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Statistics of Multivariate Extremes with Applications in Risk Management

von Dr. Rodrigo Herrera

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 Zuletzt bearbeitet: 2012-08-07 18:39:58 Hindemith Fragment, Gesichtet, Hoffman et al. 2008, Rh, SMWFragment, Schutzlevel sysop, Verschleierung

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[The first customer always chooses the first table and orders a dish. The second customer enters and decides either to sit at the first table with a probability 1/1 + α] or a new table with probability α /1 + α. When sitting at a new table the customer orders a new dish. This process continues for each new customer. The first customer sits at the first table and orders a dish. The second customer enters and decides either to sit at the first table with probability 1/1 + α or a new table with probability α /1 + α. When sitting at a new table the customer orders a new dish. This process continues for each new customer
 Anmerkungen Ein Quellenverweis fehlt. Sichter (Hindemith) Plagiatsfischer

 Zuletzt bearbeitet: 2012-08-11 19:41:00 WiseWoman Blei 2007, Fragment, Gesichtet, Rh, SMWFragment, Schutzlevel sysop, Verschleierung

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Thus, the l-th customer chooses a new unoccupied table with probability α/L — 1 + α, and an occupied table with probability nl/L — 1 + α, where nl is the number of people sitting at the table l. In the above, α is a scalar parameter of the process. One might acknowledge that the above does define a probability distribution. Let us denote by l' the number of different tables occupied after L customers have walked in. Then 1 ≦ l' ≦ L and it follows the above description that precisely l' tables are occupied. 2. The nth customer chooses the first unoccupied table with probability α / n−1+α , and an occupied table with probability c / n−1+α, where c is the number of people sitting at that table.

In the above, α is a scalar parameter of the process. One might check that the above does define a probability distribution. Let us denote by kn the number of different tables occupied after n customers have walked in. Then 1 ≦ kn ≦ n and it follows from the above description that precisely tables 1,..., kn are occupied.

 Anmerkungen Eine Quelle ist nicht angegeben. In der Quelle ist der Bruch typografisch korrekt dargestellt. Durch die Darstellung von Rh ist der Formel verfälscht, es musste α/(L — 1 + α) sein. Sichter (Hindemith), WiseWoman

 Zuletzt bearbeitet: 2012-07-30 23:31:35 Hindemith Fragment, Gesichtet, Hoffman et al. 2008, Rh, SMWFragment, Schutzlevel sysop, Verschleierung

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Seite(n): 2 (Exzerpt), Zeilen: 1. Spalte: 33ff
Notice that popular tables become less and less likely to sit down at a new table. In this representation the dishes correspond to probability density functions, and the process of ordering a dish l corresponds to drawing the parameters &#x03C6l to a probability density function, as for example a Gaussian from a prior distribution G over those parameters. The process of a customer l choosing a table cl corresponds to choosing a distribution &#x03C6cl from which to draw an observation yl. Since the structure of the process, is that customers tend to sit at tables with many other customers producing the cluster behaviour, thought it has an infinite number of mixture components to choose from. Notice that popular tables become more popular, and that as more customers come in they become less and less likely to sit down at a new table.

[...]. The “dishes” in the CRP correspond to probability density functions, and the process of “ordering” a dish k corresponds to drawing the parameters &#x03C6k to a PDF from a prior distribution H over those parameters. (For example, each dish k can be a Gaussian with parameters {&#x03BCk, &#x03A3k}= &#x03C6k ~ H.) The process of a customer t choosing a table zt corresponds to choosing a distribution &#x03C6zt from which to draw an observation yt [...]. Since customers in the CRP tend to sit at tables with many other customers, the DPMM tends to draw points from the same mixture components again and again even though it has an infinite number of mixture components to choose from.

 Anmerkungen Kein Quellenverweis zu finden, obwohl der Text weitgehend wörtlich mit der Quelle übereinstimmt (wobei die Variablen anders benannt sind). Man beachte den sinnlosen ersten Satz in der Dissertation, der aber trotzdem wörtlich mit Satzfragmenten aus der Quelle übereinstimmt. Sichter (Hindemith), WiseWoman

 Zuletzt bearbeitet: 2012-08-05 10:48:56 Hindemith Fragment, Gesichtet, Rh, SMWFragment, Schutzlevel sysop, Teh 2007, Verschleierung

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This slow growth of the number of clusters makes sense because of the rich-gets-richer phenomenon: we expect there to be large clusters thus the number of clusters l' to be far smaller than the number of observations L. Notice that α controls the number of clusters in a direct manner, with larger implying a larger number of clusters a priori. This intuition will help in the application of DPs to mixture models. This slow growth of the number of clusters makes sense because of the rich-gets-richer phenomenon: we expect there to be large clusters thus the number of clusters m to be far smaller than the number of observations n. Notice that α controls the number of clusters in a direct manner, with larger α implying a larger number of clusters a priori. This intuition will help in the application of DPs to mixture models.
 Anmerkungen keine Kennzeichnung der Übernahme; kein Quellenverweis. Man beachte das ausgelassene α in der Dissertation. Sichter (Graf Isolan), Hindemith

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