## 3 Cost Estimation Methodology by Multioutput Regression  
### Multioutput Regression  
Multioutput regression is a method of predicting two or more result values using given input data, and is used to predict x and y coordinate values or predict result values for various future points of time. Most machine learning algorithms are designed to predict a single number. However, the model can be modified and used to support multiple output values as shown in Equation (1). Predicting multiple output values using the given input data is called multioutput regression.  
\[x_{i}=\begin{bmatrix}x_{1,i}\\ x_{2,i}\\ \ldots\\ x_{1,i}\end{bmatrix}y_{i}=\begin{bmatrix}y_{0,i}&y_{1,i}&\ldots&y_{c-1,i} \end{bmatrix} \tag{1}\]  
In multioutput regression, all output values must be correlated with the input value. If there are output values that do not correlate with the input data, they should be predicted together with other output values, or individually using different models.  
The cost function for multivariate regression analysis is almost similar to the cost function for univariate regression analysis. To derive the least squares cost function, the error at the i-th point can be written as \(\hat{x}_{i}^{T}W-y_{i}\). To square this error, the square l2 vector normal must be used. Therefore, in this case, the least square cost function can be viewed as the average of the square Norm for the error at each point as shown in Equation (2)  
\[\text{g}(\text{w})=\frac{1}{p}\sum\
olimits_{i=1}^{p}\Bigl{|}\Bigl{|}\hat{x}_{ i}^{T}W-y_{i}\Bigr{|}\Bigr{|}_{2}^{2}=\frac{1}{p}\sum\
olimits_{i=1}^{p}\sum \
olimits_{c}^{c-1}\left(\hat{x}_{i}^{T}w_{c}-y_{i}\right)^{2} \tag{2}\]  
In Equation (2), if \(c=0\), it becomes the least squares cost function of multiple univariate regression  
Previous construction cost prediction studies mainly dealt with the total construction cost prediction method with various cost influencing factors. Total construction cost is divided into sub construction cost such as architecture, civil engineering, electricity, mechanic, telecommunications, and fire safe construction costs, and these sub construction costs may be required in the initial construction cost prediction.  
Since they are often carried out in the form of separate bids in the construction stage, there are cases where each construction cost must be predicted separately. In this study, it is intended to predict these detailed construction costs at once by using a single learning model using a multioutput regression method.  
First, 10 factors influencing construction costs were selected for input data, and 8 factors including total construction costs were selected as the output data, including building structure, mechanical, civil, landscape, electricity, telecommunication, and fire safe construction cost as shown in Figure 1.  
The analysis procedure in this study for predicting construction costs through multioutput regression is shown in Figure 2. First, the input/output data structure and input/output data to be used for machine learning are determined, and six cases are defined for analyzing the methodology and predictive performance for machine learning, then the ANN model required for machine learning is defined, and the predictive performance of ANN model is analyzed using K-fold. Finally, a prediction model is created for the model with the best performance, and a construction cost prediction test is conducted using this. Tensorflow, Keras library and Visual Studio Code were used for machine learning of construction cost prediction.