J. Whittaker. Graphical Models in Applied Multivariate Statistics. John Wiley, Chichester, 1990. 20  Home/Search   Document Not in Database   Summary   Related Articles   Check

....also been called a covariance selection model (Dempster [7] and a concentration graph model (Cox and Wermuth [5] we shall use the latter term in juxtaposition to the term covariance graph model to be considered in Section 6. The reader is referred to Edwards [10] Lauritzen [15] or Whittaker [25] for statistical properties Date: December 11, 2003. Key words and phrases. Graphical model selection, concentration graphs, covariance selection models, covariance graphs, Sidak s inequality, simultaneous confidence intervals. of these models, including methods for parameter estimation, model ....

J. Whittaker, Graphical Models in Applied Multivariate Statistics, John Wiley & Sons Ltd., Chichester, 1990. Department of Statistics, University of Washington, Seattle, Washington, U.S.A. E-mail address: drton@stat.washington.edu, michael@ms.washington.edu

....(1995) and Friedman and Goldszmidt (1996) Here, we consider an application that comes from a study bySewell and Shah (1968) who investigated factors that influence the intention of high school students to attend college. The data have been analyzed by several groups of statisticians, including Whittaker (1990) and Spirtes et al. 1993) all of whom have used non Bayesian techniques. Sewell and Shah (1968) measured the following variables for 10,318 Wisconsin high school seniors: Sex (SEX) male, female# Socioeconomic Status (SES) low, lower middle, upper middle, high# Intelligence Quotient (IQ) low, ....

....0.124 0.53 0.093 0.39 0.24 0.84 log ( pS D 45629 Figure 11: The a posteriori most likely network structure with a hidden variable. Probabilities shown are MAP values. Some probabilities are omitted for lack of space. in Lauritzen (1982) Verma and Pearl (1990) Frydenberg (1990) Whittaker (1990), and Richardson (1997) Bayesian methods for learning such models from data are described by Dawid and Lauritzen (1993) and Buntine (1994) Finally,several research groups havedeveloped software systems for learning graphical models. For example, Scheines et al. 1994) havedeveloped a software ....

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. John Wiley and Sons.

....total to the partial situation. For instance, there is no conceptual need to distinguish between total and partial correlation coefficients since a total correlation coefficient is simply a parcor of order . In this section, we shall first recall (and slightly extend) some results [27] 28] [29] [30] giving the covariance matrix (resp. its inverse) in terms of covariances of the random variables (resp. of the random variables ) We thus get lemmas 3.1 and 3.2, which are generalized to theorem 3.1 by considering Schur complements in and in ) Lastly ....

J. Whittaker, Graphical models in applied multivariate statistics, John Wiley and sons, Chichester, 1990.

....we first approximate it by q(x2, s2) the conditionally Gaussian distribution closest in Kullback Leibler (KL) divergence to the original mixture, where KL (p[ q) x p(x2, s2[y. 2) log 2 2 It can easily be shown that minimizing the above matching or a collapse of the mixture (see e.g. [8]) Collapse (s p(s, r, x) argmin KL qCCG p(x2, s21Y:2) q(x2, s2) KL divergence boils down to moment We will use the notation Define p(s i,r j,x) pijA (x;lij, Eij) then Collapse( sp(s,r,x) q(r,x) with q(r = j,x) pjV(x;j,j) pj = p, PI, J = PI( 15 2 3 4 Fig. 4. ....

Joe Whittaker, Graphical Models in Applied Multivariate Statistics, John Wiley & Sons, 1989.

....equality in (4.0.4) due to the presence of noise. It is possible to perform statistical tests to check whether any given conditional independence is a reasonable hypothesis about the data. The theory of such testing is beyond our scope, but is discussed in the context of graphical models Whittaker [Whi90]. Instead we will adopt a pragmatic approach. If the assumption of conditional independence gives a density estimate that works well , in whatever application we have at hand, then it is a useful assumption. How to choose a model (a collection of such assumptions) will be the subject of the next ....

J. Whittaker. Graphical models in applied multivariate statistics. Wiley, 1990.

....(1995) and Friedman and Goldszmidt (1996) Here, we consider an application that comes from a study by Sewell and Shah (1968) who investigated factors that influence the intention of high school students to attend college. The data have been analyzed by several groups of statisticians, including Whittaker (1990) and Spirtes et el. 1993) all of whom have used non Bayesian techniques. Sewell and Shah (1968) measured the following variables for 10,318 Wisconsin high school seniors: Sex (SEX) male, female; Socioeconomic Status (SES) low, lower middle, upper middle, high; Intelligence Quotient (IQ) low, ....

....high low 0.124 high high 0.53 high low 0.24 Figure 11: The a posteriori most likely network structure with a hidden variable. Proba bilities shown are MAP values. Some probabilities are omitted for lack of space. in Lauritzen (1982) Verma and Pearl (1990) Frydenberg (1990) Whittaker (1990), and Richardson (1997) Bayesian methods for learning such models from data are described by Dawid and Lauritzen (1993) and Buntine (1994) Finally, several research groups have developed software systems for learning graphical models. For example, Scheines et al. 1994) have developed a ....

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. John Wiley and Sons.

....tree, a concept we discuss in more detail in subsequent sections. As we will see, MR processes possessing such a Markov property make contact with standard Markov processes in time, with Markov random fields (MRF s) and with the large class of Bayes nets, belief networks, and graphical models [204, 170, 267, 35, 36, 169, 108, 128, 339, 89, 294, 295, 143, 168, 123, 197, 236, 337, 357, 302]. It is the exploitation of this Markovian property that leads to the e#cient algorithms that we describe. 1.3 Getting oriented A fair question to ask is: for whom is this paper written A reply that is only partially frivolous is: for the author. The reason is not self promotion (although the ....

....automatically adapt to the presence of edges, alleviating the blurring that occurs if space invariant linear filtering is performed. 3. 3 Some first ties to graphical models, time series, and matrix factorization As we have indicated, MR models on trees are a special class of graphical models [204, 170, 267, 35, 36, 169, 108, 128, 339, 89, 294, 295, 143, 168, 123, 197, 236, 337, 357, 302]. With an eye toward some of the generalizations we describe later and to lay the foundation for relating our framework to other work, we briefly summarize some of the basic graph theoretic concepts associated with this larger class of models. AgraphG = V, E) consists of a set of vertices ....

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J. Whittaker. Graphical Models in Applied Multivariate Statistics. Wiley, New York, 1990.

....and synergies, to purely qualitative statements concerning independence of variables. This range has inspired a variety of schemes for reasoning under uncertainty. Some of these schemes build on quantitative information such as belief networks [Pearl, 1988] and undirected graphical models [Whittaker, 1990]; others build on partial numerical specifi cations, allowing for interval rather than point prob abilities [Breese and Fertig, 1991; Coletti et al. 1991; Coletti, 1994; van der Gaag, 1991] or for order of magnitude estimates [Goldszmidt and Pearl, 1992] Yet other schemes are purely ....

Joe Whittaker. Graphical Models in Applied Multivariate Statistics. John Wiley & Sons., Chichester, 1990.

.... adopt for edge orientation, which is based on the notion of innovation, the graphical representation that we use here for systems of observations is similar to the one usually employed in statistics to describe Markov random fields [10, 12] or more generally, systems of coupled regression models [24]. Specifically, graphs with branches of the form (4.11c) are duals of Markov random fields graphs, i.e. we switch from one notation to the other by exchanging edges and vertices. The theory of belief networks [16, 17] in artificial intelligence relies also on directed graph representations. ....

....strive to keep aggregations to a minimum. In addition, it would be nice if the structure of the resulting tree does not depend on the order according to which observations are regrouped. An elegant solution to this problem was proposed by Lauritzen and Spiegelhalter [18] see also chapter 12 of [24]) in the context of a study of graphical models in statistics and artificial intelligence. It relies on the observation that if a graph is triangulated, i.e. if it does not contain any chordless cycle, the hypergraph formed by its maximum cliques is acyclic. A hypergraph consists of a set V ....

J. Whittaker, Graphical Models in Applied Multivariate Statistics. New York: J. Wiley, 1990.

....cases in the database and r ijjrest is the sample partial correlation of Y i and Y j adjusted for the remainder variables. This last quantity can be expressed in terms of the maximum likelihood estimates of the elements of the inverse variance matrix as r ijjrest = w ij ( w ii w jj ) 1 2 [43]. Then, the relevance measure value for each feature Y i , i = 1; n, is calculated as the average likelihood ratio test statistic for excluding an edge between Y i and any other feature in a graphical Gaussian model [38] This means that those features likely to remain conditional ....

J. Whittaker, Graphical Models in Applied Multivariate Statistics, John Wiley and Sons, Inc., Chichester, United Kingdom, 1990. 24

....all the remaining variables. These independence relationships can often be represented in terms of a graph, where the variables are associated with the nodes, and a missing edge represents a particular independence relationship (precise definitions can be found in the Appendix) See, for instance, [34, 29, 44, 12, 40, 11] for general reviews, treatments, or pointers to the large literature on this topic. The independence relationships result in the fundamental fact that the global high dimensional probability distribution P(x 1 ; xn ) over all variables, can be factored into a product of simpler local ....

....models with directed edges. Typical names for such models in the literature are Bayesian networks, belief networks, directed probabilistic independence networks, causal networks, and influence diagrams (see also [7] for a generalization) It is also possible to develop a theory for the mixed case [44], where both directed and undirected edges are present. Such mixed graphs are also called chain independence graphs. The basic theory of graphical models is reviewed in the Appendix. One fundamental observation is that in most applications the resulting graphs are sparse, with at least some ....

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J. Whittaker. Graphical models in applied multivariate statistics. John Wiley & Sons, New York, 1990.

....Histograms estimate the variables density, but may not show outliers in a large data set due to insufficient display resolution. Related patterns: Outliers. Robust estimation. Figure 1 shows a boxplot of the age distribution of respondents in a survey (the Rochdale data set referred to in Whittaker [1990]) There is a lone outlier, but what action should be taken (a) query the point and the maximum of the rest of the distribution to see if the actual values give a clue as to what is going on. The exact value of the point might suggest a mistyping. The maximum of the rest might support the ....

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics.

....Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK 1 when all the random variables in X are continuous, a graphical Gaussian model is obtained by assuming P G = N( G ) with G positive de nite and such that P G is Markov over G. For an introduction to graphical models, see for instance [7] or [11]. Typically the association structure of X is uncertain and, thus, has to be inferred from the data. This leads to the need for a statistical procedure to select the graphical models which best t the data. A Bayesian approach is particularly suited for the latter purpose, see for instance [8] It ....

....paper is as follows: after some preliminary background in Section 2, Section 3 states the required computational task whereas Section 4 contains the proposed Monte Carlo method to solve it. In Section 5 the methodology is applied to perform Bayesian model selection on Fret s data (see for instance [11]) Finally, Section 6 contains some concluding remarks. 2 2 SOME PRELIMINARY BACKGROUND Let X = X 1 ; X p ) T be a vector of p 3 continuous random variables. Denote with V = f1; pg the index set and let, for A V , XA = X a : a 2 A) be a collection of random variables. ....

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J. Whittaker, Graphical models in applied multivariate statistics, Wiley, New York, 1990.

....from work on artificial intelligence. Russell and Norvig (1995) provide a good introduction from the point of view of artificial intelligence, and Ripley (1996) gives a more mathematical treatment. For a comprehensive treatment from the statistical point of view the reader is invited to refer to Whittaker (1990) in which both directed and undirected graphs are treated. It is probably true to say that on the whole statisticians prefer to deal with undirected graphs as capturing the independence properties of the system under consideration and to avoid notions of causality as being rather abstract and not ....

Whittaker, J., 1990. "Graphical Models in Applied Multivariate Statistics". Wiley. England, Chichester.

....approximation of the optimal TSS is then found by a greedy search based on the observation biased eciency. 4. 1 ECR and Shannon entropy Recall the de nition of Shannon entropy, h(x 1 ; xN ) P N i=1 x i log x i ; where (x 1 ; xN ) de nes a probability distribution (see e.g. [7]) The entropy de nes the level of uniformness within the probability distribution, as h( reaches its maximal value of log(N) for x i = 1=N; i = 1; N, and the minimal value of 0 when x i = 1 for one i. It seems intuitive that the ECR of a TS sequence is in part a function of the ....

Joe Whittaker. Graphical models in applied multivariate statistics. Wiley, Chichester, 1990.

....i of into larger clumps. In doing so, it is desirable to keep the number of aggregations to a minimum, as well as to ensure that the structure of the resulting tree does not depend on the order in which aggregations are performed. A simple solution to this problem was presented in [21] see also [37], chapter 12) for the case of triangulated graphs. Recall that a graph is triangulated if it does not include chordless cycles. This solution relies on the fact that if a graph is triangulated, the hypergraph formed by its maximum cliques is acyclic. A hypergraph differs from a graph by the fact ....

J. Whittaker, Graphical Models in Applied Multivariate Statistics, J. Wiley, New York, 1990. 43

....: j Gamma1 ; Pr(h j = sjh j Gamma1 ; j Gamma1 ; 5) p( j j 1 ; j Gamma1 ; h 1 ; h j ; p( j j j Gamma1 ; h j ; 6) which would describe a hidden Markov relationship if neither equation depended on j Gamma1 . Figure 3 shows the directed acyclic graph (Whittaker, 1990) describing (5) and (6) Appendix A develops stochastic forward backward recursions to simulate from p(hj ; under the nonstationary hidden 8 Markov model (5) 6) Appendix B presents specific computational details customizing the recursions for the MMNHPP. Once h is drawn, the elements of ....

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley.

....Models in this class are capable of characterizing a rich set of relationships between a large number of variables, where these variables may be spatially separated and may also be of different types (e.g. classbased vs. specific collocations) The models we refer to are called graphical models (Whittaker, 1990). The probability distribution of a graphical model corresponds to a Markov field. This is because a graphical model is a probability model for multivariate random observations whose dependency structure has a valid graphical representation, a dependency graph. The dependency graph of a model is ....

Whittaker, Joe (1990). Graphical Models In Applied Multivariate Statistics. New York, NY: John Wiley & Sons.

....model must be formulated that specifies a set of relevant nonclassification variables and describes the relationships among these variables and the classification variable. In this work, we make use of a more general class of models than used in most NLP applications (graphical models, Whittaker 1990). Such models are capable of characterizing a rich set of relationships among a large number of variables, where these variables may be of different types as well as spatially separated. Graphical models are a subset of the class of hierarchical log linear models and, as such, are well suited to ....

Whittaker, J. 1990. Graphical Models In Applied Multivariate Statistics. New York, NY: John Wiley & Sons.

....underlying graphical structure of same network. Filled circles: nodes in sub domains and their common ancestors. Open circles: common children is minimized. The KL divergence is related to the di erence of the marginals of Q and PE , max i jP (S i jE) Q(S i )j r 1 2 D(Q;PE ) 2) see [24]) D(Q;PE ) depends on the numerical values of the conditional probability tables Q(S i j i ) Setting the gradient of D with respect to these parameters equal to zero, yields a coupled set of non linear equations that can be solved numerically. The quality of the approximation depends strongly ....

J. Whittaker. Graphical models in applied multivariate statistics. Wiley, Chichester, 1990.

....between the training data and the query data but not in between the training data (given the query data) con rming that the approximation in Equation 1 is reasonable. Readers not familiar with the relationship between the inverse covariance matrix and independencies are referred to the book by Whittaker (1990). 19 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 Figure 5: Left: Absolute values of the covariance matrix. White pixels are entries larger than 0.8. Right: Absolute values of the inverse covariance matrix. White ....

Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley.

....correlations allows the user to get a first look at an intimidating number of variables and see how they are related. This correlation analysis is strongly related to the graph based statistical analysis method known as conditional independence graphs, or, confusingly, graphical models (Whittaker 1990). With NicheWorks we sacrifice the conditional part of the dependency information (showing simply raw correlations) for the ability to handle large numbers of variables. This article is intended to describe both the methodology behind NicheWorks and our general approach to visualizing large, ....

Whittaker, J. (1990), Graphical Models In Applied Multivariate Statistics, Chichester: Wiley.

....this section. Belief or conditional independence networks became popular in artificial intelligence as a tool for reasoning in probabilistic expert systems [27] More in general, belief networks are effectively used in statistics for representing and manipulating complex probability distributions [28]. As a matter of fact, many learning systems, such as Boltzmann machines [29] multilayered perceptrons [30] input output) hidden Markov models [23] 24] just to mention some of them) can be easily regarded as particular graphical models. A belief network is an annotated graph in which ....

J. Whittaker, Graphical Models in Applied Multivariate Statistics. Chichester, U.K.: Wiley, 1990.

....the probabilistic relationships can be specified more naturally and compactly in terms of local probabilistic dependencies among events. Graphical models offer a language for describing a joint distribution in terms of events and the conditional dependence between them [Jensen, 1996, Pearl, 1988, Whittaker, 1990] Expert systems based on such models are among the most successful and practical products to emerge from artificial intelligence (AI) research. One of their key features is the ability to efficiently encode an otherwise unmanageably large joint distribution. Indeed, if sufficient conditional ....

....represent the full joint with O(2 q m) probabilities, instead of O(2 m ) where q is the maximum number of parents of any node in the network. A Markov network (MN) is another graphical language for modeling conditional independence and for implicitly describing a joint distribution [Whittaker, 1990, Darroch et al. 1980] Events are again associated with nodes in a graph, and edges encode probabilistic dependencies. However, as opposed to BNs, the underlying structure of a MN is an undirected graph. Given the outcomes of its direct neighbors, an event A j is conditionally independent of ....

Joe Whittaker. Graphical Models in Applied Multivariate Statistics. Wiley, Chichester, England; New York, 1990.

....log Q(x) P (x) j log Q(x) P (x) AE ; is minimised. In this paper, h: i denotes the average with respect to Q. The KL divergence is related to the difference of the probabilities of Q and P , max A jP (A) Gamma Q(A)j r 1 2 D(Q;P ) for any event A in the sample space (see (Whittaker, 1990)) In the logarithmic potential representations of P and Q, the KL divergence is D(Q;P ) X fl fl (c fl ) Gamma X ff ff (d ff ) constant ; which shows that D(Q;P ) is tractable (up to a constant) when Q is tractable and the clusters in P and Q are small. To optimise Q under ....

Whittaker, J. (1990). Graphical models in applied multivariate statistics. Wiley, Chichester.

....4. 3 Conditional independence properties of Gaussian graphical models In this section we will show that X i X j j (the rest) K ij = 0 (5) where K = 1 is the inverse covariance matrix (also called the precision matrix) of the joint distribution, and the rest means all the other nodes [Whi90, Edw95]. We can represent the joint distribution over all the nodes as (x; p; p exp 1 2 (x ) T 1 (x ) 6) where p = 2 ) jxj=2 j j 1 2 is a normalizing constant to ensure R x (x; p; 1. p, and are called the moment characteristics of the distribution. Expanding ....

J. Whittaker. Graphical Models in Applied Multivariate Statistics. Wiley, 1990.

....di erence in their independence semantics. The key problem in graphical representation of probability models is to explicate the structure of any probability distribution consistent with all the independence properties we can derive from the graph. Figure 1a) illustrates an undirected graph model [3, 46]) also known as a Markov random eld or MRF for short. For undirected graph models the ordinary graph separation of nodes is isomorphic to conditional independence statements about the variables associated with the nodes. For example, the graph in Figure 1a) states that the variables y 1 and x 2 ....

J. Whittaker. Graphical models in applied multivariate statistics. John Wiley & Sons, 1990.

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J. Whittaker. Graphical Models in Applied Multivariate Statistics. John Wiley, Chichester, 1990. 20

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Whittaker, J. (1990). Graphical models in applied multivariate statistics. Chichester, UK: John Wiley & Sons.

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Whittaker, J. (1990). Graphical models in applied multivariate statistics. Chichester: Wiley & Sons.

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J. Whittaker, Graphical models in applied multivariate statistics, John Wiley & Sons, Chichester, UK, 1990.

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J. Whittaker. Graphical Models in Applied Multivariate Statistics. Wiley, 1990.

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J. Whittaker, Graphical Models in Applied Multivariate Statistics, John Wiley & Sons, New York, 1990.

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Whittaker, J. 1990. Graphical Models in Applied Multivariate Statistics, Chichester, UK: John Wiley and Sons.

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Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley.

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Whittaker, J., `Graphical models in applied multivariate statistics', John Wiley and Sons, New York, (1990).

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J. Whittaker. Graphical Models in Applied Multivariate Statistics. Wiley, New York, 1990.

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J. Whittaker, Graphical models in applied multivariate statistics, Wiley, 1990.

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J.L. Whittaker. Graphical Models in Applied Multivariate Statistics. John Wiley and Sons, 1990.

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J. Whittaker, Graphical Models in Applied Multivariate Statistics, Wiley, 1990.

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Whittaker, J., Graphical Models in Applied Multivariate Statistics, John Wiley & Sons, 1989.

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Whittaker, (1990). Graphical Models in Applied Multivariate Statistics. Wiley, New York.

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J.L. Whittaker, Graphical Models in Applied Multivariate Statistics, Wiley, New York, 1990.

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Joe Whittaker. Graphical models in applied multivariate statistics. Wiley, Chichester, 1990.

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J. Whittaker, Graphical Models in Applied Multivariate Statistics, Wiley, Series in Probability and Mathematical Statistics, 1996

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J. Whittaker, Graphical Models in Applied Multivariate Statistics (Wiley, Chichester, 1990).

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Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, New York.

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Joe Whittaker. Graphical models in applied multivariate statistics. Wiley, Chichester, 1990.

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Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. New York: John Wiley.

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