*Article* **Constructing a Control Chart Using Functional Data**

### **Miguel Flores 1, Salvador Naya 2, Rubén Fernández-Casal 3, Sonia Zaragoza 4, Paula Raña 3 and Javier Tarrío-Saavedra 5,\***


Received: 12 October 2019; Accepted: 23 December 2019; Published: 2 January 2020

**Abstract:** This study proposes a control chart based on functional data to detect anomalies and estimate the normal output of industrial processes and services such as those related to the energy efficiency domain. Companies providing statistical consultancy services in the fields of energy efficiency; heating, ventilation and air conditioning (HVAC); installation and control; and big data for buildings, have been striving to solve the problem of automatic anomaly detection in buildings controlled by sensors. Given the functional nature of the critical to quality (CTQ) variables, this study proposed a new functional data analysis (FDA) control chart method based on the concept of data depth. Specifically, it developed a control methodology, including the Phase I and II control charts. It is based on the calculation of the depth of functional data, the identification of outliers by smooth bootstrap resampling and the customization of nonparametric rank control charts. A comprehensive simulation study, comprising scenarios defined with different degrees of dependence between curves, was conducted to evaluate the control procedure. The proposed statistical process control procedure was also applied to detect energy efficiency anomalies in the stores of a textile company in the Panama City. In this case, energy consumption has been defined as the CTQ variable of the HVAC system. Briefly, the proposed methodology, which combines FDA and multivariate techniques, adapts the concept of the control chart based on a specific case of functional data and thereby presents a novel alternative for controlling facilities in which the data are obtained by continuous monitoring, as is the case with a grea<sup>t</sup> deal of process in the framework of Industry 4.0.

**Keywords:** functional data analysis; statistical process control; control chart; data depth; nonparametric control chart; energy efficiency
