Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Vine Copulas as a Way to Describe and Main Idea: Using . . . Analyze Multi-Variate Dependence in D-Vine Copulas: Idea Econometrics: Computational Motivation C-Vine Copulas: Idea and Comparison with Bayesian Networks Vine Copulas vs. Other . . . and Fuzzy Approaches How Vine Copulas Are . . . Home Page Songsak Sriboonchitta 1 , Jainxi Liu 1 , Title Page Vladik Kreinovich 2 , and Hung T. Nguyen 1 , 3 ◭◭ ◮◮ 1 Department of Economics, Chiang Mai University ◭ ◮ Chiang Mai, Thailand, songsak@econ.chiangmai.ac.th 2 Department of Computer Science, University of Texas at El Paso Page 1 of 12 500 W. University, El Paso, TX 79968, USA, vladik@utep.edu Go Back 3 Department of Mathematical Sciences, New Mexico State University Las Cruces, New Mexico 88003, USA, hunguyen@nmsu.edu Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 1. Need for Studying Dependence in Economics Copulas • In physics, many parameters, many phenomena are in- Case of Three of More . . . dependent. Main Idea: Using . . . D-Vine Copulas: Idea • So, we can observe (and thoroughly study) simple sys- C-Vine Copulas: Idea tems by a small number of parameters. Vine Copulas vs. Other . . . • Based on these simple systems, we determine the laws How Vine Copulas Are . . . of mechanics, electrodynamics, thermodynamics, etc. Home Page • We then combine these laws to describe more complex Title Page phenomena. ◭◭ ◮◮ • In contrast, in economics, most phenomena are inter- ◭ ◮ related. Page 2 of 12 • So, in econometrics, studying dependence is of utmost Go Back importance. Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 2. Statistical Character of Economic Phenomena Copulas • Most physical processes are deterministic. Case of Three of More . . . Main Idea: Using . . . • If we repeatedly drop the same object from the Leaning D-Vine Copulas: Idea Tower of Pisa, we observe the same behavior. C-Vine Copulas: Idea • In contrast, if several very similar restaurants open in Vine Copulas vs. Other . . . the same area, some of them will survive and some not. How Vine Copulas Are . . . Home Page • It is practically impossible to predict which will sur- vive. Title Page • At best, we can predict the probability of survival. ◭◭ ◮◮ • Thus, in economics, we need to study dependence be- ◭ ◮ tween random variables. Page 3 of 12 Go Back Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 3. Copulas Copulas • The joint distribution can be described by cdf and Case of Three of More . . . marginals: Main Idea: Using . . . D-Vine Copulas: Idea def def F ( x 1 , x 2 ) = Prob( X 1 ≤ x 1 & X 2 ≤ x 2 ); F i ( x i ) = Prob( X i ≤ x i ) . C-Vine Copulas: Idea • Independence means that F ( x 1 , x 2 ) = F 1 ( x 1 ) · F 2 ( x 2 ). Vine Copulas vs. Other . . . How Vine Copulas Are . . . • A natural way to describe dependence is to describe a Home Page function C ( a, b ) such that F ( x 1 , x 2 ) = C ( F 1 ( x 1 ) , F 2 ( x 2 )) . Title Page • Such functions C ( a, b ) are called copulas . ◭◭ ◮◮ • The pdf f ( x 1 , x 2 ) can also be described in terms of ◭ ◮ copulas: Page 4 of 12 = ∂ 2 C ( a, b ) def f ( x 1 , x 2 ) = c ( F 1 ( x 1 ) , F 2 ( x 2 )) · f 1 ( x 1 ) · f 2 ( x 2 ); c ( a, b ) . ∂a ∂b Go Back Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 4. Case of Three of More Variables Copulas • F ( x 1 , . . . , x n ) = Prob( X 1 ≤ x 1 & . . . & X n ≤ x n ) can Case of Three of More . . . also be described as Main Idea: Using . . . D-Vine Copulas: Idea F ( x 1 , . . . , x n ) = C ( F 1 ( x 1 ) , . . . , F n ( x n )) . C-Vine Copulas: Idea • The copula C ( a, . . . , b ) has to be determined from the Vine Copulas vs. Other . . . data. How Vine Copulas Are . . . Home Page • To describe a function C ( a, . . . , b ) of n variables with accuracy h > 0, we need h − n values C ( i 1 · h, . . . , i n · h ). Title Page ◭◭ ◮◮ • For n ≥ 3, we usually do not have that much data. ◭ ◮ • So, we need to describe the general dependence in Page 5 of 12 terms of functions of one and two variables. Go Back Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 5. Main Idea: Using Conditional Probabilities Copulas • We started with a formal definition of independence Case of Three of More . . . Main Idea: Using . . . F ( x 1 , x 2 ) = F 1 ( x 1 ) · F 2 ( x 2 ) . D-Vine Copulas: Idea C-Vine Copulas: Idea • A more intuitive definition is F 1 | 2 ( x 1 | x 2 ) = F 1 ( x 1 ), def Vine Copulas vs. Other . . . where F 1 | 2 ( x 1 | x 2 ) = Prob( X 1 ≤ x 1 | X 2 = x 2 ) . How Vine Copulas Are . . . • In general, F 1 | 2 ( x 1 | x 2 ) = C 1 | 2 ( F 1 ( x 1 ) , F 2 ( x 2 )) , where Home Page = ∂C 12 ( a, b ) Title Page def C 1 | 2 ( a, b ) . ∂b ◭◭ ◮◮ • For densities, we have ◭ ◮ f 1 | 2 ( x 1 | x 2 ) = c 12 ( F 1 ( x 1 ) , F 2 ( x 2 )) · f 1 ( x 1 ); c 12 ( a, b ) = ∂ 2 C 12 ( a, b ) Page 6 of 12 . ∂a ∂b Go Back Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 6. D-Vine Copulas: Idea Copulas • For two variables, we have F ( x 1 , x 2 ) = C 12 ( F 1 ( x 1 ) , F 2 ( x 2 )). Case of Three of More . . . Main Idea: Using . . . • For three variables, we similarly have D-Vine Copulas: Idea F 12 | 3 ( x 2 , x 2 | x 3 ) = C 12 | 3 ( F 1 ( x 1 | x 3 ) , F 2 ( x 2 | x 3 ) , x 3 ) . C-Vine Copulas: Idea Vine Copulas vs. Other . . . • In general, for different values x 3 , we can have different How Vine Copulas Are . . . copulas C ( a, b ) = C 12 | 3 ( a, b, x 3 ). Home Page • It often makes sense to assume that the dependence Title Page between X 1 and X 2 does not depend on X 3 : ◭◭ ◮◮ F 12 | 3 ( x 1 , x 2 | x 3 ) = C 12 | 3 ( F 1 | 3 ( x 1 | x 3 ) , F 2 | 3 ( x 2 | x 3 )) . ◭ ◮ • We already know how to describe F 1 | 3 ( x 1 | x 3 ) and Page 7 of 12 F 2 | 3 ( x 2 | x 3 ) in terms of bivariate copulas and marginals. Go Back • Thus, we can describe F 12 | 3 ( x 1 , x 2 | x 3 ) (and so, Full Screen F 123 ( x 1 , x 2 , x 3 )) in terms of bivariate copulas and marginals. Close Quit
Need for Studying . . . Statistical Character . . . 7. C-Vine Copulas: Idea Copulas • Main idea: we use probability densities instead of prob- Case of Three of More . . . abilities. In general: Main Idea: Using . . . D-Vine Copulas: Idea f ( x 1 , x 2 , x 3 ) = f 1 | 23 ( x 1 | x 2 , x 3 ) · f 2 | 3 ( x 2 | x 3 ) · f 3 ( x 3 ) . C-Vine Copulas: Idea • We know that f 2 | 3 ( x 2 | x 3 ) = c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) · f 2 ( x 2 ) . Vine Copulas vs. Other . . . How Vine Copulas Are . . . • Similarly, Home Page f 1 | 23 ( x 1 | x 2 , x 3 ) = c 12 | 3 ( F 1 | 3 ( x 1 | x 3 )) , F 2 | 3 ( x 2 | x 3 ) , x 3 ) · f 1 | 3 ( x 1 | x 3 ) . Title Page • It often makes sense that assume that the correspond- ◭◭ ◮◮ ing copula does not depend on x 3 ; then: ◭ ◮ f 1 | 23 ( x 1 | x 2 , x 3 ) = c 12 | 3 ( F 1 | 3 ( x 1 | x 3 )) , F 2 | 3 ( x 2 | x 3 )) · f 1 | 3 ( x 1 | x 3 ) . Page 8 of 12 • Hence, f ( x 1 , x 2 , x 3 ) = c 12 | 3 ( F 1 | 3 ( x 1 | x 3 )) , F 2 | 3 ( x 2 | x 3 )) · Go Back f 1 | 3 ( x 1 | x 3 ) · c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) · f 2 ( x 2 ) · f 3 ( x 3 ) . Full Screen Close Quit
Need for Studying . . . Statistical Character . . . 8. Vine Copulas vs. Other Techniques Copulas • Bayesian networks assume that some conditional dis- Case of Three of More . . . tributions are independent. Main Idea: Using . . . D-Vine Copulas: Idea • Thus, the Bayesian network approach can be viewed C-Vine Copulas: Idea as a particular case of the vine copula approach. Vine Copulas vs. Other . . . • In fuzzy logic , we estimate P ( A & B ) as f & ( P ( A ) , P ( B )) How Vine Copulas Are . . . for an appropriate t-norm f & ( a, b ). Home Page • In particular, for X 1 ≤ x 1 and X 2 ≤ x 2 , we get Title Page F 12 ( x 1 , x 2 ) = f & ( F 1 ( x 1 ) , F 2 ( x 2 )); ◭◭ ◮◮ ◭ ◮ F 123 ( x 1 , x 2 , x 3 ) = f & ( F 12 ( x 1 , x 2 ) , F 3 ( x 3 )) . Page 9 of 12 • Here, we use the same operation to combine probabil- ities corresponding to different variables. Go Back • In contrast, in vine copulas, we can use different cop- Full Screen ulas for different pairs of variables. Close Quit
Need for Studying . . . Statistical Character . . . 9. Vine Copulas vs. Other Techniques (cont-d) Copulas General copulas Case of Three of More . . . ↓ Main Idea: Using . . . Vine copulas D-Vine Copulas: Idea ւ ց C-Vine Copulas: Idea Bayesian Fuzzy Vine Copulas vs. Other . . . networks techniques How Vine Copulas Are . . . Home Page • Both Bayesian networks and fuzzy techniques have nu- merous successful applications. Title Page ◭◭ ◮◮ • The more general vine copula eliminates disadvantages of both approaches: ◭ ◮ – in contrast to Bayesian techniques, vine copula can Page 10 of 12 handle dependence between variables; Go Back – in contrast to fuzzy, vine copulas use different “and”- Full Screen operations (copulas) to combine different variables. Close Quit
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