In this paper, the normalized interval regression model is proposed to analyze how the uncertainty of an output is allocated to the coefficients of input variables and which input variable is more deterministic. In the normalized upper regression model, the input variable whose coefficient has a wider spread has a less deterministic effect on the output. It implies that estimating the output by such an input variable is more unreliable. In the normalized lower regression model, the influence relationship between the input and the output variables can be found out by using the interval coefficients. The positive lower endpoint of the interval coefficient of an input variable implies that the output variable has a positive influence relationship with this input variable whereas the negative upper endpoint of the interval coefficient of an input variable shows that the output variable has a negative influence relationship with it.
In order to obtain the possibilistic functional relationship of the major data, we propose two outlier detection approaches for the upper and lower interval regression models, respectively. In the duality approach, the dual problem of the upper regression model is used to find out the potential outliers. By introducing the decrease ratio, the outliers can be efficiently identified and deleted. In the relaxation approach, by introducing the relaxation variable, the lower regression model becomes feasible. The outliers are regarded as the data corresponding to the largest value of the relaxation variable.
As a real application, we study a house pricing problem. Household size, loan ratio and annual household income are identified as the inputs and acceptable purchase price is the interval output. The data were collected from the survey in Shanghai. Since some data make the upper regression model have a huge spread and the lower regression model infeasible, by using the proposed approaches, we delete the outliers and obtain appropriate upper and lower interval regression models. Analysis results provide insights into the housing market and important policy implications in regulating urban land development.
In this paper, the normalized interval regression model is proposed to analyze how the uncertainty of an output is allocated to the coefficients of input variables and which input variable is more deterministic. In the normalized upper regression model, the input variable whose coefficient has a wider spread has a less deterministic effect on the output. It implies that estimating the output by such an input variable is more unreliable. In the normalized lower regression model, the influence relationship between the input and the output variables can be found out by using the interval coefficients. The positive lower endpoint of the interval coefficient of an input variable implies that the output variable has a positive influence relationship with this input variable whereas the negative upper endpoint of the interval coefficient of an input variable shows that the output variable has a negative influence relationship with it.In order to obtain the possibilistic functional relationship of the major data, we propose two outlier detection approaches for the upper and lower interval regression models, respectively. In the duality approach, the dual problem of the upper regression model is used to find out the potential outliers. By introducing the decrease ratio, the outliers can be efficiently identified and deleted. In the relaxation approach, by introducing the relaxation variable, the lower regression model becomes feasible. The outliers are regarded as the data corresponding to the largest value of the relaxation variable.As a real application, we study a house pricing problem. Household size, loan ratio and annual household income are identified as the inputs and acceptable purchase price is the interval output. The data were collected from the survey in Shanghai. Since some data make the upper regression model have a huge spread and the lower regression model infeasible, by using the proposed approaches, we delete the outliers and obtain appropriate upper and lower interval regression models. Analysis results provide insights into the housing market and important policy implications in regulating urban land development.
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