Optical flow: Difference between revisions

Revision as of 08:04, 20 February 2014

Hollister Sg .

The GQ Guide to 24 Hour Energy. The Cure for the Common Species of fish. The Short Order: Chef Brian LeFevre's Guide to Los Hollister Sg Angeles. And the flooding point of the film is very much likewise,abercrombie france, Sid sloth as) audience to seem forward to in twentieth Century simply by Fuchs and Blue Sky Broadcasters to work together to build the particular Hollywood blockbuster cartoon "ice age group 4" has been on July Twenty seventh in released in landmass China, those living in the actual glaciers of the animal the sea 3D adventure that children and adults are fun,abercrombie france, the film a variety of audience bursted into laughter. And features. "ice age 4" achieved great success, but also related to its subject,abercrombie portugal.

Secondly i have played both versions now my Three hundred and sixty copy and my friends playstation 3 250gb copy. The differences tend to be slight but the Ps3 types colours look more faded/less brilliant. THe game also looks extra blurry i guess due to the reduced resolution and vaseline filter where as the 360 version you can view all the detail due to the item being a sharper image.

Okay, put the pipe down home slice. The 'pyramids' on Mars had been re photographed by Mars International Surveyor about 12 yeras Nike Cortez Nm Qs ago and also were shown to be hills, along with the 'face' was shown to be nothing more than a oddly shaped hill, which usually actually with higher image resolution and correct Air Jordan 4 Singapore lighting is abundantly obvious. It's also worth pointing out the original cydonia images from the Seventies were posted by Their astronauts in the first place, and they were actually fairly good simulacra..

Whatever you consider the Cato/Kochs dispute, the fact that the legal action has now meant that Gary Leff, the actual CFO of IHS and Mercatus, has had to convey, , that the Kochs exercised no have an effect on over research at Mercatus along with IHS is. Horrifying. How could the Kochs not see the PR horror that this would bring on, it doesn't matter how justified the lawsuit could be?.

Ovenden is not the only artistic physique to face renewed scrutiny after the issues Abercrombie Fitch Ceo Plastic Surgery of child abuse as well as pedophilia have become more openly talked over in the last few decades. Case in point, the actual renowned 19th C. There've long been rumors of Carroll probable pedophilia but more recently crypticletters have come up to fuel increasing questions that the rumors may actually be true..

They may likewise try to fool you simply by touting breakthroughs or therapies, or the ads may use scientific sounding words like or even safe sounding words just like But the reality is that most of those products are useless, and at most effective a waste of money. Others are flat out dangerous to your health. Afterall, real scientific breakthroughs don't happen on obscure Website pages..

A&f Playlist you are.

Your source for dependable health information from the Federal government. Offering quick guides to healthy living, personalized health information, and tips and tools to help you and those you care about keep healthy. Includes custom tools in order to save, organize, and send search engine results to friends and family.

First Five.11, 5.12, A few.13, 5.14. Achieving a new number grade is a milestone akin to turning a fresh significant age 18, 7, 30, 50 etc. The time comes that A&f Playlist you send the path, and you technically feel just like you did the day before yet by some means, you are.

22 event. Justin Resnik was second on Carisma Z. Int'l September 25, 2013Jenni Martin McAllister ruled the afternoon at the $30,000 Los Angeles Foreign Grand Prix, earning the blue ribbon on Casseur de Corrt at the Sept. It would work. I started with lay down her in the crib and he or she cried for a couple of minutes, I traveled to rub her back. The item individual took my princess 2 nights to fall asleep by herself.

Mr. Roberts personally started several businesses prior to forming Infinium A labratory. He did so between 1996 and 2000, and to achieve success in one out of the three undertakings does not imply a series of failures. An attractive preliminary economic assessment by simply Micon International Ltd., based on a silver price of $950 per ounce above eight years, has estimated a cash cost of $208 per tonne machined, or $686 per gold oz recovered.For the company's comprehensive press release, please visit:Arguments from high consumption of gold are known: It is an unproductive advantage, it weans away financial savings, in fact it is a drain on forex trading. But there are several qualities associated with gold that make it an attractive investment decision option.Apart from the fact that yellow metal is embedded in the Indian native psyche and cultural ethos, it is almost a Air Jordan 3 perfect inflation buffer. During periods of high rising cost of living, as in the recent past, gold requirement remains high.Except that it has the storage is risk predisposed, the ability of gold to make sure against any kind of instability as well as crisis is well established.

Will there be something like this for any alternative game? No. Therefore, merely need to see how fast my personal computer is or how fast it needs to be, I will tend to rely on 3dmark. My partner and i dont care if it is some sort of "game" or not, it tells me the way it can handle graphics and i mean what I need to know when I enjoy "games"..

A quick way to navigate whenever using a health information directory is by using the hierarchical path that is certainly displayed at the top of the web Nike Blazer Vintage page. If "A" is selected, all the well being subjects beginning with "A" Nike Air Jordan Malaysia appear, and also the hierarchical path "Home > Health A Z >" shows at the top of a page. If "Anemia" is then chosen as the subject of interest, back links to Web pages presenting details on anemia appear, and the path near the top of the page reads "Home > Well being A Z > A >.In To go back to a page visited beforehand, just click on it in the ordered path (rather than clicking ones browser's "back" button).

Nike High Tops Singapore 15000+ expected CTC

We can't believe it's presently the middle of April mostly due to the fact that means Nike High Tops Singapore that March Madness seemed to be officially over a couple of weeks ago! If you had a chance to enter the bracket, we hope you enjoyed the tournament as much as all of us did. Our staff looks forward to March Madness each year, and we are excited to congratulate the particular winners of the 2012 Tournament situation Challenge. You can see the final standings below:.

Two spoke with support, while one indicated concerns about its app only to Town Center however supported the requirement for increased notice. Additional written testimony had been received just prior to the in close proximity of the record, which was analyzed at the Discussion and Instructions treatment. On Friday, January 25, 2013.

Retired pro snowboarder Mark Csincsak had a little more reality success, although it all went down away from plot lines. He won the fourth season of "The Hens," but his romantic relationship with DeAnna Pappas did not last. Csincsak proceeded to wed Ann Lueders, who showed up as a contestant on the thirteenth time of "The Bachelor." pair met while Csincsak was web hosting a reunion episode..

The Agencies Nike Jordan Malaysia have determined that loan companies appraisal policies should not favour appraisers from one or more organizations or perhaps exclude individuals based on their lack of such membership. INCLUDING THE Evaluation INSTITUTE MAICONFIDENTIALITY/PRIVILEGE NOTICE: This tranny and any attachments are intended entirely for the addressee. Pub.

Close to Raven say she GAY and has shacked on top of AzMarie Livingston, the androgynous beauty seen in Next Top Model, mentioned the site. Two are living together in Raven luxury $11,00 a month New York City condo. She with a point now where the lady doesn care what her household thinks, and she in love with AzMarie.

Place: Sprawled out on the couch. Feeling: Keep at it! Music: Abercrombie Sg Persona Some OST: Persuing My True Self applied All of this, for a little web-site! Boy oh boy, it appears as if there's never enough time to develop anything anymore. But I would worry about my job issues later, for now, I present you with ART!.

All these accounts process are done in computerized atmosphere. Current CTC: 15000+ expected CTC : 18000+ Reason : for better scope LANGUAGE PROFICIENCY Uk : Spoken and Written Hindi : Verbal and Written Marathi: Spoken as well as Written __________________________ Ramchandra R. New Jersey Tattoo design.

I Nike Air Max 90 think it's similar to OPI Also to be 25 Again, although luckily that's a polish I have been meaning to buy for about Four years, lol. I always did desire that one since Cat1104 did your NOTD pic with it on the makeupalley toe nail board years ago. This is a plummy brown lightly red that is stunning.

Air Jordan 1 lightheadedness

That's convenient in a earth where you have an infinite timeframe and/or money to try everything, but some of us (myself included) would prefer to make a choice based on the reviews. Even if I'm Korean, I've experimented with a lot of restaurants in Koreatown as well as some of them are hit or miss. I've pointed out that the ones that are recommended by The greater toronto area Life are actually the most detrimental, but I love BlogTO and all they offer and I often and then go restaurants that are researched well by BlogTO.

Arrhythmias may cause you to sense premature heartbeats, or you may feel that your heart is racing or even beating too slowly. Different signs and symptoms may be related to decreased blood output from your center. These include shortness of breath or wheezing, weakness, dizziness, lightheadedness, fainting or near fainting, in addition to chest pain or discomfort.

Type long, worth the read. When going to Las Vegas, ride your rides from places new York, New York and the Stratosphere, wager away your money in a drunken state on a few hands and wrists of poker, or please take a complete stranger back to a person's expensive hotel room. Or perhaps you TMd if you prefer a magic show Air Jordan 1 from Cirque Du Soleil or David Copperfield? I would suggest this option for your first night throughout Sin City.

Planning Abercrombie Fitch Interview a research project maybe a dissertation is really simple if you think regarding this. Unlike in project management, most of the projects come one after the other, not from the same time, making it much easier to see through. On the other hand, the vast majority of tasks need to be performed by the one and only yourself, so there's really no body else to control.

On every occasion that I are to bela dubby, the service is great. John and his team are like family to most associated with his patrons and Nike Jordan Shoes Singapore everyone enjoys coming there. I clearly am not the only one who seems this way, as it has been active each time that I have halted by.

When I do invest in my new digital system it will be small, light, and still have excellent wide angle optics and also tiny strobes, all of which I will bring in my coat pocket (Need to buy a coat of course). I'm not really going big, and to safeguard myself from marauding sharks and crocodiles I just now have to buy a large dive blade rather like the old Scubapro Cutlery and forget about using my camera. The F3 rig would have stopped a submarine..

Actually, Hannah has it dead incorrect. I have been inside the content output (Demand Studios) and I have noticed and Nike Blazer Vintage produced content below DMS. Look at the quality of content produced under Demand vs . the primarily horrendous person generated content created prior to Demand bought eHow.

Hollister Singapore |

Is It Necessary To Eat Before A Workout?With all of the marketing of activities drinks, bars, gels and also powders, this is an excellent question ought to. There's certainly no harm in feeding on and it might help you meet up with your fitness goals by giving people more energy during the specific run or elliptical exercise routine. MORE.

One recent study echoes the growing study, showing that egg consumption had no effect on the overall blood cholesterol levels of 115 healthful adults. What's more, another current study found some information that eating whole ovum increases HDL the heart safety, "good" cholesterol. An assistant clinical educator of Psychiatry at Columbia College College of Physicians Doctors, who studies the connection concerning lifestyle behaviors like eating plan and psychological health.

Haha! I don know about frustration so much as complaints re also: the headache it would get caused me. I used Squarespace Hollister Singapore myself, and investigated the item pretty thoroughly for consumers, and ultimately I stick with Wordpress blogs for the same reason I like Computers better than Macs (and I make my own PC as opposed to acquiring one from Best Buy): Nike Air Jordan Singapore I can perform whatever the hell I make sure you with them. Sure, sure, I may never need all of the extra features or potential, but I like having the possibility.

Flush with too much self-assurance, i volunteered to lead the development of a 16 foot radius EMT dome for Camp Revitalise, of which i am a spectator person. I am also learning some thing about screw in ground anchors along the way. Lastly, for sale geeky explanation of one method to modify the geometry for a larger entrance triangular shape.

It is highly recommended to have an SSL certificate. Commonly you get from your host, should you not purchase it from your host you can find it installed be hosting provider. It costs $20 to $80 per year for many of the sites, depending on your family needs..

After conducting a study involving LV, its effects on an individuals functional independence and on the disposable assistive technologies, the project's results show Nike Cortez that people with LV are still reasonably independent within the home, but outside the home this independence begins to deteriorate due to a deficit of vision assistance. The available products are expensive and narrow with application. Hence there is a absence of cheap and readily available haptic devices that could extend a LV wearer's perception of their immediate surroundings.

Ford is under this sort of microscope now. In fact, the entire Chevy 350 Engine series is extremely popular among car lovers, rapidly transforming Chevrolet directly into one of the most desired and extremely respected Hot Rods not only in the United States, but also in many different corners of the world. In only a matter of many years, the great features characteristic for the Chevy 350 Engine possess rendered Chevrolet a well known and competitive brand, setting up new trends in Hollister Singapore Store Locator building and manufacturing car motors.

Air Jordan Singapore Store Orleans Arena

I and my close friends ended up checking out the good options found on your website and then out of the blue I had an awful suspicion I never expressed respect to you for those guidelines. Most of the women happened to be consequently warmed to study all of them now have certainly been making the most of them. I appreciate you for truly getting really kind as well as for possessing variety of incredible Air Jordan Singapore Store subjects we are all really desperate to discover.

Noticeably, this particular submit is in fact the actual best about it significant design and style. We harmonise along with your findings and will thirstily research forward for your inward improvements. Saying many thanks will not you need to be adequate, for that amazing clearness within your composing.

Stand up for your family up for your good friends they may not be able to stand up for independently now that they were taken. Knowing anything that could help bring down a , bring him or her to the legal to be judged for the criminal activity he or she has committed. The time is NOW!.

Urban Dictionary . Webmonkey . Wikia . The color of an product or service can also function as a trademark. The Supreme Court held in the 1997 case of Qualitex Co. /. Impact Wrestling " July 4, 2013: HAIL Abercrombie And Fitch Factory Outlet+singapore SABIN! Rather than Much Else Impact Wrestling Particular date: July 4, 2013 Site: Orleans Arena, Las Vegas, Nevada Commentators: Mike Tenay, Taz We TMre in Sin City tonight for a taped Independence Day show. The main narrative Abercrombie & Fitch Singapore Sale is Austin Aries stole your X Division Title the other day and as of this moment grows to challenge Bully Ray fourteen days from tonight at Vacation destination X. Other than that we TMll possess a lot Abercrombie Sg Models more buildup towards Likely For Glory in the BFG Sequence as points continue to be acquired.

I actually devoted an entire couple chapters to answer these types of inquiries in my Book on Flipping Residences. Basically, to analyze an area, We look at several different metrics, like price ratios between affected and retail sales, number of months of supply, shopper activity, investor activity, etc. Similar data can be used to figure out what type of houses to focus on (you wish to be buying/selling what owner occupants in the room are buying)..

SourceForge . Transifex . TuxFamily. Paying for college has come below tremendous pressure in the past few years. Costs are rising, and the work possibilities for recent university graduates have not looked so bleak in decades. Yet, unlike when the parents of today's students went to education, loans are putting innovative graduates two steps in the opening financially before they've collateralized their first real job..

Optical flow: Difference between revisions

Revision as of 08:04, 20 February 2014

Contents

Hollister Sg .

A&f Playlist you are.

Nike High Tops Singapore 15000+ expected CTC

Air Jordan 1 lightheadedness

Hollister Singapore |

Air Jordan Singapore Store Orleans Arena

Navigation menu

@@ Line 1: / Line 1: @@
-{{distinguish|mixed model}}
+== Hollister Sg . ==
-{{Merge from|Mixture distribution|date=September 2010}}
-{{See also|Mixture distribution}}
-In [[statistics]], a '''mixture model''' is a [[probabilistic model]] for representing the presence of [[subpopulation]]s within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the [[mixture distribution]] that represents the [[probability distribution]] of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make [[statistical inference]]s about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information.
-Some ways of implementing mixture models involve steps that attribute postulated sub-population-identities to individual observations (or weights towards such sub-populations), in which case these can be regarded as types of [[unsupervised learning]] or [[cluster analysis|clustering]] procedures. However not all inference procedures involve such steps.
+The GQ Guide to 24 Hour Energy. The Cure for the Common Species of fish. The Short Order: Chef Brian LeFevre's Guide to Los [http://www.ashford.edu.sg/Chinese/define.php?a=106-Hollister-Sg Hollister Sg] Angeles. And the flooding point of the film is very much likewise,abercrombie france, Sid sloth as) audience to seem forward to in twentieth Century simply by Fuchs and Blue Sky Broadcasters to work together to build the particular Hollywood blockbuster cartoon "ice age group 4" has been on July Twenty seventh in released in landmass China, those living in the actual glaciers of the animal the sea 3D adventure that children and adults are fun,abercrombie france, the film a variety of audience bursted into laughter. And features. "ice age 4" achieved great success, but also related to its subject,abercrombie portugal. <br><br>Secondly i have played both versions now my Three hundred and sixty copy and my friends playstation 3 250gb copy. The differences tend to be slight but the Ps3 types colours look more faded/less brilliant. THe game also looks extra blurry i guess due to the reduced resolution and vaseline filter where as the 360 version you can view all the detail due to the item being a sharper image. <br><br>Okay, put the pipe down home slice. The 'pyramids' on Mars had been re photographed by Mars International Surveyor about 12 yeras [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=83-Nike-Cortez-Nm-Qs Nike Cortez Nm Qs] ago and also were shown to be hills, along with the 'face' was shown to be nothing more than a oddly shaped hill, which usually actually with higher image resolution and correct [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=9-Air-Jordan-4-Singapore Air Jordan 4 Singapore] lighting is abundantly obvious. It's also worth pointing out the original cydonia images from the Seventies were posted by Their astronauts in the first place, and they were actually fairly good simulacra.. <br><br>Whatever you consider the Cato/Kochs dispute, the fact that the legal action has now meant that Gary Leff, the actual CFO of IHS and Mercatus, has had to convey, , that the Kochs exercised no have an effect on over research at Mercatus along with IHS is. Horrifying. How could the Kochs not see the PR horror that this would bring on, it doesn't matter how justified the lawsuit could be?. <br><br>Ovenden is not the only artistic physique to face renewed scrutiny after the issues [http://www.ashford.edu.sg/Chinese/define.php?a=36-Abercrombie-Fitch-Ceo-Plastic-Surgery Abercrombie Fitch Ceo Plastic Surgery] of child abuse as well as pedophilia have become more openly talked over in the last few decades. Case in point, the actual renowned 19th C. There've long been rumors of Carroll probable pedophilia but more recently crypticletters have come up to fuel increasing questions that the rumors may actually be true.. <br><br>They may likewise try to fool you simply by touting breakthroughs or therapies, or the ads may use scientific sounding words like or even safe sounding words just like But the reality is that most of those products are useless, and at most effective a waste of money. Others are flat out dangerous to your health. Afterall, real scientific breakthroughs don't happen on obscure Website pages..<ul>
+   <li>[http://www.kxyxw.cn/bbs/forum.php?mod=viewthread&tid=263765 http://www.kxyxw.cn/bbs/forum.php?mod=viewthread&tid=263765]</li>
+   <li>[http://bbs.anjian.com/home.php?mod=spacecp&ac=blog&blogid= http://bbs.anjian.com/home.php?mod=spacecp&ac=blog&blogid=]</li>
+   <li>[http://prfinec.netne.net/node/12?page=296#comment-14839 http://prfinec.netne.net/node/12?page=296#comment-14839]</li>
+   <li>[http://advertsbook.net/forum/read.php?14,196246 http://advertsbook.net/forum/read.php?14,196246]</li>
+   <li>[http://khadakpada.com/index.php?page=item&id=354112 http://khadakpada.com/index.php?page=item&id=354112]</li>
+ </ul>
-Mixture models should not be confused with models for [[compositional data]], i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.).
+== A&f Playlist  you are. ==
-==Structure of a mixture model==
+Your source for dependable health information from the Federal government. Offering quick guides to healthy living, personalized health information, and tips and tools to help you and those you care about keep healthy. Includes custom tools in order to save, organize, and send search engine results to friends and family. <br><br>First Five.11, 5.12, A few.13, 5.14. Achieving a new number grade is a milestone akin to turning a fresh significant age 18, 7, 30, 50 etc. The time comes that [http://www.ashford.edu.sg/Chinese/define.php?a=98-A&f-Playlist A&f Playlist] you send the path, and you technically feel just like you did the day before yet by some means, you are. <br><br>22 event. Justin Resnik was second on Carisma Z. Int'l September 25, 2013Jenni Martin McAllister ruled the afternoon at the $30,000 Los Angeles Foreign Grand Prix, earning the blue ribbon on Casseur de Corrt at the Sept. It would work. I started with lay down her in the crib and he or she cried for a couple of minutes, I traveled to rub her back. The item individual took my princess 2 nights to fall asleep by herself. <br><br>Mr. Roberts personally started several businesses prior to forming Infinium A labratory. He did so between 1996 and 2000, and to achieve success in one out of the three undertakings does not imply a series of failures. An attractive preliminary economic assessment by simply Micon International Ltd., based on a silver price of $950 per ounce above eight years, has estimated a cash cost of $208 per tonne machined, or $686 per gold oz recovered.For the company's comprehensive press release, please visit:Arguments from high consumption of gold are known: It is an unproductive advantage, it weans away financial savings, in fact it is a drain on forex trading. But there are several qualities associated with gold that make it an attractive investment decision option.Apart from the fact that yellow metal is embedded in the Indian native psyche and cultural ethos, it is almost a [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=114-Air-Jordan-3 Air Jordan 3] perfect inflation buffer. During periods of high rising cost of living, as in the recent past, gold requirement remains high.Except that it has the storage is risk predisposed, the ability of gold to make sure against any kind of instability as well as crisis is well established. <br><br>Will there be something like this for any alternative game? No. Therefore, merely need to see how fast my personal computer is or how fast it needs to be, I will tend to rely on 3dmark. My partner and i dont care if it is some sort of "game" or not, it tells me the way it can handle graphics and i mean what I need to know when I enjoy "games".. <br><br>A quick way to navigate whenever using a health information directory is by using the hierarchical path that is certainly displayed at the top of the web [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=80-Nike-Blazer-Vintage Nike Blazer Vintage] page. If "A" is selected, all the well being subjects beginning with "A" [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=97-Nike-Air-Jordan-Malaysia Nike Air Jordan Malaysia] appear, and also the hierarchical path "Home > Health A Z >" shows at the top of a page. If "Anemia" is then chosen as the subject of interest, back links to Web pages presenting details on anemia appear, and the path near the top of the page reads "Home > Well being A Z > A >.In To go back to a page visited beforehand, just click on it in the ordered path (rather than clicking ones browser's "back" button).<ul>
+   <li>[http://ks35439.kimsufi.com/spip.php?article450/ http://ks35439.kimsufi.com/spip.php?article450/]</li>
+   <li>[http://verdamilio.info/org/spip.php?article573/ http://verdamilio.info/org/spip.php?article573/]</li>
+   <li>[http://bbs.zhgld.com/forum.php?mod=viewthread&tid=9138 http://bbs.zhgld.com/forum.php?mod=viewthread&tid=9138]</li>
+   <li>[http://www.s-ngo.org/forum.php?mod=viewthread&tid=466850 http://www.s-ngo.org/forum.php?mod=viewthread&tid=466850]</li>
+   <li>[http://rzazy.com/forum.php?mod=viewthread&tid=6724306&fromuid=584251 http://rzazy.com/forum.php?mod=viewthread&tid=6724306&fromuid=584251]</li>
+ </ul>
-===General mixture model===
+== Nike High Tops Singapore  15000+ expected CTC ==
-A typical finite-dimensional mixture model is a [[hierarchical Bayes model|hierarchical model]] consisting of the following components:
-*''N'' random variables corresponding to observations, each assumed to be distributed according to a mixture of ''K'' components, with each component belonging to the same [[parametric family]] of distributions (e.g., all Normal, all Zipfian, etc.) but with different parameters
+We can't believe it's presently the middle of April   mostly due to the fact that means [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=166-Nike-High-Tops-Singapore Nike High Tops Singapore] that March Madness seemed to be officially over a couple of weeks ago! If you had a chance to enter the bracket, we hope you enjoyed the tournament as much as all of us did. Our staff looks forward to March Madness each year, and we are excited to congratulate the particular winners of the 2012 Tournament situation Challenge. You can see the final standings below:. <br><br>Two spoke with support, while one indicated concerns about its app only to Town Center however supported the requirement for increased notice. Additional written testimony had been received just prior to the in close proximity of the record, which was analyzed at the Discussion and Instructions treatment. On Friday, January 25, 2013. <br><br>Retired pro snowboarder Mark Csincsak had a little more reality success, although it all went down away from plot lines. He won the fourth season of "The Hens," but his romantic relationship with DeAnna Pappas did not last. Csincsak proceeded to wed Ann Lueders, who showed up as a contestant on the thirteenth time of "The Bachelor." pair met while Csincsak was web hosting a reunion episode.. <br><br>The Agencies [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=163-Nike-Jordan-Malaysia Nike Jordan Malaysia] have determined that loan companies appraisal policies should not favour appraisers from one or more organizations or perhaps exclude individuals based on their lack of such membership. INCLUDING THE Evaluation INSTITUTE MAICONFIDENTIALITY/PRIVILEGE NOTICE: This tranny and any attachments are intended entirely for the addressee. Pub. <br><br>Close to Raven say she GAY and has shacked on top of AzMarie Livingston, the androgynous beauty seen in Next Top Model, mentioned the site. Two are living together in Raven luxury $11,00 a month New York City condo. She with a point now where the lady doesn care what her household thinks, and she in love with AzMarie. <br><br>Place: Sprawled out on the couch. Feeling: Keep at it! Music: [http://www.ashford.edu.sg/Chinese/define.php?a=57-Abercrombie-Sg Abercrombie Sg] Persona Some OST: Persuing My True Self applied All of this, for a little web-site! Boy oh boy, it appears as if there's never enough time to develop anything anymore. But I would worry about my job issues later, for now, I present you with ART!. <br><br>All these accounts process are done in computerized atmosphere. Current CTC: 15000+ expected CTC : 18000+ Reason : for better scope LANGUAGE PROFICIENCY Uk : Spoken and Written Hindi : Verbal and Written Marathi: Spoken as well as Written __________________________ Ramchandra R. New Jersey Tattoo design. <br><br>I [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=96-Nike-Air-Max-90 Nike Air Max 90] think it's similar to OPI Also to be 25 Again, although luckily that's a polish I have been meaning to buy for about Four years, lol. I always did desire that one since Cat1104 did your NOTD pic with it on the makeupalley toe nail board years ago. This is a plummy brown lightly red that is stunning.<ul>
-*''N'' corresponding random [[latent variable]]s specifying the identity of the mixture component of each observation, each distributed according to a ''K''-dimensional [[categorical distribution]]
-*A set of ''K'' mixture weights, each of which is a probability (a real number between 0 and 1 inclusive), all of which sum to 1
+   <li>[http://www.zhenerzhong.com/bbs/forum.php?mod=viewthread&tid=1887&fromuid=339 http://www.zhenerzhong.com/bbs/forum.php?mod=viewthread&tid=1887&fromuid=339]</li>
-*A set of ''K'' parameters, each specifying the parameter of the corresponding mixture component.  In many cases, each "parameter" is actually a set of parameters.  For example, observations distributed according to a mixture of one-dimensional [[Gaussian distribution]]s will have a [[mean]] and [[variance]] for each component. Observations distributed according to a mixture of ''V''-dimensional [[categorical distribution]]s (e.g., when each observation is a word from a vocabulary of size ''V'') will have a vector of ''V'' probabilities, collectively summing to 1.
+   <li>[http://mariettakaramanli.fr/spip.php?article289/ http://mariettakaramanli.fr/spip.php?article289/]</li>
+   <li>[http://bbs.lusongwang.com/forum.php?mod=viewthread&tid=25971&fromuid=3643 http://bbs.lusongwang.com/forum.php?mod=viewthread&tid=25971&fromuid=3643]</li>
+   <li>[http://111.11.181.205:8082/news/html/?38439.html http://111.11.181.205:8082/news/html/?38439.html]</li>
+   <li>[http://www.observatoiredesreligions.fr/spip.php?article11 http://www.observatoiredesreligions.fr/spip.php?article11]</li>
+ </ul>
-In addition, in a [[Bayesian inference|Bayesian setting]], the mixture weights and parameters will themselves be random variables, and [[prior distribution]]s will be placed over the variables.  In such a case, the weights are typically viewed as a ''K''-dimensional random vector drawn from a [[Dirichlet distribution]] (the [[conjugate prior]] of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors.
+== Air Jordan 1  lightheadedness ==
-Mathematically, a basic parametric mixture model can be described as follows:
+That's convenient in a earth where you have an infinite timeframe and/or money to try everything, but some of us (myself included) would prefer to make a choice based on the reviews. Even if I'm Korean, I've experimented with a lot of restaurants in Koreatown as well as some of them are hit or miss. I've pointed out that the ones that are recommended by The greater toronto area Life are actually the most detrimental, but I love BlogTO and all they offer and I often and then go restaurants that are researched well by BlogTO. <br><br>Arrhythmias may cause you to sense premature heartbeats, or you may feel that your heart is racing or even beating too slowly. Different signs and symptoms may be related to decreased blood output from your center. These include shortness of breath or wheezing, weakness, dizziness, lightheadedness, fainting or near fainting, in addition to chest pain or discomfort. <br><br>Type long, worth the read. When going to Las Vegas, ride your rides from places new York, New York and the Stratosphere, wager away your money in a drunken state on a few hands and wrists of poker, or please take a complete stranger back to a person's expensive hotel room. Or perhaps you TMd if you prefer a magic show [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=62-Air-Jordan-1 Air Jordan 1] from Cirque Du Soleil or David Copperfield? I would suggest this option for your first night throughout Sin City. <br><br>Planning [http://www.ashford.edu.sg/Chinese/define.php?a=1-Abercrombie-Fitch-Interview Abercrombie Fitch Interview] a research project maybe a dissertation is really simple if you think regarding this. Unlike in project management, most of the projects come one after the other, not from the same time, making it much easier to see through. On the other hand, the vast majority of tasks need to be performed by the one and only yourself, so there's really no body else to control. <br><br>On every occasion that I are to bela dubby, the service is great. John and his team are like family to most associated with his patrons and [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=159-Nike-Jordan-Shoes-Singapore Nike Jordan Shoes Singapore] everyone enjoys coming there. I clearly am not the only one who seems this way, as it has been active each time that I have halted by. <br><br>When I do invest in my new digital system it will be small, light, and still have excellent wide angle optics and also tiny strobes, all of which I will bring in my coat pocket (Need to buy a coat of course). I'm not really going big, and to safeguard myself from marauding sharks and crocodiles I just now have to buy a large dive blade rather like the old Scubapro Cutlery and forget about using my camera. The F3 rig would have stopped a submarine.. <br><br>Actually, Hannah has it dead incorrect. I have been inside the content output (Demand Studios) and I have noticed and [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=80-Nike-Blazer-Vintage Nike Blazer Vintage] produced content below DMS. Look at the quality of content produced under Demand vs . the primarily horrendous person generated content created prior to Demand bought eHow.<ul>
+   <li>[http://soft.zfk8.com/bbs/forum.php?mod=viewthread&tid=1387022 http://soft.zfk8.com/bbs/forum.php?mod=viewthread&tid=1387022]</li>
+   <li>[http://www.52byl.com/news/html/?191445.html http://www.52byl.com/news/html/?191445.html]</li>
+   <li>[http://hindichurch.com/index.php?option=com_kunena&view=topic&catid=3&id=200916&Itemid=326#201734 http://hindichurch.com/index.php?option=com_kunena&view=topic&catid=3&id=200916&Itemid=326#201734]</li>
+   <li>[http://aq2sw.com/forum.php?mod=viewthread&tid=2039382 http://aq2sw.com/forum.php?mod=viewthread&tid=2039382]</li>
+   <li>[http://pic.ifanxu.com/picture.php?/46/list/21,72,36,75,33,46,28,13,37,3,39,19,12,77,71&comments_order=DESC#comments http://pic.ifanxu.com/picture.php?/46/list/21,72,36,75,33,46,28,13,37,3,39,19,12,77,71&comments_order=DESC#comments]</li>
+ </ul>
-:<math>
+== Hollister Singapore | ==
-\begin{array}{lcl}
-K &=& \text{number of mixture components} \\
-N &=& \text{number of observations} \\
-\theta_{i=1 \dots K} &=& \text{parameter of distribution of observation associated with component } i \\
-\phi_{i=1 \dots K} &=& \text{mixture weight, i.e., prior probability of a particular component } i \\
-\boldsymbol\phi &=& K\text{-dimensional vector composed of all the individual } \phi_{1 \dots K} \text{; must sum to 1} \\
-z_{i=1 \dots N} &=& \text{component of observation } i \\
-x_{i=1 \dots N} &=& \text{observation } i \\
-F(x|\theta) &=& \text{probability distribution of an observation, parametrized on } \theta \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& F(\theta_{z_i})
-\end{array}
-</math>
-In a Bayesian setting, all parameters are associated with random variables, as follows:
+Is It Necessary To Eat Before A Workout?With all of the marketing of activities drinks, bars, gels and also powders, this is an excellent question ought to. There's certainly no harm in feeding on   and it might help you meet up with your fitness goals by giving people more energy during the specific run or elliptical exercise routine. MORE. <br><br>One recent study echoes the growing study, showing that egg consumption had no effect on the overall blood cholesterol levels of 115 healthful adults. What's more, another current study found some information that eating whole ovum increases HDL   the heart safety, "good" cholesterol. An assistant clinical educator of Psychiatry at Columbia College College of Physicians Doctors, who studies the connection concerning lifestyle behaviors like eating plan and psychological health. <br><br>Haha! I don know about frustration so much as complaints re also: the headache it would get caused me. I used Squarespace [http://www.ashford.edu.sg/Chinese/define.php?a=101-Hollister-Singapore Hollister Singapore] myself, and investigated the item pretty thoroughly for consumers, and ultimately I stick with Wordpress blogs for the same reason I like Computers better than Macs (and I make my own PC as opposed to acquiring one from Best Buy): [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=171-Nike-Air-Jordan-Singapore Nike Air Jordan Singapore] I can perform whatever the hell I make sure you with them. Sure, sure, I may never  need  all of the extra features or potential, but I like having the possibility. <br><br>Flush with too much self-assurance, i volunteered to lead the development of a 16 foot radius EMT dome for Camp Revitalise, of which i am a spectator person. I am also learning some thing about screw in ground anchors along the way. Lastly, for sale geeky explanation of one method to modify the geometry for a larger entrance triangular shape. <br><br>It is highly recommended to have an SSL certificate. Commonly you get from your host, should you not purchase it from your host you can find it installed be hosting provider. It costs $20 to $80 per year for many of the sites, depending on your family needs.. <br><br>After conducting a study involving LV, its effects on an individuals functional independence and on the disposable assistive technologies, the project's results show [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=4-Nike-Cortez Nike Cortez] that people with LV are still reasonably independent within the home, but outside the home this independence begins to deteriorate due to a deficit of vision assistance. The available products are expensive and narrow with application. Hence there is a absence of cheap and readily available haptic devices that could extend a LV wearer's perception of their immediate surroundings. <br><br>Ford is under this sort of microscope now. In fact, the entire Chevy 350 Engine series is extremely popular among car lovers, rapidly transforming Chevrolet directly into one of the most desired and extremely respected Hot Rods not only in the United States, but also in many different corners of the world. In only a matter of many years, the great features characteristic for the Chevy 350 Engine possess rendered Chevrolet a well known and competitive brand, setting up new trends in [http://www.ashford.edu.sg/Chinese/define.php?a=13-Hollister-Singapore-Store-Locator Hollister Singapore Store Locator] building and manufacturing car motors.<ul>
+   <li>[http://midvalleytreasurehunters.info//read.php?11,304517 http://midvalleytreasurehunters.info//read.php?11,304517]</li>
+   <li>[http://cn.wilinkedcare.com/thread-73616-1-1.html http://cn.wilinkedcare.com/thread-73616-1-1.html]</li>
+   <li>[http://bbstmp.haalee.com:8080/forum.php?mod=viewthread&tid=2612064 http://bbstmp.haalee.com:8080/forum.php?mod=viewthread&tid=2612064]</li>
+   <li>[http://yuesuoping.faceplusplus.com/forum.php?mod=viewthread&tid=211517 http://yuesuoping.faceplusplus.com/forum.php?mod=viewthread&tid=211517]</li>
+   <li>[http://121.197.1.51/forum.php?mod=viewthread&tid=3811697 http://121.197.1.51/forum.php?mod=viewthread&tid=3811697]</li>
+ </ul>
-:<math>
+== Air Jordan Singapore Store  Orleans Arena ==
-\begin{array}{lcl}
-K,N &=& \text{as above} \\
-\theta_{i=1 \dots K}, \phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
-z_{i=1 \dots N}, x_{i=1 \dots N}, F(x|\theta) &=& \text{as above} \\
-\alpha &=& \text{shared hyperparameter for component parameters} \\
-\beta &=& \text{shared hyperparameter for mixture weights} \\
-H(\theta|\alpha) &=& \text{prior probability distribution of component parameters, parametrized on } \alpha \\
-\theta_{i=1 \dots K} &\sim& H(\alpha) \\
-\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& F(\theta_{z_i})
-\end{array}
-</math>
-This characterization uses ''F'' and ''H'' to describe arbitrary distributions over observations and parameters, respectively.  Typically ''H'' will be the [[conjugate prior]] of ''F''.  The two most common choices of ''F'' are [[Gaussian distribution|Gaussian]] aka "[[normal distribution|normal]]" (for real-valued observations) and [[categorical distribution|categorical]] (for discrete observations).  Other common possibilities for the distribution of the mixture components are:
+I and my close friends ended up checking out the good options found on your website and then out of the blue I had an awful suspicion I never expressed respect to you for those guidelines. Most of the women happened to be consequently warmed to study all of them now have certainly been making the most of them. I appreciate you for truly getting really kind as well as for possessing variety of incredible [http://www.irodorijapaneserestaurant.com.sg/_old/mail/list.php?n=64-Air-Jordan-Singapore-Store Air Jordan Singapore Store] subjects we are all really desperate to discover. <br><br>Noticeably, this particular submit is in fact the actual best about it significant design and style. We harmonise along with your findings and will thirstily research forward for your inward improvements. Saying many thanks will not you need to be adequate, for that amazing clearness within your composing. <br><br>Stand up for your family up for your good friends they may not be able to stand up for independently now that they were taken. Knowing anything that could help bring down a , bring him or her to the legal to be judged for the criminal activity he or she has committed. The time is NOW!. <br><br>Urban Dictionary . Webmonkey . Wikia . The color of an product or service can also function as a trademark. The Supreme Court held in the 1997 case of Qualitex Co. /. Impact Wrestling " July 4, 2013: HAIL [http://www.ashford.edu.sg/Chinese/define.php?a=102-Abercrombie-And-Fitch-Factory-Outlet+singapore Abercrombie And Fitch Factory Outlet+singapore] SABIN! Rather than Much Else Impact Wrestling Particular date: July 4, 2013 Site: Orleans Arena, Las Vegas, Nevada Commentators: Mike Tenay, Taz We TMre in Sin City tonight for a taped Independence Day show. The main narrative [http://www.ashford.edu.sg/Chinese/define.php?a=96-Abercrombie-&-Fitch-Singapore-Sale Abercrombie & Fitch Singapore Sale] is Austin Aries stole your X Division Title the other day and as of this moment grows to challenge Bully Ray fourteen days from tonight at Vacation destination X. Other than that we TMll possess a lot [http://www.ashford.edu.sg/Chinese/define.php?a=17-Abercrombie-Sg-Models Abercrombie Sg Models] more buildup towards Likely For Glory in the BFG Sequence as points continue to be acquired. <br><br>I actually devoted an entire couple chapters to answer these types of inquiries in my Book on Flipping Residences. Basically, to analyze an area, We look at several different metrics, like price ratios between affected and retail sales, number of months of supply, shopper activity, investor activity, etc. Similar data can be used to figure out what type of houses to focus on (you wish to be buying/selling what owner occupants in the room are buying).. <br><br>SourceForge . Transifex . TuxFamily. Paying for college has come below tremendous pressure in the past few years. Costs are rising, and the work possibilities for recent university graduates have not looked so bleak in decades. Yet, unlike when the parents of today's students went to education, loans are putting innovative graduates two steps in the opening financially before they've collateralized their first real job..<ul>
-*[[Binomial distribution]], for the number of "positive occurrences" (e.g., successes, yes votes, etc.) given a fixed number of total occurrences
-*[[Multinomial distribution]], similar to the binomial distribution, but for counts of multi-way occurrences (e.g., yes/no/maybe in a survey)
+   <li>[http://4399-2.92game.net/bbs/forum.php?mod=viewthread&tid=30263 http://4399-2.92game.net/bbs/forum.php?mod=viewthread&tid=30263]</li>
-*[[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
-*[[Poisson distribution]], for the number of occurrences of an event in a given period of time, for an event that is characterized by a fixed rate of occurrence
+   <li>[http://adtrp.com/index.php?page=item&id=113438 http://adtrp.com/index.php?page=item&id=113438]</li>
-*[[Exponential distribution]], for the time before the next event occurs, for an event that is characterized by a fixed rate of occurrence
-*[[Log-normal distribution]], for positive real numbers that are assumed to grow exponentially, such as incomes or prices
+   <li>[http://www.k8383.com/news/html/?741442.html http://www.k8383.com/news/html/?741442.html]</li>
-*[[Multivariate normal distribution]] (aka [[multivariate Gaussian distribution]]), for vectors of correlated outcomes that are individually Gaussian-distributed
-*A vector of [[Bernoulli distribution|Bernoulli]]-distributed values, corresponding, e.g., to a black-and-white image, with each value representing a pixel; see the handwriting-recognition example below
+   <li>[http://enseignement-lsf.com/spip.php?article64#forum18393350 http://enseignement-lsf.com/spip.php?article64#forum18393350]</li>
-===Specific examples===
+   <li>[http://www.fortdelhermione.com/spip.php?article35 http://www.fortdelhermione.com/spip.php?article35]</li>
-====Gaussian mixture model====
+ </ul>
-[[File:nonbayesian-gaussian-mixture.svg|right|250px|thumb|Non-Bayesian Gaussian mixture model using [[plate notation]].  Smaller squares indicate fixed parameters; larger circles indicate random variables.  Filled-in shapes indicate known values.  The indication [K] means a vector of size ''K''.]]
-A typical non-Bayesian [[Gaussian distribution|Gaussian]] mixture model looks like this:
-:<math>
-\begin{array}{lcl}
-K,N &=& \text{as above} \\
-\phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
-z_{i=1 \dots N}, x_{i=1 \dots N} &=& \text{as above} \\
-\mu_{i=1 \dots K} &=& \text{mean of component } i \\
-\sigma^2_{i=1 \dots K} &=& \text{variance of component } i \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \mathcal{N}(\mu_{z_i}, \sigma^2_{z_i})
-\end{array}
-</math>
-<br style="clear:both" />
-[[File:bayesian-gaussian-mixture.svg|right|300px|thumb|Bayesian Gaussian mixture model using [[plate notation]].  Smaller squares indicate fixed parameters; larger circles indicate random variables.  Filled-in shapes indicate known values.  The indication [K] means a vector of size ''K''.]]
-A Bayesian version of a [[Gaussian distribution|Gaussian]] mixture model is as follows:
-:<math>
-\begin{array}{lcl}
-K,N &=& \text{as above} \\
-\phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
-z_{i=1 \dots N}, x_{i=1 \dots N} &=& \text{as above} \\
-\mu_{i=1 \dots K} &=& \text{mean of component } i \\
-\sigma^2_{i=1 \dots K} &=& \text{variance of component } i \\
-\mu_0, \lambda, \nu, \sigma_0^2 &=& \text{shared hyperparameters} \\
-\mu_{i=1 \dots K} &\sim& \mathcal{N}(\mu_0, \lambda\sigma_i^2) \\
-\sigma_{i=1 \dots K}^2 &\sim& \operatorname{Inverse-Gamma}(\nu, \sigma_0^2) \\
-\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \mathcal{N}(\mu_{z_i}, \sigma^2_{z_i})
-\end{array}
-</math>
-====Multivariate Gaussian mixture model====
-A Bayesian Gaussian mixture model is commonly extended to fit a vector of unknown parameters (denoted in bold), or multivariate normal distributions.  In a multivariate distribution (i.e. one modelling a vector <math>\boldsymbol{x}</math>  with ''N'' random variables) one may model a vector of parameters (such as several observations of a signal or patches within an image) using a Gaussian mixture model prior distribution on the vector of estimates given by
-:<math>
-p(\boldsymbol{\theta}) = \sum_{i=1}^K\phi_i \mathcal{N}(\boldsymbol{\mu_i,\Sigma_i})
-</math>
-where the ''i<sup>th</sup>'' vector component is characterized by normal distributions with weights <math>\phi_i</math>, means <math>\boldsymbol{\mu_i}</math> and covariance matrices <math>\boldsymbol{\Sigma_i}</math>.  To incorporate this prior into a Bayesian estimation, the prior is multiplied with the known distribution <math>p(\boldsymbol{x | \theta})</math> of the data <math>\boldsymbol{x}</math> conditioned on the parameters <math>\boldsymbol{\theta}</math> to be estimated.  With this formulation, the [[Posterior_probability|posterior distribution]] <math>p(\boldsymbol{\theta | x})</math> is ’’also’’ a Gaussian mixture model of the form
-:<math>
-p(\boldsymbol{\theta | x}) = \sum_{i=1}^K\tilde{\phi_i} \mathcal{N}(\boldsymbol{\tilde{\mu_i},\tilde{\Sigma_i}})
-</math>
-with new parameters <math>\tilde{\phi_i}, \boldsymbol{\tilde{\mu_i}}</math> and <math>\boldsymbol{\tilde{\Sigma_i}}</math> that are updated using the [[Expectation-maximization algorithm|EM algorithm]].
-<ref>
-{{cite journal
-|last=Yu |first=Guoshen
-|title=Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
-|journal=IEEE Transactions on Image Processing
-|volume=21 | year=2012|pages=2481-2499 |issue=5
-}}
-</ref>  Although EM-based parameter updates are well-established, providing the initial estimates for these parameters is currently an area of active research.  Note that this formulation yields a closed-form solution to the complete posterior distribution.  Estimations of the random variable <math>\boldsymbol{\theta}</math> may be obtained via one of several estimators, such as the mean or maximum of the posterior distribution.
-Such distributions are useful for assuming patch-wise shapes of images and clusters, for example.  In the case of image representation, each Gaussian may be tilted, expanded, and warped according to the covariance matrices <math>\boldsymbol{\Sigma_i}</math>.  One Gaussian distribution of the set is fit to each patch (usually of size 8x8 pixels) in the image.  Notably, any distribution of points around a cluster (see [[K-means clustering|''k''-means]]) may be accurately given enough Gaussian components, but scarcely over ''K''=20 components are needed to accurately model a given image distribution or cluster of data.
-====Categorical mixture model====
-[[File:nonbayesian-categorical-mixture.svg|right|250px|thumb|Non-Bayesian categorical mixture model using [[plate notation]].  Smaller squares indicate fixed parameters; larger circles indicate random variables.  Filled-in shapes indicate known values.  The indication [K] means a vector of size ''K''; likewise for [V].]]
-A typical non-Bayesian mixture model with [[categorical distribution|categorical]] observations looks like this:
-*<math>K,N:</math> as above
-*<math>\phi_{i=1 \dots K}, \boldsymbol\phi:</math> as above
-*<math>z_{i=1 \dots N}, x_{i=1 \dots N}:</math> as above
-*<math>V:</math> dimension of categorical observations, e.g., size of word vocabulary
-*<math>\theta_{i=1 \dots K, j=1 \dots V}:</math> probability for component <math>i</math> of observing item <math>j</math>
-*<math>\boldsymbol\theta_{i=1 \dots K}:</math> vector of dimension <math>V,</math> composed of <math>\theta_{i,1 \dots V};</math> must sum to 1
-The random variables:
-:<math>
-\begin{array}{lcl}
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \text{Categorical}(\boldsymbol\theta_{z_i})
-\end{array}
-</math>
-<!--
-The original version, all in LaTeX.
-:<math>
-\begin{array}{lcl}
-K,N &=& \text{as above} \\
-\phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
-z_{i=1 \dots N}, x_{i=1 \dots N} &=& \text{as above} \\
-V &=& \text{dimension of categorical observations, e.g., size of word vocabulary} \\
-\theta_{i=1 \dots K, j=1 \dots V} &=& \text{probability for component } i \text{ of observing the } j\text{th item} \\
-\boldsymbol\theta_{i=1 \dots K} &=& V\text{-dimensional vector, composed of }\theta_{i,1 \dots V} \text{; must sum to 1} \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \text{Categorical}(\boldsymbol\theta_{z_i})
-\end{array}
-</math>
--->
-<br style="clear:both" />
-[[File:bayesian-categorical-mixture.svg|right|300px|thumb|Bayesian categorical mixture model using [[plate notation]].  Smaller squares indicate fixed parameters; larger circles indicate random variables.  Filled-in shapes indicate known values.  The indication [K] means a vector of size ''K''; likewise for [V].]]
-A typical Bayesian mixture model with [[categorical distribution|categorical]] observations looks like this:
-*<math>K,N:</math> as above
-*<math>\phi_{i=1 \dots K}, \boldsymbol\phi:</math> as above
-*<math>z_{i=1 \dots N}, x_{i=1 \dots N}:</math> as above
-*<math>V:</math> dimension of categorical observations, e.g., size of word vocabulary
-*<math>\theta_{i=1 \dots K, j=1 \dots V}:</math> probability for component <math>i</math> of observing item <math>j</math>
-*<math>\boldsymbol\theta_{i=1 \dots K}:</math> vector of dimension <math>V,</math> composed of <math>\theta_{i,1 \dots V};</math> must sum to 1
-*<math>\alpha:</math> shared concentration hyperparameter of <math>\boldsymbol\theta</math> for each component
-*<math>\beta:</math> concentration hyperparameter of <math>\boldsymbol\phi</math>
-The random variables:
-:<math>
-\begin{array}{lcl}
-\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
-\boldsymbol\theta_{i=1 \dots K} &\sim& \text{Symmetric-Dirichlet}_V(\alpha) \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \text{Categorical}(\boldsymbol\theta_{z_i})
-\end{array}
-</math>
-<!--
-The (beginning of) equivalent of below, using no LaTeX.
-*''K'',''N'' = as above
-*&phi;<sub>1,...,''K''</sub>, '''&phi;''' as above
-*''z''<sub>''i''=1...''N''</sub>, ''x''<sub>''i''=1...''N''</sub> = as above
-* ''V'' = dimension of categorical observations, e.g., size of word vocabulary
--->
-<!--
-The equivalent using full LaTeX.
-:<math>
-\begin{array}{lcl}
-K,N &=& \mbox{as above} \\
-\phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
-z_{i=1 \dots N}, x_{i=1 \dots N} &=& \text{as above} \\
-V &=& \text{dimension of categorical observations, e.g., size of word vocabulary} \\
-\theta_{i=1 \dots K, j=1 \dots V} &=& \text{probability for component } i \text{ of observing the } j\text{th item} \\
-\boldsymbol\theta_{i=1 \dots K} &=& V\text{-dimensional vector, composed of }\theta_{i,1 \dots V} \text{; must sum to 1} \\
-\alpha &=& \text{shared concentration hyperparameter of } \boldsymbol\theta \text{ for each component} \\
-\beta &=& \text{concentration hyperparameter of } \boldsymbol\phi \\
-\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
-\boldsymbol\theta_{i=1 \dots K} &\sim& \text{Symmetric-Dirichlet}_V(\alpha) \\
-z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
-x_{i=1 \dots N} &\sim& \text{Categorical}(\boldsymbol\theta_{z_i})
-\end{array}
-</math>
--->
-==Examples==
-===A financial model===
-[[Image:Normal distribution pdf.png|thumb|right|250px|The [[normal distribution]] is plotted using different means and variances]]
-Financial returns often behave differently in normal situations and during crisis times. A mixture model <ref>Dinov, ID. "[http://repositories.cdlib.org/socr/EM_MM/ Expectation Maximization and Mixture Modeling Tutorial]". ''[http://repositories.cdlib.org/escholarship California Digital Library]'', Statistics Online Computational Resource, Paper EM_MM, http://repositories.cdlib.org/socr/EM_MM, December 9, 2008</ref> for return data seems reasonable. Sometimes the model used is a [[jump-diffusion model]], or as a mixture of two normal distributions.
-===House prices===
-Assume that we observe the prices of ''N'' different houses.  Different types of houses in different neighborhoods will have vastly different prices, but the price of a particular type of house in a particular neighborhood (e.g., three-bedroom house in moderately upscale neighborhood) will tend to cluster fairly closely around the mean.  One possible model of such prices would be to assume that the prices are accurately described by a mixture model with ''K'' different components, each distributed as a [[normal distribution]] with unknown mean and variance, with each component specifying a particular combination of house type/neighborhood.  Fitting this model to observed prices, e.g., using the [[expectation-maximization algorithm]], would tend to cluster the prices according to house type/neighborhood and reveal the spread of prices in each type/neighborhood. (Note that for values such as prices or incomes that are guaranteed to be positive and which tend to grow [[exponential growth|exponentially]], a [[log-normal distribution]] might actually be a better model than a normal distribution.)
-===Topics in a document===
-Assume that a document is composed of ''N'' different words from a total vocabulary of size ''V'', where each word corresponds to one of ''K'' possible topics.  The distribution of such words could be modelled as a mixture of ''K'' different ''V''-dimensional [[categorical distribution]]s.  A model of this sort is commonly termed a [[topic model]].  Note that [[expectation maximization]] applied to such a model will typically fail to produce realistic results, due (among other things) to the [[overfitting|excessive number of parameters]].  Some sorts of additional assumptions are typically necessary to get good results.  Typically two sorts of additional components are added to the model:
-#A [[prior distribution]] is placed over the parameters describing the topic distributions, using a [[Dirichlet distribution]] with a [[concentration parameter]] that is set significantly below 1, so as to encourage sparse distributions (where only a small number of words have significantly non-zero probabilities).
-#Some sort of additional constraint is placed over the topic identities of words, to take advantage of natural clustering.
-:*For example, a [[Markov chain]] could be placed on the topic identities (i.e., the latent variables specifying the mixture component of each observation), corresponding to the fact that nearby words belong to similar topics. (This results in a [[hidden Markov model]], specifically one where a [[prior distribution]] is placed over state transitions that favors transitions that stay in the same state.)
-:*Another possibility is the [[latent Dirichlet allocation]] model, which divides up the words into ''D'' different documents and assumes that in each document only a small number of topics occur with any frequency.
-===Handwriting recognition===
-The following example is based on an example in [[Christopher M. Bishop]], ''Pattern Recognition and Machine Learning''.<ref>{{cite book | last = Bishop | first = Christopher | title = Pattern recognition and machine learning | publisher = Springer | location = New York | year = 2006 | isbn = 978-0-387-31073-2 }}</ref>
-Imagine that we are given an ''N''×''N'' black-and-white image that is known to be a scan of a hand-written digit between 0 and 9, but we don't know which digit is written.  We can create a mixture model with <math>K=10</math> different components, where each component is a vector of size <math>N^2</math> of [[Bernoulli distribution]]s (one per pixel).  Such a model can be trained with the [[expectation-maximization algorithm]] on an unlabeled set of hand-written digits, and will effectively cluster the images according to the digit being written.  The same model could then be used to recognize the digit of another image simply by holding the parameters constant, computing the probability of the new image for each possible digit (a trivial calculation), and returning the digit that generated the highest probability.
-===Direct and indirect applications===
-The financial example above is one direct application of the mixture model, a situation in which we assume an underlying mechanism so that each observation belongs to one of some number of different sources or categories. This underlying mechanism may or may not, however, be observable. In this form of mixture, each of the sources is described by a component probability density function, and its mixture weight is the probability that an observation comes from this component.
-In an indirect application of the mixture model we do not assume such a mechanism. The mixture model is simply used for its mathematical flexibilities. For example, a mixture of two [[normal distribution]]s with different means may result in a density with two [[Mode (statistics)|modes]], which is not modeled by standard parametric distributions. Another example is given by the possibility of mixture distributions to model fatter tails than the basic Gaussian ones, so as to be a candidate for modeling more extreme events. When combined with [[dynamical consistency]], this approach has been applied to [[financial derivatives]] valuation in presence of the [[volatility smile]] in the context of [[local volatility]] models. This defines our application.
-===Fuzzy image segmentation===
-In image processing and computer vision, traditional [[image segmentation]] models often assign to one [[pixel]] only one exclusive pattern. In fuzzy or soft segmentation, any pattern can have certain "ownership" over any single pixel. If the patterns are Gaussian, fuzzy segmentation naturally results in Gaussian mixtures. Combined with other analytic or geometric tools (e.g., phase transitions over diffusive boundaries), such spatially regularized mixture models could lead to more realistic and computationally efficient segmentation methods.<ref>
-{{cite journal
-| last = Shen | first = Jianhong (Jackie)
-| title = A stochastic-variational model for soft Mumford-Shah segmentation
-| year = 2006
-| volume=2006
-| pages=2-16
-| journal=Int'l J. Biomedical Imaging
-| doi=10.1155/IJBI/2006/92329
-| url=http://www.hindawi.com/journals/ijbi/2006/092329/abs/
-}}</ref>
-== Identifiability ==
-Identifiability refers to the existence of a unique characterization for any one of the models in the class (family) being considered. Estimation procedure may not be well-defined and asymptotic theory may not hold if a model is not identifiable.
-=== Example ===
-Let ''J'' be the class of all binomial distributions with {{nowrap|''n'' {{=}} 2}}. Then a mixture of two members of ''J'' would have
-:<math>p_0=\pi(1-\theta_1)^2+(1-\pi)(1-\theta_2)^2</math>
-:<math>p_1=2\pi\theta_1(1-\theta_1)+2(1-\pi)\theta_2(1-\theta_2)</math>
-and {{nowrap|''p''<sub>2</sub> {{=}} 1 − ''p''<sub>0</sub> − ''p''<sub>1</sub>}}. Clearly, given ''p''<sub>0</sub> and ''p''<sub>1</sub>, it is not possible to determine the above mixture model uniquely, as there are three parameters (''π'', ''θ''<sub>1</sub>, ''θ''<sub>2</sub>) to be determined.
-=== Definition ===
-Consider a mixture of parametric distributions of the same class. Let
-:<math>J=\{f(\cdot ; \theta):\theta\in\Omega\}</math>
-be the class of all component distributions. Then the [[convex hull]] ''K'' of ''J'' defines the class of all finite mixture of distributions in ''J'':
-:<math>K=\left\{p(\cdot):p(\cdot)=\sum_{i=1}^n a_i f_i(\cdot ; \theta_i), a_i>0, \sum_{i=1}^n a_i=1, f_i(\cdot ; \theta_i)\in J\ \forall i,n\right\}</math>
-''K'' is said to be identifiable if all its members are unique, that is, given two members ''p'' and {{nowrap|''p′''}} in ''K'', being mixtures of ''k'' distributions and {{nowrap|''k′''}} distributions respectively in ''J'', we have {{nowrap|''p {{=}} p′''}} if and only if, first of all, {{nowrap|''k {{=}} k′''}} and secondly we can reorder the summations such that {{nowrap|''a<sub>i</sub> {{=}} a<sub>i</sub>''′}} and {{nowrap|''ƒ<sub>i</sub> {{=}} ƒ<sub>i</sub>''′}} for all ''i''.
-== Parameter estimation and system identification ==
-Parametric mixture models are often used when we know the distribution ''Y'' and we can sample from ''X'', but we would like to determine the ''a<sub>i</sub>'' and ''θ<sub>i</sub>'' values.  Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations.
-It is common to think of probability mixture modeling as a missing data problem.  One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data.  When we start, this membership is unknown, or missing.  The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions.
-A variety of approaches to the problem of mixture decomposition have been proposed, many of which focus on maximum likelihood methods such as [[expectation maximization]] (EM) or maximum ''a posteriori'' estimation (MAP).  Generally these methods consider separately the question of parameter estimation and system identification, that is to say a distinction is made between the determination of the number and functional form of components within a mixture and the estimation of the corresponding parameter values.  Some notable departures are the graphical methods as outlined in Tarter and Lock <ref name=tart>{{citation
-|title=Model Free Curve Estimation
-|author=Michael E. Tarter
-|year=1993
-|publisher=Chapman and Hall
-}}</ref> and more recently [[minimum message length]] (MML) techniques such as Figueiredo and Jain <ref name=Jain>{{cite journal |first1=M.A.T. |last1=Figueiredo |first2=A.K. |last2=Jain |title=Unsupervised Learning of Finite Mixture Models |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=24 |issue=3 |pages=381–396 |date=March 2002 |doi=10.1109/34.990138 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=990138
-}}</ref> and to some extent the moment matching pattern analysis routines suggested by McWilliam and Loh (2009).<ref name=mcwilli>
-{{citation
-|title=Incorporating Multidimensional Tail-Dependencies in the Valuation of Credit Derivatives (Working Paper)
-|author=N. McWilliam, K. Loh
-|year=2008
-}} [http://www.misys.com/cds-portlets/digitalAssets/4/2797_CDsAndTailDep_forPublication_final1.pdf]</ref>
-=== Expectation maximization (EM) === <!-- Linked from [[Expectation-maximization algorithm]] -->
-[[Expectation-maximization algorithm|Expectation maximization]] (EM) is seemingly the most popular technique used to determine the parameters of a mixture with an ''a priori'' given number of components. This is a particular way of implementing [[maximum likelihood]] estimation for this problem. EM is of particular appeal for finite normal mixtures where closed-form expressions are possible such as in the following iterative algorithm by Dempster ''et al.'' (1977)
-:<math> w_s^{(j+1)} = \frac{1}{N} \sum_{t =1}^N h_s^{(j)}(t) </math>
-:<math> \mu_s^{(j+1)}  =  \frac{\sum_{t =1}^N h_s^{(j)}(t) x^{(t)}}{\sum_{t =1}^N h_s^{(j)}(t)} </math>
-:<math> \Sigma_s^{(j+1)}  =  \frac{\sum_{t =1}^N h_s^{(j)}(t) [x^{(t)}-\mu_s^{(j)}][x^{(t)}-\mu_s^{(j)}]^{\top}}{\sum_{t =1}^N h_s^{(j)}(t)} </math>
-with the posterior probabilities
-:<math> h_s^{(j)}(t) = \frac{w_s^{(j)} p_s(x^{(t)}; \mu_s^{(j)},\Sigma_s^{(j)}) }{ \sum_{i = 1}^n w_i^{(j)} p_i(x^{(t)}; \mu_i^{(j)}, \Sigma_i^{(j)})}. </math>
-Thus on the basis of the current estimate for the parameters, the conditional probability for a given observation ''x''<sup>(''t'')</sup> being generated from state ''s'' is determined for each {{nowrap|''t'' {{=}} 1, …, ''N''}} ; ''N'' being the sample size.  The parameters are then updated such that the new component weights correspond to the average conditional probability and each component mean and covariance is the component specific weighted average of the mean and covariance of the entire sample.
-Dempster<ref name=dempster1977>{{cite journal |first1=A.P. |last1=Dempster |first2=N.M. |last2=Laird |first3=D.B. |last3=Rubin |title=Maximum Likelihood from Incomplete Data via the EM Algorithm |journal=Journal of the Royal Statistical Society, Series B |volume=39 |issue=1 |pages=1–38 |year=1977 |jstor=2984875 |url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.163.7580&rep=rep1&type=pdf}}
-</ref> also showed that each successive EM iteration will not decrease the likelihood, a property not shared by other gradient based maximization techniques.  Moreover EM naturally embeds within it constraints on the probability vector, and for sufficiently large sample sizes positive definiteness of the covariance iterates.  This is a key advantage since explicitly constrained methods incur extra computational costs to check and maintain appropriate values. Theoretically EM is a first-order algorithm and as such converges slowly to a fixed-point solution. Redner and Walker (1984){{full|date=November 2012}} make this point arguing in favour of superlinear and second order Newton and quasi-Newton methods and reporting slow convergence in EM on the basis of their empirical tests.  They do concede that convergence in likelihood was rapid even if convergence in the parameter values themselves was not.  The relative merits of EM and other algorithms vis-à-vis convergence have been discussed in other literature.<ref name=XuJordam>{{cite journal |first1=L. |last1=Xu |first2=M.I. |last2=Jordan |title=On Convergence Properties of the EM Algorithm for Gaussian Mixtures |journal=Neural Computation |volume=8 |issue=1 |pages=129–151 |date=January 1996 |doi=10.1162/neco.1996.8.1.129 |url=http://www.mitpressjournals.org/doi/abs/10.1162/neco.1996.8.1.129}}</ref>
-Other common objections to the use of EM are that it has a propensity to spuriously identify local maximisers,<ref name="McLaughlan_2" /> as well as displaying sensitivity to initial values.{{citation needed|date=December 2010}}  One may address these problems by evaluating EM at several initial points in the parameter space but this is computationally costly and other approaches, such as the annealing EM method of Udea and Nakano (1998) (in which the initial components are essentially forced to overlap, providing a less heterogeneous basis for initial guesses), may be preferable.
-Figueiredo and Jain <ref name="Jain" /> note that convergence to 'meaningless' parameter values obtained at the boundary (where regularity conditions breakdown, e.g., Ghosh and Sen (1985)) is frequently observed when the number of model components exceeds the optimal/true one.  On this basis they suggest a unified approach to estimation and identification in which the initial ''n'' is chosen to greatly exceed the expected optimal value.  Their optimization routine is constructed via a minimum message length (MML) criterion that effectively eliminates a candidate component if there is insufficient information to support it. In this way it is possible to systematize reductions in ''n'' and consider estimation and identification jointly.
-The [[Expectation-maximization algorithm]] can be used to compute the parameters of a parametric mixture model distribution (the ''a<sub>i</sub>'' and ''θ<sub>i</sub>'').  It is an [[iterative algorithm]] with two steps: an ''expectation step'' and a ''maximization step''. [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_2D_PointSegmentation_EM_Mixture Practical examples of EM and Mixture Modeling] are included in the [[SOCR]] demonstrations.
-==== The expectation step ====
-With initial guesses for the parameters of our mixture model, "partial membership" of each data point in each constituent distribution is computed by calculating [[expectation value]]s for the membership variables of each data point.  That is, for each data point ''x<sub>j</sub>'' and distribution ''Y<sub>i</sub>'', the membership value ''y''<sub>''i'', ''j''</sub> is:
-:<math> y_{i,j} = \frac{a_i f_Y(x_j;\theta_i)}{f_{X}(x_j)}.</math>
-==== The maximization step ====
-With expectation values in hand for group membership, [[plug-in estimates]] are recomputed for the distribution parameters.
-The mixing coefficients ''a<sub>i</sub>'' are the [[arithmetic mean|mean]]s of the membership values over the ''N'' data points.
-:<math> a_i = \frac{1}{N}\sum_{j=1}^N y_{i,j}</math>
-The component model parameters ''θ<sub>i</sub>'' are also calculated by expectation maximization using data points ''x<sub>j</sub>'' that have been weighted using the membership values.  For example, if ''θ'' is a mean ''μ''
-:<math> \mu_{i} = \frac{\sum_{j} y_{i,j}x_{j}}{\sum_{j} y_{i,j}}.</math>
-With new estimates for ''a<sub>i</sub>'' and the ''θ<sub>i</sub>'''s, the expectation step is repeated to recompute new membership values.  The entire procedure is repeated until model parameters converge.
-=== Markov chain Monte Carlo ===
-As an alternative to the EM algorithm, the mixture model parameters can be deduced using [[posterior sampling]] as indicated by [[Bayes' theorem]].  This is still regarded as an incomplete data problem whereby membership of data points is the missing data.  A two-step iterative procedure known as [[Gibbs sampling]] can be used.
-The previous example of a mixture of two [[Gaussian distribution]]s can demonstrate how the method works.  As before, initial guesses of the parameters for the mixture model are made.  Instead of computing partial memberships for each elemental distribution, a membership value for each data point is drawn from a [[Bernoulli distribution]] (that is, it will be assigned to either the first or the second Gaussian).  The Bernoulli parameter ''θ'' is determined for each data point on the basis of one of the constituent distributions.{{Vague|What does this mean?|date=March 2008}}  Draws from the distribution generate membership associations for each data point.  Plug-in estimators can then be used as in the M step of EM to generate a new set of mixture model parameters, and the binomial draw step repeated.
-=== Moment matching ===
-The method of moment matching is one of the oldest techniques for determining the mixture parameters dating back to Karl Pearson’s seminal work of 1894.
-In this approach the parameters of the mixture are determined such that the composite distribution has moments matching some given value.  In many instances extraction of solutions to the moment equations may present non-trivial algebraic or computational problems.  Moreover numerical analysis by Day <ref name=day>{{cite jstor|2334652}}</ref> has indicated that such methods may be inefficient compared to EM. Nonetheless there has been renewed interest in this method, e.g., Craigmile and Titterington (1998) and Wang.<ref name=wang>{{citation
-|title=Generating daily changes in market variables using a multivariate mixture of normal distributions
-|author=J. Wang
-|year=2001
-|journal = Proceedings of the 33rd winter conference on simulation,IEEE Computer Society
-|pages   =283–289
-}}</ref>
-McWilliam and Loh (2009) consider the characterisation of a hyper-cuboid normal mixture [[copula (statistics)|copula]] in large dimensional systems for which EM would be computationally prohibitive.  Here a pattern analysis routine is used to generate multivariate tail-dependencies consistent with a set of univariate and (in some sense) bivariate moments.  The performance of this method is then evaluated using equity log-return data with [[Kolmogorov–Smirnov]] test statistics suggesting a good descriptive fit.
-===Spectral method===
-Some problems in mixture model estimation can be solved using [[spectral method]]s.
-In particular it becomes useful if data points ''x<sub>i</sub>'' are points in high-dimensional [[real coordinate space|real space]], and the hidden distributions are known to be [[Logarithmically concave function|log-concave]] (such as [[Gaussian distribution]] or [[Exponential distribution]]).
-Spectral methods of learning mixture models are based on the use of [[Singular Value Decomposition]] of a matrix which contains data points.
-The idea is to consider the top ''k'' singular vectors, where ''k'' is the number of distributions to be learned. The projection
-of each data point to a [[linear subspace]] spanned by those vectors groups points originating from the same distribution
-very close together, while points from different distributions stay far apart.
-One distinctive feature of the spectral method is that it allows us to [[Mathematical proof|prove]] that if
-distributions satisfy certain separation condition (e.g., not too close), then the estimated mixture will be very close to the true one with high probability.
-=== Graphical Methods ===
-Tarter and Lock <ref name="tart" /> describe a graphical approach to mixture identification in which a kernel function is applied to an empirical frequency plot so to reduce intra-component variance.  In this way one may more readily identify components having differing means.  While this ''λ''-method does not require prior knowledge of the number or functional form of the components its success does rely on the choice of the kernel parameters which to some extent implicitly embeds assumptions about the component structure.
-=== Other methods ===
-Some of them can even probably learn mixtures of [[heavy-tailed distribution]]s including those with
-infinite [[variance]] (see [[#Recent Papers|links to papers]] below).
-In this setting, EM based methods would not work, since the Expectation step would diverge due to presence of
-[[outlier]]s.
-=== A simulation ===
-To simulate a sample of size ''N'' that is from a mixture of distributions ''F''<sub>''i''</sub>, ''i''=&nbsp;=&nbsp;1=&nbsp;to=&nbsp;''n'', with probabilities ''p''<sub>''i''</sub> (sum=&nbsp;''p''<sub>''i''</sub>&nbsp;=&nbsp;1):
-# Generate ''N'' random numbers from a [[categorical distribution]] of size ''n'' and probabilities ''p''<sub>''i''</sub> for ''i''=&nbsp;1=&nbsp;to&nbsp;''n''.  These tell you which of the ''F''<sub>''i''</sub> each of the ''N'' values will come from.  Denote by ''m<sub>i</sub>'' the quantity of random numbers assigned to the ''i''<sup>th</sup> category.
-# For each ''i'', generate ''m<sub>i</sub>'' random numbers from the ''F''<sub>''i''</sub> distribution.
-== Extensions ==
-In a [[Bayesian inference|Bayesian setting]], additional levels can be added to the [[graphical model]] defining the mixture model.  For example, in the common [[latent Dirichlet allocation]] [[topic model]], the observations are sets of words drawn from ''D'' different documents and the ''K'' mixture components represent topics that are shared across documents.  Each document has a different set of mixture weights, which specify the topics prevalent in that document.  All sets of mixture weights share common [[hyperparameter]]s.
-A very common extension is to connect the [[latent variable]]s defining the mixture component identities into a [[Markov chain]], instead of assuming that they are [[independent identically distributed]] random variables.  The resulting model is termed a [[hidden Markov model]] and is one of the most common sequential hierarchical models.  Numerous extensions of hidden Markov models have been developed; see the resulting article for more information.
-== History ==
-Mixture distributions and the problem of mixture decomposition, that is the identification of its constituent components and the parameters thereof, has been cited in the literature as far back as 1846 (Quetelet in McLaughlan
-,<ref name=McLaughlan_2>{{citation
-|title=Finite Mixture Models
-|first=G.J. |last=McLaughlan
-|publisher=Wiley
-|year=2000
-}}</ref> 2000) although common reference is made to the work of [[Karl Pearson]] (1894){{citation needed|date=December 2010}} as the first author to explicitly address the decomposition problem in characterising non-normal attributes of forehead to body length ratios in female shore crab populations.  The motivation for this work was provided by the zoologist [[Walter Frank Raphael Weldon]] who had speculated in 1893 (in Tarter and Lock<ref name="tart" />) that asymmetry in the histogram of these ratios could signal evolutionary divergence. Pearson’s approach was to fit a univariate mixture of two normals to the data by choosing the five parameters of the mixture such that the empirical moments matched that of the model.
-While his work was successful in identifying two potentially distinct sub-populations and in demonstrating the flexibility of mixtures as a moment matching tool, the formulation required the solution of a 9th degree (nonic) polynomial which at the time posed a significant computational challenge.
-Subsequent works focused on addressing these problems, but it was not until the advent of the modern computer and the popularisation of [[Maximum Likelihood]] (ML) parameterisation techniques that research really took off.<ref name=McLaughlan_1>{{citation
-|title=Mixture Models: inference and applications to clustering
-|first=G.J. |last=McLaughlan
-|publisher=Dekker
-|year=1988
-}}</ref>  Since that time there has been a vast body of research on the subject spanning areas such as Fisheries research, Agriculture, Botany, Economics, Medicine, Genetics, Psychology, Palaeontology, Electrophoresis, Finance, Sedimentology/Geology and Zoology.<ref name=titter_1>{{harvnb|Titterington|Smith|Makov|1985}}</ref>
-== See also ==
-=== Mixture ===
-* [[Mixture density]]
-* [[Mixture (probability)]]
-=== Hierarchical models ===
-* [[Graphical model]]
-* [[Hierarchical Bayes model]]
-=== Outlier detection ===
-* [[RANSAC]]
-{{More footnotes|date=November 2010}}
-== References ==
-{{Reflist}}
-== Further reading ==
-=== Books on mixture models ===
-*{{cite book |last1=Everitt |first1=B.S. |last2=Hand |first2=D.J. |title=Finite mixture distributions |publisher=Chapman & Hall |year=1981 |isbn=0-412-22420-8 }}
-* Lindsay B.G. (1995) "Mixture Models: Theory, Geometry, and Applications". ''NSF-CBMS Regional Conference Series in Probability and Statistics'', Vol. 5, Institute of Mathematical Statistics, Hayward.
-*{{cite book |last1=Marin |first1=J.M. |last2=Mengersen |first2=K. |last3=Robert |first3=C.P. |chapter=Bayesian modelling and inference on mixtures of distributions |chapterurl=http://www.ceremade.dauphine.fr/%7Exian/mixo.pdf |editor1-first=D. |editor1-last=Dey |editor2-first=C.R. |editor2-last=Rao |title=Essential Bayesian models |publisher=Elsevier |year=2011 |isbn=9780444537324 |pages= |url= |series=Handbook of statistics: Bayesian thinking - modeling and computation |volume=25}}
-*{{cite book |last1=McLachlan |first1=G.J. |last2=Peel |first2=D. |title=Finite Mixture Models |publisher=Wiley |year=2000 |isbn=0-471-00626-2 }}
-*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 16.1. Gaussian Mixture Models and k-Means Clustering | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=842}}
-*{{cite book |last1=Titterington |first1=D. |first2=A. |last2=Smith |first3=U. |last3=Makov |title=Statistical Analysis of Finite Mixture Distributions |publisher=Wiley |year=1985 |isbn=0-471-90763-4 |ref=harv}}
-===Application of Gaussian mixture models===
-#{{cite journal |first1=D.A. |last1=Reynolds |first2=R.C. |last2=Rose |title=Robust text-independent speaker identification using Gaussian mixture speaker models |journal=IEEE Transactions on  Speech and Audio Processing |date=January 1995 | url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=365379 |volume=3 |issue=1 |pages=72–83 |doi=10.1109/89.365379 }}
-#{{cite conference |first1=H. |last1=Permuter |first2=J. |last2=Francos |first3=I.H. |last3=Jermyn |title=Gaussian mixture models of texture and colour for image database retrieval| booktitle=  IEEE [[International Conference on Acoustics, Speech, and Signal Processing]], 2003. Proceedings (ICASSP '03)| year=2003 | url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1199538}}[http://www.sciencedirect.com/science/article/pii/S0031320305004334  The journal version]
-#{{cite book|author=Wolfgang Lemke|year=2005|title=Term Structure Modeling and Estimation in a State Space Framework|publisher=Springer Verlag|isbn=978-3-540-28342-3}}
-#{{cite conference | author=[[Damiano Brigo]] and [[Fabio Mercurio]]|title=Displaced and Mixture Diffusions for Analytically-Tractable Smile Models| booktitle=  Mathematical Finance — Bachelier Congress 2000. Proceedings | year=2001|publisher=Springer Verlag }}
-#{{cite journal |first1=Damiano |last1=Brigo |first2=Fabio |last2=Mercurio |title=Lognormal-mixture dynamics and calibration to market volatility smiles |journal=International Journal of Theoretical and Applied Finance |volume=5 |issue=4 |page=427 |date=June 2002 |doi=10.1142/S0219024902001511 |url=http://www.worldscientific.com/doi/abs/10.1142/S0219024902001511}}
-#{{cite journal |first1=Carol |last1=Alexander |title=Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects |journal=Journal of Banking & Finance |volume=28 |issue=12 |pages=2957–80 |date=December 2004 |doi=10.1016/j.jbankfin.2003.10.017 |url=http://www.carolalexander.org/publish/download/JournalArticles/PDFs/JBF2004.pdf |format=PDF}}
-#{{cite conference | author=Yannis Stylianou, Yannis Pantazis, Felipe Calderero, Pedro Larroy, Francois Severin, Sascha Schimke, Rolando Bonal, Federico Matta, and Athanasios Valsamakis |title=GMM-Based Multimodal Biometric Verification| year=2005 |url=http://www.enterface.net/enterface05/docs/results/reports/project5.pdf}}
-== External links ==
-*{{cite arxiv |first1=Frank |last1=Nielsen |title=$k$-MLE: A fast algorithm for learning statistical mixture models |date=23 March 2012 |eprint=1203.5181}}
-* The [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_2D_PointSegmentation_EM_Mixture SOCR demonstrations of EM and Mixture Modeling]
-*[http://www.csse.monash.edu.au/~dld/mixturemodel.html Mixture modelling page] (and the [http://www.csse.monash.edu.au/~dld/Snob.html Snob] program for [[Minimum Message Length]] ([[Minimum Message Length|MML]]) applied to finite mixture models), maintained by D.L. Dowe.
-*[http://www.pymix.org PyMix] — Python Mixture Package, algorithms and data structures for a broad variety of mixture model based data mining applications in Python
-*[http://scikit-learn.org/stable/modules/mixture.html scikit-learn.mixture.GMM ] — A Python package for learning Gaussian Mixture Models (and sampling from them), previously packaged with [[SciPy]] and now packaged as a [http://scikits.appspot.com/ SciKit]
-*[http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=18785&objectType=FILE GMM.m] Matlab code for GMM Implementation
-*[http://stat.duke.edu/gpustatsci/software.html GPUmix] C++ implementation of Bayesian Mixture Models using EM and MCMC with 100x speed acceleration using GPGPU.
-*[http://www.cs.ru.nl/~ali/index_files/EM.m] Matlab code for GMM Implementation using EM algorithm
-*[http://vincentfpgarcia.github.com/jMEF/] jMEF: A Java open source library for learning and processing mixtures of exponential families (using duality with Bregman divergences). Includes a Matlab wrapper.
-* Very Fast and clean C implementation of the [https://github.com/juandavm/em4gmm Expectation Maximization] (EM) algorithm for estimating [https://github.com/juandavm/em4gmm Gaussian Mixture Models] (GMMs).
-{{DEFAULTSORT:Mixture Model}}
-[[Category:Statistical models]]
-[[Category:Cluster analysis]]
-[[Category:Latent variable models]]
-[[Category:Probabilistic models]]
-[[Category:Machine learning]]

Optical flow: Difference between revisions

Revision as of 08:04, 20 February 2014

Hollister Sg .

A&f Playlist you are.

Nike High Tops Singapore 15000+ expected CTC

Air Jordan 1 lightheadedness

Hollister Singapore |

Air Jordan Singapore Store Orleans Arena

Navigation menu

Search