# Classification of Imbalanced Travel Mode Choice to Work Data Using Adjustable SVM Model
## 2 Methods and Data
### Proposed Approach
#### 2.2.1 Standard Support Vector Machine Model  
Support Vector Machine (SVM) is a broadly employed and praised ML technique for classifying data [26]. The principal purpose of creating this model was to draw the input data into high dimensional space with the aid of the kernel function in such a way that the categories can be linearly divisible [47; 48; 49]. For the binary class issue, the greatest boundary that can divide the hyperplanes is as follows:  
\[w\text{-}a+l=0 \tag{1}\]  
The decision function for SVM based on the optimal pair (\(w_{0}\), \(l_{0}\)) is expressed by:  
\[f(a)=\sum_{i\in SV}\lambda_{i}y_{i}\langle a,a_{i}\rangle+l \tag{2}\]  
where, \(\lambda_{i}\) stands for support vector, \(a_{i}\) denotes data sample and \(i\) = 1, 2,..., K. Concerning greater dimensional feature space, the value of \(\langle a.a_{i}\rangle\) is substituted by the kernel function Q \(\langle a_{i}.a_{j}\rangle\), that is:  
\[\text{Q}\big{\langle}a_{i}.a_{j}\big{\rangle}=\big{\langle}a_{i}.a_{j}\big{\rangle} \tag{3}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Variable** & **Description** & **Type** & **Mean/Mode** \\ \hline \multicolumn{4}{l}{Household information} \\ \hline drvcrnt & Number of drivers in household & Continuous & 3.003 \\ hh\_ontd & Number of household members on trip & Continuous & 1.884 \\ hhfaminc & Household income & Categorical & 7.00 \\ hhsize & Count of household members & Continuous & 4.576 \\ hhvehcnt & Count of household vehicles & Continuous & 3.647 \\ numadlt & Count of adult household members at least & Continuous & 3.194 \\ vehowned & Owned vehicle longer than a year & Flag & 1.00 \\ wrkcount & & & \\ youngchild & Number of workers in household & Continuous & 0.25 \\ \hline \multicolumn{4}{l}{Trip characteristics} \\ \hline bikemore & Reason for not biking & Nominal & 7.00 \\ timetowk & Trip time to work in & Continuous & 22.97 \\ walkmore & Reason for not walking & Nominal & 5.00 \\ Health & & & \\ health & Opinion of health & Categorical & 2.00 \\ medcond & Any condition or handicap that makes it & Flag & 2.00 \\ medcond6 & & & \\ \hline \multicolumn{4}{l}{Living environment} \\ \hline \multicolumn{4}{l}{urbansize} & Urban area size where home address is located & Categorical & 6.00 \\ wrbrur & Household in urban/rural area & Categorical & 1.00 \\ homeloc & Home location/reason to chose your current & Nominal & 1.00 \\  & home location? & Nominal & 3.00 \\ wrktrans * & Mode to work & Nominal & 3.00 \\ \hline \hline \end{tabular}  
* Target variable.  
\end{table}
Table 3: _Cont._From the regular SVM, the kernel function was selected for estimating the boundaries' space. In the beginning, the dataset \(S\) is divided into different samples, which are \(S^{1}\), \(S^{2}\), \(S^{3}\),..., \(S^{i}\), and subsequently, the kernel transformation function is implemented (Equation (4)).  
\[f(a)=\left\{\begin{array}{c}e^{-z_{i}h(a)^{2}},\,if\,a\in S^{1}\\ e^{-z_{2}h(a)^{2}},\,if\,a\in S^{2}\\.\\.\\ e^{-z_{2}h(a)^{2}},\,if\,a\in S^{K}\end{array}\right. \tag{4}\]  
where, \(h_{a}=\sum\limits_{i\in SV}\lambda_{i}y_{i}\langle a,a_{i}\rangle+l\) (where, \(\lambda_{i}\) signifies support vector), \(S^{i}\) refers to the \(i\)th sample of the training dataset, \(z_{i}\) is calculated using likelihood-ratio chi-square, which is described in the following sections.