Outline 1 What are Graphical Models? Infering Graphical Models - PDF document

Outline Outline Outline 1 What are Graphical Models? Infering Graphical Models from Time Series Graph Theory 2 Modelling, Simulating, and Inference of Complex Biological Graphs and Digraphs Systems Dags 3 Probability Theory Speaker: Sebastian Petry Association Structures Partial Covariance as a Graphical Model Institut of Statistics Multivariate Time Series and Stochastic Processes Ludwig-Maximilians-University Munich Independence of Time Series 07-14-06 Infering Grapical Models from Time Series 4 5 Summary Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Probability Theory Probability Theory Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Outline Graphical Models What are Graphical Models? 1 Graph Theory 2 The marriage (Quotation from Preface of [1]) Graphs and Digraphs Graphical models are a marriage between probability theory Dags and graph theory. They provide a natural tool for dealing with Probability Theory two problems that occur throughout applied mathematics and 3 Association Structures engineering — uncertainty and complexity — and in particular Partial Covariance as a Graphical Model they are playing an increasingly important role in design and Multivariate Time Series and Stochastic Processes analysis of machine learning. Fundamental to the idea of Independence of Time Series graphical model is the notion of modularity — a complex system is built by combining simpler parts. Infering Grapical Models from Time Series 4 Summary 5 Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Probability Theory Probability Theory Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Graphical Models The two parts of the marriage (Quotation from Preface of [1]) Probability theory provides the glue whereby the parts are combined, ensuring that the systems as whole is consistent, and providing ways to interface models to data. The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly-interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms. Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Graphs and Digraphs Graphs and Digraphs Probability Theory Probability Theory Dags Dags Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Outline Outline What are Graphical Models? What are Graphical Models? 1 1 Graph Theory Graph Theory 2 2 Graphs and Digraphs Graphs and Digraphs Dags Dags Probability Theory Probability Theory 3 3 Association Structures Association Structures Partial Covariance as a Graphical Model Partial Covariance as a Graphical Model Multivariate Time Series and Stochastic Processes Multivariate Time Series and Stochastic Processes Independence of Time Series Independence of Time Series Infering Grapical Models from Time Series Infering Grapical Models from Time Series 4 4 Summary Summary 5 5 Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series

What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Graphs and Digraphs Graphs and Digraphs Probability Theory Probability Theory Dags Dags Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Definition 1 (graph) Example 2 A graph G is a tupel G := ( V , E ) with a finite set V � = ∅ and a G = ( V , E ) subset E ⊆ V × V of two-elementic subsets of V . The elements of V are called vertices and the elements of E edges . V = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } The two vertex v i , v j ∈ V , v i � = v j , of a edge e = { v i , v j } ∈ E E = {{ 1 , 5 } , { 5 , 4 } , { 5 , 3 } , are called end vertex of e . { 4 , 7 } , { 3 , 6 } , { 2 , 4 } , { 2 , 7 } , { 6 , 7 }} We call v i and v j with e incident and v i and v j are adjacent. Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Graphs and Digraphs Graphs and Digraphs Probability Theory Probability Theory Dags Dags Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Example 4 Definition 3 (digraph)  0 0 1 0 0 1 0  A directed graph or digraph G is a tupel G := ( V , E ) with a 1 0 1 1 0 0 1 finite set V � = ∅ and a subset E ⊆ V × V of ordered paires     0 0 0 0 1 1 0 ( v i , v j ) ∈ V × V , v i � = v j and ∃ ( v i , v j ) ⇒ ∄ ( v j , v i ) . The elements     1 0 1 0 1 0 1 of V are also called vertices and the elements of E edges or     0 1 0 0 0 1 1 arcs . For an arc e = ( v i , v j ) ∈ E v i is called tail and v j called     0 0 0 0 0 0 0 head .   0 0 0 0 0 1 0 We call v i and v j with e incident or v i and v j are adjacent . Adjacenty matrix: Tails in row and heads in column. We call d in ( v ) the number of arcs with head v indegree and d out ( v ) the number of arcs with tail v outdegree . This digraph contains the directed cycle: C := { ( 2 , 4 ) , ( 4 , 5 ) , ( 5 , 2 ) } . Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Graphs and Digraphs Graphs and Digraphs Probability Theory Probability Theory Dags Dags Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Outline What are Graphical Models? 1 Definition 5 (dag) Graph Theory 2 A digraph G = ( V , E ) is called directed acyclic graph (dag) , iff Graphs and Digraphs there exists no sequence in each subset E ′ ⊆ E , with Dags E ′ = { e ′ 1 = ( v ′ 1 T , v ′ 1 H ) , ..., e ′ n = ( v ′ nT , v ′ nH ) } , 2 < n ≤ | E | , for Probability Theory 3 which Association Structures v ′ 1 T = v ′ nH , Partial Covariance as a Graphical Model v ′ iH = v ′ ( i + 1 ) T , ∀ i = 2 , ..., n − 1, Multivariate Time Series and Stochastic Processes Independence of Time Series holds. Infering Grapical Models from Time Series 4 Summary 5 Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series What are Graphical Models? What are Graphical Models? Graph Theory Graph Theory Graphs and Digraphs Graphs and Digraphs Probability Theory Probability Theory Dags Dags Infering Grapical Models from Time Series Infering Grapical Models from Time Series Summary Summary Example 8 (A dag) 0 0 1 0 0 1 0   Lemma 6 1 0 1 1 0 0 1   Let G = ( E , V ) be a dag. Then there exists at least one vertex   0 0 0 0 1 1 0   v ∈ V for which d in ( v ) = 0 holds.   1 0 1 0 1 0 1     0 0 0 0 0 1 1   By Lemma 6 and Definition 5 the following theorem holds.   0 0 0 0 0 0 0   0 0 0 0 0 1 0 Theorem 7 Every dag has a topological order. Sebastian Petry Infering Graphical Models from Time Series Sebastian Petry Infering Graphical Models from Time Series