text: 2. Geometric preliminaries A key principle in this paper, carried over from [CDMW] and [CMW], is local standardization of maps. The simplest kind of such standardization is simplicial approximation. It was shown in [CDMW] that on the right sort of subdivision, simplicial approximation can be performed without increasing Lipschitz constants too much. Let X be a simplicial complex with the standard metric. We say a subdivision of X is L-regular − if the simplices have all edge lengths between r 1L and rL, for some r which perhaps depends on dimension. The most common notion of subdivision used is barycentric subdivision, which is not regular—the simplices get progressively skinnier. However, several regular subdivision schemes are available, including the following: • Add a central vertex to each k-simplex to subdivide it into k + 1 cubes. Cubulate each such cube at scale 1/L, then break each small cube into simplices in a standard way. This method was described in [FW]. • Slice each simplex into approximately L slices of equal width along planes parallel to each face. This subdivides it into a ﬁnite number of types of polyhedra. Apply a standard simplicial √ subdivision to each. A speciﬁc such method is given in [EdGr]; its advantage is that r can be taken to be a constant 2, not depending even on dimension. Such subdivisions are useful for simplicial approximation of maps. Proposition 2.1 (Quantitative simplicial approximation theorem). For ﬁnite simplicial complexes (cid:48) K and L with piecewise linear metrics, there are constants C and C such that any k-Lipschitz map → (cid:48) f : K L has a Ck-Lipschitz simplicial approximation via a (Ck + C )-Lipschitz homotopy. The main purpose of simplicial approximation in this paper, as in [CDMW] and [CMW], is to bound the behaviors of maps on simplices. If there are only ﬁnitely many things a map can do on a simplex, we can bound its Lipschitz constant by the maximum Lipschitz constant of these restrictions. Therefore it is useful to extract this more general property and give it a name. F → Let be a ﬁnite set of maps ∆k Y , for some space Y . If X is a simplicial complex, a map →k F F F f : X Y is -mosaic if all of its restrictions to k-simplices are in . Here, is a semi-simplicial F k set whose simplices in degree k are , which can be formed naturally via restriction maps. We k can think of this shard complex as a ﬁnite subcomplex of the singular simplicial set of Y . F We refer to a collection of maps as uniformly mosaic if they are all -mosaic with respect to a F ﬁxed unspeciﬁed shard complex . The main advantage of this deﬁnition is that the property of being mosaic is preserved under → postcomposition. Thus for example if we have a homotopy equivalence ϕ : Z Y from a simplicial complex Z to a cell complex Y which contracts the 1-skeleton, then we can simplicially approximate → F F a map X Z, then compose with ϕ to get an -mosaic map for some ﬁxed whose 1-skeleton ϕ ϕ is a point. De Rham algebras exist in several variations, including smooth and piecewise polynomial. In this paper, we also use the algebra of simplexwise smooth forms on a simplicial complex. This has several advantages: such forms can be built skeleton-by-skeleton and this is the natural context for pullbacks of smooth forms on a manifold by simplexwise smooth functions. Given a simplicial ∗ complex X, this is the algebra we will denote Ω X; for a manifold with boundary the same notation will denote the smooth forms. A key step in both Gromov’s earliest arguments in [GrHED] and in this paper is quantitative ∞ − antidiﬀerentiation of forms: given an exact k-form with L -norm B, produce a (k 1)-form which 9 ##LTLine##

text: Geometric

text: preliminaries

text: A

text: key

text: principle

text: in

text: this

text: paper,

text: carried

text: over

text: from

text: [CDMW]

text: and

text: [CMW],

text: is

text: local

text: standardization

text: of

text: maps.

text: The

text: simplest

text: kind

text: of

text: such

text: standardization

text: is

text: simplicial

text: approximation.

text: It

text: was

text: shown

text: in

text: [CDMW]

text: that

text: on

text: the

text: right

text: sort

text: of

text: subdivision,

text: simplicial

text: approximation

text: can

text: be

text: performed

text: without

text: increasing

text: Lipschitz

text: constants

text: too

text: much.

text: Let

text: X

text: be

text: a

text: simplicial

text: complex

text: with

text: the

text: standard

text: metric.

text: We

text: say

text: a

text: subdivision

text: of

text: X

text: is

text: L-regular

text: if

text: the

text: simplices

text: have

text: all

text: edge

text: lengths

text: between

text: r

text: and

text: rL,

text: for

text: some

text: r

text: which

text: perhaps

text: depends

text: on

text: dimension.

text: The

text: most

text: common

text: notion

text: of

text: subdivision

text: used

text: is

text: barycentric

text: subdivision,

text: which

text: is

text: not

text: regular—the

text: simplices

text: get

text: progressively

text: skinnier.

text: However,

text: several

text: regular

text: subdivision

text: schemes

text: are

text: available,

text: including

text: the

text: following:

text: Add

text: a

text: central

text: vertex

text: to

text: each

text: k-simplex

text: to

text: subdivide

text: it

text: into

text: k

text: cubes.

text: Cubulate

text: each

text: such

text: cube

text: at

text: scale

text: then

text: break

text: each

text: small

text: cube

text: into

text: simplices

text: in

text: a

text: standard

text: way.

text: This

text: method

text: was

text: described

text: in

text: [FW].

text: Slice

text: each

text: simplex

text: into

text: approximately

text: L

text: slices

text: of

text: equal

text: width

text: along

text: planes

text: parallel

text: to

text: each

text: face.

text: This

text: subdivides

text: it

text: into

text: a

text: ﬁnite

text: number

text: of

text: types

text: of

text: polyhedra.

text: Apply

text: a

text: standard

text: simplicial

text: subdivision

text: to

text: each.

text: A

text: speciﬁc

text: such

text: method

text: is

text: given

text: in

text: [EdGr];

text: its

text: advantage

text: is

text: that

text: r

text: can

text: be

text: taken

text: to

text: be

text: a

text: constant

text: not

text: depending

text: even

text: on

text: dimension.

text: Such

text: subdivisions

text: are

text: useful

text: for

text: simplicial

text: approximation

text: of

text: maps.

text: Proposition

text: (Quantitative

text: simplicial

text: approximation

text: theorem).

text: For

text: ﬁnite

text: simplicial

text: complexes

text: K

text: and

text: L

text: with

text: piecewise

text: linear

text: metrics,

text: there

text: are

text: constants

text: C

text: and

text: C

text: such

text: that

text: any

text: k-Lipschitz

text: map

text: f

text: K

text: L

text: has

text: a

text: Ck-Lipschitz

text: simplicial

text: approximation

text: via

text: a

text: (Ck

text: C

text: )-Lipschitz

text: homotopy.

text: The

text: main

text: purpose

text: of

text: simplicial

text: approximation

text: in

text: this

text: paper,

text: as

text: in

text: [CDMW]

text: and

text: [CMW],

text: is

text: to

text: bound

text: the

text: behaviors

text: of

text: maps

text: on

text: simplices.

text: If

text: there

text: are

text: only

text: ﬁnitely

text: many

text: things

text: a

text: map

text: can

text: do

text: on

text: a

text: simplex,

text: we

text: can

text: bound

text: its

text: Lipschitz

text: constant

text: by

text: the

text: maximum

text: Lipschitz

text: constant

text: of

text: these

text: restrictions.

text: Therefore

text: it

text: is

text: useful

text: to

text: extract

text: this

text: more

text: general

text: property

text: and

text: give

text: it

text: a

text: name.

text: F

text: Let

text: be

text: a

text: ﬁnite

text: set

text: of

text: maps

text: ∆k

text: Y

text: for

text: some

text: space

text: Y

text: If

text: X

text: is

text: a

text: simplicial

text: complex,

text: a

text: map

text: →k

text: F

text: F

text: F

text: f

text: X

text: Y

text: is

text: -mosaic

text: if

text: all

text: of

text: its

text: restrictions

text: to

text: k-simplices

text: are

text: in

text: Here,

text: is

text: a

text: semi-simplicial

text: F

text: k

text: set

text: whose

text: simplices

text: in

text: degree

text: k

text: are

text: which

text: can

text: be

text: formed

text: naturally

text: via

text: restriction

text: maps.

text: We

text: k

text: can

text: think

text: of

text: this

text: shard

text: complex

text: as

text: a

text: ﬁnite

text: subcomplex

text: of

text: the

text: singular

text: simplicial

text: set

text: of

text: Y

text: F

text: We

text: refer

text: to

text: a

text: collection

text: of

text: maps

text: as

text: uniformly

text: mosaic

text: if

text: they

text: are

text: all

text: -mosaic

text: with

text: respect

text: to

text: a

text: F

text: ﬁxed

text: unspeciﬁed

text: shard

text: complex

text: The

text: main

text: advantage

text: of

text: this

text: deﬁnition

text: is

text: that

text: the

text: property

text: of

text: being

text: mosaic

text: is

text: preserved

text: under

text: postcomposition.

text: Thus

text: for

text: example

text: if

text: we

text: have

text: a

text: homotopy

text: equivalence

text: ϕ

text: Z

text: Y

text: from

text: a

text: simplicial

text: complex

text: Z

text: to

text: a

text: cell

text: complex

text: Y

text: which

text: contracts

text: the

text: then

text: we

text: can

text: simplicially

text: approximate

text: F

text: F

text: a

text: map

text: X

text: Z,

text: then

text: compose

text: with

text: ϕ

text: to

text: get

text: an

text: -mosaic

text: map

text: for

text: some

text: ﬁxed

text: whose

text: ϕ

text: ϕ

text: is

text: a

text: point.

text: De

text: Rham

text: algebras

text: exist

text: in

text: several

text: variations,

text: including

text: smooth

text: and

text: piecewise

text: polynomial.

text: In

text: this

text: paper,

text: we

text: also

text: use

text: the

text: algebra

text: of

text: simplexwise

text: smooth

text: forms

text: on

text: a

text: simplicial

text: complex.

text: This

text: has

text: several

text: advantages:

text: such

text: forms

text: can

text: be

text: built

text: skeleton-by-skeleton

text: and

text: this

text: is

text: the

text: natural

text: context

text: for

text: pullbacks

text: of

text: smooth

text: forms

text: on

text: a

text: manifold

text: by

text: simplexwise

text: smooth

text: functions.

text: Given

text: a

text: simplicial

text: complex

text: X,

text: this

text: is

text: the

text: algebra

text: we

text: will

text: denote

text: Ω

text: X;

text: for

text: a

text: manifold

text: with

text: boundary

text: the

text: same

text: notation

text: will

text: denote

text: the

text: smooth

text: forms.

text: A

text: key

text: step

text: in

text: both

text: Gromov’s

text: earliest

text: arguments

text: in

text: [GrHED]

text: and

text: in

text: this

text: paper

text: is

text: quantitative

text: antidiﬀerentiation

text: of

text: forms:

text: given

text: an

text: exact

text: k-form

text: with

text: L

text: -norm

text: B,

text: produce

text: a

text: (k

text: which

text: ##LTLine##


text: From Urban Segregation to Multifocal Pattern Detection Julien Randon-Furlinga, Madalina Olteanua,b, Antoine Lucquiauda a SAMM (EA4543), Universit´e Paris 1 Panth´eon Sorbonne b MaIAGE, INRA, Universit´e Paris-Saclay Abstract We develop a “multifocal” approach to reveal spatial dissimilarities in cities, from the most local scale to the metropolitan one. Think for instance of a statistical variable that may be measured at diﬀerent scales, eg ethnic group proportions, social housing rate, income distribution, or public transportation network density. Then, to any point in the city there corresponds a sequence of values for the variable, as one zooms out around the starting point, all the way up to the whole city – as if with a varifocal camera lens. The sequences thus produced encode in a precise manner spatial dissimilarities: how much they diﬀer from perfectly random sequences is indeed a signature of the underlying spatial structure. We introduce here a mathematical framework that allows to analyze this signature and we provide a number of illustrative examples. Furthermore, this method may be used to detect and explore dissimilarities due to hidden spatial or non-spatial structures in statistical populations other than geo-social data, eg in ecological, biological or physical data. Keywords: Dissimilarities, Pattern detection, Segregation, Random sequences, Ballot theorem Introduction Cities may be regarded as an epitome of complex systems: individual and institutional agents inter- act on multiple levels of numerous networks (both physical and virtual), leading to nontrivial collective behaviour and nontrivial patterns at many scales. Among a number of intricate questions that have emerged in the study of urban systems, that of sociospatial dissimilarities stands out as one that has aroused interest for decades, dating back at least to Schelling’s and Sakoda’s pioneering works on seg- regation [1, 2]. Urban segregation phenomena will serve here as archetypal examples to illustrate the mathematical methods and tools we are presenting. Segregation is often simply perceived as spatial separation of two or more groups, and therefore measured in terms of the relative proportions of each group in the diﬀerent neighbourhoods of a city [3–8]. However, segregation is essentially a spatial and multiscalar phenomenon [9–11]: an individual perceives segregation all the more acutely as she has to go a longer way from her home to discover what the city in its entirety might look like. Imagine the extreme case of a city where two groups A and B live in total separation, thus forming two ghettos. An individual living at the heart of one of the ghettos would have to explore the whole city to ﬁnd out that it actually comprises equal proportions of both groups. On the contrary, starting from some parts of the city that are better mixed, for instance on the boundary between the two ghettos, there would be no need to cover so large an area to come to the same realization. This basic observation has led us to imagine a mathematical framework that allows to capture and measure spatial dissimilarities as a multiscalar phenomenon across the city. Consider ever larger neighbourhoods around a starting point or areal unit. As detailed in Section 1, enlarging the area all the way up to the whole city produces for each point in the city a trajectory corresponding to the sequence of values taken by the variable under consideration from its value at the most local level around the starting point to its average value at the metropolitan level. So that, for a given starting point, how “long” it takes for the sequence to reach the city’s average indicates how “distant” the point is from the city’s global distribution. Further, the same mapping of points into trajectories allows to easily characterize a null model. In a perfectly mixed city (the “uniform” city), the aggregation process used to build the trajectories would be equivalent to adding units drawn at random (as when drawing balls in an urn without replacement). We introduce in Section 2 various ways of quantifying deviations from such random sequences. In both sections, we illustrate our ideas and methods with data available for the city of Paris. 1 From bespoke neighbourhoods to trajectories

text: From

label: title

text: Urban

text: Segregation

text: to

text: Multifocal

text: Pattern

text: Detection

text: Julien

label: author

text: Randon-Furlinga,

text: Madalina

text: Olteanua,b,

text: Antoine

text: Lucquiauda

text: a

label: paragraph

text: SAMM

text: Universit´e

text: Paris

text: Panth´eon

text: Sorbonne

text: b

text: MaIAGE,

text: INRA,

text: Universit´e

text: Paris-Saclay

text: Abstract

text: We

label: abstract

text: develop

text: a

text: “multifocal”

text: approach

text: to

text: reveal

text: spatial

text: dissimilarities

text: in

text: cities,

text: from

text: the

text: most

text: local

text: scale

text: to

text: the

text: metropolitan

text: one.

text: Think

text: for

text: instance

text: of

text: a

text: statistical

text: variable

text: that

text: may

text: be

text: measured

text: at

text: diﬀerent

text: scales,

text: eg

text: ethnic

text: group

text: proportions,

text: social

text: housing

text: rate,

text: income

text: distribution,

text: or

text: public

text: transportation

text: network

text: density.

text: Then,

text: to

text: any

text: point

text: in

text: the

text: city

text: there

text: corresponds

text: a

text: sequence

text: of

text: values

text: for

text: the

text: variable,

text: as

text: one

text: zooms

text: out

text: around

text: the

text: starting

text: point,

text: all

text: the

text: way

text: up

text: to

text: the

text: whole

text: city

text: as

text: if

text: with

text: a

text: varifocal

text: camera

text: lens.

text: The

text: sequences

text: thus

text: produced

text: encode

text: in

text: a

text: precise

text: manner

text: spatial

text: dissimilarities:

text: how

text: much

text: they

text: diﬀer

text: from

text: perfectly

text: random

text: sequences

text: is

text: indeed

text: a

text: signature

text: of

text: the

text: underlying

text: spatial

text: structure.

text: We

text: introduce

text: here

text: a

text: mathematical

text: framework

text: that

text: allows

text: to

text: analyze

text: this

text: signature

text: and

text: we

text: provide

text: a

text: number

text: of

text: illustrative

text: examples.

text: Furthermore,

text: this

text: method

text: may

text: be

text: used

text: to

text: detect

text: and

text: explore

text: dissimilarities

text: due

text: to

text: hidden

text: spatial

text: or

text: non-spatial

text: structures

text: in

text: statistical

text: populations

text: other

text: than

text: geo-social

text: data,

text: eg

text: in

text: ecological,

text: biological

text: or

text: physical

text: data.

text: Keywords:

text: Dissimilarities,

text: Pattern

text: detection,

text: Segregation,

text: Random

text: sequences,

text: Ballot

text: theorem

text: Introduction

text: Cities

text: may

text: be

text: regarded

text: as

text: an

text: epitome

text: of

text: complex

text: systems:

text: individual

text: and

text: institutional

text: agents

text: inter-

text: act

text: on

text: multiple

text: levels

text: of

text: numerous

text: networks

text: (both

text: physical

text: and

text: virtual),

text: leading

text: to

text: nontrivial

text: collective

text: behaviour

text: and

text: nontrivial

text: patterns

text: at

text: many

text: scales.

text: Among

text: a

text: number

text: of

text: intricate

text: questions

text: that

text: have

text: emerged

text: in

text: the

text: study

text: of

text: urban

text: systems,

text: that

text: of

text: sociospatial

text: dissimilarities

text: stands

text: out

text: as

text: one

text: that

text: has

text: aroused

text: interest

text: for

text: decades,

text: dating

text: back

text: at

text: least

text: to

text: Schelling’s

text: and

text: Sakoda’s

text: pioneering

text: works

text: on

text: seg-

text: regation

text: Urban

text: segregation

text: phenomena

text: will

text: serve

text: here

text: as

text: archetypal

text: examples

text: to

text: illustrate

text: the

text: mathematical

text: methods

text: and

text: tools

text: we

text: are

text: presenting.

text: Segregation

text: is

text: often

text: simply

text: perceived

text: as

text: spatial

text: separation

text: of

text: two

text: or

text: more

text: groups,

text: and

text: therefore

text: measured

text: in

text: terms

text: of

text: the

text: relative

text: proportions

text: of

text: each

text: group

text: in

text: the

text: diﬀerent

text: neighbourhoods

text: of

text: a

text: city

text: However,

text: segregation

text: is

text: essentially

text: a

text: spatial

text: and

text: multiscalar

text: phenomenon

text: an

text: individual

text: perceives

text: segregation

text: all

text: the

text: more

text: acutely

text: as

text: she

text: has

text: to

text: go

text: a

text: longer

text: way

text: from

text: her

text: home

text: to

text: discover

text: what

text: the

text: city

text: in

text: its

text: entirety

text: might

text: look

text: like.

text: Imagine

text: the

text: extreme

text: case

text: of

text: a

text: city

text: where

text: two

text: groups

text: A

text: and

text: B

text: live

text: in

text: total

text: separation,

text: thus

text: forming

text: two

text: ghettos.

text: An

text: individual

text: living

text: at

text: the

text: heart

text: of

text: one

text: of

text: the

text: ghettos

text: would

text: have

text: to

text: explore

text: the

text: whole

text: city

text: to

text: ﬁnd

text: out

text: that

text: it

text: actually

text: comprises

text: equal

text: proportions

text: of

text: both

text: groups.

text: On

text: the

text: contrary,

text: starting

text: from

text: some

text: parts

text: of

text: the

text: city

text: that

text: are

text: better

text: mixed,

text: for

text: instance

text: on

text: the

text: boundary

text: between

text: the

text: two

text: ghettos,

text: there

text: would

text: be

text: no

text: need

text: to

text: cover

text: so

text: large

text: an

text: area

text: to

text: come

text: to

text: the

text: same

text: realization.

text: This

text: basic

text: observation

text: has

text: led

text: us

text: to

text: imagine

text: a

text: mathematical

text: framework

text: that

text: allows

text: to

text: capture

text: and

text: measure

text: spatial

text: dissimilarities

text: as

text: a

text: multiscalar

text: phenomenon

text: across

text: the

text: city.

text: Consider

text: ever

text: larger

text: neighbourhoods

text: around

text: a

text: starting

text: point

text: or

text: areal

text: unit.

text: As

text: detailed

text: in

text: Section

text: enlarging

text: the

text: area

text: all

text: the

text: way

text: up

text: to

text: the

text: whole

text: city

text: produces

text: for

text: each

text: point

text: in

text: the

text: city

text: a

text: trajectory

text: corresponding

text: to

text: the

text: sequence

text: of

text: values

text: taken

text: by

text: the

text: variable

text: under

text: consideration

text: from

text: its

text: value

text: at

text: the

text: most

text: local

text: level

text: around

text: the

text: starting

text: point

text: to

text: its

text: average

text: value

text: at

text: the

text: metropolitan

text: level.

text: So

text: that,

text: for

text: a

text: given

text: starting

text: point,

text: how

text: “long”

text: it

text: takes

text: for

text: the

text: sequence

text: to

text: reach

text: the

text: city’s

text: average

text: indicates

text: how

text: “distant”

text: the

text: point

text: is

text: from

text: the

text: city’s

text: global

text: distribution.

text: Further,

text: the

text: same

text: mapping

text: of

text: points

text: into

text: trajectories

text: allows

text: to

text: easily

text: characterize

text: a

text: model.

text: In

text: a

text: perfectly

text: mixed

text: city

text: (the

text: “uniform”

text: city),

text: the

text: aggregation

text: process

text: used

text: to

text: build

text: the

text: trajectories

text: would

text: be

text: equivalent

text: to

text: adding

text: units

text: drawn

text: at

text: random

text: (as

text: when

text: drawing

text: balls

text: in

text: an

text: urn

text: without

text: replacement).

text: We

text: introduce

text: in

text: Section

text: various

text: ways

text: of

text: quantifying

text: deviations

text: from

text: such

text: random

text: sequences.

text: In

text: both

text: sections,

text: we

text: illustrate

text: our

text: ideas

text: and

text: methods

text: with

text: data

text: available

text: for

text: the

text: city

text: of

text: Paris.

text: From

text: bespoke

text: neighbourhoods

text: to

text: trajectories


text: 1.0 0.8 Social housing rate 0.40.6 0.2 0.0 0 200 400 600 800 1000 Neighbours Figure 2: Trajectories for the social housing rate, starting from some (10%) of the 936 statistical units (so-called IRIS) in Paris. The solid line corresponds to the city’s average (17.9%). Interpreting n as an index, one has for each starting unit i a sequence representing the trajectory that takes ξ from its value on unit i to its value on the whole grid. Formally, (f (n)) ≤ ≤ is what we call the i 1 n N trajectory of i for the variable ξ. Once trajectories have been built from the data available in a given city, a statistical analysis may be carried out in order to: • identify units that exhibit similar trajectories – this may be done for instance using clustering algorithms; • detect the crossing over between two regimes: the local one and the global one – in the former, ξ may take values signiﬁcantly distinct from the one on the whole grid; in the latter, it takes values very similar to the city’s average. Let us emphasize links between the ﬁrst point above and multiscalar approaches used in recent pa- pers [10, 22, 29]. In particular, [29] uses vertical slices or cross-sections of the trajectories deﬁned here, i.e. values at certain points only. But looking at full trajectories allows to examine the second point listed above: the number of aggregated neighbouring blocks needed around a starting block to get close enough to the city’s average value for the variable under consideration. We shall call this the “radius of convergence”. Geographically this plays the role of a local urban radius: it is a proxy for the area one needs to explore locally to obtain a reasonably good perception of the city as a whole. We give examples of such analysis further in this section (see 1.3). 1.2 Constructing trajectories from actual data G One of the statistical subtleties arises upon building datasets so that they are deﬁned on , the full set of all possible bespoke neighbourhoods in the city. Consider indeed the following standard situation: data is available in the form of a size factor s (eg i a number of inhabitants) for each spatial unit, as well as the value of a possibly multi- or even inﬁnite dimensional variable ξ. This could be the number of oﬃces or services of such and such type available in the unit, the social housing rate, an average income, a local density of public transportation, quantiles of a distribution, a full distribution... When grouping n units, one needs to compute the value of the variable on the new aggregate. This is (cid:88) done easily in the case when ξ is simply a number: ξ(G (i)) = ξ(j). (1) n ∈ j Gn(i) Similarly, ξ(G (i)) is readily computed when ξ(i) is a rate or an average: n 3 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Social

text: housing

text: rate

text: Neighbours

text: Figure

text: Trajectories

label: caption

text: for

text: the

text: social

text: housing

text: rate,

text: starting

text: from

text: some

text: of

text: the

text: statistical

text: units

text: (so-called

text: IRIS)

text: in

text: Paris.

text: The

text: solid

text: line

text: corresponds

text: to

text: the

text: city’s

text: average

text: Interpreting

text: n

text: as

text: an

text: index,

text: one

text: has

text: for

text: each

text: starting

text: unit

text: i

text: a

text: sequence

text: representing

text: the

text: trajectory

text: that

text: takes

text: ξ

text: from

text: its

text: value

text: on

text: unit

text: i

text: to

text: its

text: value

text: on

text: the

text: whole

text: grid.

text: Formally,

text: (f

text: (n))

text: is

text: what

text: we

text: call

text: the

text: i

text: n

text: N

text: trajectory

text: of

text: i

text: for

text: the

text: variable

text: ξ.

text: Once

text: trajectories

text: have

text: been

text: built

text: from

text: the

text: data

text: available

text: in

text: a

text: given

text: city,

text: a

text: statistical

text: analysis

text: may

text: be

text: carried

text: out

text: in

text: order

text: to:

text: identify

text: units

text: that

text: exhibit

text: similar

text: trajectories

text: this

text: may

text: be

text: done

text: for

text: instance

text: using

text: clustering

text: algorithms;

text: detect

text: the

text: crossing

text: over

text: between

text: two

text: regimes:

text: the

text: local

text: one

text: and

text: the

text: global

text: one

text: in

text: the

text: former,

text: ξ

text: may

text: take

text: values

text: signiﬁcantly

text: distinct

text: from

text: the

text: one

text: on

text: the

text: whole

text: grid;

text: in

text: the

text: latter,

text: it

text: takes

text: values

text: very

text: similar

text: to

text: the

text: city’s

text: average.

text: Let

text: us

text: emphasize

text: links

text: between

text: the

text: ﬁrst

text: point

text: above

text: and

text: multiscalar

text: approaches

text: used

text: in

text: recent

text: pa-

text: pers

text: In

text: particular,

text: uses

text: vertical

text: slices

text: or

text: cross-sections

text: of

text: the

text: trajectories

text: deﬁned

text: here,

text: i.e.

text: values

text: at

text: certain

text: points

text: only.

text: But

text: looking

text: at

text: full

text: trajectories

text: allows

text: to

text: examine

text: the

text: second

text: point

text: listed

text: above:

text: the

text: number

text: of

text: aggregated

text: neighbouring

text: blocks

text: needed

text: around

text: a

text: starting

text: block

text: to

text: get

text: close

text: enough

text: to

text: the

text: city’s

text: average

text: value

text: for

text: the

text: variable

text: under

text: consideration.

text: We

text: shall

text: call

text: this

text: the

text: “radius

text: of

text: convergence”.

text: Geographically

text: this

text: plays

text: the

text: role

text: of

text: a

text: local

text: urban

text: radius:

text: it

text: is

text: a

text: proxy

text: for

text: the

text: area

text: one

text: needs

text: to

text: explore

text: locally

text: to

text: obtain

text: a

text: reasonably

text: good

text: perception

text: of

text: the

text: city

text: as

text: a

text: whole.

text: We

text: give

text: examples

text: of

text: such

text: analysis

text: further

text: in

text: this

text: section

text: (see

text: Constructing

text: trajectories

text: from

text: actual

text: data

text: G

text: One

text: of

text: the

text: statistical

text: subtleties

text: arises

text: upon

text: building

text: datasets

text: so

text: that

text: they

text: are

text: deﬁned

text: on

text: the

text: full

text: set

text: of

text: all

text: possible

text: bespoke

text: neighbourhoods

text: in

text: the

text: city.

text: Consider

text: indeed

text: the

text: following

text: standard

text: situation:

text: data

text: is

text: available

text: in

text: the

text: form

text: of

text: a

text: size

text: factor

text: s

text: (eg

text: i

text: a

text: number

text: of

text: inhabitants)

text: for

text: each

text: spatial

text: unit,

text: as

text: well

text: as

text: the

text: value

text: of

text: a

text: possibly

text: multi-

text: or

text: even

text: inﬁnite

text: dimensional

text: variable

text: ξ.

text: This

text: could

text: be

text: the

text: number

text: of

text: oﬃces

text: or

text: services

text: of

text: such

text: and

text: such

text: type

text: available

text: in

text: the

text: unit,

text: the

text: social

text: housing

text: rate,

text: an

text: average

text: income,

text: a

text: local

text: density

text: of

text: public

text: transportation,

text: quantiles

text: of

text: a

text: distribution,

text: a

text: full

text: distribution...

text: When

text: grouping

text: n

text: units,

text: one

text: needs

text: to

text: compute

text: the

text: value

text: of

text: the

text: variable

text: on

text: the

text: new

text: aggregate.

text: This

text: is

text: done

text: easily

text: in

text: the

text: case

text: when

text: ξ

text: is

text: simply

text: a

text: number:

text: ξ(G

text: (i))

text: ξ(j).

text: n

text: j

text: Gn(i)

text: Similarly,

text: ξ(G

text: (i))

text: is

text: readily

text: computed

text: when

text: ξ(i)

text: is

text: a

text: rate

text: or

text: an

text: average:

text: n

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: 0.6 75111 0.20 75108 0.5 0.4 0.15 Social housing rate 0.3 Social housing rate 0.10 0.2 0.05 0.1 0.0 0.00 0 200 400 600 800 0 200 400 600 800 Number of neighbours Number of neighbours Figure 3: Trajectories for the social housing rate, starting from each IRIS (census block) in Paris’s 11th (left) and 8th (right) districts. Colours correspond to each diﬀerent IRIS taken as starting point. The solid ﬂat line gives the ± city’s average social housing rate (17.9%), the dashed lines correspond to 0.05 around this average. Solid vertical lines correspond to radii of convergence: the radius of convergence of a trajectory is deﬁned as the point where the ± path last enters the 0.05-interval (and therefore remains inside the interval afterwards). One observes that IRIS in the 11th district converge much faster to the city’s average than IRIS in the 8th district. city) tend to smooth patterns rather than exacerbate them. A particularly interesting feature revealed by our method is that the western (W) part of the city is “further away” from the whole city than the north-eastern (NE) part: trajectories for IRIS in the W part need 2 to 4 times as many aggregation steps to converge to the city’s average than IRIS in the NE part. Both parts correspond to extreme points for the variable in question, with a concentration of social housing in the NE part and a substantially lower than average rate in the W part (local social housing rates are also visible in Figure 4). Starting from these extremal points above and below the city’s average, with symmetrical geographical position in the city, the W and NE parts could have had similar radii of convergence to the city’s mean. However, this is clearly not the case, thus revealing a higher level of singularity, as far as social housing is concerned, in the W part of the city than in the NE part. Conversely, one observes a quasi ring of IRIS blocks with relatively short scales of convergence to the city’s mean. These are the historically socially mixed areas along the boulevards, the former faubourgs that used to be just outside the city’s walls before these were transformed into boulevards. Our method thus reveals a lasting imprint visible in terms of distance to the city’s average for a variable, the social housing rate, that may be taken as a proxy to social diversity. 1.3.2 Income distribution We now consider a second example, and build trajectories for the income distribution in Paris. As explained in section 1.1, when working with distributions available only through their quantiles, computing points for each group of blocks is a slightly more involved task. In this case, we chose to estimate the parameters of the best-ﬁtting distribution (which happened to be log-normal) given a block’s quantiles. Then, from this distribution, we simulated data corresponding to the number of households in the block. Thus we obtained a full ensemble of simulated households for every possible group of neighbouring blocks, from one block only to the whole city. Let us look at trajectories for IRIS blocks in three Parisian districts of comparable sizes and population numbers: the 13th, 16th and 20th districts – see Figure 5. The ﬁrst one corresponds to the south-western peripheral part of the city, the second one to the south-eastern peripheral part of the city, and the third one to the north-eastern peripheral part. The 16th tends to be further away from the full city’s picture, as it takes generally longer for its blocks to converge to the city’s distribution than for blocks in the 13th or in the 20th districts. The 13th district exhibits an interesting behaviour, with some trajectories that ﬁrst come close to the city’s mean but then bounce up again further from it. This can be understood as follows: some IRIS blocks in the 13th district belong to relatively well-mixed neighbourhoods, so initially their trajectory approaches the city’s average distribution. But the aggregation process around them leads to incorporate much less wealthy blocks on the border of the city, at a stage where wealthier blocks towards the center of the city and beyond have not yet been aggregated in large enough numbers to counterbalance the former. This sends trajectories 5 ##LTFigure## ##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##

text: Social

text: housing

text: rate

text: Social

text: housing

text: rate

text: Number

text: of

text: neighbours

text: Number

text: of

text: neighbours

text: Figure

text: Trajectories

text: for

text: the

text: social

text: housing

text: rate,

text: starting

text: from

text: each

text: IRIS

text: (census

text: block)

text: in

text: Paris’s

text: (left)

text: and

text: (right)

text: districts.

text: Colours

text: correspond

text: to

text: each

text: diﬀerent

text: IRIS

text: taken

text: as

text: starting

text: point.

text: The

text: solid

text: ﬂat

text: line

text: gives

text: the

text: city’s

text: average

text: social

text: housing

text: rate

text: the

text: dashed

text: lines

text: correspond

text: to

text: around

text: this

text: average.

text: Solid

text: vertical

text: lines

text: correspond

text: to

text: radii

text: of

text: convergence:

text: the

text: radius

text: of

text: convergence

text: of

text: a

text: trajectory

text: is

text: deﬁned

text: as

text: the

text: point

text: where

text: the

text: path

text: last

text: enters

text: the

text: (and

text: therefore

text: remains

text: inside

text: the

text: interval

text: afterwards).

text: One

text: observes

text: that

text: IRIS

text: in

text: the

text: district

text: converge

text: much

text: faster

text: to

text: the

text: city’s

text: average

text: than

text: IRIS

text: in

text: the

text: district.

text: city)

text: tend

text: to

text: smooth

text: patterns

text: rather

text: than

text: exacerbate

text: them.

text: A

text: particularly

text: interesting

text: feature

text: revealed

text: by

text: our

text: method

text: is

text: that

text: the

text: western

text: (W)

text: part

text: of

text: the

text: city

text: is

text: “further

text: away”

text: from

text: the

text: whole

text: city

text: than

text: the

text: north-eastern

text: (NE)

text: part:

text: trajectories

text: for

text: IRIS

text: in

text: the

text: W

text: part

text: need

text: to

text: times

text: as

text: many

text: aggregation

text: steps

text: to

text: converge

text: to

text: the

text: city’s

text: average

text: than

text: IRIS

text: in

text: the

text: NE

text: part.

text: Both

text: parts

text: correspond

text: to

text: extreme

text: points

text: for

text: the

text: variable

text: in

text: question,

text: with

text: a

text: concentration

text: of

text: social

text: housing

text: in

text: the

text: NE

text: part

text: and

text: a

text: substantially

text: lower

text: than

text: average

text: rate

text: in

text: the

text: W

text: part

text: (local

text: social

text: housing

text: rates

text: are

text: also

text: visible

text: in

text: Figure

text: Starting

text: from

text: these

text: extremal

text: points

text: above

text: and

text: below

text: the

text: city’s

text: average,

text: with

text: symmetrical

text: geographical

text: position

text: in

text: the

text: city,

text: the

text: W

text: and

text: NE

text: parts

text: could

text: have

text: had

text: similar

text: radii

text: of

text: convergence

text: to

text: the

text: city’s

text: mean.

text: However,

text: this

text: is

text: clearly

text: not

text: the

text: case,

text: thus

text: revealing

text: a

text: higher

text: level

text: of

text: singularity,

text: as

text: far

text: as

text: social

text: housing

text: is

text: concerned,

text: in

text: the

text: W

text: part

text: of

text: the

text: city

text: than

text: in

text: the

text: NE

text: part.

text: Conversely,

text: one

text: observes

text: a

text: quasi

text: ring

text: of

text: IRIS

text: blocks

text: with

text: relatively

text: short

text: scales

text: of

text: convergence

text: to

text: the

text: city’s

text: mean.

text: These

text: are

text: the

text: historically

text: socially

text: mixed

text: areas

text: along

text: the

text: boulevards,

text: the

text: former

text: faubourgs

text: that

text: used

text: to

text: be

text: just

text: outside

text: the

text: city’s

text: walls

text: before

text: these

text: were

text: transformed

text: into

text: boulevards.

text: Our

text: method

text: thus

text: reveals

text: a

text: lasting

text: imprint

text: visible

text: in

text: terms

text: of

text: distance

text: to

text: the

text: city’s

text: average

text: for

text: a

text: variable,

text: the

text: social

text: housing

text: rate,

text: that

text: may

text: be

text: taken

text: as

text: a

text: proxy

text: to

text: social

text: diversity.

text: Income

text: distribution

text: We

text: now

text: consider

text: a

text: second

text: example,

text: and

text: build

text: trajectories

text: for

text: the

text: income

text: distribution

text: in

text: Paris.

text: As

text: explained

text: in

text: section

text: when

text: working

text: with

text: distributions

text: available

text: only

text: through

text: their

text: quantiles,

text: computing

text: points

text: for

text: each

text: group

text: of

text: blocks

text: is

text: a

text: slightly

text: more

text: involved

text: task.

text: In

text: this

text: case,

text: we

text: chose

text: to

text: estimate

text: the

text: parameters

text: of

text: the

text: best-ﬁtting

text: distribution

text: (which

text: happened

text: to

text: be

text: log-normal)

text: given

text: a

text: block’s

text: quantiles.

text: Then,

text: from

text: this

text: distribution,

text: we

text: simulated

text: data

text: corresponding

text: to

text: the

text: number

text: of

text: households

text: in

text: the

text: block.

text: Thus

text: we

text: obtained

text: a

text: full

text: ensemble

text: of

text: simulated

text: households

text: for

text: every

text: possible

text: group

text: of

text: neighbouring

text: blocks,

text: from

text: one

text: block

text: only

text: to

text: the

text: whole

text: city.

text: Let

text: us

text: look

text: at

text: trajectories

text: for

text: IRIS

text: blocks

text: in

text: three

text: Parisian

text: districts

text: of

text: comparable

text: sizes

text: and

text: population

text: numbers:

text: the

text: and

text: districts

text: see

text: Figure

text: The

text: ﬁrst

text: one

text: corresponds

text: to

text: the

text: south-western

text: peripheral

text: part

text: of

text: the

text: city,

text: the

text: second

text: one

text: to

text: the

text: south-eastern

text: peripheral

text: part

text: of

text: the

text: city,

text: and

text: the

text: third

text: one

text: to

text: the

text: north-eastern

text: peripheral

text: part.

text: The

text: tends

text: to

text: be

text: further

text: away

text: from

text: the

text: full

text: city’s

text: picture,

text: as

text: it

text: takes

text: generally

text: longer

text: for

text: its

text: blocks

text: to

text: converge

text: to

text: the

text: city’s

text: distribution

text: than

text: for

text: blocks

text: in

text: the

text: or

text: in

text: the

text: districts.

text: The

text: district

text: exhibits

text: an

text: interesting

text: behaviour,

text: with

text: some

text: trajectories

text: that

text: ﬁrst

text: come

text: close

text: to

text: the

text: city’s

text: mean

text: but

text: then

text: bounce

text: up

text: again

text: further

text: from

text: it.

text: This

text: can

text: be

text: understood

text: as

text: follows:

text: some

text: IRIS

text: blocks

text: in

text: the

text: district

text: belong

text: to

text: relatively

text: well-mixed

text: neighbourhoods,

text: so

text: initially

text: their

text: trajectory

text: approaches

text: the

text: city’s

text: average

text: distribution.

text: But

text: the

text: aggregation

text: process

text: around

text: them

text: leads

text: to

text: incorporate

text: much

text: less

text: wealthy

text: blocks

text: on

text: the

text: border

text: of

text: the

text: city,

text: at

text: a

text: stage

text: where

text: wealthier

text: blocks

text: towards

text: the

text: center

text: of

text: the

text: city

text: and

text: beyond

text: have

text: not

text: yet

text: been

text: aggregated

text: in

text: large

text: enough

text: numbers

text: to

text: counterbalance

text: the

text: former.

text: This

text: sends

text: trajectories

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text: [5] Botticella, M.T., Pastorello, A., Smartt, S.J., et al., MNRAS, 398, 1041 (2009) [6] Cheng, Y.-C., Chen, T.-W., Prentice, S., Astron. Tel., 10374, 1 (2017) [7] Dong, S., Stanek, K.Z., Astron. Tel., 10372, 1 (2017) [8] Efremov, Yu.N., Afanasiev, V.L., Egorov, O.V., Astrophys. Bull., 66, 304 (2011) [9] Kasen, D., Woosley, S.E., Astrophys.J., 703, 2205 (2009) [10] Khan, R., Astron. Tel., 10373, 1 (2017) [11] Litvinova, I.Yu., Nadyozhin, D.K., Sov. Astr. Lett., 11, 25 (1985) [12] Maguire, K., Di Carlo, E., Smartt, S.J., et al., MNRAS, 403, 981 (2010) [13] Misra, K., Pooley, D., Chandra, P., et al., MNRAS, 381, 280 (2007) [14] Potanin, S.A., Gorbunov, I.A., Dodin, A.V., Savvin, A.D., Safonov, B.S., Shatsky, N.I., Astron. Rep., 61, 715 (2017) [15] Richardson, D., Jenkins, R.L., Wright, J., Maddox, L., Astron. J., 147, 118 (2014) [16] Sarneczky, K., Vida, K., Vinko, J., Szalai, T., Astron. Tel., 10381, 1 (2017) [17] Sclaﬂy, E.F., Finkbeiner, D.P., Astrophys. J., 737, 103 (2011) [18] Tomasella, L., Benetti, S., Cappellaro, E., et al., Astron. Tel., 10377, 1 (2017) [19] Utrobin, V.P., Chugai, N.N., Astron. Astrophys., 506, 829 (2009) [20] Van Dyk, S.D., Filippenko, A.V., Fox, O.D., et al., Astron. Tel., 10378, 1 (2017) 15

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text: Carlo,

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text: Smartt,

text: S.J.,

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text: Misra,

text: K.,

text: Pooley,

text: D.,

text: Chandra,

text: P.,

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text: MNRAS,

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text: S.A.,

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text: Dodin,

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text: B.S.,

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text: Filippenko,

text: A.V.,

text: Fox,

text: O.D.,

text: et

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text: Astron.

text: Tel.,


text: Table 1: Continued. − JD U σ B σ V σ R σ I σ Tel. U B V R I 2450000 7934.30 14.46 0.03 13.22 0.02 12.24 0.04 S100 7934.43 15.25 0.05 14.42 0.03 13.23 0.02 12.71 0.02 12.24 0.02 T60 7940.32 14.52 0.02 13.27 0.02 12.71 0.02 12.22 0.02 S100 7942.40 14.56 0.02 13.28 0.01 12.72 0.02 12.23 0.02 S100 7942.42 14.47 0.04 13.26 0.03 12.69 0.04 12.20 0.05 C70 7944.55 14.59 0.03 13.28 0.02 12.71 0.02 12.21 0.02 S100 7956.46 14.68 0.03 13.30 0.03 12.72 0.03 12.23 0.03 C70 7960.50 16.14 0.05 14.75 0.02 13.33 0.02 12.71 0.02 12.21 0.02 T60 7966.33 13.38 0.02 12.73 0.02 12.21 0.03 M20 7967.56 16.31 0.04 14.82 0.02 13.37 0.02 12.73 0.02 12.23 0.02 T60 7972.28 14.92 0.02 13.41 0.02 12.79 0.03 12.22 0.06 S100 7975.57 14.90 0.05 13.43 0.02 12.79 0.03 S60 7978.50 15.05 0.02 13.51 0.01 12.80 0.02 12.27 0.03 S100 7980.47 15.11 0.03 13.55 0.01 12.83 0.02 12.30 0.02 S100 7981.36 15.10 0.04 13.58 0.02 12.87 0.02 12.32 0.03 M20 7982.53 16.91 0.05 15.16 0.02 13.59 0.01 12.88 0.02 12.34 0.02 T60 7985.54 15.18 0.02 13.63 0.02 12.89 0.02 12.36 0.03 S100 7986.29 15.23 0.08 13.67 0.02 12.92 0.04 12.38 0.03 M20 7990.37 15.42 0.03 13.80 0.02 13.02 0.03 S60 7990.48 17.24 0.07 15.45 0.04 13.81 0.01 13.05 0.02 12.49 0.02 T60 7991.38 15.44 0.02 13.86 0.01 13.09 0.02 12.52 0.03 C60 7992.49 15.48 0.02 13.88 0.01 13.11 0.02 12.53 0.02 C60 7992.58 15.54 0.04 13.88 0.01 13.11 0.01 12.51 0.03 S60 7993.53 15.53 0.02 13.94 0.02 13.13 0.02 12.55 0.02 C60 7994.46 15.60 0.03 13.99 0.01 13.18 0.02 12.60 0.02 C60 7995.37 13.99 0.02 13.22 0.03 12.61 0.03 P40 7995.45 17.69 0.06 15.76 0.03 14.02 0.01 13.22 0.03 12.62 0.03 T60 7997.30 15.74 0.04 14.07 0.02 13.26 0.03 12.68 0.03 P40 7997.33 15.87 0.05 14.15 0.03 13.32 0.03 12.69 0.04 M20 7998.36 15.81 0.03 14.14 0.02 13.34 0.02 12.81 0.03 P40 7999.44 15.84 0.05 14.14 0.05 12.87 0.06 C70 8001.55 13.53 0.02 12.97 0.03 P40 8004.29 16.48 0.02 14.77 0.01 13.80 0.02 13.14 0.02 C60 8005.37 16.61 0.02 14.90 0.01 13.93 0.02 13.25 0.02 C60 8006.32 16.75 0.02 15.03 0.01 14.03 0.02 13.34 0.02 C60 8007.26 16.88 0.03 15.15 0.01 14.14 0.02 13.45 0.02 C60 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Continued.

text: JD

text: U

text: σ

text: B

text: σ

text: V

text: σ

text: R

text: σ

text: I

text: σ

text: Tel.

text: U

text: B

text: V

text: R

text: I

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VII. AKNOWLEDGEMENTS We are grateful to the technical staﬀ at CNBC for excellent support. Work at Lawrence Berkeley National Laboratory and UC Berkeley was supported by the Oﬃce of Science, Oﬃce of Basic Energy Sciences (BES), Materials Sciences and Engineering Division of the U.S. Department of Energy (DOE) under Contract No. DE-AC02-05-CH1231 within the Quantum Materials Program (KC2202) and BES. The work at Zhejiang University was supported by the National Basic Research Program of China (973 Program) under Grants No. 2011CBA00103 and 2012CB821404, the National Science Foundation of China (No. 11374261, 11204059) and Zhejiang Provincial Natural Science Foundation of China (No. LQ12A04007). M.S.L thanks MPIPKS, Dresden for hospitality. L.C.’s work was supported by CNPq (Grant No. 307487/2014-8). L.C. also thanks the Physical Chemistry Department at Technical University Dresden for hospitality. [1] C. Stock and E. E. McCabe, Journal of Physics: Condensed Matter 28, 453001 (2016). [2] G. Jin, Y. Wang, X. Dai, X. Ren, and L. He, Phys. Rev. B 94, 075150 (2016). [3] R. K. Oogarah, C. P. J. Stockdale, C. Stock, J. S. O. Evans, A. S. Wills, J. W. Taylor, and E. E. McCabe, Phys. Rev. B 95, 174441 (2017). [4] L. Craco, M. S. Laad, and S. Leoni, J. Phys.: Condensed Matter 26, 145602 (2014). [5] J.-X. Zhu, R. Yu, H. Wang, L. L. Zhao, M. D. Jones, J. Dai, E. Abrahams, E. Morosan, M. Fang, and Q. Si, Phys. Rev. Lett. 104, 216405 (2010). [6] H. Lei, H. Ryu, V. Ivanovski, J. B. Warren, A. I. Frenkel, B. Cekic, W.-G. Yin, and C. Petro- vic, Phys. Rev. B 86, 195133 (2012). [7] S. Landsgesell, K. Prokeˇs, C. Colin, D. Abou-Ras, and N. Sch¨afer, Journal of Superconduc- tivity and Novel Magnetism 28, 1111 (2015). [8] J. M. Mayer, L. F. Schneemeyer, T. Siegrist, J. V. Waszczak, and B. Van Dover, Angewandte Chemie International Edition in English 31, 1645 (1992). [9] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev. Lett. 104, 106408 (2010). [10] B. Freelon, Y. H. Liu, J.-L. Chen, L. Craco, M. S. Laad, S. Leoni, J. Chen, L. Tao, H. Wang, R. Flauca, Z. Yamani, M. Fang, C. Chang, J.-H. Guo, and Z. Hussain, Phys. Rev. B 92, 18 ##LTLine## ##LTLine## ##LTLine##

text: VII.

label: section

text: AKNOWLEDGEMENTS

text: We

text: are

text: grateful

text: to

text: the

text: technical

text: staﬀ

text: at

text: CNBC

text: for

text: excellent

text: support.

text: Work

text: at

text: Lawrence

text: Berkeley

text: National

text: Laboratory

text: and

text: UC

text: Berkeley

text: was

text: supported

text: by

text: the

text: Oﬃce

text: of

text: Science,

text: Oﬃce

text: of

text: Basic

text: Energy

text: Sciences

text: (BES),

text: Materials

text: Sciences

text: and

text: Engineering

text: Division

text: of

text: the

text: U.S.

text: Department

text: of

text: Energy

text: (DOE)

text: under

text: Contract

text: No.

text: within

text: the

text: Quantum

text: Materials

text: Program

text: and

text: BES.

text: The

text: work

text: at

text: Zhejiang

text: University

text: was

text: supported

text: by

text: the

text: National

text: Basic

text: Research

text: Program

text: of

text: China

text: Program)

text: under

text: Grants

text: No.

text: and

text: the

text: National

text: Science

text: Foundation

text: of

text: China

text: (No.

text: and

text: Zhejiang

text: Provincial

text: Natural

text: Science

text: Foundation

text: of

text: China

text: (No.

text: M.S.L

text: thanks

text: MPIPKS,

text: Dresden

text: for

text: hospitality.

text: L.C.’s

text: work

text: was

text: supported

text: by

text: CNPq

text: (Grant

text: No.

text: L.C.

text: also

text: thanks

text: the

text: Physical

text: Chemistry

text: Department

text: at

text: Technical

text: University

text: Dresden

text: for

text: hospitality.

text: C.

text: Stock

text: and

text: E.

text: E.

text: McCabe,

text: Journal

text: of

text: Physics:

text: Condensed

text: Matter

text: G.

text: Jin,

text: Y.

text: Wang,

text: X.

text: Dai,

text: X.

text: Ren,

text: and

text: L.

text: He,

text: Phys.

text: Rev.

text: B

text: R.

text: K.

text: Oogarah,

text: C.

text: P.

text: J.

text: Stockdale,

text: C.

text: Stock,

text: J.

text: S.

text: O.

text: Evans,

text: A.

text: S.

text: Wills,

text: J.

text: W.

text: Taylor,

text: and

text: E.

text: E.

text: McCabe,

text: Phys.

text: Rev.

text: B

text: L.

text: Craco,

text: M.

text: S.

text: Laad,

text: and

text: S.

text: Leoni,

text: J.

text: Phys.:

text: Condensed

text: Matter

text: J.-X.

text: Zhu,

text: R.

text: Yu,

text: H.

text: Wang,

text: L.

text: L.

text: Zhao,

text: M.

text: D.

text: Jones,

text: J.

text: Dai,

text: E.

text: Abrahams,

text: E.

text: Morosan,

text: M.

text: Fang,

text: and

text: Q.

text: Si,

text: Phys.

text: Rev.

text: Lett.

text: H.

text: Lei,

text: H.

text: Ryu,

text: V.

text: Ivanovski,

text: J.

text: B.

text: Warren,

text: A.

text: I.

text: Frenkel,

text: B.

text: Cekic,

text: W.-G.

text: Yin,

text: and

text: C.

text: Petro-

text: vic,

text: Phys.

text: Rev.

text: B

text: S.

text: Landsgesell,

text: K.

text: Prokeˇs,

text: C.

text: Colin,

text: D.

text: Abou-Ras,

text: and

text: N.

text: Sch¨afer,

text: Journal

text: of

text: Superconduc-

text: tivity

text: and

text: Novel

text: Magnetism

text: J.

text: M.

text: Mayer,

text: L.

text: F.

text: Schneemeyer,

text: T.

text: Siegrist,

text: J.

text: V.

text: Waszczak,

text: and

text: B.

text: Van

text: Dover,

text: Angewandte

text: Chemie

text: International

text: Edition

text: in

text: English

text: M.

text: Dzero,

text: K.

text: Sun,

text: V.

text: Galitski,

text: and

text: P.

text: Coleman,

text: Phys.

text: Rev.

text: Lett.

text: B.

text: Freelon,

text: Y.

text: H.

text: Liu,

text: J.-L.

text: Chen,

text: L.

text: Craco,

text: M.

text: S.

text: Laad,

text: S.

text: Leoni,

text: J.

text: Chen,

text: L.

text: Tao,

text: H.

text: Wang,

text: R.

text: Flauca,

text: Z.

text: Yamani,

text: M.

text: Fang,

text: C.

text: Chang,

text: J.-H.

text: Guo,

text: and

text: Z.

text: Hussain,

text: Phys.

text: Rev.

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: including RE O TM O(Ch) where TM = Co, Mn, Fe and Fe − Mn and Ch are the chalo- 2 2 2 2 1 x x gens S and Se. [18] [5, 19–23] In addition, new compounds, A F TM O(Ch) where A = 2 2 2 2 (Ba, Sr) have been the subject of recent investigation. [24] These materials were shown to order antiferromagnetically; the material was proposed to be an example of a frustrated AFM checkerboard spin lattice. [24] β-La O (Fe − Mn ) O(S, Se) and Ce O FeSe [25–27] 2 2 1 x x 2 2 2 2 2 Na Fe OSe also contains a [Fe OSe ] unit, albeit separated by alternating double layers 2 2 2 2 2 of Na+, as found in FeOCh. Na Fe OSe exhibits the onset of AFM ordering at 73 K and 2 2 2 2D short-range magnetic correlation well above T . [28] Craco et. al. predicted orbital- N selective Mott behavior [4] in the electronic behavior of Na Fe OSe and further suggested 2 2 2 that charge doping or external pressure would induce the simultaneous occurrence of Mott insulating and coherent charge carrier behaviors. Landsgesell investigated the eﬀects of F-doping in La O − F Fe Se (x from 0.0 to 0.5) 2 3 x x 2 2 using neutron diﬀraction and magnetic bulk measurements. [7] With increasing x, the size of the crystallographic unit cell decreases as well as the amplitude of the ordered mag- netic moments, while the conﬁguration of the magnetic order and the development of the resistivity on cooling remains unchanged. [7] Doping via the substitution of Mn for Fe in La O Fe − Mn OSe did not lead to observable HTSC; however, several diﬀerent AFM 2 2 1 x x 2 phases were identiﬁed over the broad doping range of x = 0.1 to 0.9. [20] [23] In this work, we present the results of a neutron diﬀraction study aimed at compar- ing the structural and magnetic properties of the iron oxychalcogenides La O Fe OS and 2 2 2 2 La O Fe OSe . Our focus is on powder materials since single-crystalline samples remain 2 2 2 2 diﬃcult to produce. To the best of our knowledge, there have been no reports of single crystalline La O Fe O(S, Se) . We employ neutron powder diﬀraction (NPD) to measure 2 2 2 2 the nuclear and magnetic Bragg scattering intensity as a function of temperature and we examine the structural distinctions between the two parent compounds at room and low temperatures. Section II gives an overview of the materials and the synthesis while Sec- tion III provides experimental details of the neutron measurements. Section IV gives the results of structural and magnetic measurements of M =(S, Se). In addition, we discuss the magnetic structure and the magnetic order parameter behavior. Section V provides a dis- cussion of our ﬁndings relative to other oxychalcogenides reports in the literature. Finally, our results are discussed in the context of theoretical ﬁndings that have been reported on La O Fe O(S, Se) systems. 2 2 2 2 4

text: including

text: RE

text: O

text: TM

text: O(Ch)

text: where

text: TM

text: Co,

text: Mn,

text: Fe

text: and

text: Fe

text: Mn

text: and

text: Ch

text: are

text: the

text: chalo-

text: x

text: x

text: gens

text: S

text: and

text: Se.

text: In

text: addition,

text: new

text: compounds,

text: A

text: F

text: TM

text: O(Ch)

text: where

text: A

text: (Ba,

text: Sr)

text: have

text: been

text: the

text: subject

text: of

text: recent

text: investigation.

text: These

text: materials

text: were

text: shown

text: to

text: order

text: antiferromagnetically;

text: the

text: material

text: was

text: proposed

text: to

text: be

text: an

text: example

text: of

text: a

text: frustrated

text: AFM

text: checkerboard

text: spin

text: lattice.

text: β-La

text: O

text: (Fe

text: Mn

text: O(S,

text: Se)

text: and

text: Ce

text: O

text: FeSe

text: x

text: x

text: Fe

text: OSe

text: also

text: contains

text: a

text: [Fe

text: OSe

text: unit,

text: albeit

text: separated

text: by

text: alternating

text: double

text: layers

text: of

text: Na+,

text: as

text: found

text: in

text: FeOCh.

text: Fe

text: OSe

text: exhibits

text: the

text: onset

text: of

text: AFM

text: ordering

text: at

text: K

text: and

text: short-range

text: magnetic

text: correlation

text: well

text: above

text: T

text: Craco

text: et.

text: al.

text: predicted

text: orbital-

text: N

text: selective

text: Mott

text: behavior

text: in

text: the

text: electronic

text: behavior

text: of

text: Fe

text: OSe

text: and

text: further

text: suggested

text: that

text: charge

text: doping

text: or

text: external

text: pressure

text: would

text: induce

text: the

text: simultaneous

text: occurrence

text: of

text: Mott

text: insulating

text: and

text: coherent

text: charge

text: carrier

text: behaviors.

text: Landsgesell

text: investigated

text: the

text: eﬀects

text: of

text: F-doping

text: in

text: La

text: O

text: F

text: Fe

text: Se

text: (x

text: from

text: to

text: x

text: x

text: using

text: neutron

text: diﬀraction

text: and

text: magnetic

text: bulk

text: measurements.

text: With

text: increasing

text: x,

text: the

text: size

text: of

text: the

text: crystallographic

text: unit

text: cell

text: decreases

text: as

text: well

text: as

text: the

text: amplitude

text: of

text: the

text: ordered

text: mag-

text: netic

text: moments,

text: while

text: the

text: conﬁguration

text: of

text: the

text: magnetic

text: order

text: and

text: the

text: development

text: of

text: the

text: resistivity

text: on

text: cooling

text: remains

text: unchanged.

text: Doping

text: via

text: the

text: substitution

text: of

text: Mn

text: for

text: Fe

text: in

text: La

text: O

text: Fe

text: Mn

text: OSe

text: did

text: not

text: lead

text: to

text: observable

text: HTSC;

text: however,

text: several

text: diﬀerent

text: AFM

text: x

text: x

text: phases

text: were

text: identiﬁed

text: over

text: the

text: broad

text: doping

text: range

text: of

text: x

text: to

text: In

text: this

text: work,

text: we

text: present

text: the

text: results

text: of

text: a

text: neutron

text: diﬀraction

text: study

text: aimed

text: at

text: compar-

text: ing

text: the

text: structural

text: and

text: magnetic

text: properties

text: of

text: the

text: iron

text: oxychalcogenides

text: La

text: O

text: Fe

text: OS

text: and

text: La

text: O

text: Fe

text: OSe

text: Our

text: focus

text: is

text: on

text: powder

text: materials

text: since

text: single-crystalline

text: samples

text: remain

text: diﬃcult

text: to

text: produce.

text: To

text: the

text: best

text: of

text: our

text: knowledge,

text: there

text: have

text: been

text: no

text: reports

text: of

text: single

text: crystalline

text: La

text: O

text: Fe

text: O(S,

text: Se)

text: We

text: employ

text: neutron

text: powder

text: diﬀraction

text: (NPD)

text: to

text: measure

text: the

text: nuclear

text: and

text: magnetic

text: Bragg

text: scattering

text: intensity

text: as

text: a

text: function

text: of

text: temperature

text: and

text: we

text: examine

text: the

text: structural

text: distinctions

text: between

text: the

text: two

text: parent

text: compounds

text: at

text: room

text: and

text: low

text: temperatures.

text: Section

text: II

text: gives

text: an

text: overview

text: of

text: the

text: materials

text: and

text: the

text: synthesis

text: while

text: Sec-

text: tion

text: III

text: provides

text: experimental

text: details

text: of

text: the

text: neutron

text: measurements.

text: Section

text: IV

text: gives

text: the

text: results

text: of

text: structural

text: and

text: magnetic

text: measurements

text: of

text: M

text: =(S,

text: Se).

text: In

text: addition,

text: we

text: discuss

text: the

text: magnetic

text: structure

text: and

text: the

text: magnetic

text: order

text: parameter

text: behavior.

text: Section

text: V

text: provides

text: a

text: dis-

text: cussion

text: of

text: our

text: ﬁndings

text: relative

text: to

text: other

text: oxychalcogenides

text: reports

text: in

text: the

text: literature.

text: Finally,

text: our

text: results

text: are

text: discussed

text: in

text: the

text: context

text: of

text: theoretical

text: ﬁndings

text: that

text: have

text: been

text: reported

text: on

text: La

text: O

text: Fe

text: O(S,

text: Se)

text: systems.


text: T = 280 K ) s nit u b. r a ( y sit n e nt I T = 4 K ) s nit u b. r a ( y sit n e nt I FIG. 3. (Color online) Rietveld reﬁnement proﬁles using (a) 280 K and (b) low-temperature, 4 K, data for La O Fe OS collected using the C2 diﬀractometer with wavelength λ =1.33 ˚A. The data 2 2 2 2 were reﬁned using the space group I4/mmm. Observed and calculated patterns are shown in red and black, respectively, with the diﬀerence proﬁle (blue) and nuclear Bragg peak positions shown as blue given vertical tick marks. (100 pm) is smaller than selenium (115 pm). The lattice parameters of La O Fe OS (a 2 2 2 2 =4.04539˚A, c = 17.9036˚A) are smaller than those of La O Fe OSe (a =4.0877˚A, c = 2 2 2 2 18.6005˚A). Compared to the Fe-Fe atomic distances d − values reported in LaFeOAs, Fe Fe the interatomic distances we obtained (in Table II) are larger by 1.2 % and nearly 1.1% for M = Se and S, respectively. Bond angles and atomic distances, extracted from Rietveld 8 ##LTFigure## ##LTFigure## ##LTFigure##

text: T

text: K

text: s

text: nit

text: u

text: b.

text: r

text: a

text: y

text: sit

text: n

text: e

text: nt

text: I

text: T

text: K

text: s

text: nit

text: u

text: b.

text: r

text: a

text: y

text: sit

text: n

text: e

text: nt

text: I

text: FIG.

text: (Color

text: online)

text: Rietveld

text: reﬁnement

text: proﬁles

text: using

text: (a)

text: K

text: and

text: (b)

text: low-temperature,

text: K,

text: data

text: for

text: La

text: O

text: Fe

text: OS

text: collected

text: using

text: the

text: diﬀractometer

text: with

text: wavelength

text: λ

text: ˚A.

text: The

text: data

text: were

text: reﬁned

text: using

text: the

text: space

text: group

text: Observed

text: and

text: calculated

text: patterns

text: are

text: shown

text: in

text: red

text: and

text: black,

text: respectively,

text: with

text: the

text: diﬀerence

text: proﬁle

text: (blue)

text: and

text: nuclear

text: Bragg

text: peak

text: positions

text: shown

text: as

text: blue

text: given

text: vertical

text: tick

text: marks.

text: pm)

text: is

text: smaller

text: than

text: selenium

text: pm).

text: The

text: lattice

text: parameters

text: of

text: La

text: O

text: Fe

text: OS

text: (a

text: c

text: are

text: smaller

text: than

text: those

text: of

text: La

text: O

text: Fe

text: OSe

text: (a

text: c

text: Compared

text: to

text: the

text: Fe-Fe

text: atomic

text: distances

text: d

text: values

text: reported

text: in

text: LaFeOAs,

text: Fe

text: Fe

text: the

text: interatomic

text: distances

text: we

text: obtained

text: (in

text: Table

text: II)

text: are

text: larger

text: by

text: and

text: nearly

text: for

text: M

text: Se

text: and

text: S,

text: respectively.

text: Bond

text: angles

text: and

text: atomic

text: distances,

text: extracted

text: from

text: Rietveld

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: direction. If the grids are dense, many blocks may not have any observed events, resulting in an estimated hazard-rate matrix with many zeros; if the grids are coarse, most blocks would have multiple events, leading to a matrix estimate with few zero entries. Namely, the estimated hazard function has large variability. In addition, a less obvious issue is the (cid:80) probable occurrence of missing values. If there is a lack of observations with long duration at a time of day (e.g., if n t = 0 for arrival interval j and waiting time interval k), i=1 ijk the MLE will be undeﬁned for that particular block. It leads to a hazard-rate matrix estimated with missing values. If we want to apply an arbitrary smoother (e.g., tF-SVD) to the estimated hazard rate matrix, we have to either impute the missing values or shorten the duration range – either way is restrictive and ad hoc. To address all the issues, below we introduce the tF-Hazards model. The tF-Hazards model assumes that the two-way hazard function h(a, t) consists of r components (or layers), each being a product of two univariate functions. Namely, · · · h(a, t) = u (a)v (t) + + u (a)v (t), (3) 1 1 r r · · · · · · where u (a), , u (a) and v (t), , v (t) are smooth univariate functions of the arrival 1 r 1 r ≤ ≤ time 0 < a α and the waiting time 0 < t τ, respectively. There is no sign restriction ≥ on the functions, other than h(a, t) 0 for any (a, t) in the domain. The multiplicative structure in each layer nicely separates the time-of-day eﬀect from the duration eﬀect on hazard rates. The sum of the r components makes the model ﬂexible enough to capture complex patterns of a hazard surface in a simple way. We remark that when r = 1, the tF-Hazards model degenerates to a multiplicative model in the following form h(a, t) = u(a)v(t). ≥ ≥ ≥ Since h(a, t) 0, without loss of generality, we have u(a) 0 and v(t) 0. In particular, 9

text: direction.

text: If

text: the

text: grids

text: are

text: dense,

text: many

text: blocks

text: may

text: not

text: have

text: any

text: observed

text: events,

text: resulting

text: in

text: an

text: estimated

text: hazard-rate

text: matrix

text: with

text: many

text: zeros;

text: if

text: the

text: grids

text: are

text: coarse,

text: most

text: blocks

text: would

text: have

text: multiple

text: events,

text: leading

text: to

text: a

text: matrix

text: estimate

text: with

text: few

text: zero

text: entries.

text: Namely,

text: the

text: estimated

text: hazard

text: function

text: has

text: large

text: variability.

text: In

text: addition,

text: a

text: less

text: obvious

text: issue

text: is

text: the

text: probable

text: occurrence

text: of

text: missing

text: values.

text: If

text: there

text: is

text: a

text: lack

text: of

text: observations

text: with

text: long

text: duration

text: at

text: a

text: time

text: of

text: day

text: (e.g.,

text: if

text: n

text: t

text: for

text: arrival

text: interval

text: j

text: and

text: waiting

text: time

text: interval

text: k),

text: ijk

text: the

text: MLE

text: will

text: be

text: undeﬁned

text: for

text: that

text: particular

text: block.

text: It

text: leads

text: to

text: a

text: hazard-rate

text: matrix

text: estimated

text: with

text: missing

text: values.

text: If

text: we

text: want

text: to

text: apply

text: an

text: arbitrary

text: smoother

text: (e.g.,

text: tF-SVD)

text: to

text: the

text: estimated

text: hazard

text: rate

text: matrix,

text: we

text: have

text: to

text: either

text: impute

text: the

text: missing

text: values

text: or

text: shorten

text: the

text: duration

text: range

text: either

text: way

text: is

text: restrictive

text: and

text: ad

text: hoc.

text: To

text: address

text: all

text: the

text: issues,

text: below

text: we

text: introduce

text: the

text: tF-Hazards

text: model.

text: The

text: tF-Hazards

text: model

text: assumes

text: that

text: the

text: two-way

text: hazard

text: function

text: h(a,

text: t)

text: consists

text: of

text: r

text: components

text: (or

text: layers),

text: each

text: being

text: a

text: product

text: of

text: two

text: univariate

text: functions.

text: Namely,

text: h(a,

text: t)

text: u

text: (a)v

text: (t)

text: u

text: (a)v

text: (t),

text: r

text: r

text: where

text: u

text: (a),

text: u

text: (a)

text: and

text: v

text: (t),

text: v

text: (t)

text: are

text: smooth

text: univariate

text: functions

text: of

text: the

text: arrival

text: r

text: r

text: time

text: a

text: α

text: and

text: the

text: waiting

text: time

text: t

text: τ,

text: respectively.

text: There

text: is

text: no

text: sign

text: restriction

text: on

text: the

text: functions,

text: other

text: than

text: h(a,

text: t)

text: for

text: any

text: (a,

text: t)

text: in

text: the

text: domain.

text: The

text: multiplicative

text: structure

text: in

text: each

text: layer

text: nicely

text: separates

text: the

text: time-of-day

text: eﬀect

text: from

text: the

text: duration

text: eﬀect

text: on

text: hazard

text: rates.

text: The

text: sum

text: of

text: the

text: r

text: components

text: makes

text: the

text: model

text: ﬂexible

text: enough

text: to

text: capture

text: complex

text: patterns

text: of

text: a

text: hazard

text: surface

text: in

text: a

text: simple

text: way.

text: We

text: remark

text: that

text: when

text: r

text: the

text: tF-Hazards

text: model

text: degenerates

text: to

text: a

text: multiplicative

text: model

text: in

text: the

text: following

text: form

text: h(a,

text: t)

text: u(a)v(t).

text: Since

text: h(a,

text: t)

text: without

text: loss

text: of

text: generality,

text: we

text: have

text: u(a)

text: and

text: v(t)

text: In

text: particular,


text: the following augmented Lagrangian function: C (cid:44) − L ρP (H, U, V, Θ) log (H) + (U, V) 2 (7) − ρ(cid:107) − (cid:107) + < Θ, H UVT > + H UVT 2 , 2 F ≥ with respect to U, V, and H under the sign constraint H 0. The third term is the La- · · grange multiplier with Θ being the Lagrange parameter matrix, with < , > denoting the matrix inner product induced by the Frobenius norm. The fourth term is the augmentation term where we particularly set ρ to be the augmentation parameter. The function (7) is not convex with respect to all variables and thus simultaneous optimization is formidable. As an alternative, we propose to use the ADMM algorithm to solve (7) iteratively. In particular, for the ith iteration, the iterative scheme alternates among the following three steps until convergence: (cid:0) (cid:1) (cid:44) C − − − H(l) arg min (H, U(l 1), V(l 1), Θ(l 1)), (8) ≥ H 0 (cid:44) C (cid:16) − (cid:17) U(l), V(l) arg min (H(l), U, V, Θ(l 1)), (9) (U,V) (cid:44) − − Θ(l) Θ(l 1) + ρ H(l) U(l)V(l)T , (10) where the ﬁrst two steps are partial optimizations of H given (U, V) and of (U, V) given H, respectively, and the third step is a dual update of the Lagrange parameter Θ. The tuning parameter ρ serves as the step size in the dual update. Below we give more details of each step. When there is no confusion, we shall drop the superscript indicating iterations. The optimization in (8) is a convex minimization problem. It can be well separated into mp univariate optimization problems, each corresponding to an entry of H. In particular, 13 ##LTLine## ##LTLine##

text: the

text: following

text: augmented

text: Lagrangian

text: function:

text: C

text: L

text: ρP

text: (H,

text: U,

text: V,

text: Θ)

text: log

text: (H)

text: (U,

text: V)

text: Θ,

text: H

text: UVT

text: H

text: UVT

text: F

text: with

text: respect

text: to

text: U,

text: V,

text: and

text: H

text: under

text: the

text: sign

text: constraint

text: H

text: The

text: third

text: term

text: is

text: the

text: La-

text: grange

text: multiplier

text: with

text: Θ

text: being

text: the

text: Lagrange

text: parameter

text: matrix,

text: with

text: denoting

text: the

text: matrix

text: inner

text: product

text: induced

text: by

text: the

text: Frobenius

text: norm.

text: The

text: fourth

text: term

text: is

text: the

text: augmentation

text: term

text: where

text: we

text: particularly

text: set

text: ρ

text: to

text: be

text: the

text: augmentation

text: parameter.

text: The

text: function

text: is

text: not

text: convex

text: with

text: respect

text: to

text: all

text: variables

text: and

text: thus

text: simultaneous

text: optimization

text: is

text: formidable.

text: As

text: an

text: alternative,

text: we

text: propose

text: to

text: use

text: the

text: ADMM

text: algorithm

text: to

text: solve

text: iteratively.

text: In

text: particular,

text: for

text: the

text: ith

text: iteration,

text: the

text: iterative

text: scheme

text: alternates

text: among

text: the

text: following

text: three

text: steps

text: until

text: convergence:

text: C

text: H(l)

text: arg

text: min

text: (H,

text: U(l

text: V(l

text: Θ(l

text: H

text: C

text: U(l),

text: V(l)

text: arg

text: min

text: (H(l),

text: U,

text: V,

text: Θ(l

text: (U,V)

text: Θ(l)

text: Θ(l

text: ρ

text: H(l)

text: U(l)V(l)T

text: where

text: the

text: ﬁrst

text: two

text: steps

text: are

text: partial

text: optimizations

text: of

text: H

text: given

text: (U,

text: V)

text: and

text: of

text: (U,

text: V)

text: given

text: H,

text: respectively,

text: and

text: the

text: third

text: step

text: is

text: a

text: dual

text: update

text: of

text: the

text: Lagrange

text: parameter

text: Θ.

text: The

text: tuning

text: parameter

text: ρ

text: serves

text: as

text: the

text: step

text: size

text: in

text: the

text: dual

text: update.

text: Below

text: we

text: give

text: more

text: details

text: of

text: each

text: step.

text: When

text: there

text: is

text: no

text: confusion,

text: we

text: shall

text: drop

text: the

text: superscript

text: indicating

text: iterations.

text: The

text: optimization

text: in

text: is

text: a

text: convex

text: minimization

text: problem.

text: It

text: can

text: be

text: well

text: separated

text: into

text: mp

text: univariate

text: optimization

text: problems,

text: each

text: corresponding

text: to

text: an

text: entry

text: of

text: H.

text: In

text: particular,

text: ##LTLine##

text: ##LTLine##


text: (cid:88) (cid:88) with respect to the (j, k)th entry h , we have the following optimization problem: jk − n n (cid:0) (cid:1) (cid:0) (cid:1) min d log h + t h ≥ ijk jk ijk jk h 0 jk i=1 i=1 − ρ − + θ h uT v + h uT v 2 , jk jk (j) (k) 2 j(cid:80)k (j) (k) (cid:80) × where θ is the jkth entry of Θ, and u and v are r 1 vectors of the jth row of U jk (j) (k) (cid:44) (cid:44) and the kth row of V respectively. We denote t· n t and d· n d for jk i=1 ijk jk i=1 ijk simplicity. Equating the ﬁrst order derivative of the objective function to zero, we obtain (cid:114) (cid:16) (cid:17) a unique closed-form solution of the optimization problem as (cid:101) − − − 2 (t· ρuT v + θ ) + t· ρuT v + θ + 4ρd· jk (j) (k) jk jk (j) (k) jk jk h = (cid:101) (cid:98) . (11) jk 2ρ (cid:101) We now compare the estimator h (11) with h = d· /t· , i.e. the MLEs under the jk jk jk jk (cid:98) piecewise constant two-way hazards model of Section . In a nutshell, our estimator h jk (cid:101) has several desirable properties and address the issues of the MLE h (discussed at the jk beginning of Section ). First, h exploits information in all the data points through u , jk (j) (cid:101) v and θ , all of which are estimated from a global view. Therefore, it is less sensitive (k) jk (cid:101) to diﬀerent choices of grids. Secondly, the value of h is similar across adjacent blocks, jk (cid:101) making it a smooth estimator. This is because h depends on u and v which have jk (j) (k) similar values in neighboring rows of U and V respectively. In particular, the estimator h jk is dominated by uT v when d· and t· are small (i.e., when we do not have enough data (j) (k) jk jk (cid:101) in the jkth block). Finally, the estimator is well deﬁned even when there is no observation − in a block. When t· = d· = 0, we have h = max(uT v θ /ρ , 0). jk jk jk (j) (k) jk Next, to obtain an estimate of (U, V) from (9), we rewrite the optimization problem equivalently as (cid:107) Θ − (cid:107) P min H + UVT 2 + (U, V), (12) (U,V) ρ F 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: respect

text: to

text: the

text: (j,

text: k)th

text: entry

text: h

text: we

text: have

text: the

text: following

text: optimization

text: problem:

text: jk

text: n

text: n

text: min

text: d

text: log

text: h

text: t

text: h

text: ijk

text: jk

text: ijk

text: jk

text: h

text: jk

text: ρ

text: θ

text: h

text: uT

text: v

text: h

text: uT

text: v

text: jk

text: jk

text: (j)

text: (k)

text: (j)

text: (k)

text: where

text: θ

text: is

text: the

text: jkth

text: entry

text: of

text: Θ,

text: and

text: u

text: and

text: v

text: are

text: r

text: vectors

text: of

text: the

text: jth

text: row

text: of

text: U

text: jk

text: (j)

text: (k)

text: and

text: the

text: kth

text: row

text: of

text: V

text: respectively.

text: We

text: denote

text: t·

text: n

text: t

text: and

text: d·

text: n

text: d

text: for

text: jk

text: ijk

text: jk

text: ijk

text: simplicity.

text: Equating

text: the

text: ﬁrst

text: order

text: derivative

text: of

text: the

text: objective

text: function

text: to

text: zero,

text: we

text: obtain

text: a

text: unique

text: closed-form

text: solution

text: of

text: the

text: optimization

text: problem

text: as

text: (t·

text: ρuT

text: v

text: θ

text: t·

text: ρuT

text: v

text: θ

text: 4ρd·

text: jk

text: (j)

text: (k)

text: jk

text: jk

text: (j)

text: (k)

text: jk

text: jk

text: h

text: jk

text: 2ρ

text: We

text: now

text: compare

text: the

text: estimator

text: h

text: with

text: h

text: d·

text: /t·

text: i.e.

text: the

text: MLEs

text: under

text: the

text: jk

text: jk

text: jk

text: jk

text: piecewise

text: constant

text: two-way

text: hazards

text: model

text: of

text: Section

text: In

text: a

text: nutshell,

text: our

text: estimator

text: h

text: jk

text: has

text: several

text: desirable

text: properties

text: and

text: address

text: the

text: issues

text: of

text: the

text: MLE

text: h

text: (discussed

text: at

text: the

text: jk

text: beginning

text: of

text: Section

text: First,

text: h

text: exploits

text: information

text: in

text: all

text: the

text: data

text: points

text: through

text: u

text: jk

text: (j)

text: v

text: and

text: θ

text: all

text: of

text: which

text: are

text: estimated

text: from

text: a

text: global

text: view.

text: Therefore,

text: it

text: is

text: less

text: sensitive

text: (k)

text: jk

text: to

text: diﬀerent

text: choices

text: of

text: grids.

text: Secondly,

text: the

text: value

text: of

text: h

text: is

text: similar

text: across

text: adjacent

text: blocks,

text: jk

text: making

text: it

text: a

text: smooth

text: estimator.

text: This

text: is

text: because

text: h

text: depends

text: on

text: u

text: and

text: v

text: which

text: have

text: jk

text: (j)

text: (k)

text: similar

text: values

text: in

text: neighboring

text: rows

text: of

text: U

text: and

text: V

text: respectively.

text: In

text: particular,

text: the

text: estimator

text: h

text: jk

text: is

text: dominated

text: by

text: uT

text: v

text: when

text: d·

text: and

text: t·

text: are

text: small

text: (i.e.,

text: when

text: we

text: do

text: not

text: have

text: enough

text: data

text: (j)

text: (k)

text: jk

text: jk

text: in

text: the

text: jkth

text: block).

text: Finally,

text: the

text: estimator

text: is

text: well

text: deﬁned

text: even

text: when

text: there

text: is

text: no

text: observation

text: in

text: a

text: block.

text: When

text: t·

text: d·

text: we

text: have

text: h

text: max(uT

text: v

text: θ

text: /ρ

text: jk

text: jk

text: jk

text: (j)

text: (k)

text: jk

text: Next,

text: to

text: obtain

text: an

text: estimate

text: of

text: (U,

text: V)

text: from

text: we

text: rewrite

text: the

text: optimization

text: problem

text: equivalently

text: as

text: Θ

text: P

text: min

text: H

text: UVT

text: (U,

text: V),

text: (U,V)

text: ρ

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: behavior in the morning may be diﬀerent from that in the afternoon. By taking into account the time-of-day variation, call center managers can evaluate the system performance more accurately, and adjust the staﬃng regime dynamically. Our simulation study ﬁxes the arrival rate to be 100, the service rate to be 1, and the number of agents to be 100. The hazard function for patience is set as the piecewise constant function estimated from the Quick&Reilly data. We particularly consider three levels of patience (i.e., low, medium, high) presumably corresponding to diﬀerent times of a day, through multipling the baseline hazard function by 1.5, 1, and 0.5, respectively. Based on Zeltyn and Mandelbaum (2005) and Aktekin and Soyer (2014), we numerically evaluate the steady-state probability of abandonment, given a customer is not served immediately upon arrival. As a result, we obtain the probability of abandonment as 0.0095 (for the low patience setting), 0.0068 (for the medium patience setting), 0.0042 (for the high patience setting), respectively. Apparently, diﬀerent patience levels lead to distinct system perfor- mances, which can be viewed as corresponding to diﬀerent times of a day. Therefore, the proposed tF-Hazards model will be helpful in practice as it can provide a more detailed characterization of the varying waiting behavior at diﬀerent times of a day. 5 Discussion Our research is motivated by eﬃcient management of call center operations, in partic- ular agent staﬃng. A crucial (but understudied) issue is empirical modeling of customer waiting behavior – estimation of hazard functions of customer patience time and/or system oﬀered wait time. For the ﬁrst time, we study the hazard functions as a two-way surface of both wait duration and time of day, to provide inputs to time-varying queueing models. We develop a two-way functional hazards model by considering a smooth and low-rank hazard function of two timescales, namely duration and time of day. To ﬁt the model, we employ 27

text: behavior

text: in

text: the

text: morning

text: may

text: be

text: diﬀerent

text: from

text: that

text: in

text: the

text: afternoon.

text: By

text: taking

text: into

text: account

text: the

text: time-of-day

text: variation,

text: call

text: center

text: managers

text: can

text: evaluate

text: the

text: system

text: performance

text: more

text: accurately,

text: and

text: adjust

text: the

text: staﬃng

text: regime

text: dynamically.

text: Our

text: simulation

text: study

text: ﬁxes

text: the

text: arrival

text: rate

text: to

text: be

text: the

text: service

text: rate

text: to

text: be

text: and

text: the

text: number

text: of

text: agents

text: to

text: be

text: The

text: hazard

text: function

text: for

text: patience

text: is

text: set

text: as

text: the

text: piecewise

text: constant

text: function

text: estimated

text: from

text: the

text: Quick&Reilly

text: data.

text: We

text: particularly

text: consider

text: three

text: levels

text: of

text: patience

text: (i.e.,

text: low,

text: medium,

text: high)

text: presumably

text: corresponding

text: to

text: diﬀerent

text: times

text: of

text: a

text: day,

text: through

text: multipling

text: the

text: baseline

text: hazard

text: function

text: by

text: and

text: respectively.

text: Based

text: on

text: Zeltyn

text: and

text: Mandelbaum

text: and

text: Aktekin

text: and

text: Soyer

text: we

text: numerically

text: evaluate

text: the

text: steady-state

text: probability

text: of

text: abandonment,

text: given

text: a

text: customer

text: is

text: not

text: served

text: immediately

text: upon

text: arrival.

text: As

text: a

text: result,

text: we

text: obtain

text: the

text: probability

text: of

text: abandonment

text: as

text: (for

text: the

text: low

text: patience

text: setting),

text: (for

text: the

text: medium

text: patience

text: setting),

text: (for

text: the

text: high

text: patience

text: setting),

text: respectively.

text: Apparently,

text: diﬀerent

text: patience

text: levels

text: lead

text: to

text: distinct

text: system

text: perfor-

text: mances,

text: which

text: can

text: be

text: viewed

text: as

text: corresponding

text: to

text: diﬀerent

text: times

text: of

text: a

text: day.

text: Therefore,

text: the

text: proposed

text: tF-Hazards

text: model

text: will

text: be

text: helpful

text: in

text: practice

text: as

text: it

text: can

text: provide

text: a

text: more

text: detailed

text: characterization

text: of

text: the

text: varying

text: waiting

text: behavior

text: at

text: diﬀerent

text: times

text: of

text: a

text: day.

text: Discussion

text: Our

text: research

text: is

text: motivated

text: by

text: eﬃcient

text: management

text: of

text: call

text: center

text: operations,

text: in

text: partic-

text: ular

text: agent

text: staﬃng.

text: A

text: crucial

text: (but

text: understudied)

text: issue

text: is

text: empirical

text: modeling

text: of

text: customer

text: waiting

text: behavior

text: estimation

text: of

text: hazard

text: functions

text: of

text: customer

text: patience

text: time

text: and/or

text: system

text: oﬀered

text: wait

text: time.

text: For

text: the

text: ﬁrst

text: time,

text: we

text: study

text: the

text: hazard

text: functions

text: as

text: a

text: two-way

text: surface

text: of

text: both

text: wait

text: duration

text: and

text: time

text: of

text: day,

text: to

text: provide

text: inputs

text: to

text: time-varying

text: queueing

text: models.

text: We

text: develop

text: a

text: two-way

text: functional

text: hazards

text: model

text: by

text: considering

text: a

text: smooth

text: and

text: low-rank

text: hazard

text: function

text: of

text: two

text: timescales,

text: namely

text: duration

text: and

text: time

text: of

text: day.

text: To

text: ﬁt

text: the

text: model,

text: we

text: employ