# Beauty production in heavy-ion collisions with ALICE at the LHC††thanks:
Presented at Quark Matter 2021, Krakow (Poland).

${\rm Xinye}$ ${\rm Peng}^{1,2}$ on behalf of the ALICE collaboration  1China
University of Geosciences (Wuhan), China  2Central China Normal University,
China

###### Abstract

In this contribution, the final measurements of the centrality dependence of
the nuclear modification factor ($R_{\rm AA}$) of non-prompt $\rm D^{0}$ in
Pb–Pb collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV will be
presented. These measurements provide important constraints to the mass
dependence of in-medium energy loss and hadronisation of the beauty quark. The
$p_{\rm T}$-integrated non-prompt $\rm D^{0}$ $R_{\rm AA}$ will be presented
for the first time and will be compared to the prompt $\rm D^{0}$ one. This
comparison will shed light on possible different shadowing effects between
charm and beauty quarks. In addition, the first measurements of non-prompt
$\rm D_{s}^{+}$ production in central and semi-central Pb–Pb collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV will be discussed. The non-
prompt $\rm D_{s}^{+}$ measurements provide additional information on the
hadronisation of beauty quarks and the production yield of $\rm B_{s}^{0}$
mesons. Finally, the first measurement of non-prompt D-meson elliptic flow in
Pb–Pb collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV will also
be discussed. These measurements can constrain the degree of thermalisation of
beauty quarks in the hot and dense QCD medium.

##  1 Introduction

Heavy quarks (charm and beauty) are produced in hard-scattering processes over
short time scales compared to the quark–gluon plasma (QGP). They probe the
whole system evolution interacting with the medium constituents. In
particular, because of the larger mass, beauty quarks are expected to lose
less energy [1, 2] and diffuse less than the charm quarks [3, 4]. Therefore,
the comparison between charm and beauty nuclear modification factor ($R_{\rm
AA}$) and elliptic flow ($v_{2}$) provides insight into the quark mass-
dependence of energy loss, as well as heavy-quark diffusion properties.

The D mesons from beauty-hadron decays (non-prompt D) are excellent probes for
beauty properties. Existing data on the production of B mesons [5], and
J/$\psi$ from beauty decays [6, 7, 8] at midrapidity are limited by large
uncertainties, while the broad correlation between the transverse momenta
($p_{\rm T}$) of the lepton and the parent beauty hadron reduces the
effectiveness of beauty-decay lepton measurements [9, 10].

In these proceedings, the measurement of open-beauty production in Pb–Pb
collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV from ALICE at
midrapidity is reported.

##  2 D-meson reconstruction

The non-prompt D mesons are reconstructed via their hadronic decay channels
${\rm D}^{0}\to{\rm K}^{-}\pi^{+}$, and ${\rm
D}_{s}^{+}\to{\rm\pi}^{+}\phi\to{\rm\pi}^{+}{\rm K}^{+}{\rm K}^{-}$. A multi-
classification BDT algorithm [11], trained using variables sensitive to the
decay topology and particle identification, is utilised to simultaneously
increase the non-prompt (${\rm b}\to{\rm D}$) fraction and suppress the
combinatorial background. After extracting the yield via an invariant mass
analysis, the ${\rm b}\to{\rm D}$ fraction is estimated by a
$\chi^{2}$-minimisation approach based on variation of ML-based selections
[11]. The measurement of the non-prompt $\rm D^{0}$ $v_{2}$ is performed with
the scalar-product (SP) method [12]. In order to obtain non-prompt $\rm D^{0}$
$v_{2}$ ($v_{2}^{\rm non-prompt}$), a linear fit of $v_{2}^{\rm sum}$ is
performed as a function of ${\rm b}\to{\rm D}$ fraction, and extrapolated to
$f_{\rm non-prompt}=1$, as described in [13].

##  3 Results

Figure 1: Left: the $R_{\rm AA}$ of non-prompt $\rm D^{0}$ mesons [14] as a
function of $p_{\rm T}$ in the 0–10% centrality classes, compared with the
$R_{\rm AA}$ of prompt $\rm D^{0}$ mesons [15]. Right: non-prompt to prompt
$\rm D^{0}$-meson $R_{\rm AA}$ ratio as a function of $p_{\rm T}$ in the 0–10%
central Pb–Pb collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV,
compared to model predictions [16, 17, 18, 19, 20] (top), and to different
modifications of LGR calculations [17, 18] (bottom).

The left panel of Fig. 1 shows the $R_{\rm AA}$ of non-prompt $\rm D^{0}$
mesons [14] as a function of $p_{\rm T}$ measured in the 0–10% most central
Pb–Pb collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV. The
result is compared with the prompt $\rm D^{0}$-meson $R_{\rm AA}$ [15]. The
non-prompt $\rm D^{0}$-meson $R_{\rm AA}$ is significantly higher than the
prompt $\rm D^{0}$ one for $p_{\rm T}>5$ GeV/$c$, indicating that non-prompt
$\rm D^{0}$ mesons are less suppressed than prompt $\rm D^{0}$ mesons, and
supporting the expectation that beauty quarks lose less energy than charm
quarks because of their larger mass. An extrapolation of the measured spectrum
is performed. The resulting non-prompt $\rm D^{0}$-meson $R_{\rm AA}$ for
$p_{\rm T}>0$ is $1.00\pm 0.10\ ({\rm stat.})\pm 0.13\ ({\rm
syst.}){}^{+0.08}_{-0.09}\ ({\rm extr.})\pm 0.02\ ({\rm norm.})$ [14] in the
0–10% centrality class, which is compatible with unity within uncertainties,
and larger than the prompt one [15] within less than $1.5\sigma$.

The non-prompt to prompt $\rm D^{0}$-meson $R_{\rm AA}$ ratio as a function of
$p_{\rm T}$ in the 0–10% central Pb–Pb collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV is presented in the right
panel of Fig. 1. The ratio can be described well by model predictions [16, 17,
18, 19, 20] that include collisional and radiative processes (top panel). In
order to further understand the data pattern at low $p_{\rm T}$ and the
enhancement with respect to unity at high $p_{\rm T}$ for the data, the ratio
is compared with different LGR model [17, 18] calculations (bottom panel),
highlighting the role of coalescence, shadowing, and mass dependence of energy
loss. The LGR model suggests that the “valley” structure at low $p_{\rm T}$ is
mainly due to the formation of prompt D mesons via charm-quark coalescence
(case iv), and the significant enhancement of the ratio at high $p_{\rm T}$ is
interpreted as the effect of the mass dependence of the in-medium energy loss
(case i).

Figure 2: The $R_{\rm AA}$ of non-prompt $\rm D_{s}^{+}$ mesons [21] divided
by the one of prompt $\rm D_{s}^{+}$ mesons [22] (left) and non-prompt $\rm
D^{0}$ mesons [14] (right) as a function of $p_{\rm T}$ for the 0–10% and
30–50% centrality classes in Pb–Pb collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV. The measurements are
compared with TAMU model predictions [19].

The $R_{\rm AA}$ of non-prompt $\rm D_{s}^{+}$ mesons divided by that of
prompt $\rm D_{s}^{+}$ mesons [22] (left panel) and non-prompt $\rm D^{0}$
mesons [14] (right panel) are shown in Fig. 2. In the 0–10% centrality class,
a hint that both ratios are larger than unity in the $4<p_{\rm T}<12$ GeV/$c$
interval is presented, suggesting an enhancement of non-prompt $\rm D_{s}^{+}$
in this interval. The trend can be explained by the interplay of mass
dependence of energy loss and recombination in the QGP medium. The ratios are
compatible with unity within uncertainties. The TAMU predictions [19]
qualitatively describe the results for central collisions. For semicentral
collisions the TAMU model overestimates the $R_{\rm AA}$ ratio values.

Figure 3: The non-prompt $\rm D^{0}$ $v_{2}$ in Pb–Pb collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV in the 30–50% centrality
class, compared to that of prompt average non-strange D mesons [23] (left),
and to the model predictions [17, 24, 25, 19, 26, 27] (right).

Figure 3 shows the first measurement of non-prompt $\rm D^{0}$ $v_{2}$ in
30–50% Pb–Pb collisions at $\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV,
compared to the average $v_{2}$ of prompt D mesons [23] (left panel), and to
model predictions [17, 24, 25, 19, 26, 27] (right panel). The non-prompt $\rm
D^{0}$ $v_{2}$ is found to be positive with $2.7$$\sigma$ significance for
$2<p_{\rm T}<12$ GeV/$c$. The significance for the difference between non-
prompt $\rm D^{0}$ $v_{2}$ and that of prompt average non-strange D mesons is
$3.2$$\sigma$ for $2<p_{\rm T}<8$ GeV/$c$, indicating a different degree of
participation to collective motion between charm and beauty quarks.
Theoretical predictions [17, 24, 25, 19, 26, 27] based on beauty-quark
transport in the hydrodynamically expanding medium can fairly describe the
data within uncertainties. Future measurements with higher accuracy will
provide important constraints to the models, and allow for accurate extraction
of the spatial diffusion coefficient with beauty quarks.

##  4 Conclusion

In this contribution, the most recent results on the production and azimuthal
anisotropy of non-prompt D mesons, measured in Pb–Pb collisions at
$\sqrt{s_{\scriptscriptstyle\rm NN}}$ = 5.02 TeV, were presented. The non-
prompt $\rm D^{0}$ $R_{\rm AA}$ provides an essential constraint on the mass
dependence of energy loss in the medium. The non-prompt $\rm D_{s}^{+}$
$R_{\rm AA}$ helps to further understand beauty-quark hadronisation in heavy-
ion collisions. The first measurement of non-prompt $\rm D^{0}$-meson $v_{2}$
indicates a different degree of participation in the collective motion between
charm and beauty quarks. With the LHC Run3, the upgraded ITS and the increased
integrated luminosity will allow for more precise beauty-hadron measurements
at midrapidity with ALICE.

##  5 Acknowledgement

This work was partly supported by the NSFC Key Grant 12061141008, the National
key research and development program of China under Grant 2018YFE0104700 and
2018YFE0104800, the National Natural Science Foundation of China (Grant No.
12175085), and Fundamental research funds for the Central Universities (No.
CCNU220N003).

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# A Non-$\mathbb{R}$-Factorizable Product
of $\mathbb{R}$-Factorizable Groups

Ol’ga Sipacheva  Department of General Topology and Geometry, Faculty of
Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Leninskie
Gory 1, Moscow, 199991 Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

An example of two $\mathbb{R}$-factorizable groups whose product is not
$\mathbb{R}$-factorizable is constructed. One of these groups is second-
countable.

###### Key words and phrases:

topological group, $\mathbb{R}$-factorizable group, product of
$\mathbb{R}$-factorizable groups

###### 2020 Mathematics Subject Classification:

22A05, 54F45

In [1] and [2] we constructed an example of two groups, one Lindelöf and the
other cosmic, whose product is not $\mathbb{R}$-factorizable. In this paper we
show that cosmic group can be made second-countable, which gives a negative
solution of Problem 8.5.2 in [3].

A topological group G is said to be _$\mathbb{R}$ -factorizable_ if, for every
continuous function $f\colon G\to\mathbb{R}$, there exists a continuous
epimorphism $h\colon G\to H$ onto a second-countable topological group $H$ and
a continuous function $g\colon H\to\mathbb{R}$ such that $f=g\circ h$. It is
known that any Lindelöf group is $\mathbb{R}$-factorizable [4, Assertion 1.1]
and that $\dim_{0}H\leqslant\dim_{0}G$ for any topological group $G$ and any
$\mathbb{R}$-factorizable subgroup $H$ of $G$.

The heart of the construction in [1] and [2] is two spaces $C_{1}$ and $C_{2}$
with the following properties:

  1. (1)

$\dim_{0}C_{i}=0$ for $i=1,2$;

  2. (2)

$\dim_{0}(C_{1}\times C_{2})>0$;

  3. (3)

$C_{1}^{n}$ is Lindelöf for each $n\in\mathbb{N}$;

  4. (4)

the underlying set of the space $C_{1}$ is the Cantor set $C$, its topology is
finer than that of $C$, and it has a base consisting of sets closed in $C$;

  5. (5)

$C_{2}$ is second-countable; in fact, $C_{2}$ is a subspace of the Cantor set
$C$.

According to Theorem 7 of [5], both spaces $C_{1}$ and $C_{2}$ are retracts of
certain topological groups $G_{1}$ and $G_{2}$, respectively. Therefore,
$C_{1}\times C_{2}$ is a retract of (and hence $C$-embedded in) $G_{1}\times
G_{2}$, which implies that $\dim_{0}(G_{1}\times G_{2})>0$. In [1] and [2] for
$G_{1}$ and $G_{2}$ the free Abelian topological groups $A(C_{1})$ and
$A(C_{2})$ were taken. It was shown that $G_{1}$ and $G_{2}$ are Lindelöf,
zero-dimensional, and topologically isomorphic to closed subgroups of groups
$M_{1}$ and $M_{2}$, each of which is a product of zero-dimensional second-
countable topological groups. Since any product of zero-dimensional second-
countable topological spaces is strongly zero-dimensional [6], it follows that
so is the product $M_{1}\times M_{2}$, and it contains $G_{1}\times G_{2}$ as
a subgroup. Therefore, $G_{1}\times G_{2}$ is not $\mathbb{R}$-factorizable,
although both $G_{1}$ and $G_{2}$ are.

In this paper we show that the group $G_{2}$ can be made second-countable. To
be more precise, we show that $C_{2}$ is a retract of a zero-dimensional
second-countable group $G_{2}$. This does not affect the above argument
(except that we can set $M_{2}=G_{2}$) but gives an example of an
$\mathbb{R}$-factorizable group which ceases to be $\mathbb{R}$-factorizable
when multiplied by a second-countable group.

Recall that the Boolean group $B(X)$ with basis $X$ is the set $[X]^{<\omega}$
of finite subsets of $X$ endowed with the operation $\vartriangle$ of
symmetric difference. Each point $x\in X$ is identified with the singleton
$\\{x\\}$. We use the notation $\mathop{+_{\scriptscriptstyle 2}}$ for the
group operation of $B(X)$ and occasionally write $\vartriangle$ instead of
$\mathop{+_{\scriptscriptstyle 2}}$. Thus, if $x\in X$, $F,G\in B(X)$, and
$\mathbf{A}\subset B(X)$, then

$\displaystyle x\mathop{+_{\scriptscriptstyle
2}}F=\\{x\\}\mathop{+_{\scriptscriptstyle 2}}F=\\{x\\}\vartriangle F,\qquad
F\mathop{+_{\scriptscriptstyle 2}}G=F\vartriangle G,$ $\displaystyle
F\mathop{+_{\scriptscriptstyle 2}}\mathbf{A}=\\{F\mathop{+_{\scriptscriptstyle
2}}A:A\in\mathbf{A}\\}=\\{F\vartriangle A:A\in\mathbf{A}\\}.$

When $X$ is a Tychonoff space, there exists a group topology on $B(X)$ such
that $B(X)$ with this topology contains $X$ as a subspace. The finest group
topology with this property is called the _free group topology_ on $B(X)$, and
the group $B(X)$ endowed with this topology is called the _free Boolean
topological group_ of $X$; see [7] for details. In what follows, we use the
same notation $B(X)$ for the abstract Boolean group with basis $X$ and for the
free Boolean topological group of $X$.

In Theorem 3 (version 2) of [5], for a non-Archimedean space $X$, a retraction
of the clopen subset

$B_{*}(X)=\\{F\in B(X):\text{$|F|$ is odd}\\}$

of the free Boolean topological group $B(X)$ onto $X$ was constructed. In the
particular case of the Cantor set $C$, the construction (slightly modified) is
as follows.

Recall that $C$ can be represented as the subset of $[0,1]$ consisting of
those numbers whose ternary expansions do not contain 1. This suggests the
natural base $\mathscr{B}$ for the topology of $C$:

$\mathscr{B}=\\{U_{n_{1}\dots n_{k}}:k\in\mathbb{N},n_{i}\in\\{0,2\\}\text{
for }i\leqslant k\\},$

where $U_{n_{1}\dots n_{k}}$ denotes the set of all numbers in $[0,1]$ whose
ternary expansions begin with $0.n_{1}\dots n_{k}$. Clearly, the elements of
$\mathscr{B}$ form a tree with respect to reverse inclusion and every element
of $\mathscr{B}$ is clopen.

There are two natural orders on the set of subsets of $C$, the order by
inclusion and the order induced by the usual order on $C\subset\mathbb{R}$. In
what follows, when writing, say, “$A<B$,” “$\min A$,” or “$A$ is on the left
of $B$,” we always mean the latter, unless otherwise is explicitly stated.

Let $F$ be any finite subset of $C$. We say that a set $A\subset C$ is _$F$
-void_ if $A\cap C=\varnothing$, _$F$ -even_ if $|A\cap F|$ is even and
positive, and _$A$ -odd_ if $|A\cap F|$ is odd. Clearly, each $F$-even element
of $\mathscr{B}$ is contained in a maximal (by inclusion) $F$-even element of
$\mathscr{B}$, and the union of these maximal $F$-even elements is equal to
the union of all $F$-even elements of $\mathscr{B}$. Moreover, this union is
itself $F$-even.

For $F\in B_{*}(C)$, we set

$r(F)=\min(F\setminus\bigcup\\{B\in\mathscr{B}:\text{$B$ is $F$-even or
$F$-void}\\})\\\ =\min(F\setminus\bigcup\\{B:\text{$B$ is an inclusion-maximal
$F$-even element of $\mathscr{B}$}\\}).$

The map $r\colon B_{*}(C)\to C$ thus defined is a retraction [5, Theorem 7
(version 1)]. Extending it by, say, $0$ to $B(C)$, we obtain a retraction
$\hat{r}\colon B(C)\to C$ (because $B_{*}(C)$ is clopen in $B(X)$). The rest
of this paper is devoted to the proof that $\hat{r}$ is continuous with
respect to a second-countable zero-dimensional group topology $\tau$ on $B(C)$
defined below.

Given a disjoint open cover $\Gamma$ of $C$, we set

$\mathbf{H}_{\Gamma}=\\{F\in B(C):\text{$|F\cap U|$ is even for each
$U\in\Gamma$}\\}.$

Clearly, $\mathbf{H}_{\Gamma}$ is a subgroup of $B(C)$. It is open in the free
group topology of $B(C)$ (see [7]).

The countable family

$\mathscr{H}=\\{\mathbf{H}_{\Gamma}:\text{$\Gamma$ is a (finite) disjoint
cover of $C$ by elements of $\mathscr{B}$}\\}$

of subgroups is a local base at zero for a group topology $\tau$ of $B(C)$,
which is coarser that the free group topology. It is easy to check that $C$ is
contained in $(B(C),\tau)$ as a subspace. Indeed, take any point $x\in C$ and
any base neighborhood $V_{x}$ of $x$. Let $\Gamma$ be a disjoint cover of $C$
consisting of $V_{x}$ and some other elements of $\mathscr{B}$. If
$F\in\mathbf{H}_{\Gamma}$ and $x\mathop{+_{\scriptscriptstyle
2}}F=\\{x\\}\vartriangle F\in C$, then either $F=\varnothing$ or
$F=\\{x,y\\}$. In the latter case, $x\mathop{+_{\scriptscriptstyle 2}}F=y$ and
by the definition of $\mathbf{H}_{\Gamma}$ the point $y$ must belong to the
same element of $\Gamma$ as $x$, that is, to $V_{x}$. Thus,
$(x\mathop{+_{\scriptscriptstyle 2}}\mathbf{H}_{\Gamma})\cap C\subset V_{x}$.
This shows that the topology induced by $\tau$ on $C$ is not coarser than the
original topology of $C$. On the other hand, it cannot be finer, because
$\tau$ is coarser than the free group topology. Obviously, $(B(C),\tau)$ is
$T_{0}$ and hence Tychonoff.

Note that all elements in any $\mathbf{H}_{\Gamma}$ are of even cardinality.
Therefore, for every $F\in B_{*}(C)$, we have $F\mathop{+_{\scriptscriptstyle
2}}\mathbf{H}_{\Gamma}=\\{F\vartriangle H:H\in\mathbf{H}_{\Gamma}\\}\subset
B_{*}(C)$.

Let us show that the map $r$ is continuous with respect to the topology
$\tau|_{B_{*}(C)}$. Suppose that $x=r(F)$ for $F\in B_{*}(X)$. By construction
$x\in F$. Take any neighborhood $U$ of $x$. Let $V_{1},\dots,V_{m}$ be all
inclusion-maximal $F$-even elements of $\mathscr{B}$; their number is finite
because they are pairwise disjoint (since $\mathscr{B}$ is a tree) and each of
them intersects the finite set $F$. None of these elements contains $x$
(because $x=r(F)$), and all of them are clopen. Choose a base neighborhood
$V_{x}\in\mathscr{B}$ of $x$ satisfying the conditions $V_{x}\subset U$,
$V_{x}\cap F=\\{x\\}$, and $V_{x}\cap V_{i}=\varnothing$ for $i\leqslant m$.
Consider the cover of $C$ consisting of the sets $V_{1},\dots,V_{m}$ and
$V_{x}$ and of all elements of $\mathscr{B}$ disjoint from them. Let $\Gamma$
be a disjoint refinement of this cover. Clearly, it is finite. We claim that
$r(F\mathop{+_{\scriptscriptstyle 2}}\mathbf{H}_{\Gamma})\subset V_{x}$.

Indeed, take an $H\in\mathbf{H}_{\Gamma}$. We must show that $r(F\vartriangle
H)\in V_{x}$. Note that an element $V$ of $\Gamma$ is ($F\vartriangle H$)-odd
if and only if it is $F$-odd, because each element of $\Gamma$ is either
$H$-even or $H$-void and a point of $F$ can be cancelled in $F\vartriangle H$
only by some point of $H$.

Let $V$ be the leftmost $F$-odd (= ($F\vartriangle H$)-odd) element of
$\Gamma$. Then $V\cap F$ is not covered by any inclusion-maximal $F$-even
elements of $\mathscr{B}$, because all such elements are included in $\Gamma$
and $V$ is not among them. By the definition of the map $r$ we have
$x=r(F)\leqslant\min(V\cap F)$, and $V_{x}$ is $F$-odd. Therefore, $V=V_{x}$.

The point $r(F\vartriangle H)$ cannot belong to an ($F\vartriangle H$)-even
element $W$ of $\Gamma$, because $\Gamma\subset\mathscr{B}$ and so all points
occurring in such a $W$ are discarded when defining $r(F\vartriangle H)$.
Therefore, the element $V$ of $\Gamma$ containing $r(F\vartriangle H)$ either
coincides with $V_{x}$ or is on the right of $V_{x}$. According to the
definition of $r$, there exists a family $\mathscr{B}^{\prime}$ of
$(F\vartriangle H)$-even elements of $\mathscr{B}$ such that
$\bigcup\mathscr{B}^{\prime}\supset\\{y\in F\vartriangle H:y<V\\}$.

Suppose that $V\neq V_{x}$. Note that $V_{x}$ is $(F\vartriangle H)$-odd,
because it contains precisely one point of $F$ and an even number (or none) of
the points of $H$. In particular, $(F\vartriangle H)\cap
V_{x}\neq\varnothing$. Let $W_{1},\dots,W_{k}$ be all inclusion-maximal
elements of $\mathscr{B}^{\prime}$ intersecting $(F\vartriangle H)\cap V_{x}$.
Each $W_{i}$, being an element of $\mathscr{B}$, either is contained in some
element of $\Gamma$ or is a union of some elements of $\Gamma$, because
$\mathscr{B}$ is a tree and $\Gamma\subset\mathscr{B}$ is a cover of $C$. By
maximality the sets $W_{1},\dots,W_{k}$ are pairwise disjoint. Therefore, if
$k\geqslant 2$, then all of them are contained in $V_{x}$ and
$V_{x}\cap(F\vartriangle H)=\bigcup_{i\leqslant k}W_{i}\cap(F\vartriangle H)$.
This is impossible, because $|V_{x}\cap(F\vartriangle H)|$ is odd and all
$|W_{i}\cap(F\vartriangle H)|$ are even. Thus, some element $W$ of
$\mathscr{B}^{\prime}$ contains $V_{x}\ni x$. Moreover, this $W$ is a union of
some elements of $\Gamma$. This means that $|W\cap H|$ is even and hence so is
$|W\cap F|$, because $W$ is $(F\vartriangle H)$-even. However, $x$ equals
$r(F)$ and hence does not belong to any $F$-even or $F$-void element of
$\mathscr{B}$. This contradiction proves that $V=V_{x}$, i.e.,
$r(F\vartriangle H)\in V_{x}$.

Thus, $r(F\mathop{+_{\scriptscriptstyle 2}}\mathbf{H}_{\Gamma})\subset V_{x}$.
We have shown that, for any $F\in B_{*}(C)$ and any neighborhood $U$ of
$x=r(F)$ in $C$, there exists an $\mathbf{H}_{\Gamma}\in\mathscr{H}$ such that
the image of the open neighborhood $F\mathop{+_{\scriptscriptstyle
2}}\mathbf{H}_{\Gamma}$ of $F$ in $(B(C),\tau)$ under $r$ is contained in $U$.
This means that $r$ is continuous with respect to the topology
$\tau|_{B_{*}(C)}$.

To obtain a retraction $\hat{r}\colon(B(C),\tau)\to C$, it remains to extend
$r$ over $B(C)\setminus B_{*}(C)$, e.g., by 0; this extension is continuous,
because $B_{*}(C)$ is clopen in $(B(C),\tau)$.

Restricting $\hat{r}$ to the subgroup $G_{2}$ of $(B(C),\tau)$ generated by
$C_{2}\subset C$, we obtain a retraction of $G_{2}$ onto $C_{2}$, because
$r(F)\in F$ for any $F\in B(C)$ and hence $r(G_{2})=C_{2}$. The group
$(B(C),\tau)$ is metrizable and zero-dimensional, since its local base at zero
consists of countably many open (and hence closed) subgroups, and it is
separable, since its topology is coarser than the separable free group
topology. Therefore, $G_{2}$ is a second-countable zero-dimensional group.

The author is very grateful to Evgenii Reznichenko, who suggested that the
original construction might be refined.

## References

  * [1] O. Sipacheva, “No product theorem for the covering dimension of topological groups,” arXiv:2207.04961 [math.GN].
  * [2] O. Sipacheva, “No subgroup theorem for the covering dimension of topological groups,” arXiv:2303.04593 [math.GN].
  * [3] A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures (Atlantis Press/World Sci., Amsterdam–Paris, 2008).
  * [4] M. G. Tkachenko, “Factorization theorems for topological groups and their applications,” Topol. Appl. 38, 21–37 (1991).
  * [5] P. M. Gartside, E. A. Reznichenko, and O. V. Sipacheva, “Mal’tsev and retral spaces,” Topol. Appl. 80, 115–129 (1997).
  * [6] K. Morita, “On the dimension of the product of Tychonoff spaces,” General Topology Appl. 3, 123–133 (1973).
  * [7] O. Sipacheva, “Free Boolean topological groups,” Axioms 4, 492–517 (2015).


#  Building Performance Simulations Can Inform IoT Privacy Leaks in Buildings

Alan Wang  University of Virginia, Link Lab, Computer Engineering,
Charlottesville, 22903, USA  Bradford Campbell  University of Virginia, Link
Lab, Computer Science, Charlottesville, 22903, USA  Arsalan Heydarian
University of Virginia, Link Lab, Engineering Systems and Environment,
Charlottesville, 22903, USA<EMAIL_ADDRESS>
###### Abstract

As IoT devices become cheaper, smaller, and more ubiquitously deployed, they
can reveal more information than their intended design and threaten user
privacy. Indoor Environmental Quality (IEQ) sensors previously installed for
energy savings and indoor health monitoring have emerged as an avenue to infer
sensitive occupant information. For example, light sensors are a known conduit
for inspecting room occupancy status with motion-sensitive lights. Light
signals can also infer sensitive data such as occupant identity and digital
screen information. To limit sensor overreach, we explore the selection of
sensor placements as a methodology. Specifically, in this proof-of-concept
exploration, we demonstrate the potential of physics-based simulation models
to quantify the minimal number of positions necessary to capture sensitive
inferences. We show how a single well-placed sensor can be sufficient in
specific building contexts to holistically capture its environmental states
and how additional well-placed sensors can contribute to more granular
inferences. We contribute a device-agnostic and building-adaptive workflow to
respectfully capture inferable occupant activity and elaborate on the
implications of incorporating building simulations into sensing schemes in the
real world.

## Introduction

Due to increased awareness of energy reduction measures in buildings over the
past two decades, numerous technological advancements have been introduced to
monitor the changes in indoor conditions. Sensors and actuators have become
increasingly integrated into buildings to reduce overall energy consumption
while improving occupant comfort [1, 2]. For example, building automation
systems can reduce a building’s energy consumption by dimming artificial
lighting when sufficient daylight is sensed in the building [3]. The building
can also utilize occupancy and air quality sensors to reduce energy demand by
Heating Ventilation and Air Conditioning (HVAC) units in anticipation of
occupants’ presence or comfort [4]. The number of sensors installed in
buildings will only grow with increasing energy prices and the known benefits
of smart environments [5]. However, numerous challenges still exist to using
sensor-collected data to improve the utility of occupants.

Firstly, and fundamentally, sensors have different data collection
frequencies, so researchers cannot simply purchase any environmental sensor
and install it to capture all the activity happening indoors. The frequency of
data collection restricts the types of occupant behaviors that can be
inferred. For instance, the Nyquist-Shannon sampling theorem demonstrates that
you need to sample at more than twice the highest frequency component of the
signal to correctly reverse it [6]. The time scale differences mean that
researchers cannot use a one-image per thirty-second camera to capture the
behavior of subjects that occur every 15 second without losing data.
Increasing the frequency, on the other end, can cause other undesirable
effects, such as signal aliasing. A researcher (designer or facility manager)
is still required to decide the installed sensor’s specifications, such as
frequency, mode, and observed variable; the very act of purchasing the
hardware restricts in perpetuity the downstream observable behavior.

Secondly, sensor placements have a large effect on the downstream utility of
the sensor’s collected data, but positioning is often overlooked or starts off
randomly then iteratively improved [7]. Sensors deployed at incorrect
positions can result in incorrect readings [8, 9], but it can feel more
convenient to install sensors and start collecting data as soon as possible.
Furthermore, prior sensors installed for evaluating a building can become
insufficient or undesireable for new uses of space. For example, some
residents can move out, rendering prior sensors in locations no longer
occupied redundant. Similarly, additional residents can move in, making
previous coverage inadequate for the new use of space. Optimizing the position
and number of sensors can result in lower energy consumption and better
readings, but it is challenging to incorporate and maintain considerations of
changing contexts and sensing objectives for a scaling number of sensors
manually.

Lastly, in the expediency of collecting ever more data, installed sensors can
inadvertently expose more information than necessary for its intended use,
leading to privacy concerns for the occupants. For instance, the “sensing by
proxy" paradigm demonstrates how proxy measurements such as CO2 can infer
occupant count and activity[10, 11]. Similar granular occupant activity
information has been observed by other sensors as well. For example, cooking
activities can be observed via the fluctuation of PM 2.5 [12]. Granular
appliance use can also be effectively disaggregated via non-intrusive load
monitoring [13]. For instance, appliances such as coffee makers and hair
dryers can have unique energy-use signatures relating to start-up processes
and the physical makeup of the appliance. By installing a load monitor at the
circuit level, the total energy use can be disaggregated from individual
contributions based on the appliance’s unique signatures, allowing for
invasive inferences of occupant activity without needing to install any
sensors inside the building itself. Other examples, such as using cameras and
motion amplification, have allowed researchers to exaggerate the vibration of
snack bags and reverse engineer decipherable sounds using visual data [14].

As more projects utilize machine learning and other computational methods to
retrieve sensitive data from the indoor environment, we instead consider if
similar computational methods exist to help reduce inferable information from
sensors and protect the privacy of building occupants. One promising avenue
that can help sensor installations navigate the potential overreach of sensing
is at the intersection of simulations and sensor placements [15]. Simulations
have traditionally been used to assess different building performance
attributes during the design phase. The orientation of buildings and placement
of windows, for example, can be explored and quantified using a score called
daylight autonomy [16]. Simulations of the weather conditions, and movement of
the sun, in conjunction with the location of the building, size, material, and
orientation of the window and room, enable architects to uncover the total
amount of time over a whole year when daylight can effectively stand in for
artificial lighting. Similar simulations and metrics can be found for HVAC,
where given the hours that the building will be occupied, the room size, the
expected occupancy, the total energy use, and required airflow can be
predicted and quantified [17]. The acoustic qualities of a building can also
be designed and tailored to better match the intended use of space, such as
longer reverberations for music halls and shorter reverberations for
classrooms [18]. The timing in the pipeline during which these simulations are
used represents a fundamental discourse in digital twins [19]. Specifically, a
simulation model can help optimize the placement of sensors to inform on
occupant activity, and it does not need to be run in real-time parallel to the
physical twin for it to have a lasting impact on the smart environment.

Compared to manual or autonomous methods of sensor position optimization,
where robots are used to sample the environment routinely and iteratively
uncover the most optimal positions for the static environmental sensors [20],
simulations enable a low-cost alternative for testing unlimited virtual
sensors positions at the cost of computational power. Furthermore, instead of
navigating protocols for institutional review boards (IRB) or logistics of the
environment (e.g., to avoid a party or speaker event in the building),
simulations enable researchers to avoid a broad range of complexity that can
cause nontrivial disruptions for both study administrators and study
participants. Researchers (designers or other decision makers) can run
unlimited what-if scenarios to see how different environmental or user-related
factors may impact the changes in the downstream signal before interacting
with the physical environment. For example, the movement of occupants can be
simulated using artificial agents to assess the ease of navigating the space
in case of emergencies [21], the sun’s movement can be simulated using
historical sky information [22], and stochastic models can be used to simulate
an occupants interaction with building controls [23].

In this paper, we demonstrate the potential of physics-based simulation models
to quantify the minimal number of positions necessary to capture sensitive
inferences. We show how a single well-placed sensor can be sufficient in
specific building contexts to holistically capture its environmental states
and how additional well-placed sensors can contribute to more granular
inferences. Specifically, we focus on lighting simulations as a test case
because of its accessibility and geometrically-dependant nature. We answer two
research questions:

  * •

RQ1: What are the challenges in translating lighting inferences found in the
real world to inferences found in the simulations? And,

  * •

RQ2: How can lighting simulation be used to inform us about how informative a
set of light sensors positions are given some assumptions about space-use?

We advocate for the use of simulation as a standard tool for 1) identifying
the ideal location for sensors and minimizing the number of sensors
distributed and 2) identifying potential information that sensors can collect
when deployed in real life by showing the capabilities of simulations to
calculate building states containing granular occupant activity exhaustively.
In other words, we show how the simulated environment allows researchers to
assess the effects that the position of sensors and the geometry of the
building can have on occupant activity inferences. The workflow demonstrates
an avenue for future researchers to verify the possible inferences of existing
sensor positions or use the “adjustment of sensor positions" as a method to
limit sensor inference overreach.

## Methods

We consider the scenario where a researcher is trying to ascertain the light
state of on and off for individual luminaires. Given the additive nature of
light, we approach decomposing the summed light contribution at a sensor point
in the building by formulating it as a Perfect Sum Problem with a noise
threshold $\epsilon$. Since light intensity diminishes equal to the inverse of
the square distance from the source, and each luminaire has photometric data,
moving a light sensor’s final placement at a variable distance away from light
sources can enable individual contributions to be disaggregated. For instance,
we can utilize the geometry and drop-off of light intensity by distance to
coordinate unique fingerprints for each luminaire in range. Furthermore,
having the ability to utilize the entire building space as potential placement
areas enable researchers to exhaustively explore the ability to utilize sets
of potential sensor positions that traditionally require repeated trial-and-
error to capture.

#### State Inferences

Let $L=[l_{0},l_{1},...l_{n-1}]$, where $L$ is a configuration vector of $n$
individual light source states $l_{i}\in\\{0,1\\}$. Then, for any location
$s$, given a configuration $L$, we see the contribution vector
$X(s,L)=[x_{0},x_{1},...,x_{n-1}]$, where the contributions from each light
source $x_{i}$ corresponds to each light state $l_{i}$ modified by distance
and obstruction. The maximum number of possible configurations is then equal
to the cardinality of the power set with each light source, or $2^{n}$.

Given a perfect sum solver, we can take a target sum K, threshold $\epsilon$,
and list of individual contributions $[x_{0},x_{1},...x_{n-1}]$ and return a
list of lists that contain all possible combination of contributions that add
up to $\text{{K}}\pm\epsilon$. Each list corresponds to a possible
configuration that fits the constraints, but since ultimately only one
configuration is correct, we calculate the accuracy for each inference using
$\textnormal{Accuracy}=\frac{|L\cap L_{\text{infer}}|}{|L\cup
L_{\text{infer}}|}$ and return the mean.

Sensor readings from multiple points do not always corroborate to the same
lighting configurations. To overcome this, we disambiguate misaligned light
configuration inferences by using a voting vector
$V(s,L)=\begin{bmatrix}v_{0},v_{1},...v_{n}\end{bmatrix}$, from each sensor,
where $v_{i}$ is 1 if the luminaire is determined to be on, -1 if the light is
determined to be off, and 0 if the light cannot be detected (i.e., the sensor
is out of range of the luminaire). The final inference is chosen based on the
summed votes. If the value is greater than zero, the luminaire is on. If the
final value is less than zero, we consider the luminaire off.

#### Distinctness Score

To generalize the real-world sensing to building simulations, we introduce an
error threshold $\tau$, and define a distinctness vector $D_{\tau}$ as:

$D_{\tau}(X(s,L))=e_{0},e_{1},...,e_{n-1},\text{where
}e_{i}=\begin{cases}1,&\text{if}\;\operatorname*{arg\,min}_{j}|x_{i}-x_{j}|>\tau,i\neq
j\\\ 0,&\text{otherwise}.\end{cases}$ (1)

For example, given a contribution, $X(s,L)=[1,2,4]$, if the error threshold
$\tau=1$, then $D_{1}(X(s,L))=[0,0,1]$ and the distinctness score
$\text{{D}}_{1}=\sum{D_{1}}=1$. However, given a $\tau=0.5$ for the same
contribution, $D_{0.5}(X(s,L))=[1,1,1]$, so $\text{{D}}_{0.5}=3$. The sum of
the distinctness vector helps us decipher the total number of detectable
lighting states at a given position and accounts for the sensor’s resolution
when assigning credit. We utilize this score in our simulations, where the
virtual light sensors in simulation do not have the added lumen degradation
and other measurement noise terms.

For our activity inferences, we account for $j=3$ door angles (i.e.,
$\\{0^{\circ},45^{\circ}$, $90^{\circ}\\}$), for $m=2$ doors. To find the
distinctness score for each given application state $a$ is then:

$\text{{D}}=\sum_{p=0}^{2^{n}-1}\sum_{q=0}^{jm-1}{D(X(s,L_{p},a_{q}))}$ (2)

Because choosing the minimum combination of sensor locations that can detect
all possible applications is a known NP-Hard problem called the Minimum Set
Cover (MSC) Problem, for simplicity, we visualize the inferable states for
only the single-sensor scenario. Finally, we show an example set of light
sensor locations that can capture our latent variable and the building light
states using a known algorithm for the MSC problem: the Greedy Set Cover
Approximation (GSCA) [24]. To see if a set of light sensors capture the
dynamic building information, we can then use the entire application states as
the universe of states to cover a threshold $<\tau$ against all other sensor
locations rows for the same application state column to inform membership, to
find the minimum subset of sensor locations required to cover the entirety of
the application states. We target door usage because of its fixed nature. The
opening and closing of doors have a large effect on perceived lighting by the
sensors, with the potential to block off sections of lighting altogether
consistently. Further, door usage is a deeper insight into space usage not
expected by light sensors. Using the bathroom, bedroom, or living room
naturally involves opening and closing lights and doors. In this case, where
the lighting does not automatically turn off, being able to discern door
movement even while all lights are on allows more granular inferences about
space use to be made.

### Experiments

As shown in Figure 1, even for a single light source, the angle of a door’s
rest state can significantly impact the final detected light signal. Using
these methods, we conducted two experiments: 1) we deployed light sensors into
the modeled residential setting to explore real-world challenges with
detecting light states and fusing inferences from multiple sensors together,
and 2) we modeled the residential building and simulated a set of dynamic
building elements to analyze the number of light states and latent states that
can be observed. The real-world study was conducted first as a sanity check;
if we can arbitrarily place sensors using only human intelligence, it is
unnecessary to utilize simulations to support the sensor placements. The
simulation study was conducted to see whether the simulation space would
corroborate with the findings found from our real-world study and also to
gauge the minimum number of sensors that can capture a set of luminously
disruptive behavior.

Figure 1: Relationships between building elements and lighting can be used to
inform about changes in the physical environment.

#### Real-world Experiment

In the real world, we retrofitted all lamps in a residential apartment
building with 800 lumens 10 watts Philips Hue A19 Lamps and grouped each set
of lights to their corresponding luminaire. For example, two to three lamps
can be associated with each light switch. We controlled each Philips Hue lamp
using the Hue API and Python to limit the need to alter the light switches
manually. We then used Raspberry Pi 4s connected with CQRobot TSL2591X light
sensors with an effective sensing range of 0 to 88,000 Lux as our sensor,
communicating using the I2C interface with the Pi. To account for the jitters,
we used the mean lux values for three seconds as the baselines, collected at
4.7 Hz after our code changed the lighting configurations configuration $L$
for three seconds because of the changes in light intensity during state
transitions. We used the proximity of the light sources as a guide to
installing the light sensors in each location and orientation, as shown in
Figure 2. Specifically, we visually looked for sensor positions on different
walls, enabling the sensors to capture light from different light sources.
Then, we moved our sensor system across each position and automated the
lighting transitions using the Raspberry Pi, exporting the final data into a
CSV file for post-processing. The final accuracy we report is explained in the
results, where the ground truth is the input command we used to automate the
light states. To install an initial set of sensors, we identified walls in the
testbed that all the luminaries can reach and then placed seven sensors on
those walls, seven feet above the floor shown in Figure 2. The idea is to see
if it is possible to install sensors in locations that avoid the noise in the
data caused by shadows in human traffic, the reflection of furniture, and
other LEDs from appliances and objects. This allows us to accurately detect
the lighting state of the building for the static open door scenario before we
dive into permutations of doors in the simulation experiment.

Figure 2: Building and Lighting Layout. The numbers denote the light sensor
positions, and the alphabets denote the light sources. The dashed lines denote
geometric boundaries we used to search for our candidate walls to install the
sensors.

#### Simulation Experiment

In a simulation, we utilized grasshopper to parameterize multiple door
movements and simulated each of the allotted door angle combinations using
Rhino [25]. We then used the grasshopper plug-in honeybee [26] as we have in
our previous work [27] to extract the lighting contributions at each sensor
point. In essence, the plug-in acts as a middleware that takes the building
geometry and photometric lighting files (IES files) that describe the
geometric intensity distribution of light and passes them into the lighting
render engine radiance [28]. The experiment was assumed to be undertaken at
night with covered windows to avoid external light sources. We used the
simulation default material for all wall, floor, ceiling, door, and window
objects. For the photometric lighting files, we used generic flush-mounted
dome lights for fixtures A, B, C, E, and F and a generic wall-scone fixture
for D. We automate the inputs to the grasshopper workflow and export text
files we convert for post-processing using Python modules inside grasshopper.
The process returns a ray-traced light rendering of the building for each
virtual sensor position defined. We extract each lighting value for post-
processing by passing it through the Perfect Sum solvers and the GSCA
algorithm. This workflow enabled us to explore potential lighting differences
in the overall environment without needing to physically place new sensors,
change which luminaires were on and off, and move the doors in the real world.

## Results

We first conducted a feasibility study to explore the potential challenges of
collecting sensor data manually. We then used the experiences we have learned
to inform our modeling of the building in simulation space. We summarize the
real-world experiment and the simulation experiment below.

#### Real-world Experiment

Figure 3 shows the spread of individual accuracy for those locations. The
median accuracy of inferences was generally above 80%, with location 6 having
the lowest median accuracy. We suspect this due to the direction the sensor is
facing being directly opposite to luminaire F and reflected lights off the
wall being less potent than the drop off of intensity due to distance.
Overall, we still found it possible to disaggregate all possible light states
using a single sensor (i.e., a light sensor located at position 4), but many
factors can contribute to the imperfect inferences. For example, when an
occupant closes the door to any room, the information about the light state in
that room is lost to sensors outside. We also experience situations where the
inferred light state of the signals does not align with each other, because
the noise in the environment and sensor was larger than the resolution
required to differentiate the states. For example, the lighting contribution
from two different light sources can add up to be the same (e.g., 2+5 and 3+4
both equal 7) leading to ambiguous readings. This led us to utilize a voting
mechanism to reduce the overall error of the system, which is further
described in the methods section.

When we automated the lighting states, we also realized that, unlike what may
happen in simulations, the switching on and off of lights in the real world is
not instantaneous. Specifically, when turning on the lights, there is a
distinguishable start-up time when the light starts dimmer after the switch is
flipped and approaches its final brightness after a delay. Furthermore, when
retrofitting the luminaires, we noticed that not all of the lights were using
the same bulbs and that likely more frequented areas had bulbs that were more
frequently replaced. This suggests that keeping track of the light usage in a
building can also be helpful to track which lights might need to be replaced
and account for the lamps’ brightness degradation over time. Finally, we found
that the number of available sensors, microcontrollers, and outlets also
impose a limit to how many positions can be tested at once. In addition to
purchasing a long extension cord to move our sensing apparatus across the
building, we also had adhered to a procedure of moving sensors, installing
sensors, running through all the light states, and uninstalling sensors to
sense each position.

Figure 3: Light state inference accuracy for each of the 7 positions for 64
possible lighting states for manually selected positions. The triangles mark
the mean accuracy for each sensor location, while the line across the middle
of the box marks the median accuracy.

|  |  |
---|---|---|---
|  |  |
|  |  |

(a) Bed Door $90^{\circ}$, Bath door $90^{\circ}$ (open-door scenario)

(b) Bed Door $45^{\circ}$, Bath door $90^{\circ}$

(c) Bed Door $0^{\circ}$, Bath door $90^{\circ}$

(d) Bed Door $90^{\circ}$, Bath door $45^{\circ}$

(e) Bed Door $45^{\circ}$, Bath door $45^{\circ}$

(f) Bed Door $0^{\circ}$, Bath door $45^{\circ}$

(g) Bed Door $90^{\circ}$, Bath door $0^{\circ}$

(h) Bed Door $45^{\circ}$, Bath door $0^{\circ}$

(i) Bed Door $0^{\circ}$, Bath door $0^{\circ}$

(j) Aggregate latent variable privacy $\text{D}_{0.01}$

(k) One light sensor can sense all 64 light states for the open-door scenario
using GSCA

(l) 31 sensors are needed to sense a total of 576 states using GSCA for the
dynamic-door scenario

Figure 4: Compilation of the distinctness score $\text{{D}}_{0.01}$ heat maps
for each building configuration. In addition to the resolution of the sensor,
the building’s physical configuration can also systemically alter the
inferable states of the sensor set. Note that the scale for Figures 4a through
4i is on a scale out of 64, while the aggregated scale on Figure 4j is out of
576.

#### Lighting Simulation Results

Utilizing the information we’ve learned from the real-world experiment, we
developed a lighting simulation that deviated from traditional lighting
simulations to explore locations where static sensor installations can give us
the most information. Specifically, in our lighting simulation, we included
walls instead of using the traditional work plane–an imaginary plane set at
the level of a desk where work is done–because we are not interested in the
utilization of the space but rather in the ability to install real static
sensors and detect different behaviors in the space (e.g., light switch
behavior). The results from the simulation are shown in Figure 4. Figures 4a
through 4i represent the different states of dynamic building elements, with
the heatmap showing the distinctness score $\text{{D}}_{0.01}$ value from 0 to
64, representing the number of collective states a sensor placed at the
location can detect. Figure 4a represents a typical light sensing simulation,
where the movement of additional building elements is not considered. In this
scenario, many positions in the middle of the room where all the lights can
reach can be used to infer the light state of all luminaires. From Figure 4b
to Figure 4i, we show that these informative middle positions diminish as the
doors close and block out lighting contributions from different sources. In
Figures 4d and 4e, when all doors are still partially open, we see a slight
reduction in areas that can still make all targeted inferences. However, in
Figures 4c, 4f, 4g, 4h, and 4i, we see the total possible inference visibly
diminish by half to three quarters.

Figure 4j represents the most informative single location accounting for all
possible door states. As much as there are informative locations in the
middle, we also observe there to be spots of lost information, dark zones that
have lower inference potential. This is a result of the clashing of light
contribution combinations that lead to ambiguous readings. Comparably, while
Figure 4i has lower number of possible inferences in the middle, there are
also less informative “dark spots" as a result of collisions. Figure 4k shows
how Greedy Set Cover Approximation (GSCA) found a single location that can
detect all light states, but Figure 4l shows that the previous best location
is no longer valid when we account for the opening and closing of doors. More
specifically, we can see a set of sensors being placed deep into rooms away
from noisy areas in the center, which help to disambiguate the readings when
the lighting signals are muddied by reflection and attenuation in the center
area.

## Discussion

#### Key Takeaways

The key takeaway from our study is that: sensor position is important, and
simulations can be used to quantify just how important position is. By
quantifying inferable information in simulation, building operators can adjust
the privacy-utility spectrum for where sensor installation should occur before
deployment. The quantity and location of sensors can be altered to
purposefully remove possible inferences based on the physical attributes of
the environment. Even after deployment, simulation elucidates where the
current installation is on this spectrum and ways to navigate this trade-off.
In our scenario, this trade-off between privacy and utility is exactly the
distinctness score, a quantifiable value between zero and the total number of
states we are considering. Towards answering RQ1: we found that the
informativeness of the sensor location also hinges on the sensor’s resolution.
Paradoxically, the more contributions from different light sources sensed by a
sensor-location pair, the more chance there can be ambiguous readings in the
perfect sum solver because of the number of possible combinations and jitters
in the sensor signals. These jitters, shown in Figure 1, signify a vital
distinction between real-world and virtual sensors in simulations. Adding more
sensors in the physical world does not, by default, increase the accuracy of
the final light state inference. The requirement for accurate sensing relies
on the majority of the votes cast being accurate. To know the lowest level of
resolution permissible for accurate sensing before purchasing a sensor,
simulations can be a useful tool to assist with planning. In the simulation,
the resolution of the virtual sensors is deterministic under the same
parameters, and adjustments can be made to simulate different sensor
resolutions by introducing additional noise terms. Further, by adding more
information about the activity, simulations can be improved to account for
different sampling frequencies by discretizing sample points based on a
continuous response function. With sufficient computational resources,
permutations of different sampling policies and sensor descriptions can be
used to optimize the inference accuracy, redundancy, and efficiency. With
simulations, researchers can achieve more intrusive inferences with fewer
sensors, fewer samples, and less energy compared to without simulations.

Towards answering RQ2: we found that utilizing lighting simulations with a
formulation of Perfect Sum Problem with the Set Cover Problem allowed us to
quantify the minimal number of light sensors that are required to capture the
light state of the building, including modification of the doors. We found
that as long as the sensor is placed in a location where all light sources can
reach and result in a different contribution, a single sensor is theoretically
enough to infer all of the possible light configurations in the building if
the resolution equivalent $\tau$ is sufficiently small. However, as doors are
introduced that can block off lighting contributions from other luminaires,
the minimum number of sensors required to sense the lighting state of the
building becomes equal to the total number of independent zones. For example,
three separate rooms require at least three sensors to detect the lighting
state, regardless of the number of luminaires in each room. The minimum number
of sensors required increases from one to 31 when we accounted for the
movement of the doors as seen by Figure 4k compared to Figure 4l. This
indicates that even simple residential buildings with no dimmers can result in
complex luminous environments if commonplace building elements such as doors
are considered. Unless researchers had thousands of sensors, placed at every
inch of the space, they wouldn’t be able to test the entirety of the space at
once.

Finally, simulations can be much faster at the search for optimal positions.
In our experiment, it took us roughly 30 minutes to test one position in the
real-world, where as in simulation space, we cover roughly one sensor location
every 0.11 minutes of simulation (about 2,800 points can be calculated every 5
minutes for the 64 different light states). This approximates a 270-times
increase in efficiency when using simulations to test positions compared to
real-world testing, not accounting for set-up time in either scenario. Because
simulations also do not require the researchers to be physically present in
space, simulations hold a distinct advantage over manual testing as an
important step to enhance physical sensor deployments. With increased
simulation scenarios that incorporate human movement and other modalities such
as noise and HVAC, the digital space will increasingly become more critical
not just for sensor positioning but also for a large myriad of selection
tasks. From what type of sensors to deploy, at what frequency to sample data,
what information they provide at what different times of day, and seasons,
with what different levels of occupant activity to expect, and with which soft
sensors[29] to combine and make inferences with, simulations will take an
increasingly important role in controlling and testing the scope of inferences
in buildings. Methods like this demonstrate that simulations have the
potential to serve as a stand-in for domain experts. If experts can digitize
the knowledge of specific sensor placements for building commissioning, for
example, they can enable accessible and code-compliant occupant privacy
protection designs while further providing availability to interact with other
simulations using the same building model.

#### Broader Impacts

The work we have completed represents both predictive model tasks, where we
anticipate the use-case of the occupant before installing the sensors, but
also a step towards reducing the gap between the digital twin and the original
twin. Sensor installations can take advantage of more than just their
placement concerning common building elements such as floors, walls, and
doors. Sensor placements can also benefit from being aware of other sensors in
the context. The workflow we demonstrate allows for explorations in designing
buildings that can be more effectively commissioned with fewer sensors.
Developing metrics to quantify possible inferences also provides an additional
avenue for designers and researchers to consider user privacy. For instance,
there could be dedicated “silent zones" where sensors cannot detect any
occupant activity as protected by the laws of physics. Simulations can be an
effective tool to compete with the scale of sensor developments because they
can protect their users from scenarios that have yet to happen and inform and
adjust models using real data. While the digital and original twins divide is
shortening, we consider their distinct identities to carry certain benefits.
For example, digital twins can be operated “offline" to explore reactive and
predictive scenarios that inform on the optimal corrective action without
interfering with the operations in the real system. However, this is not to
say that simulations will not also be an increasingly important part of the
operations in real-time systems. Decisions to navigate the potential split
incentives between the building operator and the occupant will likely depend
on circumstance and might require routine updates to support new management
and new tenants. A building operator might install light sensors in all rooms
to avoid the complexity of inference but expose information about the
occupant’s kitchen, living room, and bathroom use they consider private.
Similarly, an occupant might install one sensor to understand lighting in the
living room but accidentally leak the lighting states of other rooms to the
building operator. Decisions regarding which data should be hidden for privacy
or which data should be available for utility would require coalescing of
ideologies regarding ownership of space, ownership of data, ethics, among
other considerations. Regardless of the perspective, the first step is showing
in a data-driven and reproducible manner where a sensor installation
theoretically lies on the privacy-utility spectrum.

#### Limitations

One limitation of this study was that it was conducted on an older residential
unit, where buildings might not reflect a more modern understanding of the
efficient usage of light fixtures. The privacy implication of sensor use in
public and semi-public situations such as offices and libraries could have a
broader impact on the number of people affected. Another limitation is our
assumption of snapshot views in simulations. We did not incorporate time
(analysis of signals instead of values) such as through bulb response
functions [30]. This can limit the ability for simulations to reflect the time
in-between snapshots and the additional inferences that can be drawn as a
result of more realistic sample rates. Another limitation is that we only
collected light intensity levels but did not look further into other
properties of light, such as lighting colors. We suspect colors can be an
important avenue to disambiguate lighting signals further. For instance, the
individual lights might be able to be first filtered by color, reducing the
total number of possible combinations and collisions that could happen.
Another limitation is that the involvement of some furniture has the potential
to alter the indoor environment. For example, having a mirror on the wall can
drastically alter the luminous environment, similar to having light-absorptive
materials on the floors. Finally, our work does not address the difficulty of
constructing a representative building model, nor the potential diminishing
returns of modeling the environment in more realistic detail. While numerous
benefits can be achieved with an informative building model, the cost of
building a representative model can eventually outweigh the demand to protect
an occupant’s privacy. The cost of the building model can be further
exacerbated when more computation time is required to calculate physical
interactions in the space, such as increases in the number of bounces for
lighting simulations or number of particles in Computational Fluid Dynamics
(CFD) simulations.

## Conclusion

We demonstrate a theoretical framework for indoor activity inference selection
through simulation experiments and real-world sensor placements. We show how
simulations can quantify inferable occupant activities using a distinctness
score and how to find a mathematically minimum set of sensor positions
required to detect them by applying the concept of set cover. The resulting
metrics to quantify distinguishable activities enable future sensor
deployments to consider building geometry better and limit potential sensor
data overreach. We anticipate using sensor positioning paired with building
simulations to grow as an essential technique for researchers to navigate the
privacy-utility trade-off for the smart buildings of tomorrow.

## Data availability Statement

The datasets generated during and/or analyzed during the current study are
available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was partly supported by the Virginia Commonwealth Cyber Initiative
(CCI) grant, and the National Science Foundation (NSF) Grant No. 1823325.

## Author contributions statement

A.W., B.C., A.H. contributed to ideation. A.W. conducted the experiments and
analyzed the results. All authors reviewed the manuscript.

## Additional information

The author(s) declare no competing interests.