Multiway Analysis of Bridge Structural Types in the National Bridge Inventory (NBI)

A Tensor Decomposition Approach

By Offei A. Adarkwa

Nii Attoh-Okine (Ph.D) (IEEE Big Data Conference -10/27/2014)

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Presentation Outline

• Introduction • Basic Tensor Concepts • Analysis & Results • Conclusion & Recommendations

2

INTRODUCTION

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National Bridge Inventory (NBI)

• Largest database of bridge data in the world1

• Information on >600,000 bridges • Owned and operated by the Federal Highway Administration

(FHWA) • Status of a bridge:

– not deficient; – structurally deficient; and – functionally obsolete

• Basis for determining eligibility for Federal Bridge Funding

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Structurally Deficient Bridges

• Structurally deficient bridges have load carrying members in poor condition or significantly below design standards

• Repair, rehabilitation or replacement required • In the US, 1 in 9 bridges is structurally deficient • Laid end to end, structurally deficient bridges will span from DC

to Denver2

5 * http://www.transportationissuesdaily.com/wp-content/uploads/2011/10/Sherman-Minton-Bridge.jpg Sherman-Minton Bridge*

• Effective use of bridge rehabilitation and replacement funding

• Need to understand deterioration patterns of bridges under varying loading and environmental conditions

• NBI can serve as a major resource

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Focus of Research

• Analysis of deterioration trends of specific bridge deck structural types in the US

• Tensor Decomposition

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BASIC TENSOR CONCEPTS

8

What are Tensors?

• Generalizations of scalars, vectors and matrices

9

Samples

variables

Time

Samples

variables

Weather Samples

variables

Time

Why Multiway Data Analysis?

• Multiway data is everywhere

• Reveal hidden patterns in data

• Capture variation in dataset while preserving multidimensional nature of data

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11

x231 Mode-2

Mode-1

Mode-3

X

Mode-1 Fiber x:21

Mode-2 Fiber x1:2

Mode-3 Fiber X11:

Horizontal slice X1::

Lateral slice X:2:

Frontal slice X::3

Typical 3D dataset Subarrays

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j

i

k

j

i

k

j

i

k

i j j j j

k

j i i i i

k

k i i i i

j

Matricization (Unfolding)

𝜒 ∈ ℝ𝑖×𝑗𝑗

𝜒 ∈ ℝ𝑗×𝑖𝑗

𝜒 ∈ ℝ𝑗×𝑖𝑗

𝓐 + 𝓑 = 𝓒, where 𝓒𝑖𝑗𝑗 = 𝑎𝑖𝑗𝑗 + 𝑏𝑖𝑗𝑗

𝓧 = 𝒂 𝜊 𝒃 𝜊 𝒄 s.t. 𝑥𝑖𝑗𝑗 = 𝑎𝑖𝑏𝑗𝑐𝑗

𝑷𝐼×𝐽 ⨀ 𝑸𝐾×𝐽 = 𝑹𝐼𝐾×𝐽

𝑷𝐼×𝐽 ⨂𝑸𝐾×𝐿 = 𝑹𝐼𝐾×𝐽𝐿

Summation:

Outer Product:

Khatri-Rao Product:

Kronecker Product:

Hadamard Product: 𝑷𝐼×𝐽 ⊛𝑸𝐼×𝐽 = 𝑹𝐼×𝐽

Tensor Decomposition

• Reducing data to lower-order forms for analysis (classification, prediction, clustering)

• 2 main approaches – Canonical decomposition/Parallel Factors (CANDECOMP/PARAFAC; CP) – Tucker decomposition

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The CP Decomposition • Canonical Decomposition/ Parallel Factors

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ai, bi and ci - factor loadings R- number of components Rank of a tensor=min (R) required to approximate X

Choosing the appropriate CP model

• Alternating Least Squares Approach (ALS)

Where is the PARAFAC model

• % Variance Explained

• Core Consistency Diagnostic (CORCONDIAG) – High CORCONDIAG → stable model – Low CORCONDIAG → invalid and/or problematic model

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ANALYSIS & RESULTS

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Structural Deterioration Rate

𝑆𝑆𝑖,𝑗,𝑗 = 𝑛𝑖,𝑗,𝑗

∑ 𝑁𝑖,𝑗,𝑗𝑗

Where: 𝑆𝑆𝑖,𝑗,𝑗=structural deterioration rate in state i of structure type j in year k 𝑛𝑖,𝑗,𝑗=number of structurally deficient bridges in state i of structure type j in year k 𝑁𝑖,𝑗,𝑗=number of bridges in state i of structure type j in year k 17

Dataset • NBI database (1992-2013) • Structurally deficient bridge design types by state

18 Multidimensional Dataset for analysis

State

Deck Type

Year

Selected Bridge Design Types

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Stringer/Multibeam6 Girder and Floorbeam4 Slab5

Arch Deck7 Tee Beam8 Box Beam9

3D visualization

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0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

ALAKAZARCACOCTDEDCFLGAHIIDILINIAKSKYLAMEMDMAMIMNMSMOMTNENVNHNJNMNYNCNDOHOKORPARISCSDTNTXUTVTVAWAWVWIWYPR

Girder & Floorbeam system

Slab

Stringer/Multibeam or Girder

Arch Deck

Tee Beam

Box Beam or Girders(Multiple)

Exploration of NBI structure type data

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Slab Girder &Floorbeam

Stringer/Multibeamor girder

Tee Beam

BoxBeamor girders

Arch-Deck0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

slab stringer/multibm or girder girder & flbm Tee Bm Box Beam Arch Deck

Box plot of mean SD rates over 22 years SD rate trends for structural types

Decomposition plots

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• 2 component PARAFAC model was fitted to the data • Variance explained= 90.15%

-1 0 1 2 3 4 5 6 7-3

-2

-1

0

1

2

3

4

5

AL

AKAZ

ARCA

CO

CT

DE

DCFL

GAHI

ID

IL

IN

IAKS

KY LA

ME

MD

MA

MI

MN

MS

MOMT

NE

NV

NH

NJNM NYNC

ND

OH

OK

OR PA

RI

SC

SD

TN

TX

UT

VT

VA

WAWV

WI

WY

PR

Component 1

Com

pone

nt 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

slab

multibm or girder

gird & floorbm

tee bmbox bm

arch deck

Component 1

Com

pone

nt 2

0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.250.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

92

93

94

9596

97

98

990001

0203

04

050607

08

09

10

11

12

13

Loading plot for States Loading plot for Structure types Loading plot for years

Conclusion & Recommendations

• Demonstrates tensor decomposition as a knowledge discovery tool that has applications in bridge industry

• Detailed analysis which incorporates other deterioration factors such as age, ADT, and geographical location.

• Prediction

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References

(1) Wu N., Chase S. An Exploratory Data Analysis of National Bridge Inventory. Mid-Atlantic Universities Transportation Center (MAUTC). UVA-2009-03. May 2010

(2) Davis S.L., Goldberg D. The Fix We’re in For: The State of Our Nation’s Bridges 2013. Transportation for America. June 2013

(3) Steel Girder & Floorbeam. http://www.ncdot.gov/_graphics/site_graphics/projects_ncbridges_hb_t_steelgirder01.jpg. Accessed:10/21/2014

(4) Slab bridges. http://www.ncdot.gov/_graphics/site_graphics/projects_ncbridges_hb_t_conslab01.jpg. Accessed:10/21/2014

(5) Steel stringer or multibeam. http://www.ncdot.gov/projects/ncbridges/historic/types/?p=13#types. Accessed:10/21/2014

(6) Arch Deck. http://www.asce.org/CEMagazine/Article.aspx?id=25769811288#.VEbR8PldWL0. Accessed:10/21/2014

(7) Tee Beam. http://www.ncdot.gov/_graphics/site_graphics/projects_ncbridges_hb_t_teebeam01.jpg. Accessed:10/21/2014

(8) Box Beam bridge. https://www.michigan.gov/documents/MDOT_RR-102_143535_7.pdf. Accessed:10/21/2014

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