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) 1
Presentation Outline Introduction Basic Tensor Concepts Analysis & Results Conclusion & Recommendations 2
INTRODUCTION 3
National Bridge Inventory (NBI) Largest database of bridge data in the world 1 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 4
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 Denver 2 * http://www.transportationissuesdaily.com/wp-content/uploads/2011/10/sherman-minton-bridge.jpg Sherman-Minton Bridge * 5
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 6
Focus of Research Analysis of deterioration trends of specific bridge deck structural types in the US Tensor Decomposition 7
BASIC TENSOR CONCEPTS 8
What are Tensors? Generalizations of scalars, vectors and matrices Samples variables Samples Time Time variables variables Samples Weather 9
Multiway data is everywhere Why Multiway Data Analysis? Capture variation in dataset while preserving multidimensional nature of data Reveal hidden patterns in data 10
Typical 3D dataset Subarrays Mode-3 Mode-1 Mode-2 X x 231 Mode-1 Fiber x :21 Mode-2 Fiber x 1:2 Mode-3 Fiber X 11: Horizontal slice X 1:: Lateral slice X :2: Frontal slice X ::3 11
Matricization (Unfolding) Summation: A + B = C, where C iii = a iii + b iii k i χ R i jj i j j j j Outer Product: X = a ο b ο c s.t. x iii = a i b j c k j k Khatri-Rao Product: P I J Q K J = R II J k i χ R j ii j i i i i Kronecker Product: P I J Q K L = R II JJ k i j χ R k ii k k i i i i Hadamard Product: P I J Q I J = R I J j j 12
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 13
The CP Decomposition Canonical Decomposition/ Parallel Factors a i, b i and c i - factor loadings R- number of components Rank of a tensor=min (R) required to approximate X 14
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 15
ANALYSIS & RESULTS 16
Where: Structural Deterioration Rate SS i,j,k = n i,j,k N i,j,k j SS i,j,k =structural deterioration rate in state i of structure type j in year k n i,j,k =number of structurally deficient bridges in state i of structure type j in year k N i,j,k =number of bridges in state i of structure type j in year k 17
NBI database (1992-2013) Dataset Structurally deficient bridge design types by state Year State Deck Type Multidimensional Dataset for analysis 18
Selected Bridge Design Types Girder and Floorbeam 4 Slab 5 Stringer/Multibeam 6 Arch Deck 7 Tee Beam 8 Box Beam 9 19
3D visualization 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 PR WY WI WV WA VA VT UT TX TN SD SC RI PA OR OK OH ND NC NY NM NJ NH NV NE MT MO MS MN MI MD ME MA LA KY KS IA IN IL ID HI GA FL AK AL AR AZ CO CA DE CT DC Girder & Floorbeam system Slab Stringer/Multibeam or Girder Arch Deck Tee Beam Box Beam or Girders(Multiple) 20
Exploration of NBI structure type data slab stringer/multibm or girder girder & flbm Tee Bm Box Beam Arch Deck 0.8 0,4 0.7 0,35 0.6 0,3 0.5 0,25 0.4 0,2 0.3 0,15 0.2 0.1 0,1 0,05 0 Slab Stringer/ Multibeam or girder Girder & Floorbeam Tee Beam Box Beam or girders Box plot of mean SD rates over 22 years Arch-Deck 0 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 SD rate trends for structural types 21
Decomposition plots 2 component PARAFAC model was fitted to the data Variance explained= 90.15% 0.8 5 0.235 Component 2 0.7 0.6 0.5 0.4 0.3 0.2 box bm slab multibm or girder tee bm arch deck gird & floorbm Component 2 4 3 2 1 0-1 -2 ND OK SD AL KY TN LA HI AK CA AZ GAIA AR KS ORNV WAMT UT DC FL MD ID NE MIIL MO PA OH WV VT WI CO MN WY TX NH CT VA PR MA NM RI IN NJ NC ME SC DE NY 0.23 0.225 0.22 0.215 0.21 0.205 0.2 0.195 13 12 11 10 09 08 07 05 06 04 03 02 99 00 01 98 97 96 95 94 92 93 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Component 1 MS -3-1 0 1 2 3 4 5 6 7 Component 1 0.19 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 Loading plot for Structure types Loading plot for States Loading plot for years 22
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 23
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 24