text: 4 VALENTINA-MIRA WHEELER (cid:48)(cid:48) ω . The local unique existence of a solution in this setting is standard and has been discussed in detail in [47, 50]. We note that the uniqueness of a solution shows that the representation (3) of a solution to (1) is preserved. Our ﬁrst main result is the following. Theorem 2.1 (Long time existence). Let ω and ω be deﬁned as above. Assume (2), and that Σ 0 ∗ ∗ (4) there is no point z where ω (z ) = 0 . Σ We further assume for negative and positive inﬁnity that either one of (5) the limit lim ω (z) does not exist →±∞ Σ z (cid:40) or | | dω z > α, if α > 0, (6) there exists an α > 0 such that z Σ (z) > 0 for all dz z < α, if α < 0, × → R hold. Then there exists a global smooth solution ω : D(t) [0, T) to the problem (3) that ∞ → R converges smoothly to ω∞ : D( ) . The function ω∞ is smooth and generates a minimal surface. Remark. The class of support hypersurfaces that satisfy (6) above at both positive and negative inﬁnity are those whose derivative is monotone outside a compact subset with the correct sign. Examples of such include the catenoid. The catenoid also satisﬁes condition (5). An example that satisﬁes (6) but not (5) is − − ω (z) = 2 e z2 . Σ More generally, any ω whose derivative is monotone increasing outside a compact set and con- Σ verges at inﬁnity satisﬁes (6) but not (5). Examples that satisﬁes (5) but not (6) include the unduloids. Examples that satisfy (5) for → ∞ → −∞ z and (6) for z can be pasted together using those mentioned above; for example, (cid:55)→ − − mollify the positive part of a catenoid with the negative part of z 2 e z2. Remark. The condition (4) prevents ω from pinching on the axis of rotation. If this condition Σ is violated, we expect that some solutions to (3) develop ﬁnite-time singularities. This case is treated in detailed in subsection 2.3. At such points we do not require that ω is smooth; that is, Σ we allow cones. Remark. The condition (6) prevents the solution from shrinking and sliding off to inﬁnity. This − would happen for a solution supported in the Σ generated by ω (z) = 1 + e z2. Clearly for such Σ solutions we can not expect convergence. ##LTLine##

text: VALENTINA-MIRA

text: WHEELER

text: ω

text: The

text: local

text: unique

text: existence

text: of

text: a

text: solution

text: in

text: this

text: setting

text: is

text: standard

text: and

text: has

text: been

text: discussed

text: in

text: detail

text: in

text: We

text: note

text: that

text: the

text: uniqueness

text: of

text: a

text: solution

text: shows

text: that

text: the

text: representation

text: of

text: a

text: solution

text: to

text: is

text: preserved.

text: Our

text: ﬁrst

text: main

text: result

text: is

text: the

text: following.

text: Theorem

text: (Long

text: time

text: existence).

text: Let

text: ω

text: and

text: ω

text: be

text: deﬁned

text: as

text: above.

text: Assume

text: and

text: that

text: Σ

text: there

text: is

text: no

text: point

text: z

text: where

text: ω

text: (z

text: Σ

text: We

text: further

text: assume

text: for

text: negative

text: and

text: positive

text: inﬁnity

text: that

text: either

text: one

text: of

text: the

text: limit

text: lim

text: ω

text: (z)

text: does

text: not

text: exist

text: Σ

text: z

text: or

text: dω

text: z

text: α,

text: if

text: α

text: there

text: exists

text: an

text: α

text: such

text: that

text: z

text: Σ

text: (z)

text: for

text: all

text: dz

text: z

text: α,

text: if

text: α

text: R

text: hold.

text: Then

text: there

text: exists

text: a

text: global

text: smooth

text: solution

text: ω

text: D(t)

text: T)

text: to

text: the

text: problem

text: that

text: R

text: converges

text: smoothly

text: to

text: ω∞

text: D(

text: The

text: function

text: ω∞

text: is

text: smooth

text: and

text: generates

text: a

text: minimal

text: surface.

text: Remark.

text: The

text: class

text: of

text: support

text: hypersurfaces

text: that

text: satisfy

text: above

text: at

text: both

text: positive

text: and

text: negative

text: inﬁnity

text: are

text: those

text: whose

text: derivative

text: is

text: monotone

text: outside

text: a

text: compact

text: subset

text: with

text: the

text: correct

text: sign.

text: Examples

text: of

text: such

text: include

text: the

text: catenoid.

text: The

text: catenoid

text: also

text: satisﬁes

text: condition

text: An

text: example

text: that

text: satisﬁes

text: but

text: not

text: is

text: ω

text: (z)

text: e

text: Σ

text: More

text: generally,

text: any

text: ω

text: whose

text: derivative

text: is

text: monotone

text: increasing

text: outside

text: a

text: compact

text: set

text: and

text: con-

text: Σ

text: verges

text: at

text: inﬁnity

text: satisﬁes

text: but

text: not

text: Examples

text: that

text: satisﬁes

text: but

text: not

text: include

text: the

text: unduloids.

text: Examples

text: that

text: satisfy

text: for

text: z

text: and

text: for

text: z

text: can

text: be

text: pasted

text: together

text: using

text: those

text: mentioned

text: above;

text: for

text: example,

text: mollify

text: the

text: positive

text: part

text: of

text: a

text: catenoid

text: with

text: the

text: negative

text: part

text: of

text: z

text: e

text: Remark.

text: The

text: condition

text: prevents

text: ω

text: from

text: pinching

text: on

text: the

text: axis

text: of

text: rotation.

text: If

text: this

text: condition

text: Σ

text: is

text: violated,

text: we

text: expect

text: that

text: some

text: solutions

text: to

text: develop

text: ﬁnite-time

text: singularities.

text: This

text: case

text: is

text: treated

text: in

text: detailed

text: in

text: subsection

text: At

text: such

text: points

text: we

text: do

text: not

text: require

text: that

text: ω

text: is

text: smooth;

text: that

text: is,

text: Σ

text: we

text: allow

text: cones.

text: Remark.

text: The

text: condition

text: prevents

text: the

text: solution

text: from

text: shrinking

text: and

text: sliding

text: off

text: to

text: inﬁnity.

text: This

text: would

text: happen

text: for

text: a

text: solution

text: supported

text: in

text: the

text: Σ

text: generated

text: by

text: ω

text: (z)

text: e

text: Clearly

text: for

text: such

text: Σ

text: solutions

text: we

text: can

text: not

text: expect

text: convergence.

text: ##LTLine##


text: MEAN CURVATURE FLOW IN EMBEDDED CYLINDERS 7 Case 2. Condition (5) is satisﬁed on an unbounded component U++ of U+. As no minimal disk exists on this component, the derivative dωΣ again has a sign. If the sign is positive, then the dz → ∞ Hopf lemma applies as in Case 1 above. If the sign is negative, then as z , the function ω Σ is uniformly bounded from below (by the no pinching condition (4)) and decreasing. Therefore it converges, violating (5). −− − If condition (5) is satisﬁed on an unbounded component U of U then the derivative is negative. If it were positive, then similarly as above this is in contradiction with (5). R Therefore in either case the evolving surfaces are contained within a compact region of n+1, and so the graph function ω is uniformly bounded. × → Thus we are left with obtaining gradient estimates for the evolving graphs ω : D(t) [0, T) R (cid:113) . (cid:104) (cid:105)− Let us set M := F(Dn, t). Following [15] we consider the quantity v = ν , e 1, t (cid:16) (cid:17) Mt n+1 which is modulo a tangential diffeomorphism equal to 1 + (dω)2. The function v satisﬁes dy d − ≤ ∆ v 0 dt Mt and this allows us to apply the maximum principle. Since the problem deals with evolving hyper- surfaces with boundary we have that the maximum of the gradient is controlled by the maximum between the ini(cid:12)tial v(cid:12)alues and(cid:26)the boun(cid:12)dary(cid:12) values.(cid:12)In the graphic(cid:12)al setting(cid:12), this tran(cid:12)s(cid:27)lates to the (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) following estim(cid:12)ate:(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) dω ≤ dω dω dω sup max sup , sup 0 (r(s), s) , sup (0, s) , dy dy ∈ dy ∈ dy (0,r(t)) (0,r(0)) s [0,t] s [0,t] ∈ for all t [0, T]. We now reﬁne this by considering maxima at the boundary. At the artiﬁcial boundary point (y = 0) the gradient function vanishes due to rotational symmetry; that is, dω (0, t) = 0 . dy Lemma 2.2 gives a uniform es(cid:12)tima(cid:12)te for the(cid:26)gradient o(cid:12)n the(cid:12)N(cid:114)eumann bo(cid:27)undary. We can therefore (cid:12) (cid:12) (cid:12) (cid:12) conclude that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) dω ≤ dω 1 − sup max sup 0 , 1 . dy dy C (0,r(t)) (0,r(0)) Σ × → R Having obtained a-priori uniform C1 estimates for ω : D(t) [0, T) , the quasilinear parabolic operator (3) may be considered to be linear with bounded coefﬁcients, and for such a (cid:90) (cid:90) problem global existence is standard. Convergence to minimal hypersurfaces is guaranteed by bounded initial area: We calculate d − | | (cid:90) (cid:90) dµ = (cid:90) H(cid:12)2dµ dt Dn Dn (cid:12)(cid:12) which implies (cid:12) ∞ ≤ H2dµ dt dµ = c . | | ≥ 0 Dn Dn t=0 Finally, we have D(t) inf ω > 0 by assumption, so that ω does not vanish (c.f. Theorem Σ 2.11). Since all derivatives are uniformly bounded, we may apply a compactness theorem to → conclude that M M∞ and that the mean curvature of M∞ is identically zero. This argument t (cid:3) has been used before by many authors, see for example [27, 4, 47]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MEAN

label: paragraph

text: CURVATURE

text: FLOW

text: IN

text: EMBEDDED

text: CYLINDERS

text: Case

text: Condition

text: is

text: satisﬁed

text: on

text: an

text: unbounded

text: component

text: U++

text: of

text: U+.

text: As

text: no

text: minimal

text: disk

text: exists

text: on

text: this

text: component,

text: the

text: derivative

text: dωΣ

text: again

text: has

text: a

text: sign.

text: If

text: the

text: sign

text: is

text: positive,

text: then

text: the

text: dz

text: Hopf

text: lemma

text: applies

text: as

text: in

text: Case

text: above.

text: If

text: the

text: sign

text: is

text: negative,

text: then

text: as

text: z

text: the

text: function

text: ω

text: Σ

text: is

text: uniformly

text: bounded

text: from

text: below

text: (by

text: the

text: no

text: pinching

text: condition

text: and

text: decreasing.

text: Therefore

text: it

text: converges,

text: violating

text: If

text: condition

text: is

text: satisﬁed

text: on

text: an

text: unbounded

text: component

text: U

text: of

text: U

text: then

text: the

text: derivative

text: is

text: negative.

text: If

text: it

text: were

text: positive,

text: then

text: similarly

text: as

text: above

text: this

text: is

text: in

text: contradiction

text: with

text: R

text: Therefore

text: in

text: either

text: case

text: the

text: evolving

text: surfaces

text: are

text: contained

text: within

text: a

text: compact

text: region

text: of

text: and

text: so

text: the

text: graph

text: function

text: ω

text: is

text: uniformly

text: bounded.

text: Thus

text: we

text: are

text: left

text: with

text: obtaining

text: gradient

text: estimates

text: for

text: the

text: evolving

text: graphs

text: ω

text: D(t)

text: T)

text: R

text: Let

text: us

text: set

text: M

text: F(Dn,

text: t).

text: Following

text: we

text: consider

text: the

text: quantity

text: v

text: ν

text: e

text: t

text: Mt

text: which

text: is

text: modulo

text: a

text: tangential

text: diffeomorphism

text: equal

text: to

text: The

text: function

text: v

text: satisﬁes

text: dy

text: d

text: v

text: dt

text: Mt

text: and

text: this

text: allows

text: us

text: to

text: apply

text: the

text: maximum

text: principle.

text: Since

text: the

text: problem

text: deals

text: with

text: evolving

text: hyper-

text: surfaces

text: with

text: boundary

text: we

text: have

text: that

text: the

text: maximum

text: of

text: the

text: gradient

text: is

text: controlled

text: by

text: the

text: maximum

text: between

text: the

text: the

text: this

text: to

text: the

text: following

text: dω

text: dω

text: dω

text: dω

text: sup

text: max

text: sup

text: sup

text: (r(s),

text: s)

text: sup

text: s)

text: dy

text: dy

text: dy

text: dy

text: s

text: s

text: for

text: all

text: t

text: T].

text: We

text: now

text: reﬁne

text: this

text: by

text: considering

text: maxima

text: at

text: the

text: boundary.

text: At

text: the

text: artiﬁcial

text: boundary

text: point

text: (y

text: the

text: gradient

text: function

text: vanishes

text: due

text: to

text: rotational

text: symmetry;

text: that

text: is,

text: dω

text: t)

text: dy

text: Lemma

text: gives

text: a

text: uniform

text: for

text: We

text: can

text: therefore

text: conclude

text: that

text: dω

text: dω

text: sup

text: max

text: sup

text: dy

text: dy

text: C

text: Σ

text: R

text: Having

text: obtained

text: a-priori

text: uniform

text: estimates

text: for

text: ω

text: D(t)

text: T)

text: the

text: quasilinear

text: parabolic

text: operator

text: may

text: be

text: considered

text: to

text: be

text: linear

text: with

text: bounded

text: coefﬁcients,

text: and

text: for

text: such

text: a

text: problem

text: global

text: existence

text: is

text: standard.

text: Convergence

text: to

text: minimal

text: hypersurfaces

text: is

text: guaranteed

text: by

text: bounded

text: initial

text: area:

text: We

text: calculate

text: d

text: dµ

text: dt

text: Dn

text: Dn

text: which

text: implies

text: dt

text: dµ

text: c

text: Dn

text: Dn

text: Finally,

text: we

text: have

text: D(t)

text: inf

text: ω

text: by

text: assumption,

text: so

text: that

text: ω

text: does

text: not

text: vanish

text: (c.f.

text: Theorem

text: Σ

text: Since

text: all

text: derivatives

text: are

text: uniformly

text: bounded,

text: we

text: may

text: apply

text: a

text: compactness

text: theorem

text: to

text: conclude

text: that

text: M

text: M∞

text: and

text: that

text: the

text: mean

text: curvature

text: of

text: M∞

text: is

text: identically

text: zero.

text: This

text: argument

text: t

text: has

text: been

text: used

text: before

text: by

text: many

text: authors,

text: see

text: for

text: example

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: M. E. Bell et al 5 added this in quadrature with the aegean Gaussian errors our observations. We calculated the 2D autocorrelation func- e2 and source ﬂux de(cid:113)nsity S as follows: tion of the dynamic spectrum which is shown in Figure 4 fit i (cid:0) (cid:1) (bottom row). To parameterize the autocorrelation function we ﬁtted a 1D Gaussian in the time and frequency axes, × ei = e2fit + Si M 2. (4) respectively. We followed the deﬁnition in Cordes (1986) whereby the half-width half-maximum in the frequency di- rection deﬁnes the scintillation bandwidth. We used the e is the adjusted error on an individual pulsar ﬂux density i half width at the 1/e point to calculate the scintillation measurement S . By bootstrapping the errors in this way we i timescale. From this analysis we ﬁnd a scintillation band- set the minimum variability that we are capable of detecting width of ∆ν = 4.1 MHz and a scintillation timescale of d to that of the neighbouring sources. These are the errors ∆τ = 28.8 minutes. We note that due to our broad band- d used in the variability statistics described in Section . width (30 MHz) we expect approximately a factor of three diﬀerence in scintillation bandwidth between to the top and the bottom of our band. For all the calculations above we use the central frequency for all scalings. 4 RESULTS We scaled the scintillation bandwidth and timescale re- ported by Phillips & Clegg (1992) under the assumption We consider a source to be statistically variable if ∝ ∝ that ∆ν ν4.4 and ∆τ ν1.2 (Cordes 1986). We ﬁnd χ2 > 2.0. We made a low cut on the minimum χ2 used d d r r that at 154 MHz the predicted scintillation timescale is 21.6 to deﬁne variability as we have been conservative with our minutes and the scintillation bandwidth is 4.5 MHz. The error propagation. Of the 17 pulsars four showed signiﬁcant predicted scintillation timescale is slightly shorter than our variability and we discuss these below. A summary of our result of 28.8 minutes. The predicted and measured scin- results is given in Table 2 and pulsar light-curves are shown tillation bandwidths are in good agreement. The amplitude in Figures 1, 2 and 3. of variability is extreme, but such cases have been reported 4.1 PSR J0953+0755 (B0950+08) before e.g. Galama et al. (1997). We detected signiﬁcant variability in PSR J0953+0755, We can calculate the expected timescale for refractive with a modulation index over all epochs of M = 131.3% and scintillation (τr) using the scintillation bandwidth (∆τd) and a χ2r = 182.1 (see Figure 1). On one of the nights of observing timescale (∆νd) via the following expression from Stinebring (2015-04-14), extreme variability was detected. For approxi- & Condon (1990): (cid:18) (cid:19) mately one hour the ﬂux density of the pulsar increased and p(oPfeai2lkk.e9ind5gatcotmnS−em3tapaxlc.=1a91n66d8.4)sphJiyans(paseeleroiwoTdadbiolsefpe20r).s2.i5oPnSsRm(HeJao0sb9ub5r3se+(eD0t7Ma5l5). τr = π4 ν∆∆ντdd ∝ ν−11/5. (5) 2004a). This pulsar is known to scintillate at low frequen- Using our diﬀractive scintillation parameters we ﬁnd τ = cies, for example, Phillips & Clegg (1992) report observa- r 20.9 hours at 154 MHz. This is consistent with Gupta et al. tions consistent with diﬀractive scintillation. (1993) who measure a refractive timescale for this pulsar of Due to the high signal-to-noise ratio of the pulsar detec- 3.4 days at 74 MHz (also see Cole et al. 1970). There also tion during this time, we were able to examine the frequency appears to be a timescale of several hundred days in the structure of the variability in the image plane. We took the Gupta et al. data, the origin of which is unclear but could data from the night of 2015-04-14 and for each observation × be inhomogeneties in the ISM. we imaged the data in 30 0.97 MHz sub-bands. For each of This pulsar was recently observed with the LWA at the sub-bands in each of the time slots we used the aegean 39.4 MHz and a number of giant pulses were detected that forced ﬁt algorithm (discussed above) to ﬁt the ﬂux den- had a signal-to-noise ratio greater than 10 times that of the sity at the location of the pulsar. The dynamic spectrum mean pulses (Tsai et al. 2015). These giant pulses where resulting from these measurements is plotted in Figure 4. however typically reported to be rare with approximately 5 We note that our observations are not continuous i.e. each per hour (or 0.035% of the total number of pulse periods). of the snapshot observations integrate for 112 seconds and Singal & Vats (2012) observed this pulsar at 103 MHz with then return four minutes later to that declination. Figure 4 only 1.6 MHz of bandwidth. Scaling our results to their fre- shows four distinct events with discrete time and frequency quency, the scintillation bandwidth should be 0.7 MHz and structure. The peak of the variability seen in some of the the scintillation timescale should be 18 minutes. At certain sub-bands was even higher than in the full-bandwidth data: epochs, Singal & Vats (2012) report very strong pulses over a peak ﬂux density of 48.6 Jy was observed at 142 MHz. the course of 30 minutes. Although they interpret this as The modulation index in both frequency and time for giant pulse emission, we consider it much more likely to be all measurements in Figure 4 (left) gives 85.6%. As discussed the eﬀects of scintillation. Giant pulses are largely broad- in Narayan (1992) we may expect a modulation index of up band in nature (Tsai et al. 2016), yet we see signiﬁcant fre- to 100% for diﬀractive strong scintillation. In some of the quency structure in our observations. We therefore conclude frequency and times bins shown in Figure 4 the pulsar is that the extreme variability observed in PSR J0953+0755 undetected. The result of these non-detections would be to is consistent with diﬀractive scintillation and is not intrinsic decrease the modulation index as the ﬂux density measure- to the pulsar. ments are only upper limits. − 4.2 PSR J0437 4715 We followed the method described in Cordes (1986) to calculate a scintillation bandwidth and timescale based on With a spin period of 5.76 ms and a DM of (cid:13) c 2002 RAS, MNRAS 000, 1–15 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: M.

text: E.

text: Bell

text: et

text: al

text: added

text: this

text: in

text: quadrature

text: with

text: the

text: aegean

text: Gaussian

text: errors

text: our

text: observations.

text: We

text: calculated

text: the

text: autocorrelation

text: func-

text: and

text: source

text: ﬂux

label: equation

text: S

text: as

text: follows:

text: tion

text: of

text: the

text: dynamic

text: spectrum

text: which

text: is

text: shown

text: in

text: Figure

text: fit

text: i

text: (bottom

text: row).

text: To

text: parameterize

text: the

text: autocorrelation

text: function

text: we

text: ﬁtted

text: a

text: Gaussian

text: in

text: the

text: time

text: and

text: frequency

text: axes,

text: ei

text: Si

text: M

text: respectively.

text: We

text: followed

text: the

text: deﬁnition

text: in

text: Cordes

text: whereby

text: the

text: half-width

text: half-maximum

text: in

text: the

text: frequency

text: di-

text: rection

text: deﬁnes

text: the

text: scintillation

text: bandwidth.

text: We

text: used

text: the

text: e

text: is

text: the

text: adjusted

text: error

text: on

text: an

text: individual

text: pulsar

text: ﬂux

text: density

text: i

text: half

text: width

text: at

text: the

text: point

text: to

text: calculate

text: the

text: scintillation

text: measurement

text: S

text: By

text: bootstrapping

text: the

text: errors

text: in

text: this

text: way

text: we

text: i

text: timescale.

text: From

text: this

text: analysis

text: we

text: ﬁnd

text: a

text: scintillation

text: band-

text: set

text: the

text: minimum

text: variability

text: that

text: we

text: are

text: capable

text: of

text: detecting

text: width

text: of

text: ∆ν

text: MHz

text: and

text: a

text: scintillation

text: timescale

text: of

text: d

text: to

text: that

text: of

text: the

text: neighbouring

text: sources.

text: These

text: are

text: the

text: errors

text: ∆τ

text: minutes.

text: We

text: note

text: that

text: due

text: to

text: our

text: broad

text: band-

text: d

text: used

text: in

text: the

text: variability

text: statistics

text: described

text: in

text: Section

text: width

text: MHz)

text: we

text: expect

text: approximately

text: a

text: factor

text: of

text: three

text: diﬀerence

text: in

text: scintillation

text: bandwidth

text: between

text: to

text: the

text: top

text: and

text: the

text: bottom

text: of

text: our

text: band.

text: For

text: all

text: the

text: calculations

text: above

text: we

text: use

text: the

text: central

text: frequency

text: for

text: all

text: scalings.

text: RESULTS

text: We

text: scaled

text: the

text: scintillation

text: bandwidth

text: and

text: timescale

text: re-

text: ported

text: by

text: Phillips

text: Clegg

text: under

text: the

text: assumption

text: We

text: consider

text: a

text: source

text: to

text: be

text: statistically

text: variable

text: if

text: that

text: ∆ν

text: and

text: ∆τ

text: (Cordes

text: We

text: ﬁnd

text: χ2

text: We

text: made

text: a

text: low

text: cut

text: on

text: the

text: minimum

text: χ2

text: used

text: d

text: d

text: r

text: r

text: that

text: at

text: MHz

text: the

text: predicted

text: scintillation

text: timescale

text: is

text: to

text: deﬁne

text: variability

text: as

text: we

text: have

text: been

text: conservative

text: with

text: our

text: minutes

text: and

text: the

text: scintillation

text: bandwidth

text: is

text: MHz.

text: The

text: error

text: propagation.

text: Of

text: the

text: pulsars

text: four

text: showed

text: signiﬁcant

text: predicted

text: scintillation

text: timescale

text: is

text: slightly

text: shorter

text: than

text: our

text: variability

text: and

text: we

text: discuss

text: these

text: below.

text: A

text: summary

text: of

text: our

text: result

text: of

text: minutes.

text: The

text: predicted

text: and

text: measured

text: scin-

text: results

text: is

text: given

text: in

text: Table

text: and

text: pulsar

text: light-curves

text: are

text: shown

text: tillation

text: bandwidths

text: are

text: in

text: good

text: agreement.

text: The

text: amplitude

text: in

text: Figures

text: and

text: of

text: variability

text: is

text: extreme,

text: but

text: such

text: cases

text: have

text: been

text: reported

text: PSR

text: before

text: e.g.

text: Galama

text: et

text: al.

text: We

text: detected

text: signiﬁcant

text: variability

text: in

text: PSR

text: We

text: can

text: calculate

text: the

text: expected

text: timescale

text: for

text: refractive

text: with

text: a

text: modulation

text: index

text: over

text: all

text: epochs

text: of

text: M

text: and

text: scintillation

text: (τr)

text: using

text: the

text: scintillation

text: bandwidth

text: (∆τd)

text: and

text: a

text: (see

text: Figure

text: On

text: one

text: of

text: the

text: nights

text: of

text: observing

text: timescale

text: (∆νd)

text: via

text: the

text: following

text: expression

text: from

text: Stinebring

text: extreme

text: variability

text: was

text: detected.

text: For

text: approxi-

text: Condon

text: mately

text: one

text: hour

text: the

text: ﬂux

text: density

text: of

text: the

text: pulsar

text: increased

text: and

text: τr

text: π4

text: ν∆∆ντdd

text: This

text: pulsar

text: is

text: known

text: to

text: scintillate

text: at

text: low

text: frequen-

text: Using

text: our

text: diﬀractive

text: scintillation

text: parameters

text: we

text: ﬁnd

text: τ

text: cies,

text: for

text: example,

text: Phillips

text: Clegg

text: report

text: observa-

text: r

text: hours

text: at

text: MHz.

text: This

text: is

text: consistent

text: with

text: Gupta

text: et

text: al.

text: tions

text: consistent

text: with

text: diﬀractive

text: scintillation.

text: who

text: measure

text: a

text: refractive

text: timescale

text: for

text: this

text: pulsar

text: of

text: Due

text: to

text: the

text: high

text: signal-to-noise

text: ratio

text: of

text: the

text: pulsar

text: detec-

text: days

text: at

text: MHz

text: (also

text: see

text: Cole

text: et

text: al.

text: There

text: also

text: tion

text: during

text: this

text: time,

text: we

text: were

text: able

text: to

text: examine

text: the

text: frequency

text: appears

text: to

text: be

text: a

text: timescale

text: of

text: several

text: hundred

text: days

text: in

text: the

text: structure

text: of

text: the

text: variability

text: in

text: the

text: image

text: plane.

text: We

text: took

text: the

text: Gupta

text: et

text: al.

text: data,

text: the

text: origin

text: of

text: which

text: is

text: unclear

text: but

text: could

text: data

text: from

text: the

text: night

text: of

text: and

text: for

text: each

text: observation

text: be

text: inhomogeneties

text: in

text: the

text: ISM.

text: we

text: imaged

text: the

text: data

text: in

text: MHz

text: sub-bands.

text: For

text: each

text: of

text: This

text: pulsar

text: was

text: recently

text: observed

text: with

text: the

text: LWA

text: at

text: the

text: sub-bands

text: in

text: each

text: of

text: the

text: time

text: slots

text: we

text: used

text: the

text: aegean

text: MHz

text: and

text: a

text: number

text: of

text: giant

text: pulses

text: were

text: detected

text: that

text: forced

text: ﬁt

text: algorithm

text: (discussed

text: above)

text: to

text: ﬁt

text: the

text: ﬂux

text: den-

text: had

text: a

text: signal-to-noise

text: ratio

text: greater

text: than

text: times

text: that

text: of

text: the

text: sity

text: at

text: the

text: location

text: of

text: the

text: pulsar.

text: The

text: dynamic

text: spectrum

text: mean

text: pulses

text: (Tsai

text: et

text: al.

text: These

text: giant

text: pulses

text: where

text: resulting

text: from

text: these

text: measurements

text: is

text: plotted

text: in

text: Figure

text: however

text: typically

text: reported

text: to

text: be

text: rare

text: with

text: approximately

text: We

text: note

text: that

text: our

text: observations

text: are

text: not

text: continuous

text: i.e.

text: each

text: per

text: hour

text: (or

text: of

text: the

text: total

text: number

text: of

text: pulse

text: periods).

text: of

text: the

text: snapshot

text: observations

text: integrate

text: for

text: seconds

text: and

text: Singal

text: Vats

text: observed

text: this

text: pulsar

text: at

text: MHz

text: with

text: then

text: return

text: four

text: minutes

text: later

text: to

text: that

text: declination.

text: Figure

text: only

text: MHz

text: of

text: bandwidth.

text: Scaling

text: our

text: results

text: to

text: their

text: fre-

text: shows

text: four

text: distinct

text: events

text: with

text: discrete

text: time

text: and

text: frequency

text: quency,

text: the

text: scintillation

text: bandwidth

text: should

text: be

text: MHz

text: and

text: structure.

text: The

text: peak

text: of

text: the

text: variability

text: seen

text: in

text: some

text: of

text: the

text: the

text: scintillation

text: timescale

text: should

text: be

text: minutes.

text: At

text: certain

text: sub-bands

text: was

text: even

text: higher

text: than

text: in

text: the

text: full-bandwidth

text: data:

text: epochs,

text: Singal

text: Vats

text: report

text: very

text: strong

text: pulses

text: over

text: a

text: peak

text: ﬂux

text: density

text: of

text: Jy

text: was

text: observed

text: at

text: MHz.

text: the

text: course

text: of

text: minutes.

text: Although

text: they

text: interpret

text: this

text: as

text: The

text: modulation

text: index

text: in

text: both

text: frequency

text: and

text: time

text: for

text: giant

text: pulse

text: emission,

text: we

text: consider

text: it

text: much

text: more

text: likely

text: to

text: be

text: all

text: measurements

text: in

text: Figure

text: (left)

text: gives

text: As

text: discussed

text: the

text: eﬀects

text: of

text: scintillation.

text: Giant

text: pulses

text: are

text: largely

text: broad-

text: in

text: Narayan

text: we

text: may

text: expect

text: a

text: modulation

text: index

text: of

text: up

text: band

text: in

text: nature

text: (Tsai

text: et

text: al.

text: yet

text: we

text: see

text: signiﬁcant

text: fre-

text: to

text: for

text: diﬀractive

text: strong

text: scintillation.

text: In

text: some

text: of

text: the

text: quency

text: structure

text: in

text: our

text: observations.

text: We

text: therefore

text: conclude

text: frequency

text: and

text: times

text: bins

text: shown

text: in

text: Figure

text: the

text: pulsar

text: is

text: that

text: the

text: extreme

text: variability

text: observed

text: in

text: PSR

text: undetected.

text: The

text: result

text: of

text: these

text: non-detections

text: would

text: be

text: to

text: is

text: consistent

text: with

text: diﬀractive

text: scintillation

text: and

text: is

text: not

text: intrinsic

text: decrease

text: the

text: modulation

text: index

text: as

text: the

text: ﬂux

text: density

text: measure-

text: to

text: the

text: pulsar.

text: ments

text: are

text: only

text: upper

text: limits.

text: PSR

text: We

text: followed

text: the

text: method

text: described

text: in

text: Cordes

text: to

text: calculate

text: a

text: scintillation

text: bandwidth

text: and

text: timescale

text: based

text: on

text: With

text: a

text: spin

text: period

text: of

text: ms

text: and

text: a

text: DM

text: of

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 3102 - 1002 sraey eht rof ailartsuA rof segaw laudividni laer rof scitsitats evitpircseD :1 elbaT 3102 2102 1102 0102 9002 8002 7002 6002 5002 4002 3002 2002 1002 5421 5121 8811 8211 5011 7601 387 329 478 918 667 437 486 naeM 0011 1011 1501 8401 3001 969 207 258 308 357 217 376 616 naideM 059 619 099 968 528 887 496 866 476 546 865 195 155 .ved .dtS 5.1 9.1 7.3 0.2 8.1 0.2 0.2 5.1 9.2 7.3 1.2 0.3 1.2 ssenweks 4.7 0.31 5.73 7.51 7.11 7.21 1.11 7.8 1.72 2.04 0.61 6.62 7.51 sisotruk ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: caption

text: sraey

text: eht

text: rof

text: ailartsuA

text: rof

text: segaw

text: laudividni

text: laer

text: rof

text: scitsitats

text: evitpircseD

text: elbaT

text: naeM

text: naideM

text: .ved

text: .dtS

text: ssenweks

text: sisotruk

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Foster, J., Joel, G., and Eric, T. (1984), “A class of decomposable poverty Measures,” Econometrica, 52, 761–765. Foster, J. E. (2009), A Class of Chronic Poverty Measures, Oxford: Oxford Univer- sity Press, poverty dynamics: interdisciplinary perspectives: 59-76 ed. Geisser, S. (1980), “Discussion of Sampling and Bayes Inference in Scientiﬁc Modeling and Robustness by G. E. P. Box,” Journal of the Royal Statistical Society Series A, 143, 416–417. Genest, C. and Neslehov´a, J. (2007), “A primer on copulas for count data,” Astin Bulletin, 37, 475–515. Good, I. J. (1952), “Rational Decisions,” Journal of the Royal Statistical Society B, 14, 107–114. He, J., Li, H., Edmondson, A. C., Rader, D. J., and Li, M. (2012), “A Gaussian copula approach for the analysis of secondary phenotypes in case–control genetic association studies,” Biostatistics, 13, 497–508. Hofert, M., Machler, M., and McNeil, A. J. (2012), “Likelihood Inference for Archi- median Copulas in High Dimensions under Known Margins,” Journal of Multi- variate Analysis, 110, 133–150. Hoﬀ, P. D. (2007), “Extending the rank likelihood for semiparametric copula esti- mation,” The Annals of Applied Statistics, 265–283. Jiryaie, F., Withanage, N., Wu, B., and de Leon, A. (2016), “Gaussian copula distri- butions for mixed data, with application in discrimination,” Journal of Statistical Computation and Simulation, 86, 1643–1659. Joe, H. (2014), Dependence modeling with copulas, CRC Press. Kleiber, C. (1996), “Dagum and Singh Maddala income distributions,” Economics Letters, 53, 265–268. 30

text: Foster,

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text: Joel,

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text: Eric,

text: T.

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text: Measures,”

text: Econometrica,

text: Foster,

text: J.

text: E.

text: A

text: Class

text: of

text: Chronic

text: Poverty

text: Measures,

text: Oxford:

text: Oxford

text: Univer-

text: sity

text: Press,

text: poverty

text: dynamics:

text: interdisciplinary

text: perspectives:

text: ed.

text: Geisser,

text: S.

text: “Discussion

text: of

text: Sampling

text: and

text: Bayes

text: Inference

text: in

text: Scientiﬁc

text: Modeling

text: and

text: Robustness

text: by

text: G.

text: E.

text: P.

text: Box,”

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society

text: Series

text: A,

text: Genest,

text: C.

text: and

text: Neslehov´a,

text: J.

text: “A

text: primer

text: on

text: copulas

text: for

text: count

text: data,”

text: Astin

text: Bulletin,

text: Good,

text: I.

text: J.

text: “Rational

text: Decisions,”

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society

text: B,

text: He,

text: J.,

text: Li,

text: H.,

text: Edmondson,

text: A.

text: C.,

text: Rader,

text: D.

text: J.,

text: and

text: Li,

text: M.

text: “A

text: Gaussian

text: copula

text: approach

text: for

text: the

text: analysis

text: of

text: secondary

text: phenotypes

text: in

text: case–control

text: genetic

text: association

text: studies,”

text: Biostatistics,

text: Hofert,

text: M.,

text: Machler,

text: M.,

text: and

text: McNeil,

text: A.

text: J.

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text: Inference

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text: median

text: Copulas

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text: High

text: Dimensions

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text: Known

text: Margins,”

text: Journal

text: of

text: Multi-

text: variate

text: Analysis,

text: Hoﬀ,

text: P.

text: D.

text: “Extending

text: the

text: rank

text: likelihood

text: for

text: semiparametric

text: copula

text: esti-

text: mation,”

text: The

text: Annals

text: of

text: Applied

text: Statistics,

text: Jiryaie,

text: F.,

text: Withanage,

text: N.,

text: Wu,

text: B.,

text: and

text: de

text: Leon,

text: A.

text: “Gaussian

text: copula

text: distri-

text: butions

text: for

text: mixed

text: data,

text: with

text: application

text: in

text: discrimination,”

text: Journal

text: of

text: Statistical

text: Computation

text: and

text: Simulation,

text: Joe,

text: H.

text: Dependence

text: modeling

text: with

text: copulas,

text: CRC

text: Press.

text: Kleiber,

text: C.

text: “Dagum

text: and

text: Singh

text: Maddala

text: income

text: distributions,”

text: Economics

text: Letters,


text: • ∼ − − − Sample V F = LS 1 (ψ 1), where LS 1 denotes the inverse Laplace-Stieljes − transform of ψ 1. – For the Clayton copula, F = Γ (1/θ, 1), where Γ (c, d) denotes the Gamma (cid:16) (cid:17) ∈ ∞(cid:0) (cid:0) (cid:1)(cid:1) ∈ ∞ distribution with shape parameter c (0, ), scale parameter d (0, ) – For the Gumbel copula, F = Stable 1/θ, 1, cos π θ , 1; 1 , 2θ where Stable (α , β , γ , δ ; 1) denotes the Stable distribution with ex- st st st st ∈ ∈ − ponent α (0, 2], skewness parameter β [ 1, 1], scale parameter st st ∈ ∞ ∈ R γ [0, ), and location parameter δ (Nolan, 2007). st st • (cid:16) ∼ (cid:17) Sample iid X U [0, 1] for j = 1, ..., m j • − Set U = ψ log(Xj) , for t = 1, ..., T j V S1.3 Conditional posterior of the Gaussian Copula Param- eters At each iteration of the MCMC sampling scheme, the correlation matrix Γ of (cid:8) (cid:9) the Gaussian copula is generated conditional on the transformed Gaussian copula ∗ ∗ variables y = y ; i = 1, ..., n; j = 1, ..., m . Danaher and Smith (2011) proposed ij the following representation of Γ, − − Γ := diag (Σ) 1/2 Σ diag (Σ) 1/2 , where Σ is a non-unique positive deﬁnite matrix and diag(Σ) is a diagonal matrix comprising the leading diagonal of Σ. The matrix Σ is further decomposed into (cid:48) Σ = R R, with R an upper triangular Cholesky factor. If we set the leading diagonal − of R to ones, this leaves m (m 1) /2 unknown elements of R, matching the number of unknown elements of Γ, thus identifying the representation. The upper triangular elements of R are unconstrained. The transformation described above ensures that S4 ##LTLine## ##LTLine##

label: list

text: Sample

text: V

text: F

text: LS

text: (ψ

text: where

text: LS

text: denotes

text: the

text: inverse

text: Laplace-Stieljes

text: transform

text: of

text: ψ

text: For

text: the

text: Clayton

text: copula,

text: F

text: Γ

text: where

text: Γ

text: (c,

text: d)

text: denotes

text: the

text: Gamma

text: distribution

text: with

text: shape

text: parameter

text: c

text: scale

text: parameter

text: d

text: For

text: the

text: Gumbel

text: copula,

text: F

text: Stable

text: cos

text: π

text: θ

text: 2θ

text: where

text: Stable

text: (α

text: β

text: γ

text: δ

text: denotes

text: the

text: Stable

text: distribution

text: with

text: ex-

text: st

text: st

text: st

text: st

text: ponent

text: α

text: skewness

text: parameter

text: β

text: scale

text: parameter

text: st

text: st

text: R

text: γ

text: and

text: location

text: parameter

text: δ

text: (Nolan,

text: st

text: st

text: Sample

text: iid

text: X

text: U

text: for

text: j

text: m

text: j

text: Set

text: U

text: ψ

text: log(Xj)

text: for

text: t

text: T

text: j

text: V

text: Conditional

text: posterior

text: of

text: the

text: Gaussian

text: Copula

text: Param-

text: eters

text: At

text: each

text: iteration

text: of

text: the

text: MCMC

text: sampling

text: scheme,

text: the

text: correlation

text: matrix

text: Γ

text: of

text: the

text: Gaussian

text: copula

text: is

text: generated

text: conditional

text: on

text: the

text: transformed

text: Gaussian

text: copula

text: variables

text: y

text: y

text: i

text: n;

text: j

text: m

text: Danaher

text: and

text: Smith

text: proposed

text: ij

text: the

text: following

text: representation

text: of

text: Γ,

text: Γ

text: diag

text: (Σ)

text: Σ

text: diag

text: (Σ)

text: where

text: Σ

text: is

text: a

text: non-unique

text: positive

text: deﬁnite

text: matrix

text: and

text: diag(Σ)

text: is

text: a

text: diagonal

text: matrix

text: comprising

text: the

text: leading

text: diagonal

text: of

text: Σ.

text: The

text: matrix

text: Σ

text: is

text: further

text: decomposed

text: into

text: Σ

text: R

text: R,

text: with

text: R

text: an

text: upper

text: triangular

text: Cholesky

text: factor.

text: If

text: we

text: set

text: the

text: leading

text: diagonal

text: of

text: R

text: to

text: ones,

text: this

text: leaves

text: m

text: (m

text: unknown

text: elements

text: of

text: R,

text: matching

text: the

text: number

text: of

text: unknown

text: elements

text: of

text: Γ,

text: thus

text: identifying

text: the

text: representation.

text: The

text: upper

text: triangular

text: elements

text: of

text: R

text: are

text: unconstrained.

text: The

text: transformation

text: described

text: above

text: ensures

text: that

text: ##LTLine##

text: ##LTLine##


text: ∈ {C D } C ∩ D ∅ all x Rm, (x), (x) partitions the index set so that (x) (x) = and C ∪ D M (x) (x) = . ∼ N As a ﬁrst example, consider X = (X , X ), where X (0, 1) and X is a 1 2 1 2 mixture of an exponential distribution with parameter λ and a point mass at 0 with ∼ − E C { } ∈ probability p, i.e., X pδ + (1 p) (λ)). Then, (x , 0) = 1 for all x R and 2 0 1 1 C { } ∈ D { } ∈ (x , x ) = 1, 2 for all x R, x > 0. Similarly (x , 0) = 2 for all x R and 1 2 1 2 1 1 D ∅ (x , x ) = . 1 2 ∼ N As a second example, let X = (X , X ), where X is Bernoulli and X (0, 1). 1 2 1 2 C { } ∈ { } × D { } Then (x) = 2 for all x 0, 1 R. Similarly (x) = 1 for all x. Let U = (U , . . . , U ) be a vector of uniform random variables whose distri- 1 m − bution is given by some copula C. We assume that F 1 is the quantile function j corresponding to F (since F is not invertible when X is not absolutely continuous, j j j this corresponds to picking one possible generalized inverse function). The variables U are selected to satisfy the following criteria. If, at coordinate x , j ∈ C j (x), then u = F (x ), resulting in a deterministic one-to-one relationship when j j j ∈ D − conditioning on either U or X . Otherwise, j (x), and we require x = F 1(u ), j j j j j resulting in an inﬁnity of U corresponding to one X and spanning the interval j j − C M (F (X ), F (X )). This interval corresponds to gaps in the range of F . If (x) = j j j j j for every x, then C will be the copula of X. Otherwise, the copula structure will still create dependence between the non-continuous marginal variables but will not be unique in general. Mathematically, the above description leads to the joint density (cid:89) (cid:89) I I − (cid:54) f(x, u) := c(u) (u = F (x )) (F (cid:48)(x (cid:48)) u (cid:48) < F (cid:48)(x (cid:48))), (1) j j j j j j j j ∈C (cid:48)∈D j (x) j (x) I where c is the copula density corresponding to C and is an indicator variable. See Lemma 4, part (i), of Appendix B for a derivation of (1) and the corresponding (cid:48) measure. Notice that in (1), products over the indices j and j correspond to diﬀerent 5

text: {C

text: D

text: C

text: D

text: all

text: x

text: Rm,

font: unknown

text: (x),

text: (x)

text: partitions

text: the

text: index

text: set

text: so

text: that

text: (x)

text: (x)

text: and

text: C

text: D

text: M

text: (x)

text: (x)

text: N

text: As

text: a

text: ﬁrst

text: example,

text: consider

text: X

text: (X

text: X

text: where

text: X

text: and

text: X

text: is

text: a

text: mixture

text: of

text: an

text: exponential

text: distribution

text: with

text: parameter

text: λ

text: and

text: a

text: point

text: mass

text: at

text: with

text: E

text: C

text: probability

text: p,

text: i.e.,

text: X

text: pδ

text: p)

text: (λ)).

text: Then,

text: (x

text: for

text: all

text: x

text: R

text: and

text: C

text: D

text: (x

text: x

text: for

text: all

text: x

text: R,

text: x

text: Similarly

text: (x

text: for

text: all

text: x

text: R

text: and

text: D

text: (x

text: x

text: N

text: As

text: a

text: second

text: example,

text: let

text: X

text: (X

text: X

text: where

text: X

text: is

text: Bernoulli

text: and

text: X

text: C

text: D

text: Then

text: (x)

text: for

text: all

text: x

text: R.

text: Similarly

text: (x)

text: for

text: all

text: x.

text: Let

text: U

text: (U

text: U

text: be

text: a

text: vector

text: of

text: uniform

text: random

text: variables

text: whose

text: distri-

text: m

text: bution

text: is

text: given

text: by

text: some

text: copula

text: C.

text: We

text: assume

text: that

text: F

text: is

text: the

text: quantile

text: function

text: j

text: corresponding

text: to

text: F

text: (since

text: F

text: is

text: not

text: invertible

text: when

text: X

text: is

text: not

text: absolutely

text: continuous,

text: j

text: j

text: j

text: this

text: corresponds

text: to

text: picking

text: one

text: possible

text: generalized

text: inverse

text: function).

text: The

text: variables

text: U

text: are

text: selected

text: to

text: satisfy

text: the

text: following

text: criteria.

text: If,

text: at

text: coordinate

text: x

text: j

text: C

text: j

text: (x),

text: then

text: u

text: F

text: (x

text: resulting

text: in

text: a

text: deterministic

text: one-to-one

text: relationship

text: when

text: j

text: j

text: j

text: D

text: conditioning

text: on

text: either

text: U

text: or

text: X

text: Otherwise,

text: j

text: (x),

text: and

text: we

text: require

text: x

text: F

text: j

text: j

text: j

text: j

text: j

text: resulting

text: in

text: an

text: inﬁnity

text: of

text: U

text: corresponding

text: to

text: one

text: X

text: and

text: spanning

text: the

text: interval

text: j

text: j

text: C

text: M

text: (F

text: (X

text: F

text: (X

text: This

text: interval

text: corresponds

text: to

text: gaps

text: in

text: the

text: range

text: of

text: F

text: If

text: (x)

text: j

text: j

text: j

text: j

text: j

text: for

text: every

text: x,

text: then

text: C

text: will

text: be

text: the

text: copula

text: of

text: X.

text: Otherwise,

text: the

text: copula

text: structure

text: will

text: still

text: create

text: dependence

text: between

text: the

text: non-continuous

text: marginal

text: variables

text: but

text: will

text: not

text: be

text: unique

text: in

text: general.

text: Mathematically,

text: the

text: above

text: description

text: leads

text: to

text: the

text: joint

text: density

text: I

text: I

text: f(x,

text: u)

text: c(u)

text: (u

text: F

text: (x

text: (F

text: u

text: F

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: ∈C

text: j

text: (x)

text: j

text: (x)

text: I

text: where

text: c

text: is

text: the

text: copula

text: density

text: corresponding

text: to

text: C

text: and

text: is

text: an

text: indicator

text: variable.

text: See

text: Lemma

text: part

text: (i),

text: of

text: Appendix

text: B

text: for

text: a

text: derivation

text: of

text: and

text: the

text: corresponding

text: measure.

text: Notice

text: that

text: in

text: products

text: over

text: the

text: indices

text: j

text: and

text: j

text: correspond

text: to

text: diﬀerent


text: the correlation matrix Γ remains a positive deﬁnite matrix, regardless of the values of R. The following steps generate each element of R: 1. Generate the r ∗ element of the matrix R using a random-walk Metropolis j j ∗ ∗ step for j = 1, ..., m and j = 2, ..., m with j < j. (cid:48) 2. Compute Σ = R R − − 3. Compute the correlation matrix Γ = diag (Σ) 1/2 Σdiag (Σ) 1/2 | { \ } ∗ To explain step 1 in more detail, the conditional posterior r ∗ R r ∗ , y , y is j j j j given by (cid:26) (cid:27) | { \ } ∗ ∝ (cid:89) | ∗ ∗| (cid:0) (cid:1) p (r ∗ R r ∗ , y , y) p (y y ) p (y R) p (r ∗ ) j j j j j j ∝ n | |− −1 ∗(cid:48) − − ∗ Γ n exp y Γ 1 I y p (r ∗ ) , 2 2 i i j j i=1 ∝ { \ } with p (r ∗ ) 1 for all elements of R. Here, A B means A with the param- j j ∗ eters B omitted. First, we generate a new proposal value, r ∗ , from a candidate j j density N (r ∗ , σ), where r ∗ is the previous iterate value and σ is the pre-speciﬁed j j j j standard deviation of a normal distribution speciﬁed to obtain a reasonable accep- ∗ tance rate of 0.3-0.4. The new(cid:40)value(cid:0)r ∗ is accepted with (cid:1)p(cid:41)robability j j ∗ | { \ } ∗ p r ∗ R r ∗ , y , y α = min 1, j j| { \ j j} ∗ . p (r ∗ R r ∗ , y , y) j j j j We draw a random variable u from U (0, 1); if u < α, then the new value of r ∗ j j is accepted, otherwise the old value of r ∗ is retained. This algorithm is used to j j generate all of the upper triangular elements of R, one at a time. S5 ##LTLine## ##LTLine## ##LTLine##

text: the

text: correlation

text: matrix

text: Γ

text: remains

text: a

text: positive

text: deﬁnite

text: matrix,

text: regardless

text: of

text: the

text: values

text: of

text: R.

text: The

text: following

text: steps

text: generate

text: each

text: element

text: of

text: R:

text: Generate

text: the

text: r

text: element

text: of

text: the

text: matrix

text: R

text: using

text: a

text: random-walk

text: Metropolis

text: j

text: j

text: step

text: for

text: j

text: m

text: and

text: j

text: m

text: with

text: j

text: j.

text: Compute

text: Σ

text: R

text: R

text: Compute

text: the

text: correlation

text: matrix

text: Γ

text: diag

text: (Σ)

text: Σdiag

text: (Σ)

text: To

text: explain

text: step

text: in

text: more

text: detail,

text: the

text: conditional

text: posterior

text: r

text: R

text: r

text: y

text: y

text: is

text: j

text: j

text: j

text: j

text: given

text: by

text: p

text: (r

text: R

text: r

text: y

text: y)

text: p

text: (y

text: y

text: p

text: (y

text: R)

text: p

text: (r

text: j

text: j

text: j

text: j

text: j

text: j

text: n

text: −1

text: Γ

text: n

text: exp

text: y

text: Γ

text: I

text: y

text: p

text: (r

text: i

text: i

text: j

text: j

text: with

text: p

text: (r

text: for

text: all

text: elements

text: of

text: R.

text: Here,

text: A

text: B

text: means

text: A

text: with

text: the

text: param-

text: j

text: j

text: eters

text: B

text: omitted.

text: First,

text: we

text: generate

text: a

text: new

text: proposal

text: value,

text: r

text: from

text: a

text: candidate

text: j

text: j

text: density

text: N

text: (r

text: σ),

text: where

text: r

text: is

text: the

text: previous

text: iterate

text: value

text: and

text: σ

text: is

text: the

text: pre-speciﬁed

text: j

text: j

text: j

text: j

text: standard

text: deviation

text: of

text: a

text: normal

text: distribution

text: speciﬁed

text: to

text: obtain

text: a

text: reasonable

text: accep-

text: tance

text: rate

text: of

text: The

text: is

text: accepted

text: with

text: j

text: j

text: p

text: r

text: R

text: r

text: y

text: y

text: α

text: min

text: j

text: j|

text: j

text: j}

text: p

text: (r

text: R

text: r

text: y

text: y)

text: j

text: j

text: j

text: j

text: We

text: draw

text: a

text: random

text: variable

text: u

text: from

text: U

text: if

text: u

text: α,

text: then

text: the

text: new

text: value

text: of

text: r

text: j

text: j

text: is

text: accepted,

text: otherwise

text: the

text: old

text: value

text: of

text: r

text: is

text: retained.

text: This

text: algorithm

text: is

text: used

text: to

text: j

text: j

text: generate

text: all

text: of

text: the

text: upper

text: triangular

text: elements

text: of

text: R,

text: one

text: at

text: a

text: time.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Finally, let X be a binary random variable. 3  This results in the following        { } (cid:54)∈ { } (cid:54)∈ { } 1, 2 if x 0, 1 and x 0 1 2  { } (cid:54)∈ { } ∈ { } C  1 x 0, 1 and x 0  1 2 (x , x , x ) =  1 2 3  { } ∈ { } (cid:54)∈ { } 2 x 0, 1 and x 0 1 2 ∅ ∈ { } ∈ { } x 0, 1 and x 0 1 2 D { }\C { } ⊂ D and (x , x , x ) = 1, 2, 3 (x , x , x ). Notice that 3 (x , x , x ) always 1 2 3 1 2 3 1 2 3 holds in this example. The marginal density of X (Eq. 2 in the paper) is C { } 1. Case 1: (x , x , x ) = 1, 2 . 1 2 3 (cid:52) ·| f(x) = c{ }(b{ })f (x )f (x ) b3 C | ( b{ }) 1,2 1,2 1 1 2 2 a3 3 1,2 1,2 | − | = c{ }(b{ })f (x )f (x )(C | (b b{ }) C | (a b{ })) 1,2 1,2 1 1 2 2 3 1,2 3 1,2 3 1,2 3 1,2 C { } 2. Case 2: (x , x , x ) = 1 1 2 3 (cid:52) ·| b{ } f(x) = c (b )f (x ) 2,3 C | ( b ) 1 1 1 1 a{2,3} 2,3 1 1 × = f (x ) 1 1 | − | − | | [C | (b , b b ) C | (b , a b ) C | (a , b b ) + C | (a , a b )] 2,3 1 2 3 1 2,3 1 2 3 1 2,3 1 2 3 1 2,3 1 2 3 1 where the second line follows from c (b ) = 1 (as all one-dimensional marginals 1 1 are uniform). S7

text: Finally,

text: let

text: X

text: be

text: a

text: binary

text: random

text: variable.

text: This

text: results

text: in

text: the

text: following

text: if

text: x

text: and

text: x

text: C

text: x

text: and

text: x

text: (x

text: x

text: x

text: x

text: and

text: x

text: x

text: and

text: x

text: D

text: }\C

text: D

text: and

text: (x

text: x

text: x

text: (x

text: x

text: x

text: Notice

text: that

text: (x

text: x

text: x

text: always

text: holds

text: in

text: this

text: example.

text: The

text: marginal

text: density

text: of

text: X

text: (Eq.

text: in

text: the

text: paper)

text: is

text: C

text: Case

text: (x

text: x

text: x

text: f(x)

text: c{

text: }(b{

text: })f

text: (x

text: )f

text: (x

text: C

text: b{

text: c{

text: }(b{

text: })f

text: (x

text: )f

text: (x

text: )(C

text: (b

text: b{

text: C

text: (a

text: b{

text: C

text: Case

text: (x

text: x

text: x

text: b{

text: f(x)

text: c

text: (b

text: )f

text: (x

text: C

text: b

text: f

text: (x

text: [C

text: (b

text: b

text: b

text: C

text: (b

text: a

text: b

text: C

text: (a

text: b

text: b

text: C

text: (a

text: a

text: b

text: where

text: the

text: second

text: line

text: follows

text: from

text: c

text: (b

text: (as

text: all

text: one-dimensional

text: marginals

text: are

text: uniform).