text: that of δ is 0 sin(a) sin(b) sin(c) = 1. sin(b) sin(c) sin(a) Let us package the computation done here. (Cf. Braude [1, section 2.3]).) Lemma 2 (EZ Holonomy). The holonomy of a collection of shapes ∆(a , b , c ), . . . , ∆(a , b , c ) 1 1 1 n n n arranged in a ring around their c vertices is trivial if c +. . .+c = τ and the i 1 n ‘trailing’ angles a , . . . , a are a permutation of the ‘leading’ angles b , . . . , b . 1 n 1 n 3 Trisectors Moving from bisectors to trisectors we have Morley’s Theorem [7]. Theorem 3 (Morley’s Theorem). The diagram of Figure 2 exists. (cid:48) (cid:48) Figure 2: Morley’s Theorem. x = τ/6, a + b + c = 0 . Proof. The angles add to τ/2 in all the triangles, and the EZ Holonomy condition holds at the interior vertices (by symmetry we need only check one vertex). 4 Quadrisectors Bill Thurston observed that topologically, the diagrams for the Bisector Theorem and Morley’s Theorem are the planar maps of a tetrahedron and an icosehedron, and wondered (though not enough to think about the ques- tion himself) what the analog for an icosahedron might be. John Conway promptly supplied Figure 3, though with some misgivings since it is really 3 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: that

text: of

text: δ

text: is

text: sin(a)

text: sin(b)

text: sin(c)

text: sin(b)

text: sin(c)

text: sin(a)

text: Let

text: us

text: package

text: the

text: computation

text: done

text: here.

text: (Cf.

text: Braude

text: section

text: Lemma

text: (EZ

text: Holonomy).

text: The

text: holonomy

text: of

text: a

text: collection

text: of

text: shapes

text: ∆(a

text: b

text: c

text: ∆(a

text: b

text: c

text: n

text: n

text: n

text: arranged

text: in

text: a

text: ring

text: around

text: their

text: c

text: vertices

text: is

text: trivial

text: if

text: c

text: .+c

text: τ

text: and

text: the

text: i

text: n

text: ‘trailing’

text: angles

text: a

text: a

text: are

text: a

text: permutation

text: of

text: the

text: ‘leading’

text: angles

text: b

text: b

text: n

text: n

text: Trisectors

text: Moving

text: from

text: bisectors

text: to

text: trisectors

text: we

text: have

text: Morley’s

text: Theorem

text: Theorem

text: (Morley’s

text: Theorem).

text: The

text: diagram

text: of

text: Figure

text: exists.

text: Figure

text: Morley’s

text: Theorem.

text: x

text: a

text: b

text: c

text: Proof.

text: The

text: angles

text: add

text: to

text: in

text: all

text: the

text: triangles,

text: and

text: the

text: EZ

text: Holonomy

text: condition

text: holds

text: at

text: the

text: interior

text: vertices

text: (by

text: symmetry

text: we

text: need

text: only

text: check

text: one

text: vertex).

text: Quadrisectors

text: Bill

text: Thurston

text: observed

text: that

text: topologically,

text: the

text: diagrams

text: for

text: the

text: Bisector

text: Theorem

text: and

text: Morley’s

text: Theorem

text: are

text: the

text: planar

text: maps

text: of

text: a

text: tetrahedron

text: and

text: an

text: icosehedron,

text: and

text: wondered

text: (though

text: not

text: enough

text: to

text: think

text: about

text: the

text: ques-

text: tion

text: himself)

text: what

text: the

text: analog

text: for

text: an

text: icosahedron

text: might

text: be.

text: John

text: Conway

text: promptly

text: supplied

text: Figure

text: though

text: with

text: some

text: misgivings

text: since

text: it

text: is

text: really

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: Since we can vary c while ﬁxing b, we had better have the identity (cid:48) (cid:48)(cid:48) sin(3x) = A sin(x) sin(x ) sin(x ) τ τ = A sin(x) sin(x + ) sin(x + 2 ), 6 6 for some constant A. This follows from general principles because the two sides are entire functions of modest growth with the same roots. The trigonometric identity that we’ve proven analytically up to a constant factor is the following identity, which is key to computations of hyperbolic volume. (See Milnor [5].) Proposition 7 (Cyclotomic identity). − − 1 n 1 sin(nθ) = 2n 1 sin(θ) sin(θ + τ) . . . sin(θ + τ). 2n 2n Proof. Again we take the case n = 3 as a model. Start with the identity − − − − z3 1 = (z 1)(z ω)(z ω2), with ω = eiτ/3. Plug in z = ei2θ, take absolute values, and use the fact that ∈ for α, β R | − | | − | eiα eiβ = 2 sin((α β)/2) to get | | | | · | − | · | − | 2 sin(3θ) = 2 sin(θ) 2 sin(θ τ/6) 2 sin(θ 2τ/6 | | · | | · | | = 2 sin(θ) 2 sin(θ + 2τ/6) 2 sin(θ + τ/6) . Remove absolute values and check the sign to get the desired identity. This identity cries out for a geometric proof—see Section below. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Since

label: paragraph

text: we

text: can

text: vary

text: c

text: while

text: ﬁxing

text: b,

text: we

text: had

text: better

text: have

text: the

text: identity

text: A

text: sin(x)

text: sin(x

text: sin(x

text: τ

text: τ

text: A

text: sin(x)

text: sin(x

text: sin(x

text: for

text: some

text: constant

text: A.

text: This

text: follows

text: from

text: general

text: principles

text: because

text: the

text: two

text: sides

text: are

text: entire

text: functions

text: of

text: modest

text: growth

text: with

text: the

text: same

text: roots.

text: The

text: trigonometric

text: identity

text: that

text: we’ve

text: proven

text: analytically

text: up

text: to

text: a

text: constant

text: factor

text: is

text: the

text: following

text: identity,

text: which

text: is

text: key

text: to

text: computations

text: of

text: hyperbolic

text: volume.

text: (See

text: Milnor

text: Proposition

text: (Cyclotomic

text: identity).

text: n

text: sin(nθ)

text: sin(θ)

text: sin(θ

text: τ)

text: sin(θ

text: τ).

text: Proof.

text: Again

text: we

text: take

text: the

text: case

text: n

text: as

text: a

text: model.

text: Start

text: with

text: the

text: identity

text: (z

text: ω)(z

text: with

text: ω

text: Plug

text: in

text: z

text: take

text: absolute

text: values,

text: and

text: use

text: the

text: fact

text: that

text: for

text: α,

text: β

text: R

text: eiα

text: eiβ

text: sin((α

text: to

text: get

text: sin(θ)

text: sin(θ

text: sin(θ

text: sin(θ)

text: sin(θ

text: sin(θ

text: Remove

text: absolute

text: values

text: and

text: check

text: the

text: sign

text: to

text: get

text: the

text: desired

text: identity.

text: This

text: identity

text: cries

text: out

text: for

text: a

text: geometric

text: proof—see

text: Section

text: below.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Hamilton distributed training modes such as the highly optimized 1-bit SGD Seide et al. (2014). We provide Azure Resource Manager templates so that users can deploy these solutions. 3.5 Integrating OpenCV SparkML also lacks support for complex datatypes such as images. We have worked with the Spark community to contribute a schema for image data-types, and a collection of readers for ingesting and streaming images and binary ﬁles to the core Spark library. Building on this work, we have integrated the popular image processing library OpenCV Bradski and Kaehler (2000) as a SparkML transformer. Like the CNTKModel, the OpenCV transformer uses Java bindings for OpenCV that call into fast native C++ code. Our OpenCV transformer allows us to perform most OpenCV operations in parallel on all nodes of the cluster. Furthermore integration automatically chains all evaluations of native code so that data does not need to be Marshalled to and from the JVM between adjacent calls to OpenCV. 3.6 Automated Generation of PySpark Spark supports a wide variety of ecosystems through language bindings in Python, R, Java, C(cid:93) and F(cid:93). However, wrapper code for these libraries is mainly written “by hand”. This a diﬃcult task involving detailed knowledge of both the source and target ecosystems, and their communication platforms. Furthermore, this results in a large amount of boilerplate code, that impedes adding estimators or transformers to the spark ecosystem. To eliminate the need to manually create and maintain these language bindings, we have created an automated wrapper generation system. The system loads and inspects the types and parameters of any transformer or estimator using Scala reﬂection. It then generate a usable and idiomatic python wrapper for the class that can be used in PySpark. We also use this tool to create language bindings for the SparklyR ecosystem. We expose all of our estimators and transformers to these languages, proving the generality and reliability of this system. A schematic overview of the process is shown in Figure 1. Figure 1: Left: Architectural overview of some salient features of MMLSpark. Right: Performance of the “ImageFeaturizer” transformer that contains an OpenCV image resize, and deep featurization with ResNet-50 6 ##LTFigure## ##LTFigure##

text: Hamilton

text: distributed

text: training

text: modes

text: such

text: as

text: the

text: highly

text: optimized

text: SGD

text: Seide

text: et

text: al.

text: We

text: provide

text: Azure

text: Resource

text: Manager

text: templates

text: so

text: that

text: users

text: can

text: deploy

text: these

text: solutions.

label: section

text: Integrating

text: OpenCV

text: SparkML

text: also

text: lacks

text: support

text: for

text: complex

text: datatypes

text: such

text: as

text: images.

text: We

text: have

text: worked

text: with

text: the

text: Spark

text: community

text: to

text: contribute

text: a

text: schema

text: for

text: image

text: data-types,

text: and

text: a

text: collection

text: of

text: readers

text: for

text: ingesting

text: and

text: streaming

text: images

text: and

text: binary

text: ﬁles

text: to

text: the

text: core

text: Spark

text: library.

text: Building

text: on

text: this

text: work,

text: we

text: have

text: integrated

text: the

text: popular

text: image

text: processing

text: library

text: OpenCV

text: Bradski

text: and

text: Kaehler

text: as

text: a

text: SparkML

text: transformer.

text: Like

text: the

text: CNTKModel,

text: the

text: OpenCV

text: transformer

text: uses

text: Java

text: bindings

text: for

text: OpenCV

text: that

text: call

text: into

text: fast

text: native

text: C++

text: code.

text: Our

text: OpenCV

text: transformer

text: allows

text: us

text: to

text: perform

text: most

text: OpenCV

text: operations

text: in

text: parallel

text: on

text: all

text: nodes

text: of

text: the

text: cluster.

text: Furthermore

text: integration

text: automatically

text: chains

text: all

text: evaluations

text: of

text: native

text: code

text: so

text: that

text: data

text: does

text: not

text: need

text: to

text: be

text: Marshalled

text: to

text: and

text: from

text: the

text: JVM

text: between

text: adjacent

text: calls

text: to

text: OpenCV.

text: Automated

text: Generation

text: of

text: PySpark

text: Spark

text: supports

text: a

text: wide

text: variety

text: of

text: ecosystems

text: through

text: language

text: bindings

text: in

text: Python,

text: R,

text: Java,

text: and

text: However,

text: wrapper

text: code

text: for

text: these

text: libraries

text: is

text: mainly

text: written

text: “by

text: hand”.

text: This

text: a

text: diﬃcult

text: task

text: involving

text: detailed

text: knowledge

text: of

text: both

text: the

text: source

text: and

text: target

text: ecosystems,

text: and

text: their

text: communication

text: platforms.

text: Furthermore,

text: this

text: results

text: in

text: a

text: large

text: amount

text: of

text: boilerplate

text: code,

text: that

text: impedes

text: adding

text: estimators

text: or

text: transformers

text: to

text: the

text: spark

text: ecosystem.

text: To

text: eliminate

text: the

text: need

text: to

text: manually

text: create

text: and

text: maintain

text: these

text: language

text: bindings,

text: we

text: have

text: created

text: an

text: automated

text: wrapper

text: generation

text: system.

text: The

text: system

text: loads

text: and

text: inspects

text: the

text: types

text: and

text: parameters

text: of

text: any

text: transformer

text: or

text: estimator

text: using

text: Scala

text: reﬂection.

text: It

text: then

text: generate

text: a

text: usable

text: and

text: idiomatic

text: python

text: wrapper

text: for

text: the

text: class

text: that

text: can

text: be

text: used

text: in

text: PySpark.

text: We

text: also

text: use

text: this

text: tool

text: to

text: create

text: language

text: bindings

text: for

text: the

text: SparklyR

text: ecosystem.

text: We

text: expose

text: all

text: of

text: our

text: estimators

text: and

text: transformers

text: to

text: these

text: languages,

text: proving

text: the

text: generality

text: and

text: reliability

text: of

text: this

text: system.

text: A

text: schematic

text: overview

text: of

text: the

text: process

text: is

text: shown

text: in

text: Figure

text: Figure

text: Left:

text: Architectural

text: overview

text: of

text: some

text: salient

text: features

text: of

text: MMLSpark.

text: Right:

text: Performance

text: of

text: the

text: “ImageFeaturizer”

text: transformer

text: that

text: contains

text: an

text: OpenCV

text: image

text: resize,

text: and

text: deep

text: featurization

text: with

text: ##LTFigure##

text: ##LTFigure##


text: Hamilton and population trend. These data are incredibly important for understanding the risk of snow leopard extinction and where to target conservation eﬀorts. Furthermore, robust habitat information is necessary in order to lobby for legal protection of habitat. Snow leopards occupy a critical niche in the Himalayan ecosystem as one of the apex predators. Apex predators have been shown to help stabilize ecosystems, by nat- urally controlling populations of secondary lifeforms in the ecosystem. This allows for primary life forms, such as plant life, to thrive and actively improves soil stability. Added vegetation can introduce several new niches, increasing biodiversity and ecosys- tem stability. Stunning evidence of this ef- fect can be seen in Figure 2, which shows the eﬀect of re-introducing the Grey wolf to the Yellowstone ecosystem in the early 2000’s. The health and stability of the Himalayan ecosystem is critically important as it pro- vides water for for over 1.4 billion people Immerzeel et al. (2010). Figure 2: A comparison of vegetation in Yel- Surveying snow leopards is an extraordi- lowstone National Park before (above) and narily diﬃcult task. This task is made dif- after (below) the reintroduction of the Grey ﬁcult because leopards are incredibly rare, wolf. Figure originally from Beschta and Rip- live in remote and diﬃcult to reach areas, ple (2016) and are mainly nocturnal. As a result, the Snow Leopard Trust uses remote camera traps to survey snow leopard locations over 1,700km2 of potential habitat. These camera traps are equipped with regular and night vision settings, and take photographs when an attached motion sensor is triggered. These cameras generate a large number of images that need to be manually sorted to ﬁnd the leopards. For many images, it is diﬃcult for a human to identify hidden leopards, as they are often partially obscured and highly camouﬂaged. This task can take up to 20 thousand man hours per each million images. This task is insurmountable for most small nonproﬁts like the Snow Leopard Trust. Hence, it is critically important to automate this task. 5.2 Dataset We use a dataset from Snow Leopard Trust that contains roughly 8, 000 labeled images of snow leopards and non snow leopards as well as roughly 300, 000 unlabeled images. These images are from their most recent set of surveys in the last 5 years. The images are primarily black and white night vision and optical images. A small number 5% of images are in full color. Only around 10% of the images are Snow Leopards. The majority (90%) of the images are of grass, goats, foxes, and occasionally people. We split the labeled dataset into a training (80%) and testing (20%) set based on the cameras used in the analysis. Splitting the cameras into training and test sets is necessary, as subsequent frames in a camera burst 8 ##LTFigure##

text: Hamilton

text: and

text: population

text: trend.

text: These

text: data

text: are

text: incredibly

text: important

text: for

text: understanding

text: the

text: risk

text: of

text: snow

text: leopard

text: extinction

text: and

text: where

text: to

text: target

text: conservation

text: eﬀorts.

text: Furthermore,

text: robust

text: habitat

text: information

text: is

text: necessary

text: in

text: order

text: to

text: lobby

text: for

text: legal

text: protection

text: of

text: habitat.

text: Snow

text: leopards

text: occupy

text: a

text: critical

text: niche

text: in

text: the

text: Himalayan

text: ecosystem

text: as

text: one

text: of

text: the

text: apex

text: predators.

text: Apex

text: predators

text: have

text: been

text: shown

text: to

text: help

text: stabilize

text: ecosystems,

text: by

text: nat-

text: urally

text: controlling

text: populations

text: of

text: secondary

text: lifeforms

text: in

text: the

text: ecosystem.

text: This

text: allows

text: for

text: primary

text: life

text: forms,

text: such

text: as

text: plant

text: life,

text: to

text: thrive

text: and

text: actively

text: improves

text: soil

text: stability.

text: Added

text: vegetation

text: can

text: introduce

text: several

text: new

text: niches,

text: increasing

text: biodiversity

text: and

text: ecosys-

text: tem

text: stability.

text: Stunning

text: evidence

text: of

text: this

text: ef-

text: fect

text: can

text: be

text: seen

text: in

text: Figure

text: which

text: shows

text: the

text: eﬀect

text: of

text: re-introducing

text: the

text: Grey

text: wolf

text: to

text: the

text: Yellowstone

text: ecosystem

text: in

text: the

text: early

text: The

text: health

text: and

text: stability

text: of

text: the

text: Himalayan

text: ecosystem

text: is

text: critically

text: important

text: as

text: it

text: pro-

text: vides

text: water

text: for

text: for

text: over

text: billion

text: people

text: Immerzeel

text: et

text: al.

text: Figure

text: A

text: comparison

text: of

text: vegetation

text: in

text: Yel-

text: Surveying

text: snow

text: leopards

text: is

text: an

text: extraordi-

text: lowstone

text: National

text: Park

text: before

text: (above)

text: and

text: narily

text: diﬃcult

text: task.

text: This

text: task

text: is

text: made

text: dif-

text: after

text: (below)

text: the

text: reintroduction

text: of

text: the

text: Grey

text: ﬁcult

text: because

text: leopards

text: are

text: incredibly

text: rare,

text: wolf.

text: Figure

text: originally

text: from

text: Beschta

text: and

text: Rip-

text: live

text: in

text: remote

text: and

text: diﬃcult

text: to

text: reach

text: areas,

text: ple

text: and

text: are

text: mainly

text: nocturnal.

text: As

text: a

text: result,

text: the

text: Snow

text: Leopard

text: Trust

text: uses

text: remote

text: camera

text: traps

text: to

text: survey

text: snow

text: leopard

text: locations

text: over

text: of

text: potential

text: habitat.

text: These

text: camera

text: traps

text: are

text: equipped

text: with

text: regular

text: and

text: night

text: vision

text: settings,

text: and

text: take

text: photographs

text: when

text: an

text: attached

text: motion

text: sensor

text: is

text: triggered.

text: These

text: cameras

text: generate

text: a

text: large

text: number

text: of

text: images

text: that

text: need

text: to

text: be

text: manually

text: sorted

text: to

text: ﬁnd

text: the

text: leopards.

text: For

text: many

text: images,

text: it

text: is

text: diﬃcult

text: for

text: a

text: human

text: to

text: identify

text: hidden

text: leopards,

text: as

text: they

text: are

text: often

text: partially

text: obscured

text: and

text: highly

text: camouﬂaged.

text: This

text: task

text: can

text: take

text: up

text: to

text: thousand

text: man

text: hours

text: per

text: each

text: million

text: images.

text: This

text: task

text: is

text: insurmountable

text: for

text: most

text: small

text: nonproﬁts

text: like

text: the

text: Snow

text: Leopard

text: Trust.

text: Hence,

text: it

text: is

text: critically

text: important

text: to

text: automate

text: this

text: task.

text: Dataset

text: We

text: use

text: a

text: dataset

text: from

text: Snow

text: Leopard

text: Trust

text: that

text: contains

text: roughly

text: labeled

text: images

text: of

text: snow

text: leopards

text: and

text: non

text: snow

text: leopards

text: as

text: well

text: as

text: roughly

text: unlabeled

text: images.

text: These

text: images

text: are

text: from

text: their

text: most

text: recent

text: set

text: of

text: surveys

text: in

text: the

text: last

text: years.

text: The

text: images

text: are

text: primarily

text: black

text: and

text: white

text: night

text: vision

text: and

text: optical

text: images.

text: A

text: small

text: number

text: of

text: images

text: are

text: in

text: full

text: color.

text: Only

text: around

text: of

text: the

text: images

text: are

text: Snow

text: Leopards.

text: The

text: majority

text: of

text: the

text: images

text: are

text: of

text: grass,

text: goats,

text: foxes,

text: and

text: occasionally

text: people.

text: We

text: split

text: the

text: labeled

text: dataset

text: into

text: a

text: training

text: and

text: testing

text: set

text: based

text: on

text: the

text: cameras

text: used

text: in

text: the

text: analysis.

text: Splitting

text: the

text: cameras

text: into

text: training

text: and

text: test

text: sets

text: is

text: necessary,

text: as

text: subsequent

text: frames

text: in

text: a

text: camera

text: burst

text: ##LTFigure##


text: Flexible and Scalable Deep Learning with MMLSpark are highly correlated. If one splits a single burst into both sets, information will leak from the training set, and the ﬁnal testing performance will be overestimated. We did not have to “clean” the dataset, or remove any images, as anything that did not contain a snow leopard was marked as a negative image. At the time of publication, we use MMLSpark version 0.10, Spark version 2.2 and CNTK version 2.3. Figure 3: Schematic overview of basic transfer learning algorithm. We use ResNet50 and truncate the last layer(s). We then append a SparkML logistic regression estimator and train the model using the features from the preceding network. Feature visualizations originate from a similar convolution architecture by Lee et al. (2011). 5.3 Model and Results To learn an automated classiﬁcation system, we leverage deep transfer learning to learn a classiﬁer that can successfully identify snow leopards with high precision and recall. We improve the classiﬁer through the use of dataset augmentation and ensembling leopard probabilities over bursts of images. The simplest model uses features from ResNet50, trained on the ImageNet corpus. ResNet50 is a state of the art network that combines convolutional and recurrent components to capitalize on the spatial regularity of natural images and eﬀectively learn the optimal depth of the network He et al. (2016b). We truncate the last, image-net speciﬁc layer(s) but keep a majority of the early layers. This lets us leverage the domain-general low to mid level features and adapt the network to the new classiﬁcation task. To adapt the network, we append and train a SparkML logistic regressor that maps the features to a binary label. This eﬀectively adds an additional fully connected layer to the truncated network. A schematic overview of the algorithm is provided in Figure 3. We refer to these algorithms as “RN1” and “RN2” for ResNet50 with 1 and 2 truncated layers respectively. To improve performance of the algorithm, we augment the dataset using MMLSpark’s “ImageSetAugmenter” transformer. This Transformer enriches the dataset by ﬂipping the images about the horizontal axis. The ImageSetAugmenter doubles the training data, which improves the learned classiﬁer. At test time, the scores for both image parities are averaged together. This reduces the variance of the ﬁnal estimate and improves performance. We refer to this algorithm as “RN2+A” for the base algorithm with dataset augmentation. 9 ##LTFigure##

text: Flexible

text: and

text: Scalable

text: Deep

text: Learning

text: with

text: MMLSpark

text: are

text: highly

text: correlated.

text: If

text: one

text: splits

text: a

text: single

text: burst

text: into

text: both

text: sets,

text: information

text: will

text: leak

text: from

text: the

text: training

text: set,

text: and

text: the

text: ﬁnal

text: testing

text: performance

text: will

text: be

text: overestimated.

text: We

text: did

text: not

text: have

text: to

text: “clean”

text: the

text: dataset,

text: or

text: remove

text: any

text: images,

text: as

text: anything

text: that

text: did

text: not

text: contain

text: a

text: snow

text: leopard

text: was

text: marked

text: as

text: a

text: negative

text: image.

text: At

text: the

text: time

text: of

text: publication,

text: we

text: use

text: MMLSpark

text: version

text: Spark

text: version

text: and

text: CNTK

text: version

text: Figure

text: Schematic

label: caption

text: overview

text: of

text: basic

text: transfer

text: learning

text: algorithm.

text: We

text: use

text: and

text: truncate

text: the

text: last

text: layer(s).

text: We

text: then

text: append

text: a

text: SparkML

text: logistic

text: regression

text: estimator

text: and

text: train

text: the

text: model

text: using

text: the

text: features

text: from

text: the

text: preceding

text: network.

text: Feature

text: visualizations

text: originate

text: from

text: a

text: similar

text: convolution

text: architecture

text: by

text: Lee

text: et

text: al.

text: Model

text: and

text: Results

text: To

text: learn

text: an

text: automated

text: classiﬁcation

text: system,

text: we

text: leverage

text: deep

text: transfer

text: learning

text: to

text: learn

text: a

text: classiﬁer

text: that

text: can

text: successfully

text: identify

text: snow

text: leopards

text: with

text: high

text: precision

text: and

text: recall.

text: We

text: improve

text: the

text: classiﬁer

text: through

text: the

text: use

text: of

text: dataset

text: augmentation

text: and

text: ensembling

text: leopard

text: probabilities

text: over

text: bursts

text: of

text: images.

text: The

text: simplest

text: model

text: uses

text: features

text: from

text: trained

text: on

text: the

text: ImageNet

text: corpus.

text: is

text: a

text: state

text: of

text: the

text: art

text: network

text: that

text: combines

text: convolutional

text: and

text: recurrent

text: components

text: to

text: capitalize

text: on

text: the

text: spatial

text: regularity

text: of

text: natural

text: images

text: and

text: eﬀectively

text: learn

text: the

text: optimal

text: depth

text: of

text: the

text: network

text: He

text: et

text: al.

text: We

text: truncate

text: the

text: last,

text: image-net

text: speciﬁc

text: layer(s)

text: but

text: keep

text: a

text: majority

text: of

text: the

text: early

text: layers.

text: This

text: lets

text: us

text: leverage

text: the

text: domain-general

text: low

text: to

text: mid

text: level

text: features

text: and

text: adapt

text: the

text: network

text: to

text: the

text: new

text: classiﬁcation

text: task.

text: To

text: adapt

text: the

text: network,

text: we

text: append

text: and

text: train

text: a

text: SparkML

text: logistic

text: regressor

text: that

text: maps

text: the

text: features

text: to

text: a

text: binary

text: label.

text: This

text: eﬀectively

text: adds

text: an

text: additional

text: fully

text: connected

text: layer

text: to

text: the

text: truncated

text: network.

text: A

text: schematic

text: overview

text: of

text: the

text: algorithm

text: is

text: provided

text: in

text: Figure

text: We

text: refer

text: to

text: these

text: algorithms

text: as

text: and

text: for

text: with

text: and

text: truncated

text: layers

text: respectively.

text: To

text: improve

text: performance

text: of

text: the

text: algorithm,

text: we

text: augment

text: the

text: dataset

text: using

text: MMLSpark’s

text: “ImageSetAugmenter”

text: transformer.

text: This

text: Transformer

text: enriches

text: the

text: dataset

text: by

text: ﬂipping

text: the

text: images

text: about

text: the

text: horizontal

text: axis.

text: The

text: ImageSetAugmenter

text: doubles

text: the

text: training

text: data,

text: which

text: improves

text: the

text: learned

text: classiﬁer.

text: At

text: test

text: time,

text: the

text: scores

text: for

text: both

text: image

text: parities

text: are

text: averaged

text: together.

text: This

text: reduces

text: the

text: variance

text: of

text: the

text: ﬁnal

text: estimate

text: and

text: improves

text: performance.

text: We

text: refer

text: to

text: this

text: algorithm

text: as

text: for

text: the

text: base

text: algorithm

text: with

text: dataset

text: augmentation.

text: ##LTFigure##


text: Hamilton Finally, we leverage the natural grouping of cam- era trap images into in small temporally contiguous segments we refer to as “bursts”. We build on Spark’s primitives to implement more complex inter-node op- erations like grouping images by keys. These bursts originate from camera traps that take a series of im- ages until no motion is detected. We extract this burst information from image metadata, and use it to inform our predictions. This eﬀectively turns a single image classiﬁer into a video classiﬁer. We note that labels are often the same for all frames in cam- era bursts. By averaging the predictions of each im- age in a burst, we dramatically reduce the variance Figure 4: Receiver Operator Char- of the classiﬁer and allow the algorithm to use easy acteristic (ROC) curves for each al- classiﬁcations to inform neighboring diﬃcult classi- gorithm. Note the superior perfor- ﬁcations. This substantially reduces the number of mance of “RN2+A+E” to all other false negatives, and improves overall accuracy. We methods tested refer to this algorithm as “RN2+A+E” for the base algorithm with dataset augmentation and sequence ensembling. We compare these models to logistic regression trained on the raw pixels of the × image after scaling to 120 120 pixels. We refer to this baseline model as “LR120”. Figures 5 and 4 show the dramatic performance increases from deep featurization, augmentation, and ensembling. Figure 5: Normalized confusion matrices for each algorithm. 6. Future Work MMLSpark is a growing software ecosystem with several large ongoing areas of work. In the future, we plan to extend the CNTK Java bindings to allow for training in Java. This will allow users to construct CNTK computation graphs in Java and Scala, and will provide a way to leverage the parallelism of the spark cluster for CNTK model training in addition to evaluation. Furthermore, we plan to implement a recent advance in low communication parallel stochastic gradient descent called SymSGD Maleki et al. (2017). This will dramatically re- duce communication overhead during parallel training of networks. In addition this improve the scalability of parallel training systems as SymSGD’s method of gradient combination approximates a sequential computation. This contrasts other methods like AllReduce, Pa- 10 ##LTFigure## ##LTFigure##

text: Hamilton

text: Finally,

text: we

text: leverage

text: the

text: natural

text: grouping

text: of

text: cam-

text: era

text: trap

text: images

text: into

text: in

text: small

text: temporally

text: contiguous

text: segments

text: we

text: refer

text: to

text: as

text: “bursts”.

text: We

text: build

text: on

text: Spark’s

text: primitives

text: to

text: implement

text: more

text: complex

text: inter-node

text: op-

text: erations

text: like

text: grouping

text: images

text: by

text: keys.

text: These

text: bursts

text: originate

text: from

text: camera

text: traps

text: that

text: take

text: a

text: series

text: of

text: im-

text: ages

text: until

text: no

text: motion

text: is

text: detected.

text: We

text: extract

text: this

text: burst

text: information

text: from

text: image

text: metadata,

text: and

text: use

text: it

text: to

text: inform

text: our

text: predictions.

text: This

text: eﬀectively

text: turns

text: a

text: single

text: image

text: classiﬁer

text: into

text: a

text: video

text: classiﬁer.

text: We

text: note

text: that

text: labels

text: are

text: often

text: the

text: same

text: for

text: all

text: frames

text: in

text: cam-

text: era

text: bursts.

text: By

text: averaging

text: the

text: predictions

text: of

text: each

text: im-

text: age

text: in

text: a

text: burst,

text: we

text: dramatically

text: reduce

text: the

text: variance

text: Figure

text: Receiver

text: Operator

text: Char-

text: of

text: the

text: classiﬁer

text: and

text: allow

text: the

text: algorithm

text: to

text: use

text: easy

text: acteristic

text: (ROC)

text: curves

text: for

text: each

text: al-

text: classiﬁcations

text: to

text: inform

text: neighboring

text: diﬃcult

text: classi-

text: gorithm.

text: Note

text: the

text: superior

text: perfor-

text: ﬁcations.

text: This

text: substantially

text: reduces

text: the

text: number

text: of

text: mance

text: of

text: to

text: all

text: other

text: false

text: negatives,

text: and

text: improves

text: overall

text: accuracy.

text: We

text: methods

text: tested

text: refer

text: to

text: this

text: algorithm

text: as

text: for

text: the

text: base

text: algorithm

text: with

text: dataset

text: augmentation

text: and

text: sequence

text: ensembling.

text: We

text: compare

text: these

text: models

text: to

text: logistic

text: regression

text: trained

text: on

text: the

text: raw

text: pixels

text: of

text: the

text: image

text: after

text: scaling

text: to

text: pixels.

text: We

text: refer

text: to

text: this

text: baseline

text: model

text: as

text: Figures

text: and

text: show

text: the

text: dramatic

text: performance

text: increases

text: from

text: deep

text: featurization,

text: augmentation,

text: and

text: ensembling.

text: Figure

text: Normalized

text: confusion

text: matrices

text: for

text: each

text: algorithm.

text: Future

text: Work

text: MMLSpark

text: is

text: a

text: growing

text: software

text: ecosystem

text: with

text: several

text: large

text: ongoing

text: areas

text: of

text: work.

text: In

text: the

text: future,

text: we

text: plan

text: to

text: extend

text: the

text: CNTK

text: Java

text: bindings

text: to

text: allow

text: for

text: training

text: in

text: Java.

text: This

text: will

text: allow

text: users

text: to

text: construct

text: CNTK

text: computation

text: graphs

text: in

text: Java

text: and

text: Scala,

text: and

text: will

text: provide

text: a

text: way

text: to

text: leverage

text: the

text: parallelism

text: of

text: the

text: spark

text: cluster

text: for

text: CNTK

text: model

text: training

text: in

text: addition

text: to

text: evaluation.

text: Furthermore,

text: we

text: plan

text: to

text: implement

text: a

text: recent

text: advance

text: in

text: low

text: communication

text: parallel

text: stochastic

text: gradient

text: descent

text: called

text: SymSGD

text: Maleki

text: et

text: al.

text: This

text: will

text: dramatically

text: re-

text: duce

text: communication

text: overhead

text: during

text: parallel

text: training

text: of

text: networks.

text: In

text: addition

text: this

text: improve

text: the

text: scalability

text: of

text: parallel

text: training

text: systems

text: as

text: SymSGD’s

text: method

text: of

text: gradient

text: combination

text: approximates

text: a

text: sequential

text: computation.

text: This

text: contrasts

text: other

text: methods

text: like

text: AllReduce,

text: Pa-

text: ##LTFigure##

text: ##LTFigure##


text: MAXIMUM LIKELIHOOD ESTIMATION IN HMMS WITH INHOMOGENEOUS NOISE 11 50 θQML n θML n 5 0.5 0.1 10 20 50 100 200 500 1000 2000 5000 n Figure 4. Empirical mean of the Euclidean norm of the diﬀer- ∗ ence of θ and the estimators based on 100 i.i.d. replications of the · ∗ DHMM. Here K = 2, n = 5 103 and θ = (10, 20, 0.8, 0.1). The ∗ parameter θ determines λ ∗ = (10, 20) and the “true” transition θ matrix by P ∗(1, 1) = 0.8, P ∗(2, 1) = 0.1. The inhomogeneous θ θ − noise is driven by an intensity β = 40 n 1.01. n To obtain the desired consistency of the two estimators we need to check the conditions (P1), (P2), (C1)–(C3) and (H1)–(H4): To (P1) and (P2): By the assumptions in this scenario those conditions are satis- ﬁed. ∈ (cid:16) ∈ (cid:17) ∈  To (H1)–(H4): For θ Θ, s S and y G we have   y λ(s) |log f (s, y)| = − log θ exp(−λ(s)) = −y log(λ(s)) + log(y!) + λ(s) θ y! θ θ θ ≤ − y log(λ(s)) + y2 + λ(s). θ θ (cid:88) (cid:88) (cid:18)(cid:16) (cid:17) (cid:19) Hence Eπθ∗ [|log fθ∗(s, Y1)|] ≤ − log(λ(θs∗)) K π(s)λ(θs∗) + K π(s) λ(θs∗) 2 + λ(θs∗) + λ(θs∗) < ∞ s=1 s=1 and (H1) is veriﬁed. A similar calculation gives (H4). Condition (H2) follows simply by (log f (s, y))+ = 0. Condition (H3) follows by the continuity in the parameter θ of the probability function of the Poisson distribution and the continuity of the (cid:55)→ mapping θ (P , λ ). θ θ ∈ To (C1) – (C3): For any δ > 0 and any s S we have P | − | ≥ | P | | ≥ ≤ − P − − π ( Z Y δ X = s) = π ( ε δ) 1 π (ε = 0) = 1 exp( β ). θ n n n θ n θ n n From (10) it follows that − − O − 1 exp( β ) = (n p), n ]|*θ−nθ^|[E^ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MAXIMUM

text: LIKELIHOOD

text: ESTIMATION

text: IN

text: HMMS

text: WITH

text: INHOMOGENEOUS

text: NOISE

font: default

text: θQML

text: n

text: θML

text: n

text: n

text: Figure

text: Empirical

text: mean

text: of

text: the

text: Euclidean

text: norm

text: of

text: the

text: diﬀer-

text: ence

text: of

text: θ

text: and

text: the

text: estimators

text: based

text: on

text: i.i.d.

text: replications

text: of

text: the

text: DHMM.

text: Here

text: K

text: n

text: and

text: θ

text: The

text: parameter

text: θ

text: determines

text: λ

text: and

text: the

text: “true”

text: transition

text: θ

text: matrix

text: by

text: P

text: P

text: The

text: inhomogeneous

text: θ

text: θ

text: noise

text: is

text: driven

text: by

text: an

text: intensity

text: β

text: n

text: n

text: To

text: obtain

text: the

text: desired

text: consistency

text: of

text: the

text: two

text: estimators

text: we

text: need

text: to

text: check

text: the

text: conditions

text: and

text: To

text: and

text: By

text: the

text: assumptions

text: in

text: this

text: scenario

text: those

text: conditions

text: are

text: satis-

text: ﬁed.

text: To

text: For

text: θ

text: Θ,

text: s

text: S

text: and

text: y

text: G

text: we

text: have

text: y

text: λ(s)

text: |log

text: f

text: (s,

text: y)|

text: log

text: θ

text: exp(−λ(s))

text: −y

text: log(λ(s))

text: log(y!)

text: λ(s)

text: θ

text: y!

text: θ

text: θ

text: θ

text: y

text: log(λ(s))

text: λ(s).

text: θ

text: θ

text: Hence

text: Eπθ∗

text: [|log

text: fθ∗(s,

text: log(λ(θs∗))

text: K

text: π(s)λ(θs∗)

text: K

text: π(s)

text: λ(θs∗)

text: λ(θs∗)

text: λ(θs∗)

text: and

text: is

text: veriﬁed.

text: A

text: similar

text: calculation

text: gives

text: Condition

text: follows

text: simply

text: by

text: (log

text: f

text: (s,

text: y))+

text: Condition

text: follows

text: by

text: the

text: continuity

text: in

text: the

text: parameter

text: θ

text: of

text: the

text: probability

text: function

text: of

text: the

text: Poisson

text: distribution

text: and

text: the

text: continuity

text: of

text: the

text: mapping

text: θ

text: (P

text: λ

text: θ

text: θ

text: To

text: For

text: any

text: δ

text: and

text: any

text: s

text: S

text: we

text: have

text: P

text: P

text: P

text: π

text: Z

text: Y

text: δ

text: X

text: s)

text: π

text: ε

text: δ)

text: π

text: (ε

text: exp(

text: β

text: θ

text: n

text: n

text: n

text: θ

text: n

text: θ

text: n

text: n

text: From

text: it

text: follows

text: that

text: O

text: exp(

text: β

text: (n

text: p),

text: n

text: ]|*θ−nθ^|[E^

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MAXIMUM LIKELIHOOD ESTIMATION IN HMMS WITH INHOMOGENEOUS NOISE 13 6 4 2 0 −2 0 200 400 600 800 1000 5 2 θQML 1 n 0.5 θML n 0.1 0.02 10 20 50 100 200 500 1000 2000 5000 Figure 5. Exemplary trajectory of observations of the linear Gaussian model from Section (above) and Euclidean norm of the ∗ diﬀerence of θ and the estimators based on a single trajectory of · ∗ observations (below). Here M = 1, K = 2, n = 5 103 and θ = ∗ (0, 4, 0.5, 0.5, 0.4, 0.5). The parameter θ determines µ ∗ = (0, 4), θ Σ ∗ = (0.5, 0.5) and the “true” transition matrix by P ∗(1, 1) = θ θ 0.4, P ∗(2, 1) = 0.5. The inhomogeneous noise is driven by an θ − intensity β = 10 n 0.75. n 5 θQML n 2 θML n 0.5 0.1 10 20 50 100 200 500 1000 2000 5000 n Figure 6. Empirical mean of the Euclidean norm of the dif- ∗ ference of θ and the estimators norm based on 100 i.i.d. repli- · cations of the DHMM. Here M = 1, K = 2, n = 5 103 ∗ ∗ and θ = (0, 4, 0.5, 0.5, 0.4, 0.5). The parameter θ determines µ ∗ = (0, 4), Σ ∗ = (0.5, 0.5) and the “true” transition matrix θ θ by P ∗(1, 1) = 0.4, P ∗(2, 1) = 0.5. The inhomogeneous noise is θ θ − driven by an intensity β = 10 n 0.75. n To (P1) and (P2): By deﬁnition of the model this conditions are satisﬁed. nZ |*θ−LMQnθ| ]|*θ−nθ^|[E^ ##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##

text: MAXIMUM

text: LIKELIHOOD

text: ESTIMATION

text: IN

text: HMMS

text: WITH

text: INHOMOGENEOUS

text: NOISE

text: −2

text: θQML

text: n

text: θML

text: n

text: Figure

text: Exemplary

text: trajectory

text: of

text: observations

text: of

text: the

text: linear

text: Gaussian

text: model

text: from

text: Section

text: (above)

text: and

text: Euclidean

text: norm

text: of

text: the

text: diﬀerence

text: of

text: θ

text: and

text: the

text: estimators

text: based

text: on

text: a

text: single

text: trajectory

text: of

text: observations

text: (below).

text: Here

text: M

text: K

text: n

text: and

text: θ

text: The

text: parameter

text: θ

text: determines

text: µ

text: θ

text: Σ

text: and

text: the

text: “true”

text: transition

text: matrix

text: by

text: P

text: θ

text: θ

text: P

text: The

text: inhomogeneous

text: noise

text: is

text: driven

text: by

text: an

text: θ

text: intensity

text: β

text: n

text: n

text: θQML

text: n

text: θML

text: n

text: n

text: Figure

text: Empirical

text: mean

text: of

text: the

text: Euclidean

text: norm

text: of

text: the

text: dif-

text: ference

text: of

text: θ

text: and

text: the

text: estimators

text: norm

text: based

text: on

text: i.i.d.

text: repli-

text: cations

text: of

text: the

text: DHMM.

text: Here

text: M

text: K

text: n

text: and

text: θ

text: The

text: parameter

text: θ

text: determines

text: µ

text: Σ

text: and

text: the

text: “true”

text: transition

text: matrix

text: θ

text: θ

text: by

text: P

text: P

text: The

text: inhomogeneous

text: noise

text: is

text: θ

text: θ

text: driven

text: by

text: an

text: intensity

text: β

text: n

text: n

text: To

text: and

text: By

text: deﬁnition

text: of

text: the

text: model

text: this

text: conditions

text: are

text: satisﬁed.

text: nZ

text: |*θ−LMQnθ|

text: ]|*θ−nθ^|[E^

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 MAXIMUM LIKELIHOOD ESTIMATION IN HMMS WITH INHOMOGENEOUS NOISE A generalization of the consistency results of maximum likelihood estimation to scenarios with general state space of the hidden Markov chain might be of interest. S There are mainly two reasons why we assume that is ﬁnite: (1) Our main motivation comes from the DHMM which models the conduc- S tance levels of ion channel data with ﬁnite . (2) The requirements one needs to impose get more technical. In particular, our conditions on the “irreducibility and continuity of X” as well as the “well behaving HMM” from Section become more diﬃcult on general state spaces. It seems that the assumptions (A1)-(A6) of [11] are suﬃcient, but then in the proof of Theorem 1 we cannot argue with Lemma 6 anymore. This lemma can also be adapted to the more general scenario as in [11, Lemma 13], but then involves an additional term. 6. Proofs and auxiliary results We prove some results that specify the proximity of Y and Z. (cid:16) (cid:17) Lemma 1. Under the assumption formulated in (C1), we have P (12) ν lim m(Z , Y ) = 0 = 1. θ →∞ n n n ∈ ∈ P for any θ Θ and ν (S). (cid:88) (cid:88) (cid:88) Proof. By (C1) we obtain for any ε > 0 that ∞ ∞ P ≥ K P ≥ νθ (m(Zn, Yn) ε) = (cid:88) (cid:88) νθ (m(Zn, Yn) ε, Xn = k) n=1 n=1 k=1 ∞ K P P ≥ | = (cid:88) ν (X = k) (m(Z , Y ) ε X = k) θ n θ n n n n=1 k=1 ∞ ≤ P ≥ | ∞ max (m(Z , Y ) ε X = k) < . ∈ θ n n n k S n=1 By the Borel-Cantelli lemma we obtain the desired almost sure convergence of (cid:3) m(Z , Y ) to zero. n n In [11] the consistency of the maximum likelihood estimation for homogeneous HMMs under weak conditions is veriﬁed. We use the following result of them, which veriﬁes that the relative entropy rate exists. Theorem 2 ([11, Theorem 9]). Assume that the conditions (P1) and (H1) are satisﬁed. Then, there exists an (cid:96)(θ∗) ∈(cid:2) R, such that (cid:3) ∗ E − (13) (cid:96)(θ ) = l→im∞ πθ∗ n 1 log qθπ∗(Y1, . . . , Yn) n

text: MAXIMUM

text: LIKELIHOOD

text: ESTIMATION

text: IN

text: HMMS

text: WITH

text: INHOMOGENEOUS

text: NOISE

text: A

text: generalization

text: of

text: the

text: consistency

text: results

text: of

text: maximum

text: likelihood

text: estimation

text: to

text: scenarios

text: with

text: general

text: state

text: space

text: of

text: the

text: hidden

text: Markov

text: chain

text: might

text: be

text: of

text: interest.

text: S

text: There

text: are

text: mainly

text: two

text: reasons

text: why

text: we

text: assume

text: that

text: is

text: ﬁnite:

label: list

text: Our

text: main

text: motivation

text: comes

text: from

text: the

text: DHMM

text: which

text: models

text: the

text: conduc-

text: S

text: tance

text: levels

text: of

text: ion

text: channel

text: data

text: with

text: ﬁnite

text: The

text: requirements

text: one

text: needs

text: to

text: impose

text: get

text: more

text: technical.

text: In

text: particular,

text: our

text: conditions

text: on

text: the

text: “irreducibility

text: and

text: continuity

text: of

text: X”

text: as

text: well

text: as

text: the

text: “well

text: behaving

text: HMM”

text: from

text: Section

text: become

text: more

text: diﬃcult

text: on

text: general

text: state

text: spaces.

text: It

text: seems

text: that

text: the

text: assumptions

text: of

text: are

text: suﬃcient,

text: but

text: then

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: cannot

text: argue

text: with

text: Lemma

text: anymore.

text: This

text: lemma

text: can

text: also

text: be

text: adapted

text: to

text: the

text: more

text: general

text: scenario

text: as

text: in

text: Lemma

text: but

text: then

text: involves

text: an

text: additional

text: term.

text: Proofs

text: and

text: auxiliary

text: results

text: We

text: prove

text: some

text: results

text: that

text: specify

text: the

text: proximity

text: of

text: Y

text: and

text: Z.

text: Lemma

text: Under

text: the

text: assumption

text: formulated

text: in

text: we

text: have

text: P

text: ν

text: lim

text: m(Z

text: Y

text: θ

text: n

text: n

text: n

text: P

text: for

text: any

text: θ

text: Θ

text: and

text: ν

text: (S).

text: Proof.

text: By

text: we

text: obtain

text: for

text: any

text: ε

text: that

text: P

text: K

text: P

text: νθ

text: (m(Zn,

text: Yn)

text: ε)

text: νθ

text: (m(Zn,

text: Yn)

text: ε,

text: Xn

text: k)

text: K

text: P

text: P

text: ν

text: (X

text: k)

text: (m(Z

text: Y

text: ε

text: X

text: k)

text: θ

text: n

text: θ

text: n

text: n

text: n

text: P

text: max

text: (m(Z

text: Y

text: ε

text: X

text: k)

text: θ

text: n

text: n

text: n

text: k

text: S

text: By

text: the

text: Borel-Cantelli

text: lemma

text: we

text: obtain

text: the

text: desired

text: almost

text: sure

text: convergence

text: of

text: m(Z

text: Y

text: to

text: zero.

text: n

text: n

text: In

text: the

text: consistency

text: of

text: the

text: maximum

text: likelihood

text: estimation

text: for

text: homogeneous

text: HMMs

text: under

text: weak

text: conditions

text: is

text: veriﬁed.

text: We

text: use

text: the

text: following

text: result

text: of

text: them,

text: which

text: veriﬁes

text: that

text: the

text: relative

text: entropy

text: rate

text: exists.

text: Theorem

text: Theorem

text: Assume

text: that

text: the

text: conditions

text: and

text: are

text: satisﬁed.

text: Then,

text: there

text: exists

text: an

text: R,

text: such

text: that

text: E

text: l→im∞

text: πθ∗

text: n

text: log

text: Yn)

text: n


text: 30 MAXIMUM LIKELIHOOD ESTIMATION IN HMMS WITH INHOMOGENEOUS NOISE Appendix A. Strong consistency We follow the classical the approach of Wald, see [27], adopted to quasi likelihood P estimation. Let (Ω, F, ) be a probability space and (G, G ) be a measurable space. ⊆ R |·| Assume that Θ d and let be the d-dimensional Euclidean norm. Theorem 6 (Strong consistency). Let (W ) ∈N be a sequence of random variables P n n ∈ N × → ∞ mapping from (Ω, F, ) to (G, G ). For any n let h : Θ Gn [0, ) be a n ∗ ∈ measurable function. Assume that there exists an element θ Θ such that for any ⊂ ∗ (cid:54)∈ ∈ N closed C Θ with θ C and all n , we have (30) nl→im∞ θs∈uCp hhnn((θθ∗,,WW11,,......,,WWnn)) = 0 P-a.s. Let (θˆ ) ∈N be a sequence of random variables mapping from (Ω, F, P) to Θ such n n that (31) ∃c > 0 & n0 ∈ N ∀ n ≥(cid:12)n0 : hh(cid:12)nn((θθˆn∗,,WW11,,......,,WWnn)) ≥ c, P-a.s. (cid:12) (cid:12) (cid:12) (cid:12) Then lim θˆ − θ∗ = 0 P-a.s. →∞ n (cid:26) n (cid:12) (cid:12) (cid:27) (cid:12) (cid:12) Proof. For arbitrary ε > 0 deﬁn(cid:12) e (cid:12) (cid:40) ∈ − ∗ (cid:41) A(1) := ω Ω: lim sup θˆ (ω) θ > ε , ε →∞ n n (cid:40) ∈ h (θ, W (ω), . . . , W (ω)) (cid:41)≥ A(2) := ω Ω: lim sup sup n 1 n 1 , ε n→∞ θ:|θ−θ∗|≥ε hn(θˆn(ω), W1(ω), . . . , Wn(ω)) ∈ h (θ, W (ω), . . . , W (ω)) ≥ A(3) := ω Ω: lim sup sup n ∗ 1 n c . ε →∞ | − ∗|≥ h (θ , W (ω), . . . , W (ω)) n θ: θ θ ε n 1 n ⊆ ⊆ Note that A(ε1) A(ε2) A(ε3), w(cid:18)here the la(cid:12) st inclu(cid:12)sion (cid:19)follows by (31). Hence, by (30) we have P(A(ε3)) = 0 so that (cid:12)(cid:12) (cid:12)(cid:12) P(A(1)) = P lim sup θˆ − θ∗ > ε = 0, ε →∞ n n (cid:3) which implies the assertion. The following lemma is useful to verify condition (30). F P Lemma 7. Let (W ) ∈N be a sequence of random variables mapping from (Ω, , ) n n ∈ N × → ∞ to (G, G ) and, as in Theorem 6, for any n let h : Θ Gn [0, ) be a n ∗ ∈ measurable function. Assume that there is an element θ Θ such that for any ⊂ ∗ (cid:54)∈ closed C Θ with θ C we have (32) 1 1 ∗ P lim sup sup log h (θ, W , . . . , W ) < lim log h (θ , W , . . . , W ) -a.s. n→∞ θ∈C n n 1 n n→∞ n n 1 n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MAXIMUM

text: LIKELIHOOD

text: ESTIMATION

text: IN

text: HMMS

text: WITH

text: INHOMOGENEOUS

text: NOISE

text: Appendix

text: A.

text: Strong

text: consistency

text: We

text: follow

text: the

text: classical

text: the

text: approach

text: of

text: Wald,

text: see

text: adopted

text: to

text: quasi

text: likelihood

text: P

text: estimation.

text: Let

text: (Ω,

text: F,

text: be

text: a

text: probability

text: space

text: and

text: (G,

text: G

text: be

text: a

text: measurable

text: space.

text: R

text: Assume

text: that

text: Θ

text: d

text: and

text: let

text: be

text: the

text: d-dimensional

text: Euclidean

text: norm.

text: Theorem

text: (Strong

text: consistency).

text: Let

text: (W

text: ∈N

text: be

text: a

text: sequence

text: of

text: random

text: variables

text: P

text: n

text: n

text: N

text: mapping

text: from

text: (Ω,

text: F,

text: to

text: (G,

text: G

text: For

text: any

text: n

text: let

text: h

text: Θ

text: Gn

text: be

text: a

text: n

text: measurable

text: function.

text: Assume

text: that

text: there

text: exists

text: an

text: element

text: θ

text: Θ

text: such

text: that

text: for

text: any

text: N

text: closed

text: C

text: Θ

text: with

text: θ

text: C

text: and

text: all

text: n

text: we

text: have

text: nl→im∞

text: θs∈uCp

text: P-a.s.

text: Let

text: (θˆ

text: ∈N

text: be

text: a

text: sequence

text: of

text: random

text: variables

text: mapping

text: from

text: (Ω,

text: F,

text: P)

text: to

text: Θ

text: such

text: n

text: n

text: that

text: ∃c

text: N

text: n

text: c,

text: P-a.s.

text: Then

text: lim

text: θˆ

text: θ∗

text: P-a.s.

text: n

text: n

text: Proof.

text: For

text: arbitrary

text: ε

text: e

text: ω

text: Ω:

text: lim

text: sup

text: θˆ

text: (ω)

text: θ

text: ε

text: ε

text: n

text: n

text: h

text: (θ,

text: W

text: (ω),

text: W

text: (ω))

text: ω

text: Ω:

text: lim

text: sup

text: sup

text: n

text: n

text: ε

text: n→∞

text: θ:|θ−θ∗|≥ε

text: hn(θˆn(ω),

text: Wn(ω))

text: h

text: (θ,

text: W

text: (ω),

text: W

text: (ω))

text: ω

text: Ω:

text: lim

text: sup

text: sup

text: n

text: n

text: c

text: ε

text: h

text: (θ

text: W

text: (ω),

text: W

text: (ω))

text: n

text: θ:

text: θ

text: θ

text: ε

text: n

text: n

text: Note

text: that

text: the

text: st

text: by

text: Hence,

text: by

text: we

text: have

text: so

text: that

text: P

text: lim

text: sup

text: θˆ

text: θ∗

text: ε

text: ε

text: n

text: n

text: which

text: implies

text: the

text: assertion.

text: The

text: following

text: lemma

text: is

text: useful

text: to

text: verify

text: condition

text: F

text: P

text: Lemma

text: Let

text: (W

text: ∈N

text: be

text: a

text: sequence

text: of

text: random

text: variables

text: mapping

text: from

text: (Ω,

text: n

text: n

text: N

text: to

text: (G,

text: G

text: and,

text: as

text: in

text: Theorem

text: for

text: any

text: n

text: let

text: h

text: Θ

text: Gn

text: be

text: a

text: n

text: measurable

text: function.

text: Assume

text: that

text: there

text: is

text: an

text: element

text: θ

text: Θ

text: such

text: that

text: for

text: any

text: closed

text: C

text: Θ

text: with

text: θ

text: C

text: we

text: have

text: P

text: lim

text: sup

text: sup

text: log

text: h

text: (θ,

text: W

text: W

text: lim

text: log

text: h

text: (θ

text: W

text: W

text: -a.s.

text: n→∞

text: θ∈C

text: n

text: n

text: n

text: n→∞

text: n

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MAXIMUM LIKELIHOOD ESTIMATION IN HMMS WITH INHOMOGENEOUS NOISE 31 provided that the limit on the right hand-side exists. Then condition (30) is satisﬁed. (cid:32) (cid:20) (cid:21) (cid:33) Proof. Obviously (32) implies h (θ, W , . . . , W ) 1/n log lim sup sup n ∗ 1 n < 0. n→∞ θ∈C(cid:20) hn(θ , W1, . . . , Wn(cid:21)) This leads to h (θ, W , . . . , W ) 1/n lim sup sup n ∗ 1 n < 1 →∞ ∈ h (θ , W , . . . , W ) n θ C n 1 n (cid:3) from which (30) follows. References [1] E. Neher, B. Sakmann, The patch clamp technique, Scientiﬁc American 266 (3) (1992) 44–51. [2] T. Hotz, O. M. Schu¨tte, H. Sieling, T. Polupanow, U. Diederichsen, C. Steinem, A. Munk, Idealizing ion channel recordings by a jump segmentation multiresolution ﬁlter, IEEE Trans- actions on Nanobioscience 12 (4) (2013) 376–386. [3] F. Qin, A. Auerbach, F. Sachs, Hidden Markov modeling for single channel kinetics with ﬁltering and correlated noise, Biophysical Journal 79 (4) (2000) 1928–1944. [4] A. M. VanDongen, A new algorithm for idealizing single ion channel data containing multiple unknown conductance levels, Biophysical Journal 70 (3) (1996) 1303–1315. [5] I. Siekmann, L. E. Wagner, D. Yule, C. Fox, D. Bryant, E. J. Crampin, J. Sneyd, MCMC estimation of Markov models for ion channels, Biophysical Journal 100 (8) (2011) 1919–1929. [6] L. Venkataramanan, F. J. Sigworth, Applying hidden Markov models to the analysis of single ion channel activity., Biophysical Journal 82 (4) (2002) 1930–1942. [7] R. Briones, C. Weichbrodt, L. Paltrinieri, I. Mey, S. Villinger, K. Giller, A. Lange, M. Zweck- stetter, C. Griesinger, S. Becker, C. Steinem, B. L. de Groot, Voltage dependence of confor- mational dynamics and subconducting states of VDAC-1, Biophysical Journal 111 (6) (2016) 1223–1234. [8] C. Danelon, E. M. Nestorovich, M. Winterhalter, M. Ceccarelli, S. M. Bezrukov, Interaction of zwitterionic penicillins with the OmpF channel facilitates their translocation, Biophysical Journal 90 (5) (2006) 1617–1627. [9] L. E. Baum, T. Petrie, Statistical inference for probabilistic functions of ﬁnite state Markov chains, Ann. Math. Statist. 37 (6) (1966) 1554–1563. [10] L. E. Baum, T. Petrie, G. Soules, N. Weiss, A maximization technique occuring in the statis- tical analysis of probabilistic functions of Markov chains, Annals of Mathematical Statistics 41 (1970) 164–171. [11] R. Douc, E. Moulines, J. Olsson, R. van Handel, Consistency of the maximum likelihood estimator for general hidden Markov models, The Annals of Statistics 39 (1) (2011) 474–513. [12] B. G. Leroux, Maximum-likelihood estimation for hidden Markov models, Stochastic Pro- cesses and their Applications 40 (1) (1992) 127–143. [13] R. Douc, C. Matias, Asymptotics of the maximum likelihood estimator for general hidden Markov models, Bernoulli 7 (3) (2001) 381–420. [14] R. Douc, E. Moulines, T. Ryden, Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime, The Annals of Statistics 32 (5) (2004) 2254– 2304. [15] V. Genon-Catalot, C. Laredo, Leroux’s method for general hidden Markov models, Stochastic Processes and their Applications 116 (2) (2006) 222–243. [16] F. LeGland, L. Mevel, Asymptotic properties of the MLE in hidden Markov models, in: 1997 European Control Conference (ECC), 1997, pp. 3440–3445. [17] R. D. H. Heymans, J. R. Magnus, Consistent maximum-likelihood estimation with dependent observations : the general (non-normal) case, Journal of Econometrics 32 (2) (1986) 253–285. [18] P. Ailliot, F. P`ene, Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models, ESAIM: Probability and Statistics 19 (2015) 268–292. ##LTLine## ##LTLine##

text: MAXIMUM

text: LIKELIHOOD

text: ESTIMATION

text: IN

text: HMMS

text: WITH

text: INHOMOGENEOUS

text: NOISE

text: provided

text: that

text: the

text: limit

text: on

text: the

text: right

text: hand-side

text: exists.

text: Then

text: condition

text: is

text: satisﬁed.

text: Proof.

text: Obviously

text: implies

text: h

text: (θ,

text: W

text: W

text: log

text: lim

text: sup

text: sup

text: n

text: n

text: n→∞

text: hn(θ

text: This

text: leads

text: to

text: h

text: (θ,

text: W

text: W

text: lim

text: sup

text: sup

text: n

text: n

text: h

text: (θ

text: W

text: W

text: n

text: θ

text: C

text: n

text: n

text: from

text: which

text: follows.

text: References

text: E.

text: Neher,

text: B.

text: Sakmann,

text: The

text: patch

text: clamp

text: technique,

text: Scientiﬁc

text: American

text: T.

text: Hotz,

text: O.

text: M.

text: Schu¨tte,

text: H.

text: Sieling,

text: T.

text: Polupanow,

text: U.

text: Diederichsen,

text: C.

text: Steinem,

text: A.

text: Munk,

text: Idealizing

text: ion

text: channel

text: recordings

text: by

text: a

text: jump

text: segmentation

text: multiresolution

text: ﬁlter,

text: IEEE

text: Trans-

text: actions

text: on

text: Nanobioscience

text: F.

text: Qin,

text: A.

text: Auerbach,

text: F.

text: Sachs,

text: Hidden

text: Markov

text: modeling

text: for

text: single

text: channel

text: kinetics

text: with

text: ﬁltering

text: and

text: correlated

text: noise,

text: Biophysical

text: Journal

text: A.

text: M.

text: VanDongen,

text: A

text: new

text: algorithm

text: for

text: idealizing

text: single

text: ion

text: channel

text: data

text: containing

text: multiple

text: unknown

text: conductance

text: levels,

text: Biophysical

text: Journal

text: I.

text: Siekmann,

text: L.

text: E.

text: Wagner,

text: D.

text: Yule,

text: C.

text: Fox,

text: D.

text: Bryant,

text: E.

text: J.

text: Crampin,

text: J.

text: Sneyd,

text: MCMC

text: estimation

text: of

text: Markov

text: models

text: for

text: ion

text: channels,

text: Biophysical

text: Journal

text: L.

text: Venkataramanan,

text: F.

text: J.

text: Sigworth,

text: Applying

text: hidden

text: Markov

text: models

text: to

text: the

text: analysis

text: of

text: single

text: ion

text: channel

text: activity.,

text: Biophysical

text: Journal

text: R.

text: Briones,

text: C.

text: Weichbrodt,

text: L.

text: Paltrinieri,

text: I.

text: Mey,

text: S.

text: Villinger,

text: K.

text: Giller,

text: A.

text: Lange,

text: M.

text: Zweck-

text: stetter,

text: C.

text: Griesinger,

text: S.

text: Becker,

text: C.

text: Steinem,

text: B.

text: L.

text: de

text: Groot,

text: Voltage

text: dependence

text: of

text: confor-

text: mational

text: dynamics

text: and

text: subconducting

text: states

text: of

text: Biophysical

text: Journal

text: C.

text: Danelon,

text: E.

text: M.

text: Nestorovich,

text: M.

text: Winterhalter,

text: M.

text: Ceccarelli,

text: S.

text: M.

text: Bezrukov,

text: Interaction

text: of

text: zwitterionic

text: penicillins

text: with

text: the

text: OmpF

text: channel

text: facilitates

text: their

text: translocation,

text: Biophysical

text: Journal

text: L.

text: E.

text: Baum,

text: T.

text: Petrie,

text: Statistical

text: inference

text: for

text: probabilistic

text: functions

text: of

text: ﬁnite

text: state

text: Markov

text: chains,

text: Ann.

text: Math.

text: Statist.

text: L.

text: E.

text: Baum,

text: T.

text: Petrie,

text: G.

text: Soules,

text: N.

text: Weiss,

text: A

text: maximization

text: technique

text: occuring

text: in

text: the

text: statis-

text: tical

text: analysis

text: of

text: probabilistic

text: functions

text: of

text: Markov

text: chains,

text: Annals

text: of

text: Mathematical

text: Statistics

text: R.

text: Douc,

text: E.

text: Moulines,

text: J.

text: Olsson,

text: R.

text: van

text: Handel,

text: Consistency

text: of

text: the

text: maximum

text: likelihood

text: estimator

text: for

text: general

text: hidden

text: Markov

text: models,

text: The

text: Annals

text: of

text: Statistics

text: B.

text: G.

text: Leroux,

text: Maximum-likelihood

text: estimation

text: for

text: hidden

text: Markov

text: models,

text: Stochastic

text: Pro-

text: cesses

text: and

text: their

text: Applications

text: R.

text: Douc,

text: C.

text: Matias,

text: Asymptotics

text: of

text: the

text: maximum

text: likelihood

text: estimator

text: for

text: general

text: hidden

text: Markov

text: models,

text: Bernoulli

text: R.

text: Douc,

text: E.

text: Moulines,

text: T.

text: Ryden,

text: Asymptotic

text: properties

text: of

text: the

text: maximum

text: likelihood

text: estimator

text: in

text: autoregressive

text: models

text: with

text: Markov

text: regime,

text: The

text: Annals

text: of

text: Statistics

text: V.

text: Genon-Catalot,

text: C.

text: Laredo,

text: Leroux’s

text: method

text: for

text: general

text: hidden

text: Markov

text: models,

text: Stochastic

text: Processes

text: and

text: their

text: Applications

text: F.

text: LeGland,

text: L.

text: Mevel,

text: Asymptotic

text: properties

text: of

text: the

text: MLE

text: in

text: hidden

text: Markov

text: models,

text: in:

text: European

text: Control

text: Conference

text: (ECC),

text: pp.

text: R.

text: D.

text: H.

text: Heymans,

text: J.

text: R.

text: Magnus,

text: Consistent

text: maximum-likelihood

text: estimation

text: with

text: dependent

text: observations

text: the

text: general

text: (non-normal)

text: case,

text: Journal

text: of

text: Econometrics

text: P.

text: Ailliot,

text: F.

text: P`ene,

text: Consistency

text: of

text: the

text: maximum

text: likelihood

text: estimate

text: for

text: non-homogeneous

text: Markov–switching

text: models,

text: ESAIM:

text: Probability

text: and

text: Statistics

text: ##LTLine##

text: ##LTLine##


text: Viscous ﬂow regimes in a square. Part 1. Lid-driven cavity By F. Lam In the present paper, we examine the viscous ﬂow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vor- ticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn deﬁnes the Dirichlet value for the stream func- tion. We assert that the corner singularity in the cavity ﬂow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with ex- periments is found because of the converged palinstrophy ﬁeld. Our calculations demonstrate that asymmetrical shears initiate the formation of the vortices with counter-rotating strained cores, while the consecutive re-birth and the subsequent disintegration of the developed eddies proliferate the shears into smaller scales. This production process is particularly intense in close proximity to the solid surfaces. Keywords: Planar Navier-Stokes Equations; Vorticity; Turbulence; Diﬀusion; Viscosity; Randomness; Enstrophy; Palinstrophy 1. Introduction The most important aspect of ﬂow motion in the presence of hydraulically smooth solid boundaries is to quantify the vorticity production at the surfaces. The problem can be rephrased as the study of the convection and diﬀusion of the vorticity ﬁeld in connection with energy dissipation. As the ﬂuid viscosity becomes small or Reynolds number increases, experimental observation shows that the pro- duction processes are actively operational in a turbulent shear-laden region in the solid vicinity, known as a boundary layer. In this viscous near-wall region, physics dictates that momentum transfer must occur, as there is an increase in the wall shear stress, giving rise to enhanced energy consumption. Thus understanding of the near-wall ﬁne-scale vorticity structures is of importance. To avoid the contro- versy due to approximations, such as an analytical singularity in the boundary layer theory, it is best to work with the full set of the Navier-Stokes equations, so as to identify the genuine dynamics eﬀects. Surprisingly, our endeavour to underpin a self-consistent and eﬃcient numerical method for 2D turbulence is a matter of con- tinuing debate. To bypass the main diﬃculties in assigning the wall vorticity, many of the previous works on rectangles concentrate on periodic boundary conditions which are non-physical. Naturally, any Navier-Stokes ﬂow, developed from arbi- trary initial condition, must be inhomogeneous, anisotropic and highly unsteady. We believe that certain fundamental aspects on the governing diﬀerential equations have long been overlooked, leading to inconsistency among the proposed numerical schemes. This view is particularly justiﬁable for the vorticity-stream function for- mulation. For an introduction to planar vorticity dynamics and its computation, see Gresho (1991). Article published on arXiv

text: Viscous

label: title

text: ﬂow

text: regimes

text: in

text: a

text: square.

text: Part

text: Lid-driven

text: cavity

text: By

text: F.

label: author

text: Lam

text: In

label: abstract

text: the

text: present

text: paper,

text: we

text: examine

text: the

text: viscous

text: ﬂow

text: evolution

text: in

text: a

text: square

text: cavity.

text: Coupled

text: with

text: the

text: stream

text: function,

text: the

text: initial-boundary

text: value

text: problem

text: of

text: the

text: vor-

text: ticity

text: is

text: numerically

text: solved

text: by

text: a

text: method

text: of

text: iteration.

text: The

text: only

text: boundary

text: condition

text: is

text: the

text: wall

text: velocity

text: which

text: in

text: turn

text: deﬁnes

text: the

text: Dirichlet

text: value

text: for

text: the

text: stream

text: func-

text: tion.

text: We

text: assert

text: that

text: the

text: corner

text: singularity

text: in

text: the

text: cavity

text: ﬂow

text: is

text: in

text: fact

text: a

text: theoretical

text: artefact.

text: By

text: adopting

text: suitably

text: regulated

text: lid

text: velocity,

text: excellent

text: comparison

text: with

text: ex-

text: periments

text: is

text: found

text: because

text: of

text: the

text: converged

text: palinstrophy

text: ﬁeld.

text: Our

text: calculations

text: demonstrate

text: that

text: asymmetrical

text: shears

text: initiate

text: the

text: formation

text: of

text: the

text: vortices

text: with

text: counter-rotating

text: strained

text: cores,

text: while

text: the

text: consecutive

text: re-birth

text: and

text: the

text: subsequent

text: disintegration

text: of

text: the

text: developed

text: eddies

text: proliferate

text: the

text: shears

text: into

text: smaller

text: scales.

text: This

text: production

text: process

text: is

text: particularly

text: intense

text: in

text: close

text: proximity

text: to

text: the

text: solid

text: surfaces.

text: Keywords:

text: Planar

text: Navier-Stokes

text: Equations;

text: Vorticity;

text: Turbulence;

text: Diﬀusion;

text: Viscosity;

text: Randomness;

text: Enstrophy;

text: Palinstrophy

text: Introduction

text: The

text: most

text: important

text: aspect

text: of

text: ﬂow

text: motion

text: in

text: the

text: presence

text: of

text: hydraulically

text: smooth

text: solid

text: boundaries

text: is

text: to

text: quantify

text: the

text: vorticity

text: production

text: at

text: the

text: surfaces.

text: The

text: problem

text: can

text: be

text: rephrased

text: as

text: the

text: study

text: of

text: the

text: convection

text: and

text: diﬀusion

text: of

text: the

text: vorticity

text: ﬁeld

text: in

text: connection

text: with

text: energy

text: dissipation.

text: As

text: the

text: ﬂuid

text: viscosity

text: becomes

text: small

text: or

text: Reynolds

text: number

text: increases,

text: experimental

text: observation

text: shows

text: that

text: the

text: pro-

text: duction

text: processes

text: are

text: actively

text: operational

text: in

text: a

text: turbulent

text: shear-laden

text: region

text: in

text: the

text: solid

text: vicinity,

text: known

text: as

text: a

text: boundary

text: layer.

text: In

text: this

text: viscous

text: near-wall

text: region,

text: physics

text: dictates

text: that

text: momentum

text: transfer

text: must

text: occur,

text: as

text: there

text: is

text: an

text: increase

text: in

text: the

text: wall

text: shear

text: stress,

text: giving

text: rise

text: to

text: enhanced

text: energy

text: consumption.

text: Thus

text: understanding

text: of

text: the

text: near-wall

text: ﬁne-scale

text: vorticity

text: structures

text: is

text: of

text: importance.

text: To

text: avoid

text: the

text: contro-

text: versy

text: due

text: to

text: approximations,

text: such

text: as

text: an

text: analytical

text: singularity

text: in

text: the

text: boundary

text: layer

text: theory,

text: it

text: is

text: best

text: to

text: work

text: with

text: the

text: full

text: set

text: of

text: the

text: Navier-Stokes

text: equations,

text: so

text: as

text: to

text: identify

text: the

text: genuine

text: dynamics

text: eﬀects.

text: Surprisingly,

text: our

text: endeavour

text: to

text: underpin

text: a

text: self-consistent

text: and

text: eﬃcient

text: numerical

text: method

text: for

text: turbulence

text: is

text: a

text: matter

text: of

text: con-

text: tinuing

text: debate.

text: To

text: bypass

text: the

text: main

text: diﬃculties

text: in

text: assigning

text: the

text: wall

text: vorticity,

text: many

text: of

text: the

text: previous

text: works

text: on

text: rectangles

text: concentrate

text: on

text: periodic

text: boundary

text: conditions

text: which

text: are

text: non-physical.

text: Naturally,

text: any

text: Navier-Stokes

text: ﬂow,

text: developed

text: from

text: arbi-

text: trary

text: initial

text: condition,

text: must

text: be

text: inhomogeneous,

text: anisotropic

text: and

text: highly

text: unsteady.

text: We

text: believe

text: that

text: certain

text: fundamental

text: aspects

text: on

text: the

text: governing

text: diﬀerential

text: equations

text: have

text: long

text: been

text: overlooked,

text: leading

text: to

text: inconsistency

text: among

text: the

text: proposed

text: numerical

text: schemes.

text: This

text: view

text: is

text: particularly

text: justiﬁable

text: for

text: the

text: vorticity-stream

text: function

text: for-

text: mulation.

text: For

text: an

text: introduction

text: to

text: planar

text: vorticity

text: dynamics

text: and

text: its

text: computation,

text: see

text: Gresho

text: Article

text: published

text: on

text: arXiv


text: Navier-Stokes Equations in Bounded Domain 11 ∇× Solenoidal velocity from di-vorticity ζ Instead of the stream function, the velocity can also be found from (1.11) by inverting the Laplac(cid:90)ian: (cid:90) (cid:12) (cid:12) (cid:90)− (cid:48) (cid:48) (cid:48) − 1 (cid:48) (cid:48) u(x) = ∂y(cid:48)G(x, x ) ζ(x )dx ∂y(cid:48)G y(cid:48)=1 uLid(x ) dx , Σ 0 (3.8) (cid:48) (cid:48) (cid:48) v(x) = ∂ (cid:48)G(x, x ) ζ(x )dx , x Σ where Green’s function G satisfying homogeneous Dirichlet condition is given by (cid:40) (cid:88) ∞ (cid:48) − (cid:48) (cid:48) (cid:48) sinh(kπy ) sinh(kπ(1 y)), y < y, G(x, x ) = 2 sin(kπx) sin(kπx ) − (cid:48) (cid:48) (3.9) sinh(kπy) sinh(kπ(1 y )), y > y. k=1 The ﬂow ﬁeld is well-determined by solving the parabolic-elliptic system of (1.7) and (1.11). Note that two Poisson equations must be inverted at every time step. Equivalence principle At the conclusion of iteration or convergence, the ﬂow-ﬁeld solutions given by ∇× − −∇× ζ = u, u = (∂ ψ, ∂ ψ), ∆u = ζ, (3.10) y x must be self-consistent. This requirement deﬁnes an equivalence principle: nu- merical values evaluated from the vorticity-stream function formulation must agree with those from the di-vorticity, and vice versa. The principle may serve as a com- patibility check in the numerical implementation. The dynamics in (3.1) can be transformed into the canonical form containing (cid:90) (cid:90) quadratic non-linearity, t (cid:48) (cid:48) (cid:48) ζ(x, t) = η(x, t) + K(x, x , t, s) ζ2(x , s) dx ds, 0 Σ where the kernel K = K(G, N). In general, its solutions are of turbulence. Numerical method To seek the solutions of (1.7) or (3.1) by iteration, we subdivide the cavity into equally-spaced grids, where the vorticity equation is solved using any numerical schemes suitable for convection-diﬀusion equations. The grid or mesh is denoted by ≡ n grid points or an n2 mesh. Given the starting value ζ0 ζ , the implicit Euler 0 scheme of time discretisation of the transport (1.7) may be symbolically written as determinacy. Strictly speaking, the computational results from these formulations must be treated with caution as they can be misleading. The confusing state of aﬀairs can be attributed to the view that solid surfaces in incompressible viscous ﬂows identify with streamlines whose analytic exposition, i.e., the stream function, plays a unique and primary role in isolation. Article published on arXiv

text: Navier-Stokes

text: Equations

text: in

text: Bounded

text: Domain

text: Solenoidal

text: velocity

text: from

text: di-vorticity

text: ζ

text: Instead

text: of

text: the

text: stream

text: function,

text: the

text: velocity

text: can

text: also

text: be

text: found

text: from

text: by

text: inverting

text: the

label: equation

text: u(x)

text: x

text: ζ(x

text: )dx

text: uLid(x

text: dx

text: Σ

text: v(x)

text: x

text: ζ(x

text: )dx

text: x

text: Σ

text: where

text: Green’s

text: function

text: G

text: satisfying

text: homogeneous

text: Dirichlet

text: condition

text: is

text: given

text: by

text: sinh(kπy

text: y)),

text: y

text: y,

text: G(x,

text: x

text: sin(kπx)

text: sin(kπx

text: sinh(kπy)

text: y

text: y

text: y.

text: The

text: ﬂow

text: ﬁeld

text: is

text: well-determined

text: by

text: solving

text: the

text: parabolic-elliptic

text: system

text: of

text: and

text: Note

text: that

text: two

text: Poisson

text: equations

text: must

text: be

text: inverted

text: at

text: every

text: time

text: step.

text: Equivalence

text: principle

text: At

text: the

text: conclusion

text: of

text: iteration

text: or

text: convergence,

text: the

text: ﬂow-ﬁeld

text: solutions

text: given

text: by

text: ζ

text: u,

text: u

text: ψ,

text: ψ),

text: ∆u

text: ζ,

text: y

text: x

text: must

text: be

text: self-consistent.

text: This

text: requirement

text: deﬁnes

text: an

text: equivalence

text: principle:

text: nu-

text: merical

text: values

text: evaluated

text: from

text: the

text: vorticity-stream

text: function

text: formulation

text: must

text: agree

text: with

text: those

text: from

text: the

text: di-vorticity,

text: and

text: vice

text: versa.

text: The

text: principle

text: may

text: serve

text: as

text: a

text: com-

text: patibility

text: check

text: in

text: the

text: numerical

text: implementation.

text: The

text: dynamics

text: in

text: can

text: be

text: transformed

text: into

text: the

text: canonical

text: form

text: containing

text: quadratic

text: non-linearity,

text: t

text: ζ(x,

text: t)

text: η(x,

text: t)

text: K(x,

text: x

text: t,

text: s)

text: s)

text: dx

text: ds,

text: Σ

text: where

text: the

text: kernel

text: K

text: K(G,

text: N).

text: In

text: general,

text: its

text: solutions

text: are

text: of

text: turbulence.

text: Numerical

text: method

text: To

text: seek

text: the

text: solutions

text: of

text: or

text: by

text: iteration,

text: we

text: subdivide

text: the

text: cavity

text: into

text: equally-spaced

text: grids,

text: where

text: the

text: vorticity

text: equation

text: is

text: solved

text: using

text: any

text: numerical

text: schemes

text: suitable

text: for

text: convection-diﬀusion

text: equations.

text: The

text: grid

text: or

text: mesh

text: is

text: denoted

text: by

text: n

text: grid

text: points

text: or

text: an

text: mesh.

text: Given

text: the

text: starting

text: value

text: ζ0

text: ζ

text: the

text: implicit

text: Euler

text: scheme

text: of

text: time

text: discretisation

text: of

text: the

text: transport

text: may

text: be

text: symbolically

text: written

text: as

text: determinacy.

text: Strictly

text: speaking,

text: the

text: computational

text: results

text: from

text: these

text: formulations

text: must

text: be

text: treated

text: with

text: caution

text: as

text: they

text: can

text: be

text: misleading.

text: The

text: confusing

text: state

text: of

text: aﬀairs

text: can

text: be

text: attributed

text: to

text: the

text: view

text: that

text: solid

text: surfaces

text: in

text: incompressible

text: viscous

text: ﬂows

text: identify

text: with

text: streamlines

text: whose

text: analytic

text: exposition,

text: i.e.,

text: the

text: stream

text: function,

text: plays

text: a

text: unique

text: and

text: primary

text: role

text: in

text: isolation.

text: Article

text: published

text: on

text: arXiv