text: → → For χχ φ gg, qq¯, the corresponding R factors are given by XX¯ 192m8 64m6 β (4m2 ) R = χ , R = χ q χ , (35) gg M8T (m /M) qq¯ M6β (M2)T (m /M) χ q χ − ∝ ∝ where T (x) = 1 4x2 + 12x4. Since in vector DM model R (m /M)8 and R gg χ qq¯ (m /M)6, it is expected that a very large Γ/M is required. For gg channel, the minimally χ required width is given by (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:104) (cid:105) Γ (cid:38) M 4 62 GeV 3 σv 1/2 3.3 × − gg − , (36) M 750 GeV m 1.96 10 26 cm3s 1 vector,gg χ and for qq¯ channel (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:104) (cid:105) Γ (cid:38) M 3 46 GeV 2 σv 1/2 0.73 × − gg − . (37) M 750 GeV m 1.42 10 26 cm3s 1 fermion,b¯b χ ¯ The results for the cc¯ channel is similar to that in the bb channel. Thus in all the cases, the required φ width are too large and already ruled out by the current Run-2 data, which indicates that the vector DM model can not provide a consistent explanation to the LHC Run-2 diphoton excess and the GCE. 5 Conclusions In summary, we have investigate the conditions for a consistent explanation for possible the LHC diphoton excess and GCE, especially the requirement on total width of φ in a wide range of DM models where the DM particle can be scalar, fermionionic and vector, ¯ and φ can be generated by s-channel gluon fusion or quark-antiquark annihilation (bb and cc¯) at parton level. We have shown that the required Γ/M is determined by a single parameter proportional to (m /M)n. We have found that three models can explain the χ two excesses successfully: i) scalar DM model with φ coupling dominantly with qq¯. the O − minimally required Γ/M can be as low as (10 3); ii) scalar DM model with φ coupling O − dominantly with gg, the required Γ/M is about (10 2); iii) fermionic DM model with O − coupling dominantly with qq¯, the required Γ/M reaches (10 2). Other models such as the vector DM model requires larger Γ/M of order one which is already disfavoured by the current data. For the same DM model, the required width of φ is always smaller in 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

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text: 1100 11 1100 11 Scalar DM, bb, Γ/M=0.006 68%CL Scalar DM, bb, Γ/M=0.006 68%CL 95%CL 95%CL 1100 22 1100 22 GCE GCE 1100 33 diphoton 1100 33 Run I γγ limit Run I γγ limit 1100 44 1100 44 diphoton 1100 55 1100 55 1100 55 1100 44 1100 33 1100 22 1100 11 1100 55 1100 44 1100 33 1100 22 1100 11 Γ /M Γ /M bb χχ 1100 11 1100 11 Scalar DM, cc, Γ/M=0.006 68%CL Scalar DM, cc, Γ/M=0.006 68%CL 95%CL 95%CL 1100 22 1100 22 GCE GCE 1100 33 diphoton 1100 33 Run I γγ limit Run I γγ limit 1100 44 1100 44 diphoton 1100 55 1100 55 1100 55 1100 44 1100 33 1100 22 1100 11 1100 55 1100 44 1100 33 1100 22 1100 11 Γ /M Γ /M cc χχ FIG. 4: The same as Fig. ??, but for Γ/M = 0.006. 30 M/ Γ M/ Γ γγ γγ M/ Γ M/ Γ γγ γγ ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 7. CONCLUSIONS streaming-programming-guide.html. In the last decade, signiﬁcant research has been performed [6] H. Andrade, B. Gedik, K.-L. Wu, and P. Yu. Scale-up to create a system that can handle Big Data. The strategies for processing high-rate data streams in MapReduce paradigm is able to oﬀer a solution for Big system s. In Data Engineering, 2009. ICDE ’09. IEEE Data and many solutions revolves around it. A solution 25th International Conference on, pages 1375–1378, based on MapReduce is suitable for many problems but not March 2009. appropriate for many others. Previous research has been [7] Apache. Storm - Distributed and fault-tolerant paired to ﬁnd solutions which would be optimal for emerging realtime computation. http://storm.apache.org/. Big Data trends. The stream computing paradigm appears [8] C. Ballard. IBM Infosphere Streams harnessing data to be a solution to the emerging Big Data trends. in motion. Vervante, S.l, 2010. [9] A. Biem, E. Bouillet, H. Feng, A. Ranganathan, The research community is primarily focused on A. Riabov, O. Verscheure, H. Koutsopoulos, and development of solution or mining of large data sets. C. Moran. Ibm infosphere streams for scalable, The research on providing solution is divided into the real-time, intelligent transportation services. In selection of the programming model or data model for a Proceedings of the 2010 ACM SIGMOD International system. The selection of processing model for a system Conference on Management of Data, SIGMOD ’10, varies from batch processing to micro-batch processing. pages 1093–1104, New York, NY, USA, 2010. ACM. Considering the availability of MapReduce as successful [10] A. Biem, B. Elmegreen, O. Verscheure, D. Turaga, paradigm, many solutions for streaming are inﬂuenced H. Andrade, and T. Cornwell. A streaming approach by this paradigm. Some solutions also explore the Actor to radio astronomy imaging. In Acoustics Speech and model to have stream processing solution. Solutions such Signal Processing (ICASSP), 2010 IEEE International as Storm provide a sub-second latency and S4 does not Conference on, pages 1654–1657, March 2010. provide persistent state and complete fault tolerance. [11] M. Blount, M. Ebling, J. Eklund, A. James, Spark Streaming has mixed processing model and exactly C. McGregor, N. Percival, K. Smith, and D. Sow. once mechanism for record delivery which might aﬀect Real-time analysis for intensive care: Development processing. and deployment of the artemis analytic system. Engineering in Medicine and Biology Magazine, IEEE, A fundamental set of questions exists, which should be 29(2):110–118, March 2010. addressed before selecting any programming model or data [12] J. Chauhan, S. A. Chowdhury, and D. Makaroﬀ. model for stream processing. The design choices and Performance evaluation of yahoo! s4: A ﬁrst look. In challenges aﬀect system latency and throughput. The Proceedings of the 2012 Seventh International challenges linked to applications and processing system Conference on P2P, Parallel, Grid, Cloud and require elaborate reasoning and inspection of requirements Internet Computing, 3PGCIC ’12, pages 58–65, to create a stream processing system for heterogeneous data Washington, DC, USA, 2012. IEEE Computer Society. set. The stream processing paradigm requires solution which can provide low latency, high throughput, fault tolerance [13] L. Daldorﬀ, S. M. Mohammadi, J. Bergman, B. Thide, along with scalability and versatility. The system requires A. Biem, B. Elmegreen, D. Turaga, O. Verscheure, extensibility to plug and play diﬀerent components to and W. Puccio. Novel data stream techniques for real provide analytics for in-stream processing and stored stream time hf radio weather statistics and forecasting. In data in a persistent storage. Ionospheric radio Systems and Techniques, 2009.(IRST 2009). The Institution of Engineering and 8. ACKNOWLEDGEMENTS Technology 11th International Conference on, pages 1–3. IET, 2009. I sincerely thank the reviewers for their comments and [14] C. Harrison, B. Eckman, R. Hamilton, P. Hartswick, suggestions. This survey paper has been supported by the J. Kalagnanam, J. Paraszczak, and P. Williams. University of Helsinki as a learning initiative under Seminar Foundations for smarter cities. IBM Journal of course on distributed computing frameworks for Big Data. Research and Development, 54(4):1–16, July 2010. [15] J. M. Hern´andez-Mun˜oz, J. B. Vercher, L. Mun˜oz, 9. REFERENCES J. A. Galache, M. Presser, L. A. H. Go´mez, and [1] T. Akidau, A. Balikov, K. Bekiroglu, S. Chernyak, J. Pettersson. Smart cities at the forefront of the J. Haberman, R. Lax, S. McVeety, D. Mills, future internet. Springer, 2011. P. Nordstrom, and S. Whittle. Millwheel: [16] IBM Corporation. Exploiting Big Data in Fault-tolerant stream processing at internet scale. In telecommunications to increase revenue, reduce Very Large Data Bases, pages 734–746, 2013. customer churn and operating costs. [2] Amazon. Amazon Kinesis. http://www-01.ibm.com/software/data/bigdata/ http://aws.amazon.com/kinesis/. industry-telco.html. [3] Amazon. Amazon Kinesis Product Details. [17] R. Kienzler, R. Bruggmann, A. Ranganathan, and http://aws.amazon.com/kinesis/details/. N. Tatbul. Large-scale dna sequence analysis in the [4] Amazon. AWS Case Study: Supercell. cloud: A stream-based approach. In Proceedings of the 2011 International Conference on Parallel Processing - http://aws.amazon.com/solutions/case- studies/supercell/. Volume 2, Euro-Par’11, pages 467–476, Berlin, Heidelberg, 2012. Springer-Verlag. [5] U. B. AMPLab. Spark Streaming. [18] G. D. F. Morales. Distributed stream processing http://spark.incubator.apache.org/docs/latest/

label: section

text: CONCLUSIONS

text: streaming-programming-guide.html.

label: reference

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text: decade,

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text: has

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text: performed

text: H.

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text: P.

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text: Scale-up

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text: The

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text: in

text: MapReduce

text: paradigm

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text: IBM

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text: the

text: emerging

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text: trends.

text: in

text: motion.

text: Vervante,

text: S.l,

text: A.

text: Biem,

text: E.

text: Bouillet,

text: H.

text: Feng,

text: A.

text: Ranganathan,

text: The

text: research

text: community

text: is

text: primarily

text: focused

text: on

text: A.

text: Riabov,

text: O.

text: Verscheure,

text: H.

text: Koutsopoulos,

text: and

text: development

text: of

text: solution

text: or

text: mining

text: of

text: large

text: data

text: sets.

text: C.

text: Moran.

text: Ibm

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text: The

text: research

text: on

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text: solution

text: is

text: divided

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text: real-time,

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text: services.

text: In

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text: the

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text: or

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text: model

text: for

text: a

text: Proceedings

text: of

text: the

text: ACM

text: SIGMOD

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text: system.

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text: selection

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text: processing

text: model

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text: Conference

text: on

text: Management

text: of

text: Data,

text: SIGMOD

text: varies

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text: batch

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text: to

text: micro-batch

text: processing.

text: pages

text: New

text: York,

text: NY,

text: USA,

text: ACM.

text: Considering

text: the

text: availability

text: of

text: MapReduce

text: as

text: successful

text: A.

text: Biem,

text: B.

text: Elmegreen,

text: O.

text: Verscheure,

text: D.

text: Turaga,

text: paradigm,

text: many

text: solutions

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text: inﬂuenced

text: H.

text: Andrade,

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text: provide

text: persistent

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text: complete

text: fault

text: tolerance.

text: M.

text: Blount,

text: M.

text: Ebling,

text: J.

text: Eklund,

text: A.

text: James,

text: Spark

text: Streaming

text: has

text: mixed

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text: and

text: exactly

text: C.

text: McGregor,

text: N.

text: Percival,

text: K.

text: Smith,

text: and

text: D.

text: Sow.

text: once

text: mechanism

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text: record

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text: A

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text: March

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text: J.

text: Chauhan,

text: S.

text: A.

text: Chowdhury,

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text: Makaroﬀ.

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text: Parallel,

text: Grid,

text: Cloud

text: and

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text: and

text: inspection

text: of

text: requirements

text: Internet

text: Computing,

text: pages

text: to

text: create

text: a

text: stream

text: processing

text: system

text: for

text: heterogeneous

text: data

text: Washington,

text: DC,

text: USA,

text: IEEE

text: Computer

text: Society.

text: set.

text: The

text: stream

text: processing

text: paradigm

text: requires

text: solution

text: which

text: can

text: provide

text: low

text: latency,

text: high

text: throughput,

text: fault

text: tolerance

text: L.

text: Daldorﬀ,

text: S.

text: M.

text: Mohammadi,

text: J.

text: Bergman,

text: B.

text: Thide,

text: along

text: with

text: scalability

text: and

text: versatility.

text: The

text: system

text: requires

text: A.

text: Biem,

text: B.

text: Elmegreen,

text: D.

text: Turaga,

text: O.

text: Verscheure,

text: extensibility

text: to

text: plug

text: and

text: play

text: diﬀerent

text: components

text: to

text: and

text: W.

text: Puccio.

text: Novel

text: data

text: stream

text: techniques

text: for

text: real

text: provide

text: analytics

text: for

text: in-stream

text: processing

text: and

text: stored

text: stream

text: time

text: hf

text: radio

text: weather

text: statistics

text: and

text: forecasting.

text: In

text: data

text: in

text: a

text: persistent

text: storage.

text: Ionospheric

text: radio

text: Systems

text: and

text: Techniques,

text: The

text: Institution

text: of

text: Engineering

text: and

text: ACKNOWLEDGEMENTS

text: Technology

text: International

text: Conference

text: on,

text: pages

text: IET,

text: I

text: sincerely

text: thank

text: the

text: reviewers

text: for

text: their

text: comments

text: and

text: C.

text: Harrison,

text: B.

text: Eckman,

text: R.

text: Hamilton,

text: P.

text: Hartswick,

text: suggestions.

text: This

text: survey

text: paper

text: has

text: been

text: supported

text: by

text: the

text: J.

text: Kalagnanam,

text: J.

text: Paraszczak,

text: and

text: P.

text: Williams.

text: University

text: of

text: Helsinki

text: as

text: a

text: learning

text: initiative

text: under

text: Seminar

text: Foundations

text: for

text: smarter

text: cities.

text: IBM

text: Journal

text: of

text: course

text: on

text: distributed

text: computing

text: frameworks

text: for

text: Big

text: Data.

text: Research

text: and

text: Development,

text: July

text: J.

text: M.

text: Hern´andez-Mun˜oz,

text: J.

text: B.

text: Vercher,

text: L.

text: Mun˜oz,

text: REFERENCES

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text: IBM

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text: Mon. Not. R. Astron. Soc. 000, 1–?? (2002) Printed 29 May 2020 (MN LATEX style ﬁle v2.2) Properties of massive star-forming clumps with infall motions † Yu-Xin He1,3(cid:63), Jian-Jun Zhou1,2 , Jarken Esimbek1,2, Wei-Guang Ji1,2, Gang Wu1,2,3, Xin-Di Tang1,2,4, Toktarkhan Komesh5, Ye Yuan1,2, Da-Lei Li1,3 and W.A. Baan6 1Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, P. R. China 2Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Urumqi 830011, P. R. China 3University of the Chinese Academy of Sciences, Beijing 100080, P. R. China 4Max-Planck-Institut fu¨r Radioastronomie, Bonn 53121, Germany 5Department of Solid State Physics and Non-linear Physics, Faculty of Physics and Technology, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan 6Netherlands Institute for Radio Astronomy, 7991 PD Dwingeloo, The Netherlands 29 May 2020 ABSTRACT In this work, we aim to characterise high-mass clumps with infall motions. We selected 327 clumps from the Millimetre Astronomy Legacy Team 90-GHz (MALT90) survey, and identiﬁed 100 infall candidates. Combined with the results of He et al. (2015), we obtained a sample of 732 high-mass clumps, including 231 massive infall candidates and 501 clumps where infall is not detected. Objects in our sample were classiﬁed as pre-stellar, proto-stellar, HII or photo-dissociation region (PDR). The detection rates of the infall candidates in the pre-stellar, proto-stellar, HII and PDR stages are 41.2%, 36.6%, 30.6% and 12.7%, respectively. The infall candidates have a higher H column 2 density and volume density compared with the clumps where infall is not detected at every stage. For the infall candidates, the median values of the infall rates at the × − × − × − pre-stellar, proto-stellar, HII and PDR stages are 2.6 10 3, 7.0 10 3, 6.5 10 3 and × − − 5.5 10 3 M(cid:12) yr 1, respectively. These values indicate that infall candidates at later evolutionary stages are still accumulating material eﬃciently. It is interesting to ﬁnd that both infall candidates and clumps where infall is not detected show a clear trend of increasing mass from the pre-stellar to proto-stellar, and to the HII stages. The ± ± power indices of the clump mass function (ClMF) are 2.04 0.16 and 2.17 0.31 for the infall candidates and clumps where infall is not detected, respectively, which agree well with the power index of the stellar initial mass function (2.35) and the cold Planck cores (2.0). − − Key words: stars: formation ISM: kinematics and dynamics ISM: molecules − radio lines: ISM. 1 INTRODUCTION outside the gravitational radius and that no material is added to the core after it forms. The ﬁnal mass of a mas- High-mass star formation is still poorly understood sive star is decided by the amount of mass that is necessary (Tan et al. 2014). Core accretion (McKee & Tan 2002, 2003) to be self-gravitating, which is thought to be supported ei- and competitive accretion (Bonnell et al. 2001; Bonnell, ther by thermal pressure, turbulence or magnetic ﬁelds (Mc- Vine, & Bate 2004) are two contending theories underlying Kee & Tan 2002, 2003), and is not greatly inﬂuenced by a current investigations. star’s environment. The core accretion model imposes a di- The core accretion model essentially assumes that the rect link between core and star, and thus the core mass func- evolution of the core is isolated from environmental eﬀects tion (CMF) must have a similar shape to the stellar initial mass function (IMF) in that scenario. (e.g. Motte, Andre, & Neri 1998). (cid:63) E-mail:heyuxin@xao.ac.cn † Email: zhoujj@xao.ac.cn In the competitive accretion model, the mass of a star (cid:13) c 2002 RAS

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: Properties

label: title

text: of

text: massive

text: star-forming

text: clumps

text: with

text: infall

text: motions

label: author

text: Yu-Xin

text: Jian-Jun

text: Jarken

text: Wei-Guang

text: Gang

text: Xin-Di

text: Toktarkhan

text: Ye

text: Da-Lei

text: and

text: W.A.

text: Astronomical

text: Observatory,

text: Chinese

text: Academy

text: of

text: Sciences,

text: Urumqi

text: P.

text: R.

text: China

text: Laboratory

text: of

text: Radio

text: Astronomy,

text: Chinese

text: Academy

text: of

text: Sciences,

text: Urumqi

text: P.

text: R.

text: China

text: of

text: the

text: Chinese

text: Academy

text: of

text: Sciences,

text: Beijing

text: P.

text: R.

text: China

text: fu¨r

text: Radioastronomie,

text: Bonn

text: Germany

text: of

text: Solid

text: State

text: Physics

text: and

text: Non-linear

text: Physics,

text: Faculty

text: of

text: Physics

text: and

text: Technology,

text: Al-Farabi

text: Kazakh

text: National

text: University,

text: Almaty

text: Kazakhstan

text: Institute

text: for

text: Radio

text: Astronomy,

text: PD

text: Dwingeloo,

text: The

text: Netherlands

text: May

text: ABSTRACT

text: In

text: this

text: work,

text: we

text: aim

text: to

text: characterise

text: high-mass

text: clumps

text: with

text: infall

text: motions.

text: We

text: selected

text: clumps

text: from

text: the

text: Millimetre

text: Astronomy

text: Legacy

text: Team

text: survey,

text: and

text: identiﬁed

text: infall

text: candidates.

text: Combined

text: with

text: the

text: results

text: of

text: He

text: et

text: al.

text: we

text: obtained

text: a

text: sample

text: of

text: high-mass

text: clumps,

text: including

text: massive

text: infall

text: candidates

text: and

text: clumps

text: where

text: infall

text: is

text: not

text: detected.

text: Objects

text: in

text: our

text: sample

text: were

text: classiﬁed

text: as

text: pre-stellar,

text: proto-stellar,

text: HII

text: or

text: photo-dissociation

text: region

text: (PDR).

text: The

text: detection

text: rates

text: of

text: the

text: infall

text: candidates

text: in

text: the

text: pre-stellar,

text: proto-stellar,

text: HII

text: and

text: PDR

text: stages

text: are

text: and

text: respectively.

text: The

text: infall

text: candidates

text: have

text: a

text: higher

text: H

text: column

text: density

text: and

text: volume

text: density

text: compared

text: with

text: the

text: clumps

text: where

text: infall

text: is

text: not

text: detected

text: at

text: every

text: stage.

text: For

text: the

text: infall

text: candidates,

text: the

text: median

text: values

text: of

text: the

text: infall

text: rates

text: at

text: the

text: pre-stellar,

text: proto-stellar,

text: HII

text: and

text: PDR

text: stages

text: are

text: and

text: yr

text: respectively.

text: These

text: values

text: indicate

text: that

text: infall

text: candidates

text: at

text: later

text: evolutionary

text: stages

text: are

text: still

text: accumulating

text: material

text: eﬃciently.

text: It

text: is

text: interesting

text: to

text: ﬁnd

text: that

text: both

text: infall

text: candidates

text: and

text: clumps

text: where

text: infall

text: is

text: not

text: detected

text: show

text: a

text: clear

text: trend

text: of

text: increasing

text: mass

text: from

text: the

text: pre-stellar

text: to

text: proto-stellar,

text: and

text: to

text: the

text: HII

text: stages.

text: The

text: power

text: indices

text: of

text: the

text: clump

text: mass

text: function

text: (ClMF)

text: are

text: and

text: for

text: the

text: infall

text: candidates

text: and

text: clumps

text: where

text: infall

text: is

text: not

text: detected,

text: respectively,

text: which

text: agree

text: well

text: with

text: the

text: power

text: index

text: of

text: the

text: stellar

text: initial

text: mass

text: function

text: and

text: the

text: cold

text: Planck

text: cores

text: Key

text: words:

text: stars:

text: formation

text: ISM:

text: kinematics

text: and

text: dynamics

text: ISM:

text: molecules

text: radio

text: lines:

text: ISM.

text: INTRODUCTION

text: outside

text: the

text: gravitational

text: radius

text: and

text: that

text: no

text: material

text: is

text: added

text: to

text: the

text: core

text: after

text: it

text: forms.

text: The

text: ﬁnal

text: mass

text: of

text: a

text: mas-

text: High-mass

text: star

text: formation

text: is

text: still

text: poorly

text: understood

text: sive

text: star

text: is

text: decided

text: by

text: the

text: amount

text: of

text: mass

text: that

text: is

text: necessary

text: (Tan

text: et

text: al.

text: Core

text: accretion

text: (McKee

text: Tan

text: to

text: be

text: self-gravitating,

text: which

text: is

text: thought

text: to

text: be

text: supported

text: ei-

text: and

text: competitive

text: accretion

text: (Bonnell

text: et

text: al.

text: Bonnell,

text: ther

text: by

text: thermal

text: pressure,

text: turbulence

text: or

text: magnetic

text: ﬁelds

text: (Mc-

text: Vine,

text: Bate

text: are

text: two

text: contending

text: theories

text: underlying

text: Kee

text: Tan

text: and

text: is

text: not

text: greatly

text: inﬂuenced

text: by

text: a

text: current

text: investigations.

text: star’s

text: environment.

text: The

text: core

text: accretion

text: model

text: imposes

text: a

text: di-

text: The

text: core

text: accretion

text: model

text: essentially

text: assumes

text: that

text: the

text: rect

text: link

text: between

text: core

text: and

text: star,

text: and

text: thus

text: the

text: core

text: mass

text: func-

text: evolution

text: of

text: the

text: core

text: is

text: isolated

text: from

text: environmental

text: eﬀects

text: tion

text: (CMF)

text: must

text: have

text: a

text: similar

text: shape

text: to

text: the

text: stellar

text: initial

text: mass

text: function

text: (IMF)

text: in

text: that

text: scenario.

text: (e.g.

text: Motte,

text: Andre,

text: Neri

text: E-mail:heyuxin@xao.ac.cn

text: Email:

text: zhoujj@xao.ac.cn

text: In

text: the

text: competitive

text: accretion

text: model,

text: the

text: mass

text: of

text: a

text: star

text: c

text: RAS


text: Singular ridge regression: generalization error with estimated parameters 19 (cid:2) (cid:3) (cid:0) (cid:1) ∼ U − ∈ M where E ξi and σ ξi for ξ ( 1, 1) are deﬁned in (3.30). The sample matrix X(2) (cid:62) ∈ R p,N2 and vector y(2) N2 are obtained by horizontal concatenation of the N2 realizations of (y(2), x(2)). In order to evaluate the conditional total testing error in (3.17), we need the ﬁrst and second order ∼ U moments of the random testing sample. Fir(cid:0)st,(cid:2)we(cid:3)notic(cid:2)e t(cid:3)h(cid:1)at since ξ(2) (a, b), we can conclude that: 1 − ∈ { } (µ(2)) = E ζi E ξi , i 1, . . . , p , (3.41) x i σ (ξi) (cid:2) (cid:3) where − bi+1 ai+1 (cid:2)E (cid:3)ζi = (cid:0) (cid:1) − (3.42) (i + 1)(b a) ∈ { } and where for each i 1, . . . , p , E ξi and σ ξi are given by (3.30). Additionally, the ﬁrst order moment of y(2) is given by 1 − − µ(2) = − (eb ea) k. (3.43) y b a The second order moments o(cid:90)f the testing sample are: (cid:0) (cid:2) (cid:3)(cid:1) (cid:0) (cid:2) (cid:3)(cid:1) 1 b − − − ∈ { } (Σ(2)) = − zi E ξi zj E ξj dz (µ(2)) (µ(2)) , i, j 1, . . . , p , x i,j (b a)σ(ξi)σ(cid:90)(ξj) x i x j (cid:0) a (cid:2) (cid:3)(cid:1) (3.44) 1 b − − − ∈ { } (Σ(2)) = − zi E ξi (ez k)dz (µ(2)) µ(2), i 1, . . . , p , (3.45) xy i (b a)σ(ξi) x i y a 1 − − 1 − (σ2)(2) = − (e2b e2a) − (eb ea)2 + σ2. (3.46) y 2(b a) (b a)2 ε These relations allow for the explicit evaluation of the conditional total testing error that we now numerically study for two particular choices for the values of a and b that determine the support of the distribution of ζ. − Numerical illustration. Case I. Overﬁtting. Consider ﬁrst the case in which [a, b] = [ 1, 1] and hence ζ = ξ. This means that the testing sample is generated in the same way as the training one. In this setup, an increase in the number of covariates p amounts to an increase in the degree of the polynomial that we are using to ﬁt the sample. When this degree is too high, the ﬁt may lead to a match with the noise in the sample rather than with the underlying deterministic signal that we are trying to model. Figure 1 shows the evolution of the conditional training and total errors for this case (for a given training covariates sample X and a ﬁxed regularization strength reported in the caption). 1 As we anticipated, the conditional total training error is a decreasing function with p but, in turn, the generalization error decreases only up to a point and when overﬁtting to the training sample starts to occur, we observe a monotonous increase in the testing error. In this case, this turning point takes place for p = 9, a value that seems to oﬀer the optimal tradeoﬀ between the quality of training and out-of-sample performance. Another way to state this observation is that, with this noise level, the best polynomial approximation (in the mean square sense) of the the exponential function in the interval − [ 1, 1] is obtained when using polynomials of degree 9. Numerical illustration. Case II. Generalization performance. In this case we allow for the support of distribution for ζ used to generate the testing sample to be diﬀerent from the one that ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Singular

text: ridge

text: regression:

text: generalization

text: error

text: with

text: estimated

text: parameters

text: U

text: M

text: where

text: E

text: ξi

text: and

text: σ

text: ξi

text: for

text: ξ

text: are

text: deﬁned

text: in

text: The

text: sample

text: matrix

text: R

text: and

text: vector

text: are

text: obtained

text: by

text: horizontal

text: concatenation

text: of

text: the

text: realizations

text: of

text: In

text: order

text: to

text: evaluate

text: the

text: conditional

text: total

text: testing

text: error

text: in

text: we

text: need

text: the

text: ﬁrst

text: and

text: second

text: order

text: U

text: moments

text: of

text: the

text: random

text: testing

text: sample.

label: equation

text: since

text: (a,

text: b),

text: we

text: can

text: conclude

text: that:

text: E

text: ζi

text: E

text: ξi

text: i

text: p

text: x

text: i

text: σ

text: (ξi)

text: where

text: (i

text: a)

text: and

text: where

text: for

text: each

text: i

text: p

text: E

text: ξi

text: and

text: σ

text: ξi

text: are

text: given

text: by

text: Additionally,

text: the

text: ﬁrst

text: order

text: moment

text: of

text: is

text: given

text: by

text: (eb

text: ea)

text: k.

text: y

text: b

text: a

text: The

text: second

text: order

text: moments

text: the

text: testing

text: sample

text: are:

text: b

text: zi

text: E

text: ξi

text: zj

text: E

text: ξj

text: dz

text: i,

text: j

text: p

text: x

text: i,j

text: (b

text: x

text: i

text: x

text: j

text: a

text: b

text: zi

text: E

text: ξi

text: (ez

text: k)dz

text: i

text: p

text: xy

text: i

text: (b

text: a)σ(ξi)

text: x

text: i

text: y

text: a

text: (eb

text: y

text: a)

text: (b

text: ε

text: These

text: relations

text: allow

text: for

text: the

text: explicit

text: evaluation

text: of

text: the

text: conditional

text: total

text: testing

text: error

text: that

text: we

text: now

text: numerically

text: study

text: for

text: two

text: particular

text: choices

text: for

text: the

text: values

text: of

text: a

text: and

text: b

text: that

text: determine

text: the

text: support

text: of

text: the

text: distribution

text: of

text: ζ.

text: Numerical

text: illustration.

text: Case

text: I.

text: Overﬁtting.

text: Consider

text: ﬁrst

text: the

text: case

text: in

text: which

text: [a,

text: b]

text: and

text: hence

text: ζ

text: ξ.

text: This

text: means

text: that

text: the

text: testing

text: sample

text: is

text: generated

text: in

text: the

text: same

text: way

text: as

text: the

text: training

text: one.

text: In

text: this

text: setup,

text: an

text: increase

text: in

text: the

text: number

text: of

text: covariates

text: p

text: amounts

text: to

text: an

text: increase

text: in

text: the

text: degree

text: of

text: the

text: polynomial

text: that

text: we

text: are

text: using

text: to

text: ﬁt

text: the

text: sample.

text: When

text: this

text: degree

text: is

text: too

text: high,

text: the

text: ﬁt

text: may

text: lead

text: to

text: a

text: match

text: with

text: the

text: noise

text: in

text: the

text: sample

text: rather

text: than

text: with

text: the

text: underlying

text: deterministic

text: signal

text: that

text: we

text: are

text: trying

text: to

text: model.

text: Figure

text: shows

text: the

text: evolution

text: of

text: the

text: conditional

text: training

text: and

text: total

text: errors

text: for

text: this

text: case

text: (for

text: a

text: given

text: training

text: covariates

text: sample

text: X

text: and

text: a

text: ﬁxed

text: regularization

text: strength

text: reported

text: in

text: the

text: caption).

text: As

text: we

text: anticipated,

text: the

text: conditional

text: total

text: training

text: error

text: is

text: a

text: decreasing

text: function

text: with

text: p

text: but,

text: in

text: turn,

text: the

text: generalization

text: error

text: decreases

text: only

text: up

text: to

text: a

text: point

text: and

text: when

text: overﬁtting

text: to

text: the

text: training

text: sample

text: starts

text: to

text: occur,

text: we

text: observe

text: a

text: monotonous

text: increase

text: in

text: the

text: testing

text: error.

text: In

text: this

text: case,

text: this

text: turning

text: point

text: takes

text: place

text: for

text: p

text: a

text: value

text: that

text: seems

text: to

text: oﬀer

text: the

text: optimal

text: tradeoﬀ

text: between

text: the

text: quality

text: of

text: training

text: and

text: out-of-sample

text: performance.

text: Another

text: way

text: to

text: state

text: this

text: observation

text: is

text: that,

text: with

text: this

text: noise

text: level,

text: the

text: best

text: polynomial

text: approximation

text: (in

text: the

text: mean

text: square

text: sense)

text: of

text: the

text: the

text: exponential

text: function

text: in

text: the

text: interval

text: is

text: obtained

text: when

text: using

text: polynomials

text: of

text: degree

text: Numerical

text: illustration.

text: Case

text: II.

text: Generalization

text: performance.

text: In

text: this

text: case

text: we

text: allow

text: for

text: the

text: support

text: of

text: distribution

text: for

text: ζ

text: used

text: to

text: generate

text: the

text: testing

text: sample

text: to

text: be

text: diﬀerent

text: from

text: the

text: one

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Singular ridge regression: generalization error with estimated parameters 8 (cid:98) (cid:98) (cid:98) (cid:98) − (cid:62) − (cid:62) B := B = Σ 1Σ = (XA X ) 1XA Y . (2.26) 0 x xy N N (cid:98) It is important to underline that the ordinary least squares estimator (2.26) is available only when the sample covariance matrix Σ in (2.6) is invertible which, as we already pointed out in the introduction, x i(cid:98)s one of its main deﬁciencies. (cid:98) The following lemma, whose proof is in App(cid:98)endix , provides the relation between the ridge regression B and the ordinary least squares estimator B. We obviously restrict ourselves to the case in which the λ evaluation of (2.26) is feasible, that is, when Σ is invertible. The relations that we now state are not x new and are well established in the literature [Hoer 70]. We provide them for future reference and in order to complete Example A. (cid:98) Lemma 2.4 In the conditions of Example A the following statements hold true: (cid:98) (i) The relation between the ﬁnite sample ridge estimator B of B and the ﬁnite sample ordinary least λ λ squares estimator B of B is given by (cid:98) (cid:98) B = Z B (2.27) λ λ where (cid:62) Z := R XA X , (2.28) λ λ N (cid:62) I − R := (XA X + λN ) 1 (2.29) λ N p with A as in (2.8). Expression (2.27) is the ﬁnite sample based analogue of the ﬁrst relation in N (2.21). (ii) The multiplier Z in (2.28) can be equivalently expressed as λ I − Z = λNR , (2.30) λ p λ (cid:0) (cid:1) or Z = I + λN(XA X(cid:62) −1)−1, (2.31) λ p N with R in (2.30) deﬁned as in (2.29). λ I − (cid:62) Remark 2.5 The relations Z = λNR in (2.30) and Z = R XA X in (2.28) hold in a general λ p λ λ λ (cid:62)N context beyond the speciﬁc conditions of Example A and even if XX is singular or ill-conditioned. They are used later on in the paper. We now show how the classical results on the ridge estimator properties in Section 4a of [Hoer 70] that assume the existence of a least squares estimate follow as a corollary of Theorem 2.3. The proof of the following result is provided in Appendix .

text: Singular

text: ridge

text: regression:

text: generalization

text: error

text: with

text: estimated

text: parameters

text: B

text: B

text: Σ

text: 1Σ

text: (XA

text: X

text: Y

text: x

text: xy

text: N

text: N

text: It

text: is

text: important

text: to

text: underline

text: that

text: the

text: ordinary

text: least

text: squares

text: estimator

text: is

text: available

text: only

text: when

text: the

text: sample

text: covariance

text: matrix

text: Σ

text: in

text: is

text: invertible

text: which,

text: as

text: we

text: already

text: pointed

text: out

text: in

text: the

text: introduction,

text: x

text: one

text: of

text: its

text: main

text: deﬁciencies.

text: The

text: following

text: lemma,

text: whose

text: proof

text: is

text: in

text: provides

text: the

text: relation

text: between

text: the

text: ridge

text: regression

text: B

text: and

text: the

text: ordinary

text: least

text: squares

text: estimator

text: B.

text: We

text: obviously

text: restrict

text: ourselves

text: to

text: the

text: case

text: in

text: which

text: the

text: λ

text: evaluation

text: of

text: is

text: feasible,

text: that

text: is,

text: when

text: Σ

text: is

text: invertible.

text: The

text: relations

text: that

text: we

text: now

text: state

text: are

text: not

text: x

text: new

text: and

text: are

text: well

text: established

text: in

text: the

text: literature

text: [Hoer

text: We

text: provide

text: them

text: for

text: future

text: reference

text: and

text: in

text: order

text: to

text: complete

text: Example

text: A.

text: Lemma

text: In

text: the

text: conditions

text: of

text: Example

text: A

text: the

text: following

text: statements

text: hold

text: true:

text: (i)

text: The

text: relation

text: between

text: the

text: ﬁnite

text: sample

text: ridge

text: estimator

text: B

text: of

text: B

text: and

text: the

text: ﬁnite

text: sample

text: ordinary

text: least

text: λ

text: λ

text: squares

text: estimator

text: B

text: of

text: B

text: is

text: given

text: by

text: B

text: Z

text: B

text: λ

text: λ

text: where

text: Z

text: R

text: XA

text: X

text: λ

text: λ

text: N

text: I

text: R

text: (XA

text: X

text: λN

text: λ

text: N

text: p

text: with

text: A

text: as

text: in

text: Expression

text: is

text: the

text: ﬁnite

text: sample

text: based

text: analogue

text: of

text: the

text: ﬁrst

text: relation

text: in

text: N

text: (ii)

text: The

text: multiplier

text: Z

text: in

text: can

text: be

text: equivalently

text: expressed

text: as

text: λ

text: I

text: Z

text: λNR

text: λ

text: p

text: λ

text: or

text: Z

text: I

text: λN(XA

text: λ

text: p

text: N

text: with

text: R

text: in

text: deﬁned

text: as

text: in

text: λ

text: I

text: Remark

text: The

text: relations

text: Z

text: λNR

text: in

text: and

text: Z

text: R

text: XA

text: X

text: in

text: hold

text: in

text: a

text: general

text: λ

text: p

text: λ

text: λ

text: λ

text: context

text: beyond

text: the

text: speciﬁc

text: conditions

text: of

text: Example

text: A

text: and

text: even

text: if

text: XX

text: is

text: singular

text: or

text: ill-conditioned.

text: They

text: are

text: used

text: later

text: on

text: in

text: the

text: paper.

text: We

text: now

text: show

text: how

text: the

text: classical

text: results

text: on

text: the

text: ridge

text: estimator

text: properties

text: in

text: Section

text: of

text: [Hoer

text: that

text: assume

text: the

text: existence

text: of

text: a

text: least

text: squares

text: estimate

text: follow

text: as

text: a

text: corollary

text: of

text: Theorem

text: The

text: proof

text: of

text: the

text: following

text: result

text: is

text: provided

text: in

text: Appendix


text: Singular ridge regression: generalization error with estimated parameters 9 R (cid:62) Corollary 2.6 (Section 4a in [Hoer 70]) Consider the q-valued linear regression Y = B X + E obtained out of random samples of length N that satisfy the relation (2.20). Given that, by hypothesis, ∼ ∈ M the error term satisﬁes ε N(0 , Σ ), it is easy to see that the residual random matrix E ob- q ε ∼ O I q,N tained by horizontal concatenation, is matrix normal distributed as E MN ( , Σ , ). The following q ε N statements hold: (i) The ridge regression matrix est(cid:98)imator is conditionally distributed as − | ∼ − (B B) X MN( λNR B, Z R , Σ ), (2.32) λ λ λ λ ε (cid:98) where R and Z are deﬁned in (2.18) and in (2.19), respectively. λ λ (ii) If the matrix Σx deﬁned(cid:98)in (2.6) is invertible, then (2.32) can be rewritten as − | ∼ − (cid:62) − (B B) X MN( λNR B, Z (XA X ) 1Z , Σ ), (2.33) λ λ λ N λ ε (cid:74) with Z deﬁned as in (2.19) or, equivalently, as in (2.31). λ (cid:73) Example B. Nonlinear one-dimensional model with additive stochastic disturbance term. Consider the following data generating process for the variable y y = f(ξ) + ε, (2.34) R → R ∼ where f : is a continuous function, ε D (0, σ2) represents a disturbance term and ξ is ε ε distributed with zero mean, variance σ2, and probability density function g . The random variables ε ξ ξ and ξ are assumed to be independent and subjected to (A1). We now approximate the model (2.34) using a univariate ridge regression in which the p explanatory { } variables are the ﬁrst p elements φ ∈{ } of a given ordered generating set of the function space i i 1,...,p to which the function f in (2.34) belongs, all of them evaluated at ξ. More explicitly, consider an approximate model of the following form: (cid:62) ∼ y = B x + ε , ε D (µ , σ2 ), (2.35) λ λ λ ελ ελ ελ (cid:16) (cid:17) ∈ R where the vector(cid:101)of regre(cid:101)ssors x p is co(cid:101)nstructed as (cid:62) − ξ E [ξ ] ∈ { } x := ξ , . . . , ξ , with x = ξ := i i , ξ = φ (ξ), i 1, . . . , p . (2.36) 1 p i i σ(ξ ) i i i ∈ { } We emphasize that for each i 1, . . . , p , the explanatory variable x is constructed out of the i standardized version of the evaluation ξ = φ (ξ) of the generating function associated φ using the (cid:2) (cid:3) i i i standard deviation: − ∈ { } σ(ξ ) := (E ξ2 E [ξ ]2)1/2, for each i 1, . . . , p . (2.37) i i i We now provide all the elements associated to the approximate model (2.35). First, (2.2) determines, ∈ R ∈ S for any regularization strength λ +, the ridge regression matrix B once the central moments Σ ∈ R λ x p and Σ p have been computed. It is easy to see that xy ##LTLine##

text: Singular

text: ridge

text: regression:

text: generalization

text: error

text: with

text: estimated

text: parameters

text: R

text: Corollary

text: (Section

text: in

text: [Hoer

text: Consider

text: the

text: q-valued

text: linear

text: regression

text: Y

text: B

text: X

text: E

text: obtained

text: out

text: of

text: random

text: samples

text: of

text: length

text: N

text: that

text: satisfy

text: the

text: relation

text: Given

text: that,

text: by

text: hypothesis,

text: M

text: the

text: error

text: term

text: satisﬁes

text: ε

text: Σ

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: residual

text: random

text: matrix

text: E

text: ob-

text: q

text: ε

text: O

text: I

text: q,N

text: tained

text: by

text: horizontal

text: concatenation,

text: is

text: matrix

text: normal

text: distributed

text: as

text: E

text: MN

text: Σ

text: The

text: following

text: q

text: ε

text: N

text: statements

text: hold:

text: (i)

text: The

text: ridge

text: regression

text: matrix

text: is

text: conditionally

text: distributed

text: as

text: (B

text: B)

text: X

text: MN(

text: λNR

text: B,

text: Z

text: R

text: Σ

text: λ

text: λ

text: λ

text: λ

text: ε

text: where

text: R

text: and

text: Z

text: are

text: deﬁned

text: in

text: and

text: in

text: respectively.

text: λ

text: λ

text: (ii)

text: If

text: the

text: matrix

text: Σx

text: is

text: invertible,

text: then

text: can

text: be

text: rewritten

text: as

text: (B

text: B)

text: X

text: MN(

text: λNR

text: B,

text: Z

text: (XA

text: X

text: Σ

text: λ

text: λ

text: λ

text: N

text: λ

text: ε

text: with

text: Z

text: deﬁned

text: as

text: in

text: or,

text: equivalently,

text: as

text: in

text: λ

text: Example

text: B.

text: Nonlinear

text: one-dimensional

text: model

text: with

text: additive

text: stochastic

text: disturbance

text: term.

text: Consider

text: the

text: following

text: data

text: generating

text: process

text: for

text: the

text: variable

text: y

text: y

text: f(ξ)

text: ε,

text: R

text: R

text: where

text: f

text: is

text: a

text: continuous

text: function,

text: ε

text: D

text: represents

text: a

text: disturbance

text: term

text: and

text: ξ

text: is

text: ε

text: ε

text: distributed

text: with

text: zero

text: mean,

text: variance

text: and

text: probability

text: density

text: function

text: g

text: The

text: random

text: variables

text: ε

text: ξ

text: ξ

text: and

text: ξ

text: are

text: assumed

text: to

text: be

text: independent

text: and

text: subjected

text: to

text: We

text: now

text: approximate

text: the

text: model

text: using

text: a

text: univariate

text: ridge

text: regression

text: in

text: which

text: the

text: p

text: explanatory

text: variables

text: are

text: the

text: ﬁrst

text: p

text: elements

text: φ

text: of

text: a

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text: ordered

text: generating

text: set

text: of

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text: function

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text: i

text: i

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text: of

text: them

text: evaluated

text: at

text: ξ.

text: More

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text: of

text: the

text: following

text: form:

text: y

text: B

text: x

text: ε

text: ε

text: D

text: (µ

text: σ2

text: λ

text: λ

text: λ

text: ελ

text: ελ

text: ελ

text: R

text: where

text: the

text: x

text: p

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text: as

text: ξ

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text: [ξ

text: x

text: ξ

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text: x

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text: (ξ),

text: i

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text: i

text: i

text: σ(ξ

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text: We

text: emphasize

text: that

text: for

text: each

text: i

text: p

text: the

text: explanatory

text: variable

text: x

text: is

text: constructed

text: out

text: of

text: the

text: i

text: standardized

text: version

text: of

text: the

text: evaluation

text: ξ

text: φ

text: (ξ)

text: of

text: the

text: generating

text: function

text: associated

text: φ

text: using

text: the

text: i

text: i

text: i

text: standard

text: deviation:

text: σ(ξ

text: (E

text: ξ2

text: E

text: [ξ

text: for

text: each

text: i

text: p

text: i

text: i

text: i

text: We

text: now

text: provide

text: all

text: the

text: elements

text: associated

text: to

text: the

text: approximate

text: model

text: First,

text: determines,

text: R

text: S

text: for

text: any

text: regularization

text: strength

text: λ

text: the

text: ridge

text: regression

text: matrix

text: B

text: once

text: the

text: central

text: moments

text: Σ

text: R

text: λ

text: x

text: p

text: and

text: Σ

text: p

text: have

text: been

text: computed.

text: It

text: is

text: easy

text: to

text: see

text: that

text: xy

text: ##LTLine##


text: 6 1 2 3 7 8 9 Exit 4 5 Figure 1: The loading/unloading graph for the auto-carrier in Figure ??. Ramp is the exit ramp. 9 to a collection of sets): RHc (t) ≡ collection of sets of ramps that limit the height of a loaded auto-carrier of type t. (e.g., RHc (t) = {{1}, {2, 3}, {4, 7}, {5, 8}, {6, 9}} for auto-carrier in Figure ??). RLc (t) ≡ collection of sets of ramps in an auto-carrier of type t, that have a limitation on the allowed length (e.g., RLc (t) = {{1, 2}, {3}, {4, 5, 6}, {7, 8, 9}} for auto-carrier in Figure ??). RSc P(t) ≡ collection of sets of ramps where each set of ramps can be combined to accommodate a single vehicle in an auto-carrier of type t. (e.g., RSc P(t) = {{4, 5}, {7, 8}, {5, 6}} for an auto-carrier in Figure ??). We refer to RSc P(t) as the collection of split ramps for an auto-carrier of type t. Another aspect of LDP is a loading/unloading sequence for each ramp in an auto-carrier. The sequence in which the vehicles are loaded or unloaded from the ramps of an auto-carrier is unique and is dictated by a loading/unloading graph (see Figure 1). The vertices of the graph represent the ramps of an auto-carrier, and they denote the set of ramps which should be empty or unloaded before a vehicle in a particular ramp is unloaded. To illustrate, if a vehicle in ramp 4 should be unloaded then the vehicles in the ramps 7, 8, and 9 should be unloaded or empty. A ramp in the graph without any outgoing edges is referred as an exit ramp (9 in Figure 1). For every ramp of an auto-carrier i ∈ R(t) of type , the path from to the exit ramp is unique, and this gives an unloading sequence for the t i vehicle in the ramp i. Hence, for every ramp i ∈ R(t), we deﬁne qt(i) as the ordered set of ramps in which the vehicles should be unloaded before unloading the vehicle in ramp . i qt(i) ≡ ordered set of ramps providing the unloading sequence of ramp i in an auto-carrier of type t. For each auto-carrier, we also assume a ﬁxed route, i.e, an ordered sequence of dealer locations that each auto-carrier is going to visit. The set is an ordered set of dealers’ locations is obtained S by the route heuristics explained in the previous section. For any , represents s , s ∈ S s (cid:22) s 1 2 1 2 that location will be visited by the auto-carrier before visiting the location , and represents s s (cid:23) 1 2 vice-versa. We now deﬁne for each , as the dealer location. v ∈ V lo(v) dealer location for a vehicle lo(v) ≡ v ∈ V , lo(v) ∈ S. Now, we will describe various conﬁgurations in which a vehicle can be loaded on a ramp in an auto-carrier. Suppose that a vehicle v is loaded on a ramp i ∈ R(t) ∪ RSc P(t) then v can either be positioned in the same direction as that of an auto-carrier (vehicles positioned in the ramps 2, 3, 4, 5, 6 and 9 in Figure ??) or in the opposite direction (vehicles positioned in the ramps 1, 7 and 8 2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Exit

text: Figure

text: The

label: caption

text: loading/unloading

text: graph

text: for

text: the

text: auto-carrier

text: in

text: Figure

text: Ramp

text: is

text: the

text: exit

text: ramp.

text: to

text: a

text: collection

text: of

text: sets):

text: RHc

text: (t)

text: collection

text: of

text: sets

text: of

text: ramps

text: that

text: limit

text: the

text: height

text: of

text: a

text: loaded

text: auto-carrier

text: of

text: type

text: t.

text: (e.g.,

text: RHc

text: (t)

text: for

text: auto-carrier

text: in

text: Figure

text: RLc

text: (t)

text: collection

text: of

text: sets

text: of

text: ramps

text: in

text: an

text: auto-carrier

text: of

text: type

text: t,

text: that

text: have

text: a

text: limitation

text: on

text: the

text: allowed

text: length

text: (e.g.,

text: RLc

text: (t)

text: for

text: auto-carrier

text: in

text: Figure

text: RSc

text: P(t)

text: collection

text: of

text: sets

text: of

text: ramps

text: where

text: each

text: set

text: of

text: ramps

text: can

text: be

text: combined

text: to

text: accommodate

text: a

text: single

text: vehicle

text: in

text: an

text: auto-carrier

text: of

text: type

text: t.

text: (e.g.,

text: RSc

text: P(t)

text: for

text: an

text: auto-carrier

text: in

text: Figure

text: We

text: refer

text: to

text: RSc

text: P(t)

text: as

text: the

text: collection

text: of

text: split

text: ramps

text: for

text: an

text: auto-carrier

text: of

text: type

text: t.

text: Another

text: aspect

text: of

text: LDP

text: is

text: a

text: loading/unloading

text: sequence

text: for

text: each

text: ramp

text: in

text: an

text: auto-carrier.

text: The

text: sequence

text: in

text: which

text: the

text: vehicles

text: are

text: loaded

text: or

text: unloaded

text: from

text: the

text: ramps

text: of

text: an

text: auto-carrier

text: is

text: unique

text: and

text: is

text: dictated

text: by

text: a

text: loading/unloading

text: graph

text: (see

text: Figure

text: The

text: vertices

text: of

text: the

text: graph

text: represent

text: the

text: ramps

text: of

text: an

text: auto-carrier,

text: and

text: they

text: denote

text: the

text: set

text: of

text: ramps

text: which

text: should

text: be

text: empty

text: or

text: unloaded

text: before

text: a

text: vehicle

text: in

text: a

text: particular

text: ramp

text: is

text: unloaded.

text: To

text: illustrate,

text: if

text: a

text: vehicle

text: in

text: ramp

text: should

text: be

text: unloaded

text: then

text: the

text: vehicles

text: in

text: the

text: ramps

text: and

text: should

text: be

text: unloaded

text: or

text: empty.

text: A

text: ramp

text: in

text: the

text: graph

text: without

text: any

text: outgoing

text: edges

text: is

text: referred

text: as

text: an

text: exit

text: ramp

text: in

text: Figure

text: For

text: every

text: ramp

text: of

text: an

text: auto-carrier

text: i

text: R(t)

text: of

text: type

text: the

text: path

text: from

text: to

text: the

text: exit

text: ramp

text: is

text: unique,

text: and

text: this

text: gives

text: an

text: unloading

text: sequence

text: for

text: the

text: t

text: i

text: vehicle

text: in

text: the

text: ramp

text: i.

text: Hence,

text: for

text: every

text: ramp

text: i

text: R(t),

text: we

text: deﬁne

text: qt(i)

text: as

text: the

text: ordered

text: set

text: of

text: ramps

text: in

text: which

text: the

text: vehicles

text: should

text: be

text: unloaded

text: before

text: unloading

text: the

text: vehicle

text: in

text: ramp

text: i

text: qt(i)

text: ordered

text: set

text: of

text: ramps

text: providing

text: the

text: unloading

text: sequence

text: of

text: ramp

text: i

text: in

text: an

text: auto-carrier

text: of

text: type

text: t.

text: For

text: each

text: auto-carrier,

text: we

text: also

text: assume

text: a

text: ﬁxed

text: route,

text: i.e,

text: an

text: ordered

text: sequence

text: of

text: dealer

text: locations

text: that

text: each

text: auto-carrier

text: is

text: going

text: to

text: visit.

text: The

text: set

text: is

text: an

text: ordered

text: set

text: of

text: dealers’

text: locations

text: is

text: obtained

text: S

text: by

text: the

text: route

text: heuristics

text: explained

text: in

text: the

text: previous

text: section.

text: For

text: any

text: represents

text: s

text: s

text: S

text: s

text: s

text: that

text: location

text: will

text: be

text: visited

text: by

text: the

text: auto-carrier

text: before

text: visiting

text: the

text: location

text: and

text: represents

text: s

text: s

text: vice-versa.

text: We

text: now

text: deﬁne

text: for

text: each

text: as

text: the

text: dealer

text: location.

text: v

text: V

text: lo(v)

text: dealer

text: location

text: for

text: a

text: vehicle

text: lo(v)

text: v

text: V

text: lo(v)

text: S.

text: Now,

text: we

text: will

text: describe

text: various

text: conﬁgurations

text: in

text: which

text: a

text: vehicle

text: can

text: be

text: loaded

text: on

text: a

text: ramp

text: in

text: an

text: auto-carrier.

text: Suppose

text: that

text: a

text: vehicle

text: v

text: is

text: loaded

text: on

text: a

text: ramp

text: i

text: R(t)

text: RSc

text: P(t)

text: then

text: v

text: can

text: either

text: be

text: positioned

text: in

text: the

text: same

text: direction

text: as

text: that

text: of

text: an

text: auto-carrier

text: (vehicles

text: positioned

text: in

text: the

text: ramps

text: and

text: in

text: Figure

text: or

text: in

text: the

text: opposite

text: direction

text: (vehicles

text: positioned

text: in

text: the

text: ramps

text: and

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: S.

text: Marchesini,

text: H.

text: He,

text: H.

text: N.

text: Chapman,

text: S.

text: P.

text: Hau-Riege,

text: A.

text: Noy,

text: J.

text: Gulden,

text: O.

text: M.

text: Yefanov,

text: A.

text: P.

text: Mancuso,

text: V.

text: V.

text: Abramova,

text: M.

text: R.

text: Howells,

text: U.

text: Weierstall,

text: and

text: J.

text: C.

text: H.

text: Spence,

text: Phys.

text: Rev.

text: J.

text: Hilhorst,

text: D.

text: Byelov,

text: I.

text: Snigireva,

text: A.

text: Snigirev,

text: A.

text: V.

text: Petukhov,

text: B

text: and

text: I.

text: A.

text: Vartanyants,

text: Phys.

text: Rev.

text: B

text: L.

text: V.

text: Woodcock,

text: Nature

text: J.

text: Gulden,

text: O.

text: M.

text: Yefanov,

text: A.

text: P.

text: Mancuso,

text: R.

text: Dronyak,

text: A.

text: V.

text: Petukhov,

text: I.

text: P.

text: Dolbnya,

text: D.

text: G.

text: A.

text: L.

text: Aarts,

text: G.

text: J.

text: Vroege,

text: A.

text: Singer,

text: V.

text: Berna´tova´,

text: A.

text: Burkhardt,

text: O.

text: Polozhentsev,

text: and

text: H.

text: N.

text: W.

text: Lekkerkerker,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Soldatov,

text: M.

text: Sprung,

text: and

text: I.

text: A.

text: Vartanyants,

text: Opt.

text: Express

text: I.

text: P.

text: Dolbnya,

text: A.

text: V.

text: Petukhov,

text: D.

text: G.

text: A.

text: L.

text: Aarts,

text: G.

text: J.

text: Vroege,

text: and

text: H.

text: N.

text: W.

text: Lekkerkerker,

text: EPL

text: A.

text: V.

text: Zozulya,

text: S.

text: Bondarenko,

text: A.

text: Schavkan,

text: F.

text: Westermeier,

text: “See

text: appendix

text: for

text: the

text: experimental

text: detailes

text: and

text: stacking

text: se-

text: G.

text: Gru¨bel,

text: and

text: M.

text: Sprung,

text: Opt.

text: Express

text: quence

text: determination.”.

text: J.

text: W.

text: Goodman,

text: Introduction

text: to

text: Fourier

text: Optics

text: (Roberts

text: and

text: R.

text: Hettel,

text: J.

text: Synchrotron

text: Radiat.

text: Company

text: Publishers,

text: P.

text: Krishna,

text: D.

text: Pandey,

text: and

text: C.

text: A.

text: Taylor,

text: Close-packed

text: struc-

text: J.-M.

text: Meijer,

text: A.

text: Shabalin,

text: R.

text: Dronyak,

text: O.

text: M.

text: Yefanov,

text: tures

text: (IUCr,

text: A.

text: Singer,

text: R.

text: P.

text: Kurta,

text: U.

text: Lorenz,

text: O.

text: Gorobstov,

text: D.

text: Dzhigaev,

text: J.-M.

text: Meijer,

text: V.

text: W.

text: A.

text: de

text: Villeneuve,

text: and

text: A.

text: V.

text: Petukhov,

text: J.

text: Gulden,

text: D.

text: V.

text: Byelov,

text: A.

text: V.

text: Zozulya,

text: M.

text: Sprung,

text: I.

text: A.

text: Var-

text: Langmuir

text: tanyants,

text: and

text: A.

text: V.

text: Petukhov,

text: J.

text: Appl.

text: Crystallogr.

text: N.

text: Denkov,

text: O.

text: Velev,

text: P.

text: Kralchevski,

text: I.

text: Ivanov,

text: H.

text: Yoshimura,

text: and

text: K.

text: Nagayama,

text: Langmuir

text: J.

text: R.

text: Fienup,

text: Appl.

text: Opt.

text: K.

text: Ayyer,

text: O.

text: M.

text: Yefanov,

text: D.

text: Oberthu¨r,

text: S.

text: Roy-Chowdhury,

text: S.

text: Marchesini,

text: Rev.

text: Sci.

text: Instrum.

text: L.

text: Galli,

text: V.

text: Mariani,

text: S.

text: Basu,

text: J.

text: Coe,

text: C.

text: E.

text: Conrad,

text: R.

text: Fromme,

text: et

text: al.,

text: Nature