# While we are taking full responsibility for any remaining er

While we are taking full responsibility for any remaining errors and shortcomings of the paper, we would like to thank Dr. Jong-Hyeon Jeong, Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, for offering us insight and literature pointers useful in our quest to decompose the CPH model and Krzysztof Nowak, for fitting of Noisy-MAX parameters to Recidivism data in our experiments. Jirka Vomlel offered the proof of Theorem 1. We also thank the anonymous reviewers for the PGM conferences and for IJAR for their valuable input that has greatly improved the quality of this paper.

Introduction
Data-driven models are widely used to investigate the coefficient of performance (COP) of chiller systems and identify opportunities for optimization, efficiency improvement, and fault detection and diagnosis (FDD) (Wang et al., 2018b, Wei et al., 2014). The COP is defined as the cooling CC-4047 output divided by the electric power input of chillers. A higher COP means that the chiller is more energy efficient. For the system COP, the input is the total electric power of chillers and their auxiliary devices like chilled water pumps, condenser water pumps and cooling tower fans. The development of data-driven models is based purely on system operating data which reflect the unique design and operating characteristics of existing chiller systems with various scales and complexities (Afroz et al., 2018). This helps eliminate the use of physical-based models requiring the development of mathematical equations and sophisticated solving algorithms.
Various types of modelling parameters and approaches have been used to analyze specific issues of the operating performance of chiller systems. Basic chiller performance models like Gordon and Ng\'s models (Gordon et al., 1995) correlated the COP (i.e. the output) with the inputs of the chiller load and temperatures at the evaporator and condenser sides. Another COP model developed by Wang (2017) considered pressures and temperatures at the chilled water side and temperatures at the heat rejection side. Baillie and Bollas (2017) developed a model for a whole chiller system with chillers, primary loop chilled water pumps, condenser water pumps and cooling towers which operated in response to varying building cooling loads. The model had sufficient accuracy to analyze an efficiency change due to condenser fouling and failing to open a valve actuator.
Some studies used different mathematical techniques to investigate the optimal number and loading of operating chillers to reduce their overall electricity consumption. The techniques involved an exchange market algorithm used normally to optimize the trading of shares on a stock market (Sohrabi et al., 2018), an improved invasive weed optimization algorithm (Zheng and Li, 2018), an uncertainty analysis of the cooling load profile and COP (Huang et al., 2018), a holistic optimization on the load distribution, the condenser water set point and the cooling tower energy consumption (Liao et al., 2018), an optimized load distribution based on the total cooling capacity of operating chillers and the condenser water set point (Huang et al., 2016), the construction of a near optimal performance map (Wang et al., 2018), an improved firefly algorithm (Coelho and Mariani, 2013), a particle swarm optimization with neural network (Chen et al., 2014), a cuckoo search algorithm (Coelho et al., 2014), a branch and bound algorithm to search for optimal loading points based on adjusted cooling capacities (Liu et al., 2017), and a differential evolution algorithm (Lee et al., 2011).
Another stream of chiller system models focused on FDD. Yan et al. (2018) used a back-tracing sequential forward feature selection algorithm to select the most significant features for FDD and then used the support vector machine for measuring accuracy. Li and Hu (2018) considered the density-based spatial clustering of operating data with noise to produce sub-models by the principal component analysis in order to improve the sensitivity and reliability of FDD and accuracy estimation. Huang et al. (2018) established associative classifiers to interpret different types of chiller fault which helped enhance the average correct fault diagnosis ratio. Tran et al. (2016) found that the radial basis function combined with the exponentially weighted moving average residual control chart gave an accurate model with a sensitive fault detection technique to achieve the high diagnosis performance. In Wang et al.’s study (Wang et al., 2017), the distance rejection technique and multi-source non-sensor information were merged into the Bayesian network to enhance the capability of identifying new sources of fault and to increase diagnostic accuracy. In Li et al.’s study (Li et al., 2016), the linear discriminant analysis with 2-stage fault classification was used to enhance the effectiveness of FDD.