Wedge Copla This docment explains the constrction and properties o a particlar geometrical copla sed to it dependency data rom the edram case stdy done at Portland State University. The probability density nction o the Wedge copla reslting copla is shown in Figre. Figre Probability density nction o Wedge copla or c =. (best it to Nominal skew o DRAM case stdy). A wedge-shaped copla C is deined by starting with a psedo-copla A(,v), consisting o a shaded wedge-shaped region, symmetrical abot the (,)/(,) diagonal, shown in Figre. This shaded region has area ( - c)/c where c is deined in the igre. I the wedge is a region o niorm probability density normalized to nity within the nit sqare, and the probability density otside the wedge vanishes, then the wedge s niorm probability density is c/( - c). C. Glenn Shirley, October. Wedge Copla.docx
, 5 c cot c, v c v v c,v v c cv cv cv Figre The base psedo-copla A(,v) is a nction on [,] which is deined by a wedgeshaped region o niorm probability density, normalized to nity within the nit sqare. Regions otside the wedge have vanishing probability density. The size o the region is controlled by the parameter c which ranges rom c = or perect correlation and to c = or independence,, Geometrical considerations show that the probability density enclosed by the ble (,)/(,v) rectangle in Figre or (,v) anywhere in [,], is c v A(, v) v min, cv, v min v, c c c c () On the margins, this is A(,) ( ) A(, v) ( v) () where (z is a dmmy argment) ( c) z zc ( z) ( z) z c z ( c ) (3) which is a monotonically increasing nction. Notice that A(,v) satisies the reqirements o a copla, except that the marginal distribtions, Eqs. (), are not niorm. To constrct a wedge copla, with niorm margins we need the inverse o Eq. (3): z c z ( z) c c c c c c z z c ( ) ( )( ) () So the wedge copla corresponding to Figre is C( x, y) A ( x), ( y) (5) Wedge Copla.docx
which satisies the reqirement that it have a niorm marginal distribtion becase C( x,) A ( x), () A ( x), ( x) x (6) and the same or y, since x and y are cds o niorm distribtions. Since x ( ) y v () (7) and since the lower bondary o the probability area o A is deined by = cv then the corresponding bondary o C is given by y c x x ( ) (8) Similarly, the pper bondary o the probability area o A is deined by c c v c (9) and the corresponding pper bondary o the probability area o C is c ( x) x ( c ) / c y ( c ) / c x () The zone o non-vanishing probability o the copla C is deined by simltaneos satisaction o y c x x ( ) () c ( x) x ( c ) / c y ( c ) / c x () For the test application the important region is the region near the origin. In this region the bondaries o the regions with inite probability given by Eqs. (8) and () are straight lines. For the lower bondary c (3) c y c x x and or the pper bondary c () c y c x x The low tail dependence o the wedge copla is LT C( x, x) ( x) x lim lim lim x x x x x x c c (5) Wedge Copla.docx 3
So the wedge copla has asymptotic tail dependence except in the limit o independence (c ). In the opposite limit (c ), the asymptotic dependence becomes nity becase C becomes M, the Frechet pper bond, corresponding to perect correlation. Points o the wedge copla may easily be synthesized by generating niormly distribted (,v) points in the wedgeshaped inite niorm probability density area o the base psedo-copla in Figre, and then mapping them to the space o the copla sing x = (), and y = (v) where is given by Eq.(). The algorithm or generating (, v) points is given in the Appendix. The probability density maps o the wedge copla in Figre 3 were generated sing this method. Notice that it is possible to restrict the generation o synthesized points to sb-domains o the copla, particlarly the region near the origin. c = 5. Ta =.38 Censor =. Points = 5 c =.5 Ta =.585 Censor =. Points = 5 c =.5 Ta =.936 Censor =. Points = 5.8.8.8.6.6.6.........6.8...6.8...6.8 c =.5 Ta =.593 Censor =. Points = 5 c =.5 Ta =.59 Censor =. Points = 5 c =.5 Ta =.595 Censor =.6 Points = 5.8.8.8.6.6.6.........6.8...6.8...6.8 Figre 3 Synthesized probability maps o the wedge copla. Kendall s ta was compted rom the synthesized data. Top Row: The wedge copla spans independence (c ) to perect correlation (c ). Bottom Row: Synthesized points can be concentrated near the origin. Vales o Kendall s ta given in Figre 3 or the synthesized data were estimated to good precision rom synthesized data. It is also possible to derive an analytical expression or Kendall s ta or the copla as a nction o the parameter c, and o the censor raction a, sing the expression or ta o a trncated part o a copla. Sbpoplation a a We( x, y; c) (6) ( c, a) dx dywe( x, y; c) We ( a, a; c) x y The reslt, derived in the Appendix is c ( ca, ) 3c. Sbpoplation (7) v C( x, y) (, v) dxdyc( x, y) C (, v) xy Wedge Copla.docx
The inverse o Eq. (7) is 3 c. (8) 3 The wedge copla has the attractive property that the sbpoplation ta is independent o a, the degree o censoring. This is not tre o coplas in general. The small variation o ta in Figre 3 or constant c is de to sampling variation o the Monte-Carlo estimate o ta. Since ta may be compted directly rom data, Eq. (7) provides a way to estimate the parameter c o the copla. Since it is convenient i the sbpoplation ta is independent o the degree o censoring, it is sel to know more general conditions nder which this holds so that coplas with this property can be identiied. A sicient condition or this to be tre is that a copla be expressed as C(x,y) = A[(x),(y)] where A satisies, or a, A(a,a v) = a A(, v). This is shown in the Appendix. The geometrical interpretation is that all sb-regions [,a] o the base psedo-copla are geometrically sel-similar. Wedge Copla, e c i ( )ˆ I, v (a) ĵ ˆ ˆ e c i j II Figre A Basis vectors (red) or sampling the area o niorm probability in the psedo-copla, A(,v). A sbset o the space, sch as the green area, may be sampled by scaling the basis vectors. î, (a), To place a random point r in triangle I in the igre in (,v) space, sample and independently rom the niorm distribtion on [,], rejecting the sample i >. r e e [ ( ) / c]ˆ i ˆj (9) In Eq. (9) e and e are basis vectors which span triangle I in the igre. Decomposition into the orthogonal nit vectors spanning the nit sqare in the igre gives the second eqation. By symmetry, to place a random point r point in triangle II in the igre, reject the sample i >, and place the point at r iˆ [ ( ) / c] ˆj () To sample a sbpoplation o the copla sch as the sb area shown in Figre A, the niorm random variables and shold be independently sampled rom [,(a)]. So the algorithm to sample a region x y [, a] [, a] o the wedge copla is (a = samples the entire copla):. Sample two independent niorm nmbers, and rom a niorm distribtion on [,(a)]. Wedge Copla.docx 5
. I then place a point at x y ( ) c () 3. Else i < then place a point at x ( ) y c () where - is given by Eq. (3). Sbpoplation Ta We seek an expression or ta o a sbpoplation o a copla in the region J = [, a ] [,v b ]. To do this, we write down an expression or the copla o the sbpoplation, and sbstitte into the ormla or ta, The probability density nction or the region J is C(, v) D'(, v) (3) C(, v ) a b This may be converted into a copla by sing the marginal distribtion nctions C(, vb) C( a, v) ' ( ) v ' g( v) C(, v ) C(, v ) () a b a b Notice that these are not niorm, (), and g(v) v, becase a and v b. The copla or the region J, in terms o the transormed variables, is C(, v) C ( '), g ( v ') v (5) D( ', v ') [ ', '] [,] C(, v ) C(, v ) So the sbpoplation ta is a b a b Sbpoplation D( ', v ') D( ', v ') d ' dv ' ' v' D( ', v ') v D( ', v ') d ' dv ' v ' v ' v a b D( ', v ') D( ', v ') ddv v v a b C(, v) C(, v) ddv C ( a, vb) v (6) Wedge Copla.docx 6
Sicient Condition or Censor-Independence o Sbpoplation Ta Sppose that a copla is expressed as C( x, y) A ( x), ( y) (7) where A is a psedo-copla satisying, or a, A a, av a A(, v) (8) The sbpoplation ta o C is Sbpoplation C(, v) d dvc(, v) C (, ) v A ( ), ( ) ( ) ( ) ( ) ( ) A ( ), ( v) ddva ( ), ( v) v v A( x, y) dx dy A( x, y) ( ) A(,) xy v A( x, y) dx dya( x, y) ( ) xy (9) Now set x ( ) x and y ( ) y, so A ( ) x, ( ) y ( ) ( ) ( ( ), ( ) ) ( ) ( ) Sbpoplation dx dy A x y xy ( ) A( x, y) ( ) ( ) ( ) (, ) ( ) ( ) dx dy A x y xy A( x, y) dxdya( x, y) xy (3) which is independent o the degree o censoring, QED. Moreover, the sbpoplation ta or any degree o censoring is the same as the poplation ta. Ta o Wedge Copla Kendall s ta or the wedge copla is We( x, y; c) ( c) dx dywe( x, y; c) I xy (3) Since We( x, y) A(, v) ( x), v ( y) (3) we have Wedge Copla.docx 7
I We( x, y) dxdywe( x, y) xy dx dy A, v d dv d dv A, v d dv v dx dy A(, v) ddva(, v) v (33) so the evalation o ta can be done entirely in terms o the psedo copla A. Evalation o the integral is acilitated by noticing: ) The second mixed derivative o A(,v) is the pd o A, which is a constant eqal to c/(c-) inside the wedge and zero elsewhere, and ) By symmetry across the diagonal, the desired integral is twice the integral o the shaded zone in Figre A. 3) Becase A(,v) vanishes otside the wedge, the v-integration limits may be changed so that or each, v ranges rom /c to. So c I d dva(, v) c (3) c / Inside the wedge, where the argment o Eq. (3) is evalated, we have, rom Eq. () (, ) c v A v v (Inside wedge.) c c c (35),, v v cv Figre A Integration limits or I. v / c So,, c a b c (36) c / c I d dv v ( v ) I I I c c c where 3 c 8 c c / c / Ia d dv v d v d (37) c 3 c / c / Ib d dv d v d c c c c 8c c (38) Wedge Copla.docx 8
3 3 3 c / 3 3 c / Ic d dv v d v d c c 6c c c c (39) So Ia Ib Ic 3 8 c 8c c c c 3 3 c c c c c 8c 3 3 3 3c c c ( c ) (3c c ) c () and rom Eq. (36) 3c c I () c Sbstittion o Eqs. () into Eq. (3) gives Sbpoplation c () 3c Probability Density Fnction The probability density is non-vanishing only or x,y pairs which satisy y c x x ( ) (3) c ( x) x ( c ) / c y ( c ) / c x () Inside this region we need the probability density nction given by C A v c c( x, y) xy v x y c x y (5) From Eq. () we have c l ( x) x ( c) x c x c c h ( x) x ( c ) ( c )( x) c (6) Wedge Copla.docx 9