text:
State SenatorKevin Kelly
Proudly Serving the 21st District
Legislative Management Committee, Ranking Member
Higher Education & Employment Advancement Committee, Ranking Member
Executive & Legislative Nominations Committee
Regulation Review Committee
State Senator Kevin C. Kelly was elected Senate Republican Leader in 2020. He represents the 21st Senatorial District which includes the towns of Monroe, Seymour, Shelton and Stratford.
Senator Kelly is the Senate Republican Ranking Member of the Higher Education Committee and the Legislative Management Committee. He also serves on the Executive and Legislative Nominations Committee and the Regulations Review Committee. He has previously served as Ranking Member of the Committees on Aging, Children, Human Services, and Insurance & Real Estate.
Senator Kelly is an advocate for common sense solutions to make Connecticut more affordable for middle-class families. He prioritizes environmental stewardship, health care access, and issues pertaining to Connecticut's senior citizens.
Senator Kelly has made an impact in many substantive areas of law dealing with health insurance and hospitals, Medicaid, aging in place initiatives, family caregivers, women's health care, school safety, and consumer protection.
Senator Kelly has worked to improve the lives of seniors by strengthening elder abuse laws and improving aging in place initiatives which encourage independent living with dignity. In 2021 Senator Kelly successfully passed legislation to strengthen protections for nursing home residents and their loved ones in response to the pain and suffering experienced during the height of the COVID-19 pandemic. He also successfully advocated for and passed legislation to increase the personal needs allowance for nursing home residents and to fight age discrimination in the workplace. In 2015, Senator Kelly played an important role the in passage of the CARE Act which requires hospitals to provide caregivers the important information and services they need to care for their loved ones at home upon discharge.
Senator Kelly has worked to increase the access, quality, and affordability of health care in Connecticut. He was central to the enactment of An Act Concerning Life Insurance Producer Licenses which protects insurance consumers from unethical stockbrokers. He led efforts to pass historic legislation that establishes pregnancy as a qualifying life event, so pregnant women can obtain health insurance during their pregnancy and access needed prenatal care.
Senator Kelly has been recognized for his extensive work on health care access and affordability. He was named a 2022 Legislative Champion for Oral Health by the Connecticut Oral Health Initiative in recognition of his work as a champion for oral health access in Connecticut, and in particular his support and success in passing legislation to extend dental coverage to dependents up to age 26 through their family dental plans. He was named the 2021 Champion of the Year by the Greater Connecticut/Western Massachusetts Chapter of the Juvenile Diabetes Research Foundation (JDRF) in recognition of his work on bipartisan legislation addressing insulin affordability. In 2020, Senator Kelly was named a Legislative Champion by the Radiological Society of Connecticut in recognition of his strong support and advocacy for preventive breast cancer screenings and patient protections. He also has received the award for Outstanding Commitment to Quality Health Care and Patient Access from the Radiological Society of Connecticut (RSC) for his work during the 2012 legislative session during which he co-sponsored legislation that focused on improving access to breast cancer screening.
Additionally, Senator Kelly has worked to make Connecticut a more competitive state as it relates to the captive insurance industry, and in 2016 he was recognized as a 2016 Legislative Champion by the Connecticut Captive Insurance Association.
Senator Kelly holds multiple awards from AARP including a national "Capitol Caregiver" award. He also has been recognized with awards from the Connecticut Association of Optometrists, the Bridgeport Regional Business Council, the Connecticut League of Conservation Voters and the Connecticut Marine Trades Association.
Senator Kelly is an elder law attorney and the owner of Kevin Kelly & Associates, PC; a Stratford law firm.
Prior to working in private practice, he worked as an investigator for the Department of Social Services for over 13 years.
Senator Kelly earned his bachelor's degree at Assumption College (B.A. 1982), his master's degree at Fairfield University (M.A. 1985), and his law degree from the University of Connecticut School of Law (J.D. 1997).
He resides in Stratford with his spouse Cindy. Their family includes four children and seven grandchildren.
Awards include:
Legislative Champion for Oral Health (2022) by the Connecticut Oral Health Initiative
2021 Champion of the Year by the Greater Connecticut/Western Massachusetts Chapter of the Juvenile Diabetes Research Foundation (JDRF)
Legislative Champion, Connecticut Captive Insurance Association (2016)
AARP National Capitol Caregiver award
AARP Legislative Achievement Awards (2011, 2012, 2013, 2015 and 2016)
Connecticut Association of Optometrists Legislative Award (2013)
Advocate of School Nursing Award (2013)
Radiological Society of Connecticut's Outstanding Commitment to Quality Health Care and Patient Access Award (2012) recognizing his work on legislation that focused on improving access to breast cancer screening, and Legislative Champion Award (2020) recognizing his advocacy for Preventative Breast Cancer Screenings
The Bridgeport Regional Business Council (BRBC) Regional Legislator Impact Award (2015) for being a consistent voice for a pro jobs agenda
The Connecticut League of Conservation Voters Legislative Leader (2015) in recognition of supporting key environmental issues
Honorary Crew Member Award (2020) from the Connecticut Marine Trades Association for his support businesses along the state's shoreline
State Senator Kevin Kelly


Just uploaded a new version of Automated Serendipity which fixes a couple of things, and I've created version 0.2 of my Odeon Grease Monkey script. It adds code from Matthew Somerville's script, so that pages not covered by the changes I've made will still work in Firefox.
Earlier this week I was having a conversation with a mate of mine regarding the recently Slashdotted Grease Monkey Firefox extension. It went something like.
So I did, and you can find that script on my new Grease Monkey scripts page.
– I don't want to get accused of anything by Odeon lawyers. There are probably more of them than me, and I don't think they have a sense of humour.
My script totally re-writes the layout of the Odeon site pages, however it seems poetic justice that I've been beaten to writing this script by 6 days by the original Odeon site hacker – Matthew Somerville – who just made the original design work!. I would never have thought of that!
I've just uploaded another PHP script, this time it's a binary media aggregation tool – the recently blooming genre of web applications most commonly known as "iPodders". It takes a list of URLs, sniffs out all the MP3s referenced on them and then plops the ones you've not seen yet into a "today" folder ready for perusal.
It's not strictly an RSS reader. You can send it to any web page, playlist file or XML feed and it will just scrape off what it finds quite efficiently.
Just had a Super Group photo call in City of Heroes last night. Hello to all fellow members of Shadows Fall that have wandered, blinking out of Paragon City to my site to see the pics. They're right at the bottom of my City Of Heroes Screenshots folder.
I've just put my PHP-based IRC bot online for download. He spent a few weeks with the residents of #edge learning about video games and sexual uncertainty. Now I feel he is ready to be freed.
I've included the MySQL database structure with the PHP files, but I've left off the quite frankly harrowing information he picked up during his stay at #edge.
I've also just written a fun little dynamically generated image / forum avatar thing. It changes over time, roughly mirroring the UK day/night cycle, except looking uncannily like the landscape from Super Mariokart.


text:
\section{Introduction}
	Over the past several decades, astronomers have identified
	commonalities in clusters of stars ranging from small groups  to
	large galaxies.  Hot stellar systems (HSS) are a class of
	celestial objects consisting  of globular clusters, nuclear
	star clusters, compact ellpitical galaxies, giant elliptical
	galaxies, ultra compact dwarf elliptical galaxies, nuclear
	star clusters and so on. These HSS are very important to
	understand processes such as the formation of stars or
	black holes, evolution of galaxies and so on.
	Indeed, the physical properties of these objects have been
	directly linked to galaxy interactions and have been
	extensively studied over the past many years. One of the most useful
	concepts~\citep{bursteinetal97, bernardietal03,kormendyetal09,
		misgeldandhilker11} in studying these objects is  the set of fundamental
	plane relations~\citep{brosche73}. These planes are
	typically constructed with parameters such as luminosity,
	surface brightness, stellar magnitude or central velocity
	dispersion and help understand important
	properties of these stellar
	systems. Different HSS subgroups typically have different
        fundamental plane relations, photometric properties and other kinds of
        interpretations~\citep{kormendy85, djorgovski95,
          mclaughlinetal05, meylanetal01, forbesetal08, harrisetal95,
          webbink85, ichikawa86, kormendyetal09, benderetal92}. But
        the origins of these relations and the underlying evolutionary
        processes are not well understood. The statistical reliability
        of such results is also unknown. All these issues provide
        significant challenges. A huge volume of data has been
        collected and catalogued in the last decade or so to address
        these questions. For example  \citet{jordanetal08} catalogued
        and studied 12763 	candidate globular clusters (GCs) from
        the Virgo Cluster 	Survey during the eleventh Hubble
        Space Telescope observation
	cycle. A more comprehensive catalogue compiled by~\citet{misgeldandhilker11}  included 693 additional HSS along with those of \citet{jordanetal08}. \citet{chattopadhyayandkarmakar13} pointed out that several studies
	have previously compared stellar systems
	such as globular clusters and dwarf spheroidals using two-point correlations  between different
	projections of the fundamental plane of galaxies which points to the fact that several techniques have been published both statistical and astronomical to analyze this huge volume of data. Given our lack of understanding about the HSS groups, their evolution
	and relationships, an important step forward is a firm understanding of
	which groups are demonstrably different in observational parameter spaces,
	and which may be more closely related. These are statistical
        questions for which we have observations on several variables
        and for which multivariate analysis can provide comprehensive
        answers. To this end, \citet{chattopadhyayandkarmakar13} used
	$k$-means clustering with the jump
	statistic~\citep{sugarandjames03} to find five and four
	homogeneous groups in the 693 non-candidate and the larger
        13456 HSS datasets.

	Clustering~\citep{kettenring06, xuandwunsch09, everitt11}  is
	an  unsupervised learning technique that groups observations
	without a response variable. While there are many kinds of
	clustering algorithms, they are broadly categorized into
	hierarchical and non-hierarchical approaches. Hierarchical
	clustering algorithms result in a tree-like grouping hierarchy
	while non-hierarchical algorithms such as $k$-means or model-based
	clustering (MBC) typically optimize an objective
	function using iterative greedy algorithms -- these algorithms
	typically require a specified number of groups. The objective
	function is often multimodal and requires careful
	initialization \citep{maitra09}. For a detailed review on
	MBC, see, for instance,
	\citet{melnykovandmaitra10}. Even though MBC  is an
	improvement over traditional hierarchical and other
        non-hierarchical approaches to partitioning, 	they still are
        limited to regular structures for each of their groups, for
        example, the most common technique of Gaussian-mixtures
	MBC (GMMBC) assumes spherical or ellipsoidally-dispersed
        groups. Very often, however, the underlying groups in
	a dataset are irregular in shape or structure and do not all
        neatly fit the assumptions
	underlying MBC. Thus advanced methods are necessary to identify
	and analyze irregular shaped groups. \citet{riveraandmaitra20}
	proposed a novel way of identifying complex-structured and
        irregular-shaped groups
	called syncytial clustering which incorporates the results from
	clustering methods that
	find regular-shaped groups to reveal the complex irregular-structured
	groups. Specifically, their approach reveals a multi-layered group
        structure that provides insight into not just the groups but
        also their underlying sub-group structure, which may regular
        structure at some level.  We analyze the 13456 HSS of
        \citet{misgeldandhilker11} that is briefly summarized in
        Section~\ref{data} using syncytial clustering methods in
        (introduced in Section~\ref{method}) with
	initial clustering results obtained using $t$-mixtures
	MBC ($t$MMBC). We analyze the HSS using these methods in
        Section~\ref{cl:analysis}.
	Further, in all such investigations involving astronomical
        data, parameter estimation in statistical clustering
        algorithms is accompanied by uncertainty in those estimates
        and careful 	assessment of these estimates is often
        required to judge their results
	and their impact on the obtained groupings. We analyze this
	uncertainty by using a nonparametric boostrap procedure~\citep{efron79}
	to calculate the confidence of classification of each data point into
	a group. The paper concludes with some discussion about the
        physical properties of the obtained groups and to pointers for
        future work.

	\section{The HSS Dataset}
	\label{data}
	The dataset used in our analysis was compiled from different sources
	by~\citet{misgeldandhilker11} and  contains measurements on 13456 HSS of different types: globular clusters (GCs), giant
	ellipticals (gEs), compact ellipticals (cEs), ultra compact
	dwarf galaxies (UCDs), dwarf globular transition objects
	(DGTOs), nuclear of star clusters (NuScs), bulges of spiral
	galaxies (Sbuls) and nuclei of nucleated dwarf galaxies (dE,Ns). There are
	12763 HSS in this dataset that are  candidate GCs (GC\_VCCs) from the
	Virgo Globular Cluster
	survey meaning that there is ambiguity on whether these systems are GCs or
	not. \citet{chattopadhyayandkarmakar13} primarily excluded these
	candidate GCs from their analysis,
	focusing mainly on the confirmed GCs and other different types of stellar systems. We call these 693 systems non-candidate HSS. The main parameters  in \citet{misgeldandhilker11} are
	stellar mass ($M_s$), effective radius ($R_e$), mass surface
	density averaged over projected effective radius ($S_e$) and
	absolute magnitude in the V band ($M_\nu$). Following
	\cite{chattopadhyayandkarmakar13} we use the logarithm (base 10) of
	$M_s$, $R_e$ and mass-to-luminosity ratio ($M_s / L_\nu$) in
	solar luminosity ($M_{s,\odot}$) units in our analysis. The
	$M_s/L_\nu$ ratio were obtained using the standard magnitude-luminosity
	relation.
	\begin{equation}
		\label{ml:eq}
		\frac{M_s}{M_{s,\odot}} = -2.5\Big(\frac{L_\nu}{L_{\nu,\odot}}\Big)
	\end{equation}
	where $L_{\nu,\odot}$ denotes the luminosity of the sun. The $\log_{10}S_e$ is taken into account while interpreting the results but are not used in the statistical clustering mechanism. Figure~\ref{fig:ggpairs} provides the pairwise scatterplots of the three parameters for both candidate and non-candidate stellar systems along with the density plot of the parameters for both candidates and non-candidates used in clustering.
	\begin{figure}
		\centering
		\includegraphics[width=0.5\textwidth]{ggpairs-hss-crop}
		\caption{Pairwise scatterplots, estimated densities
			and correlation coefficient of the logarithm of the
			parameters in our dataset. Here Ms denotes mass, Re
			effective radius and R the mass-luminosity ratio. In
			the scatterplot, orange indicates the non candidates
			and green the candidates. For the density plots the
			blue curves represent the non candidates. Note that
			the calculated correlations are for all 13456 HSS in
			the dataset (including the non-candidates and
			candidates).}
		\label{fig:ggpairs}
	\end{figure}
	From Figure~\ref{fig:ggpairs} we see that the densities of the
	non-candidates display moderate univariate bimodality, and especially
	for the data on effective radius and mass-luminosity ratio. There is
	evidence of grouping, and certainly so at the univariate level.
	Also, the  correlation values indicate that mass
	and effective radius are  moderately positively-correlated, as
        generally typical for stellar systems. On the other hand,
        mass-luminosity ratio has a weak linear association with both
        mass and effective radius. There is evidence therefore of
        complex structure  in this dataset so         we develop
        methodology in the next section to uncover this complex
        structure and to analyze the HSS dataset and understand the
        implications of our analysis from an astronomical perspective.

	\section{Statistical Methodology}
	\label{method}
	This section briefly describes our methodology for
	analyzing the 13456 HSS. Our approach is a model-based
        analogue of 	the syncytial clustering algorithm of
	\citet{riveraandmaitra20}. Our procedure involves an initial
	clustering of the dataset using $t$MMBC followed by merging of the
	$t$MMBC mixture components using
	pairwise overlap measure and generalized overlap, calculated using
	Monte Carlo methods, as described next.
	\subsection{Initial MBC using a mixture of multivariate-$t$ densities}
	\label{mbc}
	MBC provides a principled way to find homogeneous regular-shaped
	groups in a given dataset. It scores
	over classical clustering algorithms like $k$-means due to its
	ability to better model the heterogenity that may be
	present in the groups. As pointed out by
	\citet{chattopadhyayandmaitra17} the assumption of spherically
	dispersed homogeneous groups when such assumption is not valid
        can yield erroneous results. In
	MBC~\citep{melnykovandmaitra10,mclachlanandpeel00}
	the  observations $X_1,X_2,\ldots,X_n$ are assumed to be
	realizations from a \textit{G}-component  mixture
	model~\citep{mclachlanandpeel00} with probability density function (PDF)
	\begin{equation}
		f(x;\theta) = \sum_{g=1}^{\textit{G}}\pi_g f_g(x; \eta_g)
		\label{mixedmodel}
	\end{equation}
	where $f_g(\cdot; \eta_g)$ is the density of the $g$th group,
	$\eta_g$ the vector of unknown parameters and $\pi_g  = Pr[x_i \in
	\mathcal{G}_g]$ is the
	mixing proportion of the $g$th group, $g=1,2,\ldots,G$,  and
	$\sum_{g=1}^{\textit{G}}\pi_g  = 1$. The component density
	$f_g(\cdot; \eta_g)$ can be chosen according to the specific needs of
	the application -- the most popular choice being the multivariate
	Gaussian density~\citep{chattopadhyayandmaitra17,
		fraleyandraftery98, fraleyandraftery02}. A more flexible family of mixture models proposed by
	\citet{mclachlanandpeel98} specifies $f_g(\cdot;
	\eta_g)$ to be the multivariate-$t$ density. That is,
	\begin{equation}
		\label{mult:t}
		\begin{split}
			f_g(z; &\mu_g    , \Sigma_g, \nu_g)\\
			& = \frac{\Gamma (\nu_g +
				p)/2}{\Gamma(\nu_g/2) {{\nu_g}^{\frac{p}{2}}}{{\pi_g}^{\frac{p}{2}}}{|\Sigma_g|}^\frac{1}{2}}
			\Big[1+ \frac{1}{\nu_g}(z-\mu_g)^T\Sigma_g^{-1}(z-\mu_g)\Big],
		\end{split}
	\end{equation}
	for $z\in\mathbb{R}^p$, where $\mu_g$ denotes the mean vector,
	$\Sigma_g$ the scale matrix and $\nu_g$ the degrees of
	freedom, all for the $g$th mixture component, $g=1,2,\ldots,G$.
	Our analysis uses the more flexible multivariate-$t$ mixture ($t$MM) model
	(instead of the Gaussian mixture model) because the
	multivariate-$t$MM better allows for thicker tails in the
	component mixture densities.
	Subsequent steps in clustering involve obtaining maximum
	likelihood estimates of the parameters
	$\eta_g$, $g = 1, 2,\ldots,G$ using the Expectation Maximization
	algorithm~\citep{dempsteretal77, mclachlanandkrishnan08,
		chattopadhyayandmaitra17, chattopadhyayandmaitra18} and
	assigning each individual observation
	based on the maximum posterior probability that it belongs to
	a given group. We use
	Bayesian Information Criterion (BIC)~\citep{kashyap77,schwarz78,chattopadhyayandmaitra17} to decide
	$G$.

	\subsection{Merging mixture components using overlap}
	\label{merge:full}
	After obtaining the parameter estimates of \eqref{mixedmodel},
	mixture components can be merged using several criterion described in
	\citet{melnykov16}. Here we use the pairwise overlap
	measure~\citep{maitraandmelnykov10,melnykovandmaitra11,melnykovetal12}. The
	pairwise overlap provides us with a sense of the distinctiveness
	between groups obtained using a particular clustering method. These
	values lie inside $[0,1]$ with values closer to 1 indicating that the
	two groups are poorly-separated. We now provide details on how to
	obtain the overlap measure.
	\subsubsection{Pairwise overlap between simple groups}
	\label{ol:simple}
	A simple group~\citep{riveraandmaitra20} is a group from the initial
	classifications obtained, in our case, by $t$MMBC.
	For a $G$ component $t$MM (\ref{mixedmodel}), the probability
        that an observation $X$ originally having a $m$th component
        distribution (alternatively, belonging to the $m$th group) gets misclassified to the $n$th group is
	\begin{equation}
		\omega_{n|m} = P(\pi_nf_n(x; \mu_n, \Sigma_n, \nu_n) > \pi_mf_m(x; \mu_m, \Sigma_m, \nu_m))
		\label{miss:prob}
	\end{equation}
	where $f_g(x; \mu_g, \Sigma_g, \nu_g)$ has density as in~\eqref{mult:t}. The pairwise overlap~\citep{maitraandmelnykov10} between the $m$th and $n$th group is then defined as
	\begin{equation}
		\omega_{mn} = \omega_{n|m} + \omega_{m|n}
		\label{overlap:form}
	\end{equation}
	Unlike in the case of Gaussian mixture models, the misclassification
	probability $\omega_{n|m}$ can not be readily calculated using
	analytical methods, so we use Monte Carlo methods as follows:
	\begin{enumerate}
		\item Generate N samples $x_i^*$, $i=1,2,\ldots N$ from $f_m( ;\mu_m, \Sigma_m, \nu_m)$.
		\item $\omega_{n|m}$ can then be approximated as
		\begin{equation}
			\hat{\omega}_{n|m} = \frac{1}{N}\sum_{i=1}^{N}I\{\pi_nf_n(x_i^*; \mu_n, \Sigma_n, \nu_n) > \pi_mf_m(x_i^*; \mu_m, \Sigma_m, \nu_m)\}
			\label{miss:prob:est}
		\end{equation}
		where $I(\cdot)$ is the indicator function, in this case takng 1 if $\pi_nf_n(x_i^*; \mu_n, \Sigma_n, \nu_n) > \pi_mf_m(x_i^*; \mu_m, \Sigma_m, \nu_m)$ and 0 otherwise.
	\end{enumerate}

	The pairwise overlap provides us with a sense of the distinctiveness
	between groups obtained using a particular clustering method. These
	values lie inside $[0,1]$ with values closer to 1 indicating that the
	two groups are poorly-separated. \citet{melnykovandmaitra11} used
	methods developed in \citet{maitra10} to define the generalized
	overlap measure as a measure $\ddot{\omega}$ to summarize the $G
	\times G$ matrix $\Omega$ of pairwise overlaps with $\ddot{\omega}=\frac{\lambda_{\Omega}(1)-1}{G-1}$ where $\lambda_{\Omega}(1)$ is the
	largest eigen value of $\Omega$. Typically smaller (larger) values of
	$\ddot{\omega}$ indicate more (less) distinctive groupings.

	\subsubsection{Pairwise overlap between composite groups}
	\label{ol:comp}
	Let $\mathcal{G}_p$ and $\mathcal{G}_q$ be two composite groups formed by merging one or more components of \eqref{mixedmodel} having probability densities $\sum_{h \epsilon \mathcal{G}_p }\pi_hf_h(x; \mu_h, \Sigma_h, \nu_h)$ and $\sum_{r \epsilon \mathcal{G}_q }\pi_rf_h(x; \mu_r, \Sigma_r, \nu_r)$ respectively. The probability that an observation $X$ originally belonging to $\mathcal{G}_p$ getting misclassified to $\mathcal{G}_q$ is
	\begin{equation}
		\resizebox{0.45\textwidth}{!}{$\omega_{\mathcal{G}_q|\mathcal{G}_p} = P\Bigg(\frac{\sum_{r \epsilon \mathcal{G}_q }\pi_rf_r(x; \mu_r, \Sigma_r, \nu_r)}{\sum_{h \epsilon \mathcal{G}_p }\pi_hf_h(x; \mu_h, \Sigma_h, \nu_h)} > 1\Bigg)$}
		\label{miss:prob:comp}
	\end{equation}
	where $f_g(x; \mu_g, \Sigma_g, \nu_g)$ has density given by \eqref{mult:t}.
	The pairwise overlap between $\mathcal{G}_p$ and $\mathcal{G}_q$,
	denoted by $\omega_{\mathcal{G}_p \mathcal{G}_q}$, is then computed as $\omega_{\mathcal{G}_q|\mathcal{G}_p} + \omega_{\mathcal{G}_p|\mathcal{G}_q}$. The misclassification probability $\omega_{\mathcal{G}_q|\mathcal{G}_p}$ can be estimated as follows:
	\begin{enumerate}
		\item Generate N samples $x_i^*$, $i=1,2,\ldots N$ from $\sum_{h \epsilon \mathcal{G}_p }\pi_h^*f_r(x; \mu_h, \Sigma_h, \nu_h)$,  where $\pi_h^*$ is a standardized proability obtained as $\pi_h^* = \pi_h/\sum_{h \epsilon \mathcal{G}_p } \pi_h$
		\item$\omega_{\mathcal{G}_q|\mathcal{G}_p}$ can then be approximated as
		\begin{equation}
			\resizebox{0.45\textwidth}{!}{$\hat{\omega}_{\mathcal{G}_q|\mathcal{G}_p} = \sum_{i=1}^NI\Big\{\frac{\sum_{r \epsilon \mathcal{G}_q }\pi_rf_r(x_i^*; \mu_r, \Sigma_r, \nu_r)}{\sum_{h \epsilon \mathcal{G}_p}\pi_hf_h(x_i^*; \mu_h, \Sigma_h, \nu_h)} > 1\Big\}$}
			\label{miss:prob:est:comp}
		\end{equation}
		where $I(\cdot)$ is the indicator function that takes 1 if \resizebox{0.4\textwidth}{!}{$\sum_{r \epsilon \mathcal{G}_q }\pi_rf_r(x_i^*; \mu_r, \Sigma_r, \nu_r) >
			\sum_{h \epsilon \mathcal{G}_p}\pi_hf_h(x_i^*; \mu_h, \Sigma_h, \nu_h)$} and 0 otherwise.
	\end{enumerate}
	\subsection{The model-based syncytial clustering algorithm}
	Our syncytial clustering algorithm consists of three phases, the $t$MMBC phase, the initial overlap calculation phase and the merging phase. Each of the three phases are described below.
	\begin{enumerate}
		\item \textit{The initial clustering phase}: This phase fits
		a $G$-component $t$MM to the data using the EM
		algorithm~\citep{mclachlanandpeel00}, with $G$ chosen using
		BIC~\citep{schwarz78}.
		\item \textit{Merging Phase}: This phase is triggered only
		if the generalized overlap $\ddot{\omega}^{(1)}$ of the
		original clustering solution is not negligible, that is if
		$\ddot{\omega}^{(1)} \not\approx 0$ and at least one pairwise
		overlap of the original clustering solution is greater than
		$\ddot{\omega}^{(1)}$. Specifically this phase involves the
		following steps:
		\begin{enumerate}
			\item Find the pairwise overlaps between $i$th and $j$th group $\omega_{ij}$ for the $G(G-1)/2$ pairs of groups using the method described in Section \ref{ol:simple}. Also find the generalized overlap $\ddot{\omega}$.
			\item Merge the $i$th and $j$th groups if
                          $\omega_{ij} > \kappa\ddot{\omega}$ and
                          relabel the newly-formed (merged) composite groups.
			\item Calculate the updated pairwise overlaps of the composite groups using the method described in Section \ref{ol:comp}. Also calculate the generalized overlap  of the new clustering solution. Repeat Step (b) with the updated pairwise overlaps and the generalized overlap.
		\end{enumerate}
		\item\textit{Termination Phase}: The algorithm
                  terminates in Step (ii) if
		the pairwise overlap of the  clustering solution in
                the current phase
		$\ddot{\omega}^{(l)} \approx \ddot{\omega}^{(l-1)}$ or $\ddot{\omega}^{(l)}$ is negligible (in this paper defined to be  $\leq 10^{-3}$) where $\ddot{\omega}^{(l-1)}$ is the generalized overlap of the previous phase clustering solution.
	\end{enumerate}
	\paragraph*{Selection of $\kappa$}: The parameter $\kappa$ determines the
	speed of merging and therefore the characteristic of the composite
	groups formed at each merging phase. With larger values of $\kappa$,
	few pairs merge at each phase while smaller values of $\kappa$ mean
	that many components get merged simultaneously at each phase. A
	data-driven approach proposed by \citet{riveraandmaitra20} for
	selecting $\kappa$, that we also adopt, runs the algorithm for several
	values of $\kappa$ and uses the final clustering solution with the
	smallest terminating $\ddot\omega$.
	\paragraph*{Choice of Mixture model}: The description of our algorithm
	uses $t$MMBC in the initial clustering phase, however, our algorithm
	applies to other mixture models with appropriate
	modifications. Indeed, in our paper, we have used both GMMBC and
	$t$MMBC and then decided on the initial clustering solution with the
	higher BIC.
	\subsection{An Illustrative Example}
	We illustrate our syncytial clustering algorithm on the synthetic
	 2D {\em Bananas-Arcs} dataset in~\citet{riveraandmaitra20}. From
	Figure~\ref{fig:illus} it is evident that this data has four complex
	clusters. We fit both GMMBC and $t$MMBC to this dataset using
	the methods in~\citet{chattopadhyayandmaitra17}
	and~\citet{chattopadhyayandmaitra18} and selected the best clustering
	solution as well as the number of groups using BIC. For this dataset,
	GMMBC with 15 groups provided the best solution
	(Figure~\ref{fig:illus:em}) with a generalized overlap of $8.2\times 10^{-3}$.
	The first of the merging phases (with the best $\kappa=3$) provided
	the five clusters of Figure~\ref{fig:illus:ph1} with a generalized
	overlap of $1.4\times 10^{-4}$. Therefore this merged solution provides a more
	distinctive grouping than the Gaussian mixture model solution.
	The second phase of merging resulted in the four groups of
	Figure~\ref{fig:illus:ph2} with a generalized overlap of
	$1.368\times 10^{-4}$. Since the generalized overlap of this phase is very close
	to that at the end of the first phase of merging, the merging
	terminates at this phase. This example illustrates the
	effectiveness of syncytial clustering over regular methods in
	capturing complex-structure and irregular-shaped
	clusters. Additionally, the multi-staged solution provides us with the
	ability to understand each of these complex structures in terms of
	their component (simpler) homogeneous groups. Our data-driven approach
	results in a terminating $\ddot\omega$ that is much smaller than the
	original MBC solution indicating a preference for the
	complex-structured solution in providing us with distinctive
	groups. We now analyze the HSS dataset using our syncytial clustering
	algorithm.

	\begin{figure}
		\mbox{
			\subfloat[]{\label{fig:illus:em}\includegraphics[width=0.33\columnwidth]{ba-emclus-notext-crop}}
			\subfloat[]{\label{fig:illus:ph1}\includegraphics[width=0.33\columnwidth]{ba-int-notext-crop}}
			\subfloat[]{\label{fig:illus:ph2}\includegraphics[width=0.33\columnwidth]{ba-final-notext-crop}}
		}
		\caption{From left to right: (a) Original Gaussian mixture model clustering solution according to BIC. (b) Five clusters obtained after first phase merging. (c) Final clustering solution obtained after second phase merging.}
		\label{fig:illus}
	\end{figure}
	\section{Characterization of HSS}
	\label{cl:analysis}
	\subsection{Cluster Analysis}
	\subsubsection{The initial clustering phase}
	\label{init:clus}
	We performed $G$-component $t$MMBC, for $G \in \{1, 2,\ldots,9\}$ on $M_s$, $R_e$ and the
	$M_s/L_\nu$, all on the $\log_{10}$-scale, of the 13456 HSS.
	Figure \ref{fig:bic-hss} indicates that a eight-component $t$MMBC
	provides a optimal fit as per BIC. The 8-component $t$MMBC solution
	was better, as per BIC, than the optimal GMMBC solution. Our results
	here are different from that of \citet{chattopadhyayandkarmakar13}
	who, upon using $k$-means with the Jump
	statistic~\citep{sugarandjames03}, found five groups when excluding
	the candidate GCs of \citet{jordanetal08} and four
	groups on the entire HSS dataset.
	\begin{figure}
		\centering
		\includegraphics[width=\columnwidth]{bic-hss-final-half-crop}
		\caption{BIC for each $G$ upon performing $G$-component $t$MMBC with 13456 HSS from \citet{misgeldandhilker11}.}
		\label{fig:bic-hss}
	\end{figure}

	\begin{figure}
		\centering
		\includegraphics[width=\columnwidth]{ol_or_crop}
		\caption{Pairwise overlap measures between any two
			groups obtained by our eight-component $t$MMBC
			solutions.}
		\label{fig:hss-overlap}
	\end{figure}

	\begin{figure*}
		\mbox{
			\centering
			\fbox{\subfloat[]{\label{fig:full-pos1}\includegraphics[height=0.455\textwidth]{or_pos1_crop}}}
			\fbox{\subfloat[]{\label{fig:full-pos2}\includegraphics[width=0.43\textwidth]{or_pos2_crop}}}
		}
		\mbox{
			\fbox{\subfloat[]{\label{fig:full-pos3}\includegraphics[width=\textwidth]{or_pos3_crop}}}
		}
		\caption{Three viewing angles of the scatterplot of the full dataset with different colors representing different groups and intensity of the color signifying the underlying confidence in that particular grouping. Darker shades indicate  higher confidence of classification of that particular object.}
		\label{fig:scatter:all}
	\end{figure*}
	Figure \ref{fig:hss-overlap} shows the pairwise overlaps of the eight
	groups obtained using $t$MMBC. A 3D scatterplot of the eight $t$MMBC
	groups is also provided in Figure \ref{fig:scatter:all}. Since $t$MMBC
	assumes ellipsoidally dispersed groups (with fatter tails) some of the
	groups exhibit substantial overlap (Figure
	\ref{fig:hss-overlap}). These large overlap values points to the
	possibility of complex group structure in the dataset that is not
	accounted for using $t$MMBC or GMMBC  or $k$-means of
        \citet{chattopadhyayandkarmakar13}. We therefore explore if we
        can employ our algorithm to reveal this complex structure.
	\subsubsection{The merging phases}
	\paragraph{The first phase:}
	\label{ph1:res}
	Based on the pairwise overplap map of Figure~\ref{fig:hss-overlap}, and
	for $\kappa=1$ (determined to be the solution with the lowest
	terminating $\ddot\omega$), Groups 1-4, 7 and 8 all merge into one
	group at this stage. Thus, at the end of this merging step, we end up
	with three groups and a generalized overlap of $\ddot\omega =0.13$. To
	facilitate easy reference, we relabel the erstwhile Groups 5 and 6 as
	the merged (first phase) Groups (i)-5 and (ii)-6 respectively and the
	merged entity as Group (iii), with the component individual groups as
	Group (iii)-1 through Group (iii)-4 and Group (iii)-7 and Group
	(iii)-8. A 3D scatterplot of the three first-phase merged groups is
	provided in Figure~\ref{fig:scatter:ph1}. The pairwise overlap between
	Groups (i) and (ii) is 0.01 and between Groups (ii) and (iii) is
	0.1. Between Groups (i) and (iii) the pairwise overlap is negligible. Figure
	\ref{fig:scatter:ph1} immediately points out how the first
        phase mergers using the pairwise overlaps have been able to capture the
	non-ellipsoidal structure present in the data. But the
        magnitude of the pairwise
	overlaps and the generalized overlap (and the logic of our method)
	suggest that there might be additional structure which might have
	been missed in the first phase. So we proceed with another
	merging phase.

	\begin{figure*}
		\mbox{
			\centering
			\fbox{\subfloat[]{\label{fig:mj1-ph1}\includegraphics[height=0.32\textwidth]{or_ph1_pos1_f_crop}}}
			\fbox{\subfloat[]{\label{fig:mj1-ph2}\includegraphics[width=0.615\textwidth]{or_ph1_pos2_f_crop}}}
		}
		\caption{Two viewing angles of the scatterplot of the full dataset after first merge with different colors representing different groups and intensity of the color signifying the underlying confidence in that particular grouping. Darker shades indicate  higher confidence of classification of that particular object.}
		\label{fig:scatter:ph1}
	\end{figure*}

	\begin{table}
		\centering
		\begin{tabular}{rrrrrr} \hline\hline
			Group & $M_s$ & $R_e$ & $M_{\nu}$ & $M_\nu/L_\nu$ & $S_e$\\
			\hline
			{\color{Grph1} (i)-5} & 8.817 & 3.032 & -16.049 & 0.465 & 1.954\\
			{\color{Grph2} (ii)-6} & 5.803 & 0.991 & -9.711 & -0.013 & 3.023\\
			{\color{Grph3} (iii)} & 5.404 & 0.438 & -7.557 & 0.450 & 3.731\\
			\hline
		\end{tabular}
		\caption{Mean parameter values for each of the three groups after phase one merging. The third group is the merged group.}
		\label{hss:mean:ph1}
	\end{table}

	\begin{table}
		\centering
		\begin{tabular}{rrrrr} \hline\hline
			& Group {\color{Grph1} (i)-5} & Group {\color{Grph2} (ii)-6} &  Group {\color{Grph3} (iii)}\\ \hline
			cE & 6 &  & 1 \\
			dE & 212 &  &  \\
			dE,N &  & 15 & 30  \\
			DGTO & 4 &   & 1 \\
			Dwarf & 22 & 2 &  \\
			GC & 2 & 58 & 94 \\
			GC\_VCC &  & 10 & 12753\\
			gE & 150 &  &   \\
			NuSc & 1 & 12 & 9 \\
			Sbul & 18 &  &  \\
			UCD & 3 & 30 & 23 \\ \hline\hline
			Total & 418 & 127 & 12911\\ \hline

		\end{tabular}
		\caption{Types of objects in each of the three groups after phase one merge.}
		\label{hss:no:type:ph1}
	\end{table}

	\paragraph{The second phase:}
	In the second phase, Groups (ii)-5 and (iii) merge yielding two
	complex-structured groups.  We label the
	newly-merged group as
	Group II -- with component merged sub-groups as Groups II-(ii) and
	II-(iii), and correspondingly down the hierarchy -- and the original
	group as Group I (with sub-groups classification of Group
        I-(i) and further down Group I-(i)-5, to indicate the
        three-stages of classifications) . A 3D
	scatterplot of the  two new groups is given in Figure
	\ref{fig:scatter:ph2}.  The scatterplot clearly depicts that the
	second merge has done a good job
	in capturing the general-shaped group structures in the data and
	captured to a good extent the two groups that are also visually
	supported by the scatterplot.  The pairwise overlap between these two
	groups is 0.0013. Since the pairwise overlap between Groups I and II is very close to $10^{-3}$ the algorithm terminates at this stage.

	\subsection{Measuring uncertainty in the obtained groupings through bootstrapping}
	\label{uncer:hss}
	We assess the uncertainty in the parameter estimates in each phase and the
	corresponding classification using a nonparametric bootstrap
	technique where we  repeatedly resample from the dataset with
	replacement and each resample is analyzed to obtain estimates
	of the parameters of  interest.  For a detailed description of
        bootstrap theory and methodology, refer to~\citet{efron79}.

	In our scenario, the parameters of interest are the group
	memberships and a stable cluster analysis is expected to produce
	the same group memberships for most of the observations for each
	bootstrap replication. The following procedure  describes our
	steps for obtaining the confidence of classification for each
	observation for the original clustering solution using $t$MMBC.
	\begin{enumerate}
		\item[(i)] First we perform $t$MMBC on the original
		13456 HSS as described in Section \ref{data}.  This matches the analysis so far and yields 8 $t$MMBC groups.

		\item[(ii)] Sample 13456 HSS from the data with
                  replacement and again cluster the sampled data using
                  the classifications obtain in step (i) as the
                  initial estimates for the groupings. Repeat this
                  procedure B times to obtain B sets of classification
                  estimates. Let us denote these B sets by $C_1,
                  C_2,\ldots,C_{B}$. Also let $C_{ij}$ be 1 if the
                  classification of the $j$th observation from the
                  $i$th bootstrap replication is the same as the
                  original (dataset) classification of the $j$th
                  observation and 0 otherwise, where $i = 1, 2,\ldots
                  B$ and $j = 1, 2,\ldots,13456$.\item[(iii)] The
                  boostrap-estimated classification probability for the $j$th observation is
		\begin{equation}
			\label{class:prob}
			p_j = \frac{1}{B}\sum_{i=1}^{B} C_{ij}
		\end{equation}
		where $j = 1, 2,\ldots,13456$. In our analysis, we
		took $B=1000$, that is, we resampled 1000 times from
		the dataset.
	\end{enumerate}
	In each merging phase, the confidence of
	classification is obtained by relabeling each of the
	$B$ bootstrap replicates to the solution obtained from
	the original                clustering solution and
	then repeating the steps
	(i)-(iii) with the relabelled bootstrap replicates.
	Tables \ref{tab:ph1:conf} and \ref{tab:ph2:conf} summarize the
        frequencies of HSS having classification confidence in each of seven
	intervals (0, 0.30], (0.30, 0.60], (0.60, 0.85],
	(0.85, 0.90], (0.90, 0.95] and (0.95. 1].  The fact
	that majority of HSS have a high confidence of
	classification in both the stages indicates that we
	have been able to achieve stable  clustering results for
	the 13456 HSS.

	We end our discussion in this section by noting that we
	could have also obtained an estimate of the confidence
	of classification based on the posterior probability of
	classification. However, we use a nonparametric bootstrap
	technique for estimating the classification probabilities in
	order to account for the uncertainty in the modeling and to
	guard against possible model misspecification. The choice of
	the nonparametric bootstrap technique provides us
	with more robust estimates of the confidence of classification
	compared to using posterior probability because it accounts
	for modeling errors to be included in the calculation of the
	classification confidence.

	\begin{table*}
		\label{tab:conf}
		\mbox{\subfloat[Number of candidates (C) and non-candidates (NC) in having confidences in each of the seven intervals after first merging phase.]{\label{tab:ph1:conf}{
					\centering
					\begin{tabular}{ccccccccccccccccc}
						\hline\hline
						Group & \multicolumn{2}{c}{(0, 0.30]} & \multicolumn{2}{c}{(0.30, 0.60]} & \multicolumn{2}{c}{(0.60, 0.80]} & \multicolumn{2}{c}{(0.80, 0.85]} & \multicolumn{2}{c}{(0.85, 0.90]} & \multicolumn{2}{c}{(0.90, 0.95]} & \multicolumn{2}{c}{(0.95, 1]} \\
						& C & NC & C & NC & C & NC & C & NC & C & NC & C & NC & C & NC\\\hline
						{\color{Grph1} (i)-5} &  &  &  &  & 1 &  &  &  & 1 &  & 1 &  & 415 & \\
						{\color{Grph2} (ii)-6} &  &  & 8 & 5 & 13 & 2 & 11 & 1 & 28 &  & 12 &  & 46 & 1\\
						{\color{Grph3} (iii)} &  &  & 5 & 4 & 7 & 3 & 1 & 3 & 3 & 3 & 4 & 3 & 138 & 12737\\\hline
					\end{tabular}
				}
			}
		}

		\mbox{\subfloat[Number of candidates (C) and non-candidates (NC) in having confidences in each of the seven intervals after the second merging phase.]{\label{tab:ph2:conf}{
					\centering
					\begin{tabular}{ccccccccccccccccc}
						\hline\hline
						Group & \multicolumn{2}{c}{(0, 0.30]} & \multicolumn{2}{c}{(0.30, 0.60]} & \multicolumn{2}{c}{(0.60, 0.80]} & \multicolumn{2}{c}{(0.80, 0.85]} & \multicolumn{2}{c}{(0.85, 0.90]} & \multicolumn{2}{c}{(0.90, 0.95]} & \multicolumn{2}{c}{(0.95, 1]} \\
						& C & NC & C & NC & C & NC & C & NC & C & NC & C & NC & C & NC\\\hline
						{\color{Grph1} I} &  &  &  &  & 1 &  &  &  & 1 &  & 1 &  & 415 & \\
						{\color{Grph3} II} &  &  & 13 & 20 & 5 & 12 & 4 & 30 & 4 & 16 & 3 & 3 & 184 & 12738\\\hline
					\end{tabular}
				}
			}
		}

		\caption{Number of candidate HSS (C) and
			non-candidate HSS (NC) having confidences in each of the seven
			intervals for each of the (a) three groups after first phase of merging and (b) two groups after the second merging phase. Here $(\alpha, \beta]$
			denotes the interval with left endpoint $\alpha$ (not included)
			and  right endpoint $\beta$ (included). Entries in the table are left blank when there are no members in that group.}

	\end{table*}

	\begin{figure*}
		\mbox{
			\centering
			\fbox{\subfloat[]{\label{fig:mj2-ph1}\includegraphics[height=0.39\textwidth]{or_ph2_pos1_f_crop}}}
			\fbox{\subfloat[]{\label{fig:mj2-ph2}\includegraphics[width=0.55\textwidth]{or_ph2_pos2_f_crop}}}
		}
		\caption{Two viewing angles of the scatterplot of the full dataset after third merge with different colors representing different groups and intensity of the color signifying the underlying confidence in that particular grouping. Darker shades indicate  higher confidence of classification of that particular object.}
		\label{fig:scatter:ph2}
	\end{figure*}

	\begin{table}
		\centering
		\begin{tabular}{rrrrrr} \hline\hline
			Group & $M_s$ & $R_e$  & $M_{\nu}$& $M_\nu/L_\nu$ & $S_e$\\
			\hline
			{\color{Grph1} I} & 8.617 & 3.032 & -16.049 & 0.465 & 1.954\\
			{\color{Grph3} II} & 5.408 & 0.443 & -7.577 &  0.445 & 3.724\\
			\hline
		\end{tabular}
		\caption{Mean parameter values for each of the two groups after the second phase merging. The second group is the newly merged group.}
		\label{hss:mean:ph2}
	\end{table}

	\begin{table}
		\centering
		\begin{tabular}{rrrrr} \hline\hline
			& Group {\color{Grph1} I} & Group {\color{Grph3} II} \\ \hline
			cE & 6 & 1\\
			dE & 212 &   \\
			dE,N &  & 45  \\
			DGTO & 4 & 1  \\
			Dwarf & 22 & 2 \\
			GC & 2 & 152 \\
			GC\_VCC & 0 & 12763 \\
			gE & 150 & \\
			NuSc & 1 & 21 \\
			Sbul & 18 &  \\
			UCD & 3 & 53 \\ \hline\hline
			Total & 418 & 13038 \\ \hline
		\end{tabular}
		\caption{Types of objects in each of the two groups after the second phase merging.}
		\label{hss:no:type:ph2}
	\end{table}

	\subsection{Analysis of results}
	\subsubsection{Properties of Identified Groups}
	Tables \ref{hss:no:type:ph1} and \ref{hss:no:type:ph2} give
        the number of different type of HSSs in each group at each
        stage of the merge. The mean value of the parameters for each
        group in each of the merging phases is given in Tables
        \ref{hss:mean:ph1} and \ref{hss:mean:ph2}. Additionally
        different colors in scatterplots \ref{fig:scatter:all},
        \ref{fig:scatter:ph1} and \ref{fig:scatter:ph2} indicate the
        group to which each observation belong with different shades
        indicating the confidence of that particular
        observation. Darker shades indicate higher confidences for
        that particular HSS. From the scatterplots and Tables
        \ref{tab:ph1:conf} and \ref{tab:ph2:conf} it is clear that
        most of the HSS have high confidences indicating that the
        groupings with ellipsoidal structure has good support in the
        dataset and certainly more so than in the initial clustering
        obtained using $t$MMBC. Notably, the first merging phase
        results in Groups (i)-5 and (iii) which contains most of the
        HSS. Most of the GC\_VCCs are in  Group (iii) after the first
        merge and only ten in Group (ii)-6. The stellar systems in
        Group (ii)-6 have lower confidence compared to the other two
        groups due to which inferences about the HSS in this group
        will be more uncertain compared to the stellar systems in
        other groups.  From Table \ref{hss:no:type:ph2} we see that
        after the second phase all the candidates (GC\_VCCs) were
        classified to Group II. Apart from this all the GCs also got
        classified to this group along with a good number of UCDs and
        dE,Ns. This along with the group means can help us infer about
        the kinematic properties of these stellar systems.  After the
        final merging step, we see that Group I HSSs have a larger
        stellar mass compared to the HSSs in         Group II. On the other hand, the effective
        magnitude is also substantially larger for HSSs in the second
        group. The mass over luminosity ratio is quite similar for
        both groups but the effective radius is higher for the
        second group compared to the first group.
	\subsubsection{Interpretations}
	Here we understand the physical and evolutionary properties of
        the objects in each of the two groups after the final merge
        using stellar mass surface density and absolute magnitude . To
        analyze the two complex-structured groups at the end of the
        final merging step, it is important that we carefully look at
        the group structures at each merging step to obtain deeper
        insight into the evolutionary and physical         properties
        of these celestial objects.

	Group I at the end of final merge is essentially Group 5 of
        the $t$MMBC solution.  From Table \ref{hss:mean:ph2} we see
        that the stellar systems in this group have higher mass and
        effective radius compared to the second group. The
        mass-luminosity ratio for this group is slightly higher
        compared to the second group which indicates that the stellar
        systems in this group have probably lost most of their massive
        stars and are mostly composed of low mass stars which further
        indicates that these stellar systems  are typically older
        compared to that of second group. Also since the ratio is much
        less than thirty there is not strong evidence of dark matter
        in the stellar systems in this group. Since this group is
        similar to Group 5 from the $t$MMBC solution and did not
        change during the merging phases so there is no complex
        structure present in this group as compared to Group II.  The
        conclusions for this group may be a little uncertain for one
        HSS that is seen to have low confidence of classification.

	Group II is much larger in size than that of Group I and
        contains all the GC\_VCCs and GCs. Apart from this all the
        dE,Ns and most of the UCDs belong to this group. Quite a few
        objects in this group have low confidence of classification
        mainly because this group is formed by merging Groups (ii)-6
        and (iii). Because of this, it is better to analyze  this group
        and understand its properties by separately looking at the
        properties of Groups (ii)-6 and         (iii).

	For Group (iii) at the end of phase one it can be inferred
        that the GC\_VCCs in this group (and also Group (ii)-6) are
        typically smaller HSSs compared to that of Group I (or Group
        (i)-5). Also the stellar systems in this groups have a higher
        surface density compared to the other groups formed after
        phase one which indicates that these objects are typically
        brighter compared to objects in the first group. Similar to
        Group I there is little evidence of dark matter in these HSSs
        since the mass-luminosity ratio is much less than
        thirty. These inferences are a little uncertain for the
        sixteen HSSs that have a low confidence of classification.

	The stellar systems in Group (ii)-6 have a slightly higher
        mass compared to Group (iii). The effective radius is also
        higher for the HSSs in this group and indicates that these
        stellar systems  are larger compared to the objects in Group
        (iii). The mass-luminosity ratio is substantially lower than
        the other two groups from the phase one merging which indicates that these HSSs are relatively younger compared to other stellar systems. A marginally low surface density compared to that of Group (iii) indicates that these stellar systems are not as bright as those in Group (iii) although these objects are much brighter compared to those of Group (i)-5. Finally the low mass-luminosity ratio indicates that there is negligible evidence of dark matter for the objects in this group. These inferences are highly uncertain since a good number of stellar systems in this group have a low confidence of classification.

	\section{Conclusions}
	\citet{chattopadhyayandkarmakar13} performed statistical cluster analysis on 693
	HSS from \citet{misgeldandhilker11} to determine the homogeneous
	groups present in the data. Using the popular $k$-means clustering
	algorithm and the jump statistic for selection of optimal
	number of groups they arrived at five optimal groups whose
	properties were explored using fundamental plane relations and
	other physical parameters. In their main analysis they
	excluded the candidate GCs (GC\_VCCs), claiming that the candidate
	GCs would render the data unfit for clustering. Their analysis hinges on the homogeneous
	spherically-dispersed-groups assumption that underlies
	the $k$-means algorithm, which may not be appropriate and can
	lead to erroneous results when there is a complex structure present in the underlying statistical groups found through our analysis . Our analysis initially grouped all 13456 HSS of
	\citet{misgeldandhilker11}
	using $t$MMBC and used BIC to optimally find eight groups. Motivated from a syncytial clustering technique proposed by \citet{riveraandmaitra20} we merged certain components of the $t$MMBC solution which revealed two complex groups. Using a nonparametric boostrap technique we further determined the confidence of classification of each of the 13456 HSS in each stage of merging to understand the quality of our results and also to ascertain how correctly the stellar systems have been assigned to their correct groups. We then studied the physical and evolutionary properties of the objects in each of the two groups by analyzing their mean stellar mass, surface density and effective radius.

        Out findings show that our 	Group I (or Group 5 from
        the original $t$MMBC solution) consists of typically large and
        old stellar systems with very small evidence of the presence
        of dark matter given their small $M_\nu/L_\nu$ ratio. On the
        other hand, Group II formed by merging Groups 1-4, 6-8 of the
        original $t$MMBC solution consists of younger HSSs
        that are brighter than those of Group I and also smaller in
        size compared to those in Group I. All the GC\_VCCs got
        classified to this group along with all the GCs and some other
        stellar systems.
	These results point to the fact that the GCs are indeed
        different than elliptical galaxies including dEs . These
        results support assertions or conclusions of several authors
        from the early 1980s onwards~\citep{kormendy85, djorgovski95,
          mclaughlinetal05, meylanetal01, forbesetal08, harrisetal95,
          webbink85, ichikawa86, kormendyetal09, benderetal92}. Apart from this, our
        results confirm that the GC\_VCCs are most likely to be GCs
        and there is very less chance that these will be dwarfs.

	There are a number of issues that merit further
	investigation. For example it would be useful to explore if
	the logarithmic transformation used on the parameters is
	plausible or is obfuscating the clusters. There some other
	transformation that might lead to better separated
	groups. This analysis might also be extended to large and more comprehensive data sets than in this study to analyze whether similar results hold for the
	HSS for the different datasets from different sources. Also
	with the availability of information like temperature and
	color indices of the stars present in the system it might be
	possible to obtain  deeper understanding of the evolution of
	these systems possibly through HR diagrams. Apart from this it would also be interesting to analyze how fundamental plane relations hold in conjunction to these results. Thus we see that while our analysis has provided some interesting insight into
	the different types of HSS, additional issues remain that
	deserve further attention.

	\section*{Data Availability Statement}
	The data underlying this article are available as Supplementary Data at   https://doi.org/10.1111/j.1365-2966.2011.18669.x.

	\section*{Acknowledgements}
	We sincerely thank Professor Tanuka Chattopadhyay for providing us with the dataset of \citet{chattopadhyayandkarmakar13}. We also are very grateful to C. Struck whose comments on an earlier draft of this manuscript greatly improved its content.

	\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


When cash flow is restricted, the choice between paying the bills and building a business becomes stark. Growth for individual healthcare businesses can be stymied even in the midst of growth for the entire health industry.
And the opportunity is there. Healthcare is the largest -- and fastest growing -- industry in the United States. Nearly 20% of the economy is devoted to healthcare (1) and one in eight Americans work in the industry (2). By 2024, healthcare will be the largest employer in the country and will create 80% of all new jobs (2).
Despite the size and strength of the industry, individual healthcare businesses often struggle to finance the investments needed to accommodate this growth while also staying current with staff and suppliers. Expenditures on electronic health records and other requirements of the Affordable Care Act have constrained their ability to invest in business needs and, sometimes, to meet payroll.
The pressure to run a successful healthcare business has never been greater, making access to working capital – the life blood of companies – more important than ever. Yet reimbursements are slowing and banks are less likely to lend, making access to working capital an unnecessary challenge.
Recent legislative and regulatory actions have slowed or even stopped Medicare and Medicaid reimbursements. Illinois and Pennsylvania are extreme examples, where payments to physicians, hospitals and clinics have been delayed months due to budget impasses. This leaves many healthcare companies providing services and paying their bills with little or no revenue coming in. As Accounts Receivable continue to grow and cash becomes scarce, many clinics and practices face the dire possibility of closing their doors or selling their businesses to raise funds.
Meanwhile, banks are far more risk averse in the fallout of the financial crisis. Their inflexible underwriting often requires onerous paperwork and personal guarantees that many business owners cannot – or do not want to – provide.
Layer on the changes to billing due to ICD-10 and healthcare businesses are facing unprecedented strains on their finances.
Months-long delays in bill collection from insurers and government entities mean putting off investing in people, buying a competitor, or updating facilities. Loans, factoring, and lines of credit offer access to working capital for healthcare companies looking to grow or free up cash.
Will you have to give up ownership of any assets?
What is the impact on cash flow?
Will this solution improve cash flow for at least 2 years?
Is there flexibility in payments?
Most businesses have company reserves they can tap into when necessary. In this case, using that cash may seem like the best and easiest option, as long as reserves are not too depleted to handle large or unexpected expenses. Another area where it can get tricky is using the personal savings of the executive team. One company founder considered loaning their own savings to the business, but the company's accountant argued against it for tax and credit reporting reasons.
Most businesses have established relationships with traditional lenders through their commercial bank accounts. These known entities can be easily contacted to explore financing options. Small business loans can usually be handled with relative ease (minus the ubiquitous paperwork), though large loans might require personal guarantees or even a mortgage on an executive's home. Similarly, banks have traditionally shied away from lending to new businesses, even if they are offshoots of a successful parent company.
Factoring is a process where a business sells its assets (usually Accounts Receivable) to another company for a discounted price. That second company then tries to collect against the Accounts Receivable. To be profitable, the price paid to the healthcare business must be low enough to cover the risk of not collecting all of the accounts. It is common for factoring companies to pay less than half the value of the assets.
delaying the principal repayment until the end of the borrowing period. A line of credit can be an excellent option for a company that anticipates increasing future revenues, especially if the funds from the line of credit are used to invest in the growth of the business.


Excerpt from: Description of a Singular Appearance seen in the Sun on September 1, 1859.
Monthly Notices of the Royal Astronomical Society, vol. 20, 13-15, 1860.
I saw I was an unprepared witness of a very different affair. I therefore noted down the time by the chronometer, and seeing the outburst to be very rapidly on the increase, and being somewhat flurried by the surprise, I hastily ran to call some one to witness the exhibition with me, and on returning within 60 seconds, was mortified to find that it was already much changed and enfeebled. Very shortly afterwards the last trace was gone. In this lapse of 5 minutes, the two patches of light traversed a space of about 35,000 miles.
Click here for a text of Carrington's full article, including illustration.
...And for scientists: a contemporary article on Carrington's flare, The extreme magnetic storm of 1-2 September 1859 by Bruce T. Tsurutani, Walter D. Gonzalez, G.S. Lakhina and S. Alex, Journal of Geophysical Research, issue A7, article SSH-1, July 2003.