## 3 Real-Building Case Study
### Missing Data Description of Selected Sensors  
In Section 3.2, missing data characteristics, including missing data size and missing data randomness, are introduced. In terms of missing data size, Table 2 shows the total number of missing data and maximum holes (or gaps, meaning length of consecutive missing data) for each sensor. Table 2 indicates a varieties of missing data size for the selected sensors. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Equipment** & **Sensor Type** & **Original Sensor Name from BAS** & **Sensor Description** \\ \hline
1 & & Energy meter & peco\_meter/elec\_usage\_chiller & Chiller electricity consumption \\
2 & & Flow meter & chws\_control/chw\_flow & Chilled water flow rate \\
3 & Chiller & Thermometer & chws\_control/chws\_temp & Chilled water supply temperature \\
4 & & Differential pressure sensors & chws\_control/delta\_press & Chilled water supply and return \\  & & & pressure difference \\
5 & & Differential pressure sensors & chws\_control/cw\_delta\_press & Cooling water supply and return \\
6 & Cooling tower & Thermometer & chws\_control/cwr\_temp & pressure difference \\
7 & & Valve position & chws\_control/cw\_vlr\_fdbk & Cooling water return temperature \\
8 & & Fan status & ct\_control/ct\_fan\_status/trend\_log & Cooling tower fan status \\
9 & & Thermometer & ahul1/ra\_temp/trend\_log & AHU 1 return air temperature \\
10 & & Valve position & ahul/econ & AHU 1 economizer valve position \\
11 & & Valve position & ahul/chw\_valve & AHU1 cooling coil valve position \\
12 & AHU & Humidity sensor & ahul/ra\_humidity & AHU1 return air humidity \\
13 & & Ampere meter & ahul/i\_fan\_amps & AHU 1 supply air fan ampere \\
14 & & Valve position & ahul/h\_stem\_vlr\_1\_3 & AHU 1 steam valve position \\
15 & & Differential pressure sensors & ahul/stat\_press & AHU 1 supply air static pressure \\
16 & & Thermometer & ahul/pht\_temp & AHU 1 preheat coil water \\
17 & & Flow meter & vv-1-b-1/air\_flow/flow\_input & temperature \\
18 & VAV & Thermometer & vv-1-b-1/da\_temp/trend\_log & VAV 1 discharge air temperature \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sensors used in the case study of the developed framework. Figure 2: Appearance of Nesbitt Hall. Most sensor data listed in Table 1 are systematically missed. The best example is the chiller energy meter. Figure 3 shows the Kernel Density Estimation plot reflecting density distribution of missing data happening time (hour of day). It shows that data are more frequently missed during the chiller's start-up time (around 7-8 am) of the HVAC system, where energy suddenly increases. This indicates that the missing mechanism of chiller energy meter data is somehow systematic and not random. In this study, the systematic missing of each sensor are quantified by the randomness of time indicators for the time when data are missing, including hour of day, day of week, month of year, and weekday indicator (0 for weekday and 1 for weekend). For example, the randomness for the energy meter in terms of hour of day is low because data are missed following a pattern of fault happening time, or data are systematically missing in terms of hour of day in energy meters. We use a statistical test to quantify the randomness of missing data in this study. The null hypothesis is that the data are generated in a random manner, and the equation of randomness is shown in Equation (1), which can be found in the book [19]. In Equation (2), R is the number of runs. The concept of run is defined as a series of consecutive positive (or negative) values of the difference between one data point and the median of all data. n1 is the number of data that is greater than the median of all data, and n2 is the number of data that is less than the median.