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text: al.

text: ApJ,


text: days, is shown in the bottom panel of the ﬁgure. No peri- Table 1 odic brightness variations resulting from rotational mod- SUMMARY OF PHOTOMETRIC OBSERVATIONS FOR ulation in the visibility of active regions and starspots WASP-39 can be seen above the noise level in the frequency spec- Observing Da−te Range Sigma Seasonal Mean trum. In particular, there is no brightness variability Season N (HJD 2,400,000) (mag) (mag) at the planetary orbital frequency, marked by an arrow obs (1) (2) (3) (4) (5) in the frequency spectrum. This corresponds to the or- − ± 2011-12 96 55904–56100 0.0059 −0.5128 ± 0.0006 bital period of 4.055259 days determined by Faedi et 2012-13 84 56256–56470 0.0050 −0.5150 ± 0.0005 al. (2011) in their discovery paper. A least-squares sine 2013-14 118 56623–56836 0.0058 −0.5125 ± 0.0005 ﬁt on th±e orbital period gives a semi-amplitude of only 2014-15 93 56989–57187 0.0061 0.5105 0.0006 0.00050 0.00033 mag. The lack of rotational modula- tion is consistent with WASP-39 being an old, late, G- precision. type dwarf (Faedi et al. 2011). The lack of any brightness The individual photometric observations are plotted variability on the radial velocity period gives additional in the top panel of Figure 2, where we have removed conﬁrmation that the radial velocity variations are due the 0.0018 mag scatter in the seasonal means by nor- solely to the planetary companion. maDlrizaifntgvtehresiofnouMrasye2a8s,on20s20to have the same mean. A frePqrueperninctytysppeseectturusinmg LoATvEeXr stthylee ermaunlgateeaopfj v0..0081/2t2o/009.99 cy- cles/day, corresponding to a period range of 1 to 100 HST HOT-JUPITER TRANSMISSION SPECTRAL SURVEY: CLEAR SKIES FOR COOL SATURN WASP-39b Patrick D. Fischer1, Heather A. Knutson1, David K. Sing2, Gregory W. Henry3, Michael W. Williamson3, Jonathan J. Fortney4, Adam S. Burrows5, Tiffany Kataria2, Nikolay Nikolov2, Adam P. Showman6, Gilda E. Ballester6, Jean-Michel De´sert7, Suzanne Aigrain8, Drake Deming9, Alain Lecavelier des Etangs10, Alfred Vidal-Madjar10 1Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA 2Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK 3Center of Excellence in Information Systems, Tennessee State University, Nashville, TN 37209, USA 4Department of Astronomy and Astrophysics, University of California Santa Cruz, CA 95064, USA 5Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA 6Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA 7CASA, Department of Astrophysical and Planetary Sciences, University of Colorado, 389-UCB, Boulder, CO 80309, USA 8Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 9Department of Astronomy, University of Maryland, College Park, MD 20742, USA and 10CNRS, Institut dAstrophysique de Paris, UMR 7095, 98bis boulevard Arago, F-75014 Paris, France Subject headings: planetary systems — planets and satellites: atmospheres — stars: individual (WASP-39) — techniques: spectroscopic 3. ANALYSIS for each observation: Tc (transit center time), i (incli- nation), a/R∗, and R /R∗, where a is the semi-major We calculate transit light curves using the BAT- p orbital distance, R is the radius of the planet, and R∗ MAN package of Kreidberg (2015). We calculate the p is the radius of the star. Photometric time series are wavelength-dependent four-parameter nonlinear limb- modeled by instrument systematic noise and transit light darkening coeﬃcients for each bandpass using an ATLAS curves as described in the following sub-sections. stellar model with an eﬀective temperature of 5500 K and 3.1. White-Light Curves log(g) of 4.5, which is the closest match to the WASP-39 stellar parameters reported by Faedi et al. (2011). As in For the STIS observations, we ﬁrst construct a white- recent work (e.g. Nikolov et al. 2015; Sing et al. 2015) light curve by summing the spectra across all wave- we choose to rely on limb-darkening coeﬃcients deter- lengths. We extract spectra from the images using a mined from stellar models to reduce the number of free ﬁxed aperture in the cross-dispersion direction. We test a parameters in the ﬁt and to avoid the degeneracies be- range of cross-dispersion aperture widths between 1 and tween limb-darkening parameters and transit depth. We 30 pixels for each STIS observation and choose the one note that the values of the limb-darkening coeﬃcients that minimizes the RMS of the residuals in the resulting for our HST observations are not well constrained by white-light curves; this yielded optimal widths of 8, 10, our observations, which only sample times near the cen- and 9 pixels for the three observations respectively. We ter of transit. We ﬁnd consistent results using quadratic use these white-light curves to determine optimal values coeﬃcients, further conﬁrming our lack of sensitivity to for the wavelength-independent parameters (transit ceter the assumed limb-darkening model. The orbital period is time T , inclination i, normalized semi-major axis a/R∗). c obtained from Faedi et al. (2011) and is also held ﬁxed. As in previous studies (e.g. Huitson et al. 2013; Vidal- The shape of the transit light curve then depends on Madjar et al. 2013; Sing et al. 2013; Nikolov et al. 2014, four physical parameters that are ﬁtted simultaneously 2015; Sing et al. 2015), we ﬁnd that the STIS sensitivity ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: days,

label: paragraph

text: is

text: shown

text: in

text: the

text: bottom

text: panel

text: of

text: the

text: ﬁgure.

text: No

text: peri-

text: Table

text: odic

text: brightness

text: variations

text: resulting

text: from

text: rotational

text: mod-

text: SUMMARY

text: OF

text: PHOTOMETRIC

text: OBSERVATIONS

text: FOR

text: ulation

text: in

text: the

text: visibility

text: of

text: active

text: regions

text: and

text: starspots

text: can

text: be

text: seen

text: above

text: the

text: noise

text: level

text: in

text: the

text: frequency

text: spec-

text: Observing

text: Da−te

text: Range

text: Sigma

text: Seasonal

text: Mean

text: trum.

text: In

text: particular,

text: there

text: is

text: no

text: brightness

text: variability

text: Season

text: N

text: (HJD

text: (mag)

text: (mag)

text: at

text: the

text: planetary

text: orbital

text: frequency,

text: marked

text: by

text: an

text: arrow

text: obs

text: in

text: the

text: frequency

text: spectrum.

text: This

text: corresponds

text: to

text: the

text: or-

text: bital

text: period

text: of

text: days

text: determined

text: by

text: Faedi

text: et

text: al.

text: in

text: their

text: discovery

text: paper.

text: A

text: least-squares

text: sine

text: ﬁt

text: on

text: th±e

text: orbital

text: period

text: gives

text: a

text: semi-amplitude

text: of

text: only

text: mag.

text: The

text: lack

text: of

text: rotational

text: modula-

text: tion

text: is

text: consistent

text: with

text: being

text: an

text: old,

text: late,

text: G-

text: precision.

text: type

text: dwarf

text: (Faedi

text: et

text: al.

text: The

text: lack

text: of

text: any

text: brightness

text: The

text: individual

text: photometric

text: observations

text: are

text: plotted

text: variability

text: on

text: the

text: radial

text: velocity

text: period

text: gives

text: additional

text: in

text: the

text: top

text: panel

text: of

text: Figure

text: where

text: we

text: have

text: removed

text: conﬁrmation

text: that

text: the

text: radial

text: velocity

text: variations

text: are

text: due

text: the

text: mag

text: scatter

text: in

text: the

text: seasonal

text: means

text: by

text: nor-

text: solely

text: to

text: the

text: planetary

text: companion.

text: have

text: the

text: same

text: mean.

text: A

text: frePqrueperninctytysppeseectturusinmg

text: LoATvEeXr

text: stthylee

text: ermaunlgateeaopfj

text: cy-

text: cles/day,

text: corresponding

text: to

text: a

text: period

text: range

text: of

text: to

text: HST

text: HOT-JUPITER

text: TRANSMISSION

text: SPECTRAL

text: SURVEY:

text: CLEAR

text: SKIES

text: FOR

text: COOL

text: SATURN

text: Patrick

text: D.

text: Heather

text: A.

text: David

text: K.

text: Gregory

text: W.

text: Michael

text: W.

text: Jonathan

text: J.

text: Adam

text: S.

text: Tiffany

text: Nikolay

text: Adam

text: P.

text: Gilda

text: E.

text: Jean-Michel

text: Suzanne

text: Drake

text: Alain

text: Lecavelier

text: des

text: Alfred

text: of

text: Geological

text: and

text: Planetary

text: Sciences,

text: California

text: Institute

text: of

text: Technology,

text: Pasadena,

text: CA

text: USA

text: Group,

text: School

text: of

text: Physics,

text: University

text: of

text: Exeter,

text: Stocker

text: Road,

text: Exeter,

text: UK

text: of

text: Excellence

text: in

text: Information

text: Systems,

text: Tennessee

text: State

text: University,

text: Nashville,

text: TN

text: USA

text: of

text: Astronomy

text: and

text: Astrophysics,

text: University

text: of

text: California

text: Santa

text: Cruz,

text: CA

text: USA

text: of

text: Astrophysical

text: Sciences,

text: Peyton

text: Hall,

text: Princeton

text: University,

text: Princeton,

text: NJ

text: USA

text: and

text: Planetary

text: Laboratory,

text: University

text: of

text: Arizona,

text: Tucson,

text: AZ

text: USA

text: Department

text: of

text: Astrophysical

text: and

text: Planetary

text: Sciences,

text: University

text: of

text: Colorado,

text: Boulder,

text: CO

text: USA

text: of

text: Physics,

text: University

text: of

text: Oxford,

text: Denys

text: Wilkinson

text: Building,

text: Keble

text: Road,

text: Oxford

text: UK

text: of

text: Astronomy,

text: University

text: of

text: Maryland,

text: College

text: Park,

text: MD

text: USA

text: and

text: Institut

text: dAstrophysique

text: de

text: Paris,

text: UMR

text: boulevard

text: Arago,

text: Paris,

text: France

text: Subject

text: headings:

text: planetary

text: systems

text: planets

text: and

text: satellites:

text: atmospheres

text: stars:

text: individual

text: techniques:

text: spectroscopic

text: ANALYSIS

text: for

text: each

text: observation:

text: Tc

text: (transit

text: center

text: time),

text: i

text: (incli-

text: nation),

text: a/R∗,

text: and

text: R

text: /R∗,

text: where

text: a

text: is

text: the

text: semi-major

text: We

text: calculate

text: transit

text: light

text: curves

text: using

text: the

text: BAT-

text: p

text: orbital

text: distance,

text: R

text: is

text: the

text: radius

text: of

text: the

text: planet,

text: and

text: R∗

text: MAN

text: package

text: of

text: Kreidberg

text: We

text: calculate

text: the

text: p

text: is

text: the

text: radius

text: of

text: the

text: star.

text: Photometric

text: time

text: series

text: are

text: wavelength-dependent

text: four-parameter

text: nonlinear

text: limb-

text: modeled

text: by

text: instrument

text: systematic

text: noise

text: and

text: transit

text: light

text: darkening

text: coeﬃcients

text: for

text: each

text: bandpass

text: using

text: an

text: ATLAS

text: curves

text: as

text: described

text: in

text: the

text: following

text: sub-sections.

text: stellar

text: model

text: with

text: an

text: eﬀective

text: temperature

text: of

text: K

text: and

text: White-Light

text: Curves

text: log(g)

text: of

text: which

text: is

text: the

text: closest

text: match

text: to

text: the

text: stellar

text: parameters

text: reported

text: by

text: Faedi

text: et

text: al.

text: As

text: in

text: For

text: the

text: STIS

text: observations,

text: we

text: ﬁrst

text: construct

text: a

text: white-

text: recent

text: work

text: (e.g.

text: Nikolov

text: et

text: al.

text: Sing

text: et

text: al.

text: light

text: curve

text: by

text: summing

text: the

text: spectra

text: across

text: all

text: wave-

text: we

text: choose

text: to

text: rely

text: on

text: limb-darkening

text: coeﬃcients

text: deter-

text: lengths.

text: We

text: extract

text: spectra

text: from

text: the

text: images

text: using

text: a

text: mined

text: from

text: stellar

text: models

text: to

text: reduce

text: the

text: number

text: of

text: free

text: ﬁxed

text: aperture

text: in

text: the

text: cross-dispersion

text: direction.

text: We

text: test

text: a

text: parameters

text: in

text: the

text: ﬁt

text: and

text: to

text: avoid

text: the

text: degeneracies

text: be-

text: range

text: of

text: cross-dispersion

text: aperture

text: widths

text: between

text: and

text: tween

text: limb-darkening

text: parameters

text: and

text: transit

text: depth.

text: We

text: pixels

text: for

text: each

text: STIS

text: observation

text: and

text: choose

text: the

text: one

text: note

text: that

text: the

text: values

text: of

text: the

text: limb-darkening

text: coeﬃcients

text: that

text: minimizes

text: the

text: RMS

text: of

text: the

text: residuals

text: in

text: the

text: resulting

text: for

text: our

text: HST

text: observations

text: are

text: not

text: well

text: constrained

text: by

text: white-light

text: curves;

text: this

text: yielded

text: optimal

text: widths

text: of

text: our

text: observations,

text: which

text: only

text: sample

text: times

text: near

text: the

text: cen-

text: and

text: pixels

text: for

text: the

text: three

text: observations

text: respectively.

text: We

text: ter

text: of

text: transit.

text: We

text: ﬁnd

text: consistent

text: results

text: using

text: quadratic

text: use

text: these

text: white-light

text: curves

text: to

text: determine

text: optimal

text: values

text: coeﬃcients,

text: further

text: conﬁrming

text: our

text: lack

text: of

text: sensitivity

text: to

text: for

text: the

text: wavelength-independent

text: parameters

text: (transit

text: ceter

text: the

text: assumed

text: limb-darkening

text: model.

text: The

text: orbital

text: period

text: is

text: time

text: T

text: inclination

text: i,

text: normalized

text: semi-major

text: axis

text: a/R∗).

text: c

text: obtained

text: from

text: Faedi

text: et

text: al.

text: and

text: is

text: also

text: held

text: ﬁxed.

text: As

text: in

text: previous

text: studies

text: (e.g.

text: Huitson

text: et

text: al.

text: Vidal-

text: The

text: shape

text: of

text: the

text: transit

text: light

text: curve

text: then

text: depends

text: on

text: Madjar

text: et

text: al.

text: Sing

text: et

text: al.

text: Nikolov

text: et

text: al.

text: four

text: physical

text: parameters

text: that

text: are

text: ﬁtted

text: simultaneously

text: Sing

text: et

text: al.

text: we

text: ﬁnd

text: that

text: the

text: STIS

text: sensitivity

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Figure 5. HST STIS G430L spectral light curves for the observations on UT 2013 Feb 8, raw (left) and corrected (middle) where we have divided out the best-ﬁt instrumental noise model for each visit. Color corresponds to wavelength, with the shortest wavelengths plotted in purple and the longest wavelengths plotted in light green. Light curves are oﬀset vertically by multiples of 0.01. Open circles in the left panel show the model points. The right panel shows the residuals after removing both the instrumental model and transit light curves. Dashed lines show one standard deviation. licity (new run of the Fortney et al. 2010 model). We ﬁt tively. We note that these models show a degeneracy be- for a vertical oﬀset as the only free parameter, totaling tween the eﬀects of depleted abundances and enhanced 35 data points and 34 degrees of freedom for each ﬁt. scattering. As a justiﬁcation for the subsolar metallic- We ﬁnd that the observed transmission spectrum is ity model, we note that there are two possible routes best matched by the sub-solar metallicity model (blue), by which we might deplete the Na and K abundances which has a χ2 of 28.0. It is also consistent with the in WASP-39b’s upper atmosphere. If the night side or clear solar metallicity model (cyan), which has a χ2 of deep atmosphere temperature-pressure proﬁles cross the 35.8, and the weak haze model (orange), which has a χ2 condensation curves for these elements, then circulation of 41.8. Strong hazes such as those seen in HD 189733b of gas between these regions and the upper atmosphere or WASP-6b (Pont et al. 2013; Nikolov et al. 2015) are would naturally result in their depletion (Kataria et al. × × ruled out; 100 and 1000 enhanced Rayleigh models 2016). Alternatively, the planet might also have accreted produce poor ﬁts with χ2 values of 80.3 and 130, respec- a metal-poor atmosphere from the primordial gas disk; ##LTFigure## ##LTFigure##

text: Figure

text: HST

label: caption

text: STIS

text: spectral

text: light

text: curves

text: for

text: the

text: observations

text: on

text: UT

text: Feb

text: raw

text: (left)

text: and

text: corrected

text: (middle)

text: where

text: we

text: have

text: divided

text: out

text: the

text: best-ﬁt

text: instrumental

text: noise

text: model

text: for

text: each

text: visit.

text: Color

text: corresponds

text: to

text: wavelength,

text: with

text: the

text: shortest

text: wavelengths

text: plotted

text: in

text: purple

text: and

text: the

text: longest

text: wavelengths

text: plotted

text: in

text: light

text: green.

text: Light

text: curves

text: are

text: oﬀset

text: vertically

text: by

text: multiples

text: of

text: Open

text: circles

text: in

text: the

text: left

text: panel

text: show

text: the

text: model

text: points.

text: The

text: right

text: panel

text: shows

text: the

text: residuals

text: after

text: removing

text: both

text: the

text: instrumental

text: model

text: and

text: transit

text: light

text: curves.

text: Dashed

text: lines

text: show

text: one

text: standard

text: deviation.

text: licity

text: (new

text: run

text: of

text: the

text: Fortney

text: et

text: al.

text: model).

text: We

text: ﬁt

text: tively.

text: We

text: note

text: that

text: these

text: models

text: show

text: a

text: degeneracy

text: be-

text: for

text: a

text: vertical

text: oﬀset

text: as

text: the

text: only

text: free

text: parameter,

text: totaling

text: tween

text: the

text: eﬀects

text: of

text: depleted

text: abundances

text: and

text: enhanced

text: data

text: points

text: and

text: degrees

text: of

text: freedom

text: for

text: each

text: ﬁt.

text: scattering.

text: As

text: a

text: justiﬁcation

text: for

text: the

text: subsolar

text: metallic-

text: We

text: ﬁnd

text: that

text: the

text: observed

text: transmission

text: spectrum

text: is

text: ity

text: model,

text: we

text: note

text: that

text: there

text: are

text: two

text: possible

text: routes

text: best

text: matched

text: by

text: the

text: sub-solar

text: metallicity

text: model

text: (blue),

text: by

text: which

text: we

text: might

text: deplete

text: the

text: and

text: K

text: abundances

text: which

text: has

text: a

text: χ2

text: of

text: It

text: is

text: also

text: consistent

text: with

text: the

text: in

text: upper

text: atmosphere.

text: If

text: the

text: night

text: side

text: or

text: clear

text: solar

text: metallicity

text: model

text: (cyan),

text: which

text: has

text: a

text: χ2

text: of

text: deep

text: atmosphere

text: temperature-pressure

text: proﬁles

text: cross

text: the

text: and

text: the

text: weak

text: haze

text: model

text: (orange),

text: which

text: has

text: a

text: χ2

text: condensation

text: curves

text: for

text: these

text: elements,

text: then

text: circulation

text: of

text: Strong

text: hazes

text: such

text: as

text: those

text: seen

text: in

text: HD

text: of

text: gas

text: between

text: these

text: regions

text: and

text: the

text: upper

text: atmosphere

text: or

text: (Pont

text: et

text: al.

text: Nikolov

text: et

text: al.

text: are

text: would

text: naturally

text: result

text: in

text: their

text: depletion

text: (Kataria

text: et

text: al.

text: ruled

text: out;

text: and

text: enhanced

text: Rayleigh

text: models

text: Alternatively,

text: the

text: planet

text: might

text: also

text: have

text: accreted

text: produce

text: poor

text: ﬁts

text: with

text: χ2

text: values

text: of

text: and

text: respec-

text: a

text: metal-poor

text: atmosphere

text: from

text: the

text: primordial

text: gas

text: disk;

text: ##LTFigure##

text: ##LTFigure##


text: Figure 7. Same description as in Fig. 5, 6 for observations with the G750L grism on UT 2013 Mar 17. K abundance, optical baseline radius z , and infrared 4.9σ for K. Although we conclude that both species are 0 baseline radius z (the ﬁtted infrared baseline radius is clearly present in this planet’s transmission spectrum, we ir a proxy for the unknown molecular abundances, deter- note that their abundances can increase by an order of mined from the transmission spectrum in the infrared). magnitude depending on whether or not we allow for the We also ﬁt the analytical model while ﬁxing either the possibility of a weak haze at the shortest wavelengths. Na abundance or K abundance to zero, giving three free Thus, the Na and K abundances in the clear-atmosphere parameters and 32 d.o.f. in these ﬁts. Fitting for both scenario, where the blue scattering slope is due to molec- Na abundance and K abundance as free parameters si- ular hydrogen, can be considered lower limits. However − ± multaneously gives χ2 = 33.4, with ln((cid:15) ) = 15.6 0.6 the Na/K abundance ratio is not dependent on P (Sing × − ± ×Na ref (0.1 solar), ln((cid:15) ) = 17.6 0.5 (0.2 solar), in refer- et al. 2015), and can be accurately measured independent K ence to the solar abundance values from Asplund et al. of whether the atmosphere is clear or contains a weak (2009). Fixing either Na abundance or K abundance to haze. We ﬁn±d a Na to K abundance ratio for WASP-39b zero (Figure 10 green and red lines) yields χ2 values of that is 45% 31% of the solar value, in reference to the 47.4 or 57.7 respectively. A likelihood ratio test of nested solar abundance values from Asplund et al. (2009). models gives a detection signiﬁcance of 3.7σ for Na and ##LTFigure## ##LTFigure##

text: Figure

text: Same

text: description

text: as

text: in

text: Fig.

text: for

text: observations

text: with

text: the

text: grism

text: on

text: UT

text: Mar

text: K

text: abundance,

text: optical

text: baseline

text: radius

text: z

text: and

text: infrared

text: for

text: K.

text: Although

text: we

text: conclude

text: that

text: both

text: species

text: are

text: baseline

text: radius

text: z

text: (the

text: ﬁtted

text: infrared

text: baseline

text: radius

text: is

text: clearly

text: present

text: in

text: this

text: planet’s

text: transmission

text: spectrum,

text: we

text: ir

text: a

text: proxy

text: for

text: the

text: unknown

text: molecular

text: abundances,

text: deter-

text: note

text: that

text: their

text: abundances

text: can

text: increase

text: by

text: an

text: order

text: of

text: mined

text: from

text: the

text: transmission

text: spectrum

text: in

text: the

text: infrared).

text: magnitude

text: depending

text: on

text: whether

text: or

text: not

text: we

text: allow

text: for

text: the

text: We

text: also

text: ﬁt

text: the

text: analytical

text: model

text: while

text: ﬁxing

text: either

text: the

text: possibility

text: of

text: a

text: weak

text: haze

text: at

text: the

text: shortest

text: wavelengths.

text: abundance

text: or

text: K

text: abundance

text: to

text: zero,

text: giving

text: three

text: free

text: Thus,

text: the

text: and

text: K

text: abundances

text: in

text: the

text: clear-atmosphere

text: parameters

text: and

text: d.o.f.

text: in

text: these

text: ﬁts.

text: Fitting

text: for

text: both

text: scenario,

text: where

text: the

text: blue

text: scattering

text: slope

text: is

text: due

text: to

text: molec-

text: abundance

text: and

text: K

text: abundance

text: as

text: free

text: parameters

text: si-

text: ular

text: hydrogen,

text: can

text: be

text: considered

text: lower

text: limits.

text: However

text: multaneously

text: gives

text: χ2

text: with

text: the

text: Na/K

text: abundance

text: ratio

text: is

text: not

text: dependent

text: on

text: P

text: (Sing

text: ×Na

text: ref

text: solar),

text: solar),

text: in

text: refer-

text: et

text: al.

text: and

text: can

text: be

text: accurately

text: measured

text: independent

text: K

text: ence

text: to

text: the

text: solar

text: abundance

text: values

text: from

text: Asplund

text: et

text: al.

text: of

text: whether

text: the

text: atmosphere

text: is

text: clear

text: or

text: contains

text: a

text: weak

text: Fixing

text: either

text: abundance

text: or

text: K

text: abundance

text: to

text: haze.

text: We

text: ﬁn±d

text: a

text: to

text: K

text: abundance

text: ratio

text: for

text: zero

text: (Figure

text: green

text: and

text: red

text: lines)

text: yields

text: χ2

text: values

text: of

text: that

text: is

text: of

text: the

text: solar

text: value,

text: in

text: reference

text: to

text: the

text: or

text: respectively.

text: A

text: likelihood

text: ratio

text: test

text: of

text: nested

text: solar

text: abundance

text: values

text: from

text: Asplund

text: et

text: al.

text: models

text: gives

text: a

text: detection

text: signiﬁcance

text: of

text: for

text: and

text: ##LTFigure##

text: ##LTFigure##


text: Resonant spectral gaps 1 1.

text: Resonant

text: spectral

text: gaps


text: 6 N. Do, P. Kuchment, B. Ong Thus, we will be looking at graphs G with boundary B such that the problem 1.4 has solution only for zero Dirichlet data φ. Rather than trying to describe all graphs that have this property, we will provide (for any size d of the boundary B) constructions when this does happen, which will be suﬃcient for our purpose of gap opening. Theorem 2.2. Let l > 0 and n be an odd natural number. Suppose that the pair 0 G, B satisﬁes the following conditions: (1) Graph G contains a cycle3 Z consisting of an odd number of edges of the length l ; 0 ∈ (2) Each boundary vertex v B either belongs to Z, or is connected to a vertex of Z by a path of edges of length l each. 0 Then, for λ = (nπ/l )2, there exist a neighborhood U of λ and a constant C such 0 0 ∈ 0 that the inequality (2.1) holds for any φ and λ U. ∈ WLOG, let us assume that Z is non-self-intersecting and consider an edge e Z (cid:18) (cid:19) (cid:18) (cid:19) (of length l , as all edges in the cycle). The solution of (1.4) for λ = λ on this 0 0 edge has the form nπ nπ u = a cos x + b sin x . l l 0 0 At the endpoints x = 0, l , the second term vanishes. Since n is odd, the ﬁrst term 0 − changes its value from a to a at these points. Going around an odd cycle, one concludes that this is possible only if a = 0, and thus u vanishes at all vertices of Z. In particular, it vanishes at all boundary vertices that belong to Z. For ∈ \ v B Z, as there exists a path of edges of length l from v to a vertex from Z, 0 where we know that u vanishes, the same consideration shows that u vanishes at all vertices of the path. Therefore, the problem (1.4) does not have solution for non-zero φ, and thus, due to Theorem 2.1, the inequality (2.1) follows. 3. The main result We are ready now to formulate and sketch the proof of the main result of this article: Theorem 3.1. Let l > 0 and n be an odd natural number. Let also the d-regular 0 | | graph Γ satisfy (1.1), and ﬁnite graph G with boundary B, B = d and the 0 decorated graph Γ are deﬁned as before. Suppose that the following conditions are satisﬁed: ∈ (1) λ = (nπ/l )2 / Σ , with dist(λ , Σ ) = r > 0; 0 0 D 0 D (2) The decoration (resonator) G, B satisﬁes the conditions of Theorem 2.2. 3which can be assumed non-self-intersecting. ##LTLine## ##LTLine## ##LTLine##

text: N.

text: Do,

text: P.

text: Kuchment,

text: B.

text: Ong

text: Thus,

text: we

text: will

text: be

text: looking

text: at

text: graphs

text: G

text: with

text: boundary

text: B

text: such

text: that

text: the

text: problem

text: has

text: solution

text: only

text: for

text: zero

text: Dirichlet

text: data

text: φ.

text: Rather

text: than

text: trying

text: to

text: describe

text: all

text: graphs

text: that

text: have

text: this

text: property,

text: we

text: will

text: provide

text: (for

text: any

text: size

text: d

text: of

text: the

text: boundary

text: B)

text: constructions

text: when

text: this

text: does

text: happen,

text: which

text: will

text: be

text: suﬃcient

text: for

text: our

text: purpose

text: of

text: gap

text: opening.

text: Theorem

text: Let

text: l

text: and

text: n

text: be

text: an

text: odd

text: natural

text: number.

text: Suppose

text: that

text: the

text: pair

text: G,

text: B

text: satisﬁes

text: the

text: following

text: conditions:

label: list

text: Graph

text: G

text: contains

text: a

text: Z

text: consisting

text: of

text: an

text: odd

text: number

text: of

text: edges

text: of

text: the

text: length

text: l

text: Each

text: boundary

text: vertex

text: v

text: B

text: either

text: belongs

text: to

text: Z,

text: or

text: is

text: connected

text: to

text: a

text: vertex

text: of

text: Z

text: by

text: a

text: path

text: of

text: edges

text: of

text: length

text: l

text: each.

text: Then,

text: for

text: λ

text: (nπ/l

text: there

text: exist

text: a

text: neighborhood

text: U

text: of

text: λ

text: and

text: a

text: constant

text: C

text: such

text: that

text: the

text: inequality

text: holds

text: for

text: any

text: φ

text: and

text: λ

text: U.

text: WLOG,

text: let

text: us

text: assume

text: that

text: Z

text: is

text: non-self-intersecting

text: and

text: consider

text: an

text: edge

text: e

text: Z

text: (of

text: length

text: l

text: as

text: all

text: edges

text: in

text: the

text: cycle).

text: The

text: solution

text: of

text: for

text: λ

text: λ

text: on

text: this

text: edge

text: has

text: the

text: form

text: nπ

text: nπ

text: u

text: a

text: cos

text: x

text: b

text: sin

text: x

text: l

text: l

text: At

text: the

text: endpoints

text: x

text: l

text: the

text: second

text: term

text: vanishes.

text: Since

text: n

text: is

text: odd,

text: the

text: ﬁrst

text: term

text: changes

text: its

text: value

text: from

text: a

text: to

text: a

text: at

text: these

text: points.

text: Going

text: around

text: an

text: odd

text: cycle,

text: one

text: concludes

text: that

text: this

text: is

text: possible

text: only

text: if

text: a

text: and

text: thus

text: u

text: vanishes

text: at

text: all

text: vertices

text: of

text: Z.

text: In

text: particular,

text: it

text: vanishes

text: at

text: all

text: boundary

text: vertices

text: that

text: belong

text: to

text: Z.

text: For

text: v

text: B

text: Z,

text: as

text: there

text: exists

text: a

text: path

text: of

text: edges

text: of

text: length

text: l

text: from

text: v

text: to

text: a

text: vertex

text: from

text: Z,

text: where

text: we

text: know

text: that

text: u

text: vanishes,

text: the

text: same

text: consideration

text: shows

text: that

text: u

text: vanishes

text: at

text: all

text: vertices

text: of

text: the

text: path.

text: Therefore,

text: the

text: problem

text: does

text: not

text: have

text: solution

text: for

text: non-zero

text: φ,

text: and

text: thus,

text: due

text: to

text: Theorem

text: the

text: inequality

text: follows.

text: The

text: main

text: result

text: We

text: are

text: ready

text: now

text: to

text: formulate

text: and

text: sketch

text: the

text: proof

text: of

text: the

text: main

text: result

text: of

text: this

text: article:

text: Theorem

text: Let

text: l

text: and

text: n

text: be

text: an

text: odd

text: natural

text: number.

text: Let

text: also

text: the

text: d-regular

text: graph

text: Γ

text: satisfy

text: and

text: ﬁnite

text: graph

text: G

text: with

text: boundary

text: B,

text: B

text: d

text: and

text: the

text: decorated

text: graph

text: Γ

text: are

text: deﬁned

text: as

text: before.

text: Suppose

text: that

text: the

text: following

text: conditions

text: are

text: satisﬁed:

text: λ

text: (nπ/l

text: Σ

text: with

text: dist(λ

text: Σ

text: r

text: D

text: D

text: The

text: decoration

text: (resonator)

text: G,

text: B

text: satisﬁes

text: the

text: conditions

text: of

text: Theorem

text: can

text: be

text: assumed

text: non-self-intersecting.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Resonant spectral gaps 7 Then there exists a punctured neighborhood of λ , depending on G, topology of Γ , 0 0 and r only, which does not belong to the spectrum σ(H). To understand the idea of the proof, let us assume ﬁrst that the graph Γ (and ∈ 0 thus Γ) is ﬁnite. Then the spectrum of H is discrete. Thus, if λ σ(H), there exists a non-zero eigenfunction u. Assume that the neighborhood of λ has radius 0 less than r < r. Then it contains no elements of Σ . 1 D Removing all internal edges and vertices from each of the decorations in Γ, one ∈ gets a disjoint union of edges of Γ , since each former vertex v Γ is replaced by 0 0 d vertices v , . . . , v , see Fig 3. 1 d Figure 3. Decoration removed from a former vertex of degree 4. (cid:101) We denote this new graph Γ. Denoting (cid:48) φ := (u(v ), . . . , u(v ))t, φ := (du/dx (v ), . . . , du/dx (v ))t, v 1 d v e1 1 ed d where e , . . . , e are the(cid:101)edges formerly adjacent to vertex v and coordinates x 1 d ej increase from the v(cid:40)alue zero at vertex v, one can rewrite the equation for u on Γ as the following one on Γ: − (cid:48)(cid:48) u = λu on each edge, (cid:48) − ∈ (3.1) φ = Λ(λ)φ at each vertex v V (Γ ). v v 0 Since λ is at a qualiﬁed distance from the Dirichlet spectrum Σ , the resolvent 0 (cid:88) (cid:88) D estimates for self-adjoint operators together with embedding theorems show that (cid:107) (cid:107) ≤ (cid:107) (cid:107) φ 2 M φ 2, (3.2) H2(e) v e v ##LTFigure##

text: Resonant

text: spectral

text: gaps

text: Then

text: there

text: exists

text: a

text: punctured

text: neighborhood

text: of

text: λ

text: depending

text: on

text: G,

text: topology

text: of

text: Γ

text: and

text: r

text: only,

text: which

text: does

text: not

text: belong

text: to

text: the

text: spectrum

text: σ(H).

text: To

text: understand

text: the

text: idea

text: of

text: the

text: proof,

text: let

text: us

text: assume

text: ﬁrst

text: that

text: the

text: graph

text: Γ

text: (and

text: thus

text: Γ)

text: is

text: ﬁnite.

text: Then

text: the

text: spectrum

text: of

text: H

text: is

text: discrete.

text: Thus,

text: if

text: λ

text: σ(H),

text: there

text: exists

text: a

text: non-zero

text: eigenfunction

text: u.

text: Assume

text: that

text: the

text: neighborhood

text: of

text: λ

text: has

text: radius

text: less

text: than

text: r

text: r.

text: Then

text: it

text: contains

text: no

text: elements

text: of

text: Σ

text: D

text: Removing

text: all

text: internal

text: edges

text: and

text: vertices

text: from

text: each

text: of

text: the

text: decorations

text: in

text: Γ,

text: one

text: gets

text: a

text: disjoint

text: union

text: of

text: edges

text: of

text: Γ

text: since

text: each

text: former

text: vertex

text: v

text: Γ

text: is

text: replaced

text: by

text: d

text: vertices

text: v

text: v

text: see

text: Fig

text: d

text: Figure

text: Decoration

text: removed

text: from

text: a

text: former

text: vertex

text: of

text: degree

text: We

text: denote

text: this

text: new

text: graph

text: Γ.

text: Denoting

text: φ

text: (u(v

text: u(v

text: ))t,

text: φ

text: (du/dx

text: (v

text: du/dx

text: (v

text: ))t,

text: v

text: d

text: v

text: ed

text: d

text: where

text: e

text: e

text: are

text: formerly

text: adjacent

text: to

text: vertex

text: v

text: and

text: coordinates

text: x

text: d

text: ej

text: increase

text: from

text: the

text: zero

text: at

text: vertex

text: v,

text: one

text: can

text: rewrite

text: the

text: equation

text: for

text: u

text: on

text: Γ

text: as

text: the

text: following

text: one

text: on

text: Γ:

text: u

text: λu

text: on

text: each

text: edge,

text: φ

text: Λ(λ)φ

text: at

text: each

text: vertex

text: v

text: V

text: (Γ

text: v

text: v

text: Since

text: λ

text: is

text: at

text: a

text: qualiﬁed

text: distance

text: from

text: the

text: Dirichlet

text: spectrum

text: Σ

text: the

text: resolvent

text: D

text: estimates

text: for

text: self-adjoint

text: operators

text: together

text: with

text: embedding

text: theorems

text: show

text: that

text: φ

text: M

text: φ

text: v

text: e

text: v

text: ##LTFigure##


text: Resonant spectral gaps 9 Figure 4. A degree 4 vertex replaced by a “spider” with edges of lengths l . 0 (4) Besides gap’s existence at a given location, its size is of importance. It depends on the value of the constant C in (2.1). Our construction allows for a variety of decorations achieving the gap at the same location. It would be interesting to analyse the dependence of C on the decoration, to pick the most eﬀective designs. (5) More general vertex conditions can be considered and Kirchhoﬀ conditions are used just for simplicity. Acknowledgment The work of the ﬁrst two authors was partially supported by the NSF DMS grant # 1517938. The authors express their gratitude to the NSF for the support. They are also grateful to the referees for useful comments. References [1] N. W. Ashcroft and N. D. Mermin, Solid state physics, Holt, Rinehart and Winston, New York-London, 1976. [2] G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208 [3] P. Exner, J. P. Keating, P. Kuchment, T. Sunada, and A. Teplyaev (eds.), Analy- sis on graphs and its applications, Proceedings of Symposia in Pure Mathematics, vol. 77, American Mathematical Society, Providence, RI, 2008, Papers from the program held in Cambridge, January 8–June 29, 2007. MR 2459860 (2009i:00008) [4] P. D. Hislop and O. Post, Anderson localization for radial tree-like quantum graphs., Waves in Random Complex Media 19 (2009), no. 2, 216–261. ##LTFigure##

text: Resonant

text: spectral

text: gaps

text: Figure

text: A

text: degree

text: vertex

text: replaced

text: by

text: a

text: “spider”

text: with

text: edges

text: of

text: lengths

text: l

text: Besides

text: gap’s

text: existence

text: at

text: a

text: given

text: location,

text: its

text: size

text: is

text: of

text: importance.

text: It

text: depends

text: on

text: the

text: value

text: of

text: the

text: constant

text: C

text: in

text: Our

text: construction

text: allows

text: for

text: a

text: variety

text: of

text: decorations

text: achieving

text: the

text: gap

text: at

text: the

text: same

text: location.

text: It

text: would

text: be

text: interesting

text: to

text: analyse

text: the

text: dependence

text: of

text: C

text: on

text: the

text: decoration,

text: to

text: pick

text: the

text: most

text: eﬀective

text: designs.

text: More

text: general

text: vertex

text: conditions

text: can

text: be

text: considered

text: and

text: Kirchhoﬀ

text: conditions

text: are

text: used

text: just

text: for

text: simplicity.

text: Acknowledgment

label: section

text: The

text: work

text: of

text: the

text: ﬁrst

text: two

text: authors

text: was

text: partially

text: supported

text: by

text: the

text: NSF

text: DMS

text: grant

text: The

text: authors

text: express

text: their

text: gratitude

text: to

text: the

text: NSF

text: for

text: the

text: support.

text: They

text: are

text: also

text: grateful

text: to

text: the

text: referees

text: for

text: useful

text: comments.

text: References

text: N.

text: W.

text: Ashcroft

text: and

text: N.

text: D.

text: Mermin,

text: Solid

text: state

text: physics,

text: Holt,

text: Rinehart

text: and

text: Winston,

text: New

text: York-London,

text: G.

text: Berkolaiko

text: and

text: P.

text: Kuchment,

text: Introduction

text: to

text: quantum

text: graphs,

text: Mathematical

text: Surveys

text: and

text: Monographs,

text: vol.

text: American

text: Mathematical

text: Society,

text: Providence,

text: RI,

text: MR

text: P.

text: Exner,

text: J.

text: P.

text: Keating,

text: P.

text: Kuchment,

text: T.

text: Sunada,

text: and

text: A.

text: Teplyaev

text: (eds.),

text: Analy-

text: sis

text: on

text: graphs

text: and

text: its

text: applications,

text: Proceedings

text: of

text: Symposia

text: in

text: Pure

text: Mathematics,

text: vol.

text: American

text: Mathematical

text: Society,

text: Providence,

text: RI,

text: Papers

text: from

text: the

text: program

text: held

text: in

text: Cambridge,

text: January

text: MR

text: P.

text: D.

text: Hislop

text: and

text: O.

text: Post,

text: Anderson

text: localization

text: for

text: radial

text: tree-like

text: quantum

text: graphs.,

text: Waves

text: in

text: Random

text: Complex

text: Media

text: no.

text: ##LTFigure##


text: ∗ K.-H. Chiang, C.-W. Chou, C.-L. Lee*, P.-Y. Lai, and Y.-F. Chen* † ‡ In this work we perform theoretical analysis about a coupled RC circuit with constant driven currents. Starting from stochastic diﬀerential equations, where voltages are subject to thermal noises, we derive time-correlation functions, steady-state distributions and transition probabilities of the system. The validity of the ﬂuctuation theorem (FT)[1, 2] is examined for scenarios with complete and incomplete descriptions. I. THEORETICAL STUDY: THE COMPLETE DESCRIPTION We consider the coupled RC circuit as shown in Fig. 1. Two RC circuits of resistances and capacitances (R , C ) 1 1 and (R , C ), respectively, are coupled through a third capacitance C . The two RC circuits are subject to constant 2 2 c driven currents I and I , and the voltage diﬀerences across the resistors are denoted as V and V , respectively. The 1 2 1 2 equation of state of this circuit is (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) M(cid:19)ˆ V(cid:126)˙ + V(cid:126) − ξ(cid:126) =(cid:18) V(cid:126) , (cid:19) (1) d − ≡ V ≡ ξ ≡ I R ≡ R (C + C ) R C where V(cid:126) 1 , ξ(cid:126) 1 , V(cid:126) 1 1 , and Mˆ 1− 1 c 1 c . The two resistors are V ξ d I R R C R (C + C ) 2 2 2 2 2 c 2 2 c thermalized at temperature T, while voltages across the resistors ﬂuctuate due to the Johnson-Nyquist (thermal) noises ξ and ξ . The noises are assumed to be uncorrelated and Gaussian white, and they satisfy the ﬂuctuation-dissipation 1 2 relation (cid:18) (cid:19) (cid:104) (cid:48) (cid:105) − (cid:48) ≡ Γ 0 ξ(cid:126)(s)ξ(cid:126) T (s ) = Γˆδ(s s ) , Γˆ 1 , (2) 0 Γ 2 ≡ with Γ 2R k T for m = 1, 2, where k is Boltzmann’s constant. m m B B FIG. 1. Illustration of the coupled RC circuit. Via the change of variables V(cid:126) (cid:48) ≡ V(cid:126) − V(cid:126) , the equation of state can be rewritten as Mˆ V(cid:126)˙ (cid:48) + V(cid:126) (cid:48) = ξ(cid:126), which is d mathematically identical to that of a non-driven circuit. The solution of the stochastic equation is ∗ † ‡ ##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##

text: K.-H.

label: author

text: Chiang,

text: C.-W.

text: Chou,

text: C.-L.

text: Lee*,

text: P.-Y.

text: Lai,

text: and

text: Y.-F.

text: Chen*

text: In

label: abstract

text: this

text: work

text: we

text: perform

text: theoretical

text: analysis

text: about

text: a

text: coupled

text: RC

text: circuit

text: with

text: constant

text: driven

text: currents.

text: Starting

text: from

text: stochastic

text: diﬀerential

text: equations,

text: where

text: voltages

text: are

text: subject

text: to

text: thermal

text: noises,

text: we

text: derive

text: time-correlation

text: functions,

text: steady-state

text: distributions

text: and

text: transition

text: probabilities

text: of

text: the

text: system.

text: The

text: validity

text: of

text: the

text: ﬂuctuation

text: theorem

text: is

text: examined

text: for

text: scenarios

text: with

text: complete

text: and

text: incomplete

text: descriptions.

text: I.

text: THEORETICAL

text: STUDY:

text: THE

text: COMPLETE

text: DESCRIPTION

text: We

text: consider

text: the

text: coupled

text: RC

text: circuit

text: as

text: shown

text: in

text: Fig.

text: Two

text: RC

text: circuits

text: of

text: resistances

text: and

text: capacitances

text: (R

text: C

text: and

text: (R

text: C

text: respectively,

text: are

text: coupled

text: through

text: a

text: third

text: capacitance

text: C

text: The

text: two

text: RC

text: circuits

text: are

text: subject

text: to

text: constant

text: c

text: driven

text: currents

text: I

text: and

text: I

text: and

text: the

text: voltage

text: diﬀerences

text: across

text: the

text: resistors

text: are

text: denoted

text: as

text: V

text: and

text: V

text: respectively.

text: The

text: equation

text: of

text: state

text: of

text: this

text: circuit

text: is

text: d

text: V

text: ξ

text: I

text: R

text: R

text: (C

text: C

text: R

text: C

text: where

text: and

text: Mˆ

text: 1−

text: c

text: c

text: The

text: two

text: resistors

text: are

text: V

text: ξ

text: d

text: I

text: R

text: R

text: C

text: R

text: (C

text: C

text: c

text: c

text: thermalized

text: at

text: temperature

text: T,

text: while

text: voltages

text: across

text: the

text: resistors

text: ﬂuctuate

text: due

text: to

text: the

text: Johnson-Nyquist

text: (thermal)

text: noises

text: ξ

text: and

text: ξ

text: The

text: noises

text: are

text: assumed

text: to

text: be

text: uncorrelated

text: and

text: Gaussian

text: white,

text: and

text: they

text: satisfy

text: the

text: ﬂuctuation-dissipation

text: relation

text: Γ

text: T

text: (s

text: Γˆδ(s

text: s

text: Γˆ

text: Γ

text: with

text: Γ

text: k

text: T

text: for

text: m

text: where

text: k

text: is

text: Boltzmann’s

text: constant.

text: m

text: m

text: B

text: B

text: FIG.

text: Illustration

text: of

text: the

text: coupled

text: RC

text: circuit.

text: Via

text: the

text: change

text: of

text: variables

text: the

text: equation

text: of

text: state

text: can

text: be

text: rewritten

text: as

text: Mˆ

text: which

text: is

text: d

text: mathematically

text: identical

text: to

text: that

text: of

text: a

text: non-driven

text: circuit.

text: The

text: solution

text: of

text: the

text: stochastic

text: equation

text: is

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: 1 Recursive Distributed Detection for Composite Hypothesis Testing: Nonlinear Observation Models in Additive Gaussian Noise Anit Kumar Sahu, Student Member, IEEE and Soummya Kar, Member, IEEE Abstract This paper studies recursive composite hypothesis testing in a network of sparsely connected agents. The network objective is to test a simple null hypothesis against a composite alternative concerning the state of the ﬁeld, modeled as a vector of (continuous) unknown parameters determining the parametric family of probability measures induced on the agents’ observation spaces under the hypotheses. Speciﬁcally, under the alternative hypothesis, each agent sequentially observes an independent and identically distributed time-series consisting of a (nonlinear) function of the true but unknown parameter corrupted by Gaussian noise, whereas, under the null, they obtain noise only. Two distributed recursive generalized likelihood ratio test type algorithms of the consensus+innovations form are proposed, namely CIGLRT − L CIGLRT − NL and , in which the agents estimate the underlying parameter and in parallel also update their test decision statistics by simultaneously processing the latest local sensed information and information CIGLRT − NL obtained from neighboring agents. For , for a broad class of nonlinear observation models and under a global observability condition, algorithm parameters which ensure asymptotically decaying probabilities CIGLRT − L of errors (probability of miss and probability of false detection) are characterized. For , a linear observation model is considered and upper bounds on large deviations decay exponent for the error probabilities are obtained. Index Terms Distributed Detection, Consensus, Generalized Likelihood Ratio Tests, Hypothesis Testing, Large Deviations 1. INTRODUCTION A. Background and Motivation The focus of this paper is on distributed composite hypothesis testing in multi-agent networks in which the goal is not only to estimate the state (possibly high dimensional) of the environment but also detect as to which hypothesis is in force based on the sensed information across all the agents at all times. To be speciﬁc, we Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA (email: anits@andrew.cmu.edu, soummyak@andrew.cmu.edu). This work was supported in part by NSF under grants CCF-1513936 and ECCS-1408222.

text: Recursive

label: title

text: Distributed

text: Detection

text: for

text: Composite

text: Hypothesis

text: Testing:

text: Nonlinear

text: Observation

text: Models

text: in

text: Additive

text: Gaussian

text: Noise

text: Anit

text: Kumar

text: Sahu,

text: Student

text: Member,

text: IEEE

text: and

text: Soummya

text: Kar,

text: Member,

text: IEEE

text: Abstract

text: This

text: paper

text: studies

text: recursive

text: composite

text: hypothesis

text: testing

text: in

text: a

text: network

text: of

text: sparsely

text: connected

text: agents.

text: The

text: network

text: objective

text: is

text: to

text: test

text: a

text: simple

text: hypothesis

text: against

text: a

text: composite

text: alternative

text: concerning

text: the

text: state

text: of

text: the

text: ﬁeld,

text: modeled

text: as

text: a

text: vector

text: of

text: (continuous)

text: unknown

text: parameters

text: determining

text: the

text: parametric

text: family

text: of

text: probability

text: measures

text: induced

text: on

text: the

text: agents’

text: observation

text: spaces

text: under

text: the

text: hypotheses.

text: Speciﬁcally,

text: under

text: the

text: alternative

text: hypothesis,

text: each

text: agent

text: sequentially

text: observes

text: an

text: independent

text: and

text: identically

text: distributed

text: time-series

text: consisting

text: of

text: a

text: (nonlinear)

text: function

text: of

text: the

text: true

text: but

text: unknown

text: parameter

text: corrupted

text: by

text: Gaussian

text: noise,

text: whereas,

text: under

text: the

text: null,

text: they

text: obtain

text: noise

text: only.

text: Two

text: distributed

text: recursive

text: generalized

text: likelihood

text: ratio

text: test

text: type

text: algorithms

text: of

text: the

text: consensus+innovations

text: form

text: are

text: proposed,

text: namely

text: CIGLRT

text: L

text: CIGLRT

text: NL

text: and

text: in

text: which

text: the

text: agents

text: estimate

text: the

text: underlying

text: parameter

text: and

text: in

text: parallel

text: also

text: update

text: their

text: test

text: decision

text: statistics

text: by

text: simultaneously

text: processing

text: the

text: latest

text: local

text: sensed

text: information

text: and

text: information

text: CIGLRT

text: NL

text: obtained

text: from

text: neighboring

text: agents.

text: For

text: for

text: a

text: broad

text: class

text: of

text: nonlinear

text: observation

text: models

text: and

text: under

text: a

text: global

text: observability

text: condition,

text: algorithm

text: parameters

text: which

text: ensure

text: asymptotically

text: decaying

text: probabilities

text: CIGLRT

text: L

text: of

text: errors

text: (probability

text: of

text: miss

text: and

text: probability

text: of

text: false

text: detection)

text: are

text: characterized.

text: For

text: a

text: linear

text: observation

text: model

text: is

text: considered

text: and

text: upper

text: bounds

text: on

text: large

text: deviations

text: decay

text: exponent

text: for

text: the

text: error

text: probabilities

text: are

text: obtained.

text: Index

text: Terms

text: Distributed

text: Detection,

text: Consensus,

text: Generalized

text: Likelihood

text: Ratio

text: Tests,

text: Hypothesis

text: Testing,

text: Large

text: Deviations

text: INTRODUCTION

text: A.

text: Background

text: and

text: Motivation

text: The

text: focus

text: of

text: this

text: paper

text: is

text: on

text: distributed

text: composite

text: hypothesis

text: testing

text: in

text: multi-agent

text: networks

text: in

text: which

text: the

text: goal

text: is

text: not

text: only

text: to

text: estimate

text: the

text: state

text: (possibly

text: high

text: dimensional)

text: of

text: the

text: environment

text: but

text: also

text: detect

text: as

text: to

text: which

text: hypothesis

text: is

text: in

text: force

text: based

text: on

text: the

text: sensed

text: information

text: across

text: all

text: the

text: agents

text: at

text: all

text: times.

text: To

text: be

text: speciﬁc,

text: we

text: Copyright

text: (c)

text: IEEE.

text: Personal

text: use

text: of

text: this

text: material

text: is

text: permitted.

text: However,

text: permission

text: to

text: use

text: this

text: material

text: for

text: any

text: other

text: purposes

text: must

text: be

text: obtained

text: from

text: the

text: IEEE

text: by

text: sending

text: a

text: request

text: to

text: pubs-permissions@ieee.org.

text: The

text: authors

text: are

text: with

text: the

text: Department

text: of

text: Electrical

text: and

text: Computer

text: Engineering,

text: Carnegie

text: Mellon

text: University,

text: Pittsburgh,

text: PA

text: USA

text: (email:

text: anits@andrew.cmu.edu,

text: soummyak@andrew.cmu.edu).

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: NSF

text: under

text: grants

text: and


text: 12 (cid:80) ∗ ∈ R × Hnθ , where Hn Mn M the monotonicity condition is trivially satisﬁed by the positive deﬁniteness of the (cid:62) − matrix N H Σ 1H . n=1 n n n { } { } We make the following assumption on the weight sequences α and β : t t Assumption A5. a b α = β = , (15) t (t + 1) t (t + 1)τ2 ≥ { } { } where ac 1 with c is as deﬁned in Assumption A4. The weight sequences α ≥ and β ≥ are given 1 1 t t 0 t t 0 by a b α = β = , (16) t (t + 1) t (t + 1)τ2 ≥ where ac 1 with c is as deﬁned in Assumption A4. 1 1 where 0 < τ < 1/2, b > 0. 2 CIGLRT − L Decision Statistic Update. The algorithm generates a scalar-valued decision statistic sequence { } zn(t) at the n-th agent according to the distributed recursive scheme    (cid:88)        z (t + 1) = t z (t) − δ(cid:124) (z (cid:123)((cid:122)t) − z (t)(cid:125)) + (cid:124) 1 log f(cid:123)θ(cid:122)n(t)(yn(t))(cid:125), (17) n t + 1 n ∈ n l t + 1 f0(yn(t)) l Ωn local innovation neighborhood consensus H H where f (.) and f (.) represent the likelihoods under and respectively, θ 0 (cid:18) 1 (cid:19)0 ∈ 2 δ 0, , (18) λ (L) N and (cid:62) − f (y (t)) (cid:62) − − h (θ (t))Σ 1h (θ (t)) log θn(t) n = h (θ (t))Σ 1y (t) n n n n n , (19) f (y (t)) n n n n 2 0 n which follows due to the Gaussian noise assumption in the observation model in (3). However, we speciﬁcally choose δ = 2 for subsequent analysis. λ2(L)+λN(L) The decision statistic update in (17) can be written in a compact manner as follows: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: Hnθ

text: where

text: Hn

text: Mn

text: M

text: the

text: monotonicity

text: condition

text: is

text: trivially

text: satisﬁed

text: by

text: the

text: positive

text: deﬁniteness

text: of

text: the

text: matrix

text: N

text: H

text: Σ

text: n

text: n

text: n

text: We

text: make

text: the

text: following

text: assumption

text: on

text: the

text: weight

text: sequences

text: α

text: and

text: β

text: t

text: t

text: Assumption

text: a

label: equation

text: b

text: α

text: β

text: t

text: (t

text: t

text: (t

text: where

text: ac

text: with

text: c

text: is

text: as

text: deﬁned

text: in

text: Assumption

text: The

text: weight

text: sequences

text: α

text: and

text: β

text: are

text: given

text: t

text: t

text: t

text: t

text: by

text: a

text: b

text: α

text: β

text: t

text: (t

text: t

text: (t

text: where

text: ac

text: with

text: c

text: is

text: as

text: deﬁned

text: in

text: Assumption

text: where

text: τ

text: b

text: CIGLRT

text: L

text: Decision

text: Statistic

text: Update.

text: The

text: algorithm

text: generates

text: a

text: scalar-valued

text: decision

text: statistic

text: sequence

text: zn(t)

text: at

text: the

text: n-th

text: agent

text: according

text: to

text: the

text: distributed

text: recursive

text: scheme

text: z

text: (t

text: t

text: z

text: (t)

text: (z

text: z

text: log

text: n

text: t

text: n

text: n

text: l

text: t

text: l

text: Ωn

text: local

text: innovation

text: neighborhood

text: consensus

text: H

text: H

text: where

text: f

text: and

text: f

text: represent

text: the

text: likelihoods

text: under

text: and

text: respectively,

text: θ

text: δ

text: λ

text: (L)

text: N

text: and

text: f

text: (y

text: (t))

text: h

text: (θ

text: (t))Σ

text: (θ

text: (t))

text: log

text: θn(t)

text: n

text: h

text: (θ

text: (t))Σ

text: (t)

text: n

text: n

text: n

text: n

text: n

text: f

text: (y

text: (t))

text: n

text: n

text: n

text: n

text: n

text: which

text: follows

text: due

text: to

text: the

text: Gaussian

text: noise

text: assumption

text: in

text: the

text: observation

text: model

text: in

text: However,

text: we

text: speciﬁcally

text: choose

text: δ

text: for

text: subsequent

text: analysis.

text: The

text: decision

text: statistic

text: update

text: in

text: can

text: be

text: written

text: in

text: a

text: compact

text: manner

text: as

text: follows:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 28 (cid:16) (cid:17) (cid:16) (cid:17)√ − ∗ − − 2Nσ2η + N h2(θ )2 1 N Nrk 1 1 √ η = , (75) 2 − 4h2 1 + N Nrk 1 c is deﬁned as 3 (cid:88) (cid:89) (cid:13) (cid:13) (cid:13) (cid:13) − − t1 1 t1 1 − − − (cid:62) c = α2 I β L α G Σ 1G , (76) 3 v N u u H H v=0 u=v+1 and t deﬁned as 1 { } t = max t , t , (77) 1 2 3 ∀ ≥ where t is such that, t t , 3 3 (cid:0) (cid:1) − (cid:62) λmin L + GHΣ 1GH αt < 1, (78) ∀ ≥ and t is such that9, t t , 2 2 h2 β λ (L) + α < 1. (79) t N t σ2 CIGLRT − L Theorem 5.1. Let the Assumptions B1-B5 and B7 hold. Consider, the decision statistic update of the ∗ algorithm in (31). For all θ which satisfy the following condition (cid:18) (cid:19) √ − | ∗| ≥ 1 + N√Nrk 1 Nσ2 2Nα2h2 θ 2 − − + 0 − , (80) 1 N Nrk 1 N1h2 N1σ2 (2c1α0 1) we have the following range of feasible thresholds, (cid:16) (cid:17) (cid:16) (cid:17) √ √ − ∗ − − 1 + (cid:16)Nrk 1 N (cid:17) N (hθ )2 1 N Nrk 1 N 1 1 < η < 2 √ 2Nσ2 − α2h4 1 + N Nrk 1 − 0 − , (81) σ4 (2c α 1) 1 0 → 9It is to be noted that t2 and t3 exist as αt,βt 0. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: N

text: Nrk

text: η

text: N

text: Nrk

text: c

text: is

text: deﬁned

text: as

text: c

text: α2

text: I

text: β

text: L

text: α

text: G

text: Σ

text: v

text: N

text: u

text: u

text: H

text: H

text: and

text: t

text: deﬁned

text: as

text: t

text: max

text: t

text: t

text: where

text: t

text: is

text: such

text: that,

text: t

text: t

text: λmin

text: L

text: GHΣ

text: αt

text: and

text: t

text: is

text: such

text: t

text: t

text: β

text: λ

text: (L)

text: α

text: t

text: N

text: t

text: σ2

text: CIGLRT

text: L

text: Theorem

text: Let

text: the

text: Assumptions

text: and

text: hold.

text: Consider,

text: the

text: decision

text: statistic

text: update

text: of

text: the

text: algorithm

text: in

text: For

text: all

text: θ

text: which

text: satisfy

text: the

text: following

text: condition

text: N√Nrk

text: θ

text: N

text: Nrk

text: we

text: have

text: the

text: following

text: range

text: of

text: feasible

text: thresholds,

text: N

text: N

text: (hθ

text: N

text: Nrk

text: N

text: η

text: N

text: Nrk

text: σ4

text: α

text: is

text: to

text: be

text: noted

text: that

text: and

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text: 4 CIGLRT − L , we focus on a linear observation setup, where we not only characterize thresholds and other algorithm parameters which ensure exponentially decaying probabilities of error, but also analyze the upper bounds of the associated large deviations exponent of the probabilities of error under global observability as functions of the network and model parameters. Related Work: Existing work in the literature on distributed detectors can be broadly classiﬁed into three classes. The ﬁrst class includes architectures which are characterized by presence of a fusion center and all the agents transmit their decision or local measurements or test statistics or its quantized version to the fusion center (see, for example [9], [10]) and subsequently the estimation and detection schemes are conducted by the fusion center. The second class consists of consensus schemes (see, for example [11], [12]) with no fusion center and in which in the ﬁrst phase the agents collect information over a long period of time from the environment followed by the second phase, in which agents exchange information (through consensus or gossip type procedures [11], [13], [14]) in their respective neighborhoods which are in turn speciﬁed by a pre-assigned communication graph or a sequence of possibly sparse time-varying communication graphs satisfying appropriate connectivity conditions. The third class consists of schemes which perform simultaneous assimilation of information obtained from sensing and communication (see, for example [15]–[17]). Distributed inference has been studied extensively in the literature. In particular, the diffusion and consensus+innovations schemes have been extensively used for various distributed inference problems, which include distributed parameter estimation (see, for example [18]–[20]), distributed detection (see, for example [16], [17], [21], [22]), distributed reinforcement learning (see, for example [23]), distributed information processing in presence of faulty agents or imperfect model information (see, for example [24]) and multi-task learning (see, for example [25]) to name a few. More relevant to the proposed algorithms in this paper, are distributed detection algorithms from the third class of detectors described above, which can be further sub- categorized to three classes, namely the running consensus approach [26], [27], the diffusion approach [3], [3], [17], [28]–[32] and the consensus+innovations approach [15], [16], [21]. These works address important questions pertaining to binary simple hypothesis and also characterize the fundamental limits of the detection scheme through large deviations analysis. Other relevant recent work include [33]–[35]. However, there is a fundamental difference between the algorithms proposed in this work and the algorithms discussed above. To be speciﬁc, the objective of the detection scheme in this paper is to decide between a simple null hypothesis and a composite alternative which is parameterized by a vector parameter which can take values in a continuous space, in contrast with other distributed detection schemes, where the hypotheses involved are either binary simple hypothesis (see, for example [15], [16], [21]) or multiple simple hypothesis or composite testing scenarios involving ﬁnite parametric alternatives (see, for example [33]–[36]). In a similar vein, we note that consensus or gossip type strategies have been developed for other distributed testing tasks such as in change detection [37] and quickest detection [38] problems in multi-agent networks. In the context of distributed composite hypothesis testing, the proposed algorithm involves a recursive parameter estimation update and a decision statistic update running in parallel. The proposed algorithms, CIGLRT − NL CIGLRT − L and involve parameter estimate updates in a non-linear observation model and a linear observation model setting respectively. The linear parameter estimation scheme of the consensus+innovations CIGLRT − L form which is employed in the algorithm has been studied before in [39]. Non-linear parameter

text: CIGLRT

text: L

text: we

text: focus

text: on

text: a

text: linear

text: observation

text: setup,

text: where

text: we

text: not

text: only

text: characterize

text: thresholds

text: and

text: other

text: algorithm

text: parameters

text: which

text: ensure

text: exponentially

text: decaying

text: probabilities

text: of

text: error,

text: but

text: also

text: analyze

text: the

text: upper

text: bounds

text: of

text: the

text: associated

text: large

text: deviations

text: exponent

text: of

text: the

text: probabilities

text: of

text: error

text: under

text: global

text: observability

text: as

text: functions

text: of

text: the

text: network

text: and

text: model

text: parameters.

text: Related

text: Work:

text: Existing

text: work

text: in

text: the

text: literature

text: on

text: distributed

text: detectors

text: can

text: be

text: broadly

text: classiﬁed

text: into

text: three

text: classes.

text: The

text: ﬁrst

text: class

text: includes

text: architectures

text: which

text: are

text: characterized

text: by

text: presence

text: of

text: a

text: fusion

text: center

text: and

text: all

text: the

text: agents

text: transmit

text: their

text: decision

text: or

text: local

text: measurements

text: or

text: test

text: statistics

text: or

text: its

text: quantized

text: version

text: to

text: the

text: fusion

text: center

text: (see,

text: for

text: example

text: and

text: subsequently

text: the

text: estimation

text: and

text: detection

text: schemes

text: are

text: conducted

text: by

text: the

text: fusion

text: center.

text: The

text: second

text: class

text: consists

text: of

text: consensus

text: schemes

text: (see,

text: for

text: example

text: with

text: no

text: fusion

text: center

text: and

text: in

text: which

text: in

text: the

text: ﬁrst

text: phase

text: the

text: agents

text: collect

text: information

text: over

text: a

text: long

text: period

text: of

text: time

text: from

text: the

text: environment

text: followed

text: by

text: the

text: second

text: phase,

text: in

text: which

text: agents

text: exchange

text: information

text: (through

text: consensus

text: or

text: gossip

text: type

text: procedures

text: in

text: their

text: respective

text: neighborhoods

text: which

text: are

text: in

text: turn

text: speciﬁed

text: by

text: a

text: pre-assigned

text: communication

text: graph

text: or

text: a

text: sequence

text: of

text: possibly

text: sparse

text: time-varying

text: communication

text: graphs

text: satisfying

text: appropriate

text: connectivity

text: conditions.

text: The

text: third

text: class

text: consists

text: of

text: schemes

text: which

text: perform

text: simultaneous

text: assimilation

text: of

text: information

text: obtained

text: from

text: sensing

text: and

text: communication

text: (see,

text: for

text: example

text: Distributed

text: inference

text: has

text: been

text: studied

text: extensively

text: in

text: the

text: literature.

text: In

text: particular,

text: the

text: diffusion

text: and

text: consensus+innovations

text: schemes

text: have

text: been

text: extensively

text: used

text: for

text: various

text: distributed

text: inference

text: problems,

text: which

text: include

text: distributed

text: parameter

text: estimation

text: (see,

text: for

text: example

text: distributed

text: detection

text: (see,

text: for

text: example

text: distributed

text: reinforcement

text: learning

text: (see,

text: for

text: example

text: distributed

text: information

text: processing

text: in

text: presence

text: of

text: faulty

text: agents

text: or

text: imperfect

text: model

text: information

text: (see,

text: for

text: example

text: and

text: multi-task

text: learning

text: (see,

text: for

text: example

text: to

text: name

text: a

text: few.

text: More

text: relevant

text: to

text: the

text: proposed

text: algorithms

text: in

text: this

text: paper,

text: are

text: distributed

text: detection

text: algorithms

text: from

text: the

text: third

text: class

text: of

text: detectors

text: described

text: above,

text: which

text: can

text: be

text: further

text: sub-

text: categorized

text: to

text: three

text: classes,

text: namely

text: the

text: running

text: consensus

text: approach

text: the

text: diffusion

text: approach

text: and

text: the

text: consensus+innovations

text: approach

text: These

text: works

text: address

text: important

text: questions

text: pertaining

text: to

text: binary

text: simple

text: hypothesis

text: and

text: also

text: characterize

text: the

text: fundamental

text: limits

text: of

text: the

text: detection

text: scheme

text: through

text: large

text: deviations

text: analysis.

text: Other

text: relevant

text: recent

text: work

text: include

text: However,

text: there

text: is

text: a

text: fundamental

text: difference

text: between

text: the

text: algorithms

text: proposed

text: in

text: this

text: work

text: and

text: the

text: algorithms

text: discussed

text: above.

text: To

text: be

text: speciﬁc,

text: the

text: objective

text: of

text: the

text: detection

text: scheme

text: in

text: this

text: paper

text: is

text: to

text: decide

text: between

text: a

text: simple

text: hypothesis

text: and

text: a

text: composite

text: alternative

text: which

text: is

text: parameterized

text: by

text: a

text: vector

text: parameter

text: which

text: can

text: take

text: values

text: in

text: a

text: continuous

text: space,

text: in

text: contrast

text: with

text: other

text: distributed

text: detection

text: schemes,

text: where

text: the

text: hypotheses

text: involved

text: are

text: either

text: binary

text: simple

text: hypothesis

text: (see,

text: for

text: example

text: or

text: multiple

text: simple

text: hypothesis

text: or

text: composite

text: testing

text: scenarios

text: involving

text: ﬁnite

text: parametric

text: alternatives

text: (see,

text: for

text: example

text: In

text: a

text: similar

text: vein,

text: we

text: note

text: that

text: consensus

text: or

text: gossip

text: type

text: strategies

text: have

text: been

text: developed

text: for

text: other

text: distributed

text: testing

text: tasks

text: such

text: as

text: in

text: change

text: detection

text: and

text: quickest

text: detection

text: problems

text: in

text: multi-agent

text: networks.

text: In

text: the

text: context

text: of

text: distributed

text: composite

text: hypothesis

text: testing,

text: the

text: proposed

text: algorithm

text: involves

text: a

text: recursive

text: parameter

text: estimation

text: update

text: and

text: a

text: decision

text: statistic

text: update

text: running

text: in

text: parallel.

text: The

text: proposed

text: algorithms,

text: CIGLRT

text: NL

text: CIGLRT

text: L

text: and

text: involve

text: parameter

text: estimate

text: updates

text: in

text: a

text: non-linear

text: observation

text: model

text: and

text: a

text: linear

text: observation

text: model

text: setting

text: respectively.

text: The

text: linear

text: parameter

text: estimation

text: scheme

text: of

text: the

text: consensus+innovations

text: CIGLRT

text: L

text: form

text: which

text: is

text: employed

text: in

text: the

text: algorithm

text: has

text: been

text: studied

text: before

text: in

text: Non-linear

text: parameter