In this ABM framework, we assumed that agents are active and intelligent enough to find the best spatial positions for themselves. They are short-memory entities characterized by the Markov property. Individuals remember their specific spatial positions at a present time step and use them to move to the next place. To make decisions about their own movements, agents use a Gaussian process classifier (GPC) to identify positioning suitability based on the information of geometrical room characteristics and other agents’ space occupancy.

The Gaussian process (GP) is a machine-learning extension of the multivariate Gaussian in which the probability function values correspond to random output. GP assumes that the output data are jointly normally distributed using a non-parametric Bayesian model. Posterior inference is made without prescribed model parameters but only by random functions drawn from the prior. GP makes sense in cases where it is likely to assume normally distributed data association for inference, especially when predictions must be made with little information about data.

Suppose that we have a dataset

D of

n observations,

D $=\left\{\left({\mathit{x}}_{p},{f}_{p}\right)\right|p=1,\dots ,n\}$, where

**x** is a

m-variate input vector,

$\mathit{x}\in {\mathbb{R}}^{n\times m}$ and

f $\in {\mathbb{R}}^{n}$ denotes a target output from a function

$f$(

**x**). Given the training dataset, GP would make a prediction for a new input

${\mathit{x}}_{*}$. The joint distribution of the training outputs,

$\mathit{f}$, and the test outputs,

${f}_{*}$, is represented using the covariance matrix,

$K$, such that

where

$y$ =

f(

**x**) and

${y}_{*}$ =

f(

${\mathit{x}}_{*}$), and

$K=K\left(X,X\right),{K}_{*}=K\left({X}_{*},X\right),{K}_{**}=K\left({X}_{*},{X}_{*}\right)$. For

$\mathit{x}$, the three types of covariance matrices are defined by

where

k is the covariance or kernel function used to evaluate the similarity between data points. The kernel is the core ingredient of GP and what makes it different from the general multivariate normal distribution. Rather than relying on a linear product of the deviations of input pairs

$x$ and

${x}_{*}$, GP calculates joint probability or covariance by mapping it into implicit feature space using kernel functions. One of the most frequently used GP kernels is the squared exponential, also known as the radial-basis function (RBF),

where

$\mathit{x}-{\mathit{x}}_{*}{}^{2}$ is the squared Euclidian distance between the two feature (input) vectors,

$l$ is the length scale, and

${\sigma}_{f}$ is the variance of the output

**f**. In the kernel,

${\sigma}_{f}$ is a scale factor by which the covariance is limited to

${\sigma}_{f}^{2}$ at maximum. For multivariate input, a kernel value between two vector points can be obtained by

Based on the above GP regression scheme, a Gaussian process classifier (GPC) can be obtained by mapping

${f}_{*}$ on a sigmoid function,

$\pi $, such as the logistic function; i.e.,

${\pi}_{*}=\pi \left({f}_{*}\right)=\mathsf{\Phi}\left({y}_{*}\right)$. An expected mean value of the probability of class membership

$\overline{{\pi}_{*}}$ is expressed as

The GPC provides a prediction to describe building occupant behavior probabilistically with little prior information. The GPC is also efficient at predicting continuous time-series data of unknown occupant activities. For supervised GP modeling, building room designs with different types of space occupancy were generated to create a dataset. As

Figure 6 shows, seven features representing space geometry and distances between groups (

**x**_{p1}~**x**_{p 7}) were scaled to [0,1] and matched with binary classification labels (0: not occupiable, 1: occupiable).

On top of this, a simple space transition rule can then be applied. It is assumed that each agent decides whether to stay or to leave at any one time step. Then, we have a state space Ω with two labels such that Ω = {1: stay in the room, 2: leave the room}. Introducing the probability of a transition pair based on a hypothesis, a matrix for occupancy determination (

**A**) can be prepared such that

**A** = (a

_{ij}) = [0.75, 0.25; 0.5, 0.5], where a

_{ij} is the probability of an agent to be in state

i after being in state

j. Combining above rules and model parameters, this eventually characterizes a sort of simple “model-based reflex agent” [

17]. According to spatial and environmental information, agents’ perceptual activities to maintain specific occupancy states are driven by the GPC model. Unobserved aspects of occupancy are reflected with the establishment of the AI model. The whole ABM scheme is illustrated in

Figure 7.