## 3 Methods and Data
### Assumptions for Qualitative Analysis--Application of Machine Learning  
With the use of data that describes the circumstances, characteristics, and features of construction works execution, it is possible to explore the prediction of efficiency. Machine learning provides a wide variety of tools that can be applied in this area. Fundamental knowledge and principles of machine learning can be found in the relevant literature, such as [41; 42; 43; 44]. During the initial studies, the option of efficiency predictions based on solving both regression and classification problems was investigated. However, it was found that the developed regression models' performance was not satisfactory when using the collected data. As a result, the decision was made to focus on classification problems. This is also an explanation for why this part of the study was called qualitative analysis. The main idea was to develop a machine learning-based tool that can differentiate cases describing construction works execution and classify them based on their efficiency, either greater or lower than a certain assumed critical value. The critical value of efficiency can be determined based on previously fitted continuous distributions, and the records in the dataset can be labelled for supervised training. There are several methods that can be applied to this problem, such as neural networks, classification trees, naive Bayes method, k-nearest neighbours, support vector machines (SVM), and others. SVM is a machine learning tool whose theory is based on the principles of statistical learning, which are broadly presented in [45; 46]. SVM has the capability of learning from experience hidden in the data and knowledge generalisation. The fundamentals, description, and methodology of the tools are discussed in many works, e.g., [47; 48; 49]. Due to the advantages of SVM, it was decided to apply this particular method in the study. In particular, SVM is a supervised training algorithm that has been shown to have good generalisation performance and is applicable to many problems. SVM is known to perform well with limited datasets. Another important advantage to mention is the low number of free parameters in the learning machine, so the model architecture does not need to be found by experimentation. * Let's assume that:
* \(x_{i}\) is an \(n\)-dimensional vector of independent variables,
* \(y_{i}\) takes values 1 or \(-1\) denoting the class to which the \(i\)-th point belongs, respectively,
* \(w\) is the weight vector,
* \(b\) is the bias. * During the SVM training process, the weight vector and bias are computed. To allow solving problems which are not linearly separable:
* slack variables \(\xi_{i}\), which aim is to measure the degree of misclassification, are introduced,
* the use of a kernel function \(\varphi\) (which may be polynomial, Gaussian radial basis or sigmoid) allows nonlinear mapping of the nonlinear input space to the high-dimensional feature linear space. * SVM classification can be performed using two variants--\(C\)-classification or \(\upsilon\)-classification [50]. In the case of \(C\)-classification (later referred to as classification type 1), optimisation of the following objective function takes place:  
\[\frac{1}{2}w^{T}w+C\sum_{i}\xi_{i}\to min \tag{7}\]  
subject to following constraints:  
\[\left\{ \begin{matrix}y_{i}\big{(}w^{T}\varphi(x_{i})+b\big{)}\geq 1-\xi_{i}\\ \xi_{i}>0,\ \ i=1,\ \ldots,\ n\end{matrix} \right. \tag{8}\]  
where \(C\) is a constant called capacity--a parameter that determines the trade-off between the margin size and the amount of error in training. The parameter \(C>0\), which means it may take any positive value. For \(\upsilon\)-classification (later referred to as classification type 2) objective function is given:  
\[\frac{1}{2}w^{T}w-\upsilon\rho+\frac{1}{n}\sum_{i}\xi_{i}\to min \tag{9}\]  
subject to:  
\[\left\{ \begin{matrix}y_{i}\big{(}w^{T}\varphi(x_{i})+b\big{)}\geq\rho-\xi_{i} \\ \xi_{i}>0,\ \ i=1,\ \ldots,\ n\\ \rho\geq 0\end{matrix} \right.