Modeling Dependency with Copula: Implications to Engineers and Planners
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Modeling Dependency with Copula: Implicationsto Engineers and Planners
Haizhong Wang, Ph.D, Assistant ProfessorSchool of Civil and Construction Engineering
Oregon State University, Corvallis, ORHaizhong.Wang@oregonstate.edu
Seminar at Portland State UniversityPortland, OR
September 28, 2012
Haizhong Wang (OSU) Copula Modeling September 28, 2012 1 / 31
Philosophy
Modeling Philosophy - By George Box
Essentially, all models are wrong,but some are useful.
In Empirical Model-Building and Responses Surfaces (1987), GeorgeE. P. Box and Norman R. Draper, p242, ISBN: 0471810339.
In Box’s paper, Robustness in the Strategy of Scientific ModelBuilding, in Robustness in Statistics: Proceedings of a Workshop(May 1979) edited by Launer and GN Wilkinson.
Haizhong Wang (OSU) Copula Modeling September 28, 2012 2 / 31
Philosophy
Deterministic vs. Stochastic
Haizhong Wang (OSU) Copula Modeling September 28, 2012 3 / 31
Philosophy
Monte Carlo Simulation
Source: “A Practical Guide to Monte Carlo Simulation”, by Jon Wittwer
Haizhong Wang (OSU) Copula Modeling September 28, 2012 4 / 31
Philosophy
Dependence measure
How do we measure dependence between random variables?
Correlation coefficient: a measure of linear dependence betweenrandom variables
Concordance: if “large” values of one random variable tend to beassociated with “large” values of the other and “small” values of onewith “small” values of the other.
Discordance: vice versa
Haizhong Wang (OSU) Copula Modeling September 28, 2012 5 / 31
Philosophy
Measure of Dependence
Concordance
Kendall’s tau
Spearman’s rho
Linear correlation: nonelliptical distributions
Haizhong Wang (OSU) Copula Modeling September 28, 2012 6 / 31
Philosophy
Dependence of random variables
Spatial dependencies: the dependence between a number of variablesat the same time
Temporal dependencies: the inter-temporal dependence structure of aprocess
The fact: Covariance only captures the linear dependence relationships forspecial classes of distributions such as normal distribution
The question: Is there a possibility to capture the whole dependencestructure without any disturbing effects coming from the marginaldistributions?
Haizhong Wang (OSU) Copula Modeling September 28, 2012 7 / 31
Philosophy
Copula
What is a Copula?
A Latin noun that means “a link, tie, bond”
Copulas are used to describe dependence between random variables
An Introduction to Copulas, Second Edition, Roger Nelson, 2005.
Haizhong Wang (OSU) Copula Modeling September 28, 2012 8 / 31
Philosophy
Statistics about Copula
Google 2003: Copula → 10,000 results
Google 2005: Copula → 650,000 results
Google 2012: Copula → 1950,000 results
Copulas: Tales and Facts, Thomas Mikosch, 2005. Citation: 140
Haizhong Wang (OSU) Copula Modeling September 28, 2012 9 / 31
Philosophy
The first appearance
1940/1941: Hoeffding studied nonparametric measures of associationsuch as Spearman’s rho in multivariate distributions
1959: The word copula appears for the first time (Sklar, 1959)
1999: Introduced to financial applications (Embrechts et al., 1999)
2008: Widely adopted in insurance, finance, energy, hydrology,survival analysis, etc.
Source: Daniel Berg, Using Copulas: an Introduction toPractitioners
Haizhong Wang (OSU) Copula Modeling September 28, 2012 10 / 31
Philosophy
Foundation
Definition and Sklar Theorem (1959)
Sklar theorem describes “join together one-dimensional distributionfunctions to form multivariate distribution functions”
Let H be a joint distribution function with margins F1, . . . ,Fd . Then thereexists a copula C : [0, 1]d → [0, 1] such that
H(x1, . . . , xd) = C (F1(x1), . . . ,Fd(xd))
Theoretically, C captures all aspects of dependence and Fi captures allaspects of marginal distributions
Haizhong Wang (OSU) Copula Modeling September 28, 2012 11 / 31
Philosophy
Applications
Civil engineering- reliability of analysis of highway bridges
Climate and weather related research
Analysis of extrema in financial assets and returns
Failure of paired organs in health science
Human mortality in insurance (actuarial science)
Mortalities of spouses
Mortalities of parents and children twins (identical or nonidentical)
Haizhong Wang (OSU) Copula Modeling September 28, 2012 12 / 31
Implications to Planners
Applications - Choice Modeling
Chandra R. Bhat and Naveen Eluru (2009), A Copula-Based Approachto Accommodate Residential Self-Selection Effects in Travel BehaviorModeling, Transportation Research Part B, Vol. 43, No. 7, pp.749-765.
Erisa Spissu, Abdul R. Pinjari, Ram M. Pendyala, Chandra R. Bhat(2009),A Copula-based Joint Multinomial Discrete-Continuous Modelof Vehicle Type Choice and Miles of Travel, Transportation, Vol. 36,No. 4, pp. 403-422.
Naveen Eluru, Rajesh Paleti, Ram M. Pendyala, Chandra R. Bhat(2010), Modeling Injury Severity of Multiple Occupants of Vehicles:Copula-Based Multivariate Approach, Transportation ResearchRecord, Vol. 2165, pp. 1-11.
Haizhong Wang (OSU) Copula Modeling September 28, 2012 13 / 31
Implications to Planners
Applications - Behaviour Modeling
, Ipek N. Sener, Chandra Bhat (2011), A Copula-Based SampleSelection Model of Telecommuting Choice and Frequency,Environment and Planning A, Vol. 43, No. 1, pp. 126-145.
Jeffrey J. LaMondia, Chandra R. Bhat (2012), A Conceptual andMethodological Framework of Leisure Activity Loyalty Accommodatingthe Travel Context, Transportation, Vol. 39, No. 2, pp. 321-349.
A. Portoghese, E. Spissu, C. R. Bhat, N. Eluru, and I. Meloni (2010),A Copula-Based Joint Model of Commute Mode Choice and Numberof Non-Work Stops during the Commute, Technical Report.
Haizhong Wang (OSU) Copula Modeling September 28, 2012 14 / 31
Implications to Planners
Applications - Hydrology
G. Salvadori and C. De Michele, On the Use of Copula in Hydrology:Theory and Practice, Journal of Hydrology Engineering, Vol. 12, No.4, July 1, 2007.
Amir AghaKouchak, Andras Bardossy and Emad Habib, Copula-basedUncertainty Modeling: Application to Multisensor Precipitationestimates, Hydrological Processes, 24, pp. 2111 - 2124 (2010).
Pranesh Kumar, Copula Functions: Characterizing Uncertainty inProbabilistic Systems, Applied Mathematical Sciences, Vol. 5, 2011,no. 30, 1459 - 1472.
Haizhong Wang (OSU) Copula Modeling September 28, 2012 15 / 31
Implications to Planners
Copula Model and Simulation
Create a copula model for the distribution of (X1, · · · ,Xd) generally takestwo steps
Model
Set a model for marginal distribution Fi
Set a model for copula C
C is the cdf of a random vector (U1, · · · ,Ud) with uniform margins
Simulation
Draw a sample (U1, · · · ,Ud) ≈ C
Set (X1, · · · ,Xd) = (F−11 (U1), · · · ,F−1
d (Ud)
Haizhong Wang (OSU) Copula Modeling September 28, 2012 16 / 31
Implications to Planners
Gaussian Copula
Figure: Bivariate Gaussian copula with varying parameters ρ
Haizhong Wang (OSU) Copula Modeling September 28, 2012 17 / 31
Implications to Planners
Student t Copula
Figure: Bivariate Student copula with varying parameters ρ and ν
Haizhong Wang (OSU) Copula Modeling September 28, 2012 18 / 31
Implications to Planners
Example - Entrance Ramp Flow Dependency
Haizhong Wang (OSU) Copula Modeling September 28, 2012 19 / 31
Implications to Planners
Example - Entrance Ramp Flow Dependency
Haizhong Wang (OSU) Copula Modeling September 28, 2012 20 / 31
Implications to Planners
Example - Morning Peak Hour
4005- 101 102 103 104 105 005 006 008 009 010
101 1.0000 0.0503 -0.1529 0.0680 0.0209 -0.0284 0.0767 0.2414 -0.0061 -0.4539
102 0.0503 1.0000 -0.0108 0.1021 0.1678 0.0901 0.1178 -0.0376 0.1488 -0.0591
103 -0.1529 -0.0108 1.0000 -0.0548 0.2614 0.0699 0.0104 -0.2420 0.0231 0.1339
104 0.0680 0.1021 -0.0548 1.0000 0.2458 0.1749 0.1910 0.1267 0.0948 -0.0287
105 0.0209 0.1678 0.2614 0.2458 1.0000 0.3052 -0.0164 0.0759 0.5157 0.0111
005 -0.0284 0.0901 0.0699 0.1749 0.3052 1.0000 0.0297 0.1201 0.1346 0.1041
006 0.0767 0.1178 0.0104 0.1910 -0.0164 0.0297 1.0000 -0.0344 -0.1562 0.0034
008 0.2414 -0.0376 -0.2420 0.1267 0.0759 0.1201 -0.0344 1.0000 -0.0087 0.0309
009 -0.0061 0.1488 0.0231 0.0948 0.5157 0.1346 -0.1562 -0.0087 1.0000 -0.0006
010 -0.4539 -0.0591 0.1339 -0.0287 0.0111 0.1041 0.0034 0.0309 -0.0006 1.0000
Haizhong Wang (OSU) Copula Modeling September 28, 2012 21 / 31
Implications to Planners
Example - Afternoon Peak Hour
4005- 101 102 103 104 105 005 006 008 009 010
101 1.0000 0.4030 0.2727 0.0914 0.0827 0.3679 0.3815 -0.0474 0.0044 0.0204
102 0.4030 1.0000 0.4563 0.1567 0.0937 0.5894 0.5233 0.0439 -0.1558 0.4893
103 0.2727 0.4563 1.0000 0.2309 -0.0606 0.4834 0.3606 -0.0407 -0.2626 0.3998
104 0.0914 0.1567 0.2309 1.0000 0.0008 0.2485 0.1592 -0.0808 -0.0927 0.1064
105 0.0827 0.0937 -0.0606 0.0008 1.0000 0.0024 0.1417 -0.1339 0.2076 -0.0106
005 0.3679 0.5894 0.4834 0.2485 0.0024 1.0000 0.5952 0.0249 -0.2089 0.5994
006 0.3815 0.5233 0.3606 0.1592 0.1417 0.5952 1.0000 -0.0083 0.0249 0.5276
008 -0.0474 0.0439 -0.0407 -0.0808 -0.1339 0.0249 -0.0083 1.0000 -0.0616 0.1039
009 0.0044 -0.1558 -0.2626 -0.0927 0.2076 -0.2089 0.0249 -0.0616 1.0000 -0.0787
010 0.0204 0.4893 0.3998 0.1064 -0.0106 0.5994 0.5276 0.1039 -0.0787 1.0000
Haizhong Wang (OSU) Copula Modeling September 28, 2012 22 / 31
Implications to Planners
Example - Entrance Ramp Flow Dependency
(a) 4005101/102 (b) 4005102/103
(c) 4005103/104 (d) 4005104/105
Figure: The joint probability density contour through a Gaussian copulas for themorning peak hour (01/02/2003) dependency among entrance-ramps southboundof GA400Haizhong Wang (OSU) Copula Modeling September 28, 2012 23 / 31
Implications to Planners
Example - Entrance Ramp Flow Dependency
(a) (b)
(c) (d)
Figure: The morning peak (01/02/2003) dependency surface throughnonparametric bivariate copulas for entrance-ramps southbound of GA400including 4005101 to 4005105Haizhong Wang (OSU) Copula Modeling September 28, 2012 24 / 31
Implications to Planners
Example - Day-to-Day Analysis
(a) 4005101/102 (b) 4005102/103
(c) 4005103/104 (d) 4005104/105
Figure: The joint probability density contour through a 2d student t copulas forthe morning peak hour (01/07/2003) dependency among entrance-rampssouthbound of GA400Haizhong Wang (OSU) Copula Modeling September 28, 2012 25 / 31
Implications to Planners
Student-t Copula
(a) 4005101/102 (b) 4005102/103
(c) 4005103/104 (d) 4005104/105
Figure: The joint probability density contour through a 2d student t copulas forthe morning peak hour (01/07/2003) dependency among entrance-rampssouthbound of GA400Haizhong Wang (OSU) Copula Modeling September 28, 2012 26 / 31
Implications to Planners
Result Analysis - Student t copula
(a) 4005005/006 (b) 4005006/008
(c) 4005008/009 (d) 4005009/010
Figure: The joint probability density contour through a 2d student t copulas forthe afternoon peak hour (01/09/2003) dependency among entrance-rampsnorthbound of GA400Haizhong Wang (OSU) Copula Modeling September 28, 2012 27 / 31
Implications to Planners
Simulation
(a) (b)
Figure: The dependency structure between ramp flow in 2 and 3 dimensions
Haizhong Wang (OSU) Copula Modeling September 28, 2012 28 / 31
Implications to Planners
Computing/Fitting with Copula
Matlab - Built-in-functions
R - Copula package
Haizhong Wang (OSU) Copula Modeling September 28, 2012 29 / 31
Implications to Planners
Summary - Attractive Features (Daniel Berg (2008))
The copula contains all the information about the dependencebetween random variablesCopulas provide an alternative and often more useful representation ofmultivariate distribution functions compared to traditional approachessuch as multivariate normalityMost traditional representations of dependence are based on thelinear correlation coefficient - restricted to multivariate ellipticaldistributions. Copula representations of dependence are free of suchlimitations.Copulas enable us to model marginal distributions and thedependence structure separatelyCopulas provide greater modeling flexibility, given a copula we canobtain many multivariate distributions by selecting different marginsA copula is invariant under strictly increasing transformationsMost traditional measures of dependence are measures of pairwisedependence. Copulas measure the dependence between all d randomvariablesHaizhong Wang (OSU) Copula Modeling September 28, 2012 30 / 31
Q & A
Questions and Comments?
Thanks!
Jia Li at University of California Davis
Haizhong Wang (OSU) Copula Modeling September 28, 2012 31 / 31
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