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Fa ulteit Wetens happenVakgroep Toegepaste Wiskunde, Informati a en StatistiekRegular-vinesKevin BOCKSTAELPromoter: Prof. dr. David Vyn keMasterproef ingediend tot het behalen van de a ademis he graad van master in dewiskunde, afstudeerri hting toegepaste wiskunde.A ademiejaar 2012-2013
A knowledgmentsForemost I'd like to thank my advisor, Prof. dr. David Vyn ke, for the support andfruitful dis ussions in the last months.A spe ial word of gratitude goes out to Karen and Jorrit for the brainstormingsessions, the many laughs and the en ouragements during this year. Ofourse my hat'so to Benjamin, Jonathan and Rutger for providing me with helpfulomments on avariety of subje ts regarding this work.I would like to thank the inhabitants of deFysi asa , their support, en ouragement and friendship were undeniably the bedro kupon whi h the past years have been built.I thank my parents, Betty and Jan, for their guidan e and faith in me.De auteur geeft de toelating deze masterproef vooronsultatie bes hikbaar te stellenen delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruikvalt onder de beperkingen van het auteursre ht, in het bijzonder met betrekking totde verpli hting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uitdeze masterproef.Kevin Bo kstaelGent, 1 juni 2013i
Contents1 Introdu tion12 Regular-vine based models32.1Copulas2.2Paironstru tions . . . . . . . . . . . . . . . . . . . . . . . . . .42.3Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.4Regular-vines82.5Regular-vine2.5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .opula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .opulas. . . . . . . . . . . . . . . . . . . . . . . . . . . .Evaluation of the joint Regular-vine density. . . . . . . . . . .3 Regular-vine-matri es3.1De nition3.1.13.2Evaluation of the joint Regular-vine density. . . . . . . . . . .In-depth dis ussion on the de nition of an R-vine-matrix . . . . . . . .ondition forT13.2.1Tree3.2.2Proximity3.2.3Putting it all together. . . . . . . . . . . . . . . . . . . . . . . .ondition for R-vine-matri es4.1Storing Regular-vineRelabeling Regular-vine4.51419293031. . . . . . . . . . . . .31. . . . . . . . . . . . . . . . . . . . . . .33354.24.41319. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Algorithms4.33opula spe i ation . . . . . . . . . . . . . . . . .35. . . . . . . . . . . . . . . . . . . . . . . . . .364.2.1Reordering of Regular-vine-matrix . . . . . . . . . . . . . . . . .364.2.2Reordering of a ve toru. . . . . . . . . . . . . . . . . . . . . .Equivalen e between R-vine and R-vine-matrix. . . . . . . . . . . . .38404.3.1Building an R-vine-matrix from an R-vine. . . . . . . . . . . .404.3.2Building an R-vine from an R-vine-matrix. . . . . . . . . . . .47. . . . . . . . . . . .54. . . . . . . . . . . . . . . . . . . . . . . . .54Constru tion of Regular-vineopula spe i ation4.4.1Algorithm of Prim4.4.2The sequential estimation method.Evaluation of the joint Regular-vine density. . . . . . . . . . . . . . . .54. . . . . . . . . . . . . . .604.5.1Density w.r.t. real observations. . . . . . . . . . . . . . . . . .604.5.2De nite version . . . . . . . . . . . . . . . . . . . . . . . . . . .674.5.3Evaluating the log likelihood . . . . . . . . . . . . . . . . . . . .68ii
CONTENTSiii5 Implementation5.15.1.15.270Constru tion of Regular-vineopula spe i ationStoring a Regular-vine5.1.2Dependen ies5.1.3Code and explanationDensity. . . . . . . . . . . .70opula spe i ation . . . . . . . . . . . .70. . . . . . . . . . . . . . . . . . . . . . . . . . . .70. . . . . . . . . . . . . . . . . . . . . . .71. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .735.2.1Dependen ies. . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2.2Code and explanation. . . . . . . . . . . . . . . . . . . . . . .735.2.3Log likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . .746 Theoreti al ba kbone of the study73756.1Portfolio sele tion in investments. . . . . . . . . . . . . . . . . . . . .756.2Mathemati al ne essities . . . . . . . . . . . . . . . . . . . . . . . . . .786.2.1Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . .786.2.2Modeling of time series data . . . . . . . . . . . . . . . . . . . .816.3Sele ting the Regular-vine model. . . . . . . . . . . . . . . . . . . . .836.4Simulation study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .857 Simulation study7.1Data from 8 sto ks87(2001 2013). . . . . . . . . . . . . . . . . . . . .7.1.1Sele ting the Regular-vine model7.1.2Simulation study7.1.3Markowitz e ient frontier7.2Choosing a C-vine7.3Data from 8 sto ks87. . . . . . . . . . . . . . . . .87. . . . . . . . . . . . . . . . . . . . . . . . . .94. . . . . . . . . . . . . . . . . . . .94. . . . . . . . . . . . . . . . . . . . . . . . . . . . .96(2007 2013). . . . . . . . . . . . . . . . . . . . .100. . . . . . . . . . . . . . . . . . . . . . . . . .1087.3.1Simulation study7.3.2Markowitz e ient frontier. . . . . . . . . . . . . . . . . . . .7.4Add xed in ome to portfolio, data from7.5Add xed in ome to portfolio, data from(2001 2013)(2007 2013)108. . . . . . . . .109. . . . . . . . .1108 Con lusion and outlook1129 Samenvatting114Bibliography116A Optimizing the portfolio return118B R ode120B.1Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120B.2Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
Chapter 1Introdu tionMultivariate modeling is the pro ess of nding a stru ture in high-dimensional dataand spe ifying this stru ture with a theoreti al base.Althoughommonly used forthis purpose, the multivariate normal and student t distributions show some potentialdisadvantages. For example, su h distributions demand a single parametriits variables and do not allow forfamily foromplex dependen ies. Regarding the rst drawba k,an impulse towards exible multivariate models was given with Sklar's theorem (1959)in whi h a multivariateopula is used to inter onne t a multivariate distribution withits marginal distributions.well-known multivariateare available to apairopulasWith this, a new problem arises sin e there are only fewopulas. However, bivariateertain extent.an be used as building blo ks toThis approa h isalled a pairopulas with desirable propertiesDue to Bedford and Cooke (2001), these so- alledopulaonstru t higher dimensionalopulas.onstru tion (PCC). Sin e the number of PCCis vast in extent using higher dimensional data, a proper approa h tolassify all su honstru tions is desired. A solution for this is obtained using the Regular-vine.Therefore, in the attempt to auratelyworld multivariate data, a re ent growthapture the dependen e stru tures in real-an be observed in the interest in Regular-vinebased models. Resear h so far has been mainlyD-vines, whi h are two spe ialon entrating on so- alled C-vines andases of Regular-vines. Although these models allow asmooth mathemati al handling, their exibility in stru ture is narrowed downomparedto Regular-vines.1These Regular-vine models allow for modeling of multivariate data, using bivariateand univariate building blo ks. The basis for this is given by Bedford and Cooke (2001)whi h states that the distribution of multiple variablesthe univariate margins, bivariatean beonstru ted using onlyopulas and a Regular-vine stru ture, all of whi hallow for a straightforward approa h. This indi ates that the Regular-vineopula spe -i ation is being merged from two (entirely di erent)opulas andRegular-vines. The former aon epts, bivariateumulative distribution fun tion spe ifying the pairwisedistributions, the latter a nested set of trees sele ting whi h pairwise distributions tobe estimated.This thesis is founded on the Regular-vine1 Weopula spe i ation, and although bothwill denote them as Regular-vine opula spe i ations.1
CHAPTER 1.INTRODUCTIONRegular-vines and2opulas will be approa hed properly, the s rutiny is mainly onRegular-vines.The remainder of this thesis is stru tured as follows.approa h of both Regular-vines andopulas.use in the de nition of a Regular-vineChapter 2 gives a detailedThese subje ts are then being put toopula spe i ation. Thishapter ison ludedwith an important result that in orporates the Regular-vine into the joint density fun tion. However, the approa h in thishapter does not allow for a smooth handling inalgorithms and therefore the Regular-vine-matrix is introdu ed in Chapter 3 followingDiÿmann (2010). Sin e we observed somera ks in this de nition, a detailed dis ussionon it is given in Se tion 3.2. Bearing in mind that the Regular-vine-matrix was introdu ed for aonvenient handling in algorithms, in Chapter 4 a series of algorithms arestated. The rst few allow for a better insight in the stru ture of a Regular-vine-matrixwhile the last ones bring theonstru tion of a Regular-vine model to suessfulon-lusion and empower us to evaluate the multivariate density. Subsequently, in Chapter5, a detailed dis ussion is given on the implementation of these algorithms using thelanguage R.In order to obtain an insight in the appli ations of these models in real-world multivariate data, Chapter 6 gives the ne essary tools for analyzing and redu ing raw datain order that the Regular-vineopula spe i ationan be used. These models are thenapplied in Chapter 7, where series of 8 nan ial variables are observed in di erent timeperiods. In order to address the mind's eye using higher dimensional data, we apply theapproa h from Markowitz (1952) that visualizes areturn and risk.ertain portfolio through its expe ted
Chapter 2Regular-vine based modelsIn thisvinehapter we dis uss the two entirely di erent elements used for de ning Regular-opula spe i ations, i.e.opulas and Regular-vines. An outline for the theory ofopulas is given in se tion 2.1 and 2.2 where we use Nelsen (2006) as main referen e.Subsequently, in se tion 2.3, a detailed dis ussion on Regular-vines is started andtinued in se tion 2.4. Finally, in se tion 2.5, weRegular-vineon-ombine theseon epts and de ne theis a multivariateumulative distributionopula spe i ation.2.1 CopulasDe nition 2.1.Annfun tion on [0, 1] ,n-dimensional opula CC : [0, 1]n [0, 1]with uniform distributed margins.For the next lemma we need the following.De nition 2.2.fun tion ( df )LetFX .XThebe an arbitrary random variable withinverse dfumulative distributionthen isFX 1 (p) inf {FX (x) p} .xLemma 2.3. Let X be an arbitrary random variable, then Y FX (X) is uniformlydistributed on [0, 1]. Proof. FY (y) P (Y y) P (FX (X) y) P X FX 1 (y) FX FX 1 (y) yTherefore we see thatfY (y) FY (y) y 1,so[0, 1].WeallY FX (X)theY FX (X)is uniformly distributed ontransformed random variable of X .3
CHAPTER 2.REGULAR-VINE BASED MODELS4Theorem 2.4. For every multivariate distribution F , with marginalsF1 (x1 ) , F2 (x2 ) , . . . , Fn (xn ),writethere exists an n-dimensional opula C for whi h we anF1,.,n (x1 , x2 , . . . , xn ) C1,.,n (F1 (x1 ) , F2 (x2 ) , . . . , Fn (xn )) .The nameopularefers to the fa t that itouples(2.1.1)the multivariate distribution toits margins. This theorem rst appeared in Sklar (1959) and is used extensively inopula onstru tions1.pair2.2 Pair opula onstru tionsWeal ulate the partial derivative of equation (2.1.1), F1,.,n (x1 , . . . , xn ) C1,.,n (F1 (x1 ) , . . . , Fn (xn )) x1 x1nX C1,.,n (u1 , . . . , un ) Fl (xl ) ul x1l 1 C1,.,n (u1 , . . . , un )f1 (x1 ) u1 2 F1,.,n (x1 , . . . , xn ) C1,.,n (u1 , . . . , un ) f1 (x1 ) x1 x2 x2 u1 C1,.,n (u1 , . . . , un ) f1 (x1 ) x2 ul#" nX 2 C1,.,n (u1 , . . . , un ) Fl (xl ) f1 (x1 ) u x1 ul2l 1 2 C1,.,n (u1, . . . , un ) · f1 (x1 ) · f2 (x1 ) u1 u2Repeating this, we arrive at n F1,.,n (x1 , . . . , xn ) x1 . . . xnn C1,.,n (u1 , . . . , un )· f1 (x1 ) · · · fn (xn ) u1 . . . unf1.n (x1 , . . . , xn ) and nallyf1,.,n (x1 , . . . , xn ) c1.n (F1 (x1 ) , . . . , Fn (xn )) · f1 (x1 ) . . . fn (xn ) ,1 Abbreviatedas PCC.(2.2.1)
CHAPTER 2.c1.nwhereREGULAR-VINE BASED MODELSis the density fun tion aThe de nition ofompanying5C1.n .onditional densities isf (xn x1 , . . . , xn 1 ) · f (x1 , . . . , xn 1 ) f (x1 , . . . , xn )and enables us to fa torize the densityf,f (x1 , . . . , xn ) f (x1 ) · f (x2 x1 ) · f (x3 x1 , x2 ) . . . f (xn x1 , . . . , xn 1 )The se ond fa tor(2.2.2)an be rewritten asf (x1 , x2 )f (x1 )c12 (F1 (x1 ) , F2 (x2 )) · f (x1 ) · f (x2 ) f (x1 ) c12 (F1 (x1 ) , F2 (x2 )) f (x2 ) .f (x2 x1 ) In the same way wean fa torize the third fa tor,f (x1 , x2 , x3 )f (x1 , x2 )f (x3 x1 , x2 ) f (x2 ,x3 x1 )f (x1 )f (x2 x1 )f (x1 ) As before, we c23 1 F2 1 (x2 x1 ) , F3 1 (x3 x1 ) · f (x2 x1 ) · f (x3 x1 ) f (x2 x1 ) c23 1 F2 1 (x2 x1 ) , F3 1 (x3 x1 ) · f (x3 x1 )an de omposef (x3 x1 ),whi h will result in f (x3 x1 , x2 ) c23 1 F2 1 (x2 x1 ) , F3 1 (x3 x1 ) c31 (F3 (x3 ) , F1 (x1 )) f (x3 ) .Repeating this, wevariateon lude that equation (2.2.2)an be written as a produ t of bi-opulas and univariate margins. The remaining of this work is based on thisprin iple of pairthe sake ofopulaonstru tions and provides a ba kbone for generalizing it. Forompleteness we state the following theorem.Theorem 2.5. The margins of the distribution fun tion FX,Y Z (x, y z) are FX Z (x z)and FY Z (y z).Proof.We need to prove thatfX Zis the marginal density offX Z (x z) ˆfX,Y Z , fX,Y Z (x, u z) du.i.e.
CHAPTER 2.REGULAR-VINE BASED MODELS6The right-hand side isˆ fX,Y,Z (x, u, z)dufZ (z) ˆ 1fX,Y,Z (x, u, z) du fZ (z) 1fX,Z (x, z) fZ (z) fX Z (x z)fX,Y Z (x, u z) du ˆ2.3 GraphsDe nition 2.6.graph is a pair G (N, E) of sets so that E {{n1 , n2 } n1 , n2 N}.The elements of N are the nodes of the graph G, the elements of E are its edges. Thenumber of nodes of a graph G is its order.AIn the above de nition there is no restri tion on the set of nodes. We will howeveralways usenodes1N {1, . . . , n} where n is the order2 will then be written as {1, 2}.of the graph. The edgeonne ting theand3576124Figure 2.1: An example graphG (N, E) with nodes N {1, 2, 3, 4, 5, 6, 7}E {{1, 2} , {1, 5} , {2, 5} , {5, 7} , {6, 6} , {3, 4}}Figure 2.1 shows a graphsetDe nition 2.7.N1 N2andG1 (N1 , E1 ) and G2 (N2 , E2 )E1 E2 , then G2 is a subgraph of G1 .Letand edgebe two (non-empty) graphs, if
CHAPTER 2.REGULAR-VINE BASED MODELS7De nition 2.8. Let G (N, E) be a (non-empty) graph, two nodes n1 , n2 N areneighbors if {n1 , n2 } is an edge of G, i.e. {n1 , n2 } E . The set of neighbors ofa node n in G is denoted by NG (n) {v N {n, v} E} . The degree of a noden is #NG (n). The number minn N d(n) is the minimum degree of G. Likewise ismaxn N d(n) its maximum degree.De nition 2.9.Apath is a non-empty graph G (N, E) of the form N {x1 , . . . , xn },E {{x1 , x2 } , {x2 , x3 } , . . . , {xn 1 , xn }}. Let G be a graph,with a 6 b are onne ted if there is a path from a to b. Ane tedif every node istwo nodesgraph isy le is aonne ted to every other node. Aa, b Nalledon-path of the form{{x1 , x2 } , {x2 , x3 } , . . . , {xn 1 , x1 }}.We now introdu e trees.De nition 2.10.The followingtree is a graph T (N, E) that isAonne ted and has noy les.an be found in West (2000).Theorem 2.11. Let T (N, E) be a graph, the following are equivalent.1.Tis a tree.2. Any two nodes in3.TisTareonne ted and hasDe nition 2.12.is a tree and itAonne ted by an unique path.N 1edges.spanning tree G′of a graphonne ts all the nodes ofG.Gis a subgraph ofA single graphGso thatG′an have many di erentspanning trees.Example 2.13.Consider the graphof spanning trees ofG in Figure 2.2, then G1G.123456Figure 2.2:Aonne ted graphGandG2are two examples
CHAPTER 2.REGULAR-VINE BASED MODELS8112323454566Figure 2.3: Two examples of spanning treesG1andG2 .We might want to assign a weight to ea h edge in the graph, whi h is a numberrepresenting how (un)favorable it is. In this way, wetree byan assign a weight to a spanningomputing the sum of the weights of the edges in that spanning tree and de nethe following.De nition 2.14.Amaximum (minimum) spanning tree is a spanning tree withtotal weight less (more) than or equal to the weight of every other spanning tree. Wewill use the abbreviationRemark2.15.MST for maximum spanning tree2 .Note that we need to spe ify whi h weights areonsidered for sele tingthe MST.2.4 Regular-vinesFor the sake ofompleteness we in lude the following de nition.De nition 2.16.ALetA and Bbe two sets, the relativeomplement of two setsBandisA\B {x A x / B} .We introdu e the Regular-vine following Diÿmann (2010).De nition 2.17.ofn 11.T12. for2 NoteARegular-vine (R-vine) V (T1 , ., Tn 1 ) on n elements is a settrees su h thatis a tree with nodesi 2, ., n 1, TiN1 {1, ., n}and edge set denotedis a tree with nodesNi Ei 1E1 ,and edge setEi ,that the problem of nding a Minimum Spanning Tree orresponds to nding a solution tothe Traveling Salesman Problem.
CHAPTER 2.3. (REGULAR-VINE BASED MODELS9proximity ondition) for i 2, ., n 1 and edge {a, b} Ei with a {a1 , a2 }andb {b1 , b2 }.it must hold thatRemark2.18only beonne ted if these edges share a# (a b) 1.ondition ensures that two nodes (a andThe proximityommon node in treeTi 1b)in treeTianas is shown in Figure2.4.Figure 2.4:a1a {a1 , a2 }.aThe proximitythese share an 2edges. ThisT1b {b1 , b2 }{a, b}ondition: the nodesommon node inFollowing Theorem 2.11,a2 b1and inTian only beonne ted ifTi 1 .hasn 1edges, thusT2hasn 1nodes and therefore hasan only be repeated until there is but one edge left, and sothe last tree in the R-vine. We now de ne two spe ialDe nition 2.19.b(Ti 1 )(Ti ).ba.b2Tn 1isases of Regular-vines.An R-vine is aD-vine (drawable) if eaC-vine ( anoni al)n i.h node inif ea h treeNote that an edge in the rst treeT1T1has degree of at most 2 (path stru ture).Ti , i 1, .n 1,has one unique node of degreeis a set of nodes, while edges in the se ond treeare a set of a set of nodes, and so on. Obvious enough, we mustonvenient notation.ome up with a more
CHAPTER 2.REGULAR-VINE BASED MODELS{1, 2}12{2, 3}34{4, 5}5{2, 4}{1, 2}{{1, 2}, {2, 3}}{2, 3}10{{2, 3}, {2, 4}}{2, 4}(T1 ){{2, 4}, {4, 5}}(T2 ){4, 5}{{1, 2}, {2, 3}}{{{1, 2}, {2, 3}}, {{2, 3}, {2, 4}}}{{2, 3}, {2, 4}}{{{2, 3}, {2, 4}}, {{2, 4}, {4, 5}}}(T3 ){{2, 4}, {4, 5}}{{{1, 2}, {2, 3}}, {{2, 3}, {2, 4}}}{{{{1, 2}, {2, 3}}, {{2, 3}, {2, 4}}}, {{{2, 3}, {2, 4}}, {{2, 4}, {4, 5}}}}(T4 ){{{2, 3}, {2, 4}}, {{2, 4}, {4, 5}}}Figure 2.5: An example of a 5-dimensional Regular-vine.De nition 2.20.1. For anyLetV {T1 , ., Tn 1 }ei Ei , i n 1,thebe an R-vine.omplete unionUei {n N1 e1 , e2 , . . . , ei 12. For3. Theei {a, b} Ei ,theonditioned setofeiomplete unionUeiis the set ofCei Cei a Cei b ei .The followingis the set de ned byisDei Ua Ub .withCeia Ua \DeiandCei b ei Ei , i n 1 we have Uei {1, 2, ., n}. Thenodes of N1 that are in some way onne ted to thean easily be seen that for alledgeisein e1 e2 . . . ei 1 ei } .onditioning set of eiUb \Dei .Itwithofan be found in Diÿmann (2010).
CHAPTER 2.REGULAR-VINE BASED MODELS11Lemma 2.21. Let V be a Regular-vine on n elements, then, for all i 1, ., n 1,and ei Ei , the onditioned sets asso iated with ei are singletons, #Uei i 1 and#Dei i 1.De nition 2.22.Theonstraint setofVis the setCV {(Cei , Dei ) ei {a, b} Ei , i 1, . . . , n 1} .Example 2.23.In the Regular-vine from Figure 2.5, we use the edgee {{{1, 2}, {2, 3}}, {{2, 3}, {2, 4}}} E3 .We nd that1. Theomplete unionUeis1234omposed of {1, 2} {{1, 2}, {2, 3}} e,{1, 2} {{1, 2}, {2, 3}} e,.and thereforeUe {1, 2, 3, 4} .2. The edgeimplies{a, b} ewitha {{1, 2}, {2, 3}}andDe Ua Ub {1,
Statistiek Regular-vines Kevin AEL BOCKST Promoter: Prof. dr. vid Da e k ync V Masterpro ef ingediend tot het b ehalen an v de he academisc graad master in wiskunde, ting h afstudeerric to egepaste wiskunde. cademiejaar A 2012-2013. ts wledgmen kno c A oremost F I'd e lik to thank y m advisor, Prof. dr. vid Da e, k ync V for the supp ort and .
Prof. Dr. Luc Van Bortel Ghent University Prof. Dr. Johan Van De Voorde Ghent University Prof. Dr. Dirk Vogelaers Ghent University Katrien Hertegonne University Hospital Ghent Dpt. of Respiratory Medicine and Sleep Medicine Centre De Pintelaan 185 9000 Gent, België Tel 3293322611 Fax 3293322341 Katrien.Hertegonne@UGent.be
2003 Second M.Sc., Academic Teaching Training, Ghent University 2000-2002 M.Sc. in Physical Education (Kinesiology), Ghent University, Grade: magna cum laude 1998-2000 B.Sc. in Physical Education (Kinesiology), Ghent University, Grade: cum laude Positions 2015-Present Postdoctoral Research Fellow, Harvard University
auditory word recognition ASTER DIJKGRAAF Department of Experimental Psychology, Ghent University ROBERT J. HARTSUIKER Department of Experimental Psychology, Ghent University WOUTER DUYCK Department of Experimental Psychology, Ghent University (Received: April 7, 2015; final revision received: April 1, 2016; accepted: April 1, 2016)
Ghent University is one of the major universities in Belgium. With eleven faculties housing more than 120 departments, Ghent University’s research ranges across all disciplinary areas. It extends from (Veterinary) Medicine to Business and Economics, from Psychology to Literature and Philosophy,
Ghent University ponuja možnost učenja nizozemščine in drugih jezikov. Ghent University organizira “welcome days” (informatvno predavanje, stojnice, šport, tečaj nizozemščine, žur itd). ESN prireja zabave, izlete. Wanderlust student trips. Buddy program: na fakultet t, v kolikor želiš,
Ghent University - Faculty of Veterinary Medicine Research Group Veterinary Public Health and Zoonoses – Laboratory of Chemical Analysis Salisburylaan, 133 B-9820 Merelbeke Belgium T 32-9-264.74.60 S1 . Julie Vanden Bussche†*, Massimo Marzorati‡, Debby Laukens , and Lynn Vanhaecke†
Faculty of Medicine and Health Sciences, Ghent University, Ghent, Belgium 2 . Academic Network for Sexual and Reproductive Health and Rights Policy (ANSER), Ghent, Belgium 3 Department of Womens and Childrens Health, Karolinska Institutet, Stockholm, Sweden 4 Department of Womens and Childrens Health, Uppsala University, Stockholm, Sweden
The SRD is the ultimate axial pile capacity that is experienced during the dynamic conditions of pile driving. Predictions of the SRD are usually calculated by modifying the calculation for the ultimate static axial pile capacity in compression. API RP 2A and ISO 19002 refer to several methods proposed in the literature.
