# 1. Introduction

Occupancy information is useful for building management to operate lighting and ventilation systems with a view to reduce operational cost [1,2]. Furthermore, real-time occupancy monitoring can assume paramount importance during emergency evacuation [3].

Additional equipment and procedures are required to obtain occupancy information in a given space; such equipment may include video cameras, radio-frequency identification tags, and passive infrared sensors. However, these devices can potentially disturb privacy and are limited only to observing certain areas. Carbon dioxide sensors used for indoor environmental monitoring purposes may have better performance in eliminating the aforementioned drawbacks although they have high uncertainty and slow response characteristics.

Therefore, various post-processing methods have been developed to
estimate the number of occupants. Mumma [4] used CO_{2} measurement
to determine the number of occupants for variable air volume (VAV)
applications. Based on a mass-balance equation for the contaminant,
a transient equation was used to accurately detect occupants, and only
the steady-state terms were used when the transient method results
in oscillation of the estimated occupancy. Methods based on neural
networks [5] also work well to estimate the number of occupants but
require a large amount of data for the preliminary training phases.

Our aim is to develop a Bayesian method that relies on a
mathematical model and does not require a training phase. However,
detailed parameters concerning input models such as ventilation
rates are required. In this study, the occupancy profiles in rooms
with immeasurable ventilation are estimated based upon CO_{2}-
concentration profiles. The CO_{2} concentration profiles obtained over
seven days were also used to obtain the appraisal ventilation rate by
decay and sum-up methods. We investigate the effect of both the
ventilation rate methods upon the estimated number of occupants.

# 2. Methods

## 2.1 Model development

The model space is a seminar room with a total volume of 159 m^{3}.
The room is located on the third floor of the engineering building at
the Kookmin University. One side of the room’s wall is exposed to
the outdoor environment with four operable windows that were kept
closed during the measurement. There is no mechanical ventilation
system for the room. Adjacent rooms are present on both sides and
the hallway is connected to the fourth side of the room, as illustrated
in Figure 1.

## 2.2 Bayesian Markov-chain Monte Carlo (MCMC) approach

The Bayesian MCMC approach is a stochastic simulation technique for computing inferential quantities. According to the Bayesian method, the posterior probability is computed on the basis of the prior probability and the likelihood function is derived from a probability model for the data to be observed. The posterior-distribution relation is given by a combination of the prior distribution and the likelihood as

where *π(N)* is the prior probability of proposed number of
occupants and *f(C|N)* is the likelihood of observing CO_{2} given number
of occupants. *π(N|C)* is the posterior probability of the number of
occupants. Carbon concentration in the room is generated by the CO_{2}
mass-balance equation as follows:

where *V* is the volume of the room, *m* is the CO_{2}-generation rate per
person, *N* represents the number of occupants, *Q* is ventilationrate,
and C_{out} is outdoor CO_{2} concentration. The dynamic model as a
solution of Eq. (2), which is used to generate the likelihood function,
can be expressed as

The concentration at the current time step, *C _{t}*, is determined
by the concentration at the previous time step,

*C*and the steady concentration,

_{t-Δt}*i*, and the time step,

*Δt*. We used the informative Bayesian prior of N with a calculation time step of 3 min.

We input the prior information based on the most likely occurrences
in the observed system, which should approximate the true values
as closely as possible. In this study, the prior of *N* was uniformly
distributed with minimum and maximum levels of 0 and 25 persons,
respectively. The remaining prior information was assumed to have
a Gaussian distribution. The mean of the prior probability, *m*, was
assumed to be 0.553 g/min, equivalent to sitting with a standard
deviation of 30%. We set the mean and the standard deviation of the
prior *Q* according to the measured steady-state-ventilation rate in
section 2.3. The means of the prior concentrations of CO_{2} outdoors
and CO_{2} background is 480 PPM at 5% standard deviation. The
MH-MCMC algorithm for each time step is shown in Figure 2. We
performed 10,000 iterations to collect 5,000 posterior samples after a
burn-in period comprising the first 5,000 iterations.

## 2.3 Quantification of the ventilation rate

The prior information used to calculate the Bayesian model
should be identified with high specificity in order to avoid over- and
underestimation. However, some parameters cannot be measured
accurately due to uncontrollable characteristics. For instance, there
could be difficulties measuring and controlling the ventilation rate
in a space without a ducting system. Infiltration takes place through
cracks in the walls and windows depending on the outdoor-wind
conditions. In this study, the ventilation rate is determined using two
methods: the concentration decay method and the sum-up method.
Both the methods evaluate CO_{2} levels with respect to time. CO_{2}
concentration was recorded minutely for seven days during typical
weekdays in the observed room. The alteration of CO_{2} concentration
is illustrated in Figure 3.

The concentration decay method is commonly used to measure the
ventilation rate by applying a tracer gas technique [6]. We utilize the
CO_{2} naturally generated by humans as a tracer gas. The ventilation
rate is estimated by analyzing the concentration decay rate at the end
of each day.

The space concentration decays down to the background concentration after unoccupied time due to dilution of outdoor concentration. Assuming that the space concentration is mixed uniformly and the ventilation rate is constant, the exponential decay from Figure 4 can be observed. The decay concentration is normalized from the observed period as

where C_{O} is the initial concentration at the start of the decay
observation period. The log-normalized concentration during the
selected period is illustrated in Figure 4. The slopes of linear fitting
in the figure represent the air-exchange rates summarized in Table
1. The steady-state-ventilation rate can be obtained by averaging the
seven observed air-exchange rates and then multiplying the result by
the given space volume. The calculated ventilation rate was defined to
have 0.60 ACH (1.6 ±0.74 m^{3}/min) with a standard deviation of 0.74
m^{3}/min or 46% of the mean.

The sum-up method integrating total CO_{2} generation rate and
number of occupant over the concentration deference between indoor
and outdoor as formulated in Eq. 5. A summary of the appraised
ventilation rates obtained by the sum-up method is shown in Table
2. The average of ventilation rate is 1.52 m^{3}/min and the standard
deviation is 0.30 m^{3}/min or 20% from the mean:

# 3. Occupancy Estimation

Each ventilation rate appraisal is used as a prior input for Bayesian estimation. The mean values of Q obtained from the decay and sumup methods are similar, but the standard deviation between them has a large difference. The occupancy estimation using the Q prior of the decay method is shown in Figure 5. A wide standard deviation in the Q prior produces severe fluctuation in the occupation time. Figure 6 shows occupancy estimation using the Q prior according to the sumup method. Estimation with this method fluctuates less compared to the decay method. Error estimates using the Q priors of the decay and sum-up methods are 138% and 130%, respectively. The high error shown by both methods may result from variation of the ventilation rate at each time step. However, both methods produce highly valuable information concerning space-ventilation characteristics and the estimation profile also showed good agreement with the real occupancy schedule.

# 4. Conclusion

Occupancy estimation based on the measured CO_{2} concentration
has been successfully conducted using the Bayesian MCMC approach.
The CO_{2}-concentration profile is also used to quantify the ventilation
rate by the concentration decay and sum-up methods. The mean
and standard deviation obtained from seven samples of ventilation
measurement are used as the input priors for the Bayesian MCMC.
Wide standard deviations of the ventilation rates measured using the
decay method reduce the accuracy of Bayesian estimation through
severe fluctuations and a narrow standard deviation produced by
the sum-up method can offset the estimation. Quantification of the
ventilation rate using the concentration decay and sum-up approaches
produces reasonable Bayesian estimation results with similar error
values.

# Competing Interests

The authors declare that they have no competing interests.