# OUTformation: Distributed Data-Gathering with Feedback under Unknown
Environment and Communication Delay Constraints

SooJean Han1∗, Michelle Effros1, Richard M. Murray1 1SooJean Han, Michelle
Effros, and Richard M. Murray are with the Division of Engineering and Applied
Science at California Institute of Technology, Pasadena CA.∗Corresponding
author. Email<EMAIL_ADDRESS>paper is based on work supported by the
National Science Foundation Graduate Research Fellowship under Grant No.
DGE‐1745301.

###### Abstract

Towards the informed design of large-scale distributed data-gathering
architectures under real-world assumptions such as nonzero communication
delays and unknown environment dynamics, this paper considers the effects of
allowing feedback communication from the central processor to external
sensors. Using simple but representative state-estimation examples, we
investigate fundamental tradeoffs between the mean-squared error (MSE) of the
central processor’s estimate of the environment state, and the total power
expenditure per sensor under more conventional architectures without feedback
(INformation) versus those with broadcast feedback (OUTformation). The primary
advantage of enabling feedback is that each sensor’s understanding of the
central processor’s estimate improves, which enables each sensor to determine
when and what parts of its current observations to transmit. We use theory to
demonstrate conditions in which OUTformation maintains the same MSE as
INformation with less power expended on average, and conditions in which
OUTformation obtains less MSE than INformation at additional power cost. These
performance tradeoffs are also considered under settings where environments
undergo less variation, and sensors implement random backoff times to prevent
transmission collisions. Our results are supported via numerical studies,
which show that the properties derived in theory still hold even when some of
the simplifying assumptions are removed.

##  I Introduction

In previous literature, a variety of architectures for distributed data-
gathering in large-scale cyberphysical network applications has been proposed
with the aim of reducing unnecessary transmissions of redundant information
and excess computation; see [1, 2] for surveys. However, many methods consider
only structural variations in the communication among lower-level nodes such
as external sensors (e.g., hierarchical clustering [3, 4]), and have rarely
challenged the imposition that data flows only in a single direction from
sensors on the network’s edge to the higher-level processor at the network’s
center. Some architectures allowing feedback communication in the reverse
direction, from the central processor to the external sensors, are developed
under a diversity of simplifying assumptions. In some feedback schemes for
redundant transmission reduction (e.g. [5]), communication delays are omitted;
this disregards issues pertaining to outdated data, which is important to
distributed state estimation in real-world large-scale networks. In others
(e.g. [6]), communication delays are included, but the approach for redundant
transmission reduction assumes full knowledge of the environment dynamics.

While in practice, the best choice of architecture to use for distributed
data-gathering varies by setting, imposing simplifying but impractical
assumptions makes it difficult to properly motivate feedback, and make fair
comparisons between architectures which employ feedback and those which do
not. Feedback can provide at least the following two advantages in large-scale
distributed systems. First, by nature of the distributed architecture, data
gathering and data processing are sequential actions; having a larger delay
between the sensor transmission times and central processor receipt times can
potentially lead to large estimation errors. Second, for large-scale systems,
a large number of sensors may be needed to survey the environment. But
independent sensors surveying a shared environment are likely to collect data
which exhibits redundancies, and transmitting redundant information to the
central processor is inefficient.

With these motivations in mind, the contributions of our paper are as follows.
We employ a modular framework for distributed state estimation under two
important real-world constraints: nonzero communication delays and limited
knowledge of the environment dynamics. Because the type of feedback we
consider broadcasts data “out” from the central processor to the sensors, we
refer to this architecture as the OUTformation architecture for data-
gathering, as opposed to the INformation architecture where all information
flows from the sensors “in” to the central processor. We use a modular
framework representation of the INformation and OUTformation architectures to
take a first step towards characterizing their relative performances with
respect to two specific metrics: 1) the mean-squared error (MSE) of the
central processor’s estimate of the environment state, and 2) the total power
expenditure of each sensor. The importance of such a characterization arises
from allowing users to determine which type of architecture is more suitable
for use based on hardware specifications and limited knowledge of the
environment, as well as providing essential insights to make parameter design
choices for optimal performance in both MSE and power expenditure.

We theoretically analyze simple but insightful case studies demonstrating
fundamental tradeoffs between the two performance metrics, and consider how
the tradeoffs vary under different environment statistics and parameter design
choices. In particular, we show that the main advantage of enabling feedback
is improving each sensor’s estimate of the central processor’s estimate, which
allows each sensor to make more informed decisions about when and what to
transmit, potentially reducing both 1) its uplink power expenditure, and 2)
the MSE by minimizing the delays after which the central processor receives
its transmissions. We employ numerical simulations to break some of the
simplifying assumptions made in our theoretical analysis, and show that the
properties derived in theory still hold, which suggests that the implications
of OUTformation architectures extend beyond the framework treated rigorously
by the theory.

##  II INformation and OUTformation

###  II-A Problem Setup

We consider environments which evolve as a random walk given by

$\displaystyle{\bf x}((c+1)\Delta t)={\bf x}(c\Delta
t)+\sum\limits_{i=1}^{n}\delta_{i}(c){\bf d}(c)$ (1)

Here, $c\in\mathbb{N}$, and $\Delta t{\,\in\,}\mathbb{R}^{+}$ is the
environment’s transition period. For each $i{\,\in\,}\\{1,\cdots,n\\}$ and
$c\in\mathbb{N}$, random variable $\delta_{i}(c)$ takes value $1$ with
probability $p/2$, $-1$ with probability $p/2$, and $0$ with probability $1-p$
for some $p{\,\in\,}(0,1)$, and ${\bf d}(c){\,\in\,}\mathbb{R}^{n}$ such that
$d_{i}(c){\,\sim\,}\mathcal{U}[\underline{d},\overline{d}]$ is a uniformly-
distributed random variable stepsize between constant values
$0<\underline{d}<\overline{d}$.

######  Definition 1 (Unknown Environment).

The titular “unknown environment” refers to the fact that the parameters
$\Delta t,p,\underline{d},\overline{d}$ are unknown to the central processor
and all the sensors.

We focus on architectures where a single central processor communicates with
$M{\,\in\,}\mathbb{N}$ independent external sensors. Each sensor
$j\in\\{1,\cdots,M\\}$ employs a linear measurement equation perturbed by
additive, white Gaussian noise ${\bf
w}_{j}(t){\,\sim\,}\mathcal{N}(\textbf{0},\sigma^{2}I_{n})$ where
$I_{n}{\,\in\,}\mathbb{R}^{n\times n}$ is identity:

$\displaystyle{\bf y}_{j}(a\tau_{j})=C_{j}{\bf x}(a\tau_{j})+{\bf
w}_{j}(a\tau_{j})$ (2)

Here, $C_{j}{\,\in\,}\\{0,1\\}^{m_{j}\times n}$,
$m_{j}\in\mathbb{N},m_{j}{\,<\,}n$, such that each row contains exactly one 1.
Each sensor $j$ makes one observation of the environment with sampling period
$\tau_{j}$ timesteps, and $a\in\mathbb{N}$ is any number.

Over some experiment duration
$[0,T_{\text{sim}}),T_{\text{sim}}\in\mathbb{R}^{+}$, the central processor’s
objective is to use the transmissions of each sensor to construct an estimate
$\hat{{\bf x}}(t)$ of the environment state ${\bf x}(t)$. Each sensor $j$’s
objective is to determine when and which components
$y_{jk}(t),k{\,\in\,}\\{1,\cdots,m_{j}\\}$ of ${\bf y}_{j}(t)$ it should
transmit; we show that this is contingent upon how accurately the sensor can
track the central processor’s estimate $\hat{{\bf x}}(t)$.

######  Assumption 1 (Framework Assumptions).

Each sensor $j$ operates on limited power sources (e.g. batteries).
Communications in both uplink and downlink channels are noiseless and able to
precisely encode/decode real numbers. To prevent potential collisions of
transmissions, each sensor $j$ schedules random backoff times
$B_{jk}(t){\,\mathbin{\overset{d}{\kern 0.0pt\sim}}\,}B$ for some random
variable $B{\,\in\,}[0,\infty)$ distributed according to the cdf
$F_{B}(s){\,\triangleq\,}\mathbb{P}(B{\,\leq\,}s)$ for each component
$k{\,\in\,}\\{1,\cdots,m_{j}\\}$ and time $t$. The $B_{jk}(t)$ are distributed
independently of each other and of the noise process ${\bf w}_{j}(t)$. The
central processor performs fusion by averaging over the most recent sensor
observations it received to construct an estimate $\hat{x}_{i}(t)$ for each
component $i\in\\{1,\cdots,n\\}$.

######  Remark 1.

We emphasize that our goal is not to model any particular distributed data-
gathering architecture in fine detail, but to provide insights into the
distinctions of architectures with and without feedback from the central
processor to the sensors. Thus, while more complex functionalities (e.g.,
optimized random backoff strategies [7], consensus-based fusion [8], change-
detection [9]) may be used than those described in Assumption 1, we focus
specifically on functionalities which need to be implemented differently based
on whether an architecture has enabled feedback or not.

###  II-B Modular Framework

Under Assumption 1, we distinguish between traditional INformation and
OUTformation architectures in the following way. The presence or absence of
broadcast feedback (which pushes data “outward” from the central processors to
the sensors) changes how accurately each sensor can track the central
processor’s estimate $\hat{{\bf x}}(t)$.

Figure 1: A modular representation of an OUTformation architecture for
distributed state estimation under unknown environments and communication
delay constraints, using a single central processor and a network of $M$
external sensors. For each sensor, the verification rule (Definition 5) is
highlighted in green, and the transmission rule (Definition 6) is in blue. The
broadcast rule (Definition 4) is in red; INformation architectures are
depicted by removing the red broadcast arrows.

######  Definition 2 (INformation and OUTformation).

Under an INformation architecture, each sensor’s estimate of
$\hat{x}_{i}(t),i\in\\{1,\cdots,n\\}$ is its own previous transmission to the
central processor. Under an OUTformation architecture, each sensor’s estimate
of $\hat{x}_{i}(t)$ is the more recent of either its own previous transmission
or the latest broadcast received. A modular representation of the distributed
data-gathering OUTformation architecture, which we use throughout the paper,
is illustrated in Figure 1.

######  Notation 1.

Parameters are assigned the superscript ${(I)}$ for INformation architectures,
and $(O)$ for OUTformation architectures. Let $\chi$ be a placeholder variable
such that $\chi{\,=\,}I$ for an INformation architecture, and $\chi{\,=\,}O$
for OUTformation. We henceforth let $\hat{{\bf
x}}^{(\chi)}{\,\in\,}\mathbb{R}^{n}$ be the central processor’s estimate under
architecture $\chi\in\\{I,O\\}$.

######  Notation 2.

For each sensor $j{\,\in\,}\\{1,\cdots,M\\}$ and component
$k{\,\in\,}\\{1,\cdots,m_{j}\\}$, we let $i_{k}{\,\in\,}\\{1,\cdots,n\\}$
correspond to the “full state vector component” from which observation
$y_{jk}(t)$ is made, i.e. $x_{i_{k}}(t){\,\triangleq\,}C_{j,(k,\cdot)}{\bf
x}(t)$, where $C_{j,(k,\cdot)}$ denotes the $k$th row of $C_{j}$ from (2).

######  Definition 3 (Shared vs. Unshared Components).

For each sensor $j\in\\{1,\cdots,M\\}$, let $k{\,\in\,}\\{1,\cdots,m_{j}\\}$
be a component and $i_{k}\in\\{1,\cdots,n\\}$ be the corresponding full-vector
component (see Notation 2). Then $k$ is said to be a shared component if
$x_{i_{k}}(t)$ is sensed by other sensors, and an unshared component if
$x_{i_{k}}(t)$ is sensed by only $j$. Define the set
$\mathcal{U}_{j}{\,\subseteq\,}\\{1,\cdots,m_{j}\\}$ to be the set of sensor
$j$’s unshared components, and $\mathcal{S}_{j}\triangleq\mathcal{U}_{j}^{c}$
to be the set of sensor $j$’s shared components.

######  Definition 4 (Broadcast Rule).

OUTformation architectures employ the following broadcast rule (red in Figure
1): a component $i\in\\{1,\cdots,n\\}$ is scheduled to be broadcast if
$\hat{x}_{i}^{(O)}(t)\neq\hat{x}_{i}^{(O)}(s)$, where $s<t$ is the time of the
previous broadcast. All scheduled values $\hat{x}_{i}^{(O)}(t)$ are collected
to form the dimension-reduced broadcast vector $\hat{{\bf
x}}^{(b)}(t)\in\cup_{i=1}^{n}\mathbb{R}^{i}$ and sent to the sensors at once.

######  Assumption 2 (Components Being Broadcast).

Broadcasts are made only for components which are shared in the sense of
Definition 3.

######  Definition 5 (Verification Rule).

For $\chi{\,\in\,}\\{I,O\\}$ as in Notation 1 and each sensor
$j{\,\in\,}\\{1,\cdots,M\\}$, we define $T_{j}^{(\chi)}$, a multiple of the
sampling period $\tau_{j}$ from (2), to be the verification period under
architecture $\chi$. At time $t{\,\triangleq\,}aT_{j}^{(\chi)}$,
$a\in\mathbb{N}$, the verification rule schedules the transmission of
$y_{jk}(t)$, $k{\,\in\,}\\{1,\cdots,m_{j}\\}$ for time $t{\,+\,}B_{jk}(t)$ if
$\beta_{jk}^{(\chi)}(t)=1$, where $B_{jk}(t)$ is the random backoff time from
Assumption 1. Under an INformation architecture, we use:

$\displaystyle\beta_{jk}^{(I)}(t)=\mathds{1}\left\\{\lvert
y_{jk}(s_{jk})-y_{jk}(t)\rvert\geq\epsilon\right\\}$ (3)

while under an OUTformation architecture, we use:

$\displaystyle\beta_{jk}^{(O)}(t)=\begin{cases}\mathds{1}\\{\lvert\hat{x}^{(b)}_{i_{k}}(t)-y_{jk}(t)\rvert\geq\epsilon\\}&\text{
if }s^{*}\geq s_{jk}\\\ \beta_{jk}^{(I)}(t)&\text{ else}\end{cases}$ (4)

Here, $s_{jk}{\,<\,}t$ is the time of sensor $j$’s previous transmission of
component $k$, and $s^{*}{\,<\,}t$ is the time the broadcast value
$\hat{x}^{(b)}_{i_{k}}(t)$ was created. Threshold parameter $\epsilon{\,>\,}0$
is chosen by design. The special case of event-triggered verification occurs
when the verification rule is checked with every observation sensor $j$
generates, i.e. $T_{j}^{(I)}{\,=\,}T_{j}^{(O)}{\,=\,}\tau_{j}$.

######  Definition 6 (Transmission Rules).

Let $a{\,\in\,}\mathbb{N}$, and let $\chi{\,\in\,}\\{I,O\\}$ be the
placeholder variable defined in Notation 1. For each sensor $j$, let
$T_{j}^{(\chi)}$ be the verification period from Definition 5. At time
$t{\,+\,}B_{jk}(t)$ where $t{\,\triangleq\,}aT_{j}^{(\chi)}$ and $B_{jk}(t)$
is the random backoff time from Assumption 1, the transmission rule transmits
$y_{jk}(t)$ for every component $k{\,\in\,}\\{1,\cdots,m_{j}\\}$, if
$\beta_{jk}^{(\chi)}(t){\,=\,}1$ and
$\gamma_{jk}^{(\chi)}(t,B_{jk}(t)){\,=\,}1$. Essentially, the transmission
rule is a re-checking of the verification rule in case the sensor receives
more up-to-date data about the central processor’s estimate. Because each
sensor only has their own local observations under an INformation
architecture, we have:

$\displaystyle\gamma_{jk}^{(I)}(t,B_{jk}(t))$
$\displaystyle=\beta_{jk}^{(I)}(t)$ (5)

while under an OUTformation architecture, we use:

$\displaystyle\gamma_{jk}^{(O)}(t,B_{jk}(t))=\mathds{1}\\{\beta_{jk}^{(O)}(t)=1\\}\wedge$
$\displaystyle\hskip
14.22636pt\begin{cases}\mathds{1}\\{\lvert\hat{x}^{(b)}_{i_{k}}(t+B_{jk}(t))-y_{jk}(t)\rvert\geq\epsilon\\}&\text{
if }s^{**}\geq s_{jk}\\\ \mathds{1}\left\\{\lvert
y_{jk}(s_{jk})-y_{jk}(t)\rvert\geq\epsilon\right\\}&\text{ else}\end{cases}$
(6)

Here, $s_{jk}{\,<\,}t$ is the time of sensor $j$’s previous transmission of
component $k$, and $s^{**}{\,<\,}t{\,+\,}B_{jk}(t)$ is the time the broadcast
value $\hat{x}^{(b)}_{i_{k}}(t{\,+\,}B_{jk}(t))$ was created. Threshold
parameter $\epsilon>0$ is chosen by design.

######  Notation 3.

Under Assumption 1, the three types of architectures we compare are specified
below:

  1. 1.

Pure INformation architecture $IN_{0}$: each sensor $j$ implements
verification rule (3) with verification period $T_{j}^{(I)}$, and transmission
rule (5) with zero threshold $\epsilon{\,=\,}0$.

  2. 2.

Absolute-Difference INformation architecture $IN(\epsilon)$: each sensor $j$
implements verification rule (3) with verification period $T_{j}^{(I)}$, and
transmission rule (5) with fixed threshold $\epsilon{\,>\,}0$.

  3. 3.

OUTformation architecture $OUT(\epsilon)$: each sensor $j$ implements
verification rule (4) with verification period $T_{j}^{(O)}$, transmission
rule (6) with fixed threshold $\epsilon{\,>\,}0$, and the central processor
uses the broadcast rule from Definition 4.

######  Definition 7 (Rates per Component).

For sensor $j$ under architecture $\chi{\,\in\,}\\{I,O\\}$, let
$U_{jk}^{(\chi)}(s{\,:\,}t),D_{jk}^{(\chi)}(s{\,:\,}t){\,\in\,}\mathbb{N}$ be
the cumulative number of transmissions and number of broadcasts received,
respectively, for component $k\in\\{1,\cdots,m_{j}\\}$ over the interval of
time $[s,t)$. The communication delay incurred per component $k$ is denoted
$\Delta t^{(u)}\in\mathbb{R}^{+}$ for uplink transmission, $\Delta
t^{(d)}\in\mathbb{R}^{+}$ for downlink broadcast. For each sensor, the rate of
power expended per component $k$ is $P_{U}\in\mathbb{R}^{+}$ for transmission
and $P_{D}\in\mathbb{R}^{+}$ for receipt.

######  Definition 8 (Performance Metrics).

The mean-squared error (MSE) of the environment state vector over experiment
duration $[0,T_{\text{sim}})$ under architecture $\chi\in\\{I,O\\}$ is given
by

$\displaystyle\frac{1}{T_{\text{sim}}}\sum\limits_{t=0}^{T_{\text{sim}}}\sum\limits_{i=1}^{n}\left\lvert
x_{i}(t)-\hat{x}^{(\chi)}_{i}(t)\right\rvert^{2}$ (7)

The total amount of power expenditure per sensor $j$ under architecture $\chi$
is given by:

$\displaystyle
R_{j}^{(\chi)}(0:T_{\text{sim}})=P_{U}U_{j}^{(\chi)}(0:T_{\text{sim}})+P_{D}D_{j}^{(\chi)}(0:T_{\text{sim}})$
(8)

where the power expenditure rates $P_{U}$,$P_{D}$ are from Definition 7, and
$U_{j}^{(\chi)}(s{\,:\,}t){\,\triangleq\,}\sum_{k=1}^{m_{j}}U_{jk}^{(\chi)}(s{\,:\,}t)$,
likewise
$D_{j}^{(\chi)}(s{\,:\,}t)\triangleq\sum_{k=1}^{m_{j}}D_{jk}^{(\chi)}(s{\,:\,}t)$.

##  III Theoretical Analysis

With the setup described above, we now characterize the tradeoff space between
the MSE and sensor power expenditure metrics (see Definition 8) for the three
architectures in Notation 3. In this section, we introduce theory to analyze
$IN(\epsilon)$ and $OUT(\epsilon)$ under the following simplified version of
the original problem setup in Section II-A. Later, in Section IV, we
numerically study all three architectures for the original setup of Section
II-A.

######  Setting 1 (Two Sensors over One-Component Change Environment).

We consider the true environment dynamics (1) when the vector of indicators
$[\delta_{1}(c),\cdots,\delta_{n}(c)]^{T}$ can only take values from the set
$\\{\textbf{0}_{n},{\bf e}_{1},\cdots,{\bf e}_{n},-{\bf e}_{1},\cdots,-{\bf
e}_{n}\\}$ for all $c{\,\in\,}\mathbb{N}$, where $\textbf{0}_{n}$ denotes the
zero vector in $\mathbb{R}^{n}$. Both types of architectures operate with
$M{\,=\,}2$ sensors, where each sensor $j{\,\in\,}\\{1,2\\}$ has
$m_{j}{\,\in\,}\mathbb{N}$ components. Here,
$m^{\prime}{\,<\,}\min(m_{1},m_{2})$ components are shared in the sense of
Definition 3
($\mathcal{S}{\,\triangleq\,}\mathcal{S}_{1}{\,=\,}\mathcal{S}_{2},\lvert\mathcal{S}\rvert{\,=\,}m^{\prime}$)
and the remaining $m_{j}{\,-\,}m^{\prime}$ are unshared
($\lvert\mathcal{U}_{j}\rvert{\,=\,}m_{j}{\,-\,}m^{\prime}$). The sensors have
the same sampling rate $\tau{\,\triangleq\,}t_{1}{\,=\,}t_{2}$ (see (2)) such
that $\Delta t/\tau{\,\in\,}\mathbb{N}$, and use event-triggered verification
$T_{j}^{(I)}{\,=\,}T_{j}^{(O)}{\,=\,}\tau$ (see Definition 5). Because there
are only two sensors in the network, sensor $1$ transmits immediately
($B_{1k}(t){\,=\,}0$) while sensor $2$ performs random backoff according to
the cdf $F_{B}$ prescribed in Assumption 1.

######  Notation 4.

For each sensor $j\in\\{1,2\\}$ and each component $k\in\\{1,\cdots,m_{j}\\}$,
let
$p_{jk}^{(\chi)}(t,B_{jk}(t))\triangleq\mathbb{P}(\gamma_{jk}^{(\chi)}(t,B_{jk}(t))=1)$
be the probability that component $k$ is transmitted uplink by sensor $j$ at
time $t$ under architecture $\chi$, using (5) for $\chi{\,=\,}I$ and (6) for
$\chi{\,=\,}O$. Furthermore, for simplicity of expression, every interval
$[c\Delta t,(c+1)\Delta t)$, where $c\in\mathbb{N}$ and $\Delta t$ is from
(1), is referred to as environment interval $c$.

######  Lemma 1 (Power from Unshared Components).

For sensor $j{\,\in\,}\\{1,2\\}$ under Setting 1, the expected total power
expended by unshared components over environment interval $c\in\mathbb{N}$ is
equivalent under $IN(\epsilon)$ or $OUT(\epsilon)$, and given by

$\displaystyle\mathbb{E}[R_{j,\mathcal{U}_{j}}^{(\chi)}(c\Delta t:(c+1)\Delta
t)]=P_{U}\sum\limits_{k\in\mathcal{U}_{j}}\bigg{(}\sum\limits_{h=0}^{\Delta
t/\tau}$ $\displaystyle\hskip 14.22636pt(1-F_{B}(\Delta t-h\tau-\Delta
t^{(u)}))p_{jk}^{(I)}(t_{c-1}(h),B_{jk}(t_{c-1}(h)))$ $\displaystyle\hskip
14.22636pt+\sum\limits_{h=1}^{\Delta t/\tau-1}F_{B}(\Delta t-h\tau-\Delta
t^{(u)})p_{jk}^{(I)}(t_{c}(h),B_{jk}(t_{c}(h)))\bigg{)}$ (9)

where $t_{c}(h)\triangleq c\Delta t+ht$ for any $c,h{\,\in\,}\mathbb{N}$ is a
placeholder, and $R_{j,\mathcal{U}_{j}}^{(\chi)}(c\Delta t{\,:\,}(c+1)\Delta
t)$ is the part of the power (8) contributed by unshared components.

###### Proof.

Under Definition 4, no broadcasts are made for unshared components and (6) is
equivalent to (5) when $k{\,\in\,}\mathcal{U}_{j}$. Hence, $IN(\epsilon)$ and
$OUT(\epsilon)$ have the same uplink communications for unshared components,
and their contributed total power expenditures are equivalent. For placeholder
$a{\,\in\,}\\{c-1,c\\}$, let $X_{jk}^{(\chi)}(a,h)$,
$h{\,\in\,}\\{0,\cdots,\Delta t/\tau-1\\}$, be the indicator denoting whether
or not $y_{jk}(a\Delta t+h\tau)$ is received during environment interval $c$
under architecture $\chi{\,\in\,}\\{I,O\\}$. By Assumption 1,
$y_{jk}((c{\,-\,}1)\Delta t{\,+\,}h\tau)$ is received during the interval with
probability $1{\,-\,}F_{B}(\Delta t{\,-\,}h\tau{\,-\,}\Delta t^{(u)})$, and
$y_{jk}(c\Delta t{\,+\,}h\tau)$ with probability $F_{B}(\Delta
t{\,-\,}h\tau{\,-\,}\Delta t^{(u)})$. The result follows by definition of
$p_{jk}^{(\chi)}(t,B_{jk}(t))$ in Notation 4 and
$\mathbb{E}[U_{jk}^{(\chi)}(c\Delta t{\,:\,}(c+1)\Delta t)]=\sum_{h=0}^{\Delta
t/\tau}\mathbb{E}[X_{jk}(c-1,h)]{\,+\,}\sum_{h=1}^{\Delta
t/\tau-1}\mathbb{E}[X_{jk}(c,h)]$. ∎

Because both $IN(\epsilon)$ and $OUT(\epsilon)$ have the same uplink
communications for unshared components, the central processor has the same
amount of data about the environment at each time $t$. Thus, we also have the
following result.

######  Lemma 2 (MSE from Unshared Components).

Using the setup of Lemma 1, suppose $\hat{{\bf x}}^{(\chi)}(t)$ is constructed
via the fusion rule from Assumption 1 under $IN(\epsilon)$ when $\chi{\,=\,}I$
and $OUT(\epsilon)$ when $\chi{\,=\,}O$. Then

$\displaystyle\sum\limits_{i\in\mathcal{N}}\left\lvert
x_{i}(t)-\hat{x}^{(I)}_{i}(t)\right\rvert^{2}=\sum\limits_{i\in\mathcal{N}}\left\lvert
x_{i}(t)-\hat{x}^{(O)}_{i}(t)\right\rvert^{2}$ (10)

where
$\mathcal{N}{\,\triangleq\,}\\{i_{k}{\,\in\,}\\{1,\cdots,n\\}\,|\,k{\,\in\,}\mathcal{U}_{1}{\,\cup\,}\mathcal{U}_{2}\\}$
with $i_{k}$ denoting the full state vector component corresponding to
component $k$ (see Notation 2).

The more interesting comparison arises for shared components
$k{\,\in\,}\mathcal{S}$. To obtain concrete mathematical expressions, we
present the next three results over a specific chosen interval of
communications $[T_{0},T_{f}){\,\subset\,}[0,T_{\text{sim}})$ such that 1)
$IN(\epsilon)$ and $OUT(\epsilon)$ begin from the same performance metric
values at time $T_{0}$, and 2) the sample path of communications for shared
component $k$ behaves exactly like the noiseless case. By deriving conditions
of performance improvement for $OUT(\epsilon)$ over $IN(\epsilon)$ during
$[T_{0},T_{f})$, we remark the implication that this improvement cascades over
the longer interval $[T_{0},T_{\text{sim}})$.

######  Setting 2 (Interval of Communications I).

Under Setting 1, choose the specific interval $[T_{0},T_{f})$ to be
environment interval $c$ (i.e., $T_{0}{\,\triangleq\,}c\Delta t$,
$T_{f}{\,\triangleq\,}(c+1)\Delta t$) such that the following occurs. The
environment evolves such that $[\delta_{1}(c),\cdots,\delta_{n}(c)]={\bf
e}_{i_{k}}$ for full state vector component $i_{k}{\,\in\,}\\{1,\cdots,n\\}$
defined in Notation 2 for a component $k{\,\in\,}\mathcal{S}$ which is shared
in the sense of Definition 3. The sample path of observations
$\mathcal{Y}_{k}\triangleq\\{y_{jk}(t),t\in[0,(c{\,+\,}1)\Delta
t),j\in\\{1,2\\}\\}$ for $k$ is such that for $\chi\in\\{I,O\\}$,
$\beta_{jk}^{(\chi)}(c\Delta t){\,=\,}1$ and $\beta_{jk}^{(\chi)}(c\Delta
t{\,+\,}h\tau){\,=\,}0$ for all $h{\,\leq\,}H-1$; here, $H{\,\in\,}\mathbb{N}$
is defined in Setting 1, and $\beta_{jk}^{(\chi)}$ is the verification rule
from Definition 5.

######  Theorem 1 (Power from Shared Components).

Suppose Setting 2 holds for shared component $k{\,\in\,}\mathcal{S}$ over
environment interval $c{\,\in\,}\mathbb{N}$ defined in Notation 4. Then the
expected difference between the power expended under $IN(\epsilon)$ and
$OUT(\epsilon)$ is given by

$\displaystyle\mathbb{E}[R^{(I)}(c\Delta t:(c+1)\Delta t)-R^{(O)}(c\Delta
t:(c+1)\Delta t)]$ (11) $\displaystyle=(P_{U}-P_{D})(1-F_{B}(\Delta
t^{(u)}+\Delta t^{(d)}))\mathbb{P}(\lvert W_{1}-W_{2}\rvert\geq\epsilon)$

where $W_{1},W_{2}{\,\sim\,}\mathcal{N}(0,\sigma^{2})$ are independent random
variables with $\sigma$ defined in (2), $\Delta t^{(u)}$, $\Delta t^{(d)}$,
$P_{U}$, $P_{D}$ are the communication delays and power expenditure rates in
Definition 7, $F_{B}$ is the random backoff time distribution from Setting 1,
and $R^{(\chi)}(c\Delta t{\,:\,}(c+1)\Delta t)$ is the power expended over
both sensors.

###### Proof.

By the random backoff strategy of Setting 1 and communications of Setting 2,
sensor $1$ transmits $y_{1k}(c\Delta t)$ at time $c\Delta t$ while sensor $2$
observes $y_{2k}(c\Delta t)$ and schedules to transmit at time $c\Delta
t+B_{2k}(c\Delta t)$. Sensor $2$’s scheduled transmission is cancelled if only
the central processor broadcasts $y_{1k}(c\Delta t)$ before time $c\Delta
t+B_{2k}(c\Delta t)$, which occurs with probability $1-F_{B}(\Delta
t^{(u)}+\Delta t^{(d)})$ under Setting 1, and if $\lvert y_{1k}(c\Delta
t)-y_{2k}(c\Delta t)\rvert{\,\geq\,}\epsilon$, which occurs with probability
$\mathbb{P}(\lvert W_{1}-W_{2}\rvert{\,\geq\,}\epsilon)$ by (2). The result
follows from the fact that $P_{U}-P_{D}$ is the difference between
transmitting and broadcasting an extra component (see Definition 7). ∎

######  Theorem 2 (MSE from Shared Components).

Suppose Setting 2 holds for shared component $k{\,\in\,}\mathcal{S}$ over
environment interval $c{\,\in\,}\mathbb{N}$. Let $\hat{{\bf x}}^{(\chi)}(t)$
be constructed via the fusion rule from Assumption 1, and suppose that
$\hat{x}_{i_{k}}(c\Delta t){\,\triangleq\,}\hat{x}_{i_{k}}^{(I)}(c\Delta
t){\,=\,}\hat{x}_{i_{k}}^{(O)}(c\Delta t)$. Then the probability that the MSE
contributed by component $i_{k}$, defined in Notation 2, under $OUT(\epsilon)$
is less than that under $IN(\epsilon)$ is given by:

$\displaystyle(1-F_{B}(\Delta t^{(u)}+\Delta t^{(d)}))$ (12)
$\displaystyle\hskip
14.22636pt*\mathbb{P}\left(\frac{1}{2}W_{1}W_{2}+W_{2}^{2}-\frac{3}{4}W_{1}^{2}>0,\hskip
2.84544pt\lvert W_{1}-W_{2}\rvert<\epsilon\right)$

where $\Delta t^{(u)},\Delta t^{(d)}$ are the communication delays in
Definition 7, $F_{B}$ is the random backoff time distribution from Setting 1,
and $W_{1},W_{2}{\,\sim\,}\mathcal{N}(0,\sigma^{2})$ are independent random
variables with $\sigma$ from (2).

###### Proof.

For simplicity, we exclude the factor of $1/T_{\text{sim}}$ in (7) throughout
our proof. Under Setting 2, the MSE of component $i_{k}$ during environment
interval $c$ under $IN(\epsilon)$ is:

$\displaystyle(\hat{x}_{i_{k}}(c\Delta t)-x_{i_{k}}(c\Delta t))^{2}\Delta
t^{(u)}$ $\displaystyle\hskip 19.91684pt+(y_{1k}(c\Delta t)-x_{i_{k}}(c\Delta
t))^{2}(c\Delta t+B_{2k}(c\Delta t))$ $\displaystyle\hskip
19.91684pt+\left(\frac{1}{2}(y_{1k}(c\Delta t)+y_{2k}(c\Delta
t))-x_{i_{k}}(c\Delta t)\right)^{2}$ $\displaystyle\hskip 56.9055pt*(\Delta
t-B_{2k}(c\Delta t)-\Delta t^{(u)})$ (13)

Under $OUT(\epsilon)$, the MSE is the same as (III) if sensor $2$ transmits
$y_{2k}(t)$; this occurs when

$\displaystyle\\{B_{2k}(c\Delta t)<\Delta t^{(u)}+\Delta t^{(d)}\\}\cup$ (14)
$\displaystyle\\{B_{2k}(c\Delta t)\geq\Delta t^{(u)}+\Delta t^{(d)},\lvert
y_{1k}(c\Delta t)-y_{2k}(c\Delta t)\rvert\geq\epsilon\\}$

When sensor $2$ does not transmit, the MSE is:

$\displaystyle(\hat{x}_{i_{k}}(c\Delta t)-x_{i_{k}}(c\Delta t))^{2}\Delta
t^{(u)}$ $\displaystyle\hskip 14.22636pt+(y_{1k}(c\Delta t)-x_{i_{k}}(c\Delta
t))^{2}((c+1)\Delta t-\Delta t^{(u)})$ (15)

From (2), $(1/2)(y_{1k}(t)+y_{2k}(t))-x_{i_{k}}(t)=(1/2)(W_{1}+W_{2})$ for any
$t$ and two independent $W_{1},W_{2}\sim\mathcal{N}(0,\sigma^{2})$; likewise,
$y_{1k}(t)-x_{i_{k}}(t)=W_{1}$. Hence, the difference between (III) and (III)
yields $(\Delta t-B_{2k}(c\Delta t)-\Delta
t^{(u)})((1/4)(W_{1}+W_{2})^{2}-W_{1}^{2})$, which is positive under Setting 1
if $(1/4)(W_{1}+W_{2})^{2}{\,-\,}W_{1}^{2}{\,>\,}0$. Combining this with (14)
yields our result. ∎

Figure 2: A simple empirical illustration of the results of Theorems 1 and 2
when $\sigma{\,=\,}0$ over environment (1): time-evolution of the
communications between a central processor and $M{\,=\,}2$ sensors under
Absolute-Difference INformation architecture $IN(\epsilon)$ [Top] and
OUTformation architecture $OUT(\epsilon)$ [Bottom] and event-triggered
verification. Sensor $1$ transmissions are in blue, sensor $2$ in green, and
central processor broadcasts in red. Under $OUT(\epsilon)$, if the central
processor broadcasts an update from sensor $1$, sensor $2$ may cancel its
scheduled backoff transmission (e.g., $t{\,=\,}40,60,\cdots,120,160,\cdots$).
The slopes of each line vary because communication delays are proportional to
the number of components transmitted/broadcast (see Definition 7).

######  Remark 2 (Implications of Theorems 1 and 2).

In Figure 2, we demonstrate the cascading effect of the performance
improvements derived in Theorems 1 and 2 by considering a longer interval of
time than Setting 2. For significance, the original setting of multiple
simultaneous component changes (see Section II-A) with no measurement noise
$(\sigma{\,=\,}0)$ is plotted. For a general number $M{\,\geq\,}2$ of sensors
with small $\sigma$ in (2), multiple sensors may decrease in both uplink and
downlink power at the expense of additional downlink power for one sensor, and
Theorem 1 suggests an overall decrease in total power expenditure using
$OUT(\epsilon)$ over $IN(\epsilon)$. Consequently, Theorem 2 and Figure 2
suggest that the difference in MSE performance between $OUT(\epsilon)$ and
$IN(\epsilon)$ is always zero when $\sigma=0$ in (2); when $\sigma\neq 0$,
$OUT(\epsilon)$ yields less MSE than $IN(\epsilon)$ with probability (12).

An interesting MSE advantage of $OUT(\epsilon)$ over $IN(\epsilon)$ arises
when we use larger verification periods $T_{j}^{(I)}{\,>\,}\tau_{j}$, defined
in Definition 5. Here, instead of Setting 2, we use the following interval of
communications instead.

######  Setting 3 (Interval of Communications II).

We consider the original dynamics (1) with nonzero random backoff time for
both sensors (see Assumption 1). For sampling time $\tau_{j}$ defined in (2)
and verification periods defined in Definition 5, let
$T_{j}^{(I)}{\,=\,}T_{j}^{(O)}{\,=\,}T_{j}{\,>\,}\tau_{j}$ for sensor
$j{\,\in\,}\\{1,2\\}$. Choose interval $[T_{0},T_{f})$,
$T_{0}{\,\triangleq\,}(a_{1}{\,-\,}1)T_{1}$ and
$T_{f}{\,\triangleq\,}a_{1}T_{1}$ such that
$(a_{1}-1)T_{1}{\,<\,}a_{2}T_{2}{\,<\,}a_{1}T_{1}$ for some
$a_{1},a_{2}{\,\in\,}\mathbb{N}$. The environment (1) evolves such that 1) one
shared component $k^{\prime}\in\mathcal{S}$ changes value once by magnitude
$d^{\prime}$ during $((a_{1}-1)T_{1},a_{2}T_{2}]$ and remains constant during
$(a_{2}T_{2},a_{1}T_{1}]$, and 2) one unshared component
$k_{1}\in\mathcal{U}_{1}$ of sensor $1$ remains constant during
$((a_{1}-1)T_{1},a_{2}T_{2}]$ and changes value once by magnitude $d_{1}$
during $(a_{2}T_{2},a_{1}T_{1}]$, and 3) no other full-state components
$i\in\\{1,\cdots,n\\}/\\{i_{k_{1}},i_{k^{\prime}}\\}$ (see Notation 2) change
value over $[T_{0},T_{f})$. Here,
$d_{1},d^{\prime}{\,\sim\,}\mathcal{U}[\underline{d},\overline{d}]$ via the
stepsize distribution from (1). The sample path of observations
$\mathcal{Y}\triangleq\\{{\bf
y}_{j}(t),t{\,\in\,}[0,(a_{1}-1)T_{1})\cup[T_{0},T_{f}),j{\,\in\,}\\{1,2\\}\\}$
is such that $\beta_{1k_{1}}^{(\chi)}(a_{1}T_{1}){\,=\,}1$ and
$\beta_{jk^{\prime}}^{(\chi)}(a_{j}T_{j}){\,=\,}1$ for $j\in\\{1,2\\}$, where
$\beta_{jk}^{(\chi)}$ is the verification rule from Definition 5.

######  Theorem 3 (MSE with Longer Verification Period).

Consider the interval defined in Setting 3. Let ${\bf x}^{(\chi)}(t)$ be
constructed via the fusion rule from Assumption 1 such that
$\hat{x}_{i_{k}}(T_{0})\triangleq\hat{x}_{i_{k}}^{(I)}(T_{0})=\hat{x}_{i_{k}}^{(O)}(T_{0})$
where $T_{0}$ and $k\in\\{k_{1},k^{\prime}\\}$ are defined in Setting 3. Then
the MSE contributed by component $i_{k_{1}}$ under $OUT(\epsilon)$ is less
than that under $IN(\epsilon)$ during interval $[T_{0},T_{f})$ with
probability:

$\displaystyle\mathbb{P}\left(\lvert d_{1}\rvert>0,\lvert
W_{2}^{\prime}-W_{1}^{\prime}\rvert<\epsilon\right)*$ (16)
$\displaystyle\hskip 14.22636pt(1-G_{B_{1},B_{2}}(a_{1}T_{1}-a_{2}T_{2}+\Delta
t^{(u)}+\Delta t^{(d)})$

Here,
$G_{B_{1},B_{2}}(s){\,\triangleq\,}\mathbb{P}(B_{1}{\,-\,}B_{2}{\,\leq\,}s)$
for two independent $B_{1},B_{2}$ distributed according to $F_{B}$ from
Assumption 1,
$W_{1},W_{1}^{\prime},W_{2},W_{2}^{\prime}\sim\mathcal{N}(0,\sigma^{2})$ are
independent random variables with $\sigma>0$ from (2), and the communication
delays $\Delta t^{(u)},\Delta t^{(d)}$ are in Definition 7.

###### Proof.

As in the proof of Theorem 2, we exclude the factor of $1/T_{\text{sim}}$ from
(7). Under $OUT(\epsilon)$ in Setting 3, sensor $1$ receives a broadcast
update $y_{2k^{\prime}}(a_{2}T_{2})$ about component $k^{\prime}$ if

$\displaystyle\\{B_{2k^{\prime}}(a_{2}T_{2})-B_{1k^{\prime}}(a_{1}T_{1})\geq$
$\displaystyle\hskip 28.45274pta_{1}T_{1}-a_{2}T_{2}+\Delta t^{(u)}+\Delta
t^{(d)}\\}\cap$ $\displaystyle\hskip 56.9055pt\\{\lvert
y_{2k^{\prime}}(a_{2}T_{2})-y_{1k^{\prime}}(a_{1}T_{1})\rvert<\epsilon\\}$
(17)

If (III) holds, sensor $1$ only transmits for component $k_{1}$. Under
$IN(\epsilon)$, sensor $1$ transmits both components $k^{\prime}$ and $k_{1}$.
The central processor receives $y_{1k_{1}}(a_{1}T_{1})$ with an additional
uplink delay of $\Delta t^{(u)}$ compared to $OUT(\epsilon)$. Thus, the
difference in MSE of component $i_{k_{1}}$ between $OUT(\epsilon)$ and
$IN(\epsilon)$ during interval $[T_{0},T_{f})$ is:

$\displaystyle\Delta t^{(u)}((y_{1k_{1}}(a_{1}T_{1})-x_{i_{k_{1}}}(c\Delta
t))^{2}$ $\displaystyle\hskip
28.45274pt-(\hat{x}_{i_{k_{1}}}(a_{1}T_{1})-x_{i_{k_{1}}}(c\Delta t))^{2})$
(18)

where $c{\,\in\,}\mathbb{N}$ is such that $c\Delta t{\,\leq\,}a_{1}T_{1}$ is
the time of the last true change in component $k_{1}$ which was not
transmitted to the central processor, with $\Delta t$ defined in (1). By
Setting 3 and (2), we have $y_{1k_{1}}(a_{1}T_{1})-x_{i_{k_{1}}}(c\Delta
t)=W_{1}$, $\hat{x}_{i_{k_{1}}}(a_{1}T_{1})-x_{i_{k_{1}}}(c\Delta
t)=d_{1}+W_{1}$, and
$y_{2k^{\prime}}(a_{2}T_{2})-y_{1k^{\prime}}(a_{1}T_{1})=W_{2}^{\prime}-W_{1}^{\prime}$
for some independent
$W_{1},W_{1}^{\prime},W_{2}^{\prime}{\,\sim\,}\mathcal{N}(0,\sigma^{2})$.
Combined with (III), the result follows. ∎

Figure 3: [Top] A version of Figure 2 under original dynamics (1) with longer verification periods $T_{1}^{(\chi)}{\,=\,}23,T_{2}^{(\chi)}{\,=\,}41$, $\chi\in\\{I,O\\}$. All three architectures $IN_{0}$, $IN(\epsilon)$, and $OUT(\epsilon)$ of Notation 3 are plotted. Note three times where the uplink communications of $OUT(\epsilon)$ are less than $IN(\epsilon)$: 1) at $t{\,=\,}92,138$, where sensor $1$ cancels its scheduled transmission and 2) at time $t{\,=\,}161$, where the slope of sensor $1$’s uplink transmission is steeper. This suggests that updates from sensor observations are received more quickly on average under $OUT(\epsilon)$ than $IN(\epsilon)$. [Bottom] The evolution of components $7,10,12$ of ${\bf x}(t)$ and $\hat{{\bf x}}^{(\chi)}(t)$ over time. Since the central processor receives transmissions more quickly, $\hat{{\bf x}}^{(O)}(t)$ tracks ${\bf x}(t)$ more accurately than $\hat{{\bf x}}^{(I)}(t)$.  |
---|---
(a) $\Delta t=20$, $p=0.9$, $\mu=7$, $T_{1}^{(\chi)}=22,T_{2}^{(\chi)}=35$. | (c) $\Delta t=20$, $p=0.9$, $\mu=7$, and event-triggered verification.
|
(b) $\Delta t=45$, $p=0.9$, $\mu=7$, $T_{1}^{(\chi)}=22,T_{2}^{(\chi)}=35$. | (d) $\Delta t=20$, $p=0.9$, $\mu=20$, and event-triggered verification.
Figure 4: Plots of the MSE per timestep against power expenditure per timestep
for both sensors $1$ and $2$, averaged over $40$ Monte-Carlo simulations of
environment (1). Each row of subfigures differ by their values of $\Delta t$
and $p$ (see (1)), mean random backoff time $\mu$, and frequency of
verification (see Definition 5). The architectures compared are described in
Notation 3. The blue dot corresponds to $IN_{0}$, and the orange dot for
$IN(\epsilon)$, with dashed lines to emphasize their visibility. For
$OUT(\epsilon)$, a constant broadcast probability $p^{(b)}$ is ranged over
$\\{0,0.1,\cdots,1\\}$, and we make separate curves for the uplink, downlink,
and the total (uplink $+$ downlink) power expenditures; gold dots for
downlink, green for uplink, and black for the combined total. As in Remark 3,
$IN_{0}$ performs the worst in both metrics for all four cases. The green and
black dots for $OUT(\epsilon)$ with $p^{(b)}{\,=\,}0$ coincide with
$IN(\epsilon)$ because no broadcasts are made. Under environments with fast
variation (small $\Delta t$) relative to the verification periods
$T_{j}^{(\chi)}$, $OUT(\epsilon)$ detects changes more quickly than
$IN(\epsilon)$, providing a reduction in MSE at additional downlink power (see
(a) and Remark 3); this tradeoff becomes dampened when $\Delta t$ is large,
and the MSE performance rarely varies with power expenditure (see (b)). Under
event-triggered verification, $OUT(\epsilon)$ maintains similar MSE as
$IN(\epsilon)$ by having one sensor reduce power expenditure at the cost of
the other sensor (see (c) and Remark 2); longer mean backoff time causes
increasingly outdated data to be received, which increases the MSE (see (d)).

######  Remark 3 (Implications of Theorem 3).

In Figure 3, we demonstrate the cascading effect of the MSE advantage derived
in Theorem 3 by considering a longer interval of time than Setting 3. For
presentation clarity, we choose prime-valued $T_{j}^{(\chi)}$ and zero backoff
time. For comparison, we include the communications and MSE for the pure
INformation architecture $IN_{0}$ (see Notation 3); $IN_{0}$ transmits the
most number of components uplink, consequently incurring the longest
transmission delays and largest MSE among the three architectures. By
transmission rule (6), $OUT(\epsilon)$ always transmits the same number of
components or less than $IN(\epsilon)$. Less components are likely to be
transmitted for small measurement noise $\sigma$ relative to the distribution
of the stepsize ${\bf d}(c),c\in\mathbb{N}$ in (1), since one sensor cancels
backoff transmissions if broadcast updates from the other sensor contain
similar component values (e.g., $t{\,=\,}92,138,161$ in Figure 3). Because
$OUT(\epsilon)$ transmits less uplink packets on average among the three
architectures, it also incurs the least uplink delays, and so the central
processor under $OUT(\epsilon)$ detects changes in the environment state most
quickly. On the other hand, for large $\sigma$ or a suboptimal choice of
$\epsilon$, the event $\lvert W_{2}^{\prime}-W_{1}^{\prime}\rvert<\epsilon$
occurs with low probability.

##  IV Numerical Analysis

In this section, we supplement the theoretical insights derived from the
previous Section III by empirically plotting the tradeoff space under a
variety of dynamics and parameter choices. In particular, we implement the
original setup of Section II-A, consider longer intervals than those specified
in Settings 2 and 3, and remove specific constraints on the sample path of
observations $\mathcal{Y}\triangleq\\{{\bf
y}_{j}(t),t{\,\in\,}[0,T_{\text{sim}}),j{\,\in\,}\\{1,2\\}\\}$ and state
estimates $\hat{{\bf x}}^{(\chi)},\chi{\,\in\,}\\{I,O\\}$ imposed in Theorems
1, 2, and 3. We consider the effects of varying three parameters which
demonstrate the most interesting comparisons between the performance of the
three architectures from Notation 3: 1) the variation in the environment
$\Delta t$ (see (1)), 2) the mean $\mu$ of the random backoff time
distribution $F_{B}$ (see Assumption 1), and 3) the frequency of verification
(see Definition 5). For $OUT(\epsilon)$, we additionally vary over
$p^{(b)}\in[0,1]$, a constant probability in which the central processor makes
a broadcast upon receiving a transmission. We choose communication delays
$\Delta t^{(d)}{\,<\,}\Delta t^{(u)}$ and power expenditure rates
$P_{U}{\,<\,}P_{D}$ (see Definition 7).

The results are demonstrated in Figure 4. With a larger number $M\geq 2$ of
sensors, one may be inclined to believe that $IN_{0}$ obtains the lowest MSE
under an averaging fusion rule (see Assumption 1) by the law of large numbers,
because $IN_{0}$ transmits the largest number of independent, redundant noisy
observations. However, when communication delays and random backoff times are
involved, a larger number of transmissions will take a longer delay to be
received by the central processor, causing an increase in MSE. This effect was
demonstrated in Remark 3.

For the values chosen to generate Figure 4 (a), $OUT(\epsilon)$ is greater
than $IN(\epsilon)$ in power expenditure while $IN(\epsilon)$ is greater than
$OUT(\epsilon)$ in MSE for all $p^{(b)}\in(0,1]$. This effect was demonstrated
in Remark 3; $OUT(\epsilon)$ tracks changes in the true environment more
quickly than $IN(\epsilon)$, but with greater downlink power expenditure. In
Figure 4 (a), this tradeoff is shown explicitly: increasing downlink power (in
gold) allows for decreasing uplink power (in green) for both sensors. This
suggests that for $OUT(\epsilon)$ applied to settings similar to Figure 4 (a),
a balance between total power expenditure and MSE may be obtained by choosing
$p^{(b)}$ appropriately. On the other hand, in Figure 4 (b), transmissions are
more likely to be received by the central processor while the environment
state has not changed (i.e., $T_{j}^{(\chi)}+\mu<\Delta t,\chi\in\\{I,O\\}$).
Essentially, if the environment variation is slow relative to the sensors’
verification periods, then $OUT(\epsilon)$ maintains approximately the same
MSE as $IN(\epsilon)$ on average, aside from small variations due to noise
$\sigma$, and wastes power by broadcasting unnecessarily.

The tradeoff space in Figure 4 (c) demonstrates that Remark 2 holds more
generally for slower-changing environments (large $\Delta t$) and small random
backoff time relative to the environment (i.e., $\mu{\,<\,}\Delta t$). For
$OUT(\epsilon)$ under small $\sigma$, sensor $1$ expends power for both
transmitting and receiving broadcasts, while sensor $2$ expends mostly
downlink power for receiving broadcast updates. Because both sensors operate
on event-triggered verification, $OUT(\epsilon)$ is, on average, the same as
$IN(\epsilon)$ in the MSE; any variations such that $OUT(\epsilon)$ obtains
more or less MSE than $IN(\epsilon)$ is a result of noise. This suggests that
for $OUT(\epsilon)$ applied to settings similar to Figure 4 (c), a balance for
the total power expenditure may be obtained by choosing $p^{(b)}$
appropriately, at little change in MSE. When $\mu\geq\Delta t$, as in Figure 4
(d), the power expenditure follows the same trends as when $\mu<\Delta t$, but
the overall MSE tends to be larger. This is expected because $\mu{\,>\,}\Delta
t$ implies that on average, sensor $2$’s observations about ${\bf x}(c\Delta
t)$ are received when the true state is already changed to ${\bf
x}((c+1)\Delta t)$.

##  V Conclusion

We provided a theoretical and numerical characterization of the tradeoff space
for architectures with broadcast feedback (OUTformation) and architectures
without feedback (INformation) using two performance metrics: the mean-squared
error of the central processor’s estimate of the environment state and the
total power expenditure per sensor. Our study was motivated towards enabling
users to make informed design choices in distributed data-gathering
architectures for large-scale network environments under constraints such as
nonzero communication delays and limited knowledge of the environment
dynamics. We found that under an event-triggered verification rule (see
Definition 5), OUTformation architectures expend less uplink power on average
than INformation, and yields similar MSE when variation in the environment is
large relative to mean backoff time (see Theorem 1, Remark 2, Figure 4 (c) and
(d)). Under a periodic verification rule, OUTformation architectures enable a
reduced MSE on average compared to INformation, but with additional downlink
power that potentially increases the total power expenditure (see Theorem 2,
Theorem 3, and Remark 3). This suggests that by varying parameters (i.e., the
broadcast probability $p^{(b)}$ described in Section IV), the OUTformation
architecture can attain a better tradeoff between the MSE and the total power
expenditure (see Figure 4 (a) and (b)). In conclusion, the main advantage of
feedback is that each sensor’s understanding of the central processor’s state
estimate improves. Theoretical and numerical analysis of performance tradeoffs
for architectures with and without feedback when other parameters (e.g.,
threshold $\epsilon$, number of sensors) are varied is a natural subject of
future work.

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$1s\,8g~{}^{3}G_{4}$ | $1s\,7h~{}^{3}H_{4}$ | 1.371E+07 |  |  | 4.666E-04 |  |  | 4.696E-04
$1s\,8f~{}^{3}F_{4}$ | $1s\,7g~{}^{1}G_{4}$ | 2.983E+07 |  |  | 1.015E-03 |  |  | 1.002E-03
$1s\,8f~{}^{3}F_{2}$ | $1s\,7g~{}^{3}G_{3}$ | 1.371E+09 |  |  | 3.331E-02 |  |  | 3.326E-02
$1s\,8f~{}^{3}F_{3}$ | $1s\,7g~{}^{3}G_{4}$ | 1.321E+09 |  |  | 3.497E-02 |  |  | 3.489E-02
$1s\,8d~{}^{3}D_{2}$ | $1s\,7p~{}^{1}P_{1}$ | 1.339E+09 |  |  | 7.597E-02 |  |  | 6.308E-02
$1s\,8d~{}^{1}D_{2}$ | $1s\,7f~{}^{1}F_{3}$ | 2.391E+09 |  |  | 5.811E-02 |  |  | 5.740E-02
$1s\,8d~{}^{3}D_{1}$ | $1s\,7p~{}^{1}P_{1}$ | 2.172E+08 |  |  | 7.393E-03 |  |  | 6.279E-03
$1s\,8f~{}^{3}F_{3}$ | $1s\,7g~{}^{3}G_{3}$ | 4.935E+07 |  |  | 1.680E-03 |  |  | 1.706E-03
$1s\,8h~{}^{3}H_{5}$ | $1s\,7i~{}^{1}I_{6}$ | 7.451E+01 |  |  | 2.146E-09 |  |  |
$1s\,8g~{}^{3}G_{4}$ | $1s\,7h~{}^{1}H_{5}$ | 3.008E+02 |  |  | 8.380E-09 |  |  |
$1s\,8f~{}^{3}F_{4}$ | $1s\,7i~{}^{3}I_{5}$ | 5.624E-04 |  |  | 1.567E-14 |  |  |
$1s\,8d~{}^{3}D_{3}$ | $1s\,7f~{}^{3}F_{4}$ | 2.377E+09 |  |  | 6.297E-02 |  |  | 6.306E-02
$1s\,8d~{}^{3}D_{3}$ | $1s\,7f~{}^{1}F_{3}$ | 8.746E+07 |  |  | 2.980E-03 |  |  | 2.910E-03
$1s\,8d~{}^{3}D_{2}$ | $1s\,7f~{}^{3}F_{3}$ | 2.323E+09 |  |  | 5.653E-02 |  |  | 5.558E-02
$1s\,8f~{}^{3}F_{3}$ | $1s\,7g~{}^{1}G_{4}$ | 9.961E+04 |  |  | 2.640E-06 |  |  |
$1s\,8d~{}^{3}D_{2}$ | $1s\,7f~{}^{3}F_{2}$ | 2.191E+08 |  |  | 7.468E-03 |  |  | 7.799E-03
$1s\,8d~{}^{3}D_{1}$ | $1s\,7f~{}^{3}F_{2}$ | 2.610E+09 |  |  | 5.336E-02 |  |  | 5.346E-02
$1s\,8d~{}^{3}D_{3}$ | $1s\,7h~{}^{3}H_{4}$ | 2.347 |  |  | 6.228E-11 |  |  |
$1s\,8d~{}^{3}D_{2}$ | $1s\,7f~{}^{1}F_{3}$ | 6.570E+07 |  |  | 1.603E-03 |  |  | 2.344E-03
$1s\,8p~{}^{3}P_{2}$ | $1s\,7d~{}^{3}D_{1}$ | 4.816E+07 |  |  | 2.756E-03 |  |  | 2.805E-03
$1s\,8p~{}^{3}P_{2}$ | $1s\,7d~{}^{3}D_{2}$ | 5.523E+08 |  |  | 1.897E-02 |  |  | 1.995E-02
$1s\,8s~{}^{3}S_{1}$ | $1s\,7p~{}^{3}P_{0}$ | 1.303E+09 |  |  | 1.343E-01 |  |  | 1.327E-01
$1s\,8s~{}^{3}S_{1}$ | $1s\,7p~{}^{3}P_{1}$ | 3.725E+09 |  |  | 1.282E-01 |  |  | 1.297E-01
$1s\,8p~{}^{3}P_{2}$ | $1s\,7d~{}^{3}D_{3}$ | 4.166E+09 |  |  | 1.026E-01 |  |  | 1.028E-01
$1s\,8p~{}^{3}P_{2}$ | $1s\,7d~{}^{1}D_{2}$ | 1.853E+08 |  |  | 6.398E-03 |  |  | 5.445E-03
$1s\,8p~{}^{3}P_{1}$ | $1s\,7d~{}^{3}D_{1}$ | 1.220E+09 |  |  | 4.220E-02 |  |  | 4.216E-02
$1s\,8p~{}^{3}P_{1}$ | $1s\,7d~{}^{3}D_{2}$ | 3.559E+09 |  |  | 7.382E-02 |  |  | 7.482E-02
$1s\,8p~{}^{3}P_{2}$ | $1s\,7g~{}^{3}G_{3}$ | 8.694E+01 |  |  | 2.149E-09 |  |  |
$1s\,8p~{}^{3}P_{0}$ | $1s\,7d~{}^{3}D_{1}$ | 5.087E+09 |  |  | 5.870E-02 |  |  | 5.914E-02
$1s\,8s~{}^{1}S_{0}$ | $1s\,7p~{}^{1}P_{1}$ | 1.151E+10 |  |  | 1.333E-01 |  |  | 1.312E-01
$1s\,8p~{}^{3}P_{1}$ | $1s\,7d~{}^{1}D_{2}$ | 3.013E+08 |  |  | 6.284E-03 |  |  | 5.200E-03
$1s\,8s~{}^{3}S_{1}$ | $1s\,7p~{}^{3}P_{2}$ | 6.665E+09 |  |  | 1.391E-01 |  |  | 1.393E-01
$1s\,8s~{}^{3}S_{1}$ | $1s\,7p~{}^{1}P_{1}$ | 1.805E+08 |  |  | 6.430E-03 |  |  | 5.494E-03
$1s\,8s~{}^{3}S_{1}$ | $1s\,7f~{}^{3}F_{2}$ | 1.631E+03 |  |  | 3.492E-08 |  |  |
$1s\,2p~{}^{3}P_{1}$ | $1s\,2s~{}^{3}S_{1}$ | 2.893E+08 |  | 3.070E+08 | 1.204E-02 |  | 1.270E-02 | 1.213E-02
$1s\,2p~{}^{3}P_{0}$ | $1s\,2s~{}^{3}S_{1}$ | 2.584E+08 | 2.588E+08 | 2.740E+08 | 3.953E-03 | 3.954E-03 | 4.200E-03 | 3.953E-03
$1s\,2p~{}^{1}P_{1}$ | $1s\,2s~{}^{1}S_{0}$ | 1.802E+08 |  | 1.910E+08 | 3.148E-02 |  | 3.370E-02 | 3.148E-02
$1s\,3p~{}^{1}P_{1}$ | $1s\,3s~{}^{3}S_{1}$ | 1.056E+07 |  |  | 1.610E-03 |  |  |
$1s\,3p~{}^{3}P_{2}$ | $1s\,3s~{}^{3}S_{1}$ | 7.683E+07 |  |  | 4.417E-02 |  |  | 4.455E-02
$1s\,3p~{}^{3}P_{1}$ | $1s\,3s~{}^{3}S_{1}$ | 3.717E+07 |  |  | 2.017E-02 |  |  | 2.046E-02
$1s\,3p~{}^{3}P_{0}$ | $1s\,3s~{}^{3}S_{1}$ | 3.316E+07 |  |  | 6.657E-03 |  |  |
$1s\,3d~{}^{1}D_{2}$ | $1s\,3p~{}^{3}P_{1}$ | 1.895E+06 |  |  | 2.093E-03 |  |  |
$1s\,3p~{}^{1}P_{1}$ | $1s\,3s~{}^{1}S_{0}$ | 2.728E+07 |  |  | 5.529E-02 |  |  | 5.572E-02
$1s\,3d~{}^{3}D_{1}$ | $1s\,3p~{}^{3}P_{0}$ | 9.389E+06 |  |  | 2.228E-02 |  |  |
$1s\,4f~{}^{3}F_{2}$ | $1s\,4s~{}^{3}S_{1}$ | 1.997E+01 |  |  | 2.882E-08 |  |  |
$1s\,4p~{}^{1}P_{1}$ | $1s\,4s~{}^{3}S_{1}$ | 2.577E+06 |  |  | 2.311E-03 |  |  |
$1s\,3d~{}^{3}D_{1}$ | $1s\,3p~{}^{3}P_{1}$ | 5.612E+06 |  |  | 5.040E-03 |  |  |
$1s\,3d~{}^{3}D_{2}$ | $1s\,3p~{}^{3}P_{1}$ | 9.326E+06 |  |  | 1.407E-02 |  |  |
$1s\,3d~{}^{1}D_{2}$ | $1s\,3p~{}^{3}P_{2}$ | 6.051E+05 |  |  | 7.756E-04 |  |  |
$1s\,3d~{}^{3}D_{3}$ | $1s\,3p~{}^{3}P_{2}$ | 7.143E+06 |  |  | 1.407E-02 |  |  |
$1s\,3d~{}^{3}D_{1}$ | $1s\,3p~{}^{3}P_{2}$ | 1.163E+05 |  |  | 1.384E-04 |  |  |
$1s\,3d~{}^{3}D_{2}$ | $1s\,3p~{}^{3}P_{2}$ | 7.384E+05 |  |  | 1.481E-03 |  |  |
$1s\,4p~{}^{3}P_{2}$ | $1s\,4s~{}^{3}S_{1}$ | 1.829E+07 |  |  | 6.130E-02 |  |  | 6.164E-02
$1s\,4p~{}^{3}P_{1}$ | $1s\,4s~{}^{3}S_{1}$ | 8.709E+06 |  |  | 2.782E-02 |  |  |
$1s\,5f~{}^{3}F_{2}$ | $1s\,5s~{}^{3}S_{1}$ | 1.054E+01 |  |  | 5.897E-08 |  |  |
$1s\,5p~{}^{1}P_{1}$ | $1s\,5s~{}^{3}S_{1}$ | 8.533E+05 |  |  | 2.981E-03 |  |  |
$1s\,4p~{}^{3}P_{0}$ | $1s\,4s~{}^{3}S_{1}$ | 7.765E+06 |  |  | 9.203E-03 |  |  |
$1s\,4d~{}^{1}D_{2}$ | $1s\,4p~{}^{3}P_{1}$ | 4.808E+05 |  |  | 3.033E-03 |  |  |
$1s\,4p~{}^{1}P_{1}$ | $1s\,4s~{}^{1}S_{0}$ | 6.715E+06 |  |  | 7.727E-02 |  |  |
$1s\,4d~{}^{3}D_{1}$ | $1s\,4p~{}^{3}P_{0}$ | 2.858E+06 |  |  | 3.941E-02 |  |  |
$1s\,4d~{}^{3}D_{1}$ | $1s\,4p~{}^{3}P_{1}$ | 1.701E+06 |  |  | 8.893E-03 |  |  |
$1s\,4d~{}^{3}D_{2}$ | $1s\,4p~{}^{3}P_{1}$ | 2.928E+06 |  |  | 2.555E-02 |  |  |
$1s\,4d~{}^{1}D_{2}$ | $1s\,4p~{}^{3}P_{2}$ | 1.697E+05 |  |  | 1.252E-03 |  |  |
$1s\,5p~{}^{3}P_{2}$ | $1s\,5s~{}^{3}S_{1}$ | 5.993E+06 |  |  | 7.800E-02 |  |  |
$1s\,4d~{}^{3}D_{3}$ | $1s\,4p~{}^{3}P_{2}$ | 2.134E+06 |  |  | 2.474E-02 |  |  |
$1s\,6f~{}^{3}F_{2}$ | $1s\,6s~{}^{3}S_{1}$ | 5.188 |  |  | 8.747E-08 |  |  |
$1s\,6p~{}^{1}P_{1}$ | $1s\,6s~{}^{3}S_{1}$ | 3.444E+05 |  |  | 3.633E-03 |  |  |
$1s\,4d~{}^{3}D_{1}$ | $1s\,4p~{}^{3}P_{2}$ | 3.440E+04 |  |  | 2.428E-04 |  |  |
$1s\,4d~{}^{3}D_{2}$ | $1s\,4p~{}^{3}P_{2}$ | 2.295E+05 |  |  | 2.704E-03 |  |  |
$1s\,5p~{}^{3}P_{1}$ | $1s\,5s~{}^{3}S_{1}$ | 2.833E+06 |  |  | 3.530E-02 |  |  |
$1s\,5p~{}^{3}P_{0}$ | $1s\,5s~{}^{3}S_{1}$ | 2.525E+06 |  |  | 1.169E-02 |  |  |
$1s\,5d~{}^{1}D_{2}$ | $1s\,5p~{}^{3}P_{1}$ | 1.588E+05 |  |  | 3.853E-03 |  |  |
$1s\,5p~{}^{1}P_{1}$ | $1s\,5s~{}^{1}S_{0}$ | 2.227E+06 |  |  | 9.849E-02 |  |  |
$1s\,5d~{}^{3}D_{1}$ | $1s\,5p~{}^{3}P_{0}$ | 1.024E+06 |  |  | 5.472E-02 |  |  |
$1s\,5d~{}^{3}D_{2}$ | $1s\,5p~{}^{3}P_{1}$ | 1.061E+06 |  |  | 3.583E-02 |  |  |
$1s\,5d~{}^{3}D_{1}$ | $1s\,5p~{}^{3}P_{1}$ | 6.079E+05 |  |  | 1.234E-02 |  |  |
$1s\,6p~{}^{3}P_{2}$ | $1s\,6s~{}^{3}S_{1}$ | 2.407E+06 |  |  | 9.450E-02 |  |  |
$1s\,5g~{}^{3}G_{3}$ | $1s\,5p~{}^{3}P_{2}$ | 2.591E-02 |  |  | 8.722E-10 |  |  |
$1s\,7f~{}^{3}F_{2}$ | $1s\,7s~{}^{3}S_{1}$ | 2.673 |  |  | 1.143E-07 |  |  |
$1s\,7p~{}^{1}P_{1}$ | $1s\,7s~{}^{3}S_{1}$ | 1.597E+05 |  |  | 4.270E-03 |  |  |
$1s\,5d~{}^{1}D_{2}$ | $1s\,5p~{}^{3}P_{2}$ | 5.872E+04 |  |  | 1.671E-03 |  |  |
$1s\,2s~{}^{1}S_{0}$ | $1s\,2p~{}^{3}P_{1}$ | 1.742E+03 | 1.489E+03 |  | 1.668E-05 | 1.586E-05 |  |
$1s\,5d~{}^{3}D_{3}$ | $1s\,5p~{}^{3}P_{2}$ | 7.581E+05 |  |  | 3.424E-02 |  |  |
$1s\,6p~{}^{3}P_{1}$ | $1s\,6s~{}^{3}S_{1}$ | 1.134E+06 |  |  | 4.273E-02 |  |  |
$1s\,6p~{}^{3}P_{0}$ | $1s\,6s~{}^{3}S_{1}$ | 1.010E+06 |  |  | 1.415E-02 |  |  |
$1s\,3p~{}^{1}P_{1}$ | $1s\,3d~{}^{3}D_{2}$ | 1.027E+04 |  |  | 2.674E-04 |  |  |
$1s\,6d~{}^{1}D_{2}$ | $1s\,6p~{}^{3}P_{1}$ | 6.383E+04 |  |  | 4.650E-03 |  |  |
$1s\,6p~{}^{1}P_{1}$ | $1s\,6s~{}^{1}S_{0}$ | 8.972E+05 |  |  | 1.192E-01 |  |  |
$1s\,3p~{}^{1}P_{1}$ | $1s\,3d~{}^{3}D_{1}$ | 6.384E+02 |  |  | 2.921E-05 |  |  |
$1s\,5d~{}^{3}D_{2}$ | $1s\,5p~{}^{3}P_{2}$ | 8.274E+04 |  |  | 3.802E-03 |  |  |
$1s\,5d~{}^{3}D_{1}$ | $1s\,5p~{}^{3}P_{2}$ | 1.216E+04 |  |  | 3.360E-04 |  |  |
$1s\,6d~{}^{3}D_{1}$ | $1s\,6p~{}^{3}P_{0}$ | 4.293E+05 |  |  | 6.917E-02 |  |  |
$1s\,8f~{}^{3}F_{2}$ | $1s\,8s~{}^{3}S_{1}$ | 1.472 |  |  | 1.396E-07 |  |  |
$1s\,8p~{}^{1}P_{1}$ | $1s\,8s~{}^{3}S_{1}$ | 8.304E+04 |  |  | 4.907E-03 |  |  |
$1s\,7p~{}^{3}P_{2}$ | $1s\,7s~{}^{3}S_{1}$ | 1.114E+06 |  |  | 1.109E-01 |  |  |
$1s\,6d~{}^{3}D_{2}$ | $1s\,6p~{}^{3}P_{1}$ | 4.475E+05 |  |  | 4.550E-02 |  |  |
$1s\,6d~{}^{3}D_{1}$ | $1s\,6p~{}^{3}P_{1}$ | 2.546E+05 |  |  | 1.558E-02 |  |  |
$1s\,6g~{}^{3}G_{3}$ | $1s\,6p~{}^{3}P_{2}$ | 1.947E-02 |  |  | 1.969E-09 |  |  |
$1s\,4f~{}^{1}F_{3}$ | $1s\,4d~{}^{3}D_{2}$ | 7.999E+02 |  |  | 9.406E-05 |  |  |
$1s\,6d~{}^{1}D_{2}$ | $1s\,6p~{}^{3}P_{2}$ | 2.418E+04 |  |  | 2.070E-03 |  |  |
$1s\,7p~{}^{3}P_{1}$ | $1s\,7s~{}^{3}S_{1}$ | 5.235E+05 |  |  | 5.013E-02 |  |  |
$1s\,6d~{}^{3}D_{3}$ | $1s\,6p~{}^{3}P_{2}$ | 3.165E+05 |  |  | 4.322E-02 |  |  |
$1s\,7p~{}^{3}P_{0}$ | $1s\,7s~{}^{3}S_{1}$ | 4.660E+05 |  |  | 1.660E-02 |  |  |
$1s\,7d~{}^{1}D_{2}$ | $1s\,7p~{}^{3}P_{1}$ | 2.956E+04 |  |  | 5.450E-03 |  |  |
$1s\,7p~{}^{1}P_{1}$ | $1s\,7s~{}^{1}S_{0}$ | 4.135E+05 |  |  | 1.395E-01 |  |  |
$1s\,8p~{}^{3}P_{2}$ | $1s\,8s~{}^{3}S_{1}$ | 5.839E+05 |  |  | 1.285E-01 |  |  |
$1s\,7d~{}^{3}D_{1}$ | $1s\,7p~{}^{3}P_{0}$ | 2.034E+05 |  |  | 8.313E-02 |  |  |
$1s\,6d~{}^{3}D_{2}$ | $1s\,6p~{}^{3}P_{2}$ | 3.480E+04 |  |  | 4.834E-03 |  |  |
$1s\,6d~{}^{3}D_{1}$ | $1s\,6p~{}^{3}P_{2}$ | 5.063E+03 |  |  | 4.238E-04 |  |  |
$1s\,7d~{}^{3}D_{2}$ | $1s\,7p~{}^{3}P_{1}$ | 2.125E+05 |  |  | 5.483E-02 |  |  |
$1s\,7d~{}^{3}D_{1}$ | $1s\,7p~{}^{3}P_{1}$ | 1.205E+05 |  |  | 1.872E-02 |  |  |
$1s\,4f~{}^{3}F_{2}$ | $1s\,4d~{}^{3}D_{2}$ | 1.716E+03 |  |  | 2.822E-04 |  |  |
$1s\,4f~{}^{3}F_{2}$ | $1s\,4d~{}^{3}D_{1}$ | 1.253E+04 |  |  | 3.453E-03 |  |  |
$1s\,4f~{}^{3}F_{3}$ | $1s\,4d~{}^{3}D_{2}$ | 1.177E+04 |  |  | 2.972E-03 |  |  |
$1s\,7g~{}^{3}G_{3}$ | $1s\,7p~{}^{3}P_{2}$ | 1.208E-02 |  |  | 3.094E-09 |  |  |
$1s\,8p~{}^{3}P_{1}$ | $1s\,8s~{}^{3}S_{1}$ | 2.761E+05 |  |  | 5.820E-02 |  |  |
$1s\,7d~{}^{1}D_{2}$ | $1s\,7p~{}^{3}P_{2}$ | 1.133E+04 |  |  | 2.458E-03 |  |  |
$1s\,8p~{}^{3}P_{0}$ | $1s\,8s~{}^{3}S_{1}$ | 2.458E+05 |  |  | 1.927E-02 |  |  |
$1s\,8d~{}^{1}D_{2}$ | $1s\,8p~{}^{3}P_{1}$ | 1.525E+04 |  |  | 6.290E-03 |  |  |
$1s\,7d~{}^{3}D_{3}$ | $1s\,7p~{}^{3}P_{2}$ | 1.495E+05 |  |  | 5.190E-02 |  |  |
$1s\,8p~{}^{1}P_{1}$ | $1s\,8s~{}^{1}S_{0}$ | 2.137E+05 |  |  | 1.594E-01 |  |  |
$1s\,4p~{}^{1}P_{1}$ | $1s\,4d~{}^{3}D_{2}$ | 2.228E+03 |  |  | 3.782E-04 |  |  |
$1s\,4p~{}^{1}P_{1}$ | $1s\,4d~{}^{3}D_{1}$ | 1.830E+02 |  |  | 5.217E-05 |  |  |
$1s\,8d~{}^{3}D_{1}$ | $1s\,8p~{}^{3}P_{0}$ | 1.058E+05 |  |  | 9.683E-02 |  |  |
$1s\,5f~{}^{1}F_{3}$ | $1s\,5d~{}^{3}D_{2}$ | 4.490E+02 |  |  | 1.976E-04 |  |  |
$1s\,8d~{}^{3}D_{2}$ | $1s\,8p~{}^{3}P_{1}$ | 1.106E+05 |  |  | 6.390E-02 |  |  |
$1s\,8d~{}^{3}D_{1}$ | $1s\,8p~{}^{3}P_{1}$ | 6.264E+04 |  |  | 2.181E-02 |  |  |
$1s\,7d~{}^{3}D_{2}$ | $1s\,7p~{}^{3}P_{2}$ | 1.651E+04 |  |  | 5.828E-03 |  |  |
$1s\,7d~{}^{3}D_{1}$ | $1s\,7p~{}^{3}P_{2}$ | 2.388E+03 |  |  | 5.088E-04 |  |  |
$1s\,4f~{}^{1}F_{3}$ | $1s\,4d~{}^{3}D_{3}$ | 2.239E+02 |  |  | 8.021E-05 |  |  |
$1s\,4f~{}^{3}F_{4}$ | $1s\,4d~{}^{3}D_{3}$ | 3.952E+03 |  |  | 2.037E-03 |  |  |
$1s\,8g~{}^{3}G_{3}$ | $1s\,8p~{}^{3}P_{2}$ | 7.316E-03 |  |  | 4.188E-09 |  |  |
$1s\,8d~{}^{1}D_{2}$ | $1s\,8p~{}^{3}P_{2}$ | 5.854E+03 |  |  | 2.838E-03 |  |  |
$1s\,8d~{}^{3}D_{3}$ | $1s\,8p~{}^{3}P_{2}$ | 7.769E+04 |  |  | 6.042E-02 |  |  |
$1s\,5f~{}^{3}F_{2}$ | $1s\,5d~{}^{3}D_{1}$ | 6.366E+03 |  |  | 6.440E-03 |  |  |
$1s\,5f~{}^{3}F_{2}$ | $1s\,5d~{}^{3}D_{2}$ | 8.653E+02 |  |  | 5.298E-04 |  |  |
$1s\,5f~{}^{3}F_{3}$ | $1s\,5d~{}^{3}D_{2}$ | 5.850E+03 |  |  | 5.486E-03 |  |  |
$1s\,8d~{}^{3}D_{2}$ | $1s\,8p~{}^{3}P_{2}$ | 8.606E+03 |  |  | 6.808E-03 |  |  |
$1s\,8d~{}^{3}D_{1}$ | $1s\,8p~{}^{3}P_{2}$ | 1.240E+03 |  |  | 5.926E-04 |  |  |
$1s\,6f~{}^{1}F_{3}$ | $1s\,6d~{}^{3}D_{2}$ | 2.281E+02 |  |  | 2.968E-04 |  |  |
$1s\,4f~{}^{1}F_{3}$ | $1s\,4d~{}^{1}D_{2}$ | 9.867E+02 |  |  | 1.333E-03 |  |  |
$1s\,5p~{}^{1}P_{1}$ | $1s\,5d~{}^{3}D_{1}$ | 6.486E+01 |  |  | 7.310E-05 |  |  |
$1s\,5p~{}^{1}P_{1}$ | $1s\,5d~{}^{3}D_{2}$ | 6.990E+02 |  |  | 4.786E-04 |  |  |
$1s\,5f~{}^{1}F_{3}$ | $1s\,5d~{}^{3}D_{3}$ | 1.169E+02 |  |  | 1.503E-04 |  |  |
$1s\,5g~{}^{1}G_{4}$ | $1s\,5f~{}^{3}F_{3}$ | 4.071E-02 |  |  | 7.027E-08 |  |  |
$1s\,5f~{}^{3}F_{4}$ | $1s\,5d~{}^{3}D_{3}$ | 2.073E+03 |  |  | 3.840E-03 |  |  |
$1s\,3d~{}^{1}D_{2}$ | $1s\,3p~{}^{1}P_{1}$ | 1.623E+02 |  |  | 4.213E-04 |  |  |
$1s\,6f~{}^{3}F_{2}$ | $1s\,6d~{}^{3}D_{1}$ | 3.089E+03 |  |  | 9.147E-03 |  |  |
$1s\,6f~{}^{3}F_{2}$ | $1s\,6d~{}^{3}D_{2}$ | 4.183E+02 |  |  | 7.554E-04 |  |  |
$1s\,6f~{}^{3}F_{3}$ | $1s\,6d~{}^{3}D_{2}$ | 2.809E+03 |  |  | 7.754E-03 |  |  |
$1s\,6h~{}^{3}H_{4}$ | $1s\,6d~{}^{3}D_{3}$ | 2.843E-06 |  |  | 7.634E-12 |  |  |
$1s\,4f~{}^{3}F_{2}$ | $1s\,4d~{}^{3}D_{3}$ | 1.427 |  |  | 2.163E-06 |  |  |
$1s\,7f~{}^{1}F_{3}$ | $1s\,7d~{}^{3}D_{2}$ | 1.199E+02 |  |  | 3.912E-04 |  |  |
$1s\,4f~{}^{3}F_{3}$ | $1s\,4d~{}^{3}D_{3}$ | 1.201E+01 |  |  | 3.641E-05 |  |  |
$1s\,5g~{}^{3}G_{3}$ | $1s\,5f~{}^{3}F_{3}$ | 1.697E+01 |  |  | 5.393E-05 |  |  |
$1s\,4d~{}^{1}D_{2}$ | $1s\,4p~{}^{1}P_{1}$ | 2.337E+02 |  |  | 1.275E-03 |  |  |
$1s\,6p~{}^{1}P_{1}$ | $1s\,6d~{}^{3}D_{1}$ | 2.772E+01 |  |  | 9.347E-05 |  |  |
$1s\,6p~{}^{1}P_{1}$ | $1s\,6d~{}^{3}D_{2}$ | 2.809E+02 |  |  | 5.814E-04 |  |  |
$1s\,5g~{}^{3}G_{4}$ | $1s\,5f~{}^{3}F_{3}$ | 2.946E+02 |  |  | 1.365E-03 |  |  |
$1s\,5f~{}^{1}F_{3}$ | $1s\,5d~{}^{1}D_{2}$ | 4.733E+02 |  |  | 2.434E-03 |  |  |
$1s\,6f~{}^{1}F_{3}$ | $1s\,6d~{}^{3}D_{3}$ | 5.749E+01 |  |  | 2.139E-04 |  |  |
$1s\,5g~{}^{3}G_{3}$ | $1s\,5f~{}^{3}F_{2}$ | 2.471E+02 |  |  | 1.357E-03 |  |  |
$1s\,6g~{}^{1}G_{4}$ | $1s\,6f~{}^{3}F_{3}$ | 4.040E-02 |  |  | 2.080E-07 |  |  |
$1s\,6f~{}^{3}F_{4}$ | $1s\,6d~{}^{3}D_{3}$ | 1.022E+03 |  |  | 5.483E-03 |  |  |
$1s\,7f~{}^{3}F_{2}$ | $1s\,7d~{}^{3}D_{1}$ | 1.583E+03 |  |  | 1.168E-02 |  |  |
$1s\,7f~{}^{3}F_{2}$ | $1s\,7d~{}^{3}D_{2}$ | 2.139E+02 |  |  | 9.670E-04 |  |  |
$1s\,7f~{}^{3}F_{3}$ | $1s\,7d~{}^{3}D_{2}$ | 1.431E+03 |  |  | 9.878E-03 |  |  |
$1s\,4p~{}^{3}P_{1}$ | $1s\,4s~{}^{1}S_{0}$ | 6.109 |  |  | 9.182E-05 |  |  |
$1s\,8f~{}^{1}F_{3}$ | $1s\,8d~{}^{3}D_{2}$ | 6.662E+01 |  |  | 4.828E-04 |  |  |
$1s\,7h~{}^{3}H_{4}$ | $1s\,7d~{}^{3}D_{3}$ | 2.930E-06 |  |  | 1.954E-11 |  |  |
$1s\,3p~{}^{3}P_{1}$ | $1s\,3s~{}^{1}S_{0}$ | 1.569 |  |  | 2.533E-05 |  |  |
$1s\,5f~{}^{3}F_{2}$ | $1s\,5d~{}^{3}D_{3}$ | 8.386E-01 |  |  | 4.233E-06 |  |  |
$1s\,5g~{}^{1}G_{4}$ | $1s\,5f~{}^{3}F_{4}$ | 2.518 |  |  | 2.056E-05 |  |  |
$1s\,7p~{}^{1}P_{1}$ | $1s\,7d~{}^{3}D_{1}$ | 1.376E+01 |  |  | 1.143E-04 |  |  |
$1s\,7p~{}^{1}P_{1}$ | $1s\,7d~{}^{3}D_{2}$ | 1.350E+02 |  |  | 6.916E-04 |  |  |
$1s\,7f~{}^{1}F_{3}$ | $1s\,7d~{}^{3}D_{3}$ | 2.970E+01 |  |  | 2.736E-04 |  |  |
$1s\,5d~{}^{1}D_{2}$ | $1s\,5p~{}^{1}P_{1}$ | 1.320E+02 |  |  | 2.075E-03 |  |  |
$1s\,6g~{}^{3}G_{3}$ | $1s\,6f~{}^{3}F_{3}$ | 1.058E+01 |  |  | 1.002E-04 |  |  |
$1s\,5g~{}^{3}G_{5}$ | $1s\,5f~{}^{3}F_{4}$ | 7.029E+01 |  |  | 8.248E-04 |  |  |
$1s\,5f~{}^{3}F_{3}$ | $1s\,5d~{}^{3}D_{3}$ | 7.459 |  |  | 7.281E-05 |  |  |
$1s\,8f~{}^{3}F_{2}$ | $1s\,8d~{}^{3}D_{1}$ | 8.638E+02 |  |  | 1.410E-02 |  |  |
$1s\,8f~{}^{3}F_{2}$ | $1s\,8d~{}^{3}D_{2}$ | 1.166E+02 |  |  | 1.170E-03 |  |  |
$1s\,7g~{}^{1}G_{4}$ | $1s\,7f~{}^{3}F_{3}$ | 2.895E-02 |  |  | 3.757E-07 |  |  |
$1s\,7f~{}^{3}F_{4}$ | $1s\,7d~{}^{3}D_{3}$ | 5.292E+02 |  |  | 7.027E-03 |  |  |
$1s\,6h~{}^{1}H_{5}$ | $1s\,6g~{}^{3}G_{4}$ | 6.714E-05 |  |  | 8.511E-10 |  |  |
$1s\,6g~{}^{3}G_{4}$ | $1s\,6f~{}^{3}F_{3}$ | 1.833E+02 |  |  | 2.530E-03 |  |  |
$1s\,8f~{}^{3}F_{3}$ | $1s\,8d~{}^{3}D_{2}$ | 7.778E+02 |  |  | 1.190E-02 |  |  |
$1s\,6f~{}^{1}F_{3}$ | $1s\,6d~{}^{1}D_{2}$ | 2.231E+02 |  |  | 3.422E-03 |  |  |
$1s\,5g~{}^{1}G_{4}$ | $1s\,5f~{}^{1}F_{3}$ | 5.560E+01 |  |  | 7.873E-04 |  |  |
$1s\,8h~{}^{3}H_{4}$ | $1s\,8d~{}^{3}D_{3}$ | 2.384E-06 |  |  | 3.506E-11 |  |  |
$1s\,6g~{}^{3}G_{3}$ | $1s\,6f~{}^{3}F_{2}$ | 1.544E+02 |  |  | 2.520E-03 |  |  |
$1s\,5p~{}^{3}P_{1}$ | $1s\,5s~{}^{1}S_{0}$ | 3.646 |  |  | 1.444E-04 |  |  |
$1s\,8p~{}^{1}P_{1}$ | $1s\,8d~{}^{3}D_{1}$ | 7.917 |  |  | 1.371E-04 |  |  |
$1s\,8p~{}^{1}P_{1}$ | $1s\,8d~{}^{3}D_{2}$ | 7.648E+01 |  |  | 8.198E-04 |  |  |
$1s\,6f~{}^{3}F_{2}$ | $1s\,6d~{}^{3}D_{3}$ | 4.375E-01 |  |  | 6.160E-06 |  |  |
$1s\,8f~{}^{1}F_{3}$ | $1s\,8d~{}^{3}D_{3}$ | 1.632E+01 |  |  | 3.307E-04 |  |  |
$1s\,8g~{}^{1}G_{4}$ | $1s\,8f~{}^{3}F_{3}$ | 1.916E-02 |  |  | 5.546E-07 |  |  |
$1s\,8f~{}^{3}F_{4}$ | $1s\,8d~{}^{3}D_{3}$ | 2.912E+02 |  |  | 8.509E-03 |  |  |
$1s\,7g~{}^{3}G_{3}$ | $1s\,7f~{}^{3}F_{3}$ | 5.952 |  |  | 1.420E-04 |  |  |
$1s\,6g~{}^{1}G_{4}$ | $1s\,6f~{}^{3}F_{4}$ | 1.579 |  |  | 3.820E-05 |  |  |
$1s\,7i~{}^{3}I_{5}$ | $1s\,7f~{}^{3}F_{4}$ | 3.031E-10 |  |  | 9.202E-15 |  |  |
$1s\,6d~{}^{1}D_{2}$ | $1s\,6p~{}^{1}P_{1}$ | 6.426E+01 |  |  | 2.766E-03 |  |  |
$1s\,7h~{}^{1}H_{5}$ | $1s\,7g~{}^{3}G_{4}$ | 4.442E-05 |  |  | 1.419E-09 |  |  |
$1s\,6h~{}^{3}H_{4}$ | $1s\,6g~{}^{3}G_{4}$ | 7.363E-01 |  |  | 1.963E-05 |  |  |
$1s\,6f~{}^{3}F_{3}$ | $1s\,6d~{}^{3}D_{3}$ | 3.997 |  |  | 1.071E-04 |  |  |
$1s\,7g~{}^{3}G_{4}$ | $1s\,7f~{}^{3}F_{3}$ | 1.029E+02 |  |  | 3.580E-03 |  |  |
$1s\,7f~{}^{1}F_{3}$ | $1s\,7d~{}^{1}D_{2}$ | 1.124E+02 |  |  | 4.344E-03 |  |  |
$1s\,6g~{}^{3}G_{5}$ | $1s\,6f~{}^{3}F_{4}$ | 4.411E+01 |  |  | 1.533E-03 |  |  |
$1s\,7g~{}^{3}G_{3}$ | $1s\,7f~{}^{3}F_{2}$ | 8.691E+01 |  |  | 3.568E-03 |  |  |
$1s\,6h~{}^{3}H_{5}$ | $1s\,6g~{}^{3}G_{4}$ | 2.175E+01 |  |  | 8.133E-04 |  |  |
$1s\,6g~{}^{1}G_{4}$ | $1s\,6f~{}^{1}F_{3}$ | 3.458E+01 |  |  | 1.460E-03 |  |  |
$1s\,6h~{}^{3}H_{4}$ | $1s\,6g~{}^{3}G_{3}$ | 1.880E+01 |  |  | 7.999E-04 |  |  |
$1s\,6p~{}^{3}P_{1}$ | $1s\,6s~{}^{1}S_{0}$ | 1.650 |  |  | 1.829E-04 |  |  |
$1s\,7f~{}^{3}F_{2}$ | $1s\,7d~{}^{3}D_{3}$ | 2.339E-01 |  |  | 7.980E-06 |  |  |
$1s\,8g~{}^{3}G_{3}$ | $1s\,8f~{}^{3}F_{3}$ | 3.402 |  |  | 1.810E-04 |  |  |
$1s\,8i~{}^{3}I_{5}$ | $1s\,8f~{}^{3}F_{4}$ | 1.720E-10 |  |  | 1.159E-14 |  |  |
$1s\,7i~{}^{1}I_{6}$ | $1s\,7h~{}^{3}H_{5}$ | 5.700E-06 |  |  | 3.775E-10 |  |  |
$1s\,6h~{}^{1}H_{5}$ | $1s\,6g~{}^{3}G_{5}$ | 1.577E-01 |  |  | 9.091E-06 |  |  |
$1s\,8h~{}^{1}H_{5}$ | $1s\,8g~{}^{3}G_{4}$ | 2.604E-05 |  |  | 1.853E-09 |  |  |
$1s\,8g~{}^{3}G_{4}$ | $1s\,8f~{}^{3}F_{3}$ | 5.874E+01 |  |  | 4.556E-03 |  |  |
$1s\,7g~{}^{1}G_{4}$ | $1s\,7f~{}^{3}F_{4}$ | 8.908E-01 |  |  | 5.413E-05 |  |  |
$1s\,8f~{}^{1}F_{3}$ | $1s\,8d~{}^{1}D_{2}$ | 6.082E+01 |  |  | 5.228E-03 |  |  |
$1s\,7f~{}^{3}F_{3}$ | $1s\,7d~{}^{3}D_{3}$ | 2.171 |  |  | 1.397E-04 |  |  |
$1s\,8g~{}^{3}G_{3}$ | $1s\,8f~{}^{3}F_{2}$ | 4.971E+01 |  |  | 4.542E-03 |  |  |
$1s\,7d~{}^{1}D_{2}$ | $1s\,7p~{}^{1}P_{1}$ | 2.986E+01 |  |  | 3.308E-03 |  |  |
$1s\,7h~{}^{3}H_{4}$ | $1s\,7g~{}^{3}G_{4}$ | 5.475E-01 |  |  | 3.674E-05 |  |  |
$1s\,6h~{}^{3}H_{6}$ | $1s\,6g~{}^{3}G_{5}$ | 6.663 |  |  | 5.382E-04 |  |  |
$1s\,7g~{}^{3}G_{5}$ | $1s\,7f~{}^{3}F_{4}$ | 2.490E+01 |  |  | 2.173E-03 |  |  |
$1s\,6h~{}^{1}H_{5}$ | $1s\,6g~{}^{1}G_{4}$ | 5.750 |  |  | 5.233E-04 |  |  |
$1s\,7h~{}^{3}H_{5}$ | $1s\,7g~{}^{3}G_{4}$ | 1.617E+01 |  |  | 1.522E-03 |  |  |
$1s\,7g~{}^{1}G_{4}$ | $1s\,7f~{}^{1}F_{3}$ | 1.941E+01 |  |  | 2.064E-03 |  |  |
$1s\,7h~{}^{3}H_{4}$ | $1s\,7g~{}^{3}G_{3}$ | 1.399E+01 |  |  | 1.497E-03 |  |  |
$1s\,8f~{}^{3}F_{2}$ | $1s\,8d~{}^{3}D_{3}$ | 1.323E-01 |  |  | 9.751E-06 |  |  |
$1s\,7p~{}^{3}P_{1}$ | $1s\,7s~{}^{1}S_{0}$ | 6.514E-01 |  |  | 2.037E-04 |  |  |
$1s\,8k~{}^{3}K_{6}$ | $1s\,8g~{}^{3}G_{5}$ | 3.145E-12 |  |  | 4.541E-16 |  |  |
$1s\,8i~{}^{1}I_{6}$ | $1s\,8h~{}^{3}H_{5}$ | 4.834E-06 |  |  | 7.152E-10 |  |  |
$1s\,8g~{}^{1}G_{4}$ | $1s\,8f~{}^{3}F_{4}$ | 5.113E-01 |  |  | 6.900E-05 |  |  |
$1s\,8f~{}^{3}F_{3}$ | $1s\,8d~{}^{3}D_{3}$ | 1.241 |  |  | 1.714E-04 |  |  |
$1s\,7h~{}^{1}H_{5}$ | $1s\,7g~{}^{3}G_{5}$ | 1.174E-01 |  |  | 1.703E-05 |  |  |
$1s\,8h~{}^{3}H_{4}$ | $1s\,8g~{}^{3}G_{4}$ | 3.501E-01 |  |  | 5.228E-05 |  |  |
$1s\,7i~{}^{3}I_{5}$ | $1s\,7h~{}^{3}H_{5}$ | 5.800E-02 |  |  | 8.848E-06 |  |  |
$1s\,8g~{}^{3}G_{5}$ | $1s\,8f~{}^{3}F_{4}$ | 1.430E+01 |  |  | 2.771E-03 |  |  |
$1s\,4d~{}^{1}D_{2}$ | $1s\,4f~{}^{3}F_{3}$ | 2.552E-02 |  |  | 2.950E-06 |  |  |
$1s\,8h~{}^{3}H_{5}$ | $1s\,8g~{}^{3}G_{4}$ | 1.034E+01 |  |  | 2.167E-03 |  |  |
$1s\,7h~{}^{3}H_{6}$ | $1s\,7g~{}^{3}G_{5}$ | 4.962 |  |  | 1.008E-03 |  |  |
$1s\,8d~{}^{1}D_{2}$ | $1s\,8p~{}^{1}P_{1}$ | 1.202E+01 |  |  | 3.533E-03 |  |  |
$1s\,7i~{}^{3}I_{6}$ | $1s\,7h~{}^{3}H_{5}$ | 2.581 |  |  | 5.386E-04 |  |  |
$1s\,8g~{}^{1}G_{4}$ | $1s\,8f~{}^{1}F_{3}$ | 1.110E+01 |  |  | 2.629E-03 |  |  |
$1s\,8h~{}^{3}H_{4}$ | $1s\,8g~{}^{3}G_{3}$ | 8.945 |  |  | 2.130E-03 |  |  |
$1s\,7h~{}^{1}H_{5}$ | $1s\,7g~{}^{1}G_{4}$ | 4.281 |  |  | 9.801E-04 |  |  |
$1s\,7i~{}^{3}I_{5}$ | $1s\,7h~{}^{3}H_{4}$ | 2.298 |  |  | 5.293E-04 |  |  |
$1s\,8p~{}^{3}P_{1}$ | $1s\,8s~{}^{1}S_{0}$ | 4.069E-01 |  |  | 2.472E-04 |  |  |
$1s\,8k~{}^{1}K_{7}$ | $1s\,8i~{}^{3}I_{6}$ | 5.574E-07 |  |  | 1.500E-10 |  |  |
$1s\,7i~{}^{1}I_{6}$ | $1s\,7h~{}^{3}H_{6}$ | 1.603E-02 |  |  | 4.659E-06 |  |  |
$1s\,8h~{}^{1}H_{5}$ | $1s\,8g~{}^{3}G_{5}$ | 7.530E-02 |  |  | 2.425E-05 |  |  |
$1s\,8i~{}^{3}I_{5}$ | $1s\,8h~{}^{3}H_{5}$ | 4.886E-02 |  |  | 1.665E-05 |  |  |
$1s\,7i~{}^{3}I_{7}$ | $1s\,7h~{}^{3}H_{6}$ | 9.564E-01 |  |  | 3.815E-04 |  |  |
$1s\,7i~{}^{1}I_{6}$ | $1s\,7h~{}^{1}H_{5}$ | 8.492E-01 |  |  | 3.724E-04 |  |  |
$1s\,8h~{}^{3}H_{6}$ | $1s\,8g~{}^{3}G_{5}$ | 3.185 |  |  | 1.435E-03 |  |  |
$1s\,8i~{}^{3}I_{6}$ | $1s\,8h~{}^{3}H_{5}$ | 2.174 |  |  | 1.014E-03 |  |  |
$1s\,8h~{}^{1}H_{5}$ | $1s\,8g~{}^{1}G_{4}$ | 2.748 |  |  | 1.397E-03 |  |  |
$1s\,8i~{}^{3}I_{5}$ | $1s\,8h~{}^{3}H_{4}$ | 1.935 |  |  | 9.960E-04 |  |  |
$1s\,5g~{}^{3}G_{3}$ | $1s\,5f~{}^{3}F_{4}$ | 2.921E-04 |  |  | 9.898E-08 |  |  |
$1s\,5d~{}^{1}D_{2}$ | $1s\,5f~{}^{3}F_{3}$ | 1.327E-02 |  |  | 5.894E-06 |  |  |
$1s\,8i~{}^{1}I_{6}$ | $1s\,8h~{}^{3}H_{6}$ | 1.351E-02 |  |  | 8.769E-06 |  |  |
$1s\,8k~{}^{3}K_{6}$ | $1s\,8i~{}^{3}I_{6}$ | 7.030E-03 |  |  | 4.610E-06 |  |  |
$1s\,8k~{}^{3}K_{7}$ | $1s\,8i~{}^{3}I_{6}$ | 4.402E-01 |  |  | 3.873E-04 |  |  |
$1s\,8i~{}^{3}I_{7}$ | $1s\,8h~{}^{3}H_{6}$ | 8.054E-01 |  |  | 7.181E-04 |  |  |
$1s\,8k~{}^{3}K_{6}$ | $1s\,8i~{}^{3}I_{5}$ | 3.999E-01 |  |  | 3.808E-04 |  |  |
$1s\,5f~{}^{1}F_{3}$ | $1s\,5g~{}^{3}G_{4}$ | 1.579E-04 |  |  | 1.002E-07 |  |  |
$1s\,8i~{}^{1}I_{6}$ | $1s\,8h~{}^{1}H_{5}$ | 7.150E-01 |  |  | 7.007E-04 |  |  |
$1s\,4f~{}^{3}F_{2}$ | $1s\,4d~{}^{1}D_{2}$ | 1.565E-03 |  |  | 1.396E-06 |  |  |
$1s\,8k~{}^{1}K_{7}$ | $1s\,8i~{}^{3}I_{7}$ | 2.359E-03 |  |  | 2.670E-06 |  |  |
$1s\,6g~{}^{3}G_{3}$ | $1s\,6f~{}^{3}F_{4}$ | 1.966E-04 |  |  | 1.883E-07 |  |  |
$1s\,8k~{}^{3}K_{8}$ | $1s\,8i~{}^{3}I_{7}$ | 1.896E-01 |  |  | 2.894E-04 |  |  |
$1s\,8k~{}^{1}K_{7}$ | $1s\,8i~{}^{1}I_{6}$ | 1.718E-01 |  |  | 2.833E-04 |  |  |
$1s\,6d~{}^{1}D_{2}$ | $1s\,6f~{}^{3}F_{3}$ | 6.451E-03 |  |  | 8.624E-06 |  |  |
$1s\,6f~{}^{1}F_{3}$ | $1s\,6g~{}^{3}G_{4}$ | 1.029E-04 |  |  | 1.986E-07 |  |  |
$1s\,7g~{}^{3}G_{3}$ | $1s\,7f~{}^{3}F_{4}$ | 1.152E-04 |  |  | 2.702E-07 |  |  |
$1s\,5f~{}^{3}F_{2}$ | $1s\,5d~{}^{1}D_{2}$ | 5.903E-04 |  |  | 2.294E-06 |  |  |
$1s\,6h~{}^{3}H_{4}$ | $1s\,6g~{}^{3}G_{5}$ | 6.068E-06 |  |  | 2.255E-08 |  |  |
$1s\,7d~{}^{1}D_{2}$ | $1s\,7f~{}^{3}F_{3}$ | 3.284E-03 |  |  | 1.117E-05 |  |  |
$1s\,6g~{}^{1}G_{4}$ | $1s\,6h~{}^{3}H_{5}$ | 4.578E-06 |  |  | 1.951E-08 |  |  |
$1s\,5g~{}^{3}G_{4}$ | $1s\,5f~{}^{3}F_{4}$ | 1.831E-04 |  |  | 9.818E-07 |  |  |
$1s\,7f~{}^{1}F_{3}$ | $1s\,7g~{}^{3}G_{4}$ | 5.993E-05 |  |  | 2.933E-07 |  |  |
$1s\,8g~{}^{3}G_{3}$ | $1s\,8f~{}^{3}F_{4}$ | 6.991E-05 |  |  | 3.509E-07 |  |  |
$1s\,7h~{}^{3}H_{4}$ | $1s\,7g~{}^{3}G_{5}$ | 4.679E-06 |  |  | 4.272E-08 |  |  |
$1s\,8d~{}^{1}D_{2}$ | $1s\,8f~{}^{3}F_{3}$ | 1.654E-03 |  |  | 1.328E-05 |  |  |
$1s\,6f~{}^{3}F_{2}$ | $1s\,6d~{}^{1}D_{2}$ | 2.439E-04 |  |  | 3.044E-06 |  |  |
$1s\,6g~{}^{3}G_{4}$ | $1s\,6f~{}^{3}F_{4}$ | 1.504E-04 |  |  | 1.997E-06 |  |  |
$1s\,7g~{}^{1}G_{4}$ | $1s\,7h~{}^{3}H_{5}$ | 3.281E-06 |  |  | 3.618E-08 |  |  |
$1s\,8f~{}^{1}F_{3}$ | $1s\,8g~{}^{3}G_{4}$ | 3.349E-05 |  |  | 3.782E-07 |  |  |
$1s\,8h~{}^{3}H_{4}$ | $1s\,8g~{}^{3}G_{5}$ | 3.237E-06 |  |  | 6.241E-08 |  |  |
$1s\,7h~{}^{1}H_{5}$ | $1s\,7i~{}^{3}I_{6}$ | 3.423E-07 |  |  | 7.012E-09 |  |  |
$1s\,7i~{}^{3}I_{5}$ | $1s\,7h~{}^{3}H_{6}$ | 3.089E-07 |  |  | 7.195E-09 |  |  |
$1s\,7g~{}^{3}G_{4}$ | $1s\,7f~{}^{3}F_{4}$ | 9.743E-05 |  |  | 2.963E-06 |  |  |
$1s\,8g~{}^{1}G_{4}$ | $1s\,8h~{}^{3}H_{5}$ | 1.922E-06 |  |  | 5.012E-08 |  |  |
$1s\,7f~{}^{3}F_{2}$ | $1s\,7d~{}^{1}D_{2}$ | 1.148E-04 |  |  | 3.750E-06 |  |  |
$1s\,8h~{}^{1}H_{5}$ | $1s\,8i~{}^{3}I_{6}$ | 2.896E-07 |  |  | 1.322E-08 |  |  |
$1s\,8g~{}^{3}G_{4}$ | $1s\,8f~{}^{3}F_{4}$ | 6.730E-05 |  |  | 4.019E-06 |  |  |
$1s\,8i~{}^{3}I_{5}$ | $1s\,8h~{}^{3}H_{6}$ | 2.614E-07 |  |  | 1.356E-08 |  |  |
$1s\,8f~{}^{3}F_{2}$ | $1s\,8d~{}^{1}D_{2}$ | 7.204E-05 |  |  | 4.714E-06 |  |  |
$1s\,8k~{}^{3}K_{6}$ | $1s\,8i~{}^{3}I_{7}$ | 4.025E-08 |  |  | 3.225E-09 |  |  |
$1s\,8i~{}^{1}I_{6}$ | $1s\,8k~{}^{3}K_{7}$ | 2.308E-08 |  |  | 2.521E-09 |  |  |
$1s\,6h~{}^{3}H_{5}$ | $1s\,6g~{}^{3}G_{5}$ | 5.944E-07 |  |  | 1.580E-07 |  |  |
$1s\,7h~{}^{3}H_{5}$ | $1s\,7g~{}^{3}G_{5}$ | 5.843E-07 |  |  | 3.246E-07 |  |  |
$1s\,8h~{}^{3}H_{5}$ | $1s\,8g~{}^{3}G_{5}$ | 6.309E-07 |  |  | 5.500E-07 |  |  |
$1s\,6g~{}^{1}G_{4}$ | $1s\,6h~{}^{3}H_{4}$ | 7.242E-08 |  |  | 7.908E-08 |  |  |
$1s\,5f~{}^{1}F_{3}$ | $1s\,5g~{}^{3}G_{3}$ | 3.167E-08 |  |  | 5.467E-08 |  |  |
$1s\,7h~{}^{1}H_{5}$ | $1s\,7i~{}^{3}I_{5}$ | 3.447E-08 |  |  | 6.657E-08 |  |  |
$1s\,7g~{}^{1}G_{4}$ | $1s\,7h~{}^{3}H_{4}$ | 2.867E-08 |  |  | 1.200E-07 |  |  |
$1s\,8h~{}^{1}H_{5}$ | $1s\,8i~{}^{3}I_{5}$ | 2.911E-08 |  |  | 1.255E-07 |  |  |
$1s\,8k~{}^{3}K_{7}$ | $1s\,8i~{}^{3}I_{7}$ | 9.508E-09 |  |  | 4.606E-08 |  |  |
$1s\,7i~{}^{3}I_{6}$ | $1s\,7h~{}^{3}H_{6}$ | 2.970E-09 |  |  | 2.915E-08 |  |  |
$1s\,6f~{}^{1}F_{3}$ | $1s\,6g~{}^{3}G_{3}$ | 4.007E-09 |  |  | 5.947E-08 |  |  |
$1s\,8i~{}^{3}I_{6}$ | $1s\,8h~{}^{3}H_{6}$ | 2.518E-09 |  |  | 5.499E-08 |  |  |
$1s\,8g~{}^{1}G_{4}$ | $1s\,8h~{}^{3}H_{4}$ | 2.068E-09 |  |  | 8.247E-08 |  |  |
$1s\,7f~{}^{1}F_{3}$ | $1s\,7g~{}^{3}G_{3}$ | 1.155E-09 |  |  | 6.727E-08 |  |  |
$1s\,8g~{}^{3}G_{3}$ | $1s\,8f~{}^{1}F_{3}$ | 1.923E-10 |  |  | 5.671E-08 |  |  |
$1s\,8k~{}^{3}K_{6}$ | $1s\,8i~{}^{1}I_{6}$ | 5.984E-13 |  |  | 1.846E-09 |  |  |