditional LASSO estimates the regression coefficient \(\hat{\mathbf{\beta}}\) using the following formula:  
\[\hat{\mathbf{\beta}}=\text{argmin}\left\{\|\mathbf{Y}-\mathbf{X}\beta\|_{2}^{ 2}+,\lambda,\sum_{j=1}^{n}\left|\beta_{j}\right|\right\} \tag{2}\]  
In Equation (2), \(\lambda\) is the tuning parameter, and the penalty is defined as the absolute sum of \(\beta\). The amount of shrinkage is generally defined as the dot product of these two terms. The priority of the candidate variables is determined by controlling this shrinkage parameter. Although LASSO successfully identifies the explanatory variables using rigorous evidence, the result varies depending on the inspection year. Equation (2) is modified to use the inspection data from multiple years, such that \(\mathbf{X}=\left[ \begin{array}{cc}w_{1}\mathbf{x}_{1}^{\text{T}}&w_{2} \mathbf{x}_{2}^{\text{T}}&\cdots&w_{s}\mathbf{x}_{s}^{\text{T}}\end{array}  \right]^{\text{T}}\) and \(\mathbf{Y}=\left[ \begin{array}{cc}w_{1}\mathbf{y}_{1}^{\text{T}}&w_{2} \mathbf{y}_{2}^{\text{T}}&\cdots&w_{s}\mathbf{y}_{s}^{\text{T}}\end{array}  \right]^{\text{T}}\), where \(s\) denotes the total number of previous inspection years. The weighting parameter \(w_{i}\) is introduced to estimate the contribution of the inspections of previous years for LASSO regression and is assumed to be half for each inspection cycle. \begin{table}
\begin{tabular}{c c} \hline
**Index No.** & **Candidate Variables** \\ \hline
1 & Route signing prefix \\
2 & Highway agency district \\
3 & Base highway network \\
4 & Maintenance responsibility \\
5 & Functional classification of inventory route \\
6 & Year built (age) \\
7 & Lanes on the structure \\
8 & Lanes under the structure \\
9 & Average daily traffic \\
10 & Design load \\
11 & Skew \\
12 & Type of service on bridge \\
13 & Type of service under bridge \\
14 & Kind of material and/or design \\
15 & Type of design and/or construction \\
16 & Number of spans in main unit \\
17 & Inventory route, total horizontal clearance \\
18 & Length of maximum span \\
19 & Structure length \\
20 & Bridge roadway width \\
21 & Deck width \\
22 & Deck structure type \\
23 & Type of wearing surface \\
24 & Type of membrane \\
25 & Deck protection \\
26 & Average daily truck traffic \\
27 & Designated national network \\ \hline \end{tabular}
\end{table} 
Table 1: List of candidate variables. The elements of Wyoming bridges were analyzed in this study to investigate the consistency of explanatory variable selection using the weighting factor in LASSO regression. The binary function used to quantify the difference in the ranking system for adjacent years is given by  
\[f_{ij}=\left\{ \begin{array}{l}1\text{ if }v_{i}=v_{i-1}\\ 0\text{ if }v_{i}\
eq v_{i-1}\end{array} \right. \tag{3}\]  
In Equation (3), \(v_{i}\) is the \(i\)th element of a list of \(n\) explanatory variables in the \(j\)th year.