text:
(15789) 1993 SC es la designación provisional de un cuerpo menor del sistema solar; un objeto transneptuniano, más concretamente, un plutino.

Su descubrimiento fue hecho en 1993 en el Observatorio de La Palma con el telescopio Isaac Newton.

Sobre este objeto se sabe muy poco, aunque su órbita ya está establecida con la suficiente fiabilidad como para habérsele asignado un número del Centro de Planetas Menores definitivo.

Enlaces externos
 Lista de objetos transneptunianos y sus dimensiones estimadas (en inglés).

Plutinos
Objetos astronómicos descubiertos por Iwan Williams
Objetos astronómicos descubiertos por Alan Fitzsimmons
Objetos astronómicos descubiertos desde el Observatorio del Roque de los Muchachos
Objetos astronómicos descubiertos en 1993
Objetos astronómicos por denominación provisional


This fits your. These rainbow acrylic tumblers are beautiful, fun and real conversation pieces. They are stylish enough for the dining room and practical enough for patio and pool. Solid, shatterproof and pleasing to hold. Hand wash only. NOTE: Box reads 12 fl. oz. but glasses are 14 fl. oz.


The Dutch agency heading the international probe into Malaysia Airlines MH17 crash in eastern Ukraine has contacted RT over the footage used in our recent documentary on the tragedy. RT's documentary discovered fragments of the plane still in Donetsk.
The RT Documentary film, titled "MH17: A year without truth," showed fragments of the crashed Boeing and pieces of luggage still scattered in the area at the time of filming. The RTD crew collected the parts of the plane's exterior they spotted, bringing them to the administration of the nearby town of Petropavlovsk.
"With great interest we watched your documentary, 'MH17: A year without truth,'" Dutch Safety Board spokesperson Sara Vernooij wrote to RT. "In this film, RT shows parts of the cockpit roof which were found near Petropavlivka. We would like to gather those pieces and bring them over to the Netherlands so the Dutch Safety Board can use them for the investigation and the reconstruction."
In the documentary, the RTD crew talked to witnesses, experts and family members of flight MH17 passengers in a bid to understand whether the truth of what caused the tragedy will ever be established. You can watch the film here.
The documentary also features relatives of those who perished in the MH17 crash who say they have no possibility to get information about the course of the investigation, and believe it is not objective.
RT's Roman Kosarev has recently visited Petropavlovsk and talked to the town's mayor, Natalya Voloshina, who coordinates the collection of fragments in the regional administration. She explained that all the parts of the wreckage found by the locals are stored, awaiting their retrieval by Dutch investigators.
The investigators advised locals to save any plane parts found from the crash at a meeting with Natalya Voloshina in May, adding that they would try to collect them later. They have since been in touch via email and phone. While no date to pick up the newly found objects has been set, the investigators asked for photographs of the finds.
Moving the objects can be difficult and local residents who find them sometimes bring the parts themselves if they are small enough, but otherwise the district has created a team that collects the wreckage.
Voloshina said she was puzzled that no one had instructed local officials what to do with the wreckage in the aftermath of the crash, and when the fragments were being collected by the experts. She finally established contact remotely and received instructions.
For people, she said, it is not just a duty – many volunteered to comb the fields for remains and plane parts – but a psychological issue, as the wreckage is constant reminder of the tragedy and of the war in general. With the town itself still recovering after fighting in the area, the woman says she hopes the story of MH17 will finally reach its conclusion.
Malaysia Airlines Flight MH17 crashed in Donetsk Region on July 17, 2014, killing all 283 passengers and 15 crew members aboard. With the sides of the Ukrainian conflict still blaming each other for the tragedy with no conclusive proof, the world awaits the results of the Dutch-led international probe, which are due to be published in October. The criminal investigation into the crash, however, will continue after the data is made public.


text:
require 'rails_helper'

describe PagePolicy, type: :policy do
  subject { described_class }

  permissions :update? do
    it "should allow update if user in page_admins" do
      committee = build(:committee, slug: Page.page_admins.first)
      user = build(:user)
      user.committees = [committee]
      page = build(:page)
      expect(subject).to permit(user, page)
    end

    it "should disallow update if user not in page_admins" do
      committee = build(:committee)
      user = build(:user)
      user.committees = [committee]
      page = build(:page)
      expect(subject).not_to permit(user, page)
    end
  end
end

redpajama_set_name: RedPajamaGithub


Guruvayur Sri Krishna Temple is a Hindu temple dicated to the god Krishna (an avathar of the god Vishnu), located in the town of Guruvayur Kerala, India. It is one of the most important places of worship for Hindus of Kerala and is often referred to as "Bhuloka Vaikunta" which translates to the "Holy Abode of Vishnu on Earth".
This offering will be conducted at Chorun Hall (located at first floor of Ticket Counter building).Take the ticket for chorun from ticket counter before enetring the chorun Hall.This offerings starts daily at 5am till 1pm and also in the evening.The choroonu for kids will not be conducted on Ekadasi and Vavu Night. The choroonu will not be conducted on Guruvayur Ekadasi Day (Entire day).


text:
Аніл Менон ( 1964) — індійський письменник-фантаст, а також інформатик, доктор наук.

Життєпис
Він працював в університеті Сіракузи та є автором наукових робіт, також редагував книги з тематики еволюційних алгоритмів. Дослідження вченого стосувалося математичних основ реплікаторних систем, мажоризації та реконструкції імовірнісних баз даних у співпраці з професорами Кішаном Мехротром, Молу Хілукурі та Санджай Ранка.

Пропрацювавши декілька років комп'ютерним науковцем, він спрямував свою творчу енергію на художню літературу. Його оповідання та рецензії з'явилися в антології серії «Exotic Gothic», «Strange Horizons», «Interzone», «Lady Churchill's Rosebud Wristlet», «Chiaroscuro», «Sybil's Garage», «Apex Digest» та інших журналах.

У 2009 році провідна в Індії феміністична преса Zubaan Books опублікувала дебютний роман письменника для підлітків «Звір за дев'ять мільярдів футів». Він увійшов до списку номіннтів премії "Vodafone Crossword Book Award 2010 " . У 2009 році спільно з Ванданою Сінгх та Сучітра Матхур він допоміг організувати першу в Індії виставку художньої літератури в ІІТ-Канпурі.

Літературні твори

Оповідання
 «Кохання в жаркому кліматі» у збірці: Історії (ред.) Джей Лейк, Wheatland Press, 2005.
 «Архіпелаг» у журналі «Дивні горизонти», квітень 2005 року. Шорт-лист: 2006 р. Премія Товариства Карла Брендона Parallax Society .
 «Юстас Альберт» у « Часі для Бедлама» (ред.) Роджер Арбукл, Saltboy Publisher, 2005.
 «Стандартне відхилення» у журналі «Чіароскуро», березень 2005 року. Передруковано в журналі «Калкіон», липень 2010 року. Почесне згадування: Найкраща фентезі та жах року (ред.) Еллен Датлоу), 2006 р.
 «Діалетея» у журналі « Новий жанр», випуск № 5, весна 2007 року.
 «Невидима рука» у Наручна розетка леді Черчілль, № 20, червень 2007 року.
 «Небо, повне констант» в Albedo One, випуск 33, 2007.
 «Вермільйон» журналу «Інтернова», випуск № 10, січень 2007 року.
 «Харріс про свиню: практичні підказки для свинарника» у «From The Trenches» (ред.) JP Haines & S. Henderson), Carnifax Press, 2006. . Передруковано в журналі Apex, грудень 2008 року.
 «В ніч» в часописі Інтерзона, січень 2008 року. Передруковано у The Apex Book Of World SF (ред. Lavie Tidhar), 2008 р. Передрукований Apex Digest, листопад 2008 року. Передруковано «Галактики», січень 2010 року.
 «Скло, що горить» у Відповідь на Ворона (ред.) Марія Грація Кавічіоллі), 2009 р.
 «Сутра Пуанкаре» в «Гараж. Сибіла», 7-й випуск 2011 року. Висунута на премію «Параллакс» Товариства Карла Брендона у 2010 році та на премію «Карл Брендон» .
 «Невпевнена година» у «Заплутаному банку»: кохання, диво та еволюція (ред.) Кріс Лінч), лютий 2011 року.
 «Хавелі» в «Екзотична готика» 5, т. II (ред.) Данель Олсон), PS Publishing, 2012.

Дитяча фантастика
 «Лід» у Shockwave & Other Cyber Stories (ред.) Вацала Каул), Книги пінгвінів, Індія, 2007. .
 «До і після» у Діалозі гідності, серпень 2010 року.
 «Крикнат» у «Шепіт у класі», «Голоси на полі» (ред.) Річа Джа), липень 2011 року.
 «Різна балетка» у «Спортивних історіях» (ред.) Хіманджалі Санкар), Схоластична Індія, 2011 р.
 «Більше немає», журнал Hoot, липень 2011 року.

Романи

 Звір за дев'ять мільярдів футів, видавництва Zubaan (Індія), 2009 р., . Увійшов до списку вручення премії Vodafone-Crossword 2010 року.

Антології

 Розбиття лука: спекулятивна фантастика, натхненна Рамаяною (ред. Аніл Менон і Вандана Сінгх), книги Zubaan, грудень 2011 року.

Статті

 «Полювання на Снарка: на слід регіонального індійського СФ».  Переклад: Хуан Мадрігал  , Literatura Prospectiva 5 травня 2010 року.
 «Будівництво світу в умовах жаркого клімату», редакція журналу World SF, 19 травня 2010 року
 «Сире та приготоване» , редакція блогу World SF, 25 листопада 2009 року.

Інтерв'ю

 Інтерв'ю з SF-сигналом 02 листопада 2009 року.
 Hindustan Times, інтерв'ю Сумеєт Каул, 06 грудня 2009 року.
 Індуїстський, метро «Кочі», інтерв'ю Према Манманхана, 15 січня 2010 року.
 Інтернет- інтерв'ю Kalkion від Swapnil Bhartiya, 15 лютого 2010 року.
 Блог літератури на санскриті , інтерв'ю Венеції Енселл, 4 березня 2010 року.
 Журнал OPEN: Зміна епічних традицій , інтерв'ю Суда Г. Тілак, 2 серпня 2010 року.
 NRK (Норвезька телерадіомовна корпорація) , інтерв'ю Ана Летісії Сігварцен, 16 червня 2011 року.

Примітки

Посилання

Наукові фантасти Індії

redpajama_set_name: RedPajamaWikipedia


text:
San Felipe Neri en Eurosia es un título cardenalicio diaconal de la Iglesia Católica. Fue instituido por el papa Pablo VI en 1967 con la constitución apostólica Ea sollicitudine affecti.

Titulares
 Alfred Bengsch; título presbiteral pro illa vice (26 de junio de 1967 - 13 de diciembre de 1979)
 Vacante (1979 - 2003)
 Attilio Nicora (21 de octubre de 2003 - 12 de junio de 2014); título presbiteral pro illa vice (12 de junio de 2014 - 22 de abril de 2017)

Referencias

 GCatholic

Enlaces externos

Felipe Neri en Eurosia


text:
Ilya Anatolyevich Zimin (, May 1, 1972 – February 26, 2006) was a Russian journalist and television reporter who worked for the NTV network. Winner of the prize TEFI (2002) — the reporter.  He was killed by a youth who turned out to be his sexual partner.

Death
On February 26, 2006, Zimin was murdered under unclear circumstances. His death was caused by a cranial trauma. The police did not consider robbery as the cause of murder, neither did they connect Zimin's death with his professional activity. They suspected that the murder was the result of a domestic dispute. However, the main suspect, Igor Velchev, arrested on June 23, 2006 in Moldova, denied guilt. He stated in his defense that Zimin had approached him "offering sexual relations", so Velchev "pushed him hard and walked out."

On December 25, 2007, the Ocnița District Court of Moldova fully acquitted the accused.

Ilya Zimin is buried at the Central Cemetery in Khabarovsk.

References

External links
Russian TV reporter killed under mysterious circumstances
 Страница журналиста на сайте программы «Профессия-репортёр»

1972 births
Mass media people from Vladivostok
2006 deaths
Journalists killed in Russia
Violence against men in Europe
20th-century Russian journalists


text:
Krivulja ampersand je ravninska krivulja četrte stopnje (kvartična), ki je dana z enačbo

Spada med algebrske krivulje, ima rod enak 0. Ima tudi tri dvojne točke.

Zunanje povezave
 Krivulja ampersand na MathWorld
 Krivulja ampersand na 2dcurves.com
 Krivulja ampersand na WolframAlpha

Krivulje
Algebrske krivulje


text:
On Morals, Ethics and Screenwriting: An Interview with Jimmy McGovern
Steven M. Maras
Discipline of Media and Communication
Ethics in Screenwriting
New Perspectives
Steven Maras
Palgrave Studies in Screenwriting
Maras, S. M. (2016). On Morals, Ethics and Screenwriting: An Interview with Jimmy McGovern. In S. M. (Ed.), Ethics in Screenwriting: New Perspectives (pp. 77-95). (Palgrave Studies in Screenwriting). Palgrave Macmillan. https://doi.org/10.1057/978-1-137-54493-3_4
Maras, Steven M. / On Morals, Ethics and Screenwriting : An Interview with Jimmy McGovern. Ethics in Screenwriting: New Perspectives. editor / Steven Maras. UK : Palgrave Macmillan, 2016. pp. 77-95 (Palgrave Studies in Screenwriting).
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title = "On Morals, Ethics and Screenwriting: An Interview with Jimmy McGovern",
author = "Maras, {Steven M.}",
series = "Palgrave Studies in Screenwriting",
publisher = "Palgrave Macmillan",
editor = "{Steven Maras}",
booktitle = "Ethics in Screenwriting",
Maras, SM 2016, On Morals, Ethics and Screenwriting: An Interview with Jimmy McGovern. in SM (ed.), Ethics in Screenwriting: New Perspectives. Palgrave Studies in Screenwriting, Palgrave Macmillan, UK, pp. 77-95. https://doi.org/10.1057/978-1-137-54493-3_4
On Morals, Ethics and Screenwriting : An Interview with Jimmy McGovern. / Maras, Steven M.
Ethics in Screenwriting: New Perspectives. ed. / Steven Maras. UK : Palgrave Macmillan, 2016. p. 77-95 (Palgrave Studies in Screenwriting).
T1 - On Morals, Ethics and Screenwriting
T2 - An Interview with Jimmy McGovern
AU - Maras, Steven M.
T3 - Palgrave Studies in Screenwriting
BT - Ethics in Screenwriting
A2 - null, Steven Maras
PB - Palgrave Macmillan
CY - UK
Maras SM. On Morals, Ethics and Screenwriting: An Interview with Jimmy McGovern. In SM, editor, Ethics in Screenwriting: New Perspectives. UK: Palgrave Macmillan. 2016. p. 77-95. (Palgrave Studies in Screenwriting). https://doi.org/10.1057/978-1-137-54493-3_4

redpajama_set_name: RedPajamaCommonCrawl


This Saturday, April 21st, 5-8pm, there'll be an opening reception for a new group of bookshelf paintings I've been working on the last couple years. I go back to this motif every few years as a way to gauge where I'm at. Spending hours looking at the order/disarray of my books lets me reflect on what I've been doing and what the path onward might be. These pictures are some of the most abstract but personal ones I've done.
Come by and say hello if you're free.
and an Algren to replace the one that someone took home with them. Collect 'em all right here!
p.s. I was a guest on a new podcast called Ridesharing Stories to talk about the driving racket, among other things.


text:
BUSINESS AND SUSTAINABILITY BLOWFIELD PDF
Yozshunris Request an Inspection Copy. Monday — Friday, — UK time. Reviews This is an admirable piece of work that provides a good overview of the debate and critically examines how business can respond to sustainability challenges. Climate change, the resource constrained economy, and sustainability in general are amongst the hottest and most problematic topics for contemporary businesses. Innovation, Bisiness, and Design 7.
Author: Voodoogul Faegor
Uploader: Zukinos
Fourth Edition Michael Blowfield and Alan Murray Comprehensive coverage of the interaction between business and society reveals how CSR is defined, negotiated, and managed in the twenty-first century.
An objective, critical approach evaluates the theories and key drivers of CSR, as well as looking to the future for this ever-evolving subject. A range of discussion and case study questions promote reflection, debate, and critical thinking, challenging students to find ways for businesses to act in a truly responsible way.
New to this Edition: Updated and extended coverage of climate change, globalization, the gig economy, and the digitalization of business ensure all the latest issues to affect CSR are explored through a critical lens. Corporate Social Responsibility Michael Blowfield and Alan Murray Description In its journey from the margins to the mainstream, corporate social responsibility has become a significant part of the business agenda. Whilst society has always held expectations of business that go beyond maximising profits, the backdrop against which businesses now operate - characterised by financial crisis, climate change, political shifts, and population growth - has seen CSR become increasingly central to the ability of businesses to address global concerns.
The engaging, accessible style supports a critical perspective, while a forward-thinking outlook encourages you to consider how the interactions of business and society will continue to evolve. This book is supported by online resources.
For students: Additional case study sources - useful links for where to source further case studies and examples. Further reading - a comprehensive list of research to aid research. Suggested films - a list of films that illustrate issues relevant to CSR managers.
Sources on regulations and governance - links to important legislation and guidelines affecting CSR Web exercises - links to relevant websites direct you to valuable sources of information.
Journal club - links to seminal journal articles along with commentary on the paper, critical questions, and key points to consider.
For lecturers:.
BOATWRIGHT ROMANS PDF
Whilst society has always held expectations of business that go beyond wealth creation, the backdrop against which businesses now operate -characterised by financial crisis, climate change, political shifts, and population growth - has seen corporate responsibility becoming increasingly central to the ability of businesses to address global concerns. In a world where prosperity is measured in terms of economic growth, the pressure on business to maximise profits whilst also being held publically accountable for its social and environmental record, continues to increase. Drawing on numerous case studies, theories, and perspectives, this textbook explores what corporate responsibility reveals about the changing role of business in the twenty-first century. The third edition has been updated to reflect the full impact of the financial crisis and new examples of actual business practice have been presented throughout. These include: Google and their struggles with the Chinese government over censorship; Trafigura and their illegal dumping oftoxic waste; and Proctor and Gamble and their charitable work to provide clean water to poor communities.
3420G MCI PDF
Blowfield: Business and Sustainability
Additional Resources Description Climate change, the resource constrained economy, and sustainability in general are amongst the hottest and most problematic topics for contemporary businesses. He has worked as an academic and consultant in the field of corporate social and environmental responsibility with a particular focus on the socio-political context of corporate responsibility, and the role of business in society. Building on extensive experience in international development, including living and working in twenty different countries, he began a critical examination of corporate responsibility in global supply chains in the late s. He was principal investigator and programme manager for a portfolio of government and corporate-funded projects that analyzed the field of corporate responsibility in the context of developing countries. It is well supported by both theory and case studies and is written in a clear and compelling manner. It will be an invaluable resource for students on our programmes, since the structure is well aligned with our curriculum.
BAIXAR COMPACTADOR DE ARQUIVOS PDF
ULTRASONIC TESTING OF MATERIALS KRAUTKRAMER PDF
AKTA PERHUBUNGAN PERUSAHAAN 1967 PDF
DOSTOR MAROC 2011 PDF
CESSNA 172R POH PDF
DAVID SHENK DATA SMOG PDF
FIZYKA I ASTRONOMIA ZBIR ZADA MATURALNYCH MATURA NA 100 PDF


In high energy astronomy, every photon counts, and background events have been as big a mystery as the objects in the foreground. In x-ray astronomy in particular, the unexplained 'noise' takes the form of an apparently diffuse and fairly significant background wherever the detector points.
One component emanates uniformly from all directions in the sky, while another correlates strongly with the disk of the Milky Way galaxy, so we're likely seeing two distinct phenomena and have at least two mysteries to solve. Recently, astronomers have the use of a new tool in the form of the Chandra X-ray Observatory, which has brought to the investigation the sharpest x-ray vision ever used.
The first mystery is relatively close at hand, meaning on the order of 'only' tens of thousands of light-years away, within our own Milky Way. Astronomers see that the thin disk of our galaxy shines brightly in high energy, or 'hard', x-rays. Until now, no x-ray telescope could tell if this emission came from many individual stellar sources, such as neutron stars, or if it were spread through space.
An international team led by Dr. Ken Ebisawa of NASA's Goddard Space Flight Center pointed Chandra for a long look at a region of the galactic plane in the Scutum constellation. What they see in the highest resolution x-ray image ever taken is that 90% of the total emission is truly diffuse and coming from something spread through the plane of the galaxy. That something is most likely to be hot ionized gas, or plasma, where 'hot' is putting it mildly; to radiate in the x-ray band, the plasma would likely have to be at tens of millions of degrees.
But this only leads to another set of questions: Where did the plasma come from, and why does it stay confined to the narrow disk? Gas that hot has more than enough energy to escape the gravitational pull of the galaxy, so something must be confining it. Not knowing where it comes from or what's keeping it there means that astronomers are missing out on one of the most energetic processes going on in our own neighborhood. But they are never short on theories. That the gas is ionized may be a big hint. Dr. Ebisawa points out that ionized gas, where outer electrons have been stripped off the atoms leaving a net positive charge, could then be confined by a magnetic field.
Furthermore, though Chandra shows the emission as diffuse, it is by no means uniform. It is clumpy, and some patches of higher intensity are suspiciously close to suspected supernova remnants. These powerful explosions could well be the source of the plasma, but none of the many models of supernova evolution has yet distinguished itself from the pack by providing a testable explanation. The new data have settled only an observational uncertainty, leaving the real physics behind it to be discovered.
The unaccounted for 10% in their data appears to be far behind the gas in the galactic plane, and it is part of the second mystery, the component which is uniform in every direction. Unlike the cosmic microwave background (in brief, the diffuse heat left over from the Big Bang), this x-ray background is not explained by any theory of the origin of the universe. Nor can it be explained by assuming a uniform distribution of objects similar to what we have already resolved nearby; there aren't enough sources to explain the amount of radiation we see in the distance.
So two independent teams of astronomers recently pointed Chandra toward a couple of apparently empty pieces of sky for deep exposures of 500,000 and one million seconds respectively. Now known as the Chandra Deep Fields South and North, each is effectively a window from our own galaxy looking out to the more distant universe.
In this case, by contrast, the emission turns out to be due to many individual sources of x-rays, which have been identified as distant quasars by their signature energy spectra. Quasars are the very bright cores of some galaxies which spew out enormous amounts of high energy radiation, so much radiation that it can only be due to a black hole many millions of times the mass of the Sun. The average distance of these sources implies that they depict the universe as it was when much younger, only a few billion years old out of its current estimated age of 15 billion years.
This result has surprising implications for astronomers' understanding of the evolution of galaxies. What it says is that there were many more quasars in the past than are seen now, raising the big question: What happened to them? The implication is that quasars may be a stage of galaxy evolution, one through which even apparently ordinary galaxies like ours may have passed.
These galaxies would still harbor at their centres the supermassive black holes, only now the black holes are quiescent, perhaps because after a few billion years of gobbling up everything within reach, there is no more material near enough to feed them. The Chandra data strengthens considerably this evolutionary view that most normal galaxies may be essentially 'fossil quasars'.
But again, the solution to the observational mystery has raised yet more questions for astronomers. Foremost among them is the question of which came first, the galaxy or the black hole? Was the black hole born in the dense centre of an existing galaxy, or did the galaxy form around a 'seed' black hole? If the latter, where did the seed black holes come from? Though an observational uncertainty has been put to rest about the x-ray background, astronomers still have a lot to learn from the 'noise'.


text:
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It's the sequel no-one wanted. Lockdown 2. The latest instalment in a long-running saga for which no finale has yet been scripted. The shutters are going up once again, restrictions tightened, and freedoms limited. There can be no question, this has implications for us all. COVID-19 – and all that it entails – is a...
The technology that connects us also controls us
One newspaper in the UK has described it as "the most important documentary of our times." There's no doubt – in our minds, at least – that it should be compulsory viewing. For young people, parents and adults alike. For social media users and those online, no matter their age. For you. For us. For...
The coronavirus experience – Part 2: the positives
The coronavirus experience. Good or bad? How has it been for you? These are questions we posed on our blog a little earlier this month, the first of a two-part post looking at life in lockdown, with the focus on the problems posed for parents during the most difficult days in quarantine, with classrooms closed,...
The coronavirus experience – Part 1: the challenges
The coronavirus experience. Good or bad? How has it been for you? These might seem like strange questions, for at first glance, there can be few positives to be found in a pandemic. COVID-19 has impacted upon us all on a global level and, the obvious implications for health aside, lockdown has changed our lives...
Is worry infectious?
Ever thought about anxiety? Where it comes from? How it influences and affects those closest to us? Is it, for instance, genetic or a product of our environment? Should we be thinking about such things in altogether different terms? Most people, we think it's fair to say, don't have such thoughts or ever ask such...
Living in challenging times
These are challenging times for many. Children and young people with pre-existing mental health issues, are brought into sharp focus this week, following new research from YoungMinds. Having surveyed more than 2,000 under-25s with a mental health history, the charity's findings are striking. The picture painted underlines that COVID-19 isn't just affecting our bodies. Increased...
Parenting and coronavirus: you can't do more than your best
These are difficult times, for children and adults alike. Schools and colleges closed for the foreseeable future, workplaces relocated and the UK in lockdown, coronavirus (COVID-19) is having an impact on all our lives. It is unprecedented and unsettling. Our physical health might be foremost under the microscope. But don't forget the toll that such...
Mental health and the Coronavirus crisis
Concerned about Coronavirus (COVID-19)? You're far from alone. It is dominating debate and discussion – the sole talking point, and the subject on everybody's lips. On news bulletins. On social media. In the school playground. This seems certain to be the case for days, weeks and months to come. These are troubling times that are...
Coronavirus – our update
This is an unsettling time. Children and young people who are already experiencing stress and anxiety, can find it especially difficult. Our aim is to continue to provide a safe, effective and supportive mental health service whilst protecting our patients, their families, staff and our associates. We are monitoring the developing Coronavirus situation on...
Is social media normalising self harm?
More often than not, self-harm is an issue that affects adolescent girls. The common perception is that boys don't engage in such practices. Yet a concerning trend is beginning to emerge. No longer can self-harm be considered an issue that is exclusive to females. Far from it, in fact. This is the main finding following...
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Rabbie's, the award winning tour company are supporting Ayr Pipe Band Society and have kindly donated a voucher for 2 providing the winner and a guest to experience a Day Tour on them.
The tickets are valid from 29th September 2019 and below are some of the many tours available for you and your guest to choose from….
There are many, many more to experience….


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text:
Chance Pe Dance Movie First Look
Chance Pe Dance Movie First Look November 6, 2009
Also known as (AKA) / Was earlier
titled Star Yahoo
Banner UTV Motion Pictures
Color C
Language Hindi
Producer Ronnie Screwvala
Director Ken Ghosh
Star Cast Shahid Kapoor Genelia Dsouza
Cassettes and CD's on T-Series
Lyricist Irfan Siddique Amitabh Bhattacharya
Music Director Adnan Sami Sandeep Shirodkar Ken Ghosh
Publicity Designs Marching Ants
Movie Wallpapers 1 2 3 4 5 6 7 Shahid Kapoor
Photos 0 1234 5 6 7 8 9 10 Shahid Kapoor Wallpaper
Actors Genelia Dsouza and Shahid Kapoor
Director Ken Ghosh's new film 'Chance pe dance' is still filming but the first promo has already been leaked online.
Chance Pe Dance Movie Cast Crew
Chance Pe Dance Movie Cast Crew and First Look
The film, which was earlier titled 'Yahoo' stars the recently metamorphosed Shahid Kapoor and Genelia Dsouza. Not much is known about the film, but the official tagline says that 'Chance pe dance' is the story of a guy who has to make use of the one important chance he gets to prove his mettle.
Chance pe Dance will reportedly showcase the dancing skills of Shahid and Genelia, and rumours suggest that the film is a remake of Hollywood's teenage dance flick 'Step Up' which starred Channing Tatum and Jenna Dewan. Step up was the story of the disadvantaged Tyler Gage (Channing Tatum) and the privileged modern dancer Nora Clark (Jenna Dewan), who find themselves paired up in a showcase that determines both of their futures. Realizing that they only have one chance, they finally work together.
It remains to be seen how director Ken Ghosh turns the plotline into a desi one for 'Chance pe Dance'. The film will cash in on Shahid Kapoor's recent rise to fame ala Kaminey. Genelia, who is already an established actress in the south became a household name last year when her 'Jaane Tu ya Jaane na' hit theaters. Reportedly, 'Chance pe dance' earlier starred Jiah Khan in the lead role, however she was replaced with Genelia after producers weren't too happy with her acting skills. But is this move justifiable? Jiah Khan is an accomplished dancer skilled in Salsa, Lambada, Samba and Kathak; and Genelia's dancing skills are questionable.
'Chance pe dance' will reteam Ken Ghosh and Shahid Kapoor for the third time, after the mega successful 'Ishq Vishq' and box office bomb 'Fida'.
Posted by Mahendra at 12:27 AM 0 comments


text:
Milwaukee Community Journal: Cong. Moore Introduces Bill to Keep U.S. Tax Dollars from Being Used to Build Wall
Thursday, February 7, 2019 | comments
Milwaukee Community Journal - In response to President Trump's State of the Union address Tuesday, Wisconsin Cong. Gwen Moore has introduced a bill blocking the president from using taxpayer dollars to construct a border wall.
Milwaukee Community Journal
In response to President Trump's State of the Union address Tuesday, Wisconsin Cong. Gwen Moore has introduced a bill blocking the president from using taxpayer dollars to construct a border wall.
"My constituents don't want a wall," the congresswoman said Wednesday in a press statement about her legislation. "They want effective border security, to feel safe at home and to know their hard-earned tax dollars are used responsibly."
Called the "No Taxpayer Funding for the Wall Act," the legislation blocks Trump from executing what she called his "reckless plan" and force him to honor his pledge—which he made throughout his 2016 presidential campaign— that Mexico fund the building of the wall.
During his speech, Trump renewed his demand for a border wall in a section of his speech devoted to immigration. "I call on both Democrats and Republicans to join me in my effort by supporting this common sense legislation to protect the American public's tax dollars. We must pass this bill to ensure that President Trump can no longer hold the American people hostage.
"Protecting hard-earned dollars from wasteful spending is just common sense," the congresswoman added. "It's time for the GOP to get on board." Moore said the president used the "bully pulpit" the SOTU offers the chief executive officer of the United States to "spread lies and instill fear in the hearts of the American public by painting immigrants and asylum seekers as criminals—murderers determined to take down the law-abiding residents of the United States.
"We must rise above these racist lies and see his words for what they are: a manufactured crisis, designed to scare the public into submission and garner support for his wall."
It was the Democratic controlled Congress' rejection of Trump's wall demand that lead to the president shutting down the federal government for 35 days. The historic shut down impacted not only hundreds of thousands of federal workers, but impacted the quality of life for millions of Americans as it related to federal transportation (airport security and air traffic control), parks, judiciary, and small businesses that cater or are contracted to provide services to federal government entities.
Trump's government shutdown was also noted by former Georgia gubernatorial candidate Stacy Abrams in her Democratic response to Trumps SOTU address. Touted as a rising star in the Democratic Party, Abrams, who lost a hard-fought and controversial race for Georgia governor during midterm elections, called Trump's government shutdown a "disgrace" and a "stunt" the president engineered that "defied every tenet of fairness and abandoned not just our people—but our values."
Standing in front of a racially diverse group of people, Abrams—whose SOTU response is the first ever delivered by a Black woman—also spoke on issues Trump didn't mention at all in his speech: mass shootings, LGBTQ support and voter suppression. She lost her governor's race by two points in the face of suspected voter suppression by her opponents' administration.
"Just a few weeks ago, I joined volunteers to distribute meals to furloughed federal workers. They waited in line for a box of food and a sliver of hope since they hadn't received a paycheck in weeks," Abrams said. "Making their livelihoods a pawn for political games is a disgrace."
In Georgia and around the country, people are striving for a middle class where a salary truly equals economic security. But instead, families' hopes are being crushed by Republican leadership that ignores real life or just doesn't understand it."
Interestingly, Abrams stressed she didn't want Trump to fail, but to change his leadership approach. "We may come from different sides of the political aisle, but our joint commitment to the ideals of this nation cannot be negotiable"


text:
Q: stacked density plot with gg plot in R I fail to plot correcly my stacked density chart
here is my data structure
Classes 'tbl_df', 'tbl' and 'data.frame':   8517 obs. of  3 variables:
 $ time  : num  -500 -499 -498 -497 -496 -495 -494 -493 -492 -491 ...
 $ series: Factor w/ 17 levels "V1","V2","V3",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ value : num  0.0788 0.0625 0.0455 0.0323 0.0426 ...

and here is my code
ggplot(data=data,aes(x=time, group=data$series, fill=data$series)) +
    geom_density(adjust=1.5, position="fill")

but it plots me horizontal lines

Instead I would like it plots me something like that

If anyone has any idea? I would appreciate. Thanks
here is my data structure
dput(head(data))
    structure(list(time = c(-500, -499, -498, -497, -496, -495),
        series = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = c("V1",
        "V2", "V3", "V4", "V5", "V6", "V7", "V8", "V9", "V10", "V11",
        "V12", "V13", "V14", "V15", "V16", "V17"), class = "factor"),
        value = c(0.0787576635451054, 0.0625, 0.0455255463892602,
        0.0322986577181208, 0.0426306596802365, 0.0369765758489718
        )), row.names = c(NA, -6L), class = c("tbl_df", "tbl", "data.frame"
    ))

A: Does this work for you?
ggplot(data, aes(x=time, fill = series)) +
  geom_density(adjust=1.5, position="fill")

My bad, initially I got the wrong idea, you were almost there.
Just remove data$.

A: For the sake of illustrating it with multiple factors, I changed it a tad.
df <- structure(list(time = c(-500, -499, -498, -497, -496, -495),
           series = structure(c(1L, 1L, 2L, 2L, 3L, 3L), .Label = c("V1",
                                                                    "V2", "V3", "V4", "V5", "V6", "V7", "V8", "V9", "V10", "V11",
                                                                    "V12", "V13", "V14", "V15", "V16", "V17"), class = "factor"),
           value = c(0.0787576635451054, 0.0625, 0.0455255463892602,
                     0.0322986577181208, 0.0426306596802365, 0.0369765758489718
           )), row.names = c(NA, -6L), class = c("tbl_df", "tbl", "data.frame"
           ))

ggplot(df, aes(x = time)) +
  geom_density(stat = "density", aes(fill = series), position = "stack") # or fill if you want proportions

redpajama_set_name: RedPajamaStackExchange


text:
Maelstrom - a play-by-post rpg
Maelstrom -Game Home
Recent activity for Maelstrom
Winters created a new post
LucianNepreen created a new post 23 days ago
Characters in this post
View character profile for: Captain Phineas Steelheart
View character profile for: Skippy Ganymede
Posted by : LaserSexPanther on Mar 2, 2019, 2:26am
Skippy sat in the small break room/entrance to the Gutter, alone with her meal. Max had to rush off to help the sector A group with a repair. Leaving Skippy to eat since she wouldn't have been tall enough to help anyway. Sitting at the small table with no matching legs she at her bland sandwich. Staring at it somewhat blankly, have eaten the same thing for months it wasn't as easy to choke down as it once was.
Suddenly their came a slam from the upper bulkhead. The heavy door forced open, and a clattering on the ladder. Who ever it was hadn't used the ladder before or not often at least meaning they were someone coming to bitch about cold and the heat not getting to their home. But the person on the ladder suddenly dropped with a thud. Meaning they didn't climb it's entity, and the second door suddenly flew open. Startling Skippy enough she almost fell backwards.
Phineas looked around the small room, looking past Skippy at first seeming to look for someone to ask for the repair. Well not ask, more like demand. It was obvious to her, he was either ignoring her presence or thought little enough of her, he just didn't see her.
"Little girl, where is your father. The Roost is ripping itself apart and we need it fixed now." Phineas said.
Skippy shoved the last little bit of sandwich in her mouth before speaking with her mouth somewhat full. But not so much so you couldn't understand her. "Excuse me? I work here." she snapped back, the man was intimidating yes. But she wasn't going to be pushed around by someone who needed HER help.
"Come with me, I am going to need you to help me." she said already clipping her belt back on and putting on her elbow length leather gloves. "I don't have anyone to help me right now so you want the tower to stay standing you are going to help me cycle the pumps." Skippy didn't wait to long for the man to persist, tossing him a pair of gloves.
"I get you only need to one, but we like to keep them as a pair down here. Gloves like that are a pain to replace, Mr.... Sorry I didn't catch your name." she said as the two rushed down the path to the the section the lay under the Roost. Unlike top side even with out the snow if you knew what you were doing it was easy and faster to get around down here. Someone who didn't could get hurt but with Skippy leading the way it was an easy task, so long as you didn't lean to far right or left. And someone as tall as Phineas would easy burn an ear on a pipe if he did.
"It's Phineas and skip the pleasantries girly we don't have time to waste." he said.
"That's good, because I wasn't wasting time. I was getting to know the person who, and this goes equally for you as well. Is about to put their lives in a strangers hands. So just figured knowing each others names would be fair." she snapped as the approved another large bulk head. Beside it as a odd box with what looked like a bell shaped metal piece attached to a string of wire.
"I know the Roost has one of these to. You need to call them up on the radio and tell them to shut off all the elevators and release all the clamps so we can cycle the pumps and knock the ice free." Skippy explained.
Pushing the next bulkhead open it hissed at her pushing. This room was sealed off in a way. Not to protect anyone from anything or to keep anything in, but the door just sealed from the heat differences.
Phineas picked up the radio dialing in the frequency to call the Roost. "Boys shut the tower down. Worker needs to cycle the pumps." he said.
Skippy couldn't hear what they were saying on the other side it was just mumbles and grunts. But a moment later the 4 large pistons wheezed to a stop, though out of sync. "Okay, now's the part where I need help." she said, handing him a wrench.
The roof here was surprisingly not as high as one would think Phineas could easily reach the top if he stretched a bit. "Hit the ice up there." she pointed it out it was out of her reach. Phineas glared at her. "Girl, stop testing me." he said. "Oi, I get you think you are a big bad man but down here I am queen. You want the Roost to stay standing I can't do it alone." she said, already leaning all her weight on a massive bolt as it squeaked a bit and moved. Before she quickly moved out of the way a jet of steam sprayed out, just missing her. "Do you not run these things at night?"
Continue story
< Prev : Mutations Next > : Adventure begins.


Happy to announce, a brand-new version with lot of new capabilities of BMC Helix Multi-Cloud Service Management 1811 is now GA and will be rolled out to our production customers!! We continue to add exciting features to the themes we have embarked upon. You can now turn-on Jira integration for Problem records in ITSM and collaborate using Smart IT. You can view Jira ticket information on the problem records in all Smart IT clients. Support for many more field types in Jira is delivered in addition to attachment support for Salesforce ServiceCloud. Many more refinements including ability to actually see the user who commented, drill down to Jira ticket and a bunch more!
It's amazing to note the progress we have made and how far along we have come with the 5th GA release of BMC Helix Multi-Cloud Service Management in so little time. We appreciate all the input from many of you through the journey and wishing you all a wonderful holiday season ahead. Thank you for being an important part of us!
What is BMC Helix Multi-Cloud Service Management?
Multi-cloud service management solutions enable multi-provider brokering to efficiently manage and resolve issues. When organizations embrace service management tools, it enhances the ability to secure better service from multi-cloud service providers. For instance, IT can manage tickets across multiple cloud providers utilizing one integrated service desk. It also enables direct collaboration with multi-cloud providers such as AWS or Salesforce Service Cloud to efficiently troubleshoot and resolve issues as well as proactively inform employees of planned service outages utilizing a federated calendar.
In addition to gaining control over Cloud Service Providers (CSPs) issues or incidents, service management provides the visibility and insight needed to ensure that CSPs are operating and delivering to expectations. As such, it's possible to monitor multiple cloud services against the parameters of established SLAs. At the same time, the most dynamic tools provide an integrated service desk that shields agents from the complexities of differing cloud service management systems.
Modern service management solutions, like BMC Helix, also provide out-of-the-box integration with DevOps tools such as JIRA. For the DevOps manager, these capabilities mean resolving incidents directly in JIRA including associating multiple JIRA user stories with remedy incidents.
Because of their collaborative and seamless nature, having access to service management tools enables IT to provide better customer service with less downtime due to unresolved issues. It results in cost-efficient and effective resolution of problems. Flexibility to choose the best solutions and not worry about the level of support service.
The more informed IT is on the management of the organization's cloud service provider investment, the more likely it is to provide in-house teams with the predictable high-performance environments they need to succeed.
Need more information on BMC Helix Multi-Cloud Service Management?
Please feel free to reach out if you have any additional questions.


Large Size approx. 50×35 mm, generally recommended for women over 45 years old, that gave birth to more than one child or with some incontinence problems.
Medium Size approx. 43×30 mm, generally recommended for women younger than 45 years old, that gave birth to one child or never delivered children and without known incontinence problems.
Small Size approx. 35×25 mm, generally recommended for women that already used larger sizes and are experts in the practice with the Yoni Egg or that have some vaginal muscle tensions problems (such as vaginism).
A beautiful egg, and very safely packaged in simple cardboard box and cotton pouch. I can't wait to start using it. Thank you a lot!
Love my new yoni egg! The green colour is gorgeous, the quality is great and I love the size (medium). Would buy again without hesitation!


Hellebore is inspired by the pale mauve shades of Hellebore flowers. It combines shades of silvery mauve with a range of mauve, rose, pink and lilac tones. It has a dose of bright speckles to lift the mauve and which will create epic pops of subtle colour in a knitted fabric (check my IG feed for images featuring this colour soon).
It combines well with The Botanist and Belladonna - these were all created to build on each other.
Notify me when 100g singles (360m / 393y) Hellebore is available.


text:
Home > Boonshoft School of Medicine > Master of Public Health Program > Master of Public Health Program Student Publications > 233
Master of Public Health Program Student Publications
The Association of Perfluoroalkyl Substance Exposure and Lung Function in the U.S. Population
Brenna C. Heinle, Wright State University - Main Campus
Master's Culminating Experience
Background/Aim: Perfluoroalkyl substances (PFASs) are chemical compounds used in consumer products and are linked with increase in cholesterol, thyroid disease, and pregnancy-induced hypertension. However, their association with lung function is not completely understood.
Methods: Cross sectional 2011-12 U.S. population data from the National Health and Nutrition Examination Survey (NHANES) was used (N=1,450, aged 12 to 79 years, 52% men). Serum concentration of four PFASs, perfluorononanoic acid (PFNA), perfluorooctanoic acid (PFOA), perfluorooctane sulfonic acid (PFOS), and perfluorohexane sulfonic acid (PFHxS) were assessed using mass spectrometry and were categorized into tertiles. Lung function was measured by spirometry as forced vital capacity (FVC), forced expiratory volume in one second (FEV1), and the ratio of FVC/FEV1 (%). Sex stratified adjusted linear regression analysis was used to predict lung function with PFASs tertiles.
Results: In men all four PFASs serum concentrations and lung function was higher, except FVC/FEV1 (%) which was lower than women (p
Conclusion: High PFASs exposure has a consistent significant negative impact on female lung function in unadjusted analysis and this association needs further exploration.
Heinle, B. C. (2019). The Association of Perfluoroalkyl Substance Exposure and Lung Function in the U.S. Population. Wright State University. Dayton, Ohio.
ILE Poster HeinleB_Final.pdf (920 kB)
mph-heinle-poster
Public Health Commons


The next generation A-Class is under development and will be enhanced to offer higher levels of practicality and interior appeal.
Much of the current A-Class' success has been attributed to its stylish good looks, but a facelifted version was introduced earlier this year with improvements to its design and interior quality. The next iteration will shift the goalposts, however. Spy images of the next generation A-Class, which is set to arrive in 2018, emerged last month. According to reports, Mercedes will focus on incorporating more technologies and improving the hatchback's practicality.
The current A-Class isn't the most spacious vehicle in its segment: although the driver and front passenger are seated in comfort, rear passengers aren't afforded generous head- and legroom. Benz will reportedly solve this issue by building the new A-Class on a heavily revised MFA platform (called MFA2), which will see the wheelbase (and overall length) of the A-Class increase. The roofline will likely be extended to improve rear headroom and the rear-end will be reshaped to increase the hatchback's luggage capacity. You can also expect to see new headlight and taillight clusters when the new model arrives.
The 2.1-litre turbodiesel unit in the A-Class is likely to be replaced by Mercedes' newly developed 2.0-litre turbodiesel. There's also the possibility of a new 2.0-litre turbopetrol engine joining the lineup and there's even talk of a new 3-cylinder turbodiesel motor making an appearance. A plug-in hybrid version, combining a small capacity 3-cylinder petrol engine and an electric motor driven through a new 9-speed dual clutch transmission, is also on the cards.The big engine news, however, pertains to the potent Mercedes-AMG A45 where engineers are hoping to inject a further 18 kW of power into the hot hatch, which will hike the power output of the 2.0-litre turbocharged engine up to at least 294 kW. The current 7-speed automatic transmission will make way for a new 9-speed dual-clutch 'box. Furthermore, the all-wheel drive system will be revised with the aim of achieving a 0-100 kph sprint time of less than 4 sec.
Notable interior changes are expected in the new A-Class to bring the model's layout in line with those of the C- and E-Class. A new cabin architecture with high-quality materials, a new dashboard design and enlarged display screen/s will make the interior even more appealing. The new A-Class will also be offered with numerous standard and optional safety features that may include Distronic Plus with steering assist Stop & Go Pilot and park assist.


text:
\section{Introduction}
\noindent Dirichlet process mixture (DPM) models and closely related formulations have been very widely used for flexible modeling of data and for clustering.  DPMs of Gaussians have been shown to possess frequentist optimality properties in density estimation, obtaining minimax adaptive rates of posterior concentration with respect to the true unknown smoothness of the density \cite{shen2011adaptive}.  DPMs are also very widely used for probabilistic clustering of data.  In the clustering context, it is well known the DPMs favor introducing new components at a log rate as the sample size increases, and tend to produce some large clusters along with many small clusters.  As the sample size $N$ increases, these small clusters can be introduced as an artifact even if they are not needed to characterize the true data generating process; for example, even if the true model has finitely many clusters, the DPM will continue to introduce new clusters as $N$ increases \cite{miller2013simple,argiento2009comparison,lartillot2004bayesian,
onogi2011characterization,miller2014inconsistency}.

Continuing to introduce new clusters as $N$ increases can be argued to be an appealing property. The number of `types' of individuals is unlikely to be finite in an infinitely large population, and there is always a chance of discovering new types as new samples are collected.  This rationale has motivated a rich literature on generalizations of Dirichlet processes, which have more flexibility in terms of the rate of introduction of new clusters.  For example, the two parameter Poisson-Dirichlet process (a.k.a., the Pitman-Yor process) is a generalization that instead induces a power law rate, which is more consistent with many observed data processes  \cite{perman1992size}. There has also been consideration of a rich class of Gibbs-type processes, which considerably generalize Pitman-Yor to a broad class of so-called exchangeable partition probability functions (EPPFs) \cite{gnedin2006exchangeable,lijoi2010models,de2015gibbs,
bacallado2017sufficientness,favaro2013conditional}.  Much of the emphasis in the Gibbs-type process literature has been on data in which `species' are observed directly, and the goal is predicting the number of new species in a further sample \cite{lijoi2007bayesian}. It remains unclear whether such elaborate generalizations of Dirichlet processes have desirable behavior when clusters/species are latent variables in a mixture model.

The emphasis of this article is on addressing practical problems that arise in implementing DPMs and generalizations when sample sizes and data dimensionality are moderate too large.  In such settings, it is common knowledge that the number of clusters can be too large, leading to a lack of interpretability, computational problems and other issues.  For these reasons, it is well motivated to develop {\em sparser} clustering methods that do not restrict the number of clusters to be finite {\em a priori} but instead favor deletion of small clusters that may not be needed to accurately characterize the true data generating mechanism.  With this goal in mind, we find that the usual focus on exchangeable models, and in particular EPPFs, can limit practical performance.  There has been some previous work on non-exchangeable clustering methods motivated by incorporation of predictor-dependence in clustering \cite{blei2011distance,ghosh2011spatial,soumya2014nonparametric,socher2011spectral}, but our focus is instead on providing a simple approach that tends to delete small and unnecessary clusters produced by a DPM.  Marginalizing out the random measure in the DPM specification produces a Chinese restaurant process (CRP).  We propose a simple powered modification to the CRP, which has the desired impact on clustering and develop associated inference methods.

\section{Powered Chinese restaurant process (pCRP)}
\subsection{The Chinese restaurant process (CRP)}
The Chinese restaurant process is a simple stochastic process that is exchangeable. In the analogy from which this process takes its name, customers seat themselves at a restaurant with an infinite number of tables. Each customer sits at a previously occupied table with probability proportional to the number of customers already sitting there, and at a new table with probability proportional to a concentration parameter $\alpha$. For example, the first customer enters and sits at the first table. The second customer enters and sits at the first table with probability $\frac{1}{1+\alpha}$ and at a new table with probability $\frac{\alpha}{1+\alpha}$. The $i^{th}$ customer sits at an occupied table with probability proportional to the number of customers already seated at that table, or sits at a new table with a probability proportional to $\alpha$. Formally, if $z_i$ is the table chosen by the $i^{th}$ customer, then
\begin{equation}
\begin{aligned}
&\hspace*{-0.04in} p(z_i = k | \mathbf{z}_{-i}, \alpha)=\\
& \hspace*{-0.08in}  \left\{
                \begin{array}{ll}
                  \frac{N_{k,-i}}{N+\alpha-1},  \text{ if \textit{k} is occupied, i.e. }  N_k > 0, \\
                  \frac{\alpha}{N+\alpha-1}, \text{ if \textit{k} is a new table, i.e. }  k = k^{\star} = K + 1, \\
                \end{array}
              \right.
\label{equation:crp_equation}
\end{aligned}
\end{equation}
where $\mathbf{z}_{-i} = (z_1, z_2, \ldots , z_{i-1}, z_{i+1},  \ldots, z_N)$ and $N_{k,-i}$ is the number of customers seated at table $k$ excluding customer $i$.
From the definition above, we can observe that the CRP is defined by a rich-get-richer property in which the probability of being allocated to a table increases in proportion to the number of customers already at that table.

In a CRP mixture model, each table is assigned a specific parameter in a kernel generating data at the observation level.  Customers assigned to a specific table are given the cluster index corresponding to that table, and have their data generated from the kernel with appropriate cluster/table-specific parameters.  The CRP provides a prior probability model on the clustering process, and this prior can be updated with the observed data to obtain a posterior over the cluster allocations for each observation in a data set.  The CRP provides an exchangeable prior on the partition of indices $\{1,\ldots,N\}$ into clusters; exchangeability means that the ordering of the indices has no impact on the probability of a particular configuration -- only the number of clusters $K_N$ and the size of each cluster can play a role.  The CRP implies that $\mathrm{E}[K_N|\alpha] = O(\alpha \log N)$~\cite{teh2011dirichlet}.

\subsection{Powered Chinese restaurant process}

Popular Bayesian nonparametric priors, such as the Dirichlet process \cite{ferguson1973bayesian,blackwell1973ferguson,antoniak1974mixtures}, Chinese restaurant process, Pitman-Yor process \cite{perman1992size,pitman1997two} and Indian buffet process \cite{griffiths2005infinite,thibaux2007hierarchical}, assume infinite exchangeability.  In particular, suppose we have a clustering process for an infinite sequence of data points $i=1,2,3,\ldots,\infty$.  This clustering process will induce a partition of the integers $\{1,\ldots,N\}$ into $K_N$ clusters of size $N_1,\ldots,N_{K_N}$, for $N=1,2,\ldots,\infty$. For an exchangeable clustering process, the probability of a particular partition of $\{1,\ldots,N\}$ only depends on $N_1,\ldots,N_{K_N}$ and $K_N$, and does not depend on the order of the indices $\{1,\ldots,N\}$.  In addition, the probability distributions for different choices of $N$ are {\em coherent};  the probability distribution of partitions of $\{1,\ldots,N\}$ can be obtained from the probability distribution of partitions of $\{1,\ldots,N+1\}$ by marginalizing out the cluster assignment for data point $i=N+1$.  These properties are often highly appealing computationally and theoretically, but it is nonetheless useful to consider processes that violate the infinite exchangeability assumption.  This can occur when the addition of a new data point $i=N+1$ to a sample of $N$ data points can impact the clustering of the original $N$ data points.  For example, we may re-evaluate whether data point $1$ and $2$ are clustered together in light of new information provided by a third data point, a type of {\em feedback} property.

We propose a new powered Chinese restaurant process (pCRP), which is designed to favor elimination of artifactual small clusters produced by the usual CRP by implicit incorporation of a feedback property violating the usual exchangeability assumptions.  The proposed pCRP makes the random seating assignment of the customers depend on the powered number of customers at each table (i.e. raise the number of each table to power $r$). Formally, we have
\begin{equation}
\begin{aligned}
&p(z_i = k | \mathbf{z}_{-i}, \alpha)=\\
&\left\{
                \begin{array}{ll}
                  \frac{N_{k,-i}^r}{\sum_h^K N_{h,-i}^r+\alpha}, \text{ if \textit{k} is occupied, i.e. }  N_k > 0,\\
                  \frac{\alpha}{\sum_h^K N_{h,-i}^r+\alpha}, \text{ if \textit{k} is a new table, i.e. }  k = k^{\star}=K+1,
                \end{array}
              \right.
\label{equation:powered_crp_equation}
\end{aligned}
\end{equation}
where $r>1$ and $N_{k,-i}$ is the number of customers seated at table $k$ excluding customer $i$. More generally, one may consider a $g$-CRP to generalize the CRP such that
\begin{equation}
\begin{aligned}
&p(z_i = k | \mathbf{z}_{-i}, \alpha)=\\
&\left\{
\begin{array}{ll}
\frac{g(N_{k,-i})}{\sum_h^K g(N_{h,-i})+\alpha}, \text{ if \textit{k} is occupied, i.e. }  N_k > 0,\\
\frac{\alpha}{\sum_h^K g(N_{h,-i})+\alpha}, \text{ if \textit{k} is a new table, i.e. }  k = k^{\star}=K+1,
\end{array}
\right.
\label{equation:g_crp_equation}
\end{aligned}
\end{equation}
where $g(\cdot): \mathbb{R^+} \rightarrow \mathbb{R^+} $ is an increasing function and $g(0) = 0$. We achieve shrinkage of small clusters via a rich-get-(more)-richer property by requiring $g(x) \geq x$ for $x > 1$ to`enlarge'  clusters containing more than one element. We require the $g$-CRP to maintain a \textit{proportional invariance} property:
\begin{equation}\label{eq:proportional.invariant}
\frac{g(cN_1)}{g(cN_2)} = \frac{g(N_1)}{g(N_2)}
\end{equation}
for any $c, N_1, N_2 > 0$, so that scaling cluster sizes by a constant factor has no impact on the prediction rule in \eqref{equation:g_crp_equation}.  The following Lemma~\ref{lemma:unique} shows that the pCRP in equation~\eqref{equation:powered_crp_equation} using the power function is the only $g$-CRP that satisfies the proportional invariance property.
\begin{lemma}
	\label{lemma:unique}
	If a continuous function $g(x): \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies equation~\eqref{eq:proportional.invariant}, then $g(x) = g(1) \cdot x^r$ for all $x > 0$ and some constant $r \in \mathbb{R}$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:unique}]
	It is easy to verify that  $g(x) = g(1) \cdot x^r$ for some $r > 0$ is a solution to the functional equation~\eqref{eq:proportional.invariant}. We next show its uniqueness.

	Equation~\eqref{eq:proportional.invariant} implies that $g(cN_1)/g(N_1) = g(cN_2)/g(N_2)$ for any $N_1, N_2 > 0$. Denote $f(c) = g(cN)/g(N) > 0$ for arbitrary $N > 0$. We then have $f(s t) = g(stN)/g(N) = g(stN)/g(tN) \cdot g(tN) / g(N) = f(s) f(t)$ for any $s, t > 0$. By letting $f^*(x) = f(e^x) > 0$, it follows that $\log f^*(s + t) = \log f^*(s) + \log f^*(t)$, which is the well known Cauchy functional equation and has the unique solution $\log f^*(x) = r x$ for some constant $r$. Therefore, $f(x) = f^*(\log(x)) = x^r$ which gives $g(cN) = g(N) c^r$. We complete the proof by letting $N = 1$.
\end{proof}

As a generalization of the CRP, which corresponds to the special case in which $r=1$, the proposed pCRP with $r > 1$ generates new clusters following a probability that is configuration dependent and not exchangeable. For example, for three customers $z_1, z_2, z_3$, $p(z_3 = 2 \mid z_1=1, z_2=1) < p( z_3 = 1 \mid z_1=1, z_2=2)$, where $z_i = k$ if the $i^{th}$ customer sits at table $k$.
This non-exchangeability is a critical feature of pCRP, allowing new cluster generation to learn from existing patterns. Consider two extreme configurations: (i) $K_N = N$ with one member in each cluster, and (ii) $K_N = 1$ with all members in a single cluster. The probabilities of generating a new cluster under (i) and (ii) are both $\alpha/(N + \alpha)$ in CRP, but dramatically different in pCRP: (i) $\alpha/(N + \alpha)$ and (ii) $\alpha/(N^r + \alpha)$, respectively. Therefore, if the previous customers are more spread out, there is a larger probability of continuing this pattern by creating new tables.  Similarly, if customers choose a small
number of tables, then a new customer is more likely to join the dominant clusters rather than open a new table.

The power $r$ is a critical parameter controling how much we penalize small clusters. The larger the power $r$, the greater the penalty.  We propose a method to choose $r$ in a data-driven fashion: cross validation using a proper loss function to select a fixed $r$.

\subsection{Power parameter tuning}\label{sec:pcrp_parameter_calibration}
The proportional invariance property makes it easier to define a cross validation (CV) procedure for estimating $r$.  In particular, one can tune $r$ to obtain good performance on an initial training sample and that $r$ would also be appropriate for a subsequent data set that has a very different sample size.  For other choices of $g(\cdot)$, which do not possess proportional invariance, it may be necessary to adapt $r$ to the sample size for appropriate calibration.

In evaluating generalization error, we use the following loss function based on within-cluster sum of squares:
\begin{equation}
 \sum_{k=1}^K  \sqrt{ \sum_{j: j \in C_k}^{N_k} ||\mathbf{x}_j - \overline{\mathbf{x}}_k||^2 },
\end{equation}
where $C_k$ is the data samples in the $k$th cluster and $ \overline{\mathbf{x}}_k$ is the mean vector for cluster $k$. The square root has an important impact in favoring a smaller nunber of clusters; for example, inducing a price to be paid for introducing two clusters with the same mean.  In implementing CV, we start by choosing a small value of $r$ ($r=1+\epsilon$) and then increasing until we identify an inflection point.

\begin{figure}[h!]
\centering
  \includegraphics[width=0.3\textwidth]{img_pcrp/bgmm_infinite_without_hyper.pdf}
  \caption{A Bayesian infinite GMM.}
  \label{fig:gmm_infinite_without_hyper}
\end{figure}

\subsection{Posterior inference by collapsed Gibbs sampling}
Although the proposed pCRP is generic, we focus on its application in Gaussian mixture models (GMMs) for concreteness. We develop a collapsed Gibbs sampling algorithm (Alg 3 in \cite{neal2000markov} and further introduced in \cite{murphy2012machine}) for posterior computation. Our proposed pCRP can also be easily implemented via a non-collapsed Gibbs sampling algorithm (\cite{west1993hierarchical}, i.e. Alg 2 in \cite{neal2000markov}). In addition, we permute the data at each sampling iteration to eliminate order dependence as in~\cite{socher2011spectral}.

Let $\mathcal{X}$ be the observations, assumed to follow a mixture of multivariate Gaussian distributions. We use a conjugate Normal-Inverse-Wishart (NIW) prior $p(\boldsymbol\mu, \boldsymbol\Sigma | \boldsymbol\beta)$ for the mean vector $\boldsymbol\mu$ and covariance matrix $\boldsymbol\Sigma$ in each multivariate Gaussian component, where $\boldsymbol\beta$ consists of all the hyperparameters in NIW. I.e. we will work with the following definition of Bayesian infinite Gaussian mixture model:
\begin{equation}
\begin{aligned}
\mathbf{x}_i | z_i, \{\boldsymbol\mu_k, \boldsymbol\Sigma_k \} &\sim \mathcal{N}(\boldsymbol\mu_{z_i}, \boldsymbol\Sigma_{z_i}), \\
z_i | \bm{\pi}    									&\sim \mathrm{Multinomial}(\pi_1, \ldots, \pi_K), \\
\{\boldsymbol\mu_k, \boldsymbol\Sigma_k\}					&\sim \mathrm{NIW}(\boldsymbol\beta), \\
\bm{\pi} 											&\sim \mathrm{Dirichlet}(\alpha/K, \ldots, \alpha/K),
\end{aligned}
\end{equation}
by taking the limit as $K \rightarrow \infty$ \cite{rasmussen1999infinite}, where $\mathcal{N}(\cdot, \cdot)$ is the multivariate normal distribution, $\mathrm{Multinomial}(\cdot)$ is the Multinomial distribution and $\mathrm{Dirichlet}(\cdot)$ is the Dirichlet distribution. The process is illustrated in Figure~\ref{fig:gmm_infinite_without_hyper}.

A key quantity in a collapsed Gibbs sampler is the probability of each customer $i$ sitting with table $k$: $p(z_i = k | \mathbf{z}_{-i}, \mathcal{X}, \alpha, \boldsymbol\beta)$, where $\mathbf{z}_{-i}$ are the seating assignments of all the other customers and $\alpha$ is the concentration parameter in CRP and pCRP. This probability is calculated as follows:
\begin{equation}
\begin{aligned}
&p(z_i = k| \mathbf{z}_{-i}, \mathcal{X} , \alpha, \boldsymbol\beta)  \\
& \varpropto p(z_i = k | \mathbf{z}_{-i}, \alpha, \cancel{\boldsymbol\beta})  p(\mathcal{X} |z_i = k, \mathbf{z}_{-i}, \cancel{\alpha}, \boldsymbol\beta) \\
& = p(z_i = k| \mathbf{z}_{-i}, \alpha) p(\mathbf{x}_i |\mathcal{X}_{-i}, z_i = k, \mathbf{z}_{-i}, \boldsymbol\beta)
\\
&\,\,\, \cdot p(\mathcal{X}_{-i} |\cancel{z_i = k}, \mathbf{z}_{-i}, \boldsymbol\beta)\\
& \varpropto p(z_i = k| \mathbf{z}_{-i}, \alpha) p(\mathbf{x}_i|\mathcal{X}_{-i}, z_i = k, \mathbf{z}_{-i}, \boldsymbol\beta) \\
& \varpropto p(z_i = k| \mathbf{z}_{-i}, \alpha) p(\mathbf{x}_i | \mathcal{X}_{k,-i}, \boldsymbol\beta),\\
\end{aligned}
\label{equation:pcrp_ifmm_collabsed_gibbs}
\end{equation}
where $\mathcal{X}_{k,-i}$ are the observations in table $k$ excluding the $i^{th}$ observation and the first term of the last equation above $p(z_i = k| \mathbf{z}_{-i}, \alpha)$ is the proposed pCRP in equation~\eqref{equation:powered_crp_equation}.
If $z_i = k$ is an existing component, the second term above $p(\mathbf{x}_i | \mathcal{X}_{k,-i}, \boldsymbol\beta)$ is calculated using the posterior predictive distribution at $\mathbf{x}_i$, where the posterior prediction distribution of the new data $\mathbf{x}^\star$ given the data set $\mathcal{X}$ and the prior parameter $\boldsymbol\beta$ under the NIW prior is
\begin{equation}
p(\mathbf{x}^\star | \mathcal{X}, \boldsymbol\beta) = \int_{\boldsymbol\mu}  \int_{\boldsymbol\Sigma} p(\mathbf{x}^\star | \boldsymbol\mu, \boldsymbol\Sigma) p(\boldsymbol\mu,\boldsymbol\Sigma | \mathcal{X}, \boldsymbol\beta) d\boldsymbol\mu d \boldsymbol\Sigma.
\end{equation}
When $z_i = k^\star$ is a new component then we have:
\begin{equation}
\begin{aligned}
&p(\mathbf{x}_i | \mathcal{X}_{k,-i}, \boldsymbol\beta) = p(\mathbf{x}_i | \boldsymbol\beta) \\
&= \int_{\boldsymbol\mu}  \int_{\boldsymbol\Sigma} p(\mathbf{x}_i | \boldsymbol\mu, \boldsymbol\Sigma) p(\boldsymbol\mu, \boldsymbol\Sigma|\boldsymbol\beta) d\boldsymbol\mu d \boldsymbol\Sigma,
\end{aligned}
\end{equation}
which is just the prior predictive distribution and can be calculated by the posterior predictive distribution for new data $p(\mathbf{x}^\star | \mathcal{X}, \boldsymbol\beta)$ under NIW prior but with $\mathcal{X} = \{\emptyset\}$.

Algorithm~\ref{algo:pcrp_ifmm_plain_gibbs} gives the pseudo code of the collapsed Gibbs sampler to implement pCRP in Gaussian mixture models.

\begin{algorithm}[tb]
   \caption{Collapsed Gibbs Sampling for pCRP}
   \label{alg:example}
\begin{algorithmic}
\STATE {\bfseries Input:} Choose an initial $\bm{z}$, $r$, $\alpha$, $\boldsymbol\beta$;
\FOR{$T$ iterations}
\STATE $\bullet$ Sample random permutation $\tau$ of $1, \ldots, N$;
	\FOR{$i \in (\tau(1), \ldots, \tau(N))$}
		\STATE $\bullet$ Remove $\mathbf{x}_i$'s statistics from component $z_i$;
		\FOR{$k=1$ {\bfseries to} $K$}
		\STATE $\bullet$ Calculate $p(z_i=k| \mathbf{z}_{-i}, \alpha) =  \frac{N_{k,-i}^r}{\sum_h^K N_{h,-i}^r+\alpha}$;
		\STATE $\bullet$ Calculate $p(\mathbf{x}_i | \mathcal{X}_{k,-i}, \boldsymbol\beta)$;
		\STATE $\bullet$ Calculate $p(z_i = k | \mathbf{z}_{-i}, \mathcal{X}, \alpha, \boldsymbol\beta) \propto p(z_i=k| \mathbf{z}_{-i}, \alpha) p(\mathbf{x}_i | \mathcal{X}_{k,-i}, \boldsymbol\beta)$;
		\ENDFOR
		\STATE $\bullet$ Calculate $p(z_i = k^\star | \mathbf{z}_{-i}, \alpha)= \frac{\alpha}{\sum_h^K N_{h,-i}^r+\alpha}$;
		\STATE $\bullet$ Calculate $p(\mathbf{x}_i | \boldsymbol\beta)$;
		\STATE $\bullet$ Calculate $p(z_i = k^\star | \mathbf{z}_{-i}, \mathcal{X}, \alpha, \boldsymbol\beta) \propto p(z_i = k^\star | \mathbf{z}_{-i}, \alpha) p(\mathbf{x}_i | \boldsymbol\beta)$;
		\STATE $\bullet$ Sample $k_{new}$ from $p(z_i | \mathbf{z}_{-i}, \mathcal{X}, \alpha, \boldsymbol\beta)$ after normalizing;
		\STATE $\bullet$ Add $\mathbf{x}_i$'s statistics to the component $z_i=k_{new}$ ;
	     \STATE $\bullet$ If any component is empty, remove it and decrease $K$.
	\ENDFOR
\ENDFOR
\end{algorithmic}\label{algo:pcrp_ifmm_plain_gibbs}
\end{algorithm}

\section{Experiments}
We conduct experiments to demonstrate the main advantages of the proposed pCRP using both synthetic and real data. In a wide range of scenarios across various sample sizes, pCRP reduces over-clustering of CRP, and leads to performances that are as good or better than CRP in terms of density estimation, out of sample prediction, and overall clustering results.

In all experiments, we run the Gibbs sampler 20, 000 iterations with a burn-in of 10, 000. The sampler is thinned by keeping every 5$^{th}$ draw. We use the same concentration parameter $\alpha = 1$ for both CRP and pCRP in all scenarios. In addition, we equip CRP with an unfair advantage to match the magnitude of its prior mean $\alpha \log(N)$ to the true number of clusters, termed {\it CRP-Oracle}. The power $r$ in pCRP is tuned using cross validation. In order to measure overall clustering performance, we use normalized mutual information (NMI) \cite{mcdaid2011normalized} and variation of information (VI) \cite{meilua2003comparing}, which measures the similarity between the true and estimated cluster assignments. Higher NMI and lower VI indicate better performance.  If applicable, metrics using the true clustering are calculated to provide an upper bound for all methods, coded as `Ground Truth'.

\subsection{Simulation experiments}\label{sec:simulated_examples}
We first use simulated data to assess the performance of pCRP in emptying extra components, compared to the traditional CRP.  Figure~\ref{fig:pcrp_prior_densities} shows the true data generating density, which represent the two cases of well-mixed Gaussian components and shared mean Gaussian mixture coded as Sim 1 and Sim 2, respectively.

The oracle concentration parameters in CRP-Oracle are (0.52, 0.40) in Sim 1 and (0.52, 0.40) in Sim 2, which are all smaller than the unit concentration parameter used in CRP and pCRP.  The sample sizes in the two simulation cases are respectively (300, 2000).  Figure~\ref{fig:pcrp_cross-validation} shows the cross validation curve to select $r$ in pCRP using a training data set with 200 samples. The representative cases of inflection point described in Section \ref{sec:pcrp_parameter_calibration} were observed: the loss curve for cross validation `blows up' for one particular $r$ value in both Sim 1 and Sim 2.  We can find the first stage of Sim 2 is oscillating from 14.2595 to 14.2005 which is flatter than that of Sim 1 that oscillates from 15.5260 to 14.9020. This is because the components in Sim 2 are well separated. We choose this change point as the power $r$ in either case.

\begin{figure}[h!]
\center
\subfigure[Sim 1]{\includegraphics[width=0.228\textwidth]{img_pcrp/true_density_sim2.pdf} \label{fig:true_density_sim2}}
~
\subfigure[Sim 2]{\includegraphics[width=0.228\textwidth]{img_pcrp/true_density_sim8.pdf} \label{fig:true_density_sim6}}
\center
\caption{Data generating densities for two scenarios: (a) Sim1: a mixture of three poorly separated Gaussian components; (b) Sim 2: a mixture of three well separated Gaussian components.}
\label{fig:pcrp_prior_densities}
\end{figure}

\begin{figure}[h!]
\center
\subfigure[Sim 1]{\includegraphics[width=0.228\textwidth]{img_pcrp/cv_sim2.pdf} \label{fig:pcrp_cv_sim2}}
~
\subfigure[Sim 2]{\includegraphics[width=0.228\textwidth]{img_pcrp/cv_sim8.pdf} \label{fig:pcrp_cv_sim6}}
\center
\caption{Cross validation curves to choose $r$ for Sim 1 and Sim 2. The $x$-axis is the power value, the $y$-axis is the loss. The vertical line is the chosen power $r$ value.}
\label{fig:pcrp_cross-validation}
\end{figure}

\begin{figure}[!h]
	\center
	\subfigure[ CRP-Oracle]{\includegraphics[width=0.40\textwidth]{img_pcrp/posterior_num_density/cluster_num_crp-04.pdf} \label{fig:pcrp_clusternum_3methods_traceplot_sim1_crp04}}
	\hfill
	\subfigure[ CRP]{\includegraphics[width=0.23\textwidth]{img_pcrp/posterior_num_density/cluster_num_crp-1.pdf} \label{fig:pcrp_clusternum_3methods_traceplot_sim1_crp1}}
	\hfill
	\subfigure[ pCRP]{\includegraphics[width=0.23\textwidth]{img_pcrp/posterior_num_density/cluster_num_pcrp-1.pdf} \label{fig:pcrp_clusternum_3methods_traceplot_sim1_pcrp1}}
	\center
	\caption{Traceplots of cluster numbers using the three methods in Sim 1 when $N=2000$. The $x$-axis is the sampling iteration, the $y$-axis is the number of clusters.}
	\label{fig:pcrp_clusternum_3methods_traceplot_sim1}
\end{figure}

\begin{figure*}[!h]
	\center
	\subfigure[CRP-Oracle]{\includegraphics[width=0.32\textwidth]{img_pcrp/posterior_num_density/post_components_crp-04.pdf} \label{fig:adhoc_3methods_traceplot_sim1_posterior_densities_crp_match}}
	\hfill
	\subfigure[CRP]{\includegraphics[width=0.32\textwidth]{img_pcrp/posterior_num_density/post_components_crp-1.pdf} \label{fig:adhoc_3methods_traceplot_sim1_posterior_densities_crp1}}
	\hfill
	\subfigure[pCRP]{\includegraphics[width=0.32\textwidth]{img_pcrp/posterior_num_density/post_components_pcrp-1.pdf} \label{fig:adhoc_3methods_traceplot_sim1_posterior_densities_pcrp1}}
	\center
	\caption{Posterior densities for three methods in Sim 1 when $N=2000$. The dashed lines are weighted components.

	}
	\label{fig:pcrp_3methods_traceplot_sim1_posterior_densities}
\end{figure*}

Figure~\ref{fig:pcrp_clusternum_3methods_traceplot_sim1} shows traceplots of posterior samples for the number of clusters for each of the methods in Sim 1.  Clearly pCRP places relatively high posterior probability on three clusters, which is the ground truth. In contrast, CRP has higher posterior variance, systematic over-estimation of the number of clusters, and worse computational efficiency.  The CRP-Oracle has better performance, but does clearly worse than p-CRP, and there is still a tendency for over-estimation.  This demonstrates that one cannot simply fix up or calibrate the CRP by choosing the precision to be appropriately small.
Figure~\ref{fig:posterior_num_clusters} suggests that CRP will have larger probability on larger cluster numbers especially when the sample size increases, while pCRP tends to have larger probability on the true cluster number as the sample size increases.
For example, in Sim 1, the probability of selecting three clusters increases from 0.55 to 0.68 in pCRP when $N$ increases from 300 to 2000 and the probability for all the other cluster number decreases.
However, the probability of finding four clusters stabilizes around 0.37 and 0.38 in CRP-Oracle when $N$ increases from 300 to 2000. CRP has increased probability of selecting larger number of clusters (say 5, 6, 7, 8 clusters) when $N$ increases from 300 to 2000. In fact, the proposed pCRP has the largest concentration probability on the true number of clusters among all the three methods including CRP-Oracle, and this observation is consistent between $N = 300$ and $N = 2000$.

Table~\ref{table:pcrp_simulation_posterior_summary} provides numerical summaries of this simulation. We can see all three methods lead to similar NMI, but pCRP consistently gives the highest value. Furthermore, pCRP leads to the lowest value of VI in most tests. The parsimonious effect of pCRP discussed above is further confirmed by the average and maximum number of clusters; see the columns $K$ and $K_{\max}$ in the table.

The posterior density plots in Figure~\ref{fig:pcrp_3methods_traceplot_sim1_posterior_densities} show that there is one small unnecessary cluster in CRP-Oracle and two small unnecessary clusters in CRP, while all three methods capture the general shape of the true density and thus provide good fitting performance. The over-clustering effect of CRP is much reduced by pCRP as seen in Figure~\ref{fig:adhoc_3methods_traceplot_sim1_posterior_densities_pcrp1}.

\begin{figure}[h!]
	\center
	\subfigure[Sim 1, $N$=300]{\includegraphics[width=0.22\textwidth]{img_pcrp/posterior_num_clusters_sim2_n300.pdf} \label{fig:posterior_num_sim2_n300}}
	~
	\subfigure[Sim 2, $N$=300]{\includegraphics[width=0.22\textwidth]{img_pcrp/posterior_num_clusters_sim62_n300.pdf} \label{fig:posterior_num_sim6_n300}}
	~
	\subfigure[Sim 1, $N$=2000]{\includegraphics[width=0.22\textwidth]{img_pcrp/posterior_num_clusters_sim2_n2000.pdf} \label{fig:posterior_num_sim2_n2000}}
	~
	\subfigure[Sim 2, $N$=2000]{\includegraphics[width=0.22\textwidth]{img_pcrp/posterior_num_clusters_sim62_n2000.pdf} \label{fig:posterior_num_sim6_n2000}}
	\center
	\caption{Estimated posterior of the number of clusters in observed data for CRP-Oracle (red x), CRP (blue circle) and pCRP (green star). }
	\label{fig:posterior_num_clusters}
\end{figure}

\begin{table*}[h!]
\centering
\begin{tabular}{|ccccc|}
\hline
\multicolumn{5}{|c|}{$N=300$}                                                                     \\ \hline
Method               & NMI (SE) & VI (SE) & $K$ (SE) & $K_{\max}$ \\ \hline
Ground truth (Sim 1) & 1.0                & 0.0               & 3                  & -            \\
CRP-Oracle (Sim 1)   & 0.800 (1.1$\times 10^{-3}$)        & 0.669 (4.5$\times 10^{-3}$)       & 4.2 (2.3$\times 10^{-2}$)          & 8            \\
CRP (Sim 1)          & 0.773 (1.2$\times 10^{-3}$)        & 0.795 (5.4$\times 10^{-3}$)       & 5.3 (3.3$\times 10^{-2}$)          & 12           \\
pCRP (Sim 1)         & \textbf{0.827} (0.7$\times 10^{-3}$)       & \textbf{0.580} (4.4$\times 10^{-3}$)       & \textbf{3.6} (1.7$\times 10^{-2}$)          & \textbf{7}            \\ \hline
Ground truth (Sim 2) & 1.0                & 0.0               & 2                  & -            \\
CRP-Oracle (Sim 2)   & 0.937 (0.8$\times 10^{-3}$)        & 0.210 (3.0$\times 10^{-3}$)       & 4.0 (2.3$\times 10^{-2}$)          & 9            \\
CRP (Sim 2)          & 0.917 (0.9$\times 10^{-3}$)        & 0.287 (3.7$\times 10^{-3}$)       & 4.8 (2.9$\times 10^{-2}$)          & 12           \\
pCRP (Sim 2)         & \textbf{0.963} (0.3$\times 10^{-3}$)        & \textbf{0.116} (0.8$\times 10^{-3}$)       & \textbf{3.2} (0.9$\times 10^{-2}$)          & \textbf{6}            \\ \hline \hline
\multicolumn{5}{|c|}{$N=2000$}                                                                    \\ \hline
Method               & NMI (SE) & VI (SE) & $K$ (SE) & $K_{\max}$ \\ \hline
Ground truth (Sim 1) & 1.0                & 0.0               & 3                  & -            \\
CRP-Oracle (Sim 1)   & 0.812 (5.3$\times 10^{-4}$)        & 0.610 (2.6$\times 10^{-3}$)       & 4.0 (2.3$\times 10^{-2}$)          & 10           \\
CRP (Sim 1)          & 0.782 (8.5$\times 10^{-4}$)        & 0.732 (4.0$\times 10^{-3}$)       & 5.8 (3.6$\times 10^{-2}$)          & 12           \\
pCRP (Sim 1)         & \textbf{0.823} (6.6$\times 10^{-4}$)        & \textbf{0.869} (7.3$\times 10^{-3}$)       & \textbf{3.5} (1.6$\times 10^{-2}$)          & \textbf{7}            \\ \hline
Ground truth (Sim 2) & 1.0                & 0.0               & 2                  & -            \\
CRP-Oracle (Sim 2)   & 0.962 (4.8$\times 10^{-4}$)        & 0.122 (1.7$\times 10^{-3}$)       & 4.1 (2.2$\times 10^{-2}$)          & 8            \\
CRP (Sim 2)          & 0.940 (8.2$\times 10^{-4}$)        & 0.205 (3.1$\times 10^{-3}$)       & 5.6 (3.5$\times 10^{-2}$)          & 13           \\
pCRP (Sim 2)         & \textbf{0.977} (1.2$\times 10^{-4}$)        & \textbf{0.072} (0.4$\times 10^{-3}$)      & \textbf{3.1} (0.8$\times 10^{-2}$)          & \textbf{6}            \\ \hline
\end{tabular}
\caption{Comparison of CRP and pCRP on Sim 1 and Sim 2. $K$ is the average number of found clusters. $K_{\max}$ is the maximum number of clusters during sampling. SE is the standard error of mean. Ground truth is calculated using the true assignments.}
\label{table:pcrp_simulation_posterior_summary}
\end{table*}

\begin{figure}[h!]
	\center
	\subfigure[True clustering when $N$=3000]{\includegraphics[width=0.42\textwidth]{img_pcrp/digits_posterior_clustering/true_clustering.pdf} \label{fig:pcrp_digits14_n3000_true_clustering}}
	~
	\subfigure[CRP-Oracle when $N$=3000]{\includegraphics[width=0.228\textwidth]{img_pcrp/digits_posterior_clustering/ellipse_n3000_crp_match.pdf} \label{fig:crp_digits14_n3000_crp_match}}

	~
	\subfigure[pCRP when $N$=3000]{\includegraphics[width=0.228\textwidth]{img_pcrp/digits_posterior_clustering/ellipse_n3000_pcrp1.pdf} \label{fig:crp_digits14_n3000_pcrp1}}
	\center
	\caption{Results of clustering 3000 randomly sampled digits from 1 to 4 in spectral space. Observations in the same color represent the same digit. CRP-Oracle seems to over-fit the noise (the red cluster). We omit the result of CRP as it is similar to CRP-Oracle.}
	\label{fig:pcrp_digits14}
\end{figure}

\subsection{Digits 1-4}
In this experiment, we cluster 1000 and 3000 digits of the classes 1 to 4 in MNIST data set \cite{lecun2010mnist}, where the four clusters are approximately equally distributed. From cross validation on a different set of 1000 samples, we obtain the power value $r=1.05$. The concentration parameter $\alpha$ in CRP-Oracle is calculated as 0.58 ($N = 1000$) and 0.5 $(N = 3000)$.

Figure \ref{fig:pcrp_digits14} shows the clustering result of all the three methods for $N=3000$. Both CRP and CRP-Oracle seem to over-fit the data by introducing a small cluster (in red), while pCRP gives a cleaner clustering result with four clusters. This comparison is further confirmed by Table~\ref{table:pcrp_digits_result}, where the average posterior cluster number in CRP apparently increases when $N$ grows to 3000. In contrast, pCRP is closer to the true situation by reducing the over-clustering effect, even compared to CRP-Oracle; see the columns of $K$ and $K_{\max}$. All methods lead to similar NMI but pCRP gives lower VI.

\begin{table*}[h!]
\centering
\begin{tabular}{|ccccc|}
\hline
\multicolumn{5}{|c|}{$N=1000$}                                                                    \\ \hline
Method               & NMI (SE) & VI (SE) & $K$ (SE) & $K_{\max}$   \\ \hline
Ground truth         & 1.0                & 0                 & 4                  & -            \\
CRP-Oracle           & \textbf{0.651} (3.3$\times 10^{-4}$)        & \textbf{1.382} (1.4$\times 10^{-3}$)       & 4.37 (1.3$\times 10^{-2}$)         & 7            \\
CRP                  & \textbf{0.651} (3.3$\times 10^{-4}$)        & 1.386 (1.4$\times 10^{-3}$)       & 4.58 (1.6$\times 10^{-2}$)         & 8            \\
pCRP                 & \textbf{0.651} (3.3$\times 10^{-4}$)        & \textbf{1.382} (1.4$\times 10^{-3}$)       & \textbf{4.08} (0.6$\times 10^{-2}$)        & \textbf{6}            \\ \hline \hline
\multicolumn{5}{|c|}{$N=3000$}                                                                    \\ \hline
Method               & NMI (SE) & VI (SE) & $K$ (SE) & $K_{\max}$   \\ \hline
Ground truth         & 1.0                & 0.0               & 4                  & -            \\
CRP-Oracle           & 0.651 (2.0$\times 10^{-4}$)        & 1.400 (1.1$\times 10^{-3}$)       & 5.17 (1.2$\times 10^{-2}$)         & 8            \\
CRP                  & 0.651 (2.0$\times 10^{-4}$)        & 1.402 (1.1$\times 10^{-3}$)       & 5.44 (1.6$\times 10^{-2}$)         & 9            \\
pCRP                 & \textbf{0.652} (1.9$\times 10^{-4}$)        & \textbf{1.389} (1.1$\times 10^{-3}$)       & \textbf{4.57} (1.2$\times 10^{-2}$)         & \textbf{7}            \\ \hline
\end{tabular}
\caption{Comparison of CRP-Oracle, CRP and pCRP on a 4 digits subset of MNIST. $K$ is the average number of discovered clusters. $K_{\max}$ is the maximum number of clusters during sampling.}
\label{table:pcrp_digits_result}
\end{table*}

\begin{figure}[!h]
	\center
	\subfigure[The distribution of Old Faithful Geyser after standardization]{\includegraphics[width=0.32\textwidth]{img_pcrp/img_oldfaithful/data-distribution.pdf} \label{fig:adhoc_oldfaithful_data_distribution}}
	~
	\subfigure[Clustering results using CRP-Oracle, CRP, pCRP and manual clustering.]{\includegraphics[width=0.32\textwidth]{img_pcrp/img_oldfaithful/bar_comparison2.pdf} \label{fig:adhoc_oldfaithful_bar_comparison}}
	\center
	\caption{Clustering result for Old Faithful Geyser}
	\label{fig:adhoc_oldfaithful}
\end{figure}

\subsection{Old Faithful Geyser}
The Old Faithful Geyser data ($N=272$) are widely used to illustrate the performance of clustering algorithms.
We use a test sample of 100 in CV leading to the power value $r=1.11$. We compare all methods on the other 172 data points. A manual clustering that consists of two Gaussian components is viewed as the ground truth. The concentration parameter is 0.39 in CRP-Oracle.  Figure~\ref{fig:adhoc_oldfaithful_bar_comparison} shows the size of each component obtained from all methods and the manual clustering. We can see that there are two mixture components in CRP-Oracle and pCRP, and four mixture components in the CRP method. In this case where the sample size is relatively small, we again see that pCRP successfully suppresses small components and generates more parsimonious results than CRP.

\pagebreak

redpajama_set_name: RedPajamaArXiv


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