text: 7 (cid:82) (cid:82) (cid:48) (cid:48) g(t) order of g and (g )2 contribution, we found that even (a) 0.5 | | ≈ − the maximum of dωP(i)(t, ω)/ dωP(0)(t, ω) 10 7, which is negligible, for the entire t in our calculation. We conclude that it suﬃces to discuss the photo-current ob- t/τ0 tained from the zeroth-order G<,(0) alone. In this regard the adiabatic method we developed to obtain the two- time Green’s function for the time-dependent LF Hamil- (b) α=0 (c) α=1 tonian is already exact at the zeroth order. Figure 1 shows the photo-current P(t, ω) at several times t throughout the adiabatic turn-oﬀ of the coupling g(t). Several sidebands, present at times long before the adiabatic turn-oﬀ process began, have their frequencies ∼ shifted by g2ω = 0.25ω as g diminishes to zero. Their 0 0 intensities diminish over time. The main peak at the en- ergy ε¯(t) also slides in frequency by g2ω = 0.25ω with 0 0 its intensity growing over time. Even for α = 0 case, − we can see that the sidebands emerge at ω = ε¯ nω 0 (n > 0). These sidebands for α = 0 are coming from the terms proportional to the g2 in Eq. (4.6). It is notable that we obtain the diminishing sidebands feature even without manifestly introducing the dissipa- tion mechanism such as the bosonic bath, explicitly [17]. In an adiabatic evolution of the quantum system such as the expanding potential well, the instantaneous energy of the system smoothly follows the ground state value of the instantaneous Hamiltonian. As the wall expands the energy also diminishes, but this is done without an explicit dissipation mechanism. The same phenomenon is happening in our Green’s function treatment of the adiabatic evolution. A. Transient behavior in the semi-classical limit The boson ﬁeld is treated as a quantized oscillator in our approach to transient dynamics. In this subsec- FIG. 1. Proﬁle of the coupling g(t) over time. Frequency tion, we ask what happens if the boson ﬁeld is treated resolved photo-current intensity P(t, ω) at various times for semi-classically, and the relevant Hilbert space is that the coherent state amplitudes (b) α = 0 and (c) α = 1. Fully of fermions only. The semi-classical limit of the time- deve−loped sideband feature at ω = ε¯+ (integer)ω0 shown at dependent LF Hamilt→onian−is ob−tained by going to the t = 30τ0 −start to slide ov(cid:48)er to the higher frequency for times interaction picture, a ae iω0(t t0), past t = 15τ where g (t) also starts to vary. The bare (cid:82) (cid:82) 0 fermion energy peak vanishes at t = 5τ . (cid:16) (cid:17) 0 HˆLF(t) = ei tt0 dt1HbHLF(t)e−i tt0 dt1Hb − ω0a†a † † − − † − | | (cid:46) = εc c + g(t)ω c c ae iω0(t t0) + a eiω0(t t0) , is fulﬁlled at all times t T . The observation time t 0 − g extends from 45τ up to 45τ in our calculation. The (4.10) 0 0 initial preparation time t is set further back at t = − 0 0 (cid:104) (cid:105) 130τ0. For the parameters of the model we choose g = and then replacing a by its average a = α, assuming 0.5 and ε = ω0, which gives the renormalized energy a coherent state of the boson: ε¯ = 0.75ω . The lesser Green’s function can be obtained 0 numerically for various choices of the coherent state α. † − † The phase angle in α can be absorbed since it always Hcl.(t) = εc c + 2g(t)ω α cos[ω (t t )]c c. (4.11) LF 0 0 0 appears as the product αeiω0t0 in the Green’s function [see Eq. (4.1)]. From the calculations, it turns out the two higher- The lesser Green’s function for the semi-classical, time- (cid:48) (cid:48) order Green’s functions in orders of g and (g )2 make dependent LF model is still(cid:2)of the form, (cid:3) negligible contributions to the photo-current P(t, ω) for tthenesigt(yt)Pc(hi)o(ste,nω)inwiEthq.su(p4e.8r)s.criFpotrit=he{1p,h2o}todceunrorteenst tihne- G<(t, t(cid:48)) = iTr ρ(t0)c†(t(cid:48))c(t) . (4.12) ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: g(t)

font: default

text: order

text: of

text: g

text: and

text: (g

text: contribution,

text: we

text: found

text: that

text: even

text: (a)

text: the

text: maximum

text: of

text: dωP(i)(t,

text: ω)/

text: ω)

text: which

text: is

text: negligible,

text: for

text: the

text: entire

text: t

text: in

text: our

text: calculation.

text: We

text: conclude

text: that

text: it

text: suﬃces

text: to

text: discuss

text: the

text: photo-current

text: ob-

text: tained

text: from

text: the

text: zeroth-order

text: alone.

text: In

text: this

text: regard

text: the

text: adiabatic

text: method

text: we

text: developed

text: to

text: obtain

text: the

text: two-

text: time

text: Green’s

text: function

text: for

text: the

text: time-dependent

text: LF

text: Hamil-

text: (b)

text: (c)

text: tonian

text: is

text: already

text: exact

text: at

text: the

text: zeroth

text: order.

text: Figure

text: shows

text: the

text: photo-current

text: P(t,

text: ω)

text: at

text: several

text: times

text: t

text: throughout

text: the

text: adiabatic

text: turn-oﬀ

text: of

text: the

text: coupling

text: g(t).

text: Several

text: sidebands,

text: present

text: at

text: times

text: long

text: before

text: the

text: adiabatic

text: turn-oﬀ

text: process

text: began,

text: have

text: their

text: frequencies

text: shifted

text: by

text: as

text: g

text: diminishes

text: to

text: zero.

text: Their

text: intensities

text: diminish

text: over

text: time.

text: The

text: main

text: peak

text: at

text: the

text: en-

text: ergy

text: ε¯(t)

text: also

text: slides

text: in

text: frequency

text: by

text: with

text: its

text: intensity

text: growing

text: over

text: time.

text: Even

text: for

text: α

text: case,

text: we

text: can

text: see

text: that

text: the

text: sidebands

text: emerge

text: at

text: ω

text: ε¯

text: nω

text: (n

text: These

text: sidebands

text: for

text: α

text: are

text: coming

text: from

text: the

text: terms

text: proportional

text: to

text: the

text: in

text: Eq.

text: It

text: is

text: notable

text: that

text: we

text: obtain

text: the

text: diminishing

text: sidebands

text: feature

text: even

text: without

text: manifestly

text: introducing

text: the

text: dissipa-

text: tion

text: mechanism

text: such

text: as

text: the

text: bosonic

text: bath,

text: explicitly

text: In

text: an

text: adiabatic

text: evolution

text: of

text: the

text: quantum

text: system

text: such

text: as

text: the

text: expanding

text: potential

text: well,

text: the

text: instantaneous

text: energy

text: of

text: the

text: system

text: smoothly

text: follows

text: the

text: ground

text: state

text: value

text: of

text: the

text: instantaneous

text: Hamiltonian.

text: As

text: the

text: wall

text: expands

text: the

text: energy

text: also

text: diminishes,

text: but

text: this

text: is

text: done

text: without

text: an

text: explicit

text: dissipation

text: mechanism.

text: The

text: same

text: phenomenon

text: is

text: happening

text: in

text: our

text: Green’s

text: function

text: treatment

text: of

text: the

text: adiabatic

text: evolution.

text: A.

text: Transient

text: behavior

text: in

text: the

text: semi-classical

text: limit

text: The

text: boson

text: ﬁeld

text: is

text: treated

text: as

text: a

text: quantized

text: oscillator

text: in

text: our

text: approach

text: to

text: transient

text: dynamics.

text: In

text: this

text: subsec-

text: FIG.

text: Proﬁle

text: of

text: the

text: coupling

text: g(t)

text: over

text: time.

text: Frequency

text: tion,

text: we

text: ask

text: what

text: happens

text: if

text: the

text: boson

text: ﬁeld

text: is

text: treated

text: resolved

text: photo-current

text: intensity

text: P(t,

text: ω)

text: at

text: various

text: times

text: for

text: semi-classically,

text: and

text: the

text: relevant

text: Hilbert

text: space

text: is

text: that

text: the

text: coherent

text: state

text: amplitudes

text: (b)

text: α

text: and

text: (c)

text: α

text: Fully

text: of

text: fermions

text: only.

text: The

text: semi-classical

text: limit

text: of

text: the

text: time-

text: deve−loped

text: sideband

text: feature

text: at

text: ω

text: ε¯+

text: shown

text: at

text: dependent

text: LF

text: Hamilt→onian−is

text: ob−tained

text: by

text: going

text: to

text: the

text: t

text: −start

text: to

text: slide

text: to

text: the

text: higher

text: frequency

text: for

text: times

text: interaction

text: picture,

text: a

text: ae

text: past

text: t

text: where

text: g

text: (t)

text: also

text: starts

text: to

text: vary.

text: The

text: bare

text: fermion

text: energy

text: peak

text: vanishes

text: at

text: t

text: 5τ

text: HˆLF(t)

text: ei

text: εc

text: c

text: g(t)ω

text: c

text: c

text: ae

text: a

text: is

text: fulﬁlled

text: at

text: all

text: times

text: t

text: T

text: The

text: observation

text: time

text: t

text: g

text: extends

text: from

text: up

text: to

text: in

text: our

text: calculation.

text: The

text: initial

text: preparation

text: time

text: t

text: is

text: set

text: further

text: back

text: at

text: t

text: For

text: the

text: parameters

text: of

text: the

text: model

text: we

text: choose

text: g

text: and

text: then

text: replacing

text: a

text: by

text: its

text: average

text: a

text: α,

text: assuming

text: and

text: ε

text: which

text: gives

text: the

text: renormalized

text: energy

text: a

text: coherent

text: state

text: of

text: the

text: boson:

text: ε¯

text: The

text: lesser

text: Green’s

text: function

text: can

text: be

text: obtained

text: numerically

text: for

text: various

text: choices

text: of

text: the

text: coherent

text: state

text: α.

text: The

text: phase

text: angle

text: in

text: α

text: can

text: be

text: absorbed

text: since

text: it

text: always

text: Hcl.(t)

text: εc

text: c

text: α

text: cos[ω

text: (t

text: t

text: )]c

text: c.

text: LF

text: appears

text: as

text: the

text: product

text: in

text: the

text: Green’s

text: function

text: [see

text: Eq.

text: From

text: the

text: calculations,

text: it

text: turns

text: out

text: the

text: two

text: higher-

text: The

text: lesser

text: Green’s

text: function

text: for

text: the

text: semi-classical,

text: time-

text: order

text: Green’s

text: functions

text: in

text: orders

text: of

text: g

text: and

text: (g

text: make

text: dependent

text: LF

text: model

text: is

text: the

text: form,

text: negligible

text: contributions

text: to

text: the

text: photo-current

text: P(t,

text: ω)

text: for

text: tihne-

text: G<(t,

text: iTr

text: ##LTFigure##

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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: a latent Conditional Auto-Regressive process that can reﬂect diﬀerent dependency patterns. We assess the performance of our model within a simulation environment and then we apply it to ChIP-seq real data. KEYWORDS: Heterogeneity; Model-based clustering; Spatial de- pendency; Multiple allocations. 1 Introduction In the last 15 years, the development of parallel massively sequencing platforms for mapping the genome has completely revolutionized the way of studying gene expression patterns. These recent technologies called Next Generation Sequencing (NGS) allow to simultaneously investigate thousands of features within a single reliable and cost-eﬀective experiment, thus repre- senting a valid alternative to microarray experiments in enhancing our under- standing of how genetic diﬀerences aﬀect health and disease (Nagalakshmi and others, 2008). Indeed, these innovative platforms have been quickly ap- plied to many genomic contexts giving rise to a large amount of available data in complex form. Data coming from such experiments are highly structured and their analysis has raised an imperative need for speciﬁc methodologies: technical or biological replicates can be observed in several experimental con- ditions (Bao and others, 2014), and the single observational units, such as genes or exons, are very likely characterized by spatial dependencies and re- lationships (Mo, 2012). Moreover, the abundance of a particular transcript is measured as a count and so the single data point is a discrete measurement. We consider in this paper the Chromatin ImmunoPrecipitation and se- quencing (ChIP-Seq) data on proteins p300 and CBP analysed by Ramos and others (2010). In this experiment, two technical replicates are observed after 30 minutes from the initial interaction of the proteins with the human genome, with the aim of discovering the binding proﬁle of these transcription factors. Figure 1 displays summarized counts for 1000 base pairs contiguous windows along chromosome 21. The plot shows segments with high read counts and segments where there is a uniformly low level of signal, thus suggesting a potential spatial eﬀect. In ChIP-Seq data like the one investigated (see Kuan and others (2011) for a review), researchers usually associate the counts to two speciﬁc compo- nents: a background level, which accounts for the noise in the process and 2

text: a

label: abstract

text: latent

text: Conditional

text: Auto-Regressive

text: process

text: that

text: can

text: reﬂect

text: diﬀerent

text: dependency

text: patterns.

text: We

text: assess

text: the

text: performance

text: of

text: our

text: model

text: within

text: a

text: simulation

text: environment

text: and

text: then

text: we

text: apply

text: it

text: to

text: ChIP-seq

text: real

text: data.

text: KEYWORDS:

text: Heterogeneity;

text: Model-based

text: clustering;

text: Spatial

text: de-

text: pendency;

text: Multiple

text: allocations.

label: section

text: Introduction

text: In

label: paragraph

text: the

text: last

text: years,

text: the

text: development

text: of

text: parallel

text: massively

text: sequencing

text: platforms

text: for

text: mapping

text: the

text: genome

text: has

text: completely

text: revolutionized

text: the

text: way

text: of

text: studying

text: gene

text: expression

text: patterns.

text: These

text: recent

text: technologies

text: called

text: Next

text: Generation

text: Sequencing

text: (NGS)

text: allow

text: to

text: simultaneously

text: investigate

text: thousands

text: of

text: features

text: within

text: a

text: single

text: reliable

text: and

text: cost-eﬀective

text: experiment,

text: thus

text: repre-

text: senting

text: a

text: valid

text: alternative

text: to

text: microarray

text: experiments

text: in

text: enhancing

text: our

text: under-

text: standing

text: of

text: how

text: genetic

text: diﬀerences

text: aﬀect

text: health

text: and

text: disease

text: (Nagalakshmi

text: and

text: others,

text: Indeed,

text: these

text: innovative

text: platforms

text: have

text: been

text: quickly

text: ap-

text: plied

text: to

text: many

text: genomic

text: contexts

text: giving

text: rise

text: to

text: a

text: large

text: amount

text: of

text: available

text: data

text: in

text: complex

text: form.

text: Data

text: coming

text: from

text: such

text: experiments

text: are

text: highly

text: structured

text: and

text: their

text: analysis

text: has

text: raised

text: an

text: imperative

text: need

text: for

text: speciﬁc

text: methodologies:

text: technical

text: or

text: biological

text: replicates

text: can

text: be

text: observed

text: in

text: several

text: experimental

text: con-

text: ditions

text: (Bao

text: and

text: others,

text: and

text: the

text: single

text: observational

text: units,

text: such

text: as

text: genes

text: or

text: exons,

text: are

text: very

text: likely

text: characterized

text: by

text: spatial

text: dependencies

text: and

text: re-

text: lationships

text: (Mo,

text: Moreover,

text: the

text: abundance

text: of

text: a

text: particular

text: transcript

text: is

text: measured

text: as

text: a

text: count

text: and

text: so

text: the

text: single

text: data

text: point

text: is

text: a

text: discrete

text: measurement.

text: We

text: consider

text: in

text: this

text: paper

text: the

text: Chromatin

text: ImmunoPrecipitation

text: and

text: se-

text: quencing

text: (ChIP-Seq)

text: data

text: on

text: proteins

text: and

text: CBP

text: analysed

text: by

text: Ramos

text: and

text: others

text: In

text: this

text: experiment,

text: two

text: technical

text: replicates

text: are

text: observed

text: after

text: minutes

text: from

text: the

text: initial

text: interaction

text: of

text: the

text: proteins

text: with

text: the

text: human

text: genome,

text: with

text: the

text: aim

text: of

text: discovering

text: the

text: binding

text: proﬁle

text: of

text: these

text: transcription

text: factors.

text: Figure

text: displays

text: summarized

text: counts

text: for

text: base

text: pairs

text: contiguous

text: windows

text: along

text: chromosome

text: The

text: plot

text: shows

text: segments

text: with

text: high

text: read

text: counts

text: and

text: segments

text: where

text: there

text: is

text: a

text: uniformly

text: low

text: level

text: of

text: signal,

text: thus

text: suggesting

text: a

text: potential

text: spatial

text: eﬀect.

text: In

text: ChIP-Seq

text: data

text: like

text: the

text: one

text: investigated

text: (see

text: Kuan

text: and

text: others

text: for

text: a

text: review),

text: researchers

text: usually

text: associate

text: the

text: counts

text: to

text: two

text: speciﬁc

text: compo-

text: nents:

text: a

text: background

text: level,

text: which

text: accounts

text: for

text: the

text: noise

text: in

text: the

text: process

text: and


text: ∼ π Beta (1, 1) i ∗ | ∗ ∗ P (z = 1 π ) = π (9) jh | ∗ ∼ NhB y z ; Θ (ψ(u , µ ); φ). jd jh h d For the other two parameters we select conjugate and ﬂat priors. More ∼ ∼ precisely, for every i and h, µ Gamma(a , b ) and φ Unif(a , b ), id µ µ hd φ φ where (a , b , a , b ) are elements of the vector ξ of hyperparameters. Given µ µ φ φ these chosen priors, full conditionals can be derived and a Gibbs sampling MCMC algorithm applied to estimate the parameters. At the implementa- tion step of the algorithm we exploit the Gamma-Poisson mixture represen- tation of a Negative Binomial distribution. We introduce a further latent variabile s, speciﬁc to each unit j in each replicate/condition d, that has a Gamma density with shape and rate parameters equal to φ . It follows that: hd (cid:82) f(s ) = Gamma(φ , φ ) jd hd hd | p(y s ) = Pois(ψ(u , µ )s ) jd | jd Nh Bd jd p(y ) = p(y s )f(s )ds = (ψ(u ; µ ), φ ) jd jd jd jd h d hd which allows us to use a Gibbs sampler for the mean parameters µ . i 2.4 Modelling spatial correlation with a CAR struc- ture The units/observations (in our case, genes or genomic locations) are not independent but spatially correlated. We introduce the spatial correlation by allowing the primary weights π to be j-varying and we denote them as i π . The spatial relationship is taken into account by allowing the weights ij of the mixture in Eq. 4 to vary from one gene to another. The way we for- mulate it is inspired by the work proposed by Ferna´ndez and Green (2002). They ﬁrst introduced k independent p-dimensional latent variables with a Markov random-ﬁeld distribution. The weights are then a non-linear function of these latent variables and spatial relationships are expressed in terms of neighborhood relationships. This could be restrictive because the correlation between two genes should decrease as their distance increases. We extend 11

label: equation

text: π

text: Beta

text: i

text: P

text: (z

text: π

text: π

text: jh

text: NhB

text: y

text: z

text: Θ

text: (ψ(u

text: µ

text: φ).

text: jd

text: jh

text: h

text: d

text: For

text: the

text: other

text: two

text: parameters

text: we

text: select

text: conjugate

text: and

text: ﬂat

text: priors.

text: More

text: precisely,

text: for

text: every

text: i

text: and

text: h,

text: µ

text: Gamma(a

text: b

text: and

text: φ

text: Unif(a

text: b

text: id

text: µ

text: µ

text: hd

text: φ

text: φ

text: where

text: (a

text: b

text: a

text: b

text: are

text: elements

text: of

text: the

text: vector

text: ξ

text: of

text: hyperparameters.

text: Given

text: µ

text: µ

text: φ

text: φ

text: these

text: chosen

text: priors,

text: full

text: conditionals

text: can

text: be

text: derived

text: and

text: a

text: Gibbs

text: sampling

text: MCMC

text: algorithm

text: applied

text: to

text: estimate

text: the

text: parameters.

text: At

text: the

text: implementa-

text: tion

text: step

text: of

text: the

text: algorithm

text: we

text: exploit

text: the

text: Gamma-Poisson

text: mixture

text: represen-

text: tation

text: of

text: a

text: Negative

text: Binomial

text: distribution.

text: We

text: introduce

text: a

text: further

text: latent

text: variabile

text: s,

text: speciﬁc

text: to

text: each

text: unit

text: j

text: in

text: each

text: replicate/condition

text: d,

text: that

text: has

text: a

text: Gamma

text: density

text: with

text: shape

text: and

text: rate

text: parameters

text: equal

text: to

text: φ

text: It

text: follows

text: that:

text: hd

text: f(s

text: Gamma(φ

text: φ

text: jd

text: hd

text: hd

text: p(y

text: s

text: Pois(ψ(u

text: µ

text: )s

text: jd

text: jd

text: Nh

text: Bd

text: jd

text: p(y

text: p(y

text: s

text: )f(s

text: )ds

text: (ψ(u

text: µ

text: φ

text: jd

text: jd

text: jd

text: jd

text: h

text: d

text: hd

text: which

text: allows

text: us

text: to

text: use

text: a

text: Gibbs

text: sampler

text: for

text: the

text: mean

text: parameters

text: µ

text: i

text: Modelling

text: spatial

text: correlation

text: with

text: a

text: CAR

text: struc-

text: ture

text: The

text: units/observations

text: (in

text: our

text: case,

text: genes

text: or

text: genomic

text: locations)

text: are

text: not

text: independent

text: but

text: spatially

text: correlated.

text: We

text: introduce

text: the

text: spatial

text: correlation

text: by

text: allowing

text: the

text: primary

text: weights

text: π

text: to

text: be

text: j-varying

text: and

text: we

text: denote

text: them

text: as

text: i

text: π

text: The

text: spatial

text: relationship

text: is

text: taken

text: into

text: account

text: by

text: allowing

text: the

text: weights

text: ij

text: of

text: the

text: mixture

text: in

text: Eq.

text: to

text: vary

text: from

text: one

text: gene

text: to

text: another.

text: The

text: way

text: we

text: for-

text: mulate

text: it

text: is

text: inspired

text: by

text: the

text: work

text: proposed

text: by

text: Ferna´ndez

text: and

text: Green

text: They

text: ﬁrst

text: introduced

text: k

text: independent

text: p-dimensional

text: latent

text: variables

text: with

text: a

text: Markov

text: random-ﬁeld

text: distribution.

text: The

text: weights

text: are

text: then

text: a

text: non-linear

text: function

text: of

text: these

text: latent

text: variables

text: and

text: spatial

text: relationships

text: are

text: expressed

text: in

text: terms

text: of

text: neighborhood

text: relationships.

text: This

text: could

text: be

text: restrictive

text: because

text: the

text: correlation

text: between

text: two

text: genes

text: should

text: decrease

text: as

text: their

text: distance

text: increases.

text: We

text: extend


text: the approach by considering a Gaussian conditional auto-regressive model (Pettitt and others, 2002), where the distances are directly used to model correlations instead of dummies denoting the neighborhood condition. In a Bayesian framework, this is accomplished by introducing additional layers to the hierarchy formulation shown in Eq. 9, with their own set of hyper- parameters. With reference to π, we introduce the spatial latent vectors, denoted by x , . . . , x , . . . , x , with i = 1, . . . , k: each x is a Gaussian con- 1 i k i ditional autoregressive model given by N − f(x ) = (0, Q 1) (10) i where Q is a precision (cid:26)matrix(cid:80)of order p and γ (cid:48) is some non-linear func- jj tion of δ (cid:48), which are the distances between all the units. More speciﬁcally, jj − 1 + p(cid:48) γ (cid:48) if j-th diagonal element Q = I + ∆ Γ = − j =1 jj p γ (cid:48) elsewhere jj After some algebraic steps it is possible to show that Eq. 10 is equivalent (cid:40) (cid:34) (cid:35)(cid:41) to (cid:88) (cid:88) (cid:88) p p p · −1 − f(x ) = c exp γ (cid:48)(x x (cid:48))2 + x2 (11) i 2 jj ij ij ij (cid:48) j=1 j =1 j=1 (cid:89) with c the normalization constant p − c = (2π) p/2 (1 + v )1/2. j j=1 In the previous expression, v ’s denote the eigenvalues of the spatial matrix − j ∆ Γ. Given x , . . . , x , the weights for location j can be obtained via 1 k logistic formulation exp(x /η) ij π = ij 1 + exp(x /η) ij where η is a ‘shrink-or-stretch’ tuning parameter to be estimated that pro- vides a way to exaggerate the diﬀerences in units allocation among the clus- ters. 2.5 Conditional Auto-Regressive Multiple Allocation Mixture (CAR-MAM ) 12 ##LTLine## ##LTLine##

text: the

text: approach

text: by

text: considering

text: a

text: Gaussian

text: conditional

text: auto-regressive

text: model

text: (Pettitt

text: and

text: others,

text: where

text: the

text: distances

text: are

text: directly

text: used

text: to

text: model

text: correlations

text: instead

text: of

text: dummies

text: denoting

text: the

text: neighborhood

text: condition.

text: In

text: a

text: Bayesian

text: framework,

text: this

text: is

text: accomplished

text: by

text: introducing

text: additional

text: layers

text: to

text: the

text: hierarchy

text: formulation

text: shown

text: in

text: Eq.

text: with

text: their

text: own

text: set

text: of

text: hyper-

text: parameters.

text: With

text: reference

text: to

text: π,

text: we

text: introduce

text: the

text: spatial

text: latent

text: vectors,

text: denoted

text: by

text: x

text: x

text: x

text: with

text: i

text: k:

text: each

text: x

text: is

text: a

text: Gaussian

text: con-

text: i

text: k

text: i

text: ditional

text: autoregressive

text: model

text: given

text: by

text: N

text: f(x

text: Q

text: i

text: where

text: Q

text: is

text: a

text: precision

text: order

text: p

text: and

text: γ

text: is

text: some

text: non-linear

text: func-

text: jj

text: tion

text: of

text: δ

text: which

text: are

text: the

text: distances

text: between

text: all

text: the

text: units.

text: More

text: speciﬁcally,

text: jj

text: γ

text: if

text: j-th

text: diagonal

text: element

text: Q

text: I

text: Γ

text: j

text: jj

text: p

text: γ

text: elsewhere

text: jj

text: After

text: some

text: algebraic

text: steps

text: it

text: is

text: possible

text: to

text: show

text: that

text: Eq.

text: is

text: equivalent

text: to

text: p

text: p

text: p

text: −1

text: f(x

text: c

text: exp

text: γ

text: x

text: i

text: jj

text: ij

text: ij

text: ij

text: j

text: with

text: c

text: the

text: normalization

text: constant

text: p

text: c

text: v

text: j

text: In

text: the

text: previous

text: expression,

text: v

text: ’s

text: denote

text: the

text: eigenvalues

text: of

text: the

text: spatial

text: matrix

text: j

text: Γ.

text: Given

text: x

text: x

text: the

text: weights

text: for

text: location

text: j

text: can

text: be

text: obtained

text: via

text: k

text: logistic

text: formulation

text: exp(x

text: /η)

text: ij

text: π

text: ij

text: exp(x

text: /η)

text: ij

text: where

text: η

text: is

text: a

text: ‘shrink-or-stretch’

text: tuning

text: parameter

text: to

text: be

text: estimated

text: that

text: pro-

text: vides

text: a

text: way

text: to

text: exaggerate

text: the

text: diﬀerences

text: in

text: units

text: allocation

text: among

text: the

text: clus-

text: ters.

text: Conditional

text: Auto-Regressive

text: Multiple

text: Allocation

text: Mixture

text: (CAR-MAM

text: ##LTLine##

text: ##LTLine##


text: Table 1: Misclassiﬁcation error rate (in percentage) for NegBinMix and MAM for the three scenarios of activation. Degree of activation Number of clusters Model π = 0.25 π = 0.50 π = 0.75 i i i k = 2 MAM 1.05 2.70 2.65 ∗ (i.e. k = 4) NegBinMix 1.20 2.65 2.55 k = 3 MAM 0.95 3.05 5.10 ∗ (i.e. k = 8) NegBinMix 11.00 25.05 9.15 { } ∗ with a number of groups k = 2, 3 : in the augmented k space this equals to represent a situation, as in a classical model-based clustering framework, ∗ ∗ where the actual number of groups ranges from k = 4 to k = 8. We ex- plore three degrees of clustering between primary and outward/non-primary groups by selecting three scenarios which we call low, medium and high ac- tivation: this is achieved by setting all the π equal to - respectively - 0.25, 0.50 and 0.75. We run both algorithms (MAM and NegBinMix) for 10000 MCMC iterations and a 5000 burn-in window is selected. For every µ , we id choose hyperparameters of the Gamma prior distributions equal to a = 1 µ and b = 0.001; for every dispersion parameter φ , we select the ranges of µ hd the prior uniform distributions to be [a = 100, b = 2000]. Convergence φ φ is checked for every chain and we assign to the clusters according to the maximum a posteriori rule: we compute the posterior probabilities of the ∗ allocation vectors z given the data and, for every unit, we allocate to the j component with the highest probability value (see also Section ). We choose as an overall performance indicator the misclassiﬁcation error rate (see Ta- ble 4), that is, the average number of units not correctly allocated when compared to the known true membership. The posterior means for the pa- rameters in the selected models (both the overlapping and non-overlapping mixtures) are consistent with the true values and do not show any substantial bias. We report the misclassiﬁcation error rates for the estimated models in Table 4: the percentage of misclassiﬁed units is always lower in the low activation scenario because most of the observations are allocated in the outward component, which has small variance and near zero (ﬁxed) mean, thus simplifying the clustering task. As we can see in Table 4, our model has comparable (sometimes better) classiﬁcation rates, with respect to the 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Misclassiﬁcation

label: caption

text: error

text: rate

text: (in

text: percentage)

text: for

text: NegBinMix

text: and

text: MAM

text: for

text: the

text: three

text: scenarios

text: of

text: activation.

text: Degree

label: table

text: of

text: activation

text: Number

text: of

text: clusters

text: Model

text: π

text: π

text: π

text: i

text: i

text: i

text: k

text: MAM

text: (i.e.

text: k

text: NegBinMix

text: k

text: MAM

text: (i.e.

text: k

text: NegBinMix

text: with

text: a

text: number

text: of

text: groups

text: k

text: in

text: the

text: augmented

text: k

text: space

text: this

text: equals

text: to

text: represent

text: a

text: situation,

text: as

text: in

text: a

text: classical

text: model-based

text: clustering

text: framework,

text: where

text: the

text: actual

text: number

text: of

text: groups

text: ranges

text: from

text: k

text: to

text: k

text: We

text: ex-

text: plore

text: three

text: degrees

text: of

text: clustering

text: between

text: primary

text: and

text: outward/non-primary

text: groups

text: by

text: selecting

text: three

text: scenarios

text: which

text: we

text: call

text: low,

text: medium

text: and

text: high

text: ac-

text: tivation:

text: this

text: is

text: achieved

text: by

text: setting

text: all

text: the

text: π

text: equal

text: to

text: respectively

text: and

text: We

text: run

text: both

text: algorithms

text: (MAM

text: and

text: NegBinMix)

text: for

text: MCMC

text: iterations

text: and

text: a

text: burn-in

text: window

text: is

text: selected.

text: For

text: every

text: µ

text: we

text: id

text: choose

text: hyperparameters

text: of

text: the

text: Gamma

text: prior

text: distributions

text: equal

text: to

text: a

text: µ

text: and

text: b

text: for

text: every

text: dispersion

text: parameter

text: φ

text: we

text: select

text: the

text: ranges

text: of

text: µ

text: hd

text: the

text: prior

text: uniform

text: distributions

text: to

text: be

text: [a

text: b

text: Convergence

text: φ

text: φ

text: is

text: checked

text: for

text: every

text: chain

text: and

text: we

text: assign

text: to

text: the

text: clusters

text: according

text: to

text: the

text: maximum

text: a

text: posteriori

text: rule:

text: we

text: compute

text: the

text: posterior

text: probabilities

text: of

text: the

text: allocation

text: vectors

text: z

text: given

text: the

text: data

text: and,

text: for

text: every

text: unit,

text: we

text: allocate

text: to

text: the

text: j

text: component

text: with

text: the

text: highest

text: probability

text: value

text: (see

text: also

text: Section

text: We

text: choose

text: as

text: an

text: overall

text: performance

text: indicator

text: the

text: misclassiﬁcation

text: error

text: rate

text: (see

text: Ta-

text: ble

text: that

text: is,

text: the

text: average

text: number

text: of

text: units

text: not

text: correctly

text: allocated

text: when

text: compared

text: to

text: the

text: known

text: true

text: membership.

text: The

text: posterior

text: means

text: for

text: the

text: pa-

text: rameters

text: in

text: the

text: selected

text: models

text: (both

text: the

text: overlapping

text: and

text: non-overlapping

text: mixtures)

text: are

text: consistent

text: with

text: the

text: true

text: values

text: and

text: do

text: not

text: show

text: any

text: substantial

text: bias.

text: We

text: report

text: the

text: misclassiﬁcation

text: error

text: rates

text: for

text: the

text: estimated

text: models

text: in

text: Table

text: the

text: percentage

text: of

text: misclassiﬁed

text: units

text: is

text: always

text: lower

text: in

text: the

text: low

text: activation

text: scenario

text: because

text: most

text: of

text: the

text: observations

text: are

text: allocated

text: in

text: the

text: outward

text: component,

text: which

text: has

text: small

text: variance

text: and

text: near

text: zero

text: (ﬁxed)

text: mean,

text: thus

text: simplifying

text: the

text: clustering

text: task.

text: As

text: we

text: can

text: see

text: in

text: Table

text: our

text: model

text: has

text: comparable

text: (sometimes

text: better)

text: classiﬁcation

text: rates,

text: with

text: respect

text: to

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: to the protein binding process. Since in ChIP-Seq data researchers usually associate the counts to a background group and a signal group, we run the ∗ algorithm with k = 2 (k = 4) in order to capture the two aforementioned expected clusters and potentially a better characterization of them through our multiple allocation cluster, while simultaneously taking into account the spatial dependency between the genomic regions. We choose the geometric · mean as our combination scheme ψ( ) to lessen the eﬀect (on the multi- ple allocation cluster mean values) produced by the highest counts. Out of the 4000 genomic regions, 159 are allocated in the outward cluster, which accounts for the zero-inﬂation in the data, represented by the observations around positions 17200 and 18000 (see Figure 2). This group has ﬁxed mean values equal to 0.01 for both replicates T01 and T02, while the dispersion parameters are estimated by the algorithm. The ﬁrst primary cluster i = 1, whose units are indicated by green dots in Figure 2, has posterior means equal to µ = (1.61, 1.07) and represents the background process of the pro- 1 tein binding: 3478 possibly inactive genomic regions, with very low counts, are allocated in this group. The second primary cluster, i = 2, is representa- tive of a signaling group of 48 genomic regions having a higher mean count level of µ = (19.27, 34.23) depicted with red dots in Figure 2. The multi- 2 ple allocation cluster in our analysis is interpretable as a group of 315 units involved in both the background and signal clusters: in this case, given that we are observing these counts after 30 minutes from the initial interaction of the protein with the strand of chromosome 21, the genomic regions allocated in this cluster could be thought as either being locations that were active immediately at the beginning and now not signaling anymore or locations only starting to interact with the p300 protein after 30 minutes. The mean values for this multiple allocation cluster are equal to 5.57 and 6.06 in the two replicates T01 and T02: the units belonging to it are shown as blue dots in Figure 2. Finally, the posterior probability for each unit to be allocated in the signaling group is shown in Figure 2 as a magenta solid line. We can see from the plot that this probability is higher in those segments where genomic regions with higher counts are observed; moreover, the allocation weight of the signal component follows a spatial pattern, increasing and decreasing across the analyzed strand and thus accounting for the spatial eﬀect occur- ring among the observations. For comparison, we also run the algorithm for a conventional four components Negative Binomial mixture model, ﬁxing the location parameter of the ﬁrst component to 0.01 as we do in our model.The result shows that only a total of three clusters are estimated, with one of 17

text: to

text: the

text: protein

text: binding

text: process.

text: Since

text: in

text: ChIP-Seq

text: data

text: researchers

text: usually

text: associate

text: the

text: counts

text: to

text: a

text: background

text: group

text: and

text: a

text: signal

text: group,

text: we

text: run

text: the

text: algorithm

text: with

text: k

text: (k

text: in

text: order

text: to

text: capture

text: the

text: two

text: aforementioned

text: expected

text: clusters

text: and

text: potentially

text: a

text: better

text: characterization

text: of

text: them

text: through

text: our

text: multiple

text: allocation

text: cluster,

text: while

text: simultaneously

text: taking

text: into

text: account

text: the

text: spatial

text: dependency

text: between

text: the

text: genomic

text: regions.

text: We

text: choose

text: the

text: geometric

text: mean

text: as

text: our

text: combination

text: scheme

text: ψ(

text: to

text: lessen

text: the

text: eﬀect

text: (on

text: the

text: multi-

text: ple

text: allocation

text: cluster

text: mean

text: values)

text: produced

text: by

text: the

text: highest

text: counts.

text: Out

text: of

text: the

text: genomic

text: regions,

text: are

text: allocated

text: in

text: the

text: outward

text: cluster,

text: which

text: accounts

text: for

text: the

text: zero-inﬂation

text: in

text: the

text: data,

text: represented

text: by

text: the

text: observations

text: around

text: positions

text: and

text: (see

text: Figure

text: This

text: group

text: has

text: ﬁxed

text: mean

text: values

text: equal

text: to

text: for

text: both

text: replicates

text: and

text: while

text: the

text: dispersion

text: parameters

text: are

text: estimated

text: by

text: the

text: algorithm.

text: The

text: ﬁrst

text: primary

text: cluster

text: i

text: whose

text: units

text: are

text: indicated

text: by

text: green

text: dots

text: in

text: Figure

text: has

text: posterior

text: means

text: equal

text: to

text: µ

text: and

text: represents

text: the

text: background

text: process

text: of

text: the

text: pro-

text: tein

text: binding:

text: possibly

text: inactive

text: genomic

text: regions,

text: with

text: very

text: low

text: counts,

text: are

text: allocated

text: in

text: this

text: group.

text: The

text: second

text: primary

text: cluster,

text: i

text: is

text: representa-

text: tive

text: of

text: a

text: signaling

text: group

text: of

text: genomic

text: regions

text: having

text: a

text: higher

text: mean

text: count

text: level

text: of

text: µ

text: depicted

text: with

text: red

text: dots

text: in

text: Figure

text: The

text: multi-

text: ple

text: allocation

text: cluster

text: in

text: our

text: analysis

text: is

text: interpretable

text: as

text: a

text: group

text: of

text: units

text: involved

text: in

text: both

text: the

text: background

text: and

text: signal

text: clusters:

text: in

text: this

text: case,

text: given

text: that

text: we

text: are

text: observing

text: these

text: counts

text: after

text: minutes

text: from

text: the

text: initial

text: interaction

text: of

text: the

text: protein

text: with

text: the

text: strand

text: of

text: chromosome

text: the

text: genomic

text: regions

text: allocated

text: in

text: this

text: cluster

text: could

text: be

text: thought

text: as

text: either

text: being

text: locations

text: that

text: were

text: active

text: immediately

text: at

text: the

text: beginning

text: and

text: now

text: not

text: signaling

text: anymore

text: or

text: locations

text: only

text: starting

text: to

text: interact

text: with

text: the

text: protein

text: after

text: minutes.

text: The

text: mean

text: values

text: for

text: this

text: multiple

text: allocation

text: cluster

text: are

text: equal

text: to

text: and

text: in

text: the

text: two

text: replicates

text: and

text: the

text: units

text: belonging

text: to

text: it

text: are

text: shown

text: as

text: blue

text: dots

text: in

text: Figure

text: Finally,

text: the

text: posterior

text: probability

text: for

text: each

text: unit

text: to

text: be

text: allocated

text: in

text: the

text: signaling

text: group

text: is

text: shown

text: in

text: Figure

text: as

text: a

text: magenta

text: solid

text: line.

text: We

text: can

text: see

text: from

text: the

text: plot

text: that

text: this

text: probability

text: is

text: higher

text: in

text: those

text: segments

text: where

text: genomic

text: regions

text: with

text: higher

text: counts

text: are

text: observed;

text: moreover,

text: the

text: allocation

text: weight

text: of

text: the

text: signal

text: component

text: follows

text: a

text: spatial

text: pattern,

text: increasing

text: and

text: decreasing

text: across

text: the

text: analyzed

text: strand

text: and

text: thus

text: accounting

text: for

text: the

text: spatial

text: eﬀect

text: occur-

text: ring

text: among

text: the

text: observations.

text: For

text: comparison,

text: we

text: also

text: run

text: the

text: algorithm

text: for

text: a

text: conventional

text: four

text: components

text: Negative

text: Binomial

text: mixture

text: model,

text: ﬁxing

text: the

text: location

text: parameter

text: of

text: the

text: ﬁrst

text: component

text: to

text: as

text: we

text: do

text: in

text: our

text: model.The

text: result

text: shows

text: that

text: only

text: a

text: total

text: of

text: three

text: clusters

text: are

text: estimated,

text: with

text: one

text: of


text: ACM SIGKDD international conference on Knowledge discovery in data mining, ACM, 532–537. Battle, A. and Segal, E. and Koller, D. (2005). Probabilistic Dis- covery of Overlapping Cellular Processes and their regulation. Journal of Computational Biology, 12, 909–927. Banfield, J. P. and Raftery, A. E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803–821. Bao, Y., Vinciotti, V., Wit, E. and A. C. ’t Hoen, P. (2014). Joint modeling of ChIP-seq data via a Markov random ﬁeld model. Biostatistics, 15(2), 296–310. Bezdek, James C. (1981). Pattern Recognition with Fuzzy Objective Func- tion Algorithms, Kluwer Academic Publishers, Norwell, MA, USA. Ferna´ndez, C. and Green, P.J. (2002). Modelling spatially correlated data via mixtures: a Bayesian approach. Journal of the Royal Statistical Society: Series B, 64, 805–826. Fraley, C. and Raftery, A. E. (2002). Model-based clustering, discrim- inant analysis and density estimation. Journal of the American Statistical Association, 97, 611–631. Fu, Q. and Banerjee, A. (2008). Multiplicative Mixture Models for Over- lapping Clustering. Data Mining, 2008. ICDM’08. Eighth IEEE Interna- tional Conference on. IEEE, 791–796. Green, P. J. (1995). Reversible jump Markov Chain, Monte Carlo compu- tation and Bayesian model determination. Biometrika, 82, 711-732. Heller, A.K. and Ghahramani, Z. (2007). A Nonparametric Bayesian Approach to Modeling Overlapping Clusters. International Conference on Artiﬁcial Intelligence and Statistics, 187–194. Heller, A.K. and Wiliamson, S. and Ghahramani, Z. (2008). Statis- tical Models for Partial Membership. Proceedings of the 25th international conference on Machine learning, ACM, 392–399. 20

text: ACM

label: reference

text: SIGKDD

text: international

text: conference

text: on

text: Knowledge

text: discovery

text: in

text: data

text: mining,

text: ACM,

text: Battle,

text: A.

text: and

text: Segal,

text: E.

text: and

text: Koller,

text: D.

text: Probabilistic

text: Dis-

text: covery

text: of

text: Overlapping

text: Cellular

text: Processes

text: and

text: their

text: regulation.

text: Journal

text: of

text: Computational

text: Biology,

text: Banfield,

text: J.

text: P.

text: and

text: Raftery,

text: A.

text: E.

text: Model-based

text: Gaussian

text: and

text: non-Gaussian

text: clustering.

text: Biometrics,

text: Bao,

text: Y.,

text: Vinciotti,

text: V.,

text: Wit,

text: E.

text: and

text: A.

text: C.

text: ’t

text: Hoen,

text: P.

text: Joint

text: modeling

text: of

text: ChIP-seq

text: data

text: via

text: a

text: Markov

text: random

text: ﬁeld

text: model.

text: Biostatistics,

text: Bezdek,

text: James

text: C.

text: Pattern

text: Recognition

text: with

text: Fuzzy

text: Objective

text: Func-

text: tion

text: Algorithms,

text: Kluwer

text: Academic

text: Publishers,

text: Norwell,

text: MA,

text: USA.

text: Ferna´ndez,

text: C.

text: and

text: Green,

text: P.J.

text: Modelling

text: spatially

text: correlated

text: data

text: via

text: mixtures:

text: a

text: Bayesian

text: approach.

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society:

text: Series

text: B,

text: Fraley,

text: C.

text: and

text: Raftery,

text: A.

text: E.

text: Model-based

text: clustering,

text: discrim-

text: inant

text: analysis

text: and

text: density

text: estimation.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Fu,

text: Q.

text: and

text: Banerjee,

text: A.

text: Multiplicative

text: Mixture

text: Models

text: for

text: Over-

text: lapping

text: Clustering.

text: Data

text: Mining,

text: Eighth

text: IEEE

text: Interna-

text: tional

text: Conference

text: on.

text: IEEE,

text: Green,

text: P.

text: J.

text: Reversible

text: jump

text: Markov

text: Chain,

text: Monte

text: Carlo

text: compu-

text: tation

text: and

text: Bayesian

text: model

text: determination.

text: Biometrika,

text: Heller,

text: A.K.

text: and

text: Ghahramani,

text: Z.

text: A

text: Nonparametric

text: Bayesian

text: Approach

text: to

text: Modeling

text: Overlapping

text: Clusters.

text: International

text: Conference

text: on

text: Artiﬁcial

text: Intelligence

text: and

text: Statistics,

text: Heller,

text: A.K.

text: and

text: Wiliamson,

text: S.

text: and

text: Ghahramani,

text: Z.

text: Statis-

text: tical

text: Models

text: for

text: Partial

text: Membership.

text: Proceedings

text: of

text: the

text: international

text: conference

text: on

text: Machine

text: learning,

text: ACM,


text: or multiple allocation) group (cid:89) ∗ k − − π = πuhi(1 π )1 uhi (3) h i i (cid:80) i=1 ∗ with h = 1, . . . , k . Notice that these weights are constructed so that ∗ ∗ k π = 1, and the deﬁnition comes naturally from the representation of h=1 h clusters as sets in a sample space. The Multiple Allocation Mixture model (MAM) can be deﬁned as a reparameterization of Eq. 1 in the new formula- tion (cid:88) ∗ | k ∗ | f(y Θ) = π f(y ψ(u , θ), φ ) (4) j h j h h h=1 where ψ(u , θ) is a location parameter and φ is a nuisance parameter. h h More speciﬁcally, ψ(u , θ) is a function that depends on u , which is the h-th h h row of U, and transforms the primary parameters θ into the parameters of i the multiple allocation components according to several possible schemes: • additive model: the parameters of the mixed groups could be the sum (cid:88) of the original parameters, that is (cid:80) k ψ(u , θ) = u θ + θ 1 ∗ , h hi i b [ k u =0] i=1 hi i=1 where in the second term the indicator function is introduced to account for the parameters of the outward mixture component; (cid:80) • co-dominance of order 1 : (cid:80) (cid:80) k u θ ψ(u , θ) = i=1 hi i + θ 1 ∗ ; h k u b [ ki=1 uhi=0] i=1 hi • (cid:32) (cid:33) co-dominance of order 0(cid:89): (cid:80) 1/ k u (cid:80) k i=1 hi ψ(u , θ) = θuhi + θ 1 ∗ . h i b [ k u =0] i=1 hi i=1 8 ##LTLine##

text: or

text: multiple

text: allocation)

text: group

text: k

text: π

text: π

text: uhi

text: h

text: i

text: i

text: with

text: h

text: k

text: Notice

text: that

text: these

text: weights

text: are

text: constructed

text: so

text: that

text: k

text: π

text: and

text: the

text: deﬁnition

text: comes

text: naturally

text: from

text: the

text: representation

text: of

text: h

text: clusters

text: as

text: sets

text: in

text: a

text: sample

text: space.

text: The

text: Multiple

text: Allocation

text: Mixture

text: model

text: (MAM)

text: can

text: be

text: deﬁned

text: as

text: a

text: reparameterization

text: of

text: Eq.

text: in

text: the

text: new

text: formula-

text: tion

text: k

text: f(y

text: Θ)

text: π

text: f(y

text: ψ(u

text: θ),

text: φ

text: j

text: h

text: j

text: h

text: h

text: where

text: ψ(u

text: θ)

text: is

text: a

text: location

text: parameter

text: and

text: φ

text: is

text: a

text: nuisance

text: parameter.

text: h

text: h

text: More

text: speciﬁcally,

text: ψ(u

text: θ)

text: is

text: a

text: function

text: that

text: depends

text: on

text: u

text: which

text: is

text: the

text: h-th

text: h

text: h

text: row

text: of

text: U,

text: and

text: transforms

text: the

text: primary

text: parameters

text: θ

text: into

text: the

text: parameters

text: of

text: i

text: the

text: multiple

text: allocation

text: components

text: according

text: to

text: several

text: possible

text: schemes:

text: additive

text: model:

text: the

text: parameters

text: of

text: the

text: mixed

text: groups

text: could

text: be

text: the

text: sum

text: of

text: the

text: original

text: parameters,

text: that

text: is

text: k

text: ψ(u

text: θ)

text: u

text: θ

text: θ

text: h

text: hi

text: i

text: b

text: k

text: u

text: hi

text: where

text: in

text: the

text: second

text: term

text: the

text: indicator

text: function

text: is

text: introduced

text: to

text: account

text: for

text: the

text: parameters

text: of

text: the

text: outward

text: mixture

text: component;

text: co-dominance

text: of

text: order

text: k

text: u

text: θ

text: ψ(u

text: θ)

text: hi

text: i

text: θ

text: h

text: k

text: u

text: b

text: hi

text: co-dominance

text: of

text: order

text: k

text: u

text: k

text: hi

text: ψ(u

text: θ)

text: θuhi

text: θ

text: h

text: i

text: b

text: k

text: u

text: hi

text: ##LTLine##


text: 2 Charles Curry, Kurusch Ebrahimi–Fard, Simon J.A. Malham and Anke Wiese We consider the simulation of Itˆo stochastic diﬀerential systems driven by independent L´evy processes possessing moments of all orders. Our goal is to derive a new strong numerical integration scheme for such systems that is eﬃ- cient in the following sense. The eﬃcient scheme, which is deﬁned for any order of approximation, has a strong error at leading order that is always smaller than the strong error of the corresponding stochastic Taylor approximation. This is true for any such stochastic diﬀerential systems of any size, and for all orders of approximation. Moreover, for half-integer orders, our eﬃcient in- tegrator is optimal in the sense that its global root mean-square strong error realizes its smallest possible value compared to the error of the corresponding stochastic Taylor approximation. L´evy processes are a class of stochastic pro- cesses which are continuous in probability and possess stationary increments, independent of the past. They are examples of stochastic processes with a well understood structure that incorporate jump discontinuities. Due to the L´evy decomposition of a L´evy process the systems we consider can be written in (cid:90) (cid:90) the form (cid:88) (cid:88) d t (cid:96) t yt = y0 + Vi(ys)dWsi + Vi(ys−)dJsi, i=0 0 i=d+1 0 ∈ R ∈ N for y N with N ; see for example Applebaum [2]. We assume all the t R → R governing autonomous vector ﬁelds V : N N are smooth and in general i ≡ non-commuting. Note that by convention we suppose W0 t. The system is t thus driven by the independent Wiener processes Wi for i = 1, . . . , d, and the (cid:90) (cid:90) purely discontinuous martingales Ji for i = d + 1, . . . , (cid:96) which are expressible in the form t Ji = vQi(dv, ds), t R 0 (cid:82) − where Qi(dv, ds) := Qi(dv, ds) ρi(dv)ds. The Qi are Poisson measures on R × R R with intensity measures ρi(dv)ds, where the ρi are measures on + ∧ ∞ with ρi(0) = 0 and R(1 v2)ρi(dv) < . Intuitively, Qi(B, (a, b]) counts the number of jumps of the process Ji taking values in the set B in the time interval (a, b], whilst ρi(B) measures the expected number of jumps per unit time the process Ji accrues taking values in the set B. We emphasize that we assume that the Ji possess moments of all orders. L´evy processes have many applications. Most notably though they appear in mathematical ﬁnance in the construction of models going beyond the Black– Scholes–Merton model to incorporate discontinuities in stock prices. See for example Cont & Tankov [17] and Barndorﬀ–Nielsen, Mikosch & Resnick [4] and the references therein. There are also applications to physical sciences, see for instance Barndorﬀ–Nielsen, Mikosch & Resnick [4]. Work on Euler type methods for simulating L´evy-driven stochastic diﬀerential systems in- cluding how to incorporate the jump processes can for example be found in Jacod [39], Fournier [25]. See Higham & Kloeden [35,36] for implicit methods and Protter & Talay [65] for weak approximations. The monograph by Platen ##LTLine## ##LTLine## ##LTLine##

text: Charles

label: author

text: Curry,

text: Kurusch

text: Ebrahimi–Fard,

text: Simon

text: J.A.

text: Malham

text: and

text: Anke

text: Wiese

text: We

text: consider

text: the

text: simulation

text: of

text: Itˆo

text: stochastic

text: diﬀerential

text: systems

text: driven

text: by

text: independent

text: L´evy

text: processes

text: possessing

text: moments

text: of

text: all

text: orders.

text: Our

text: goal

text: is

text: to

text: derive

text: a

text: new

text: strong

text: numerical

text: integration

text: scheme

text: for

text: such

text: systems

text: that

text: is

text: eﬃ-

text: cient

text: in

text: the

text: following

text: sense.

text: The

text: eﬃcient

text: scheme,

text: which

text: is

text: deﬁned

text: for

text: any

text: order

text: of

text: approximation,

text: has

text: a

text: strong

text: error

text: at

text: leading

text: order

text: that

text: is

text: always

text: smaller

text: than

text: the

text: strong

text: error

text: of

text: the

text: corresponding

text: stochastic

text: Taylor

text: approximation.

text: This

text: is

text: true

text: for

text: any

text: such

text: stochastic

text: diﬀerential

text: systems

text: of

text: any

text: size,

text: and

text: for

text: all

text: orders

text: of

text: approximation.

text: Moreover,

text: for

text: half-integer

text: orders,

text: our

text: eﬃcient

text: in-

text: tegrator

text: is

text: optimal

text: in

text: the

text: sense

text: that

text: its

text: global

text: root

text: mean-square

text: strong

text: error

text: realizes

text: its

text: smallest

text: possible

text: value

text: compared

text: to

text: the

text: error

text: of

text: the

text: corresponding

text: stochastic

text: Taylor

text: approximation.

text: L´evy

text: processes

text: are

text: a

text: class

text: of

text: stochastic

text: pro-

text: cesses

text: which

text: are

text: continuous

text: in

text: probability

text: and

text: possess

text: stationary

text: increments,

text: independent

text: of

text: the

text: past.

text: They

text: are

text: examples

text: of

text: stochastic

text: processes

text: with

text: a

text: well

text: understood

text: structure

text: that

text: incorporate

text: jump

text: discontinuities.

text: Due

text: to

text: the

text: L´evy

text: decomposition

text: of

text: a

text: L´evy

text: process

text: the

text: systems

text: we

text: consider

text: can

text: be

text: written

text: in

text: the

text: form

text: d

text: t

text: t

text: yt

text: Vi(ys)dWsi

text: Vi(ys−)dJsi,

text: R

text: N

text: for

text: y

text: N

text: with

text: N

text: see

text: for

text: example

text: Applebaum

text: We

text: assume

text: all

text: the

text: t

text: R

text: R

text: governing

text: autonomous

text: vector

text: ﬁelds

text: V

text: N

text: N

text: are

text: smooth

text: and

text: in

text: general

text: i

text: non-commuting.

text: Note

text: that

text: by

text: convention

text: we

text: suppose

text: t.

text: The

text: system

text: is

text: t

text: thus

text: driven

text: by

text: the

text: independent

text: Wiener

text: processes

text: Wi

text: for

text: i

text: d,

text: and

text: the

text: purely

text: discontinuous

text: martingales

text: Ji

text: for

text: i

text: d

text: which

text: are

text: expressible

text: in

text: the

text: form

text: t

text: Ji

text: vQi(dv,

text: ds),

text: t

text: R

text: where

text: Qi(dv,

text: ds)

text: Qi(dv,

text: ds)

text: ρi(dv)ds.

text: The

text: Qi

text: are

text: Poisson

text: measures

text: on

text: R

text: R

text: R

text: with

text: intensity

text: measures

text: ρi(dv)ds,

text: where

text: the

text: ρi

text: are

text: measures

text: on

text: with

text: and

text: Intuitively,

text: Qi(B,

text: (a,

text: b])

text: counts

text: the

text: number

text: of

text: jumps

text: of

text: the

text: process

text: Ji

text: taking

text: values

text: in

text: the

text: set

text: B

text: in

text: the

text: time

text: interval

text: (a,

text: b],

text: whilst

text: ρi(B)

text: measures

text: the

text: expected

text: number

text: of

text: jumps

text: per

text: unit

text: time

text: the

text: process

text: Ji

text: accrues

text: taking

text: values

text: in

text: the

text: set

text: B.

text: We

text: emphasize

text: that

text: we

text: assume

text: that

text: the

text: Ji

text: possess

text: moments

text: of

text: all

text: orders.

text: L´evy

text: processes

text: have

text: many

text: applications.

text: Most

text: notably

text: though

text: they

text: appear

text: in

text: mathematical

text: ﬁnance

text: in

text: the

text: construction

text: of

text: models

text: going

text: beyond

text: the

text: Black–

text: Scholes–Merton

text: model

text: to

text: incorporate

text: discontinuities

text: in

text: stock

text: prices.

text: See

text: for

text: example

text: Cont

text: Tankov

text: and

text: Barndorﬀ–Nielsen,

text: Mikosch

text: Resnick

text: and

text: the

text: references

text: therein.

text: There

text: are

text: also

text: applications

text: to

text: physical

text: sciences,

text: see

text: for

text: instance

text: Barndorﬀ–Nielsen,

text: Mikosch

text: Resnick

text: Work

text: on

text: Euler

text: type

text: methods

text: for

text: simulating

text: L´evy-driven

text: stochastic

text: diﬀerential

text: systems

text: in-

text: cluding

text: how

text: to

text: incorporate

text: the

text: jump

text: processes

text: can

text: for

text: example

text: be

text: found

text: in

text: Jacod

text: Fournier

text: See

text: Higham

text: Kloeden

text: for

text: implicit

text: methods

text: and

text: Protter

text: Talay

text: for

text: weak

text: approximations.

text: The

text: monograph

text: by

text: Platen

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SDEs driven by L´evy processes 11 Theorem 1 (Stochastic Taylor expansion) For a L´evy-driven equation R → R the action of the ﬂowmap ϕt on s(cid:88)uﬃciently(cid:2) smoot(cid:3)h functions f : N N can be expanded as follows ◦ ◦ ϕ f = I (t) V˜ f . t w w ∈ ∗ w A Remark 1 (Stochastic Taylor expansion convergence) For integration schemes derived from the stochastic Taylor expansion to converge, it suﬃces that the ◦ terms V˜ f included in the expansion, and those at leading order in the w remainder, satisfy global Lipschitz and linear growth conditions (see Platen & Bruti-Liberati [62]). Hereafter we assume these conditions are satisﬁed. Remark 2 (Platen and Bruti–Liberati form) The stochastic Taylor expansion derived in Platen & Bruti-Liberati [62] is an equivalent though diﬀerent rep- resentation; we show this in the electronic supplementary material. Given the stochastic Taylor expansion for the ﬂowmap in Theorem 1, we now show how we can write it in separated form. One component of the separated form are iterated integrals which are free in the sense of having no integrand. We deﬁne these abstractly for an arbitrary given alphabet for the moment, the reason for this will be apparent presently. Deﬁnition 1 (Free multiple iterated integrals) Given a collection of { } A stochastic processes Zai (cid:90)∈A, indexed by a given countable alphabet , free t ai multiple iterated integrals take the form · · · I (t) := dZa1 dZam, w ··· τ1 τm 0<τ1< <τm<t A∗ where w = a . . . a are words in . 1 m { } We need to augment W0, W1, . . . , Wd, Jd+1, . . . , J(cid:96) with a set of extended driving processes as follows. A key component in the characterization of the algebra generated by the vector space of free iterated integrals of the driving { } processes W0, W1, . .∈. ,{Wd, Jd+1(cid:90), . .}(cid:90). , J(cid:96) are th≥e compensated power brack- ets deﬁned for each i d + 1, . . . , (cid:96) and for p 2 by t Ji(p) := vp(cid:82)Qi(dv, ds). t R 0 − Equivalently we have Ji(p) = [Ji](p) t R vp ρi(dv), where [Ji](p) is the pth t t order nested quadratic covariation bracket of Ji. Importantly, the compen- t sated power brackets have the property that if Ji(p) is contained in the linear { − } ≥ span of t, Ji, Ji(2), . . . , Ji(p 1) for some p 2, then Ji(q) is also in this lin- t t ≥ t ≥ (cid:8) (cid:9) ear span for all q p. Hence inductively for p 2, we augment our family { } W0, W1, . . . , Wd, Jd+1, . . . , J(cid:96) to include the compensated power brackets − Ji(p) as long as they are not contained in the linear span of t, Ji, Ji(2), . . . , Ji(p 1) . t t t By doing so, we obtain a possibly inﬁnite family of stochastic processes. The ##LTLine## ##LTLine##

text: SDEs

text: driven

text: by

text: L´evy

text: processes

text: Theorem

text: (Stochastic

text: Taylor

text: expansion)

text: For

text: a

text: L´evy-driven

text: equation

text: R

text: R

text: the

text: action

text: of

text: the

text: ﬂowmap

text: ϕt

text: on

text: functions

text: f

text: N

text: N

text: can

text: be

text: expanded

text: as

text: follows

text: ϕ

text: f

text: I

text: (t)

text: V˜

text: f

text: t

text: w

text: w

text: w

text: A

text: Remark

text: (Stochastic

text: Taylor

text: expansion

text: convergence)

text: For

text: integration

text: schemes

text: derived

text: from

text: the

text: stochastic

text: Taylor

text: expansion

text: to

text: converge,

text: it

text: suﬃces

text: that

text: the

text: terms

text: V˜

text: f

text: included

text: in

text: the

text: expansion,

text: and

text: those

text: at

text: leading

text: order

text: in

text: the

text: w

text: remainder,

text: satisfy

text: global

text: Lipschitz

text: and

text: linear

text: growth

text: conditions

text: (see

text: Platen

text: Bruti-Liberati

text: Hereafter

text: we

text: assume

text: these

text: conditions

text: are

text: satisﬁed.

text: Remark

text: (Platen

text: and

text: Bruti–Liberati

text: form)

text: The

text: stochastic

text: Taylor

text: expansion

text: derived

text: in

text: Platen

text: Bruti-Liberati

text: is

text: an

text: equivalent

text: though

text: diﬀerent

text: rep-

text: resentation;

text: we

text: show

text: this

text: in

text: the

text: electronic

text: supplementary

text: material.

text: Given

text: the

text: stochastic

text: Taylor

text: expansion

text: for

text: the

text: ﬂowmap

text: in

text: Theorem

text: we

text: now

text: show

text: how

text: we

text: can

text: write

text: it

text: in

text: separated

text: form.

text: One

text: component

text: of

text: the

text: separated

text: form

text: are

text: iterated

text: integrals

text: which

text: are

text: free

text: in

text: the

text: sense

text: of

text: having

text: no

text: integrand.

text: We

text: deﬁne

text: these

text: abstractly

text: for

text: an

text: arbitrary

text: given

text: alphabet

text: for

text: the

text: moment,

text: the

text: reason

text: for

text: this

text: will

text: be

text: apparent

text: presently.

text: Deﬁnition

text: (Free

text: multiple

text: iterated

text: integrals)

text: Given

text: a

text: collection

text: of

text: A

text: stochastic

text: processes

text: Zai

text: indexed

text: by

text: a

text: given

text: countable

text: alphabet

text: free

text: t

text: ai

text: multiple

text: iterated

text: integrals

text: take

text: the

text: form

text: I

text: (t)

text: dZam,

text: w

text: τ1

text: τm

text: <τm<t

text: A∗

text: where

text: w

text: a

text: a

text: are

text: words

text: in

text: m

text: We

text: need

text: to

text: augment

text: Wd,

text: with

text: a

text: set

text: of

text: extended

text: driving

text: processes

text: as

text: follows.

text: A

text: key

text: component

text: in

text: the

text: characterization

text: of

text: the

text: algebra

text: generated

text: by

text: the

text: vector

text: space

text: of

text: free

text: iterated

text: integrals

text: of

text: the

text: driving

text: processes

text: ,{Wd,

text: are

text: th≥e

text: compensated

text: power

text: brack-

text: ets

text: deﬁned

text: for

text: each

text: i

text: d

text: and

text: for

text: p

text: by

text: t

text: Ji(p)

text: ds).

text: t

text: R

text: Equivalently

text: we

text: have

text: Ji(p)

text: [Ji](p)

text: t

text: R

text: vp

text: ρi(dv),

text: where

text: [Ji](p)

text: is

text: the

text: pth

text: t

text: t

text: order

text: nested

text: quadratic

text: covariation

text: bracket

text: of

text: Ji.

text: Importantly,

text: the

text: compen-

text: t

text: sated

text: power

text: brackets

text: have

text: the

text: property

text: that

text: if

text: Ji(p)

text: is

text: contained

text: in

text: the

text: linear

text: span

text: of

text: t,

text: Ji,

text: Ji(p

text: for

text: some

text: p

text: then

text: Ji(q)

text: is

text: also

text: in

text: this

text: lin-

text: t

text: t

text: t

text: ear

text: span

text: for

text: all

text: q

text: p.

text: Hence

text: inductively

text: for

text: p

text: we

text: augment

text: our

text: family

text: Wd,

text: to

text: include

text: the

text: compensated

text: power

text: brackets

text: Ji(p)

text: as

text: long

text: as

text: they

text: are

text: not

text: contained

text: in

text: the

text: linear

text: span

text: of

text: t,

text: Ji,

text: Ji(p

text: t

text: t

text: t

text: By

text: doing

text: so,

text: we

text: obtain

text: a

text: possibly

text: inﬁnite

text: family

text: of

text: stochastic

text: processes.

text: The

text: ##LTLine##

text: ##LTLine##


text: SDEs driven by L´evy processes 13 and iterating gives the separated expansion, where the operators V˜ are those ∈ { } a of the stochastic Taylor expansion for a 0, 1, . . . , d , and for a = i(m) are given by the V˜ deﬁned above. We have thus just established the following i(m) result. Theorem 2 (Separated stochastic Taylor expansion: existence) For a ◦ L´evy-driven equation suppose the terms V˜ f in the stochastic Taylor expan- R w sion for the ﬂowmap are analytic on N. Then it can be written in separated form, i.e. as a separated stochastic Taylor expansion. Remark 4 (Linear vector ﬁelds and linear diﬀeomorphisms) In this special case the separated expansion has an especially simple form; see the electronic supplementary material. Note that the identity map is a special case of a linear diﬀeomorphism. Hereafter we assume the existence of a separated Taylor expansion for the ﬂowmap, and all iterated integrals are free iterated integrals. 3 Convolution algebras and map-truncate-invert schemes We now introduce a class of numerical integration schemes we call map- truncate-invert schemes and show how they can be encoded algebraically. The algebraic structures arise naturally from the products of iterated integrals and the composition of operators appearing in the separated stochastic Taylor expansion. Consider the class of numerical integration schemes obtained from the stochastic Taylor expansion by simulating truncations of the expansion on each subinterval of a uniform discretization of a given time domain [0, T]. Deﬁnition 4 (Grading function and truncations) A grading function A∗ → N ∈ A∗ g: assigns a positive integer to each non-empty word w and zero to the empty word. A truncation is speciﬁed by a grading function and ∈ N truncation value n . We write π , π ≤ and π ≥ for the projections A∗ g=n gA∗n { g∈nA∗ } of onto the following subsets: (i) π ( ) := w : g(w) = n ; (ii) A∗ { ∈ A∗ ≤ } g=n A∗ { ∈ A∗ ≥ } π ≤ ( ) := w : g(w) n and (iii) π ≥ ( ) := w : g(w) n . g n g n A numerical integration scheme (cid:88)based on the stochastic Taylor expansion is thus given by successive applications of the approximate ﬂow I (t)V˜ w w ∈ A∗ w πg≤n( ) across the computational subintervals. Now more generally, consider a larger class of numerical schemes that are constructed as follows. For a given invert- R → R ible map f : Diﬀ( N) Diﬀ( N), we construct a series expansion for f(ϕ ) t using the stochastic Taylor expansion for ϕ . We truncate the series and simu- t late the retained iterated Itˆo integrals across each computational subinterval. − An integration scheme is obtained by computing the inverse map f 1 of the simulated truncations at each step; see Malham & Wiese [51] and Ebrahimi– Fard et al. [20]. We now develop an algebraic framework for studying such ##LTLine##

text: SDEs

text: driven

text: by

text: L´evy

text: processes

text: and

text: iterating

text: gives

text: the

text: separated

text: expansion,

text: where

text: the

text: operators

text: V˜

text: are

text: those

text: a

text: of

text: the

text: stochastic

text: Taylor

text: expansion

text: for

text: a

text: d

text: and

text: for

text: a

text: i(m)

text: are

text: given

text: by

text: the

text: V˜

text: deﬁned

text: above.

text: We

text: have

text: thus

text: just

text: established

text: the

text: following

text: i(m)

text: result.

text: Theorem

text: (Separated

text: stochastic

text: Taylor

text: expansion:

text: existence)

text: For

text: a

text: L´evy-driven

text: equation

text: suppose

text: the

text: terms

text: V˜

text: f

text: in

text: the

text: stochastic

text: Taylor

text: expan-

text: R

text: w

text: sion

text: for

text: the

text: ﬂowmap

text: are

text: analytic

text: on

text: N.

text: Then

text: it

text: can

text: be

text: written

text: in

text: separated

text: form,

text: i.e.

text: as

text: a

text: separated

text: stochastic

text: Taylor

text: expansion.

text: Remark

text: (Linear

text: vector

text: ﬁelds

text: and

text: linear

text: diﬀeomorphisms)

text: In

text: this

text: special

text: case

text: the

text: separated

text: expansion

text: has

text: an

text: especially

text: simple

text: form;

text: see

text: the

text: electronic

text: supplementary

text: material.

text: Note

text: that

text: the

text: identity

text: map

text: is

text: a

text: special

text: case

text: of

text: a

text: linear

text: diﬀeomorphism.

text: Hereafter

text: we

text: assume

text: the

text: existence

text: of

text: a

text: separated

text: Taylor

text: expansion

text: for

text: the

text: ﬂowmap,

text: and

text: all

text: iterated

text: integrals

text: are

text: free

text: iterated

text: integrals.

text: Convolution

text: algebras

text: and

text: map-truncate-invert

text: schemes

text: We

text: now

text: introduce

text: a

text: class

text: of

text: numerical

text: integration

text: schemes

text: we

text: call

text: map-

text: truncate-invert

text: schemes

text: and

text: show

text: how

text: they

text: can

text: be

text: encoded

text: algebraically.

text: The

text: algebraic

text: structures

text: arise

text: naturally

text: from

text: the

text: products

text: of

text: iterated

text: integrals

text: and

text: the

text: composition

text: of

text: operators

text: appearing

text: in

text: the

text: separated

text: stochastic

text: Taylor

text: expansion.

text: Consider

text: the

text: class

text: of

text: numerical

text: integration

text: schemes

text: obtained

text: from

text: the

text: stochastic

text: Taylor

text: expansion

text: by

text: simulating

text: truncations

text: of

text: the

text: expansion

text: on

text: each

text: subinterval

text: of

text: a

text: uniform

text: discretization

text: of

text: a

text: given

text: time

text: domain

text: T].

text: Deﬁnition

text: (Grading

text: function

text: and

text: truncations)

text: A

text: grading

text: function

text: A∗

text: N

text: A∗

text: g:

text: assigns

text: a

text: positive

text: integer

text: to

text: each

text: non-empty

text: word

text: w

text: and

text: zero

text: to

text: the

text: empty

text: word.

text: A

text: truncation

text: is

text: speciﬁed

text: by

text: a

text: grading

text: function

text: and

text: N

text: truncation

text: value

text: n

text: We

text: write

text: π

text: π

text: and

text: π

text: for

text: the

text: projections

text: A∗

text: g=n

text: gA∗n

text: g∈nA∗

text: of

text: onto

text: the

text: following

text: subsets:

text: (i)

text: π

text: w

text: g(w)

text: n

text: (ii)

text: A∗

text: A∗

text: g=n

text: A∗

text: A∗

text: π

text: w

text: g(w)

text: n

text: and

text: (iii)

text: π

text: w

text: g(w)

text: n

text: g

text: n

text: g

text: n

text: A

text: numerical

text: integration

text: scheme

text: on

text: the

text: stochastic

text: Taylor

text: expansion

text: is

text: thus

text: given

text: by

text: successive

text: applications

text: of

text: the

text: approximate

text: ﬂow

text: I

text: (t)V˜

text: w

text: w

text: A∗

text: w

text: πg≤n(

text: across

text: the

text: computational

text: subintervals.

text: Now

text: more

text: generally,

text: consider

text: a

text: larger

text: class

text: of

text: numerical

text: schemes

text: that

text: are

text: constructed

text: as

text: follows.

text: For

text: a

text: given

text: invert-

text: R

text: R

text: ible

text: map

text: f

text: Diﬀ(

text: N)

text: Diﬀ(

text: N),

text: we

text: construct

text: a

text: series

text: expansion

text: for

text: f(ϕ

text: t

text: using

text: the

text: stochastic

text: Taylor

text: expansion

text: for

text: ϕ

text: We

text: truncate

text: the

text: series

text: and

text: simu-

text: t

text: late

text: the

text: retained

text: iterated

text: Itˆo

text: integrals

text: across

text: each

text: computational

text: subinterval.

text: An

text: integration

text: scheme

text: is

text: obtained

text: by

text: computing

text: the

text: inverse

text: map

text: f

text: of

text: the

text: simulated

text: truncations

text: at

text: each

text: step;

text: see

text: Malham

text: Wiese

text: and

text: Ebrahimi–

text: Fard

text: et

text: al.

text: We

text: now

text: develop

text: an

text: algebraic

text: framework

text: for

text: studying

text: such

text: ##LTLine##


text: SDEs driven by L´evy processes 17 Remark 11 For any word w, say of length k, we can naturally truncate a series in convolutional powers of J with real coeﬃcients c , c , c and so forth, to · · · 1 2 3 c J+c J(cid:63)2 + +c J(cid:63)k. This is because the action of J(cid:63)(k+1) and subsequent 1 2 k terms in the series is, by convention, zero as a word of length k cannot be partitioned into more than k non-empty parts. − The computation of f 1(σˆ ) from σˆ is in general non-trivial. For the expo- t t nential Lie series integrator, Castell & Gaines [12,13] proposed computing exp(σˆ )(y ) by numerical approximation of the ordinary diﬀerential equation (cid:48) t 0 u = σˆ (u), subject to the initial condition u(0) = y . An approximate so- t 0 lution may then be recovered from u(1). For the sinhlog integrator in the (cid:0) (cid:1) shuﬄe product context, Malham & Wiese [51] showed that for linear con- stant coeﬃcient equations, since σˆ is a square matrix, the approximate ﬂow − exp sinh 1(σˆ ) = σˆ + id + σˆ2 1/2 may be computed using a matrix square t t t root. If the vector ﬁelds are nonlinear Malham & Wiese [51] suggested expand- § ing the square root to suﬃciently high degree terms. In 5 we introduce direct − ◦ ◦ map-truncate-invert schemes which evaluate F 1 π ≤ F(cid:63)(id) directly and −g n circumvent the step involving the computation of f 1 in general. 4 Endomorphism inner product, gradings and error analysis We establish convergence and accuracy of integration schemes at an alge- braic level. We ﬁrst deﬁne an inner product on the space of endomorphisms corresponding to the L2 inner product of the associated approximate ﬂows. We then deﬁne the mean-square grading and introduce stochastic Taylor schemes of a speciﬁc local and then global order. We subsequently consider word length grading. We show stochastic Taylor integration schemes of a given strong or- der obtained by truncating according to word length are always more accurate than those obtained by truncating according to the mean-square grade. We measure accuracy by the leading order term in the L2-norm of the remain- der after truncation. To start, to deﬁne the endomorphism inner product, we require an algebraic encoding of the expectation of the iterated integrals. ∈ A∗ Deﬁnition 8 (Expectation map) For any word w , the expectation R(cid:104)A(cid:105) → R (cid:55)→ | | | | ∈ { }∗ map E : [t] is deﬁned by E : w t w / w ! if w 0 and is zero | | { }∗ ⊂ A∗ for all other words. Here w denotes the length of the word w, 0 is R the free monoid over the letter 0 and [t] is the polynomial ring over a single R indeterminate t commuting with . Remark 12 (Expectation of iterated Itˆo integrals) The expectation map corre- sponds to the expectation of iterated Itˆo integrals as follows. First, integrals indexed by words not ending in the letter 0 are martingales and hence have zero expectation. Second, consider integrals indexed by a word with at least one non-zero letter. Fubini’s Theorem implies such integrals also have zero expectation; see Protter [64]. R(cid:104)A(cid:105) We now deﬁne an inner product on End( ∗), following Ebrahimi–Fard et al. [20]. Suppose we apply two separate functions to the ﬂowmap which are ##LTLine##

text: SDEs

text: driven

text: by

text: L´evy

text: processes

text: Remark

text: For

text: any

text: word

text: w,

text: say

text: of

text: length

text: k,

text: we

text: can

text: naturally

text: truncate

text: a

text: series

text: in

text: convolutional

text: powers

text: of

text: J

text: with

text: real

text: coeﬃcients

text: c

text: c

text: c

text: and

text: so

text: forth,

text: to

text: c

text: J+c

text: +c

text: This

text: is

text: because

text: the

text: action

text: of

text: and

text: subsequent

text: k

text: terms

text: in

text: the

text: series

text: is,

text: by

text: convention,

text: zero

text: as

text: a

text: word

text: of

text: length

text: k

text: cannot

text: be

text: partitioned

text: into

text: more

text: than

text: k

text: non-empty

text: parts.

text: The

text: computation

text: of

text: f

text: 1(σˆ

text: from

text: σˆ

text: is

text: in

text: general

text: non-trivial.

text: For

text: the

text: expo-

text: t

text: t

text: nential

text: Lie

text: series

text: integrator,

text: Castell

text: Gaines

text: proposed

text: computing

text: exp(σˆ

text: )(y

text: by

text: numerical

text: approximation

text: of

text: the

text: ordinary

text: diﬀerential

text: equation

text: t

text: u

text: σˆ

text: (u),

text: subject

text: to

text: the

text: initial

text: condition

text: y

text: An

text: approximate

text: so-

text: t

text: lution

text: may

text: then

text: be

text: recovered

text: from

text: For

text: the

text: sinhlog

text: integrator

text: in

text: the

text: shuﬄe

text: product

text: context,

text: Malham

text: Wiese

text: showed

text: that

text: for

text: linear

text: con-

text: stant

text: coeﬃcient

text: equations,

text: since

text: σˆ

text: is

text: a

text: square

text: matrix,

text: the

text: approximate

text: ﬂow

text: exp

text: sinh

text: 1(σˆ

text: σˆ

text: id

text: σˆ2

text: may

text: be

text: computed

text: using

text: a

text: matrix

text: square

text: t

text: t

text: t

text: root.

text: If

text: the

text: vector

text: ﬁelds

text: are

text: nonlinear

text: Malham

text: Wiese

text: suggested

text: expand-

text: ing

text: the

text: square

text: root

text: to

text: suﬃciently

text: high

text: degree

text: terms.

text: In

text: we

text: introduce

text: direct

text: map-truncate-invert

text: schemes

text: which

text: evaluate

text: F

text: π

text: directly

text: and

text: −g

text: n

text: circumvent

text: the

text: step

text: involving

text: the

text: computation

text: of

text: f

text: in

text: general.

text: Endomorphism

text: inner

text: product,

text: gradings

text: and

text: error

text: analysis

text: We

text: establish

text: convergence

text: and

text: accuracy

text: of

text: integration

text: schemes

text: at

text: an

text: alge-

text: braic

text: level.

text: We

text: ﬁrst

text: deﬁne

text: an

text: inner

text: product

text: on

text: the

text: space

text: of

text: endomorphisms

text: corresponding

text: to

text: the

text: inner

text: product

text: of

text: the

text: associated

text: approximate

text: ﬂows.

text: We

text: then

text: deﬁne

text: the

text: mean-square

text: grading

text: and

text: introduce

text: stochastic

text: Taylor

text: schemes

text: of

text: a

text: speciﬁc

text: local

text: and

text: then

text: global

text: order.

text: We

text: subsequently

text: consider

text: word

text: length

text: grading.

text: We

text: show

text: stochastic

text: Taylor

text: integration

text: schemes

text: of

text: a

text: given

text: strong

text: or-

text: der

text: obtained

text: by

text: truncating

text: according

text: to

text: word

text: length

text: are

text: always

text: more

text: accurate

text: than

text: those

text: obtained

text: by

text: truncating

text: according

text: to

text: the

text: mean-square

text: grade.

text: We

text: measure

text: accuracy

text: by

text: the

text: leading

text: order

text: term

text: in

text: the

text: of

text: the

text: remain-

text: der

text: after

text: truncation.

text: To

text: start,

text: to

text: deﬁne

text: the

text: endomorphism

text: inner

text: product,

text: we

text: require

text: an

text: algebraic

text: encoding

text: of

text: the

text: expectation

text: of

text: the

text: iterated

text: integrals.

text: A∗

text: Deﬁnition

text: (Expectation

text: map)

text: For

text: any

text: word

text: w

text: the

text: expectation

text: R

text: map

text: E

text: [t]

text: is

text: deﬁned

text: by

text: E

text: w

text: t

text: w

text: w

text: if

text: w

text: and

text: is

text: zero

text: A∗

text: for

text: all

text: other

text: words.

text: Here

text: w

text: denotes

text: the

text: length

text: of

text: the

text: word

text: w,

text: is

text: R

text: the

text: free

text: monoid

text: over

text: the

text: letter

text: and

text: [t]

text: is

text: the

text: polynomial

text: ring

text: over

text: a

text: single

text: R

text: indeterminate

text: t

text: commuting

text: with

text: Remark

text: (Expectation

text: of

text: iterated

text: Itˆo

text: integrals)

text: The

text: expectation

text: map

text: corre-

text: sponds

text: to

text: the

text: expectation

text: of

text: iterated

text: Itˆo

text: integrals

text: as

text: follows.

text: First,

text: integrals

text: indexed

text: by

text: words

text: not

text: ending

text: in

text: the

text: letter

text: are

text: martingales

text: and

text: hence

text: have

text: zero

text: expectation.

text: Second,

text: consider

text: integrals

text: indexed

text: by

text: a

text: word

text: with

text: at

text: least

text: one

text: non-zero

text: letter.

text: Fubini’s

text: Theorem

text: implies

text: such

text: integrals

text: also

text: have

text: zero

text: expectation;

text: see

text: Protter

text: We

text: now

text: deﬁne

text: an

text: inner

text: product

text: on

text: End(

text: following

text: Ebrahimi–Fard

text: et

text: al.

text: Suppose

text: we

text: apply

text: two

text: separate

text: functions

text: to

text: the

text: ﬂowmap

text: which

text: are

text: ##LTLine##