Hierarchical Beta Processes and the Indian Buffet Process


 Homer Fields
 3 years ago
 Views:
Transcription
1 Hierarhial Beta Proesses and the Indian Buffet Proess Romain Thibaux Dept. of EECS University of California, Berkeley Berkeley, CA 9472 Mihael I. Jordan Dept. of EECS and Dept. of Statistis University of California, Berkeley Berkeley, CA 9472 Abstrat We show that the beta proess is the de Finetti mixing distribution underlying the Indian buffet proess of [2]. This result shows that the beta proess plays the role for the Indian buffet proess that the Dirihlet proess plays for the Chinese restaurant proess, a parallel that guides us in deriving analogs for the beta proess of the many known extensions of the Dirihlet proess. In partiular we define Bayesian hierarhies of beta proesses and use the onnetion to the beta proess to develop posterior inferene algorithms for the Indian buffet proess. We also present an appliation to doument lassifiation, exploring a relationship between the hierarhial beta proess and smoothed naive Bayes models. 1 Introdution Mixture models provide a wellknown probabilisti approah to lustering in whih the data are assumed to arise from an exhangeable set of hoies among a finite set of mixture omponents. Dirihlet proess mixture models provide a nonparametri Bayesian approah to mixture modeling that does not require the number of mixture omponents to be known in advane [1]. The basi idea is that the Dirihlet proess indues a prior distribution over partitions of the data, a distribution that is readily ombined with a prior distribution over parameters and a likelihood. The distribution over partitions an be generated inrementally using a simple sheme known as the Chinese restaurant proess. As an alternative to the multinomial representation underlying lassial mixture models, fatorial models assoiate to eah data point a set of latent Bernoulli variables. The fatorial representation has several advantages. First, the Bernoulli variables may have a natural interpretation as featural desriptions of objets. Seond, the representation of objets in terms of sets of Bernoulli variables provides a natural way to define interesting topologies on lusters e.g., as the number of features that two lusters have in ommon. Third, the number of lusters representable with m features is 2 m, and thus the fatorial approah may be appropriate for situations involving large numbers of lusters. As in the mixture model setting, it is desirable to onsider nonparametri Bayesian approahes to fatorial modeling that remove the assumption that the ardinality of the set of features is known a priori. An important first step in this diretion has been provided by Griffiths and Ghahramani [2], who defined a stohasti proess on features that an be viewed as a fatorial analog of the Chinese restaurant proess. This proess, referred to as the Indian buffet proess, involves the metaphor of a sequene of ustomers tasting dishes in an infinite buffet. Let Z i be a binary vetor where Z i,k = 1 if ustomer i tastes dish k. Customer i tastes dish k with probability m k /i, where m k is the number of ustomers that have previously tasted dish k; that is, Z i,k Berm k /i. Having sampled from the dishes previously sampled by other ustomers, ustomer i then goes on to taste an additional number of new dishes determined by a draw from a Poissonα/i distribution. Modulo a reordering of the features, the Indian buffet proess an be shown to generate an exhangeable distribution over binary matries that is, P Z 1,... Z n = P Z σ1,... Z σn for any permutation σ. Given suh an exhangeability result, it is natural to inquire as to the underlying distribution that renders the sequene onditionally independent. Indeed, De Finetti s theorem states that the distribution of any
2 infinitely exhangeable sequene an be written [ ] n P Z 1,... Z n = P Z i B dp B, where B is the random element that renders the variables {Z i } onditionally independent and where we will refer to the distribution P B as the de Finetti mixing distribution. For the Chinese restaurant proess, the underlying de Finetti mixing distribution is known it is the Dirihlet proess. As this result suggests, identifying the de Finetti mixing distribution behind a given exhangeable sequene is important; it greatly extends the range of statistial appliations of the exhangeable sequene. In this paper we make the following three ontributions: 1. We identify the de Finetti mixing distribution behind the Indian buffet proess. In partiular, in Se. 4 we show that this distribution is the beta proess. We also show that this onnetion motivates a twoparameter generalization of the Indian buffet proess proposed in [3]. While the beta proess has been previously studied for its appliations in survival analysis, this result shows that it is also the natural objet of study in nonparametri Bayesian fatorial modeling. 2. In Se. 5 we exploit the link between the beta proess and the Indian buffet proess to provide a new algorithm to sample beta proesses. 3. In Se. 6 we define the hierarhial beta proess, an analog for fatorial modeling of the hierarhial Dirihlet proess [11]. The hierarhial beta proess makes it possible to speify models in whih features are shared among a number of groups. We present an example of suh a model in an appliation to doument lassifiation in Se. 7, where we also explore the relationship of the hierarhial beta proess to naive Bayes models. 2 The beta proess The beta proess was defined by Hjort [4] for appliations in survival analysis. In those appliations, the beta proess plays the role of a distribution on funtions umulative hazard funtions defined on the positive real line. In our appliations, the sample paths of the beta proess need to be defined on more general spaes. We thus develop a nomenlature that is more suited to these more general appliations. A positive random measure B on a spae Ω e.g., R is a Lévy proess, or independent inrement proess, if the masses BS 1,... BS k assigned to disjoint Draw 1 Figure 1: Top. A measure B sampled from a beta proess blue, along with its orresponding umulative distribution funtion red. The horizontal axis is Ω, here [, 1]. The tips of the blue segments are drawn from a Poisson proess with base measure the Lévy measure. Bottom. 1 samples from a Bernoulli proess with base measure B, one per line. Samples are sets of points, obtained by inluding eah point ω independently with probability B{ω}. subsets S 1,... S k of Ω are independent 1. The Lévy Khinhine theorem [5, 8] implies that a positive Lévy proess is uniquely haraterized by its Lévy measure or ompensator, a measure on Ω R +. Definition. A beta proess B BP, B is a positive Lévy proess whose Lévy measure depends on two parameters: is a positive funtion over Ω that we all the onentration funtion, and B is a fixed measure on Ω, alled the base measure. In the speial ase where is a onstant it will be alled the onentration parameter. We also all γ = B Ω the mass parameter. If B is ontinuous, the Lévy measure of the beta proess is νdω, dp = ωp 1 1 p ω 1 dpb dω 1 on Ω [, 1]. As a funtion of p, it is a degenerate beta distribution, justifying the name. ν has the following elegant interpretation. To draw B BP, B, draw a set of points ω i, p i Ω [, 1] from a Poisson proess with base measure ν see Fig. 1, and let: B = i p i δ ωi 2 where δ ω is a unit point mass or atom at ω. This 1 Positivity is not required to define Lévy proesses but greatly simplifies their study, and positive Lévy proesses are suffiient here. On Ω = R, positive Lévy proesses are also alled subordinators.
3 implies BS = i:ω i S p i for all S Ω. As this representation shows, B is disrete and the pairs ω i, p i orrespond to the loation ω i Ω and weight p i [, 1] of its atoms. Sine νω [, 1] = the Poisson proess generates infinitely many points, making 2 a ountably infinite sum. Nonetheless, as shown in the appendix, its expetation is finite if B is finite. If B is disrete 2, of the form B = i q iδ ωi, then B has atoms at the same loations B = i p iδ ωi with p i Betaω i q i, ω i 1 q i. 3 This requires q i [, 1]. If B is mixed disreteontinuous, B is the sum of the two independent ontributions. 3 The Bernoulli proess Definition. Let B be a measure on Ω. We define a Bernoulli proess with hazard measure B, written X BePB, as the Lévy proess with Lévy measure µdp, dω = δ 1 dpbdω. 4 If B is ontinuous, X is simply a Poisson proess with intensity B: X = N δ ω i where N PoiBΩ and ω i are independent draws from the distribution B/BΩ. If B is disrete, of the form B = i p iδ ωi, then X = i b iδ ωi where the b i are independent Bernoulli variables with the probability that b i = 1 equal to p i. If B is mixed disreteontinuous, X is the sum of the two independent ontributions. In summary, a Bernoulli proess is similar to a Poisson proess, exept that it an give weight at most 1 to singletons, even if the base measure B is disontinuous, for instane if B is itself drawn from a beta proess. We an intuitively think of Ω as a spae of potential features, and X as an objet defined by the features it possesses. The random measure B enodes the probability that X possesses eah partiular feature. In the Indian buffet metaphor, X is a ustomer and its features are the dishes it tastes. Conjugay. Let B BP, B, and let X i B BePB for i = 1,... n be n independent Bernoulli proess draws from B. Let X 1...n denote the set of observations {X 1,..., X n }. Applying theorem 3.3 of Kim [6] to the beta and Bernoulli proesses, the posterior distribution of B after observing X 1...n is still 2 The Lévy measure still exists, but is not as useful as in the ontinuous ase. νdp, dω = i Betaωiqi, ωi1 qidpδω dω. i a beta proess with modified parameters: B X 1...n BP + n, + n B + 1 X i. 5 + n That is, the beta proess is onjugate to the Bernoulli proess. This result an also be derived from Corollary 4.1 of Hjort [4]. 4 Connetion to the Indian buffet proess We now present the onnetion between the beta proess and the Indian buffet proess. The first step is to marginalize out B to obtain the marginal distribution of X 1. Independene of X 1 on disjoint intervals is preserved, so X 1 is a Lévy proess. Sine it gives mass or 1 to singletons its Lévy measure is of the form 4, so X 1 is still a Bernoulli proess. Its expetation is EX 1 = EEX 1 B = EB = B, so its hazard measure is B. 3 Combining this with Eq. 5 and using P X n+1 X 1...n = E B X1...n P X n+1 B gives us the following formula, whih we rewrite using the notation m n,j, the number of ustomers among X 1...n having tried dish ω j : X n+1 X 1...n BeP + n B + 1 X i + n = BeP + n B + m n,j + n δ ω j.6 j To make the onnetion to the Indian buffet proess let us first assume that is a onstant and B is ontinuous with finite total mass B Ω = γ. Observe what happens when we generate X 1...n sequentially using Eq. 6. Sine X 1 BePB and B is ontinuous, X 1 is a Poisson proess with intensity B. In partiular, the total number of features of X 1 is X 1 Ω Poiγ. This orresponds to the first ustomer trying a Poiγ number of dishes. Separating the base measure of Eq. 6 into its ontinuous and disrete parts, we see that X n+1 is the sum of two independent Bernoulli proesses: X n+1 = U + V where U BeP m n,j j +n δ ω j has an atom at ω j tastes dish j with independent probability m n,j + n and V BeP +n B is a Poisson proess with intensity γ +n B, generating a Poi number of new features new + n dishes. 3 We show that EB = B in the Appendix.
4 F 2 and G 2. By indution we get, for any n B d = ˆB n + G n where ˆBn = d Sine lim n ˆBn = B, we an use ˆBn as an approximation of B. The following iterative algorithm onstruts ˆB n starting with ˆB =. At eah step n 1: F i Figure 2: Draws from a beta proess with onentration and uniform base measure with mass γ, as we vary and γ. For eah draw, 2 samples are shown from the orresponding Bernoulli proess, one per line. This is a twoparameter, γ generalization of the Indian buffet proess, whih we reover when we let, γ = 1, α. Griffiths and Ghahramani propose suh an extension in [3]. The ustomers together try a Poinγ number of dishes, but beause they tend to try the same dishes the number of unique dishes is only Poi γ n 1 i=, roughly +i + n Poi γ + γ log This quantity beomes Poiγ if all ustomers share the same dishes or Poinγ if no sharing, justifying the names onentration parameter for and mass parameter for γ. The effet of and γ is illustrated in Fig An algorithm to generate beta proesses +1 B Eq. 5, when n = 1 and B is ontinuous, shows that B X 1 is the sum of two independent 1 beta proesses F 1 BP + 1, +1 X 1 and G 1 BP + 1,. In partiular, this lets us sample B by first sampling X 1, whih is a Poisson proess with base measure B, then sampling F 1 and G 1. Let X 1 = K 1 δ ω j. Sine the base measure of F 1 is disrete we an apply Eq. 3, with ω i = + 1 and q i = 1/ + 1. We get F 1 = K 1 p jδ ωj where p j Beta1,. G 1 has onentration + 1 and mass γ/ + 1. Sine its base measure is ontinuous, we an further deompose it via Eq. 5 into two independent beta proesses sample K n Poi γ +n 1, sample K n new loations ω j from 1 γ B independently, sample their weight p j Beta1, + n 1 independently, ˆB n = ˆB n 1 + K n p jδ ωj. The expeted mass added at step n is EF n Ω = γ +n+n 1 and the expeted remaining mass after step n is EG n Ω = γ +n. Other algorithms exist to build approximations of beta proesses. The Inverse Levy Measure algorithm of Wolpert and Ikstadt [12] is very general and an generate atoms in dereasing order of weight, but requires inverting the inomplete beta funtion at eah step, whih is omputationally intensive. The algorithm of Lee and Kim [7] bypasses this diffiulty by approximating the beta proess by a ompound Poisson proess but requires a fixed approximation level. This means that their algorithm only onverges in distribution. Our algorithm is a simple and effiient alternative. It is losely related to the stikbreaking onstrution of Dirihlet proesses [1], in that it generates the atoms of B in a sizebiased order. 6 The hierarhial beta proess The parallel with the Dirihlet proess leads us to onsider hierarhies of beta proesses in a manner akin to the hierarhial Dirihlet proesses of [11]. To motivate our onstrution, let us onsider the following appliation to doument lassifiation to whih we return in Se. 7. Suppose that our training data X is a list of douments, where eah doument is lassified by one of n ategories. We model a doument by the set of words it ontains. In partiular we do not take the number of appearanes of eah word into aount. We assume that doument X i,j is generated by inluding eah word ω independently with a probability p j ω speifi to ategory j. These probabilities form a disrete measure A j over the spae of words Ω, and we put a beta proess BP j, B prior on A j.
5 Figure 3: Left. Graphial model for the hierarhial beta proess. Right. Example draws from this model with = j = 1 and B uniform on [, 1] with mass γ = 1. From top to bottom are shown a sample for B, A j and 25 samples X 1,j,... X 25,j. If B is a ontinuous measure, implying that Ω is infinite, with probability one the A j s will share no atoms, an undesirable result in the pratial appliation to douments. For ategories to share words, B must be disrete. On the other hand, B is unknown a priori, so it must be random. This suggests that B should itself be generated as a realization of a beta proess. We thus put a beta proess prior BP, B on B. This allows sharing of statistial strength among ategories. In summary we have the following model, whose graphial representation is shown in Fig. 3. Baseline B BP, B Categories A j BP j, B j n 8 Douments X i,j BePA j i n j To lassify a new doument Y we need to ompare its probability under eah ategory: P X nj +1,j = Y X where X nj +1,j is a new doument in ategory j. The next two subsetions give a Monte Carlo inferene algorithm to do this for hierarhies of arbitrarily many levels. 6.1 The disrete part Sine all elements of our model are Lévy proesses, we an partition the spae Ω and perform inferene separately on eah part. We hoose the partition that has ells {ω} for eah of the features ω that have been observed at least one, and has a single large ell ontaining the rest of the spae. We first onsider inferene over the singletons {ω}, and return to inferene over the remaining ell in the following subsetion. Inferene for {ω} deals only with the values b = B {ω}, b = B{ω}, a j = A j {ω} and x ij = X i,j {ω}. Let x denote the set of all x ij and let a denote the set of all a j. These variables form a slie of the hierarhy of their respetive proesses, and they have the following distributions: b Beta b, 1 b a j Beta j b, j 1 b 9 x ij Bera j. Stritly speaking, if B is ontinuous, b = and so the prior over b is improper. We treat this by onsidering b to be nonzero and taking the limit as b approahes zero. This is justified under the limit onstrution of the beta proess see Theorem 3.1 of [4]. For a fixed value of b, we an average over a using onjugay. To average over b, we use rejetion sampling; we sample b from an approximation of its onditional distribution and orret for the differene by rejetion. Let m j = n j x ij. Marginalizing out a in Eq. 9 and using Γx + 1 = xγx, the log posterior distribution of b given x is up to a onstant: fb = b 1 logb + 1 b 1 log1 b m j 1 + log j b + i + i= n j m j 1 i= log j 1 b + i. 1 This posterior is log onave and has a maximum at b, 1 whih we an obtain by binary searh. Using the onavity of log j b+i for i >, log j 1 b+ i for i and log1 b tangentially at b we obtain the following upper bound on f up to a onstant: gb = α 1 logb b/β where α = b + and 1 β 1 mj> = 1 b 1 1 b + n j m j 1 i= m j 1 j j 1 b + i. j j b + i g is, up to a onstant, the log density of a Gammaα, β variable, whih we an use as a rejetion sampling proposal distribution. Setting u = 1 b in Eq. 1 maintains the form of f, only exhanging the oeffiients. Therefore we an hoose instead to approximate 1 b by a gamma variable. Among these two possible approximations we hoose the one with the lowest variane αβ 2. In the experiments of Se. 7 the aeptane
6 rate was above 9%, showing that g is a good approximation of f. This algorithm gives us samples from P b x. In partiular, with T samples, b 1,... b T, we an ompute the following approximation: P x nj +1,j = 1 x = EEa j b, x x = jeb x + m j j + n j where Eb x 1 b t. T In the end if we want samples of a we an obtain them easily. We an sample from the onditional distribution of a j whih is, by onjugay: a j b, x Beta j b + m j, j 1 b + n j m j. 6.2 The ontinuous part Let s now look at the rest of the spae, where all observations are equal to zero. We approximate B on that part of the spae by ˆB N obtained after N steps of the t γ +k 1 algorithm of Se. 5. ˆBN onsists of K k Poi atoms at eah level k = 1,... N. For eah atom ω, b out of the K k atoms of level k, we have the following hierarhy. It is similar to Eq. 9 exept that it refers to a random loation ω hosen in sizebiased order. b Beta1, + k 1 a j Beta j b, j 1 b 11 x ij Bera j We want to infer the distribution of the next observation X nj +1,j from group or ategory j, given that all other observations from all other groups are zero, that is X =. The loations of the atoms of X nj+1,j will be B /γ distributed so we only need to know the distribution of X nj +1,jΩ, the number of atoms. X = implies that all levels k have generated zero observations. Sine eah level is independent, we an reason on eah level separately, where we want the posterior distribution of K k and of the variables in Eq. 11. Let x = {x ij j n, i n j } and a = {a j j n}. Let P k be the probability over b, a and x defined by Eq. 11 and let q k = P k x =. The posterior distribution of K k is P K k = m X = qk m γ Poi m so K k X = Poi + k 1 γ + k 1 q k. Let p k = P k x nj+1,j = 1, x =, then P k x nj+1,j = 1 x = = p k /q k. Let D k be the number of atoms of X nj +1,j from level k γ D k Poi + k 1 q k p k q k = Poi γ + k 1 p k Adding the ontributions of all levels we get the following result whih is exat for N = : N γ X nj +1,jΩ Poi + k 1 p k. k=1 Using T samples b k,1,... b k,t from Eq. 11 we an ompute p k : Let rb = E k a jt 1 a j t nj b j = j b j + n j n j =1 then p k = E k [rb] 1 T 6.3 Larger hierarhies Γ j Γ j 1 b + n j Γ j 1 bγ j + n j T rb k,t. 12 t=1 We now extend model Eq. 8 to larger hierarhies suh as: Baseline B BP, B Categories A j BP j, B j n Subategories S l,j BP l,j, A j l n j Douments X i,l,j BePS l,j i n l,j To extend the algorithm of Se. 6.2 we draw samples of a from Eq. 11 and replae b by a in Eq. 12. Extending the algorithm of Se. 6.1 is less immediate sine we an no longer use onjugay to integrate out a. The Markov hain must now instantiate a and b. Se. 6.1 lets us sample a b, x, leaving us with the task of sampling b b, a. Up to a onstant the log onditional probability of b is f 2 b = b 1 logb + 1 b 1 log1 b [logγ j b + logγ j 1 b] + j b loga j /1 a j. 13 Let sx = logγx log x. Sine s is onave, f 2 itself is onave with a maximum b in, 1 whih we an obtain by binary searh. Bounding s and log1 b by their tangent yields the following upper bound g 2 of f 2 omitting the onstant: g 2 b = α 1 logb b/β where.
7 α = b + n 1 β = 1 b n 1 1 b + n n b aj j log 1 a j + [ j ψ j b j ψ j 1 b ]. We an use this Gammaα, β variable, or the one obtained by setting u = 1 b in Eq. 13 as a proposal distribution as in Se The ase of b b, a is the general ase for nodes of large hierarhies, so this algorithm an handle hierarhies of arbitrary depth. 7 Appliation to doument lassifiation Naive Bayes is a very simple yet powerful probabilisti model used for lassifiation. It models douments as lists of features, and assumes that features are independent given the ategory. Bayes rule an then be used on new douments to infer their ategory. This method has been used suessfully in many domains despite its simplisti assumptions. Naive Bayes does suffer however from several known shortomings. Consider a binary feature ω and let p j,ω be the probability that a doument from ategory j has feature ω. Estimating p j,ω as its maximum likelihood m j,ω /n j,ω leads to many features having probability or 1, making inferene impossible. To prevent suh extreme values, p j,ω is generally estimated with Laplae smoothing, whih an be interpreted as plaing a ommon Betaa, b prior on p j,ω : ˆp j,ω = m j,ω + a n j,ω + a + b. Laplae smoothing also orrets for unbalaned training data by imposing greater smoothing on the probabilities of small ategories, for whih we have low onfidene. Nonetheless, Laplae smoothing an lead to paradoxes with unbalaned data [9]. Consider the situtation where most ategories u have enough data to show with onfidene that p u,ω is lose to a very small value p. The impat on lassifiation of p j,ω is relative to p u,ω for u j so if ategory j has little data, we expet p j,ω to be lose to p for it to have little impat. Laplae smoothing however brings it lose to a a+b, very far from p, where it will have an enormous impat. This inonsisteny makes rare features hurt performane, and leads to the pratie of ombining naive Bayes with feature seletion, potentially wasting information. We propose instead to use a hierarhial beta proess hbp as a prior over the probabilities p j,ω. Suh a hierarhial Bayesian model allows sharing among ategories by shrinking the maximum likelihood probabilities towards eah other rather than towards a a+b. Suh an effet ould in priniple be ahieved using a finite model with a hierarhial beta prior; however, suh an approah would not permit new features that do not appear in the training data. The model in Eq. 8 allows the number of known features to grow with data, and the number of unknown features to serve as evidene for belonging to a poorly known ategory, one for whih we have little training data. A hbp gives a onsistent prior for varying amounts of data, whereas Laplae smoothing amounts to hanging the prior every time a new feature appears. We ompared the performane of hbp and naive Bayes on 1 posts from the 2 Newsgroups dataset 4, grouped into 2 ategories. We hose an unbalaned dataset, with the number of douments per ategory dereasing linearly from 1 to 2. We randomly seleted 4% of these papers as a test set for a lassifiation task. We enoded the douments X ij as a set of binary features representing the presene of words. All words were used without any pruning or feature seletion. We used the model in Eq. 8, setting the parameters, γ and j a priori in the following way. Sine we expet a lot of ommonalities between ategories, with differenes onentrated on a few words only, we take j to be small. Therefore douments drawn from Eq. 8 are lose to being drawn from a ommon BP, γ prior, under whih the expeted number of features per doument is γ. Estimating this value from the data gives γ = 15. Knowing γ we an solve for by mathing the expetation of Eq. 7 to the number of unique features N in the data. This an be done by interpreting Eq. 7 as a fixed point equation: F γ γ log + n/ leading to = 7. Finally we seleted the value of j by rossvalidation on a heldout portion of the training set, giving us j = 1 4. We then ran our Monte Carlo inferene proedure and lassified eah doument Y of the test set by finding the lass for whih P X nj+1,j = Y X was highest, obtaining 58% auray. By omparison, we performed a grid searh over the spae of Laplae smoothing parameters a, b [1 11, 1 7 ] for the best naive Bayes model. For a = 1 8 and b = 1 7, naive Bayes reahed 5% auray. The lassifiation of douments is often takled with a multinomial models under the bagofwords assump 4 The data are available at
8 tion. The advantage of a featurebased model is that it beomes natural to add other nontext features suh as presene of a header, the text is rightjustified, et. Sine the hbp handles rare features appropriately, we an safely inlude a muh larger set of features than would be possible for a naive Bayes model. 8 Conlusions In this paper we have shown that the beta proess originally developed for appliations in survival analysis is the natural objet of study for nonparametri Bayesian fatorial modeling. Representing data points in terms of sets of features, fatorial models provide substantial flexibility relative to the multinomial representations assoiated with lassial mixture models. We have shown that the beta proess is the de Finetti mixing distribution underlying the Indian buffet proess, a distribution on sparse binary matries. This result parallels the relationship between the Dirihlet proess and the Chinese restaurant proess. We have also shown that the beta proess an be extended to a reursivelydefined hierarhy of beta proesses. This representation makes it possible to develop nonparametri Bayesian models in whih unbounded sets of features an be shared among multiple nested groups of data. Compared to the Dirihlet proess, the beta proess has the advantage of being an independent inrements proess Brownian motion is another example of an independent inrements proess. However, some of the simplifying features of the Dirihlet proess do not arry over to the beta proess, and in partiular we have needed to design new inferene algorithms for beta proess and hierarhial beta proess mixture models rather than simply borrowing from Dirihlet proess methods. Our inferene methods are elementary and additional work on inferene algorithms will be neessary to fully exploit beta proess models. 9 Appendix Hjort [4] derives the following moments for any set S Ω. If B is ontinuous, EBS = bνdω, db = B S. i:ω i S S [,1] If B is disrete: EBS = Eb i δ ωi = b δ ωi = B S and V ar BS = S i:ω i S B dω1 B dω. ω + 1 Thus despite being disrete, B an be viewed as an approximation of B, with flutuations going to zero as. Referenes [1] C. Antoniak. Mixtures of Dirihlet proesses with appliations to Bayesian nonparametri problems. The Annals of Statistis, 26: , [2] T. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet proess. In Advanes in Neural Information Proessing Systems NIPS 18, 25. [3] T. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet proess. Tehnial Report 251, Gatsby Computational Neurosiene Unit, 25. [4] N. L. Hjort. Nonparametri Bayes estimators based on Beta proesses in models for life history data. The Annals of Statistis, 183: , 199. [5] A. Y. Khinhine. Korrelationstheorie der stationären stohastisen prozesse. Mathematishe Annalen, 19:64 615, [6] Y. Kim. Nonparametri Bayesian estimators for ounting proesses. The Annals of Statistis, 272: , [7] J. Lee and Y. Kim. A new algorithm to generate beta proesses. Computational Statistis and Data Analysis, 47: , 24. [8] P. Lévy. Théorie de l addition des variables aléatoires. GauthiersVillars, Paris, [9] J. Rennie, L. Shih, J. Teevan, and D. Karger. Takling the poor assumptions of naïve Bayes text lassifiers. In International Conferene on Mahine Learning ICML 2, 23. [1] J. Sethuraman. A onstrutive definition of Dirihlet priors. Statistia Sinia, 4:639 65, [11] Y. W. Teh, M. I. Jordan, M. Beal, and D. M. Blei. Hierarhial Dirihlet proesses. Journal of the Amerian Statistial Assoiation, 11: , 26. [12] R. Wolpert and K. Ikstadt. Simulation of Lévy random fields. In D. Dey, P. Muller, and D. Sinha, editors, Pratial Nonparametri and Semiparametri Bayesian Statistis. SpringerVerlag, New York, 1998.
Restricted Least Squares, Hypothesis Testing, and Prediction in the Classical Linear Regression Model
Restrited Least Squares, Hypothesis Testing, and Predition in the Classial Linear Regression Model A. Introdution and assumptions The lassial linear regression model an be written as (1) or (2) where xt
More informationWeighting Methods in Survey Sampling
Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiaohih Chang Ferry Butar Butar Abstrat It is said that a welldesigned survey an best prevent nonresponse. However, no matter
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationSebastiĂĄn Bravo LĂłpez
Transfinite Turing mahines SebastiĂĄn Bravo LĂłpez 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationComputer Networks Framing
Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial
More informationSupply chain coordination; A Game Theory approach
aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah JeanClaude Hennet x and Yasemin Arda xx x LI CNRUMR 668 UniversitĂ©
More informationBUILDING A SPAM FILTER USING NAĂVE BAYES. CIS 391 Intro to AI 1
BUILDING A SPAM FILTER USING NAĂVE BAYES 1 Spam or not Spam: that is the question. From: "" Subjet: real estate is the only way... gem oalvgkay Anyone an buy real estate with no
More informationAn Iterated Beam Search Algorithm for Scheduling Television Commercials. Mesut Yavuz. Shenandoah University
0080569 An Iterated Beam Searh Algorithm for Sheduling Television Commerials Mesut Yavuz Shenandoah University The Harry F. Byrd, Jr. Shool of Business Winhester, Virginia, U.S.A. myavuz@su.edu POMS 19
More informationHierarchical Clustering and Sampling Techniques for Network Monitoring
S. Sindhuja Hierarhial Clustering and Sampling Tehniques for etwork Monitoring S. Sindhuja ME ABSTRACT: etwork monitoring appliations are used to monitor network traffi flows. Clustering tehniques are
More informationA Holistic Method for Selecting Web Services in Design of Composite Applications
A Holisti Method for Seleting Web Servies in Design of Composite Appliations MÄrtiĆĆĄ Bonders, JÄnis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,
More informationImproved Vehicle Classification in Long Traffic Video by Cooperating Tracker and Classifier Modules
Improved Vehile Classifiation in Long Traffi Video by Cooperating Traker and Classifier Modules Brendan Morris and Mohan Trivedi University of California, San Diego San Diego, CA 92093 {b1morris, trivedi}@usd.edu
More informationBayesian Statistics: Indian Buffet Process
Bayesian Statistics: Indian Buffet Process Ilker Yildirim Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14627 August 2012 Reference: Most of the material in this note
More informationScalable Hierarchical Multitask Learning Algorithms for Conversion Optimization in Display Advertising
Salable Hierarhial Multitask Learning Algorithms for Conversion Optimization in Display Advertising Amr Ahmed Google amra@google.om Abhimanyu Das Mirosoft Researh abhidas@mirosoft.om Alexander J. Smola
More informationDiscovering Trends in Large Datasets Using Neural Networks
Disovering Trends in Large Datasets Using Neural Networks Khosrow Kaikhah, Ph.D. and Sandesh Doddameti Department of Computer Siene Texas State University San Maros, Texas 78666 Abstrat. A novel knowledge
More informationMirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are nonbonded.
6 HĂŒkel Theory This theory was originally introdued to permit qualitative study of the Ïeletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror
More informationLecture 5. Gaussian and GaussJordan elimination
International College of Eonomis and Finane (State University Higher Shool of Eonomis) Letures on Linear Algebra by Vladimir Chernyak, Leture. Gaussian and GaussJordan elimination To be read to the musi
More informationIntelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds
Intelligent Measurement Proesses in 3D Optial Metrology: Produing More Aurate Point Clouds Charles Mony, Ph.D. 1 President Creaform in. mony@reaform3d.om Daniel Brown, Eng. 1 Produt Manager Creaform in.
More informationSequences CheatSheet
Sequenes CheatSheet Intro This heatsheet ontains everything you should know about real sequenes. It s not meant to be exhaustive, but it ontains more material than the textbook. Definitions and properties
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear LeastSquares Fits with a Common Interept: Determination
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deisionmakers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More informationExponential Maps and Symmetric Transformations in ClusterSpin. YouGang Feng
Exponential Maps and Symmetri Transformations in ClusterSpin System for Lattie Ising Models YouGang Feng Department of Basi Sienes, College of Siene, Guizhou University, Caijia Guan, Guiyang, 550003
More information5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationFrom (2) follows, if z0 = 0, then z = vt, thus a2 =?va (2.3) Then 2:3 beomes z0 = z (z? vt) (2.4) t0 = bt + b2z Consider the onsequenes of (3). A ligh
Chapter 2 Lorentz Transformations 2. Elementary Considerations We assume we have two oordinate systems S and S0 with oordinates x; y; z; t and x0; y0; z0; t0, respetively. Physial events an be measured
More informationRobust Classification and Tracking of Vehicles in Traffic Video Streams
Proeedings of the IEEE ITSC 2006 2006 IEEE Intelligent Transportation Systems Conferene Toronto, Canada, September 1720, 2006 TC1.4 Robust Classifiation and Traking of Vehiles in Traffi Video Streams
More informationBig Data Analysis and Reporting with Decision Tree Induction
Big Data Analysis and Reporting with Deision Tree Indution PETRA PERNER Institute of Computer Vision and Applied Computer Sienes, IBaI Postbox 30 11 14, 04251 Leipzig GERMANY pperner@ibaiinstitut.de,
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More informationA ContextAware Preference Database System
J. PERVASIVE COMPUT. & COMM. (), MARCH 005. TROUBADOR PUBLISHING LTD) A ContextAware Preferene Database System Kostas Stefanidis Department of Computer Siene, University of Ioannina,, kstef@s.uoi.gr Evaggelia
More informationImproved SOMBased HighDimensional Data Visualization Algorithm
Computer and Information Siene; Vol. 5, No. 4; 2012 ISSN 19138989 EISSN 19138997 Published by Canadian Center of Siene and Eduation Improved SOMBased HighDimensional Data Visualization Algorithm Wang
More informationREDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION
C I E 17 th International Conferene on Eletriity istriution Barelona, 115 May 003 EUCTION FACTO OF FEEING LINES THAT HAVE A CABLE AN AN OVEHEA SECTION Ljuivoje opovi J.. Elektrodistriuija  Belgrade 
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More informationTheory of linear elasticity. I assume you have learned the elements of linear
Physial Metallurgy Frature mehanis leture 1 In the next two letures (Ot.16, Ot.18), we will disuss some basis of frature mehanis using ontinuum theories. The method of ontinuum mehanis is to view a solid
More informationOverview. Recognition and Tracking of Human Motion. Problem statement. Problem statement. Christian Vogler Rutgers University
Reognition and Traking of Human otion Christian Vogler Rutgers University Overview Problem statement Reognition ase study: Sign language Phonemebased modeling odeling simultaneous events Reognition with
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationImpact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities
Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary
More informationThe Quantum Theory of Radiation
A. Einstein, Physikalishe Zeitshrift 18, 121 1917 The Quantum Theory of Radiation A. Einstein (Reeived Marh, 1917) The formal similarity of the spetral distribution urve of temperature radiation to Maxwell
More informationA Keyword Filters Method for Spam via Maximum Independent Sets
Vol. 7, No. 3, May, 213 A Keyword Filters Method for Spam via Maximum Independent Sets HaiLong Wang 1, FanJun Meng 1, HaiPeng Jia 2, JinHong Cheng 3 and Jiong Xie 3 1 Inner Mongolia Normal University 2
More informationRecovering Articulated Motion with a Hierarchical Factorization Method
Reovering Artiulated Motion with a Hierarhial Fatorization Method Hanning Zhou and Thomas S Huang University of Illinois at UrbanaChampaign, 405 North Mathews Avenue, Urbana, IL 680, USA {hzhou, huang}@ifpuiuedu
More informationchapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More informationAsymmetric Error Correction and FlashMemory Rewriting using Polar Codes
1 Asymmetri Error Corretion and FlashMemory Rewriting using Polar Codes Eyal En Gad, Yue Li, Joerg Kliewer, Mihael Langberg, Anxiao (Andrew) Jiang and Jehoshua Bruk Abstrat We propose effiient oding shemes
More informationPattern Recognition Techniques in Microarray Data Analysis
Pattern Reognition Tehniques in Miroarray Data Analysis Miao Li, Biao Wang, Zohreh Momeni, and Faramarz Valafar Department of Computer Siene San Diego State University San Diego, California, USA faramarz@sienes.sdsu.edu
More informationExploiting Relative Addressing and Virtual Overlays in Ad Hoc Networks with Bandwidth and Processing Constraints
Exploiting Relative Addressing and Virtual Overlays in Ad Ho Networks with Bandwidth and Proessing Constraints Maro AurĂ©lio Spohn Computer Siene Department University of California at Santa Cruz Santa
More informationBayes Bluff: Opponent Modelling in Poker
Bayes Bluff: Opponent Modelling in Poker Finnegan Southey, Mihael Bowling, Brye Larson, Carmelo Piione, Neil Burh, Darse Billings, Chris Rayner Department of Computing Siene University of Alberta Edmonton,
More informationChapter 5 Single Phase Systems
Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude
More informationTHE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES
Proeedings of FEDSM 98 998 ASME Fluids Engineering Division Summer Meeting June 225, 998 Washington DC FEDSM98529 THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES John D. Wright Proess
More informationRELEASING MICRODATA: DISCLOSURE RISK ESTIMATION, DATA MASKING AND ASSESSING UTILITY
Setion on Survey Researh Methods JSM 008 RELEASING MICRODATA: DISCLOSURE RISK ESTIMATION, DATA MASKING AND ASSESSING UTILITY Natalie Shlomo 1 1 Southampton Statistial Sienes Researh Institute, University
More information5. Hypothesis Tests About the Mean of a Normal Population When Ï 2 is Not Known
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting 5. Hypothesis ests About the Mean of a Normal Population When Ï is Not Known In Setion 3 we used the tdistribution as
More informationProgramming Basics  FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html
CWCS Workshop May 2005 Programming Basis  FORTRAN 77 http://www.physis.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html Program Organization A FORTRAN program is just a sequene of lines of plain text.
More informationprotection p1ann1ng report
f1re~~ protetion p1ann1ng report BUILDING CONSTRUCTION INFORMATION FROM THE CONCRETE AND MASONRY INDUSTRIES Signifiane of Fire Ratings for Building Constrution NO. 3 OF A SERIES The use of fireresistive
More informationSpecial Relativity and Linear Algebra
peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory
More informationCooperative Search and Nogood Recording
Cooperative Searh and Nogood Reording Cyril Terrioux LSS  Equipe NCA (LM) 39 rue JoliotCurie F1343 Marseille Cedex 13 (Frane) email: terrioux@limunivmrsfr Abstrat Within the framework of onstraint
More informationNOMCLUST: AN R PACKAGE FOR HIERARCHICAL CLUSTERING OF OBJECTS CHARACTERIZED BY NOMINAL VARIABLES
The 9 th International Days of Statistis and Eonomis, Prague, September 101, 015 NOMCLUST: AN R PACKAGE FOR HIERARCHICAL CLUSTERING OF OBJECTS CHARACTERIZED BY NOMINAL VARIABLES ZdenÄk Ć ul Hana ĆezankovĂĄ
More informationFOOD FOR THOUGHT Topical Insights from our Subject Matter Experts
FOOD FOR THOUGHT Topial Insights from our Sujet Matter Experts DEGREE OF DIFFERENCE TESTING: AN ALTERNATIVE TO TRADITIONAL APPROACHES The NFL White Paper Series Volume 14, June 2014 Overview Differene
More informationSymmetric Subgame Perfect Equilibria in Resource Allocation
Proeedings of the TwentySixth AAAI Conferene on Artifiial Intelligene Symmetri Subgame Perfet Equilibria in Resoure Alloation Ludek Cigler Eole Polytehnique FĂ©dĂ©rale de Lausanne Artifiial Intelligene
More informationMaterial Identification of Solder Joint in Microelectronics Package Using Bayesian Approach
Material Identifiation of Solder Joint in Miroeletronis Pakage Using Bayesian Approah J.H. Gang a, J.H. Choi a, D. An a and J.W. Joo b a Shool of Aerospae and Mehanial Engineering, Korea Aerospae University
More informationCHAPTER 14 Chemical Equilibrium: Equal but Opposite Reaction Rates
CHATER 14 Chemial Equilibrium: Equal but Opposite Reation Rates 14.1. Collet and Organize For two reversible reations, we are given the reation profiles (Figure 14.1). The profile for the onversion of
More informationStatic Fairness Criteria in Telecommunications
Teknillinen Korkeakoulu ERIKOISTYĂ Teknillisen fysiikan koulutusohjelma 92002 Mat208 Sovelletun matematiikan erikoistyĂ¶t Stati Fairness Criteria in Teleommuniations Vesa Timonen, email: vesatimonen@hutfi
More informationOpen and Extensible Business Process Simulator
UNIVERSITY OF TARTU FACULTY OF MATHEMATICS AND COMPUTER SCIENCE Institute of Computer Siene Karl Blum Open and Extensible Business Proess Simulator Master Thesis (30 EAP) Supervisors: Luiano GarĂaBaĂ±uelos,
More informationSet Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos)
A.1 Set Theory and Logi: Fundamental Conepts (Notes by Dr. J. Santos) A.1. Primitive Conepts. In mathematis, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existene
More informationDirichlet Processes A gentle tutorial
Dirichlet Processes A gentle tutorial SELECT Lab Meeting October 14, 2008 Khalid ElArini Motivation We are given a data set, and are told that it was generated from a mixture of Gaussian distributions.
More informationA Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders
A Robust Optimization Approah to Dynami Priing and Inventory Control with no Bakorders Elodie Adida and Georgia Perakis July 24 revised July 25 Abstrat In this paper, we present a robust optimization formulation
More informationi_~f e 1 then e 2 else e 3
A PROCEDURE MECHANISM FOR BACKTRACK PROGRAMMING* David R. HANSON + Department o Computer Siene, The University of Arizona Tuson, Arizona 85721 One of the diffiulties in using nondeterministi algorithms
More informationState of Maryland Participation Agreement for PreTax and Roth Retirement Savings Accounts
State of Maryland Partiipation Agreement for PreTax and Roth Retirement Savings Aounts DC4531 (08/2015) For help, please all 18009666355 www.marylandd.om 1 Things to Remember Complete all of the setions
More informationNeural networkbased Load Balancing and Reactive Power Control by Static VAR Compensator
nternational Journal of Computer and Eletrial Engineering, Vol. 1, No. 1, April 2009 Neural networkbased Load Balaning and Reative Power Control by Stati VAR Compensator smail K. Said and Marouf Pirouti
More informationFundamentals of Chemical Reactor Theory
UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil & Environmental Engineering Department Fundamentals of Chemial Reator Theory Mihael K. Stenstrom Professor Diego Rosso Teahing Assistant Los Angeles, 3 Introdution
More informationParametric model of IPnetworks in the form of colored Petri net
Parametri model of IPnetworks in the form of olored Petri net Shmeleva T.R. Abstrat A parametri model of IPnetworks in the form of olored Petri net was developed; it onsists of a fixed number of Petri
More informationDeadlinebased Escalation in ProcessAware Information Systems
Deadlinebased Esalation in ProessAware Information Systems Wil M.P. van der Aalst 1,2, Mihael Rosemann 2, Marlon Dumas 2 1 Department of Tehnology Management Eindhoven University of Tehnology, The Netherlands
More informationTrade Information, Not Spectrum: A Novel TV White Space Information Market Model
Trade Information, Not Spetrum: A Novel TV White Spae Information Market Model Yuan Luo, Lin Gao, and Jianwei Huang 1 Abstrat In this paper, we propose a novel information market for TV white spae networks,
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x  2 2 ; x
More informationRecommending Questions Using the MDLbased Tree Cut Model
WWW 2008 / Refereed Trak: Data Mining  Learning April 225, 2008 Beijing, China Reommending Questions Using the MDLbased Tree Cut Model Yunbo Cao,2, Huizhong Duan, ChinYew Lin 2, Yong Yu, and HsiaoWuen
More informationDiameter growth and bounded topology of complete manifolds with nonnegative Ricci curvature
Diameter growth and bounded topology of omplete manifolds with nonnegative Rii urvature Huihong Jiang and YiHu Yang Abstrat In this Note, we show that a omplete ndim Riemannian manifold with nonnegative
More informationDSPI DSPI DSPI DSPI
DSPI DSPI DSPI DSPI Digital Signal Proessing I (879) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:
More informationGranular Problem Solving and Software Engineering
Granular Problem Solving and Software Engineering Haibin Zhu, Senior Member, IEEE Department of Computer Siene and Mathematis, Nipissing University, 100 College Drive, North Bay, Ontario, P1B 8L7, Canada
More informationarxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA
LBNL52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrene Berkeley National Laboratory
More informationHEAT EXCHANGERS2. Associate Professor. IIT Delhi Email: prabal@mech.iitd.ac.in. P.Talukdar/ MechIITD
HEA EXHANGERS2 Prabal alukdar Assoiate Professor Department of Mehanial Engineering II Delhi Email: prabal@meh.iitd.a.in Multipass and rossflow he subsripts 1 and 2 represent the inlet and outlet, respetively..
More informationJEFFREY ALLAN ROBBINS. Bachelor of Science. Blacksburg, Virginia
A PROGRAM FOR SOLUtiON OF LARGE SCALE VEHICLE ROUTING PROBLEMS By JEFFREY ALLAN ROBBINS Bahelor of Siene Virginia Polytehni Institute and State University Blaksburg, Virginia 1974 II Submitted to the Faulty
More information3 Game Theory: Basic Concepts
3 Game Theory: Basi Conepts Eah disipline of the soial sienes rules omfortably ithin its on hosen domain: : : so long as it stays largely oblivious of the others. Edard O. Wilson (1998):191 3.1 and and
More informationFIRE DETECTION USING AUTONOMOUS AERIAL VEHICLES WITH INFRARED AND VISUAL CAMERAS. J. Ramiro MartĂnezde Dios, Luis Merino and AnĂbal Ollero
FE DETECTION USING AUTONOMOUS AERIAL VEHICLES WITH INFRARED AND VISUAL CAMERAS. J. Ramiro MartĂnezde Dios, Luis Merino and AnĂbal Ollero Robotis, Computer Vision and Intelligent Control Group. University
More informationProcurement auctions are sometimes plagued with a chosen supplier s failing to accomplish a project successfully.
Deision Analysis Vol. 7, No. 1, Marh 2010, pp. 23 39 issn 15458490 eissn 15458504 10 0701 0023 informs doi 10.1287/dea.1090.0155 2010 INFORMS Managing Projet Failure Risk Through Contingent Contrats
More informationMarker Tracking and HMD Calibration for a Videobased Augmented Reality Conferencing System
Marker Traking and HMD Calibration for a Videobased Augmented Reality Conferening System Hirokazu Kato 1 and Mark Billinghurst 2 1 Faulty of Information Sienes, Hiroshima City University 2 Human Interfae
More informationAn Efficient Network Traffic Classification Based on Unknown and Anomaly Flow Detection Mechanism
An Effiient Network Traffi Classifiation Based on Unknown and Anomaly Flow Detetion Mehanism G.Suganya.M.s.,B.Ed 1 1 Mphil.Sholar, Department of Computer Siene, KG College of Arts and Siene,Coimbatore.
More informationPath Planning of Automated Optical Inspection Machines for PCB Assembly Systems
96 International Journal TaeHyoung of Control, Park, Automation, HwaJung Kim, and Systems, and Nam vol. Kim 4, no., pp. 9604, February 2006 Path Planning of Automated Optial Inspetion Mahines for PCB
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationIEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 401
IEEE TRASACTIOS O DEPEDABLE AD SECURE COMPUTIG, VOL. 9, O. 3, MAY/JUE 2012 401 Mitigating Distributed Denial of Servie Attaks in Multiparty Appliations in the Presene of Clok Drifts Zhang Fu, Marina Papatriantafilou,
More information' R ATIONAL. :::~i:. :'.:::::: RETENTION ':: Compliance with the way you work PRODUCT BRIEF
' R :::i:. ATIONAL :'.:::::: RETENTION ':: Compliane with the way you work, PRODUCT BRIEF Inplae Management of Unstrutured Data The explosion of unstrutured data ombined with new laws and regulations
More informationFindings and Recommendations
Contrating Methods and Administration Findings and Reommendations Finding 91 ESD did not utilize a formal written prequalifiations proess for seleting experiened design onsultants. ESD hose onsultants
More informationIn order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method
BEAM DESIGN In order to be able to design beams, we need both moments and shears. 1. Moment a) From diret design method or equivalent frame method b) From loads applied diretly to beams inluding beam weight
More informationHenley Business School at Univ of Reading. PreExperience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD)
MS in International Human Resoure Management For students entering in 2012/3 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date of speifiation:
More informationSinglePhase Transformers
SinglePhase Transformers SinglePhase Transformers Introdution Figure shows the transformer hemati symbol and the orresponding ommonly used Steinmetz model is shown in Figure. Therein, the model voltage
More informationA Design Environment for Migrating Relational to Object Oriented Database Systems
To appear in: 1996 International Conferene on Software Maintenane (ICSM 96); IEEE Computer Soiety, 1996 A Design Environment for Migrating Relational to Objet Oriented Database Systems Jens Jahnke, Wilhelm
More informationA ThreeHybrid Treatment Method of the Compressor's Characteristic Line in Performance Prediction of Power Systems
A ThreeHybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition of Power Systems A ThreeHybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition
More informationIn this chapter, we ll see state diagrams, an example of a different way to use directed graphs.
Chapter 19 State Diagrams In this hapter, we ll see state diagrams, an example of a different way to use direted graphs. 19.1 Introdution State diagrams are a type of direted graph, in whih the graph nodes
More informationGABOR AND WEBER LOCAL DESCRIPTORS PERFORMANCE IN MULTISPECTRAL EARTH OBSERVATION IMAGE DATA ANALYSIS
HENRI COANDA AIR FORCE ACADEMY ROMANIA INTERNATIONAL CONFERENCE of SCIENTIFIC PAPER AFASES 015 Brasov, 830 May 015 GENERAL M.R. STEFANIK ARMED FORCES ACADEMY SLOVAK REPUBLIC GABOR AND WEBER LOCAL DESCRIPTORS
More informationRATING SCALES FOR NEUROLOGISTS
RATING SCALES FOR NEUROLOGISTS J Hobart iv22 WHY Correspondene to: Dr Jeremy Hobart, Department of Clinial Neurosienes, Peninsula Medial Shool, Derriford Hospital, Plymouth PL6 8DH, UK; Jeremy.Hobart@
More informationRelativity Worked Solutions
Physis Option Problems  Relativisti Addition of Veloities (from Gianoli 6 th edition 43. (I A person on a roket traveling at.5 (with respet to the Earth observes a meteor ome from behind and pass her
More informationOptimal Sales Force Compensation
Optimal Sales Fore Compensation Matthias KrĂ€kel Anja ShĂ¶ttner Abstrat We analyze a dynami moralhazard model to derive optimal sales fore ompensation plans without imposing any ad ho restritions on the
More information4.15 USING METEOSAT SECOND GENERATION HIGH RESOLUTION VISIBLE DATA FOR THE IMPOVEMENT OF THE RAPID DEVELOPPING THUNDERSTORM PRODUCT
4.15 USNG METEOSAT SECOND GENEATON HGH ESOLUTON VSBLE DATA FO THE MPOVEMENT OF THE APD DEVELOPPNG THUNDESTOM PODUCT Oleksiy Kryvobok * Ukrainian HydroMeteorologial nstitute Kyiv, Ukraine Stephane Senesi
More informationA Survey of Usability Evaluation in Virtual Environments: Classi cation and Comparison of Methods
Doug A. Bowman bowman@vt.edu Department of Computer Siene Virginia Teh Joseph L. Gabbard Deborah Hix [ jgabbard, hix]@vt.edu Systems Researh Center Virginia Teh A Survey of Usability Evaluation in Virtual
More informationPricebased versus quantitybased approaches for stimulating the development of renewable electricity: new insights in an old debate
Priebased versus based approahes for stimulating the development of renewable eletriity: new insights in an old debate uthors: Dominique FINON, Philippe MENNTEU, MarieLaure LMY, Institut d Eonomie et
More informationAn integrated optimization model of a Closed Loop Supply Chain under uncertainty
ISSN 18166075 (Print), 18180523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 917 An integrated optimization model of a Closed Loop Supply Chain under unertainty Xiaoxia
More information