# Statistical Equivalence of Metrics for Meteor Dynamical Association

Peña-Asensio Eloy<EMAIL_ADDRESS><EMAIL_ADDRESS>Sánchez-Lozano Juan
Miguel

###### Abstract

Meteor showers, originating as a result of the activity of comets or the
disruption of large objects, provide a unique window into the composition and
dynamics of our Solar System. While modern meteor detection networks have
amassed extensive data, distinguishing sporadic meteors from those belonging
to specific meteor showers remains challenging. In this study, we
statistically evaluate and compare four orbital similarity criteria within
five-dimensional parameter space ($D_{SH}$, $D_{D}$, $D_{H}$, and
$\varrho_{2}$) to study dynamical associations using the already classified
meteors (manually by a human) in CAMS database as a benchmark. In addition, we
assess various distance metrics typically used in Machine Learning with two
different vectors: ORBIT, grounded in heliocentric orbital elements, and GEO,
predicated on geocentric observational parameters. To estimate their degree of
correlation and efficacy, the Kendall rank correlation coefficient and the
Top-k accuracy are employed. The statistical equivalence of the Top-1 results
is examined using the Kolmogorov-Smirnov test and the percentage of Top-1
agreement is calculated on an event-by-event basis. Additionally, we compute
the optimal cut-offs for all methods for distinguishing sporadic background
events. Our findings demonstrate the superior performance of the sEuclidean
metric in conjunction with the GEO vector. Within the scope of D-criteria,
$D_{SH}$ emerged as the preeminent metric, closely followed by $\varrho_{2}$.
The Bray-Curtis metric displayed an advantage compared to the other distance
metrics when paired with the ORBIT vector for Top-k accuracy, however, the
Cityblock metric is more effective when considering the sporadic background.
$\varrho_{2}$ stands out as the most equivalence to the distance metrics when
utilizing the GEO vector and the most compatible with GEO and ORBIT
simultaneously, whereas $D_{D}$ aligns more closely when using the ORBIT
vector. The stark contrast in $D_{D}$’s behavior compared to other D-criteria
highlights potential inequivalence. Our results suggest that geocentric
features provide a more robust basis than orbital elements for meteor
dynamical association. Most distance metrics associated with the GEO vector
surpass the D-criteria when differentiating the meteoroid background. Accuracy
displayed a dependence on solar longitude with a pronounced decrease around
180∘ matching an apparent increase in the meteoroid background activity,
tentatively associated with the transition from the Perseids to the Orionids.
Considering lately identified meteor showers, $\sim$27% of meteors in CAMS
would have different associations. This work unveils that Machine Learning
distance metrics can rival or even exceed the performance of tailored orbital
similarity criteria for meteor dynamical association.

###### keywords:

Meteor, Meteoroids, Meteor showers, Meteoroid streams, Statistics

††journal: Advances in Space Research

[1]organization=Department of Aerospace Science and Technology, Politecnico di
Milano, addressline=Via La Masa 34, city=Milano, postcode=20156,
state=Lombardia, country=Italy

[2]organization=Departament de Química, Universitat Autònoma de Barcelona,
addressline=Carrer dels Til·lers, city=Bellaterra, postcode=08193,
state=Catalonia, country=Spain

[3]organization=Centro Universitario de la Defensa, Universidad Politécnica de
Cartagena, addressline=C/ Coronel López Peña S/N, Base Aérea de San Javier,
city=Santiago de la Ribera, postcode=30720, country=Spain

##  1 Introduction

Within the expanse of our planetary system, remnants from its formation
provide glimpses into the early stages of our cosmic neighborhood (Bottke et
al., 2002; Walker & Cameron, 2006). Among these remnants, comets emerge as
witnesses to the dramatic events that shaped our nearby environment. These
celestial bodies can undergo processes of disruption due to various factors
such as volatile sublimation when approaching the Sun, tidal forces, or
impacts with other bodies. According to the theory of formation and evolution
of small bodies of the Solar System (Whipple, 1951; Bredikhin, 1954; Plavec,
1954; Hughes, 1986; Babadzhanov & Obrubov, 1992), meteoroid streams are formed
mainly as a result of the activity of comets or the ejection of meteoroids
from cometary nuclei with various initial velocities (Chapman, 2010; Tóth et
al., 2011; Gritsevich et al., 2012). Meteoroids exhibit a diverse composition,
including rock, metal, or a combination of both, and span a wide range of
sizes, from micrometer-scale grains to larger objects up to one meter in
diameter (Trigo-Rodríguez & Llorca, 2006, 2007; Koschny & Borovicka, 2017).
Despite their heterogeneous characteristics, these meteoroids share a common
origin, derived from a parent body, which imparts certain similarities among
them.

Additionally, though less common, asteroids can also generate meteoroid
streams as a result of catastrophic impact events. Some associations have been
found, such as the case of the potentially hazardous asteroid (3200) Phaethon
(1983TB), whose origin could be the nucleus of an extinct comet (Zhong-Yi et
al., 2020), and the Geminids meteor shower. Multiple studies have confirmed
the high probability that the Geminids are dynamically associated with such
asteroid (Whipple, 1983; Gustafson, 1989; Williams & Wu, 1993). However, as
they traverse the space, the influence of planetary perturbations and non-
gravitational forces gradually renders them indistinguishable from the
background population (Olsson-Steel, 1986; Bottke et al., 2000; Pauls &
Gladman, 2005; Brož, 2006; Koschny et al., 2019).

Eventually, the journey of meteoroids brings them into intersecting paths with
the Earth’s orbit, leading to captivating interactions with our planet (Brown
et al., 2002; Murad & Williams, 2002; Gritsevich, 2009; Trigo-Rodríguez,
2022). As these meteoroids penetrate the Earth’s atmosphere, they experience a
dramatic transformation fueled by the intense heat generated through air
molecule friction. The high-speed entry produces enormous amounts of heat,
causing the outer layers of the meteoroids to rapidly vaporize (Popova et al.,
2019). This process, known as ablation, leads to the formation of a glowing
plasma sheath surrounding the meteoroid (Ceplecha et al., 1998; Silber et al.,
2018). The energy released during atmospheric aerobraking causes the visible
phenomenon known as a meteor, which is called a fireball or bolide if its
magnitude surpasses that of the planet Venus. When a meteoroid stream
intersects the Earth’s path periodically, it gives rise to the phenomenon of
meteor showers (Jenniskens, 1994, 1998, 2006; Vaubaillon et al., 2019;
Jenniskens, 2023). The meteors within them share common features, including
their time of occurrence, apparent origin in the sky, known as the radiant,
and their geocentric impact velocity, as well as their orbital elements in an
equivalent manner.

Thanks to the existence of meteor detection networks (Jacchia & Whipple, 1956;
Ceplecha, 1957; Bland, 2004; Trigo-Rodriguez et al., 2005; Weryk et al., 2007;
Jenniskens et al., 2011; Kornoš et al., 2014; Gritsevich et al., 2014; Colas
et al., 2015; SonotaCo, 2016; Gardiol et al., 2016; Devillepoix et al., 2020;
Colas et al., 2020; Boaca et al., 2022; Borovička et al., 2022), researchers
now have access to an abundance of videos and images that play a key role in
improving our understanding of the distribution and characteristics of
meteoroid streams. The availability of this extensive dataset has
revolutionized our capacity to investigate and analyze incoming
extraterrestrial material. Through processing these videos and extracting
relevant information, we can determine essential parameters such as
trajectory, velocity, brightness, and other physical properties of meteors
(Ceplecha, 1987; Borovicka, 1990; Vida et al., 2020; Peña-Asensio et al.,
2021, 2024). This valuable information significantly contributes to our
knowledge about meteor showers, assisting in the identification and tracking
of potentially hazardous complexes that may pose a threat to Earth in the near
term (Voloshchuk & Kashcheev, 1996; Halliday, 1987; Borovička et al., 2015;
Trigo-Rodríguez et al., 2017; Trigo-Rodríguez & Blum, 2022).

In this regard, accurate classification of sporadic meteors from those
belonging to the same stream is of utmost importance, as the latter provides
valuable insights into the population density of future impactors (Wiegert &
Brown, 2004; Porubčan et al., 2004; Jopek & Williams, 2013; Dumitru et al.,
2017; Jenniskens, 2017; Vaubaillon et al., 2019; Peña-Asensio et al., 2022,
2023). Determining the point at which a meteor shower transitions from a
cohesive entity to a collection of unrelated meteoroids (sporadic background),
or establishing the criteria to accurately associate meteors with a specific
shower, poses a significant challenge. To tackle the issue of orbital
dynamical association, multiple endeavors have been undertaken to define
similarity criteria or D-criteria. These criteria aim to effectively
differentiate between events that are associated with a specific meteoroid
stream and those that are unrelated to other objects or swarms. Ultimately,
analyzing the impact features can aid in associating meteorites with their
parent bodies (Carbognani & Fenucci, 2023).

In this study, we assess the rank correlation, efficacy, and equivalence of
four five-dimensional similarity criteria designed for quantifying dynamical
associations between meteor orbits, as well as various distance metrics with
two different vectors (one shared with the D-criteria). The evaluation is
conducted using a comprehensive meteor database and extends to exploring
alternative metrics for orbit association, as well as computing the optimal
thresholds for each method. The objective is to elucidate the statistical
strengths, limitations, and similarities of each approach, thereby providing a
robust framework for future research in meteor associations with parent bodies
or meteoroid streams.

In Section 2, we detail the database utilized and the methodology applied.
Section 3 presents our findings, and Section 4 provides a summary of the key
outcomes of our study.

##  2 Data and Procedures

The methodology presented herein is designed to analyze multiple meteor
dynamical association approaches by comparing five-dimensional orbital
similarity criteria and various vector-based distance metrics typically used
in Machine Learning. For the latter, we use as a vector (1) the same
parameters utilized by the similarity criteria defined by some heliocentric
orbital elements, which we termed as ORBIT, and (2) the four-dimensional
vector proposed by Sugar et al. (2017) and named here as GEO. It should be
noted that while the term “metrics” may be appropriate to describe the
D-criteria to a certain extent, in this work, we use the term “metrics”
exclusively to refer to vector-based distance metrics, which are further
explained.

This section is subdivided into different subsections. Subsection 2.1
elaborates on the data sources utilized. Subsection 2.2 presents D-criteria
for comparing the orbital elements of two orbits. In Subsection 2.3, we
introduce the two vectors that will be used along with the distance metrics.
In Subsection 2.4 we explain the theoretical background used for calculating
the rank correlations, comparing the performances with the Top-k accuracy
method, and estimating the equivalence with the Kolmogorov-Smirnov test and
Top-1 event-by-event agreement. Finally, in Subsection 2.5, we detail our
strategy to determine the optimal thresholds for distinguishing between
sporadic background and meteor showers. All implementations of the statistical
analyses were conducted utilizing the SciPy library (Virtanen et al., 2020).

###  2.1 Databases

CAMS, short for the Cameras for All-Sky Meteor Surveillance project
(Jenniskens et al., 2011), is an international initiative sponsored by NASA
and managed by the Carl Sagan Center within the SETI Institute, located in
California, USA. Its primary objective is to monitor and map meteor activity
through nighttime optical video surveillance, employing triangulation
techniques. It annually records an average of half a million meteor orbits,
although the publication of this data stopped in 2016. The last release was
the Meteoroid Orbit Database v3.0, which includes 471,582 events registered
since 2010.

While there are other automated meteor detection networks, CAMS stands out as
the primary and most widely recognized repository of meteor data.
Nevertheless, it was noted that its performance in accurately detecting fast
meteors falls short in comparison to its detection of slower meteors (Koseki,
2017, 2022). To address this issue, we implement a filtering mechanism to
exclude lower-quality detections and to reduce spurious data, requiring a
minimum convergence angle of 15 degrees between cameras, ensuring an estimated
velocity error of no more than 10% of the nominal value, not allowing
hyperbolic orbits, and selecting perihelion distance compatible with impacts
on the Earth.

Certainly, we rely on the classification provided by CAMS as a ground truth,
which may not be infallible. However, the classification within this database
did not utilize any formal dissimilarity criteria. Instead, it depended on
human visual clustering within sun-centered ecliptic longitude-latitude
representations, with clusters manually delineated using specific coordinates
and geocentric velocity limits (Jenniskens et al., 2018). Our analyses
proceeds under the presumption that the CAMS classification is accurate, a
premise that, regardless, serves our primary objective of assessing the
equivalence between metrics and D-criteria.

For identifying meteoroid streams responsible for meteor showers, we use the
V.2 list of all known showers from the IAU Meteor Data Center, updated in
January 2024 (Jopek & Jenniskens, 2011; Jopek & Kanuchová, 2013; Jopek &
Kaňuchová, 2017; Jenniskens et al., 2020). To facilitate the association of
these meteor showers with entries in the CAMS database, we employ the IAU
numeral code. This list includes 1484 entries, 956 corresponding to unique
meteor showers. To ensure a direct comparison of association performances, we
filter both CAMS and IAU meteor shower datasets to include only identical,
unique meteor showers.

###  2.2 Orbital Similarity Criteria

Orbital elements such as inclination $i$, eccentricity $e$, longitude of the
ascending node $\Omega$, perihelion distance $q$, and argument of the
perihelion $\omega$ allow us to determine the path of any moving object
following a Keplerian trajectory in our Solar System. Likewise, it is possible
to look for the connection between a meteor shower and its parent body (or any
two objects) through the similarities of their orbits.

This search approach is not recent. The first attempts focused on measuring
the degree of similarity between orbits were designed in the second half of
the last century, they were so-called D-criteria. The first D-criteria was
introduced by Southworth & Hawkins (1963):

$D_{SH}^{2}=\left(e_{B}-e_{A}\right)^{2}+\left(q_{B}-q_{A}\right)^{2}+\left(2\sin\frac{I_{AB}}{2}\right)^{2}+\left(\frac{e_{B}+e_{A}}{2}\right)^{2}\left(2\sin\frac{\pi_{BA}}{2}\right)^{2},$
(1)

where other concepts of geometry come into play such as the angles between
their respective perihelion points ($\pi_{BA}$) and between the inclinations
of the orbits ($I_{AB}$).

Drummond (1981) not only defined the angle between the perihelion points on
each orbit ($\theta_{BA}$) by adding both the ecliptic longitude ($\lambda$)
and the perihelion latitude ($\beta$), but also weighted the terms $e$ and $q$
to provide a metric in which each term contributed equally to the overall sum.
In this way, a new variant of the $D_{SH}$ criterion, named $D_{D}$ in honor
of its creator, was developed:

$D_{D}^{2}=\left(\frac{e_{B}-e_{A}}{e_{B}+e_{A}}\right)^{2}+\left(\frac{q_{B}-q_{A}}{q_{B}+q_{A}}\right)^{2}+\left(\frac{I_{AB}}{\pi}\right)^{2}+\left(\frac{e_{B}+e_{A}}{2}\right)^{2}\left(\frac{\theta_{BA}}{\pi}\right)^{2},$
(2)

A decade later, Jopek (1993) carried out a random perturbation model of
several orbits, ignoring $i$, $\Omega$, and $\omega$, to analyze the $D_{SH}$
and $D_{D}$ criteria. He found dependency relationships of $q$ and $e$ values
for the reference orbit; $q$ in the case of $D_{SH}$ and $e$ for the criterion
$D_{D}$. To reduce these dependency relationships between orbital parameters,
Jopek proposed a new similarity criterion, $D_{H}$, defined by:

$D_{H}^{2}=\left(e_{B}-e_{A}\right)^{2}+\left(\frac{q_{B}-q_{A}}{q_{B}+q_{A}}\right)^{2}+\left(2\sin\frac{I_{AB}}{2}\right)^{2}+\left(\frac{e_{B}+e_{A}}{2}\right)^{2}\left(2\sin\frac{\pi_{BA}}{2}\right)^{2}.$
(3)

Note that these D-criteria cannot be categorized mathematically as metrics due
to their violation of the triangle inequality (Kholshevnikov et al., 2016).
Instead, they are more appropriately defined as quasimetrics, as they adhere
to a relaxed form of the triangle inequality (Milanov et al., 2019).
Contemporary functions, such as $\varrho_{2}$, enable the precise
quantification of orbital similarity through consistent mathematical
formulations:

$\varrho_{2}^{2}=\left(1+e_{1}^{2}\right)p_{1}+\left(1+e_{2}^{2}\right)p_{2}-2\sqrt{p_{1}p_{2}}\left(\cos
I+e_{1}e_{2}\cos P\right)$ (4)

with

$\displaystyle p$ $\displaystyle=a\left(1-e^{2}\right),$ (5)
$\displaystyle\cos I$ $\displaystyle=\cos i_{1}\cos i_{2}+\sin i_{1}\sin
i_{2}\cos\left(\Omega_{1}-\Omega_{2}\right),$ (6) $\displaystyle\cos P$
$\displaystyle=\sin i_{1}\sin
i_{2}\sin\omega_{1}\sin\omega_{2}+\cos\omega_{1}\cos\omega_{2}\cos\left(\Omega_{1}-\Omega_{2}\right)$
$\displaystyle\quad+\cos i_{1}\cos
i_{2}\sin\omega_{1}\sin\omega_{2}\cos\left(\Omega_{1}-\Omega_{2}\right)$
$\displaystyle\quad+\left(\cos i_{2}\cos\omega_{1}\sin\omega_{2}-\cos
i_{1}\sin\omega_{1}\cos\omega_{2}\right)\sin\left(\Omega_{1}-\Omega_{2}\right).$
(7)

The limit values of such D-criteria, also called thresholds, cut-off levels,
or upper limits, determine whether two objects may be associated. Being, for
example, $A$ and $B$ a meteor and meteor shower respectively, if the distance
$D(A,B)$ between $A$ and $B$ is greater than this limit value, the association
must be discarded. The smaller this distance is, the greater the possibility
that there is a dynamical similarity between two objects, and, therefore, the
meteoroid belongs to the meteoroid stream.

Some studies on the suitability of these criteria have already been carried
out. For example, Galligan (2001) explored the performance of four similarity
functions in the near-ecliptic region–$D_{SH}$, $D_{D}$, $D_{H}$, and $D_{N}$
(Valsecchi et al., 1999)–, resulting in $D_{N}$ criterion being the most
stable in the case of the lack of a priori information on orbital inclination
regimes, while $D_{SH}$, which is based on meteor shower dispersion
theoretical models, is more suitable with very different cut-off levels.
However, $D_{N}$ has not been adopted in our approach due to its less
straightforward application from the standard parameters provided in meteor
databases. Likewise, Moorhead (2015) analyzed such cut-off values to determine
a chosen acceptable false-positive rate and distinguish which showers are
significant within a set of sporadic meteors. Jenniskens (2008) and Rudawska
et al. (2015) introduced the four-dimensional metrics $D_{B}$ and $D_{X}$,
respectively. However, to maintain consistency within the parameter space
domain analyzed in this study, we opt not to include these criteria.

Through these values, it has been possible to associate meteor showers with
parent bodies such as the 109P (1862) III Swift–Tuttle comet and the Perseid
meteor shower, whose first connection data from the late 19th century when
Schiaparelli calculated the orbits of the Perseids and discovered their strong
similarity to that of this comet. Involved on this connection, (Sokolova et
al., 2014) calculated the cut-off level of $D_{SH}$ resulting in
$D_{SH}$$\leqslant 0.2$. Literature provides more classical examples such as
the April Lyrids, whose extremely small value of the $D_{D}$ criterion
($D_{D}$=0.009) suggests that such meteors showers have indeed come from comet
Thatcher (Arter & Williams, 1997). Other recent examples are the case of a
fireball detected in the night sky over Kyoto whose likely parent, with
$D_{SH}$= 0.0079, could be the binary near-Earth asteroid (164121) 2003 YT1
(Kasuga et al., 2020); the binary asteroid 2000 UG11 associated with
Andromedids ($D_{SH}$=0.183 and $D_{H}$=0.176) and the asteroid (4179)
Toutatis, with values of $D_{SH}$=0.180 and $D_{H}$=0.175, that postulate it
associated with October Capricornids (Dumitru et al., 2017); the meteor shower
June epsilon Ophiuchids, whose values in three D-criteria ($D_{SH}$=0.05,
$D_{D}$=0.03 and $D_{H}$= 0.06) confirm that is likely to originate from comet
300P/Catalina (Matlovic et al., 2020); or the recently observed fall and
recovery of the Traspena meteorite is posited to be linked with the
potentially hazardous asteroid 1989 QF (Minos), exhibiting
$\varrho_{2}=$0.1059 (Andrade et al., 2023). We note the absence of a cut-off
estimate works for $\varrho_{2}$, unlike the traditional D-criteria.

Although the cases mentioned above demonstrate the usefulness of the
similarity criteria, some limitations confirm the need to investigate these
metrics. For example, Galligan (2001) found that, for the case of the $D_{SH}$
criterion, it is necessary to use different upper limits depending on the
orbital inclination angle of the stream. In fact, Sokolova et al. (2014),
intending to improve the reliability of identification of the observed
objects, recommends analyzing the $D_{SH}$ threshold values independently for
each meteoroid complex. Following that approach, the study of comparison of
four similarity criteria carried out by Rudawska et al. (2012) confirmed the
difficulty in obtaining one specific value of threshold that would fit all
cases, reaching the conclusion that the ideal threshold depends on the cluster
analysis method, the meteors shower, and the sample; this latter statement is
also seconded by Jopek & Bronikowska (2017). Ye (2018) also pointed out that
the traditional D-criteria may not necessarily reflect a shared origin of two
objects due to the orbital evolution influenced by planetary perturbations.

In short, these studies are clear examples of the need to analyze the
effectiveness and equivalence of the different approaches to establish
dynamical associations of meteors.

###  2.3 Meteor Vectors and Distance Metrics

In the preceding section, we discussed five-dimensional D-criteria for
associating meteors with meteor showers. While these approaches are widely
used, they are not without limitations. It is an active research topic for
which there is no consensus on either criteria or thresholds. To search for
alternatives and compare their performance, we introduce two meteor vectors
–ORBIT and GEO– to evaluate multiple Machine Learning distance metrics in
meteor-shower association.

The ORBIT vector focuses simply on the same five heliocentric orbital elements
that are used by the above-mentioned orbital similarity criteria, which allows
for a more direct comparison of the effectiveness:

$ORBIT=\left[\begin{array}[]{c}q\\\ e\\\ i/\pi\\\ \frac{\sin(\omega)+1}{4}\\\
\frac{\cos(\omega)+1}{4}\\\ \frac{\sin(\Omega)+1}{4}\\\
\frac{\cos(\Omega)+1}{4}\\\ \end{array}\right].$ (8)

Note that the database has been filtered to minimize spurious events, ensuring
the inclusion of only non-hyperbolic orbits ($0<e\leq 1$) that intersect
Earth’s orbit, specifically with $0<q\leq 1$ au. The inclination, when
normalized by 180º, spans the range [0, 1]. For the circular components,
$\omega$ and $\Omega$, which range from [-1, 1], we normalize them to [0, 1]
and assign half the weight to each circular component. Utilizing sine and
cosine functions for the circular angle $\omega$ and $\Omega$, we effectively
account for the shortest circular distance between angles, ensuring that 358∘
is recognized as 4∘ away from 2∘, rather than 356∘. Consequently, all five
independent parameters are normalized and weighted equally, constructing a
five-dimensional space vector.

The GEO vector is based mainly on geocentric observable parameters and was
proposed by Sugar et al. (2017). This six-component vector (but in four-
dimensional space as it has only four independent parameters) inherently
addresses the issue of longitude wrapping. It normalizes the six components to
ensure that each variable contributes equally. The vector’s initial two
components represent the meteor’s position, as the meteoroid intersects the
Earth’s orbit. The subsequent three components define the unit vector opposite
to the meteor’s velocity direction. The final component represents the
magnitude of the geocentric velocity, normalized by the maximum velocity
allowed for the study population:

$GEO=\left[\begin{array}[]{c}\cos\left(\lambda_{\odot}\right)\\\
\sin\left(\lambda_{\odot}\right)\\\
\sin\left(\lambda_{\mathrm{g}}-\lambda_{\odot}\right)\cos\left(\beta_{\mathrm{g}}\right)\\\
\cos\left(\lambda_{\mathrm{g}}-\lambda_{\odot}\right)\cos\left(\beta_{\mathrm{g}}\right)\\\
\sin\left(\beta_{\mathrm{g}}\right)\\\ v_{\mathrm{g}}/72\end{array}\right].$
(9)

In this vector, $v_{\mathrm{g}}$ represents the geocentric velocity in
kilometers per second, $\lambda_{\odot}$ is the solar longitude,
$\beta_{\mathrm{g}}$ is the geocentric ecliptic latitude of the radiant, and
$\lambda_{\mathrm{g}}-\lambda_{\odot}$ being the Sun-centered ecliptic
longitude of the radiant. All components span the range [-1, 1], except for
the element related to velocity, which varies between [0, 1]. Given that
velocity measurements are subject to the greatest degree of error, the authors
allowed a reduced weight for the velocity.

Although the D-criteria are theoretically five-dimensional, the orbits of the
meteors are constrained by having impacted the Earth, virtually reducing the
dimensionality by one. Consequently, this dimensionality reduction enables a
comparison between the performances of the GEO and ORBIT vectors.

In the quest to develop a robust methodology for associating meteors with
their parent meteor showers, we explore various distance metrics typically
used in Machine Learning that can quantify the similarity between the
previously defined vectors. In Table 1, we introduce the distance metrics that
are employed in this study.

Table 1: Summary of distance metrics. Metric Name | Formula | Brief Explanation
---|---|---
Euclidean | $\sqrt{\sum_{i}(u_{i}-v_{i})^{2}}$ | Square root of sum of squared differences
sEuclidean | $\sqrt{\sum_{i}\frac{(u_{i}-v_{i})^{2}}{{V_{i}}^{2}}}$ | Normalized Euclidean by variance $V_{i}$
Cityblock (Manhattan) | $\sum_{i}\left|u_{i}-v_{i}\right|$ | Sum of absolute differences
Cosine Similarity | $1-\frac{u\cdot v}{\|u\|\cdot\|v\|}$ | Angle between vectors
Canberra | $\sum_{i}\frac{\left|u_{i}-v_{i}\right|}{\left|u_{i}\right|+\left|v_{i}\right|}$ | Weighted Cityblock distance
Bray-Curtis | $\frac{\sum_{i}\left|u_{i}-v_{i}\right|}{\sum_{i}\left|u_{i}+v_{i}\right|}$ | Normalized weighted Cityblock
Chebyshev | $\max_{i}\left|u_{i}-v_{i}\right|$ | Maximum absolute difference

###  2.4 Statistical Analysis

####  2.4.1 Rank Correlations

We select the Kendall rank correlation coefficient ($\tau$) to measure the
ordinal association between the distance metrics and D-criteria.
Mathematically, it is defined as:

$\tau=\frac{2}{n(n-1)}\sum_{i<j}sgn\left(x_{i}-x_{j}\right)sgn\left(y_{i}-y_{j}\right),$
(10)

where $(x_{1},y_{1}),...,(x_{n},y_{n})$ are a set of samples of the variables.

$\tau$ is notable for its ability to measure the strength and direction of the
relationship between two variables without requiring them to be on the same
scale. Unlike parametric correlations like Pearson’s, which assume linear
relationships and normal distribution of data, Kendall’s approach is based on
the ranking of data points, assessing concordance and discordance in their
relative ordering across two datasets. It focuses on rank rather than absolute
values obviates the need for identical scales between datasets. Consequently,
we can employ it to compare the results of the D-criteria and the distance
metric without applying any normalization. We use the asymptotic method to
compute Kendall’s tau, which provides an efficient and scalable approximation
suitable for large datasets and handles ties effectively.

The process is as follows. For each meteor in the dataset, we first compute
its similarity/closeness to every meteor shower based on predefined D-criteria
and distance metrics (both for GEO and ORBIT vectors). These calculations
yield two separate sets of rankings for every meteor: one set derived from the
D-criteria and another from the distance metrics. Each set sorts all meteor
showers from the most to the least similar to the meteor in question. Once we
obtain these rankings, the $\tau$ is computed for each meteor, comparing the
two sets of rankings to ascertain the degree of ordinal classification. For
more information into the Kendall rank correlation coefficient applied here
refer to Kendall (1938); Fenwick (1994); Hollander et al. (2013).

####  2.4.2 Top-k Accuracy

The heart of the present study centers on the evaluation of the classification
accuracy of various D-criteria and distance metrics. To address this
challenge, a unified methodology is imperative for the consistent application
of statistical tests across all approaches under consideration. Despite the
diversity in metrics and D-criteria, they converge on a singular objective: to
quantify the association between a meteor and its corresponding meteor shower.
As such, the Top-k accuracy is employed as a standardizing criterion to
compare the overall accuracy among the various methods (Xia et al., 2009).

It quantifies the frequency with which the correct label class is included
among the first k predicted labels. In the specific context, these labels
denote the meteor showers associated with individual meteoroid impacts as
classified by CAMS. For each meteor in the dataset, the similarities and
distances are calculated in relation to all reference meteor showers. These
values are subsequently sorted in ascending order to generate a ranked list. A
successful classification in the Top-1 category occurs when the meteor shower
with the minimum similarity or distance aligns with the meteor shower
associated with the meteor in the CAM dataset. Similarly, a Top-5 success is
recorded if the associated meteor shower is among the top five labels in the
ranked list, and this extends analogously for other values of k.

In the present study, multiple tests encompassing Top-1, Top-5, and Top-10
accuracy are performed to evaluate the efficacy of D-criteria and distance
metrics in associating a meteor with its originating meteor shower. This
multi-tiered approach enables both a precise assessment of the top prediction
(Top-1) and an evaluation of the model’s capacity to identify a broader set of
correct associations (Top-5 and Top-10). While one might assume that the Top-1
accuracy is paramount for meteor association, it is important to consider the
significance of conducting Top-5 and Top-10 analyses. These extended
evaluations yield insights into the efficacy of various ranking methodologies,
going beyond mere concurrence with CAMS classifications. These analyses aid in
contrasting the variability in rankings produced by different metrics. It is
distinct when two metrics diverge at the Top-1 level yet converge within the
Top-5, compared to a scenario where they diverge up to the Top-10.

####  2.4.3 Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov test (K-S test) serves as a robust, non-parametric
statistical method designed to assess the goodness-of-fit and equivalence of
continuous, one-dimensional probability distributions. The test is
particularly advantageous due to its distribution-free nature, making it
applicable to datasets without the assumption of any specific distribution.
The K-S test is employed in two primary contexts: the one-sample K-S test and
the two-sample K-S test. The two-sample K-S aims to compare two empirical
distributions and to determine if the two samples come from the same
distribution. The K-S statistic $D$ is:

$D=\sup_{x}|F_{1,n}(x)-F_{2,m}(x)|$ (11)

where $F_{1,n}(x)$ and $F_{2,m}(x)$ are the empirical distribution functions
of the two samples of sizes $n$ and $m$, respectively. Here we follow the
treatment explained in Hodges (1958).

When applying the K-S test to Top-1 test results, interpreting the results
sheds light on the comparative distributions of accuracy between
classification methods. Failing to reject the null hypothesis $H_{0}$
indicates no statistically significant difference in accuracy distributions,
but it does not affirm equivalence in method performance. Conversely,
rejecting $H_{0}$ suggests a statistically significant difference, supporting
the alternative hypothesis $H_{1}$ that the samples originate from distinct
distributions. This outcome implies that $H_{0}$ does not adequately explain
the observed data, with the decision to reject based on the significance level
$\alpha$, set here at 0.05 for 95% confidence.

####  2.4.4 Top-1 Agreement

Consider two classifiers tested on a dataset consisting of two equally sized
classes. The first classifier might excel in identifying Class A but fail to
recognize Class B, whereas the second classifier achieves the opposite,
accurately identifying Class B while mistaking instances of Class A. Despite
both classifiers reporting an overall accuracy of 50%, their distinct
performance on the individual classes reveals a divergent understanding and
representation of the underlying patterns in the data. This example
underscores the necessity of applying another test, as (1) Kendall’s
correlation assesses whether the order of rankings is similar between two sets
of observations and (2) K-S is specifically focused on the shape of accuracy
distributions rather than precise values.

For this reason, we calculate as well the percentage of Top-1 coincidence
between distance metrics and D-vectors on an event-by-event basis, which
provides a direct measure of agreement on the most preferred classification
outcome, capturing the extent to which different approaches concur on the
single best classification. This straightforward metric offers an immediate
sense of the hit-and-miss between approaches. A heatmap is an optimal
visualization tool for showcasing the pairwise agreement between
classification metrics, using a rectangular matrix to highlight the magnitude
of their coincidences.

###  2.5 Differentiating the sporadic background

The last part of our work deals with the effective discrimination of the
sporadic background from meteor events that are associated with specific
showers. We calculate the Top-1 accuracy values across the entire (non-
filtered) database and construct the Receiver Operating Characteristic (ROC)
curves for each D-criteria and distance metric, utilizing both the GEO and
ORBIT vectors, using binary labels from CAMS (0: sporadic; 1: associated). The
ROC curve represents the diagnostic ability of a binary classifier system as
its discrimination threshold is varied. Using the ROC curve output, it is
possible to quantify the optimal threshold that maximizes the classifier’s
performance with Youden’s $J$ statistic (Youden, 1950; Schisterman et al.,
2005):

$J=\frac{TP}{TP+FN}+\frac{TN}{TN+FP}-1,$ (12)

where TP represents the true positives, FN the false positives, TN the true
negatives, and FP the false positives.

To synthesize the overall performance of each classification method in
differentiating the sporadic background, we utilize the Matthews Correlation
Coefficient, usually denoted by MCC or $\phi$ (Matthews, 1975). The $\phi$
offers a measure of the quality of binary classifications, encapsulating
sensitivity, specificity, and the balance between them. It ranges from -1
(total disagreement between prediction and observation) to 1 (perfect
prediction), with 0 denoting random guessing. The $\phi$ is defined as:

$\phi=\frac{(TP\cdot TN)-(FP\cdot
FN)}{\sqrt{(TP+FP)\cdot(TP+FN)\cdot(TN+FP)\cdot(TN+FN)}}.$ (13)

##  3 Results

Similar to Section 2, where we detailed the database and methodology in
distinct subsections, the results section is also organized into subsections
for clarity and depth. Subsection 3.1 examines the dataset, Subsection 3.2
presents the rank correlation estimations, Subsection 3.3 reports on the
accuracy results, Subsection 3.4 explores the equivalence between distance
metrics and D-criteria, Subsection 3.5 offers the level of coincidence between
approaches for the Top-1 tests, and finally Subsection 3.6 provides optimal
cut-offs and false positive rates.

###  3.1 Population Analysis

Within the extensive CAMS database, 24.6% of its entries can be directly
linked to a distinct meteor shower. In contrast, 75.4% of the data points are
categorized as sporadic events, implying they are part of the broader
meteoroid background rather than specific meteor showers. After applying the
filters mentioned in Section 2, the database reduces its number to account for
102,680 orbits.

The number of unique meteor shower classifications is somewhat constrained,
amounting to 376 distinct categories. A total of 80% of these classified
meteor showers have been observed more than 10 times. A quarter of them, or
25%, boasts over 100 individual recorded meteor events. An even smaller
fraction, 5%, can claim over 1000 meteor instances. Four of the meteor showers
stand out due to their frequent documentation: the Perseids, Orionids,
Geminids, and Southern Taurids (enumerated in descending order based on their
observation frequency). Meteors belonging to these showers have been observed
more than 10,000 times.

Regarding the IAU meteor shower database, after filtering it reduces its
number to 724, having 355 unique IDs shared with the CAMS database. Note that
$\sim$30% are duplicate entries, corresponding to distinct values for the same
meteor shower estimated in different studies.

A key aspect of our analysis of associations is the parameter of solar
longitude, that correlates meteor activity with Earth’s specific orbital
locations. Such a correlation is instrumental in discerning patterns and
understanding recurring meteoritic phenomena. To visually represent this
correlation, Figure 1 offers a histogram that plots impacting meteoroid
classifications (sporadic or associated) based on solar longitude. The most
active meteor showers are annotated. It can be observed an apparent
concentration of the meteoroid background activity toward 180∘ of solar
longitude.

Fig. 1: Histogram of CAMS database as a function of the solar longitude.
Sporadic and associated meteors are depicted.

###  3.2 Degree of Rank Correlation

For each of the showers listed in the IAU database, we compute the
similarity/closeness between the shower and each meteor in the CAMS database
using the D-criteria and all distance metric combinations. We then calculate
the Kendall rank correlation between each D-criterion and each vector-metric
combination.

Figure 2 displays the Kendall rank correlation between the evaluated
D-criteria and distance metrics. Each column corresponds to a particular
distance metric, and the plots are color-coded by D-vectors. The box plots
encapsulate the quartile distribution of the samples, where each sample
denotes the rank correlation between the D-criteria and distance metrics for a
meteor with all meteor showers. The calculation is performed for each meteor
against all meteor showers, a process executed separately for both the GEO and
ORBIT vectors. Points lying outside the whiskers of the box plots are
classified as outliers, positioned more than 1.5 times the interquartile range
away from the median (Q2, depicted by the box’s central line). A homogeneous
dataset would result in a compact interquartile range, with the median
equidistant from the box’s extremes (Q1 and Q3), indicating symmetry. The span
from the plot to each whisker indicates the data’s variability or spread,
suggesting a more concentrated distribution if the span is shorter and greater
dispersion if it is extended.

The different figures reveal particular features in the Kendall rank
correlation between D-criteria and distance metrics, as delineated by the
employment of GEO and ORBIT vectors. The sEuclidean metric paired with the GEO
vector consistently demonstrates the highest median correlation across all
D-criteria, indicating a robust ordinal association. In contrast, the ORBIT
vector presents a distinctive landscape. $D_{D}$ criterion, when evaluated
with ORBIT vectors, achieves the highest correlation values. $\varrho_{2}$
criterion exhibits considerable variability in correlation, as evidenced by
notably wide box plots for some distance metrics when using the ORBIT vectors.
This behavior starkly contrasts with the other D-criteria, pointing to
$\varrho_{2}$ unique response to the parameters captured by ORBIT vectors.
While the GEO vector is characterized by a greater number of lower outliers,
indicating instances of significantly divergent rankings, ORBIT vectors show
fewer upper outliers. The results show a general tendency for the median
correlation values to either be randomly centered or skewed across both
vectors and all metrics. This variability suggests that no singular pattern of
correlation prevails universally. Additionally, the maximum whisker extension
observed with the Cosine distance metric, specifically when paired with the
GEO vector and $D_{D}$ criterion, signals instances of high variability or
dispersion in the degree of correlation.

Fig. 2: Kendall rank correlation between D-criteria and metric distances for
associated meteors in CAMS database. Each column corresponds to a unique
vector (ORBIT or GEO). Each sample symbolizes the rank correlation between the
similarity criteria and the distance metrics of each of the meteors from the
CAMS database concerning the distinct meteor showers. Outlier values surpass
1.5 times the interquartile range of the median.

###  3.3 Accuracy of Best Choices

Using both the D-criteria and the employed meteor vectors and distance
metrics, each meteor is juxtaposed with all showers, as detailed in Tables 2
and 3. This approach facilitates representing, in percentages, instances where
the associated shower from the CAMS database in terms of distance and
similarity aligns (Top-1) ranks among the five showers with the most minimal
similarity and standardized Euclidean distance values (Top-5) or falls within
the top ten showers (Top-10).

Table 2: Top-k accuracies of D-criteria for associated meteors in CAMS database. Top-k |  $\boldsymbol{D_{SH}}$ (%) |  $\boldsymbol{D_{D}}$ (%) |  $\boldsymbol{D_{H}}$ (%) |  $\boldsymbol{\varrho_{2}}$ (%)
---|---|---|---|---
1 | 86.23 | 80.07 | 85.83 | 85.56
5 | 94.90 | 93.60 | 94.99 | 95.67
10 | 97.50 | 96.63 | 97.60 | 97.93
Table 3: Top-1, Top-5, and Top-10 accuracies of distance metrics for associated meteors in CAMS database. | Euclidean | sEuclidean | Cityblock | Cosine | Canberra | Bray-Curtis | Chebyshev
---|---|---|---|---|---|---|---
|  |  |  | Top-1 (%) |  |  |
GEO | 84.68 | 87.06 | 84.79 | 84.63 | 85.89 | 84.96 | 83.41
ORBIT | 83.67 | 77.53 | 83.92 | 83.83 | 79.64 | 83.96 | 80.11
|  |  |  | Top-5 (%) |  |  |
GEO | 92.99 | 95.03 | 93.72 | 92.95 | 94.35 | 93.89 | 90.94
ORBIT | 94.17 | 92.68 | 94.05 | 93.88 | 91.06 | 94.10 | 92.29
|  |  |  | Top-10 (%) |  |  |
GEO | 95.56 | 97.06 | 96.16 | 95.52 | 96.66 | 96.23 | 94.05
ORBIT | 96.37 | 96.25 | 96.57 | 96.19 | 95.08 | 96.61 | 94.45

The optimal D-criterion for achieving Top-1 accuracy is $D_{SH}$ (86.23%),
whereas $\varrho_{2}$ excels in both Top-5 (95.67%) and Top-10 (97.93%)
categories and have a good accuracy in Top-1 (85.56%). Conversely, $D_{D}$
ranks as the least effective across all evaluations, markedly lagging behind
the others, which exhibit comparable performances. The sEuclidean when
combined with the GEO vector demonstrates superior performance (87.06%) over
the other D-criteria in achieving Top-1 accuracy and overly the rest of the
distance metrics in all Top-k tests. When paired with the ORBIT vector, the
Bray-Curtis metric delivers the highest overall accuracy (including the
$D_{D}$ criterion in all tests), except for Top-5 accuracy, where the
Euclidean metric slightly outperforms it. Across the distance metrics
evaluated, the GEO vector is found to yield better outcomes than the ORBIT
vector. The Chebyshev metric exhibits the worst results with the GEO vector,
while the sEuclidean and Canberra present the lower performances for the ORBIT
vector. Table 4 shows the mean accuracies. The distance metrics combined with
the GEO vectors offer the best overall accuracy for Top-1, while the
D-criteria outstrip in Top-5 and Top-10.

Table 4: Mean accuracies and standard deviations for Top-k tests across the D-criteria and distance metrics with GEO and ORBIT vectors. Test | All (%) | D-criteria (%) | GEO (%) | ORBIT (%)
---|---|---|---|---
Top-1 | $83.7\pm 2.5$ | $84.2\pm 2.5$ | $\boldsymbol{85.1\pm 1.1}$ | $81.8\pm 2.5$
Top-5 | $93.6\pm 1.3$ | $\boldsymbol{94.8\pm 0.7}$ | $93.4\pm 1.2$ | $93.2\pm 1.1$
Top-10 | $96.2\pm 1.0$ | $\boldsymbol{97.4\pm 0.5}$ | $95.9\pm 0.9$ | $95.9\pm 0.8$

Figure 3 illustrates the variation in Top-k test accuracy as a function of
solar longitude across different D-criteria. Similarly, Figure 4 displays the
Top-k results for the sEuclidean, Canberra, Bray-Curtis, and Chebyshev
distance metrics. Across all evaluations, the results are of the same order of
magnitude. A distinct pattern emerges: for Top-1, the accuracy variation is
irregular, whereas, for Top-10, it tends towards uniformity, except for a
notable decrease (up to 50%) around 180∘ of solar longitude. Visually, the
lower performance of $D_{D}$ is prominent, and $D_{SH}$ and $\varrho_{2}$
excel, especially at solar longitudes between 170∘ and 220∘, as well as around
70∘ in Top-5 and Top-10, and 350∘ in Top-1 (with a sudden increase of the
accuracy of $D_{H}$). Conversely, the performances of the distance metrics
generally follow the same trend, albeit less uniformly in the Top-10
distribution. Besides the common peak at 180∘, it is observed that they
struggle to associate meteors at around 310∘, where Chebyshev (with GEO
vector) and Canberra (with ORBIT vector) exhibit remarkably lower
performances.

Fig. 3: Top-k accuracies along solar longitude of the D-criteria for
associated meteors in CAMS database. Fig. 4: Top-k accuracies along solar
longitude of different distance metrics for associated meteors in CAMS
database.

The trend of minimum accuracy in meteor association is pinpointed at 180∘
solar longitude, aligning with an apparent increase in the meteoroid
background activity, as depicted in Fig. 1. This time frame also bridges the
Perseids and Orionids, meteor showers renowned for their high activity and
velocities above 60 km/s, expecting, in consequence, a diffuseness of their
parameters. Instrumental constraints correlate meteoroid velocity with
measurement inaccuracies (Hajduková & Kornoš, 2020). As a result, high-
velocity meteoroids are more challenging to accurately characterize. This is
depicted in Figure 5, showcasing a concentration of high apparent velocities
within this specific solar longitude range. It is conceivable that these
meteoroids were once part of such swarms but have lost their orbital affinity
due to temporal decoherence, making many of them challenging to distinguish.
Furthermore, the increased activity during these periods, characterized by
similar velocities, may have influenced the association process conducted by
CAMS.

Fig. 5: 2D-histogram of sporadic meteor apparent velocities and solar
longitudes at impact in the CAMS database. Darker colors denote higher
density.

###  3.4 Statistical Equivalence

Figure 6 displays classification outcomes labeled as $H_{0}$ or $H_{1}$,
corresponding to the hypothesis tested for each data comparison for the Top-1
accuracy results from the metric distance and the D-criteria. Labels are
determined based on p-values: instances where the p-value is less than 0.05
are marked as $H_{1}$, indicating the rejection of the null hypothesis
($H_{0}$) in favor of the alternative ($H_{1}$), suggesting a statistically
significant difference between the compared distributions. Conversely,
instances with a p-value greater than or equal to 0.05 retain the $H_{0}$
label, indicating insufficient evidence to reject the null hypothesis, thus
suggesting no statistically significant difference between distributions under
examination.

The figure reveals a distinct pattern in the distribution of hypothesis
testing results, particularly when evaluating the $D_{D}$ criterion with the
GEO vector. Contrary to the other D-criteria, which generally do not reject
the null hypothesis $H_{0}$ when paired with the GEO vector, $D_{D}$ stands
out by predominantly rejecting $H_{0}$ (indicated by $H_{1}$), suggesting
differences in distributions. This trend is reversed for the ORBIT vector,
where $D_{D}$ results in non-rejection of $H_{0}$, except for one the Canberra
metric. This behavior is markedly different from the other criteria tested
with the ORBIT vector, which rejects $H_{0}$ for the same three metrics
(sEuclidean, Canberra, and Chebyshev). $\varrho_{2}$ appears the most likely
compatible with both vectors at the same time.

The consistent failure to reject $H_{0}$ with GEO vector for all distance
metrics under the $\varrho_{2}$ criterion does not confirm the distributions
being identical but rather indicates the test lacked sufficient evidence to
demonstrate statistical differences. This outcome positions $\varrho_{2}$ as
the D-criterion that is most plausibly comparable to the distance metrics in
terms of meteor association when using the GEO vector. Also, the $\varrho_{2}$
metric exhibits the highest probability of being compatible with both vectors
simultaneously.

Fig. 6: K-S test comparing Top-1 accuracies of distance metrics and D-criteria
with a 95% level of confidence for associated meteors in CAMS database.
$H_{0}$ indicates no statistically significant difference between
distributions, while $H_{1}$ indicates a significant difference between the
compared distributions.

###  3.5 Event-by-Event Agreement

The heatmap on Figure 7 visualizes the agreement level between various
D-vectors and distance metrics, showcasing their comparative analysis for the
Top-1 results across the two meteor vectors. Each cell represents the
percentage of Top-1 coincidence between pairs, with GEO-related comparisons
highlighted in shades of blue for intuitive analysis, and ORBIT-related
comparisons in shades of red, enabling a clear distinction between the two
meteor vectors used. The diagonal, intentionally left blank, separates GEO and
ORBIT results for a dual analysis within a single visual representation. The
cross-accuracies of D-criteria, independent of GEO or ORBIT vectors, are
outlined with a black frame in the figure’s top left corner.

Analyzing the heatmap reveals that the D-vector $D_{SH}$ has a strong event-
by-event alignment with $D_{H}$ for Top-1 (97.43%), indicating these criteria
frequently concur on their top classifications. This is closely followed by
$\varrho_{2}$ and $D_{H}$ (94.69%). Within the GEO vector, Euclidean and
Cosine (99.66%), along with Cityblock and Bray-Curtis (99.15%), show the
highest levels of coincidence in Top-1. The sEuclidean metric generally shows
good agreement for the GEO vector across various metrics and D-criteria
($\sim$88%), except when paired with $D_{D}$ (83.37%). For the ORBIT vector,
Cityblock and Bray-Curtis (98.99%), as well as Euclidean and Cosine (95.62%),
exhibit the highest values. There is better alignment between $D_{D}$ and the
ORBIT vector (reaching $\sim$86% with various distance metrics) than seen with
the GEO vector.

The consistency observed in the heatmap resonates with the findings from
Kendall’s correlation and the K-S test. These statistical measures support the
identified patterns of agreement and discrepancy among the classifiers,
providing robustness to the analysis and confirming the reliability of these
patterns.

Fig. 7: Heatmap of cross-coincidence between D-vectors and distance metrics
using GEO (lower triangle, blue colormap) and ORBIT (upper triangle, red
colormap) vectors of Top-1 accuracies for associated meteors in CAMS database.
D-vector’s own cross-coincidences are highlighted within a black rectangle in
the top left corner.

###  3.6 Thresholds and Confusion Matrices

Table 5 presents the evaluation of D-criteria and distance metrics within the
CAMS database, considering both sporadic and associated meteor events, where
optimal thresholds and the effectiveness of different methods are
encapsulated. The standout performer among D-criteria is $D_{SH}$, distinctly
outshining others with a $\phi$ of 0.6400. Conversely, $D_{D}$ emerges as the
least effective.

When using the GEO vector, the sEuclidean metric takes precedence, exhibiting
the highest overall accuracy and a $\phi$ value of 0.6464, closely followed by
Cityblock and Bray-Curtis metrics. The scenario shifts when transitioning to
the ORBIT vector, where Cityblock edges out as the frontrunner, albeit with
Bray-Curtis and Euclidean not far behind, suggesting a competitive field with
closely matched performances. The sEuclidean metric with ORBIT vector does not
mirror its GEO vector success, hinting at vector-specific behavior that
influences metric efficacy.

Cityblock, while not outperforming other distance metrics in replicating CAMS’
associated Top-1 classifications, excelled in more effectively distinguishing
the sporadic background on average. Except for $D_{SH}$, all distance metrics
applied to the GEO vector–aside from Cosine–surpass the rest of the D-criteria
in terms of the $\phi$. Interestingly, despite generally lower performance
with the ORBIT vector, several distance metrics still exceed some D-criteria
performances. Cityblock, in particular, scores relatively close to achieving
the superior results of $D_{SH}$ and sEuclidean.

Additionally, the observed thresholds for traditional D-criteria ($D_{SH}$,
$D_{D}$, and $D_{H}$) align perfectly with values documented in the scientific
literature, reinforcing the validity of our findings.

Table 5: Threshold, accuracies, and Matthews correlation coefficients for different D-criteria and distance metrics in the CAMS database taking into account the sporadic and associated events. Method | Cut-off | TP (%) | FN (%) | FP (%) | TN (%) | Acc. (%) | $\boldsymbol{\phi}$
---|---|---|---|---|---|---|---
D-criteria
$D_{SH}$ | 0.170 | 59.51 | 8.62 | 7.21 | 24.66 | 84.17 | 0.6400
$D_{D}$ | 0.083 | 58.34 | 9.78 | 8.35 | 23.52 | 81.87 | 0.5877
$D_{H}$ | 0.176 | 58.81 | 9.32 | 7.76 | 24.12 | 82.92 | 0.6122
$\varrho_{2}$ | 0.174 | 59.56 | 8.56 | 7.97 | 23.91 | 83.47 | 0.6213
Distance metric with the GEO vector
Euclidean | 0.138 | 58.75 | 9.37 | 7.13 | 24.75 | 83.50 | 0.6279
sEuclidean | 0.370 | 59.25 | 8.88 | 6.78 | 25.10 | 84.34 | 0.6464
Cityblock | 0.252 | 59.45 | 8.67 | 7.20 | 24.67 | 84.13 | 0.6393
Cosine | 0.004 | 57.85 | 10.27 | 6.97 | 24.91 | 82.76 | 0.6154
Canberra | 0.380 | 59.98 | 8.15 | 8.22 | 23.65 | 83.63 | 0.6228
Bray-Curtis | 0.037 | 59.80 | 8.33 | 7.48 | 24.40 | 84.19 | 0.6387
Chebyshev | 0.089 | 61.01 | 7.11 | 8.83 | 23.05 | 84.06 | 0.6282
Distance metric with the ORBIT vector
Euclidean | 0.086 | 60.23 | 7.89 | 8.01 | 23.86 | 84.10 | 0.6335
sEuclidean | 0.542 | 59.80 | 8.32 | 9.71 | 22.17 | 81.97 | 0.5803
Cityblock | 0.173 | 59.63 | 8.50 | 7.46 | 24.42 | 84.04 | 0.6359
Cosine | 0.002 | 57.47 | 10.65 | 6.94 | 24.94 | 82.41 | 0.6093
Canberra | 0.431 | 55.69 | 12.43 | 8.13 | 23.74 | 79.44 | 0.5454
Bray-Curtis | 0.031 | 59.31 | 8.81 | 7.29 | 24.58 | 83.90 | 0.6343
Chebyshev | 0.061 | 59.55 | 8.57 | 8.54 | 23.34 | 82.89 | 0.6062

As an additional note to our findings, it is noteworthy that upon
incorporating the complete list of meteor showers—not limited to those used
within the CAMS database—an average of 27% of the meteor classified (Top-1) by
all D-criteria and distance metrics would align better with newly recognized
meteor showers. In future efforts, we aim to do a comparative analysis by
testing other databases such as GMN (Vida et al., 2021) and EDMOND (Kornoš et
al., 2014).

##  4 Conclusions

This study undertook a statistical evaluation of four orbital similarity
criteria (or D-criteria) within a five-dimensional parameter space to probe
the dynamical associations within meteor data. Utilizing the extensive data
compiled by the CAMS network, we have not only relied on D-criteria ($D_{SH}$,
$D_{D}$, $D_{H}$, and $\varrho_{2}$) but also ventured into distance metrics
commonly applied in Machine Learning (Euclidean, sEuclidean, Cityblock,
Cosine, Canberra, Bray-Curtis, and Chebyshev), investigated across two
distinctive meteor vectors. One vector termed ORBIT, based on heliocentric
orbital elements, is essentially shared with the D-criteria, and the other
one, GEO, based on geocentric observational parameters, was proposed by Sugar
et al. (2017). Our methodology hinged on the Kendall rank correlation
coefficient and Top-k accuracy tests to assess the correlation and performance
of these criteria and metrics. We also applied the Kolmogorov-Smirnov test and
computed the level of coincidence of individual Top-1 results for discerning
the statistical equivalence of the different approaches. Finally, we
calculated the optimal thresholds and evaluated their performances in
distinguishing the sporadic background from the meteor showers.

Our key findings can be summarized as follows:

  * 1.

The sEuclidean metric paired with the GEO vector demonstrates superior
performances than the D-criteria and the other distance metrics in achieving
Top-1 accuracy (87.06%).

  * 2.

Regarding the D-criteria, the $D_{SH}$ criterion holds the upper hand in
achieving Top-1 accuracy (86.23%), while $\varrho_{2}$ maintains dominance in
both the Top-5 (95.67%) and Top-10 (97.93%) categories (surpassed by $D_{SH}$
in Top-1 accuracy by 0.67%).

  * 3.

The Bray-Curtis metric, allied with the ORBIT vector, demonstrated a
consistent edge over other distance metrics, outperforming the $D_{D}$
criterion across all Top-k tests (83.96%, 94.10%, and 96.61%, in increasing
order of k) and only slightly beaten by the Euclidean metric in Top-5 accuracy
by a negligible difference (0.07%).

  * 4.

$D_{D}$ exhibits an opposite trend to the other D-criteria when evaluating its
equivalence against distance metrics with the GEO vector.

  * 5.

Among the D-criteria, $\varrho_{2}$ appears as the most likely similar to the
distance metrics with the GEO vector, being also the most compatible with both
GEO and ORBIT vectors at the same time.

  * 6.

In general terms, the D-criteria and the metric distances provide similar
accuracies in Top-k tests (83.7$\pm$2.5%, 93.6$\pm$1.3%, 96.2$\pm$1.0%, in
ascending order of k), with the $D_{D}$ and the metric Chebyshev performing
worse.

  * 7.

The mean highest accuracies are achieved with the GEO (85.1$\pm$1.1%,
93.4$\pm$1.2%, and 95.9$\pm$0.9%) rather than the ORBIT vector (81.8$\pm$2.5%,
93.2$\pm$1.1%, and 95.9$\pm$0.8%) within the Top-1, Top-5, and Top-10 tests,
and surpassing also the D-criteria in Top-1 (84.4$\pm$2.5%, 94.8$\pm$0.7%, and
97.4$\pm$0.5%). This suggests that geocentric parameters provide a more robust
basis than orbital elements for meteor dynamical association.

  * 8.

We observed moderate solar longitude-dependent deviations and a common
significant decrease in accuracy around 180∘ of solar longitude. We
tentatively linked these features to heightened meteoroid background activity
and the interface with two of the most active, high-velocity meteor showers:
the Perseids and the Orionids.

  * 9.

The sEuclidean metric achieves the highest overall accuracy (84.34%) and
$\phi$ of 0.6464 for GEO vector applications, excelling in distinguishing the
sporadic background. It is closely followed by Cityblock ($\phi$=0.6393 and
84.13% accuracy) and Bray-Curtis metrics ($\phi$=0.6387 and 84.19% accuracy).

  * 10.

Among D-criteria, $D_{SH}$ distinguishes itself with a $\phi$ of 0.6400,
translating to an 84.17% accuracy rate in separating the background, while
$D_{D}$ emerges as the least effective, with a $\phi$ of 0.5877 and an
accuracy of 81.87%.

  * 11.

In the ORBIT vector context, Cityblock takes the lead ($\phi$=0.6359 and
84.04% accuracy), closely challenged by Bray-Curtis ($\phi$=0.6343 and 83.90%
accuracy) and Euclidean ($\phi$=0.6335 and 84.10% accuracy).

  * 12.

Excluding Cosine, all distance metrics associated with the GEO vector surpass
the D-criteria in $\phi$ when differentiating the meteoroid background.

  * 13.

Despite the ORBIT vector’s generally lower performance, various distance
metrics still exceed certain D-criteria in effectiveness.

  * 14.

Optimal cut-offs for all D-criteria and distance metrics are provided, founded
on the CAMS database classification.

  * 15.

Based on these approaches, $\sim$27% of associated meteors in CAMS would align
with showers identified after the database’s release.

  * 16.

Future research will concentrate on studying effectiveness, equivalences, and
thresholds for a synthetic impacting population, exploring the performance and
specific attributes of the methods for each individual meteor shower.

The work culminates in the significant revelation that Machine Learning
distance metrics can rival or even outperform the specifically tailored
orbital similarity criteria for meteor dynamical association. This opens up
new pathways for the use of computational techniques in the field of meteor
science, offering an opportunity to refine our approaches to classifying
meteor showers and sporadic meteors alike.

### Acknowledgements

EP-A and JMS-L have carried out this work in the framework of the project
Fundación Seneca (22069/PI/22), Spain. EP-A thanks funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 Research and
Innovation Programme (grant agreement No. 865657) for the project “Quantum
Chemistry on Interstellar Grains” (QUANTUMGRAIN). JMS-L thanks funding from
the Ministerio de Ciencia e Innovación (Spain) (grants PID2020-112754GB-I00
and PID2021-128062NB-I00). This work was supported by the LUMIO project funded
by the Agenzia Spaziale Italiana. We acknowledge Andrés Suárez-García for his
initial involvement and contributions to this work.

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# C*-algebras where every element is a limit of products of positive elements

Leonel Robert  Department of Mathematics, University of Louisiana at
Lafayette, Lafayette, 70506, USA<EMAIL_ADDRESS>
###### Abstract.

We obtain a characterization of the unital C*-algebras with the property that
every element is a limit of products of positive elements, thereby answering a
question of Murphy and Phillips.

Let $A$ be a unital C*-algebra. Let $A_{+}$ denote the set of positive
elements of $A$. Let $P(A)$ denote the set of finite products of elements in
$A_{+}$. Quinn showed in [quinn] that if $A$ is an AF C*-algebra and has no
finite dimensional representations, then $P(A)$ is dense in $A$. In [mur-
phil], Murphy and Phillips took up the systematic investigation of the
C*-algebras with the property that $P(A)$ is dense in $A$, which they termed
the approximate positive factorization property (APFP). They obtained
necessary conditions for the APFP to hold, confirmed the APFP for various
classes of C*-algebras, and raised the following question ([mur-phil, Question
4.12]): If $A$ is an infinite dimensional simple unital C*-algebra of real
rank zero, stable rank one, and trivial $K_{1}$-group, does $A$ have the APFP?
In the theorem below we answer this question affirmatively by establishing
necessary and sufficient conditions for the APFP to hold.

######  Theorem 1.

Let $A$ be a unital C*-algebra. The following are equivalent:

  1. (i)

$P(A)$ is dense in $A$, i.e., $A$ has the approximate positive factorization
property.

  2. (ii)

$A$ has no nonzero finite dimensional representations, the stable rank of $A$
is one, the $K_{1}$-group is trivial, and the map $\rho_{A}\colon
K_{0}(A)\to\mathrm{Aff}(T(A))$, from the $K_{0}$-group of $A$ to the
continuous affine functions on the trace space of $A$, has dense range.

Let us recall the meaning of the properties in (ii).

A unital C*-algebra $A$ is said to have stable rank one if its invertible
elements form a dense set.

By trivial $K_{1}$-group we mean $K_{1}(A)=0$, where $K_{1}(A)$ is the abelian
group associated to $A$ via K-theory.

Let $T(A)$ denote the convex set of tracial states on $A$ (endowed with the
weak* topology). Let us regard traces extended to $M_{\infty}(A)$—the infinite
matrices on $A$ with finitely many nonzero entries—by setting
$\tau((a_{ij})_{i,j})=\tau(\sum_{i=1}^{\infty}a_{ii})$. Given a tracial state
$\tau$ and a $K_{0}$-group element $[p]-[q]\in K_{0}(A)$, define their pairing
as $([p]-[q],\tau)\mapsto\tau(p)-\tau(q)$. Fixing $[p]-[q]$ and varying
$\tau$, we get a continuous real-valued affine function on $T(A)$, i.e., an
element of $\mathrm{Aff}(T(A))$. Let us denote by $\rho_{A}\colon
K_{0}(A)\to\mathrm{Aff}(T(A))$ the mapping induced by this pairing:

$\rho_{A}([p]-[q])(\tau)=\tau(p)-\tau(q).$

It is known that if $A$ is a simple C*-algebra of real rank zero, then the
range of $\rho_{A}$ is dense in $\mathrm{Aff}(T(A))$ (e.g., see [Dur, Theorem
4.1]). Thus, Theorem 1 indeed answers Murphy and Phillips’ question
affirmatively.

In the second theorem of this note (which we prove first), we obtain a
description of the invertible elements in $\overline{P(A)}$ which, in a sense,
extends the description as “elements of positive determinant", valid in the
case of matrices. Let $\widetilde{P}(A)=\overline{P(A)}\cap GL(A)$, where
$GL(A)$ denotes the invertible elements of $A$. Let $U(A)$ denote the unitary
group of $A$, and $U_{0}(A)$ the connected component of 1 in $U(A)$. Let
$DU_{0}(A)$ denote the commutator subgroup of $U_{0}(A)$.

######  Theorem 2.

Let $A$ be a unital C*-algebra. Then $\widetilde{P}(A)$ is the set of
invertible elements whose unitary part in the polar decomposition belongs to
$\overline{DU_{0}(A)}$, i.e., the elements of the form $ua$, with
$u\in\overline{DU_{0}(A)}$ and $a\in A_{+}$ positive and invertible.

##  1\. Proofs of Theorems 1 and 2

We will use some facts around the de la Harpe-Skandalis determinant. We
introduce here necessary notation, but refer the reader to [delaHarpe-
skandalis] and [thomsen] for background on this topic.

Let $\overline{[A,A]}$ denote the closure of the linear span of the
commutators $[x,y]:=xy-yx$, with $x,y\in A$. Let us regard
$A/\overline{[A,A]}$ as a Banach space under the quotient norm. Let $T\colon
A\to A/\overline{[A,A]}$ denote the quotient map (or “universal trace").
Regard $T$ extended to $M_{\infty}(A)$ by setting
$T((a_{ij})_{i,j})=T(\sum_{i=1}^{\infty}a_{ii})$.

Let $GL_{0}^{\infty}(A)$ denote the inductive limit of the groups
$GL_{0}(M_{n}(A))$, $n\in\mathbb{N}$. Given a smooth path
$\alpha\colon[t_{1},t_{2}]\to GL_{0}^{\infty}(A)$, its de la Harpe-Skandalis
determinant is defined as

$\widetilde{\Delta}_{T}(\alpha)=\int_{t_{1}}^{t_{2}}T(\alpha^{\prime}(t)\alpha^{-1}(t))dt\in
A/\overline{[A,A]}.$

If two paths $(a_{1}(t))_{t}$ and $(a_{2}(t)_{t})$ both connect $1$ to an
invertible element $a\in GL_{0}^{\infty}(A)$, then the values of
$\widetilde{\Delta}((a_{1}(t))_{t})$ and $\widetilde{\Delta}((a_{2}(t))_{t})$
differ by at most an element of the abelian subgroup

$2\pi i\\{T(p)-T(q):p,q\in M_{\infty}(A)\hbox{ projections}\\}.$

Let’s denote this subgroup by $2\pi iT(K_{0}(A))$). Given an invertible
element $a\in GL_{0}^{\infty}(A)$, its de la Harpe-Skandalis determinant
$\Delta_{T}(a)$ is defined as the image of
$\widetilde{\Delta}_{T}((a(t)_{t}))$ in the quotient
$(A/\overline{[A,A]})/2\pi iT(K_{0}(A))$, where $t\mapsto a(t)$ is any path in
$GL_{0}^{\infty}(A)$ connecting $1$ to $a$.

###### Proof of Theorem 2.

Let $Q$ denote the set of invertible elements whose unitary part in the polar
decomposition belongs to $\overline{DU_{0}(A)}$. Thus, $Q$ is defined by the
condition $x|x|^{-1}\in\overline{DU_{0}(A)}$. By the continuity of the
operations involved in this condition, it is clear that $Q$ is a norm closed
subset of $GL(A)$ (in fact, of $GL_{0}(A)$).

Let us prove that $Q$ is contained in $\widetilde{P}(A)$. By the form of the
polar decomposition of the elements in $Q$, it will suffice to show that
$DU_{0}(A)$ is contained in $\widetilde{P}(A)$. In fact, we will show that
$DGL_{0}(A)$ is contained in $\widetilde{P}(A)$. It is easily checked, using
polar decompositions, that $P(A)\cap GL_{0}(A)$ is invariant under similarity,
i.e., it is a normal subgroup of $GL_{0}(A)$. Thus $\widetilde{P}(A)$ is a
closed normal subgroup of $GL_{0}(A)$. By [normal, Theorem 1.1], to show that
$\widetilde{P}(A)$ contains $DGL_{0}(A)$ it suffices to show the equality of
closed two-sided ideals

$\mathrm{Ideal}([A,A])=\mathrm{Ideal}([\widetilde{P}(A),A]).$

Let $I$ be the ideal on the right hand side. It is clearly contained on the
left hand side. Observe that the image of $\widetilde{P}(A)$ in the quotient
$A/I$ is central. Since the invertible positive elements of $A$ map onto the
invertible positive elements of $A/I$, and the latter span $A/I$, it follows
that $A/I$ is commutative. Thus, $\mathrm{Ideal}([A,A])\subseteq I$, as
desired. We thus conclude by [normal, Theorem 1.1] that
$DGL_{0}(A)\subseteq\widetilde{P}(A)$.

For the inclusion $\widetilde{P}(A)\subseteq Q$, it will suffice to show that
$Q$ is a group, as it is a closed set (in $GL_{0}(A)$) and contains all
invertible positive elements. Let $u_{1}a_{1}$ and $u_{2}a_{2}$ be in $Q$,
with $u_{1},u_{2}\in\overline{DU_{0}(A)}$ and $a_{1},a_{2}\in A_{+}$. Then,

$u_{1}a_{1}u_{2}a_{2}=u_{1}u_{2}(u_{2}^{*}a_{1}u_{2})a_{2}.$

Set $a_{1}^{\prime}=u_{2}^{*}a_{1}u_{2}$. Let
$a_{1}^{\prime}a_{2}=w|a_{1}a_{2}|$ be the polar decomposition of
$a_{1}^{\prime}a_{2}$. Then

$u_{1}a_{1}u_{2}a_{2}=u_{1}u_{2}w|a_{1}a_{2}|.$

We will be done once we have shown that $w\in\overline{DU_{0}(A)}$. That is,
we must show that given two invertible positive elements $a,b$, the unitary
$u=ab|ab|^{-1/2}$ (in the polar decomposition of $ab$) belongs to
$\overline{DU_{0}(A)}$. Write $a=e^{c}$ and $b=e^{d}$, with $c,d\in A$
selfadjoint. Define $u_{t}=e^{tc}e^{td}|e^{td}e^{tc}|^{-1/2}$, for
$t\in[0,1]$. This is a path of unitaries connecting 1 to $u$. Let us calculate
its determinant:

$\displaystyle\widetilde{\Delta}_{T}((u_{t})_{t})$
$\displaystyle=\widetilde{\Delta}_{T}((e^{tc})_{t})+\widetilde{\Delta}_{T}((e^{td})_{t})-\widetilde{\Delta}_{T}((|e^{tc}e^{td}|)_{t})$
$\displaystyle=T(c)+T(d)-\widetilde{\Delta}_{T}((|e^{tc}e^{td}|)_{t}).$

Since the path

$t\mapsto|e^{tc}e^{td}|^{2}=e^{td}e^{2tc}e^{td}$

has determinant $2(T(c)+T(d))$, the path $t\mapsto|e^{tc}e^{td}|$ has
determinant $T(c)+T(d)$. Hence, from the above calculation we get
$\widetilde{\Delta}_{T}((u_{t})_{t})=0$. By (the proof of) [delaHarpe-
skandalis, Lemma 3],

$u=\prod_{k=1}^{N}e^{ih_{k}},$

where $h_{k}$ is selfadjoint for all $k$ and
$\sum_{k=1}^{N}h_{k}\in\overline{[A,A]}$. (From the proof of [delaHarpe-
skandalis, Lemma 3], $e^{ih_{k}}=u_{t_{k-1}}^{-1}u_{t_{k}}$ for all $k$, where
$0=t_{0}<t_{1}<\cdots<t_{N}=1$ is a sufficiently fine partition of the
interval $[0,1]$.) Now $\prod_{k=1}^{N}e^{ih_{k}}$ belongs to
$\overline{DU_{0}(A)}$, by [normal, Theorem 6.2 (b)] (alternatively, by the
proof of [delaHarpe-skandalis, Proposition 4]). ∎

In the proof of Theorem 1 we rely on some results by Thomsen from [thomsen].
We isolate these facts in a lemma:

######  Lemma 1.

Let $A$ be a unital C*-algebra. If $U_{0}(A)=\overline{DU_{0}(A)}$ then
$\rho_{A}\colon K_{0}(A)\to\mathrm{Aff}(T(A))$ has dense range. Conversely, if
$\rho_{A}$ has dense range and $A$ has stable rank one, then
$U_{0}(A)=\overline{DU_{0}(A)}$.

###### Proof.

We maintain the notation used in [thomsen, Section 3]. By [thomsen, Theorem
3.2],

(1)
$U_{0}(A)/\overline{DU_{0}(A)}\cong\mathrm{Aff}(T(A))/\overline{\Delta_{1}^{0}(\pi_{1}(U_{0}(A)))},$

where $\Delta_{1}^{0}(\pi_{1}(U_{0}(A))$ is a subgroup of
$\rho_{A}(K_{0}(A))$. Thus, from $U_{0}(A)=\overline{DU_{0}(A)}$ we get that
$\Delta_{1}^{0}(\pi_{1}(U_{0}(A))$ is dense in $\mathrm{Aff}(T(A))$, and a
fortiori, $\rho_{A}$ has dense range. (The map $\Delta_{1}^{0}$ is defined in
terms of the de la Harpe-Skadalis determinant for paths recalled above. More
specifically, given a loop $\alpha$ in $U_{0}(A)$ starting and ending in 1,
one computes $h=\frac{1}{2\pi i}\widetilde{\Delta}_{T}(\alpha)\in
A/\overline{[A,A]}$, and takes the affine function
$\widehat{h}\in\mathrm{Aff}(T(A))$ induced by this element. Then
$\Delta_{1}^{0}([\alpha]):=\widehat{h}$. By the homotopy invariance of the
determinant, and the isomorphism $\pi_{1}(U_{0}^{\infty}(A))\cong K_{0}(A)$
from Bott periodicity, $\Delta_{1}^{0}([\alpha])\in\rho_{A}(K_{0}(A))$.)

Suppose now that $\rho_{A}$ has dense range and $A$ has stable rank one. By
the isomorphism (1), to show that $U_{0}(A)=\overline{DU_{0}(A)}$ it suffices
to show that $\Delta_{1}^{0}(\pi_{1}(U_{0}(A)))=\rho_{A}(K_{0}(A))$. For
C*-algebras of stable rank one, the inclusion of $U_{0}(A)$ in
$U_{0}^{\infty}(A)$ induces an isomorphism of homotopy groups
$\pi_{1}(U_{0}(A))\cong\pi_{1}(U_{0}^{\infty}(A))$. This follows from
combining [rieffel, Corollary 8.6], which says if $A$ has stable rank one then
$C(\mathbb{T},A)$ has connected stable rank at most two, and [rieffel,
Proposition 2.6] applied to $C(\mathbb{T},A)$. Hence,
$\pi_{1}(U_{0}(A))\cong\pi_{1}(U_{0}^{\infty}(A))\cong K_{0}(A)$. The subgroup
$\Delta_{1}^{0}(\pi_{1}(U_{0}(A)))$ is then precisely $\rho_{A}(K_{0}(A))$. ∎

###### Proof of Theorem 1.

(i)$\Rightarrow$(ii). In fact, this implication is already present in [mur-
phil]. Suppose that $A$ has the APFP. By [mur-phil, Proposition 2.7], $A$ has
no finite dimensional representations, and by [mur-phil, Theorem 2.4], it has
stable rank one and trivial $K_{1}$-group. (These are also fairly
straightforward consequences of the APFP.) The density of the range of
$\rho_{A}$ is essentially proven in [mur-phil, Lemma 2.8], but since that
result is stated differently, and in less generality, we include an argument
here. Let $u\in U_{0}(A)$. Then $u$ is the limit of a sequence of elements in
$\widetilde{P}(A)$. By Theorem 2, they have the form $w_{n}a_{n}$ with
$w_{n}\in\overline{DU_{0}(A)}$ and $a_{n}\in A_{+}$. Comparing unitary parts,
we get that $w_{n}\to u$. It follows that $U_{0}(A)=\overline{DU_{0}(A)}$.
Hence, by Lemma 1, $\rho_{A}$ has dense range.

(ii)$\Rightarrow$(i). Since $A$ has stable rank one, it suffices to
approximate invertible elements of $A$ by elements of $P(A)$. By polar
decomposition, every invertible element has the form $ua$, where $u$ is a
unitary and $a\in A_{+}$. By the description of $\widetilde{P}(A)$ from
Theorem 2, we will be done once we have shown that
$U(A)=\overline{DU_{0}(A)}$. Now, by the $K_{1}$-injectivity/surjectivity of
stable rank one C*-algebras ([rieffel, Theorem 2.10]), $U(A)/U_{0}(A)\cong
K_{1}(A)$. Since we have assumed that $K_{1}(A)=0$, we get that
$U(A)=U_{0}(A)$. Thus, it remains to show that
$U_{0}(A)=\overline{DU_{0}(A)}$. This follows from Lemma 1. ∎

For a nonunital C*-algebra $A$, Murphy and Phillips define the approximate
positive factorization property by asking that $1+P(A)$ be dense in $1+A$, in
the unitization of $A$. A different, weaker, property is obtained by simply
asking that $P(A)$ be dense in $A$. We have not pursued the investigation of
these approximate positive factorization properties in the nonunital setting.

## References