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UFO seen on soldier's video 25 Jun 2008
A SHAKEN soldier told how he saw THIRTEEN UFOs spinning in the skies above his military barracks.He was among three squaddies who spotted the objects while out on night patrol. He filmed them on his mobile phone and reported the close encounter to Army top brass (source the Sun):
Ufo Hunters New Episode - Storm
In May of 2008, the UK's Ministry of Defense made decades worth of classified files on UFOs freely available to the public. One month later, UFO hysteria ripped
through the country. In June, a police helicopter crew spotted unusual aircraft in Wales. South Wales Police confirmed the sighting and reported it to the Ministry
of Defense. Numerous witnesses and videos poured into Britain's top newspapers of more sightings--UFO madness had come to the UK. Former MOD investigator Nick Pope
joins the team in solving the most exciting and recent UFO sightings in the world.
Ufo Insiders - Robert Dean - Phillip Krapf
Robert Dean talks on Ufo Insiders
UFO Spotted Three Times in Shropshire
Extract from this source article
Ministry of Defence experts were making a thorough investigation into a report plus video evidence of a UFO sighting that a shocked soldier and two of his squaddie mates had witnessed at Market Drayton, Shropshire.
The three were on guard duty when they claimed to have seen some objects above the military barracks. After capturing the unusual object on a mobile phone, they took the evidence to a senior officer.
The top brass have since put the solder in question under strict silence. Two hours after the sighting, police officers in Cardiff, 80 miles away from Market Drayton, had an encounter with a strange craft in the sk
David Cameron - a vote for him = Ufo disclosure ?
If you live in the Uk then please take this into consideration when you next vote for prime minister:
David Cameron, the leader of the opposition part in the Uk has come forward playing the Ufo card saying a vote for him as leader is a vote for full Ufo disclosure:
David Cameron " If I became prime minister I would always be entirely open and frank about these things "
Below is extract from this article
David Cameron has vowed to publish any secret files that may exist on UFOs if he becomes prime minister.
Speaking at one of his regular Cameron Direct public meetings, he pledged to be "open and frank" with any government knowledge about close encounters.
The Conservative leader was responding to a question about apparent government cover-ups of UFO activity.
Mr Cameron joked: "I'm convinced we've been visited by aliens - one of them is trade secretary Peter Mandelson."
At the meeting in Tynemouth, North Tyneside, he was questioned about a spate of mysterious incidents.
A member of the audience asked: "In July last year the respected scientist and astronaut Dr Edgar Mitchell, who was the sixth man to walk on the moon, spoke on a British radio station.
"He said the American government had had contact with extra terrestrials on multiple occasions and that these were ongoing.
"He spoke about the Roswell event in 1947 where wreckage of a downed UFO was recovered and found to contain alien bodies.
"He said this event was real but was covered up by the government for many years.
"Do you agree with me that the British people have a right to know if we have been visited, and if so, when you become prime minister will you seek to lift the veil of secrecy and give the public the truth that they deserve?
read more ,.. source
Ufo Dec 21 2008
Nick pope speaks on Fox News about airforce shooting down ufos JAN 27 2009
Former Uk ministry of defense officer Nick pope is yet again in the media (on Fox)- this time over his recent comments about the Uk Airforce shooting down Ufos. I am starting to think is the Uk government sponsoring him on his Ufo disclosure campaign ? - Nick is always there when Ufo news comes out from the Uk - he is doing alot of good work and is well respected in the media and in ufo circles, he worked for the Uk government in their Ufo research program so i guess he would be the ideal person to support to get the word out - like what he is doing right now ..
Denmark releases all its Ufo Files
As the Eu starts to expand its Ufo disclosure campaign amoung member states, following Frances lead, Denmark is the most recent country to dislose all its classified Ufo Files to the Public.
Below extract from this article source
Read one of the most recent secret translated files below:
Secret File: 15-year-old then rummænd on Fyn
28. January 2009 08.56 Inland
The bell was 04.30 early morning in the summer of 1982. Observers - a 15-year-old - was cycling on the way to his work as summer successor at a nursery. Approximately 15 minutes later he spotted something very illuminating 50-60 meters contained in a field.
He rose to the bike and walked toward the bright object, he wondered about the strange phenomenon. The subject is described as two deep dishes placed on top of each other around. 2 meters high and 2 meters in diameter. There were no visible details on the surface, apart from something that resembled a "member" of one side.
10-15 meters from the subject observer could see five figures. The so human, but was not higher than 60 cm (!)., had large head and chest and small hips and legs.
To start one of the observations in Denmark's past secret ufoarkiv, which today has been put online.
Danish security has nothing to do with ufoer
Thus, Denmark has followed the United States, England and France, have already made public the archives of ufoobservationer.
One of those who have been closest to the previously confidential files of Thomas Pedersen set by Air Force Captain. He has had the Danish ufoarkiv lying on his office for many years.
- We have decided to publish it because there is nothing secret in it as such. We have continuously been approached by journalists who want to see this archive. And when we told them that they like, because there is nothing secret in it, then lose interest, he says the Go 'Morgen P3.
- Air Force has no interest in maintaining secrecy on ufoer. Our mission is to ensure the Danish security in peacetime. So we have no interest in investigating ufoer.
Danish defense hose gays against ufoer
The archive from the Air Force is in no fewer than 329 pages and range from 1978 to 2002. Since then, the association Skandinavisk UFO Information taken care of all reports of ufoer.
Most of the reports are due the Air Force has never been solved. But by observations that have been precise and detailed enough to investigate and clarify the ufoen typically proved to be either a military or civilian aircraft, a weather phenomenon, "China lamps, a bright planet or something else quite natural.
Today, the Air Force only tasked to provide pieces of the investigative work in the form of information about where and when, there have been military aircraft in Danish airspace
Captain Thomas Pedersen also rejects that the defense has ever shot against a UFO:
- If you did, I would personally think it was one of the stupidest one could do. Because if now we see a movie as Independence Day, they are the highly sophisticated with their equipment. When we come to our small catapult against a superior unit - so I think it would be a defeat, he says with a grin.
Click here for link to article source
Read on of the secret translated files below:
UFO sphere filmed Dundee, Scotland 24/1/09
ufo seen in East of Dundee, Scotland
UK Airforce chiefs gave orders to shoot down ufos - Nick Pope
The Media are all over this story which i posted on recently at the moment.
Nick pope former Uk MOD staff talks about the Uk airforces proceedure to shoot at Ufos
Some thing seen over louisiana Jan 2009
These strange objects were seen over louisiana in the USA earlier this month. were they lanterns or ufos? Please Note to all - remember its Chinese New Year now (jan 26) , the chinese let many 'lanterns' go around this time so don't confuse current sightings with these object ....
Ovni USA (part.30) (Louisiane janvier 2009)
Uploaded by ufo_ovni17
British Air Force Chiefs ordered RAF Pilots to shoot UFOs
The Ufo topic keeps Nick Pope a busy man these days people. The former Uk Ministry of defense staffer seems to be involved with many ufo reports at the moment on the TV, internet and in books, debates and documentaries - keep up the good work - the Ufo community appreciate your efforts!
He recently reported to the Sun paper in the UK (who must also be congratulated for their non stop reporting on the Ufo topic ) about how the Uk Air force is ordered to deal with Ufos - shoot them down!. Its seems like the most irresponsible thing to do - to provoke a more advanced race upon first contact - not a wise thing to do considering they can and have easily rendered our most powerful defense (nuclear weapons) useless.
If Ufos are a threat to the safety of our airspace then why doesn't our goverment admit it?. I'm sure thousands of people flying in passenger aircraft all around the world at the moment would be very concerned if they knew that there is always a chance a Ufo could hit their plane at any time - because in reality governments have & will never be able to keep airspace secure against ufos....
But then again if our most powerful weapons can't protect us from these craft i guess you wouldn't want to tell the public right? Makes sense ... there is a more serious issue at stake here guys ... its massive a national security concern - we are completely defenseles against ufos (for once something out of the governments control) - thus the secrecy:
Nick pope explains this further in the below extract from this article in the Sun Paper :
RAF pilots have tried to BLAST UFOs out of the sky under a top secret Government directive, it was claimed last night.
Nick Pope — who worked on the Ministry of Defence's UFO desk for three years — revealed the rules of engagement for the first time.
He claimed RAF pilots had fired at UFOs on several occasions — but failed to bring them down.
He added: "We know of cases where the order has been given to shoot down — with little effect to the UFO." Mr Pope said the rules of engagement were drawn up after dozens of close encounters with suspect craft in British airspace.
RAF attacks on UFOs were "not automatic but happen when something in our airspace is deemed to be a threat".
He explained: "In the case of UFOs, whether the object is causing a threat is very much a (pilot's) judgment call."
Mr Pope, 43, from London, said the "shoot down" orders had been issued under the highly-classified directive since the beginning of the 1980s. He believes a pattern has emerged in incidents already publicised in this country and abroad.
When a UFO is thought to be threatening a country's airspace, the drastic action has been taken. He said: "There was a faction in the MoD who said 'We want to shoot down a UFO and that will resolve the issue one way or another'.''
He claimed credible UFO witness statements had come from dozens of near-misses with planes, police helicopters and RAF jets in recent years.
He said: "The public won't know unless it comes down in a heavily-populated area."
Weapons He added: "I do believe we will bring one down. We're developing increasingly sophisticated weapons."
Mr Pope also rubbished the MoD's stance that UFOs pose no danger to the public, saying: "I think that's a line I wrote myself in the 1990s.
Read More .. source the SUN
Unrelated but interesting ... Pilot all over the world including this video below are often told to track and shoot down Ufos .. is this a wise thing people?
SKY News UFO footage Jan '09 Live from Pontefract Conference
This is an interview at the Pontefract Conference, it shows footage of one of the most famous ufo sightings that took place in turkey and was filmed by a security guard and witnessed by many people. the Ufo was up in the sky for about 3 hours over a 3 nights: (once again please ignore the reporters attitude)
The Prime Minister & Queen were lied to about crop circles - Colin Andrews
If the both the Prime minister and head of state can't be told the truth about crop circles then what chance do we the public have? These are the bold new accusations made from the the world most senior expert on the Crop circle phenomenon - Colin Andrews.
Andrews who has been the world wide authority on crop circles since the early 80's and has received many letters from the Queen confirming her interest crop circles and he is now making a stand on what he claims is a 'culture of cover ups' from the UK Ministry of Defense.
News reports about the letters from the Queen to Colin confirming her interest in Ufos and crop circles have hit the media (see the Sun article ) and Colin has received much pressure to back down on his claims.
In fact, after releasing sensitive documents on his own website last week he was forced (probably via legal threats) to censor and remove his own posts (see remove article in question here):
Below Comments sourced from Colin Andrews Website
"In the last 24 hours I have been instructed to remove one of the many documents posted on my site in that article. I stand by the complete contents of my allegation that Prime Minister Margaret Thatcher, the Queen and the public were lied to regarding Crop Circles and UFOs.
I also stand by my commitment to lobby for a new period of openness and honesty regarding such topics as the UFO and crop circles.
No government has the right to control the growth of human consciousness.
My article was written and support documents have been released for the first time to support this new era and to see to it that the peoples business really does bring in the tide of change that we all deserve.
Nick pope Former Uk Ministry of defense reply to Colin Andrews comments:
Colin Andrews and I have discussed this over the past few days, as his article implies I was involved ina cover-up.
Some of the issues Colin highlights have arisen because the line between official and private interest (even within government, the military and the intelligence agencies) can be blurred.
There are particular sensitivities and political difficulties
when members of the Royal Family are involved.
I agree that the MoD has downplayed its interest and involvement in the crop circle issue, as it has with the UFO phenomenon. This has been the policy for many years and it's no secret that I had to play my part in this when
I worked for the government.
Here's an article I wrote, setting out the official position i have on crop circles
Also by the way Colin says that the letter from the Queen is only the first stage of his release of documents exposing the crop circle phenomenon. The second stage will be letters from private industry and certain officials.. keep you posted on this guys..
Comments sourced from Colin Andrews Website
More media coverage on this topic here
Secretary of State Hillary Clinton & her interest in Ufos
It looks like almost everyone in the new Obama administration are both prepared for and have been ideally chosen for Ufo disclosure . As you know Hillary Clinton is the new Secretary of State and she too is aware for the pressing need to look into the phenomenon.
If you have time check out www.Hillaryclintonufo.net. This website is likely is endorsed by the Laurence Rockefeller association and looks at Hillary's connection to the Ufo topic and has interesting documents showing what Rockefeller briefed her on.
The below extract from this very interesting article looks at Hillary's
connection to the Ufo topic:
Newly confirmed Secretary of State, Hillary Clinton has long had an interest in UFOs and the existence of extraterrestrial life. She was given an informal briefing in August 1995 by billionaire philanthropist Laurence Rockefeller, on the best available UFO evidence. An August 4, 1995, memo by President Clinton's Science Advisor, Dr Jack Gibbons, confirmed that the Clinton's would be vacationing with Rockefeller at his Teton Ranch in Wyoming that summer, and they would discuss UFOs:
"You will probably see Mr Rockefeller on your vacation in the Tetons. He will want to talk with you about his interest in extrasensory perception, paranormal phenomenon, and UFOs."
In the memo, Gibbons then attempted to dissuade the Clintons from pursuing Rockefeller's agenda to disclose national security files concerning UFOs:
"He knows that we are trying to be helpful in responding to his concerns about UFO's and human potential … but I've made no secret about my conviction that we must not be too diverted from more earthly imperatives."
Both Clintons were involved in numerous attempts to force the release of national security files concerning UFOs.
Read More .... Article Source
Admins needed for new UFO discussion Forum
I am looking to re-establish our Chat Forum which taken down a few weeks ago due to ongoing attacks and spammers. Many people have requested that they would like the forum back but it must be easier to navigate and be free from spam.
To achieve this I need some hard working admins to help moderate comments to ensure the forum remains a fun and interesting place to be. This is a great opportunity to be part of the global debate on the ongoing discussion of the Ufo phenomenon.
Also if you are Ufo blogger or have a Ufo website i am willing to co-share our established forum with your site if you are able to contribute to assist in moderating the forum. This includes adding your banners & info to the forum and this will bring great exposure to your site.
Any interested people please email me, describing yourself and any previous forum moderation experience.
Matt@RealUfos
Newsflash: Queens Letter confirms the British Royal Family interest in Ufos & Crop Circles !
More huge news people. .. just keeps rolling in,
Colin Andrews the world most famous and well known Crop Circle researcher has recently been requested by the royal family to remove certain letters from the Queen to him posted posted on his website confirming the Queens interest in Crop circles and Ufos.
Click here to view the Amazing documents before they are
permanently removed.
Her Majesty Queen Elizabeth:"Has taken an interest in the crop circles for some time".
Below Extracts from this amazing article
Colin Andrews has been asked to remove details of the British Royal's family interest in crop circle research from his website. On the site he states that their interest was aroused because crop circles appeared on their land.
Colin's book 'Circular Evidence' was on the Queen's Summer Reading list in 1989 and HRH Prince Philip, Duke of Edinburgh subscribed to his Circles Phenomenon Research Newsletter. The Duke also asked Colin to keep him up-to-date with his research.
Colin states that the British public has been continually lied to in regard to the Royal Family's interest in UFOs. He thinks it is about time the public is told the truth about this matter.
Nick Pope who was formerly head of the Ministry of Defence's UFO department replies that:
"Some of the issues Colin highlights have arisen because the line between official and private interest - even within government, the military and the intelligence agencies - can be blurred. There are particular sensitivities and political difficulties when members of the Royal Family are involved.
"I agree that the MoD has downplayed its interest and involvement in the crop circle issue, as it has with the UFO phenomenon. This has been the policy for many years and it's no secret that I had to play my part in this when I worked for the government."
The Royal family's interest in UFOs actually goes back to the 1950s. Earl Louis Mountbatten was one of the first subscribers to the highly-influential "Flying Saucer Review" magazine when it began publication in 1955, and he received copies right up to the time of his death in August 1979.
The Duke of Edinburgh kept a map at Buckingham Palace showing the location of UFO sightings on his wall and he asked the British Air Ministry to send him a copy of every UFO report they received. He also allowed his equerry, Sir Peter Horsley, to discreetly study UFO cases and to bring UFO witnesses to the Palace for private discussions.
Read more .. source article
Interesting video - Colin Andrews military helicopters balls of light and crop circles
New NASA legal nightmare ? - former employees now support claims from UFO hacker Gary McKinnon
Guys more big news again ...
As you may know on Friday NASA ufo hacker Gary McKinnon won the right to a judicial review of a Home Office decision to extradite him to the US. This basically means its more and more likely that Gary will win the right to have his trial in the UK and not be sent to the USA with a harsh sentence.
In addition the new Obama administration is aware that further attention to Gary's case is negative news for them as it comes at a particularly sensitive time - considering Obama's recent decision to shut down Guantanamo bay. The Bush administrations endorsed prosecution of Gary is losing all political and public support - especially in light of the new shocking evidence brought up from former NASA employees who support Gary's claims NASA has altered or destroyed its photos containing images of UFOs ....
Extract from this article:
Evidence against NASA could scrap NASA's case against UK hacker Gary McKinnon. Credible witnesses have claimed that NASA has altered or destroyed its photos containing images of UFOs. This could become a legal and public relations concern for NASA.
The space agency is attempting to prosecute McKinnon for hacking into NASA computer files. McKinnon has stated that he saw UFO-related files in NASA's computers. But NASA has denied any "cover-up".
NASA's claim of innocence faces a serious challenge. Some of the whistleblowers are former NASA employees and contractors with inside knowledge of NASA's operation. If NASA's destruction of public property is confirmed, the alleged cost of McKinnon's hacking would be relatively insignificant.
Part of NASA's mission is to look for signs of intelligent life in outer space. So asking for more money to 'look', after they've already destroyed evidencethat they 'found', is a not going to be easy.
Video - Former NASA employee Donna Hare, and former NASA engineer John Schuessler & U.S. Air Force Sergeant Karl Wolfe
Among these whistleblowers are U.S. Air Force Sergeant Karl Wolfe, former NASA employee Donna Hare, and former NASA engineer John Schuessler. Wolfe and Hare exposed some of NASA's misdeeds at the May 9, 2001 Disclosure Project press conference in Washington, D.C. They both offered to give similar testimony before the U.S. Congress.
Donna Hare had a secret clearance while working for NASA contractor, Philco Ford. She testifies that she was shown a photo of a picture with a distinct UFO. Her colleague explained that it was his job to airbrush such evidence of UFOs out of photographs before they were released to the public. She also heard information from other Johnson Space Center employees that some astronauts had seen extraterrestrial craft and that when some of them wanted to speak out about this they were threatened." [See Hare's testimony at 100:10 minutes into the May 9, 2001 Disclosure Project press conference.
John Schuessler is a Denver area resident and retired aerospace engineer from Lockheed Martin. He worked at NASA on nearly every manned U.S. space flight since its inception. In a March 19, 2008 lecture in Lakewood, CO, Schuessler talked about seeing an unfamiliar photo from one of the Apollo moon landings.
Continued .. read more here source
UFO in Obama Inauguration Video - What ?
Ok people,
I have really hesitated to post this video on the possible Ufo seen during Obama's Inauguration because it thought it was all hype myself. However, now as this has made the news and is doing the rounds in the media - i thought it would be good to discuss the footage here to see why its causing such a stir....
As you may know the Obama administration is good news for Ufo disclosure - as explained in my recent post the new administration is going to make astonishing broad sweeping changes to ensure that the US develops an "open government" to enable Ufo disclosure. So is there any co-incidence with the recent Ufo sighting during the Inauguration - who knows?. It does sound crazy, but some people think Ufos are always here to observe our progress - infact ufos have been seen over many major events where the impacts on the future of mankind are important (like the popes funeral and new popes Inauguration) - so could this video be an example of just that?
Here's the footage in question - what do you think?
Ufo Over Bucks County Pennsylvania, 24th January, 2009
A Ufo was seen recently over Bucks County Pennsylvania on the 24th January 2009. One witness said: "It looked most like a star except for the rapidly changing color to it and it was far brighter than any other star in the sky".
Sadly the reporters have ridiculed & ruined this story - what are the media networks really scared of? - They always try to give light humour to Ufo stories because they have the old school belief that the public don't take the topic seriously and poor ratings will result if the story gets too factual ... I mean wake up Media networks - check your surveys, general concensus is that the public take these things seriously now - I mean Obama is even now aware of this - please spare us the ridicule and give us all the facts - we can handle it!
above to read more posts >>
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Everyone knows about the reported recovery of a crashed alien spaceship near Roswell, New Mexico in July 1947. However, most people are unaware that, at the time of the incident, Roswell Army Airfield was home to the world's only atomic bomber squadron, the 509th Bomb Group. Was this merely a coincidence? More on the book here
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text:
\section{Introduction}

Undirected graphical model is an important tool to capture dependence among random variables and has drawn tremendous attention in various fields including signal processing, bioinformatics and network modeling \citep{wainwright2008graphical}.
Let ${\bf z}=(z^{(1)},\dots,z^{(d)})$ be a $d$-dimensional random vector. We denote the undirected graph corresponding to ${\bf z}$ by $(V,E)$, where vertices $V$ correspond to components of ${\bf z}$ and edges $E=\{e_{ij},1\leq i,j\leq d\}$ indicate whether  node $z^{(i)}$ and $z^{(j)}$ are conditionally independent given the remaining nodes. In particular, the edge $e_{ij}$ is absent if and only if $z^{(i)}\independent z^{(j)}|{\bf z}\setminus \{z^{(i)},z^{(j)}\}$. When ${\bf z}$ follows multivariate Gaussian distribution with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$, the precision matrix $\boldsymbol{\Omega}=(w_{ij})_{d\times d}=\boldsymbol{\Sigma}^{-1}$ captures exactly this relationship; that is, $w_{ij}= 0$ if and only if $e_{ij}$ is absent \citep{lauritzen1996graphical, edwards2000introduction}. Therefore, under the Gaussian assumption, this problem reduces to the estimation of precision matrix, where a rich literature on model selection and parameter estimation can be found in both low-dimensional and high-dimensional settings, including \cite{dempster1972covariance}, \cite{drton2004model}, \cite{meinshausen2006high}, \cite{friedman2008sparse},  \cite{fan2009network}, and \cite{cai2011constrained}. While Gaussian graphical model (GGM) can be useful, the stringent requirement on normality is not always satisfied in real application where the observed data usually have fat tails or are skewed  \citep{xue2012regularized}.

To relax the Gaussian assumption, \cite{liu2009nonparanormal}  proposed the nonparanormal model, where they find transformations that marginally gaussianize  the data and then work under the Gaussian graphical model framework to estimate the network structure. Under the nonparanormal model, \cite{xue2012regularized} proposed rank-based estimators to approximate the precision matrix. The nonparanormal model, although flexible, still assume the transformed data follows a multivariate Gaussian distribution, which can also be restrictive at times.
Instead of using these nonparametric methods to find transformations and work under GGM, we would like to propose a more natural way of constructing graphs. That is, we work directly on the conditional dependence structure by introducing a measure for conditional dependence between node $i$ and $j$ given the remaining nodes. Then, we can introduce a hypothesis testing procedure to decide whether the edge $e_{ij}$ is present or not.

It is worth noting that, based on hypothesis testing, we could indeed build a general conditional dependency graph, where presence of an edge $e_{ij}$ between nodes $z^{(i)}$, $z^{(j)}$ represents that the two nodes are dependent conditional on some factors ${\bf f}$. Graphical model is one type of such graphs where ${\bf f}$ is chosen to be ${\bf z}\setminus \{z^{(i)},z^{(j)}\}$; in such cases, we call ${\bf f}$ internal factors. More generally, ${\bf f}$ could contain covariates we observe or latent factors that we do not observe, which are not necessarily part of ${\bf z}$, and we call them external factors. As an example, in Fama-French three-factor model, the return of each stock can be considered as one node and ${\bf f}$ are the chosen three-factors. This example will be further elaborated in Section \ref{sec::Realdata}.  Another interesting application is discussed in \cite{stock2002macroeconomic}, where external factors are aggregated macroeconomic variables, and the nodes are disaggregated macroeconomic variables.

Back to hypothesis testing, in economics, there has been abundant literature on different conditional independence tests.  \cite{linton1997conditional} proposed two nonparametric tests of conditional independence based on a generalization of the empirical distribution function; however, a complicated bootstrap procedure is needed to calculate critical values of the test, which leads to limited practical value. \cite{su2007consistent, su2008nonparametric, su2014testing} proposed conditional independence tests based on Hellinger distance, conditional characteristic function and empirical likelihood, respectively. However, all those tests either have tuning parameters or are computationally expensive.

As an attempt to propose a remedy for the above issues, we consider the following setup. In particular, suppose $\{({\bf x}_i, {\bf y}_i, {\bf f}_i, \boldsymbol{\epsilon}_{i,x}, \boldsymbol{\epsilon}_{i,y}), i=1,\dots, n\}$ are i.i.d. realizations of $({\bf x}, {\bf y}, {\bf f}, \boldsymbol{\epsilon}_x, \boldsymbol{\epsilon}_y)$, which are generated from the following model:
\begin{align}\label{eq::model-general}
{\bf x} = {\bf G}_x({\bf f}) +\boldsymbol{\epsilon}_{x}, \quad {\bf y}={\bf G}_y({\bf f})+\boldsymbol{\epsilon}_{y},
\end{align}
where ${\bf f}$ is the $K$-dimensional common factors, ${\bf G}_x$ and ${\bf G}_y$ are general mappings from $\mathbb{R}^K$ to $\mathbb{R}^p$ and $\mathbb{R}^q$, respectively. Note that we only observe $\{({\bf x}_i, {\bf y}_i, {\bf f}_i), i=1,\dots, n\}$.   Here, we assume independence between $(\boldsymbol{\epsilon}_{x}, \boldsymbol{\epsilon}_{y})$ and ${\bf f}$. In addition, the dimensions $p$ and $q$ are assumed to be fixed while the number of factors $K$ could diverge to infinity.

Our goal is to test whether ${\bf x}$ and ${\bf y}$ are independent given ${\bf f}$, i.e.,
\begin{align}\label{eq::indep-test-ori}
  H_0:  {\bf x}\independent{\bf y}|{\bf f}
\end{align}
Under model \eqref{eq::model-general}, the testing problem is equivalent to test whether $\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y$ are independent, i.e.,
\begin{align}\label{eq::indep-test-error}
  H_0:  \boldsymbol{\epsilon}_x\independent \boldsymbol{\epsilon}_y .
\end{align}
Note that in the case of building graphical models without Gaussian assumption, we could use \eqref{eq::model-general} by setting  ${\bf x}$ and ${\bf y}$ as the random vectors associated with a pair of nodes in the graph and ${\bf f}$ represents the rest of the nodes. Then, the  method to be developed could be used to construct a high-dimensional undirected graphs by conducting the test in \eqref{eq::indep-test-ori} for each pair of the nodes.  This gives a graphical summary of the conditional dependence structure.

Since $\{(\boldsymbol{\epsilon}_{i,x}, \boldsymbol{\epsilon}_{i,y}), i=1,\dots, n\}$ are hidden, a natural idea is to estimate them by the residuals after a projection of  ${\bf x}$ and ${\bf y}$ onto ${\bf f}$. Asymptotically, a fully nonparametric projection on ${\bf f}$ (e.g., local polynomial regression) would consistently recover the random errors when $K$ is fixed along with certain smoothness assumptions on ${\bf G}_x$ and ${\bf G}_y$. However, it becomes challenging when $K$ diverges due to the curse of dimensionality if no structural assumptions are made on ${\bf G}_x$ and ${\bf G}_y$.   As a result, we will study the case where ${\bf G}_x$ and ${\bf G}_y$ are linear functions (factor models) in Section \ref{subsec::T} and the case where ${\bf G}_x$ and ${\bf G}_y$ are additive functions in Section \ref{subsec:: Functional Projection} when $K$ diverges.

To complete our proposal, we  need to find a suitable measure of dependence between random variables/vectors. In this regards, many different measures of dependence have  been proposed. Some of them rely heavily on Gaussian assumptions, such as Pearson correlation, which measures linear dependence and the uncorrelatedness is equivalent to independence only when the joint distribution is Gaussian; or Wilks Lambda \citep{wilks1935independence}, where normality is adopted to calculate the likelihood ratio. To deal with non-linear dependence and non-Gaussian distribution, statisticians have proposed rank-based correlation measures, including Spearman's $\rho$  and Kendall's $\tau$, which are more robust than Pearson correlation against deviations from normality. However, these correlation measures are usually only effective for monotone types of dependence. In addition, under the null hypothesis that two variables are independent, no general statistical distribution of the coefficients associated with these measures has been derived. Other related works include \cite{hoeffding1948non},  \cite{blomqvist1950measure}, \cite{blum1961distribution}, and some methods described in \cite{hollander2013nonparametric} and \cite{anderson1958introduction}. Taking these into consideration, distance covariance \citep{szekely2007measuring} was introduced to address all these deficiencies. The major benefits of distance covariance are: first, zero distance covariance implies independence, and hence it is a true dependence measure. Second, distance covariance can measure the dependence between any two vectors which potentially are of different dimensions.
Due to these advantages, we will focus on  distance covariance in this paper as our measure of dependence.

The main contribution of this paper is two-fold. First, under the factor model assumption, we propose a computationally efficient conditional independence test. Both the response vectors and the common factors can be of different dimensions and the number of the factors could grow to infinity with sample size.  Second, we apply this test to build conditional dependency graph and covariates-adjusted dependency graph, as  generalizations of the Gaussian graphical model.

The rest of this paper is organized as follows. In Section \ref{sec::Methods}, we  present our new procedure for testing conditional independence via projected distance covariance (P-DCov)  and describe how to construct conditional dependency graphs based on the proposed test. Section \ref{sec::TheoreticalP} gives theoretical properties including the asymptotic distribution of the test statistic under the null hypothesis as well as the type I error guarantee. Section \ref{sec::NumericalS} contains extensive numerical studies and Section \ref{sec::Realdata} demonstrates the performance of P-DCov via two real data sets. We conclude the paper with a short discussion in Section \ref{sec::Disc}.  Several technical lemmas and all proofs are relegated to the appendix.

\section{Methods}\label{sec::Methods}
First, we introduce some notations. For a random vector ${\bf z}$, $\|{\bf z}\|$ and $\|{\bf z}\|_1$ represent its Euclidean norm and $\ell_1$ norm, respectively. A collection of $n$ i.i.d. observations of ${\bf z}$ is denoted as $\{{\bf z}_1,\dots,{\bf z}_n\}$, where ${\bf z}_k=(z^{(1)}_k,\dots,z^{(d)}_k)^T$ represents the $k$-th observation. For any matrix ${\bf M}$,  $\|{\bf M}\|_F$, $\|{\bf M}\|$ and  $\|{\bf M}\|_{\max}$ denote its Frobenius norm, operator norm and max norm, respectively. $\|{\bf M}\|_{a,b}$ is the $(a,b)$ norm defined as the $\ell_b$ norm of the vector consisting of column-wise $\ell_a$ norm of ${\bf M}$.
\subsection{A brief review of distance covariance}
As an important tool, distance covariance is briefly reviewed in this section with further details available in \cite{szekely2007measuring}. We introduce several definitions as follows.
\begin{Def}($w$-weighted $L_2$ norm)
Let $c_d=\frac{\pi^{(d+1)/2}}{\Gamma((d+1)/2)}$, for any positive integer $d$, where $\Gamma$ is the Gamma function. Then for function $\gamma$ defined on $\mathbb{R}^p\times\mathbb{R}^q$, the $w$-weighted $L_2$ norm of $\gamma$ is defined by
\[
\|\gamma(\boldsymbol{\tau},\boldsymbol{\rho})\|^2_w=\int_{\mathbb{R}^{p+q}}|\gamma(\boldsymbol{\tau},\boldsymbol{\rho})|^2 w(\boldsymbol{\tau},\boldsymbol{\rho}) d\boldsymbol{\tau} d\boldsymbol{\rho},\ \ \text{where}\ \ w(\boldsymbol{\tau},\boldsymbol{\rho})=(c_p c_q \|\boldsymbol{\tau}\|^{1+p} \|\boldsymbol{\rho}\|^{1+q})^{-1}.
\]
\end{Def}

\begin{Def} (Distance covariance) The distance covariance between random vectors ${\bf x}\in\mathbb{R}^p$ and ${\bf y}\in\mathbb{R}^q$ with finite first moments is the nonnegative number  $\mathcal{V}({\bf x},{\bf y})$ defined by
\[
\mathcal{V}^2({\bf x},{\bf y})=\|g_{x,y}(\boldsymbol{\tau},\boldsymbol{\rho})-g_{x}(\boldsymbol{\tau})g_{y}(\boldsymbol{\rho})\|^2_w,
\]
where $g_{x}$, $g_{y}$ and $g_{x,y}$ represent the characteristic functions of ${\bf x}$, ${\bf y}$ and the joint characteristic function of ${\bf x}$ and ${\bf y}$, respectively.
\end{Def}

Suppose we observe random sample $\{({\bf x}_k,{\bf y}_k):k=1,\dots,n\}$ from the joint distribution of $({\bf x},{\bf y})$. We denote ${\bf X} = ({\bf x}_1,{\bf x}_2,\dots,{\bf x}_n)$ and ${\bf Y} = ({\bf y}_1,{\bf y}_2,\dots, {\bf y}_n)$.

\begin{Def} (Empirical distance covariance) The empirical distance covariance between  samples ${\bf X}$ and ${\bf Y}$
  is the nonnegative random variable $\mathcal{V}_n({\bf X},{\bf Y})$  defined by
\[
\mathcal{V}^2_n({\bf X},{\bf Y})=S_1({\bf X},{\bf Y})+S_2({\bf X},{\bf Y})-2S_3({\bf X},{\bf Y}),
\]
where
\begin{align*}
S_1({\bf X},{\bf Y})&=\frac{1}{n^2}\sum_{k,l=1}^n \|{\bf x}_k-{\bf x}_l\|\|{\bf y}_k-{\bf y}_l\|,\ S_2({\bf X},{\bf Y})=\frac{1}{n^2}\sum_{k,l=1}^n \|{\bf x}_k-{\bf x}_l\|\frac{1}{n^2}\sum_{k.l=1}^n\|{\bf y}_k-{\bf y}_l\|,
\cr
S_3({\bf X},{\bf Y})&=\frac{1}{n^3}\sum_{k=1}^n\sum_{l,m=1}^n \|{\bf x}_k-{\bf x}_l\|\|{\bf y}_k-{\bf y}_m\|.
\end{align*}

\end{Def}

With above definitions, Lemma \ref{thm::old-consistency} depicts the consistency of $\mathcal{V}_n({\bf X},{\bf Y})$ as an estimator of $\mathcal{V}({\bf x}, {\bf y})$. Lemma \ref{thm::old-asymptotic} shows the asymptotic distribution of $\mathcal{V}_n({\bf X},{\bf Y})$ under the null hypothesis that ${\bf x}$ and ${\bf y}$ are independent. Corollary \ref{coro::old-teststat} reveals properties of the test statistic $n\mathcal{V}^2_n/S_2$ proposed in \cite{szekely2007measuring}.
\begin{lemma}\label{thm::old-consistency}(Theorem 2 in \cite{szekely2007measuring})
Assume that $\mathbb{E}(\|{\bf x}\|+\|{\bf y}\|)<\infty$, then almost surely
\[
\lim_{n\rightarrow \infty} \mathcal{V}_n({\bf X},{\bf Y})=\mathcal{V}({\bf x},{\bf y}).
\]
\end{lemma}

\begin{lemma}\label{thm::old-asymptotic}(Theorem 5 in \cite{szekely2007measuring})
Assume that ${\bf x}$ and ${\bf y}$ are independent, and $\mathbb{E}(\|{\bf x}\|+\|{\bf y}\|)<\infty$, then as $n\rightarrow \infty$,
\[
n\mathcal{V}^2_n({\bf X},{\bf Y})\stackrel{D}{\rightarrow}\|\zeta(\boldsymbol{\tau},\boldsymbol{\rho})\|^2_w,
\]
where $\stackrel{D}{\rightarrow}$ represents convergence in distribution and $\zeta(\cdot , \cdot )$ denotes a complex-valued centered Gaussian random process with covariance function
\[
R({\bf u},{\bf u}_0)=(g_{x}(\boldsymbol{\tau}-\boldsymbol{\tau}_0)-g_{x}(\boldsymbol{\tau})\overline{g_{x}(\boldsymbol{\tau}_0)})(g_{y}(\boldsymbol{\rho}-\boldsymbol{\rho}_0)-g_{y}(\boldsymbol{\rho})\overline{g_{y}(\boldsymbol{\rho}_0)}),
\]
in which ${\bf u}=(\boldsymbol{\tau},\boldsymbol{\rho})$, ${\bf u}_0=(\boldsymbol{\tau}_0,\boldsymbol{\rho}_0)$.
\end{lemma}

\begin{coro}\label{coro::old-teststat}(Corollary 2 in \cite{szekely2007measuring})
Assume that $\mathbb{E}(\|{\bf x}\|+\|{\bf y}\|)<\infty$.
\begin{enumerate}
\item If ${\bf x}$ and ${\bf y}$ are independent, then as $n\rightarrow \infty$, $n\mathcal{V}^2_n({\bf X},{\bf Y})/S_2\stackrel{D}{\rightarrow}Q$ with $Q\overset{D}{=}\sum_{j=1}^{\infty}\lambda_j Z^2_j$, where $Z_j\overset{i.i.d}{\sim}\mathcal{N}(0,1)$ and $\{
    \lambda_j\}$ are non-negative constants depending on the distribution of $({\bf x},{\bf y})$; $\mathbb{E}(Q)=1$.
\item If ${\bf x}$ and ${\bf y}$ are dependent, then  as $n\rightarrow \infty$, $n\mathcal{V}^2_n({\bf X},{\bf Y})/S_2\stackrel{P}{\rightarrow}\infty$.
\end{enumerate}
\end{coro}

\subsection{Conditional independence test via projected distance covariance (P-DCov)\label{subsec::T}}
Here, we consider the case where ${\bf G}_x$ and ${\bf G}_y$ are linear in \eqref{eq::model-general}, which leads to the  following factor model setup:
\begin{align}\label{eq::model}
{\bf x} = {\bf B}_x {\bf f} +\boldsymbol{\epsilon}_{x}, \quad {\bf y}={\bf B}_y {\bf f}+\boldsymbol{\epsilon}_{y},
\end{align}
where ${\bf B}_x$ and ${\bf B}_y$ are factor loading matrices of dimension $p\times K$ and $q\times K$ respectively, and ${\bf f}$ is the $K$-dimensional vector of common factors.  Here, the number of common factors $K$ could grow to infinity and the matrices ${\bf B}_x$ and ${\bf B}_y$ are assumed to be sparse to reflect that ${\bf x}$ and ${\bf y}$ only depend on several important factors.  As a result, we will impose regularization on the estimation of ${\bf B}_x$ and ${\bf B}_y$. Now, we are in the position to propose a test for problem \eqref{eq::indep-test-ori}. We first provide an estimate for the idiosyncratic components $\boldsymbol{\epsilon}_{x}$ and $\boldsymbol{\epsilon}_{y}$, and then calculate distance covariance between the estimates. More generally, we project  ${\bf x}$ and ${\bf y}$ onto the space orthogonal to the linear space spanned by ${\bf f}$ and evaluate the dependency between the projected vectors.
The conditional independence test is summarized in the following steps.

\noindent{\bf Step 1:}
Estimate factor loading matrices ${\bf B}_x$ and ${\bf B}_y$ by the penalized least square (PLS) estimators $\widehat{\bB}_x$ and $\widehat{\bB}_y$ defined as follows.
\begin{align}
\widehat{\bB}_x &= \arg\min_{{\bf B}} \frac{1}{2}\|{\bf X}-{\bf B}{\bf F}\|_F^2+\sum_{j,k}p_{\lambda_1}(|B_{jk}|),\\
\widehat{\bB}_y &= \arg\min_{{\bf B}}\frac{1}{2} \|{\bf Y}-{\bf B}{\bf F}\|_F^2+\sum_{j,k}p_{\lambda_2}(|B_{jk}|),
\end{align}
where ${\bf X}=({\bf x}_1, {\bf x}_2, \dots, {\bf x}_n)$, ${\bf Y}=({\bf y}_1, {\bf y}_2, \dots, {\bf y}_n)$, ${\bf F}=({\bf f}_1, {\bf f}_2, \dots, {\bf f}_n)$, $p_{\lambda}(\cdot)$ is the penalty function with penalty level $\lambda$.

\noindent{\bf Step 2:} Estimate the error vectors $\boldsymbol{\epsilon}_{i,x}$ and $\boldsymbol{\epsilon}_{i,y}$ by
\begin{align*}
\hat{\boldsymbol{\epsilon}}_{i,x} &={\bf x}_i-\widehat{\bB}_x{\bf f}_i=({\bf B}_x-\widehat{\bB}_x){\bf f}_i+\boldsymbol{\epsilon}_{i,x},\cr
\hat{\boldsymbol{\epsilon}}_{i,y} &={\bf y}_i-\widehat{\bB}_y{\bf f}_i=({\bf B}_y-\widehat{\bB}_y){\bf f}_i+\boldsymbol{\epsilon}_{i,y},\ \ i=1,\dots,n.
\end{align*}

\noindent{\bf Step 3:} Define the estimated error matrices $\widehat{\bf E}_x = (\hat\boldsymbol{\epsilon}_{1,x}, \dots, \hat\boldsymbol{\epsilon}_{n,x})$ and $\widehat{\bf E}_y = (\hat\boldsymbol{\epsilon}_{1,y}, \dots, \hat\boldsymbol{\epsilon}_{n,y})$. Calculate the empirical distance covariance between $\widehat{\bf E}_x$ and $\widehat{\bf E}_y$ as
\[
\mathcal{V}^2_n(\widehat{\bf E}_x,\widehat{\bf E}_y)=S_1(\widehat{\bf E}_x,\widehat{\bf E}_y)+S_2(\widehat{\bf E}_x,\widehat{\bf E}_y)-2S_3(\widehat{\bf E}_x,\widehat{\bf E}_y).
\]

\noindent{\bf Step 4:} Define the P-DCov test statistic as $T({\bf x},{\bf y},{\bf f})=n\mathcal{V}^2_n(\widehat{\bf E}_x,\widehat{\bf E}_y)/S_2(\widehat{\bf E}_x,\widehat{\bf E}_y)$.

\noindent{\bf Step 5:} With predetermined significance level $\alpha$, we reject the null hypothesis when $T({\bf x},{\bf y},{\bf f})>(\Phi^{-1}(1-\alpha/2))^2$.

Theoretical properties of the proposed conditional independence test will be studied in Section \ref{sec::TheoreticalP}.
In the above method, we implicitly assume that the number of variables $K$ is large so that the penalized least-squares methods are used.  When the number of variables $K$ is small, we can take $\lambda_1 = \lambda_2 = 0$ so that no penalization is imposed.

\subsection{Building graphs via conditional independence test}\label{subsec::graph-building}
Now we explore a specific application of our conditional independence test  to graphical models. To identify the conditional independence relationship in a graphical model, i.e.,  $z^{(i)}\independent z^{(j)}|{\bf z}\setminus \{z^{(i)},z^{(j)}\}$, we assume
\begin{align}\label{eq::graph-internal-factor}
z^{(i)}_k=\boldsymbol{\beta}^{\top}_{1,{ij}} {\bf f}_k+\epsilon^{(i)}_{k},\ \ z^{(j)}_k=\boldsymbol{\beta}^{\top}_{2,{ij}} {\bf f}_k+\epsilon^{(j)}_{k},\ \ k=1,\dots,n,
\end{align}
where ${\bf f}_k=({\bf z}^{(-i,-j)}_k)^{\top}$ represents all coordinates of ${\bf z}_k$ other than ${\bf z}^{(i)}_k$ and ${\bf z}^{(j)}_k$, and $\boldsymbol{\beta}_{1,{ij}}$ and $\boldsymbol{\beta}_{2,{ij}}$ are $d-2$ dimensional  regression coefficients. Under model \eqref{eq::graph-internal-factor}, we decide whether edge $e_{ij}$ will be drawn through  directly testing $z^{(i)}\independent z^{(j)}|\mathcal{L}({\bf z}^{(-i, -j)})$, where $\mathcal{L}({\bf f})$ is the linear space spanned by ${\bf f}$.

More specifically, for each node pair $\{(i,j): 1\leq i<j \leq d\}$, we define $T^{(i,j)}=T(z^{(i)},z^{(j)},{\bf z}^{(-i,-j)})$ using the same steps as in Section \ref{subsec::T} as the test for the current null hypothesis:
\begin{align}\label{eqn::edgetest}
  H_{0, ij}:  \boldsymbol{\epsilon}^{(i)}\independent \boldsymbol{\epsilon}^{(j)}.
\end{align}

We now summarize the testing results by a graph in which nodes represent variables in ${\bf z}$ and the edge $e_{ij}$ between node $i$ and node $j$ is drawn only when $H_{0, ij}$ is rejected at level $\alpha$.

In \eqref{eq::graph-internal-factor}, the factors are created internally via the observations on remaining nodes ${\bf z}\setminus \{z^{(i)},z^{(j)}\}$. In financial applications, it is often desirable to build graphs when conditioning on external factors. In such cases, it is straightforward to change the factors in \eqref{eq::graph-internal-factor} to  external factors.

We will demonstrate the two different types of conditional dependency graphs via examples in Sections \ref{sec::NumericalS} and \ref{sec::Realdata}.

\subsection{Graph estimation with FDR control}\label{subsec::FDR control}
 Through the graph building process described in Section \ref{subsec::graph-building},  we can carry out $\bar{d}= d(d-1)/2$ P-DCov tests simultaneously and we wish to control the {\it false discovery rate} (FDR) at a pre-specified level $0<\alpha<1$. Let $R_F$ and $R$ be the number of falsely rejected hypotheses and the number of total rejections, respectively. The {\it false discovery proportion }(FDP) is defined as $R_F/\max\{1, R\}$ and the FDR is the expectation of FDP.

In the literature, various procedures have been proposed for conducting large-scale multiple hypothesis testing via FDR control. In this work, we will follow the most commonly used Benjamini and Hochberg (BH) procedure developed in the seminal work of \cite{benjamini1995controlling}, where P-values of all marginal tests are compared. More specifically, let $P_{(1)}\leq P_{(2)} \leq \dots \leq P_{( \bar{d} )}$ be the ordered P-values of the $\bar{d}$ hypotheses given in (\ref{eqn::edgetest}). Let $s=\max\{ 0 \leq i \leq \bar{d} : P_{(i)}\leq  \alpha i /\bar{d} \}$, and we reject the $s$ hypotheses $H_{0, ij}$ with the smallest P-values. We will demonstrate the performance of this strategy via real data examples.

\subsection{Extension to functional projection}\label{subsec:: Functional Projection}
In the P-DCov described in Section \ref{subsec::T}, we assume the conditional dependency of ${\bf x}$ and ${\bf y}$ given  factor ${\bf f}$ is expressed via a linear form of ${\bf f}$. In other words, we are projecting ${\bf x}$ and ${\bf y}$ onto the space orthogonal  to $\mathcal{L}({\bf f})$ and evaluate the dependence between the projected vectors. Although this linear projection assumption makes the theoretical development easier and delivers the main idea of this work, a natural extension is to consider a nonlinear projection. In particular, we consider the following additive generalization \citep{stone1985additive}  of the factor model setup:
\begin{align}\label{eq::model-additive}
{\bf x} = \sum_{j=1}^K {\bf g}^x_j(f_{j})+\boldsymbol{\epsilon}_{x}, {\bf y}=  \sum_{j=1}^K {\bf g}^y_j(f_{j})+\boldsymbol{\epsilon}_{y},
\end{align}
where $\{{\bf g}^x_j(\cdot), {\bf g}^y_j(\cdot), j=1,\dots,K\}$ are unknown vector-valued functions we would like to estimate.   In \eqref{eq::model-additive}, we consider the additive space spanned by factor ${\bf f}$. By this extension, we could identify more general conditional dependency structures between ${\bf x}$ and ${\bf y}$ given ${\bf f}$. This is a special case of \eqref{eq::model-general}, but avoids the issue of curse of dimensionality.

In the high-dimensional setup where $K$ is large, we can use a penalized additive model  \citep{ravikumar2009sparse, fan2011nonparametric}  to estimate the unknown functions. The conditional independence test described in Section \ref{subsec::T} could be modified by replacing the linear regression with the (penalized) additive model regression.  We will investigate the P-DCov method coupled with the sparse additive model \citep{ravikumar2009sparse} in  numerical studies.

\section{Theoretical Results}\label{sec::TheoreticalP}
In this section, we derive the asymptotic properties of our conditional independence test. First, we introduce  several assumptions  on $\boldsymbol{\epsilon}_x$, $\boldsymbol{\epsilon}_y$ and ${\bf f}$.

\begin{condition}\label{Assump:Moments}
$\mathbb{E}\boldsymbol{\epsilon}_{x}=\mathbb{E}\boldsymbol{\epsilon}_{y}={\bf 0}$,\ \  $\mathbb{E} \|\boldsymbol{\epsilon}_{x}\|<\infty$, $\mathbb{E} \|\boldsymbol{\epsilon}_{y}\|<\infty$,\ \ $\mathbb{E} \|\boldsymbol{\epsilon}_{x}\|^2<\infty$, $\mathbb{E} \|\boldsymbol{\epsilon}_{y}\|^2<\infty$.

\end{condition}

\begin{condition}\label{Assump:Tail Condition} We denote  $h_x$ as the density function of random variable $x$. Let us assume that the densities of $\|\boldsymbol{\epsilon}_{1,x}-\boldsymbol{\epsilon}_{2,x}\|$ and $\|\boldsymbol{\epsilon}_{1,y}-\boldsymbol{\epsilon}_{2,y}\|$ are bounded on $[0,C_0]$, for some positive constant $C_0$. In other words, there exists a positive constant $M$,
\[
\max_{t\in[0,C_0]} h_{\|\boldsymbol{\epsilon}_{i,x}-\boldsymbol{\epsilon}_{j,x}\|}(t)\leq M, \ \ \ \ \max_{t\in[0,C_0]} h_{\|\boldsymbol{\epsilon}_{i,y}-\boldsymbol{\epsilon}_{j,y}\|}(t)\leq M.
\]
\end{condition}
Conditions \ref{Assump:Moments} and \ref{Assump:Tail Condition} impose mild moment and distributional assumptions on  random errors $\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y$. To better understand when the proposed method works, we give the following high-level assumption, whose justifications are noted below.

\begin{condition}\label{Assump:dn} There exist   constants $C_1>1$ and $\gamma>0$, such that for any $C_2>1$, with probability greater than $1-C_1^{-C_2}$, we have for any $n$,
\[
 \|({\bf B}_x-\widehat{\bB}_x){\bf F}\|_{2,\infty} \leq C_2 a_n,  \qquad \|({\bf B}_y-\widehat{\bB}_y){\bf F}\|_{2,\infty} \leq C_2 a_n,
\]
where the sequence $a_n=o\{(n^{(1+\gamma)}\log n)^{-1/3}\}$.

\end{condition}

\begin{condition}\label{Assump:l1}
Let ${\bf B}_{x,l}$ denote the $l$-th row of ${\bf B}_x$, and similarly we define $\widehat{\bB}_{x,l}$, ${\bf B}_{y,l}$ and $\widehat{\bB}_{y,l}$.  We assume for any fixed $l$,
\begin{align*}
\|{\bf B}_{x,l}-\widehat{\bB}_{x,l}\|_1 = O_p(e_n), \qquad \|{\bf B}_{y,l}-\widehat{\bB}_{y,l}\|_1 = O_p(e_n),
\end{align*}
where sequences $e_n$ and $a_n$ in Condition \ref{Assump:dn} satisfy $a_n e_n = o(\frac{1}{\sqrt{n}\log K })$.
\end{condition}

\begin{remark}
Conditions \ref{Assump:dn} and \ref{Assump:l1} are mild conditions that are imposed to ensure the quality of the projection and guarantee the theoretical properties regarding our conditional independence test. For example, one could directly call the results from penalized least squares for high-dimensional regression \citep{buhlmann2011statistics, hastie2015statistical} and  robust estimation \citep{belloni2011, wang2013l1,  fan2014robust}. We now discuss two special examples as follows.
\begin{enumerate}
\item ($K$ is fixed) In this fixed dimensional case, it is straightforward to verify that the projection based on ordinary least squares satisfies the two conditions.
\item (Sparse Linear Projection) Let ${\bf B}_x = ({\bf b}_1^T, {\bf b}_2^T,\dots, {\bf b}_p^T)^T$ and $\widehat{\bf B}_x = (\widehat{\bf b}_1^T, \widehat{\bf b}_2^T,\dots, \widehat{\bf b}_p^T)^T$. Note that the graphical model case corresponds to  $p=1$.   We apply the popular $L_1$-regularized least squares for each dimension of ${\bf x}$ regressing on the factor ${\bf F}$. Here, we further assume the true regression coefficient ${\bf b}_j$ is sparse for each $j$ with $S_j = \{k: ({\bf b}_j)_k\neq 0\}$, $\hat S_j = \{k: (\widehat{\bf b}_j)_k\neq 0\}$ and $|S_j|=s_j$. From Theorem 11.1, Example 11.1 and Theorem 11.3 in \cite{hastie2015statistical}, and since  $\{{\bf f}_i\}_{i=1}^n$ are i.i.d., we have with high probability,
$\|\widehat {\bf b}_j -{\bf b}_j\|\le C\sqrt{\frac{s_j\log K}{n}}$, $\hat S_j = S_j$ and $\max_i \|({\bf f}_i)_{S_j}\|\le s_j \log n$. Then, we have with high probability, for each $i=1,\dots, n$ and $j = 1,\dots,p$,
\begin{align}\label{eq::lasso.case}
	\|(\widehat {\bf b}_j -{\bf b}_j)^T{\bf f}_i\|  =  \|(\widehat {\bf b}_j - {\bf b}_j)_{S_j}^T({\bf f}_i)_{S_j}\| \le  \|(\widehat {\bf b}_j - {\bf b}_j)_{S_j}\|\|({\bf f}_i)_{S_j}\|\le Cs_{\max} \log n \sqrt{\frac{s_{\max}\log K}{n}},
\end{align}
where $s_{\max}=\max_j s_j$.
It is now easy to verify that Condition \ref{Assump:dn} and \ref{Assump:l1} are satisfied even under the ultra-high-dimensional case where $\log K=o(n^a), 0<a<1/3$. We would like to omit the details here for brevity about the specification of various constants.
\end{enumerate}
\end{remark}

\begin{thm}\label{thm::consistency}Under Conditions \ref{Assump:Moments} and \ref{Assump:dn},
\[
\mathcal{V}^2_n(\hat{\boldsymbol{\epsilon}}_x,\hat{\boldsymbol{\epsilon}}_y)\stackrel{P}{\rightarrow}\mathcal{V}^2(\boldsymbol{\epsilon}_x,\boldsymbol{\epsilon}_y).
\]

In particular, when $\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y$ are independent, $\mathcal{V}^2_n(\hat{\boldsymbol{\epsilon}}_x,\hat{\boldsymbol{\epsilon}}_y)\stackrel{P}{\rightarrow} 0$.
\end{thm}

Theorem \ref{thm::consistency} shows that the sample distance covariance between the estimated residual vectors converges to the distance covariance between the population error vectors. It enables us to use the distance covariance of the estimated residual vectors to construct the conditional independence test as described in Section \ref{subsec::T}.

\begin{thm}\label{thm::null-distribution} Under Conditions \ref{Assump:Moments}-\ref{Assump:l1}, and  the null hypothesis that $\boldsymbol{\epsilon}_x \independent \boldsymbol{\epsilon}_y$ (or equivalently ${\bf x}\independent {\bf y}|{\bf f}$),
$$
n\mathcal{V}^2_n(\hat{\boldsymbol{\epsilon}}_x,\hat{\boldsymbol{\epsilon}}_y)\stackrel{\mathcal{D}}{\rightarrow}\|\zeta\|^2,
$$
where $\zeta$ is a zero-mean Gaussian process defined analogously as in Lemma~\ref{thm::old-asymptotic}.
\end{thm}

Theorem \ref{thm::null-distribution} provides the asymptotic distribution of the test statistic $T({\bf x},{\bf y},{\bf f})$ under the null hypothesis, which is the basis of Theorem \ref{thm::TestSigLev}.

\begin{coro}\label{coro::new-teststat}
Under the same conditions of Theorem \ref{thm::null-distribution}, \[
n\mathcal{V}^2_n(\hat{\boldsymbol{\epsilon}}_x,\hat{\boldsymbol{\epsilon}}_y) /S_2(\hat{\boldsymbol{\epsilon}}_x,\hat{\boldsymbol{\epsilon}}_y) \stackrel{\mathcal{D}}{\rightarrow} Q, \ \ \text{where}\ \ Q\overset{\mathcal{D}}=\sum_{j=1}^\infty \lambda_j Z^2_j,
\]
where $Z_j\overset{i.i.d}{\sim}\mathcal{N}(0,1)$ and $\{
    \lambda_j\}$ are non-negative constants depending on the distribution of $({\bf x},{\bf y})$; $\mathbb{E}(Q)=1$.
\end{coro}

\begin{thm}\label{thm::TestSigLev}
Consider the test that rejects conditional independence when
\begin{align}\label{eq::reject-region}
\frac{n\mathcal{V}^2_n(\hat{\boldsymbol{\epsilon}}_{x},\hat{\boldsymbol{\epsilon}}_{y})}{S_2}>(\Phi^{-1}(1-\alpha/2))^2,
\end{align}
where $\Phi(\cdot)$ is the cumulative distribution function of $\mathcal{N}(0,1)$. Let $\alpha_n({\bf x},{\bf y},{\bf f})$ denote its associated type I error. Then under Conditions \ref{Assump:Moments}-\ref{Assump:l1}, for all $0<\alpha\le0.215$,
\begin{enumerate}
\item $\lim_{n\rightarrow\infty}\alpha_n({\bf x},{\bf y},{\bf f})\leq \alpha$,
\item $\sup_{\boldsymbol{\epsilon}_{x}\independent\boldsymbol{\epsilon}_{y}}\lim_{n\rightarrow \infty}\alpha_n({\bf x},{\bf y},{\bf f})=\alpha$.
\end{enumerate}
\end{thm}
Part (i) of Theorem \ref{thm::TestSigLev} indicates the proposed test with critical region \eqref{eq::reject-region} has an asymptotic significance error at most $\alpha$. Part (ii) of Theorem \ref{thm::TestSigLev} implies that there exists a pair $(\boldsymbol{\epsilon}_x,\boldsymbol{\epsilon}_y)$ such that the pre-specified significant level $\alpha$ is achieved asymptotically.  In other words, the size of testing $H_0: \boldsymbol{\epsilon}_{x}\independent\boldsymbol{\epsilon}_{y}$ is $\alpha$.

\begin{remark}
	When the sample size $n$ is small, the theoretical critical value in \eqref{eq::reject-region} could sometimes be too conservative in practice \citep{szekely2007measuring}. Therefore, we recommend using random permutation to get a reference distribution for the test statistic $T({\bf x},{\bf y},{\bf f})$ under $H_{0}$. Random permutation is used to decouple $\boldsymbol{\epsilon}_{i,x}$ and $\boldsymbol{\epsilon}_{i,y}$ so that the resulting pair  $(\boldsymbol{\epsilon}_{\pi(i),x}, \boldsymbol{\epsilon}_{i,y})$ follows the null model, where $\{\pi(1), \dots, \pi(n)\}$ are a random permutation of  indices $\{1, \dots, n\}$. Here, we set the number of permutations  $R(n)=\lfloor 200+5000/n\rfloor$ as in \cite{szekely2007measuring}.
	Consequently, we can also estimate the P-value associated with the conditional independence test based on the quantiles of the test statistics over $R(n)$ random permutations.
\end{remark}

\section{Monte Carlo Experiments}\label{sec::NumericalS}
In this section, we investigate the performance of P-DCov with five simulation examples.  In Example \ref{exmp::SimTestB}, we consider a factor model and test the conditional independence between two vectors ${\bf x}$ and ${\bf y}$ given their common factor ${\bf f}$, via P-DCov. In Examples \ref{exmp::GaussianGraph}, we investigate the classical Gaussian graphical model. In Example \ref{exmp::smu}, we consider the case of general graphical model without the Gaussian assumption. In Examples \ref{exmp::factor-graph} and \ref{exmp::smu-factor}, we consider the case of factor based dependency graph and a general graphical model with external factors, respectively.

\begin{exmp}\label{exmp::SimTestB} [High-dimensional factor model]
Let $p=5$, $q=10$ and $K=1000$. Assume the rows of ${\bf B}_x$ and rows of ${\bf B}_y$ are i.i.d. distributed as ${\bf z}_K=({\bf z}_1^T,{\bf z}_2^T)^T$, where ${\bf z}_1$ is a 3-dimensional vector with elements i.i.d. from $Unif[2,3]$ and ${\bf z}_2={\bf 0}_{K-3}$.
 $\{{\bf f}_i\}^n_{i=1}$ are i.i.d. from $\mathcal{N}(0,I_K)$. We generate $n$ $i.i.d.$ copies $\{{\bf r}_i\}_{i=1}^n$ from  log-normal distribution $\ln \mathcal{N}(0,\boldsymbol{\Sigma})$ where $\boldsymbol{\Sigma}$ is an equal correlation matrix of size $(p+q)\times(p+q)$ with $\Sigma_{jk}=\rho$ when $j \not = k$ and $\Sigma_{jj}=1$.  $\boldsymbol{\epsilon}_{i,x}$ and $\boldsymbol{\epsilon}_{i,y}$ are the centered version of the first $p$ coordinates and the last $q$ coordinates of ${\bf r}_i$. Then, $\{{\bf x}_i\}^n_{i=1}$ and $\{{\bf y}_i\}^n_{i=1}$ are generated according to ${\bf x}_i={\bf B}_x {\bf f}_i+\boldsymbol{\epsilon}_{i,x}$ and ${\bf y}_i = {\bf B}_y{\bf f}_i+\boldsymbol{\epsilon}_{i,y}$ correspondingly.
\end{exmp}

In Example \ref{exmp::SimTestB}, we consider a high-dimensional factor model with sparsity structure. Note that the errors are generated from a heavy tail distribution to demonstrate the proposed test works beyond  Gaussian errors. We assume each coordinate of ${\bf x}$ and ${\bf y}$ only depends on the first three factors.
We calculate $T({\bf x},{\bf y},{\bf f})$ in the P-DCov test, and $T_0 ({\bf x},{\bf y},{\bf f})$ in which we replace $\hat{\boldsymbol{\epsilon}}_{i,x}$ and $\hat{\boldsymbol{\epsilon}}_{i,y}$ by the true $\boldsymbol{\epsilon}_{i,x}$ and $\boldsymbol{\epsilon}_{i,y}$ as an oracle test to compare with. To get reference distributions of $T({\bf x},{\bf y},{\bf f})$ and $T_0 ({\bf x},{\bf y},{\bf f})$, we follow the permutation procedure as described in Section \ref{sec::TheoreticalP}.  In this example, we set the significance level $\alpha= 0.1$.
We vary the sample size from 100 to 200 with increment of 10 and show the empirical power based on 1000 repetitions for both $T({\bf x},{\bf y},{\bf f})$ and $T_0({\bf x},{\bf y},{\bf f})$ in Figure \ref{fig:: Example 1} for $\rho\in\{0.1,0.2,0.3,0.4\}$. In the implementation of  penalized least squares in Step 1, we use R package \pkg{glmnet} with the default tuning parameter selection method (10-fold cross-validation) and perform least square on the selected variables to reduce estimation bias of these estimated parameters.

From Figure \ref{fig:: Example 1}, it is clear that as the sample size or $\rho$ increases, the empirical power also increases in general. Also, comparing the panels (A) and (B) in Figure \ref{fig:: Example 1}, we  see that when the sample size is small, the P-DCov test has smaller power than the oracle test, however, the difference between them becomes negligible as the sample size increases.  This is consistent with our theory regarding the asymptotic distribution of the test statistics.  When $\rho=0$, Table \ref{Table::Example 1} reports the empirical type I error for both P-DCov as well as the oracle version. It is clear that the type I error of P-DCov is under good control as the sample size increases.

\begin{table}
\caption{Type I error of Example 1\label{Table::Example 1}}
\centering
\begin{tabular}{cccccccccccc}
\hline
& \multicolumn{11}{c}{Test based on $\hat{\boldsymbol{\epsilon}}_x$ and $\hat{\boldsymbol{\epsilon}}_y$}\\
\hline
$n$&100 & 110 & 120& 130 & 140& 150 & 160 & 170 & 180 & 190 & 200 \\ \hline
& 0.129 & 0.112 & 0.117 & 0.095 & 0.104 & 0.108 & 0.098 & 0.113 & 0.102 & 0.117 & 0.111 \\
\hline\hline
& \multicolumn{11}{c}{Test based on $\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y$} \\
\hline
$n$&100 & 110 & 120 & 130 & 140& 150 & 160 & 170 & 180 & 190 & 200 \\
\hline
 & 0.089 & 0.092 & 0.078 & 0.112 & 0.104 & 0.090 & 0.099 & 0.113 & 0.104 & 0.106 & 0.096 \\
\hline\hline
\end{tabular}
\end{table}

\begin{figure}[tb!]
\caption{Power-sample size graph of Example 1\label{fig:: Example 1}
}
\begin{tabular}{cc}
	\includegraphics[scale=0.4]{Figures/ex1-power-est.pdf}&\includegraphics[scale=0.4]{Figures/ex1-power-true.pdf}\\
	(A) $T({\bf x},{\bf y},{\bf f})$&(B) $T_0({\bf x},{\bf y},{\bf f})$\\
\end{tabular}

\end{figure}

\begin{exmp}\label{exmp::GaussianGraph}[Gaussian graphical model]
We consider a standard Gaussian graphical model with precision matrix $\boldsymbol{\Omega}=\boldsymbol{\Sigma}^{-1}$,  where $\boldsymbol{\Omega}$ is a tridiagonal matrix of size $d\times d$, and is associated with the autoregressive process of order one. We set $d=30$ and the $(i,j)$-element in $\boldsymbol{\Sigma}$ to be $\sigma_{i,j}=\exp(- | s_i-s_j |)$, where $s_1<s_2<\dots<s_d$. In addition,
\[
s_i-s_{i-1}\overset{\text{i.i.d}}{\sim}  \text{Uniform} \ (1,3), \ \ i=2,\dots,d.
\]

\end{exmp}

In this example, we would like to compare the proposed P-DCov with the state-of-the-art approaches for recovering Gaussian graphical models.
In terms of recovering structure $\Omega$, we compare lasso.dcov (projection by LASSO followed by distance covariance), sam.dcov (projection by sparse additive model followed by distance covariance), lasso.pearson (projection by LASSO followed by Pearson correlation), sam.pearson (projection by sparse additive model followed by Pearson correlation)  with three popular estimators corresponding to the LASSO, adaptive LASSO and SCAD penalized likelihoods (called  graphical.lasso, graphical.alasso and graphical.scad on the graph) for the precision matrix \citep{friedman2008sparse, fan2009network}. Here, lasso.dcov and sam.dcov are two examples of our P-DCov methods.  We use  R package \pkg{SAM} to fit the sparse additive model.  To evaluate the performances, we construct receiver operating characteristic (ROC) curves for each method with sample sizes $n=100$ and $n=300$. The process of constructing the ROC curves involves conducting the P-DCov test for each pair of nodes and record the corresponding P-values.  In each of the ROC curve, true positive rates (TPR) are plotted against false positive rates (FPR) at various thresholds  of those P-values (``TP" means the true entry of the precision matrix is nonzero and estimated as nonzero; ``FP" means the true entry of the precision matrix is zero but estimated as nonzero.)  We follow the implementation in  \cite{fan2009network} for the three penalized likelihood estimators.  The average results over 100 replications of different methods are reported in Figure \ref{fig:: ggm}. The associated AUC (Area Under the Curve) for each method is also displayed in the legend of the figure.

\begin{figure}[tb!]
\caption{ROC curves for Gaussian graphical models \label{fig:: ggm}}
\begin{tabular}{cc}
	\includegraphics[scale=0.43]{Figures/ggm-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/ggm-roc-n=300.pdf}\\
	(A) $n=100$&(B) $n=300$\\
\end{tabular}

\end{figure}

We observe that lasso.pearson and sam.pearson perform similarly with the penalized likelihood methods when $n=100$. On the other hand, lasso.dcov and sam.dcov lead to slightly smaller AUC value due to the use of the distance covariance, which is expected for the Gaussian model.  This shows that we do not pay a big price for using the more complicated distance covariance and sparse additive model.

\begin{exmp}\label{exmp::smu}[A general graphical model]
We consider a general graphical model with a combination of multivariate $t$ distribution  and  multivariate Gaussian distribution. The dimension of ${\bf x}$ is $d=30$.  In detail,
${\bf x}= ({\bf x}_1^T,{\bf x}_2^T)^T$ where ${\bf x}_1$ follows a 20 dimensional multivariate $t$ distribution with degrees of freedom 5, location parameter $0$ and identity covariance matrix and ${\bf x}_2$ follows the same Gaussian graphical model as in Example \ref{exmp::GaussianGraph} except the dimension is now 10. In addition, ${\bf x}_1$ and ${\bf x}_2$ are independent.
\end{exmp}
To generate a multivariate $t$-distribution, we first generate a random vector ${\bf w}_{20}$ from the standard multivariate Gaussian distribution and  an independent random variable $\tau \sim \chi^2 (5)$ and then set ${\bf x}_1 = {\bf w}/\sqrt{\tau}$. One important fact about the multivariate $t$ distribution is that the zero element in the precision matrix does not imply conditional independence like the case of Gaussian graphical models \citep{finegold2009robust}. Indeed, for ${\bf x}_1$, we actually have the fact that ${\bf x}_1^{(i)}$ and ${\bf x}_1^{(j)}$ are dependent given ${\bf x}_1^{(-i, -j)}$ for any pair $1\le i\neq j\le 20$. Consequently, the Gaussian likelihood based methods will falsely claim that all the components of ${\bf x}_1$ are independent.

The average ROC curve results are rendered in Figure \ref{fig:: smu}. As expected, by using the new projection based distance covariance method for testing conditional independence, lasso.dcov outperforms all the other methods in terms of AUC, with a more evident advantage when $n=300$. One interesting observation is that: in the region where FPR is very low, the likelihood based methods actually outperform P-DCov methods. One possible reason is that the likelihood based methods are more capable of capturing the conditional dependency structure within ${\bf x}_2$ as it follows a Gaussian graphical model.

\begin{figure}[tb!]
\caption{ROC curves for a general graphical model \label{fig:: smu}}
\begin{tabular}{cc}
	\includegraphics[scale=0.43]{Figures/smu-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/smu-roc-n=300.pdf}\\
	(A) $n=100$&(B) $n=300$\\
\end{tabular}

\end{figure}

\begin{exmp}\label{exmp::factor-graph}[Factor based dependency graph]
We consider a dependency graph with the contribution of external factors. In particular, we generate ${\bf u}\sim N(0,\boldsymbol{\Omega})$, where $\boldsymbol{\Omega}$ is the same tridiagonal matrix used in Example \ref{exmp::GaussianGraph} and ${\bf f} \sim N(0, {\bf I})$, then the observation ${\bf x} = {\bf u} + {\bf Q} g({\bf f})$ where ${\bf Q}$ is a sparse coefficient matrix that dictates how each dimension of ${\bf x}$ depends on the factor $g({\bf f})$. In particular, the generation of ${\bf Q}$ follows the setting in \cite{cai2013covariate}.  For each element $Q_{ij}$, we first generate a Bernoulli distribution with success probability 0.2 to determine whether $Q_{ij}$ is 0 or not. If $Q_{ij}$ is not 0, we then generate $Q_{ij} \sim \mbox{Uniform}\mbox{ }(0.5,1)$. Here we consider two forms of $g(\cdot)$, namely $g({\bf f})={\bf f}$ and $g({\bf f})={\bf f}^2$.
\end{exmp}
We report  results regarding the average ROC curves  for lasso.pearson, lasso.dcov, sam.pearson and sam.dcov. The results for both $g({\bf f})={\bf f}$ and $g({\bf f})={\bf f}^2$ are depicted in Figure \ref{fig:: factor}. Note that we are not building a conditional dependency graph among ${\bf x}$, but a dependency graph of ${\bf x}$ conditioning on the external factor ${\bf f}$. There are some insightful observations from the figure. First of all, by looking at the first case when $g({\bf f}) ={\bf f}$, it is clear that lasso.pearson is the best as it takes advantage of the sparse linear structure paired with the Gaussian distribution of the residual. By using the distance covariance as a dependency measure, or by using the sparse additive model as a projection method, it is reassuring that we do not lose much efficiency. Second, for the case when $g({\bf f})={\bf f}^2$, we can see a substantial advantage of the sparse additive model based methods as it can capture this nonlinear contribution of the factors to the dependency structure of ${\bf x}$.

\begin{figure}[tb!]
\caption{ROC curves for factor based dependency graph \label{fig:: factor}}
\begin{tabular}{cc}
	\includegraphics[scale=0.43]{Figures/factor-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/factor-roc-n=300.pdf}\\
	(A) $n=100$, $g({\bf f})={\bf f}$ &(B) $n=300$, $g({\bf f})={\bf f}$\\
\includegraphics[scale=0.43]{Figures/factor-add-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/factor-add-roc-n=300.pdf}\\
	(C) $n=100$, $g({\bf f})={\bf f}^2$ &(D) $n=300$, $g({\bf f})={\bf f}^2$\\

\end{tabular}

\end{figure}

\begin{exmp}\label{exmp::smu-factor}[A general graphical model with external factors]
We consider a general conditional dependency graph with the contribution of external factors by combining the ingredients of Examples \ref{exmp::smu} and \ref{exmp::factor-graph}. In particular, we generate ${\bf u}$ from Example \ref{exmp::smu} and ${\bf f} \sim N(0, I_{30})$, then set ${\bf x} = {\bf u} + {\bf Q} g({\bf f})$ where ${\bf Q}$ is the same as Example \ref{exmp::factor-graph}. We also consider $g({\bf f})={\bf f}$ and $g({\bf f}) ={\bf f}^2$.
\end{exmp}
In this example, we would like to investigate the performance of a two-step projection method. In particular, we first project ${\bf x}$ onto the space spanned by ${\bf f}$ and denote  the residual by $\hat{\bf u}$. Then we explore the conditional dependency structure of $\hat{\bf u}^{(i)}$ and $\hat{\bf u}^{(j)}$ given $\hat{\bf u}^{(-i, -j)}$ by projecting them onto the space orthogonal to the space (linearly or additively) spanned  by $\hat{\bf u}^{(-i, -j)}$. Here, we compare the performances of  methods using the external factor and those that ignore them. The average ROC curves are rendered in Figure \ref{fig:: smu-factor}.

From the figure, we  see that first of all, when $g({\bf f})={\bf f}$,  the methods using  external factors outperform their counterparts without using the information with the best method being lasso.dcov (lasso regression based projection coupled with distance covariance).  Second, when we have nonlinear factors, using the factors do not necessarily help when we only consider  linear projection. For example,  the performances of lasso.pearson and lasso.pearson.f in panel (c) illustrates this point. On the other hand, by using sparse additive model based projection, we have a substantial gain over all the remaining methods especially for $n=300$.

\begin{figure}[tb!]
\caption{ROC curves for a general graphical model with external factors \label{fig:: smu-factor}}
\begin{tabular}{cc}
	\includegraphics[scale=0.43]{Figures/smu-f-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/smu-f-roc-n=300.pdf}\\
	(A) $n=100$, $g({\bf f})={\bf f}$ &(B) $n=300$, $g({\bf f})={\bf f}$\\
\includegraphics[scale=0.43]{Figures/smu-f-add-roc-n=100.pdf}&\includegraphics[scale=0.43]{Figures/smu-f-add-roc-n=300.pdf}\\
	(C) $n=100$, $g({\bf f})={\bf f}^2$ &(D) $n=300$, $g({\bf f})={\bf f}^2$\\

\end{tabular}

\end{figure}

\section{Real Data Analysis}\label{sec::Realdata}
\subsection{Financial Data}
In the first empirical example, we consider the Fama-French three-factor model \citep{fama1993common}. We collect daily excess returns of 90 stocks among the S\&P 100 index, which are available between August 19, 2004 and August 19, 2005. We chose the starting date as Google's Initial Public Offering date, and consider one year of daily excess returns since then. In particular, we consider the following three-factor model
\[
r_{it}-r_{ft}=\beta_{i,\text{MKT}}(\text{MKT}_t-r_{ft})+\beta_{i,\text{SMB}}\text{SMB}_{t}+\beta_{i,\text{HML}}\text{HML}_t+u_{it},
\]
for $i=1,\dots,90$ and $t=1,\dots,252$. At time $t$, $r_{it}$ represents the return for stock $i$, $r_{ft}$ is the risk-free return rate, and MKT, SMB and HML constitute market, size and value factors.

We perform P-DCov test with FDR control on all pairs of stocks and study the dependence between stocks conditional on the  Fama-French three-factors. Under significance level $\alpha=0.01$, we found out that 15.46\% of the pairs of stocks are conditionally dependent given the three factors, which implies that the three factors may not be sufficient to explain the dependencies among stocks. As a comparison, we also implemented the conditional independence test with the distance covariance based test replaced by Pearson correlation based  test. It turns out the 9.34\% of the pairs are significant under the same significance level. This shows the P-DCov test is more powerful in discovering significant conditionally dependent pairs than the Pearson correlation test.

We then investigate the top 5 pairs of stocks that correspond to the largest test statistic values using the P-DCov test. They are (BHI, SLB), (CVX, XOM), (HAL,  SLB), (COP, CVX), and (BHI, HAL). Interestingly, all six stocks involved are closely related to the oil industry. This reveals the high level of dependence among oil industry stocks that cannot be well explained by the Fama-French three-factor model. In addition, we examine the stock pairs that are conditionally dependent under the P-DCov test but not under the Pearson correlation test.  The two most significant pairs are (C, USB) and (MRK, PFE). The first pair are in the financial industry (Citigroup and U.S. Bancorp) and the second pair are pharmaceutical companies (Merck \& Co. and Pfizer). This shows that by using the proposed P-DCov, some interesting conditional dependence structures could be recovered.  This is consistent with the findings that the sector correlations are still present even after adjusting Fama-French factors and 10 industrial factors \citep{fan2016incorporating}.

\subsection{Breast Cancer Data}
In this section, we explore the difference in genetic networks between breast cancer patients who achieve pathologic Complete Response (pCR) and patients who do not achieve pCR.  Achieving pCR is defined as no invasive and no in situ residuals left in breast in the surgical specimen. As studied in \cite{kuerer1999clinical} and \cite{kaufmann2006recommendations}, pCR has predicted long-term outcome in several neoadjuvant studies and hence serves as a potential surrogate marker for survival. In this study, we use the normalized gene expression data of 130 patients with stages I-III breast cancers analyzed by \cite{hess2006pharmacogenomic}. Among the 130 patients, 33 of them achieved pCR (class 1), while the other 97 patients did not achieve pCR (class 2). To construct the conditional dependence network for each class, we first perform a two-sample $t$-test between the two groups and select the  100 genes with the smallest $p$-values. Afterwards, we construct networks of these 100 selected genes for each class using P-DCov with the FDR control at  level $\alpha=0.01$. Notice, in this case, $d=100$ and the corresponding sample sizes in two groups are $n_1=33$ and $n_2=97$ respectively.

In networks, the degree of a particular node describes how many edges are connected to this node, and the average degree serves as a measure of connectivity of the graphs. In Figure \ref{fig:: Real Data: Breast Cancer-Degree}, we summarize the distribution of degrees for the genetic networks of class 1 and class 2 respectively. We see that the average degree of genetic network for class 1 is 9.7 which is much smaller than the average degree of network for class 2, which is 44.88. To look at the networks more closely, we select 7 genes among the 100 genes and draw the corresponding networks in Figure \ref{fig:: Real Data: Breast Cancer-Network}. We see that for Class 1 where pCR is achieved, gene MCCC2 is a hub and is connected with three other genes. However, in the network for Class 2, gene MCCC2 is disconnected from the other six genes. On the other hand, gene MAPT is isolated in the network for Class 1, but is connected with two other genes in class 2.

These findings imply that the two classes may have very different conditional dependence structures, and hence likely to have different  precision matrices. As a result, when classification is the target, linear discriminant analysis may be too simple to capture the actual decision boundary.

\begin{figure}[tb!]
\caption{Degree distribution of the genetic networks\label{fig:: Real Data: Breast Cancer-Degree} }
\begin{tabular}{cc}
	\includegraphics[scale=0.43]{Figures/DegreeDist1-eps-converted-to.pdf}&\includegraphics[scale=0.43]{Figures/DegreeDist2-eps-converted-to.pdf}\\
	(A) Class 1 &(B) Class 2\\

\end{tabular}

\end{figure}

\begin{figure}[tb!]
\caption{Genetic networks for the two classes based on 7 selected genes\label{fig:: Real Data: Breast Cancer-Network}}
\begin{tabular}{cc}
	\includegraphics[scale=0.3]{Figures/subgraph1fdr.png}&\includegraphics[scale=0.3]{Figures/subgraph2fdr.png}\\
	(A) Class 1 &(B) Class 2\\

\end{tabular}

\end{figure}

\section{Discussion}\label{sec::Disc}
In this work, we proposed a general framework for testing conditional independence via projection and showed a new way to create dependency graphs. The current theoretical results assume that contribution of factors is sparse linear. How to extend the theory to the case of sparse additive model projection would be an interesting future work. Another interesting direction is to use the proposed test to create dependency graphs among groups of nodes, which could have applications in genetics.

An R package \pkg{pgraph} for implementing the proposed methodology is available on CRAN.

redpajama_set_name: RedPajamaArXiv


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text:
Campylobacter (altgriechisch καμπὐλος kampylos = krumm, βακτήρια bakteria = Stab) (vereinzelt auch Camphylobacter) ist eine Gattung korkenzieherförmiger (Spirillen) Bakterien, die Krankheiten wie Campylobacter-Enteritis (Campylobacteriose) hervorrufen kann. Sie sind gramnegativ, mikroaerophil und polar begeißelt. Der Gattungsname wurde 1963 von Sebald und Veron geschaffen. Früher wurden Vertreter der Gattung als "mikroaerophile Vibrionen" bezeichnet und zu der Familie Vibrionaceae gestellt.

Das Genom der folgenden Arten ist vollständig sequenziert: Campylobacter concisus, C. curvus, C. fetus, C. hominis und C. jejuni

Merkmale
Die Zellgröße liegt im Bereich von 0,2–0,8 × 0,5–5 Mikrometer. Sie sind entweder mit jeweils einer einzelnen Geißel unipolar an einem Ende oder bipolar an beiden Enden der Zelle begeißelt. Die Zellen können sich im Laufe der Kultur von korkenzieherförmig zu kokkenförmig ändern. Die meisten Arten von Campylobacter sind Katalase- und Oxidase-positiv, zu den Katalase-negativen zählen beispielsweise C. sputorum, C. concisus, C. mucosalis und C. helveticus. Die medizinisch wichtigen C. fetus subsp. fetus, C. coli, C. jejun subsp. jejuni sind Katalase-positiv.

Stoffwechsel
Die Arten der Gattung Campylobacter sind chemoorganotroph, und zwar Nitrat-Atmer. Das bedeutet, sie haben einen oxidativen Energiestoffwechsel und verwenden Nitrat als Oxidans, anstelle des bei der "normalen" aeroben Atmung verwendeten Sauerstoffs (O2). Elektronendonatoren sind Aminosäuren und Zwischenprodukte des Tricarbonsäurezyklus, die durch Nitrat oxidiert werden. Kohlenhydrate werden nicht verwendet.

Pathogenität
Viele Campylobacter spezies sind für den Menschen krankheitserregend, andere sind ungefährlich und natürliche Mitbewohner des Menschen und verschiedener Tiere.

Beim Menschen ausgelöste Erkrankungen sind meist lebensmittelbedingt, die pathogenen Bakterien der Campylobacter-Gruppe gehören zur Gruppe der Zoonoseerreger. Sie können von Tieren auf den Menschen übertragen werden und dort zu einer Erkrankung führen. Sie befinden sich auch oft auf rohem Geflügelfleisch (Stand 2017: Durchseuchungsrate in Großbritannien bei 50 %). Bei Tieren wie Schwein oder Rind bleibt die Infektion häufig unentdeckt, weil diese meist keine Krankheitssymptome zeigen. Im Haushalt hingegen, insbesondere bei Geflügelfleisch, ist auf gute Küchenhygiene zu achten, um Kreuzkontaminationen zu vermeiden. Die Behördliche Lebensmittelüberwachung in Deutschland führt daher epidemiologische Studien durch. Sie empfiehlt Verbrauchern die Produkte gut durchzugaren. Wesentlich zu dieser Empfehlung trägt bei, dass (anders als andere Bakterien) Campylobacter sich nicht im oder auf dem Lebensmittel vermehrt, es besteht also meist die realistische Chance diese Mikroorganismen durch Hitze abzutöten, so dass sie sich nicht im Darm eines Wirtes vermehren können und somit keine Durchfallerkrankung bedingen können. Hersteller von Lebensmitteln werden im Sinne einer vorbeugenden Produkthaftung dazu angehalten, entsprechende Zubereitungshinweise aufzudrucken. Seit 1. Januar 2018 gilt eine gesetzliche Untersuchungspflicht für Masthähnchen innerhalb der Europäischen Union. Die Umstände der Schlachthygiene sind jedoch nach wie vor besorgniserregend.

Die Erreger C. jejuni und C. coli können beim Menschen eine entzündliche Durchfallerkrankung (Campylobacter-Enteritis) auslösen. Campylobacter ist, noch vor den Salmonellen, der häufigste bakteriellen Durchfallerreger - kein bakterieller Erreger überhaupt wird so oft gemeldet. Der meistnachgewiesene Vertreter ist C. jejuni. Für den Menschen weniger wichtige Durchfallerreger sind C. lari, C. fetus und C. upsaliensis. Sehr selten tritt auch C. helveticus als Erreger auf. Einige Arten wie C. curvus, C. showae und C. gracilis können an Parodontitis beteiligt sein.

C. fetus (Synonym: Vibrio fetus) verursacht neben dem Durchfall beim Menschen auch einen enzootischen Campylobacter-Abort bei Rindern und Schafen. C. hyointestinalis und C. mucosalis sind Verursacher einer nekrotisierende Enterokolitis bei Schweinen.

Meldepflicht
In Deutschland ist der direkte oder indirekte Nachweis von darmpathogenen Campylobacter sp. namentlich meldepflichtig nach  des Infektionsschutzgesetzes (IfSG), soweit der Nachweis auf eine akute Infektion hinweist. Die Meldepflicht betrifft in erster Linie die Leitungen von Laboren ( IfSG).

In der Schweiz ist der positive und negative laboranalytische Befund von Campylobacter spp. für Laboratorien meldepflichtig und zwar nach dem Epidemiengesetz (EpG) in Verbindung mit der Epidemienverordnung und  der Verordnung des EDI über die Meldung von Beobachtungen übertragbarer Krankheiten des Menschen.

Literatur
 George M. Garrity: Bergey's Manual of Systematic Bacteriology. 2. Auflage, Band 2: The Proteobacteria. Part C: The Alpha-, Beta-, Delta-, and Epsilonproteabacteria. Springer, New York 2005, ISBN 0-387-24145-0.
 Werner Köhler (Hrsg.): Medizinische Mikrobiologie. 8. Auflage, München/Jena 2001, ISBN 978-3-437-41640-8.
 Björn-Martin Werth: Vergleichende Untersuchungen zu Schnellmethoden für den Nachweis thermophiler Campylobacter spp. in natürlich und artifiziell kontaminiertem Geflügelfleisch und – innereien sowie in Geflügelkot. Mensch-und-Buch-Verlag, Berlin 2006, ISBN 3-86664-113-3.

Einzelnachweise

Weblinks
 Bundesinstitut für Risikobewertung: Campylobacter
 Robert Koch-Institut: Campylobacter-Infektionen

Campylobacterales
Campylobacterales (Ordnung)
Lebensmittelmikrobiologie
Meldepflichtiger Erreger

redpajama_set_name: RedPajamaWikipedia


text:
Q: Notation for "vectorized" function Is there a notation for expressing vectorization of a function as implemented by Mathematica's Map
In[1] = Map[f,{a,b,c,d,e}]
Out[1] = {f[a],f[b],f[c],f[d],f[e]}

or Haskell's fmap?
ghci> fmap (+10) [1,2,3]
[11,12,13]

That is, given a function $x_i \mapsto f(x_i)$, is there a common typographic idiom of decorating or marking $f$ to express the function $g(\boldsymbol{x})\mapsto (g(x_0),g(x_1),\ldots,g(x_n))$, for $\boldsymbol{x}=(x_0,x_1,\ldots,x_n)$?
I've seen something like
$$\boldsymbol{f}(\boldsymbol{x})$$
used occasionally; but it's not clear that this would be understood without having stated that $\boldsymbol{f}:=g$.

A: Maybe just $\big( f(x_i) \big)_i$? But mentioned above, I think what you search for is usually written out or explained explicitly.

A: I suppose you could do something like
\begin{equation}
\boldsymbol{f}_g(\boldsymbol{x}),
\end{equation}
but you would have to define it; it's not a notation I've seen before (nor can I remember to have ever needed it). (The definition of the above would be:
\begin{equation}
\boldsymbol{f}_g(\boldsymbol{x}) \equiv (g(x_1),\,g(x_2),\,\dots,(g(x_n)),
\end{equation}
).

redpajama_set_name: RedPajamaStackExchange


NuCalm earned a perfect 5.0 peer-to-peer product evaluation score from Dental Product Shopper – published in the November 2016 issue.
In Dental Product Shopper's 10-year history, they have peer-evaluated 3,154 dental products. Of these, only 10% have received the "BEST PRODUCT" designation with a 4.2 score or higher and only one other product has earned a perfect 5.0, which was a GPS locator app back in 2011.
NuCalm continues to help dentists solve the problem of patient fear, stress, and anxiety with the world's only patented technology clinically proven to lower stress and improve sleep quality without drugs.


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text:
Q: how to get imei number in phonegap app with imeiplugin Can anyone explain to me how to get the device's IMEI number from Phonegap?
I am using the IMEI plugin in my app it is not showing anything.
To install, I have tried both:
phonegap plugin add https://github.com/zho/phonegap-imeiplugin.git

-- OR --
cordova plugin add imeiplugin

Example Usage:
window.plugins.imeiplugin.getImei(callback);

function callback(imei) {
    console.log("My Android IMEI :" + imei);

I want the output to be displayed when user opens the app, but it is not showing.

A: here is the solution that how can you  use imeiplugin
   Index.html
<!DOCTYPE html>

Index.js
var app = {
        // Application Constructor
        initialize: function () {
            app.bindEvents();
        },
        bindEvents: function () {
            document.addEventListener('deviceready', app.onDeviceReady, false);
        },
        onDeviceReady: function () {
            window.plugins.imeiplugin.getImei(callback);
        }
    };
    function callback(imei) {
        var element=document.getElementById("demo");
        element.value=imei;

    }

A: There is something wrong with the plugin you provided it showing an error like imeiplugin is unefined  at window.plugins.imeiplugin.getImei(callback);
so i tried with the below and its worked for me
cordova plugin add org.hygieiasoft.cordova.uid

and call the finction on device ready
 document.addEventListener('deviceready', onDeviceReady, false);
    function onDeviceReady() {
        console.log(cordova.plugins.uid.IMEI);
    }

check this for more info


Older Americans may be the ones contributing to the current housing shortage, an analysis by economists at Freddie Mac has found.
The study, which was first reported by Realtor.com, found that more than 1.1 million homes have been kept off the market by owners born between 1931 and 1941. In previous years, more people of the same age would have already listed or passed on their family homes as they retire and move to other places.
Freddie Mac further found that 300,000 more homes than usual are being held on to by Americans born between 1942 and 1947. Those born between 1948 and 1958 are occupying 250,000 more homes than previous generations had at the same age.
"We estimate that approximately 1.6 million more senior households are staying in place than would have been the case if they had behaved like previous generations of homeowners," said Sam Khater, chief economist at Freddie Mac, in a prepared statement.
That said, the situation is likely to change itself on its own. As baby boomers age, most will start to sell their homes – some are just waiting to do it later.
"We believe the additional demand for homeownership from seniors aging in place will increase the relative price of owning versus renting, making renting more attractive to younger generations," Khater said.