In software development, identifying software faults is an important task. The presence of faults not only reduces the quality of the software, but also increases the cost of development life cycle. Fault identification can be performed by analysing the characteristics of the buggy source codes from the past and predict the present ones based on the same characteristics using statistical or machine learning models. Many studies have been conducted to predict the fault proneness of software systems. However, most of them provide either inadequate or insufficient information and thus make the fault prediction task difficult. In this paper, we present a novel set of software metrics called Error-type software metrics, which provides prediction models with information about patterns of different types of Java runtime error. Particular, in this study, the ESM values consist of information of three common Java runtime errors which are Index Out Of Bounds Exception, Null Pointer Exception, and Class Cast Exception. Also, we proposed a methodology for modelling, extracting, and evaluating error patterns from software modules using Stream X-Machine (a formal modelling method) and machine learning techniques. The experimental results showed that the proposed Error-type software metrics could significantly improve the performances of machine learning models in fault-proneness prediction.