I found it is hard for me to understand the perspective/contour visualisation graphs of different copula cdf, counter-monotonicity, independence, co-monotonicity.. anyone can share his/her understanding? thanks!
I'm not good in 3D either. David Wilmot did an excellent job in the tutorial - perhaps he will be able to help you.
Thanks for the kind feedback Rioch! I'm just about to leave to fly to Dublin to run a block tutorial so don't have time to respond immediately. I will aim to do so one evening this week.
hi, hope this helps but this is my interpretation of the contours (on page 18 of the ch14 of the course notes): *each of the contours are representative of 2 variables, U1 and U2 the numbers on the lines (on the contour) indicate the probability of the outcome, and any point on the "perspective" surface is a cumulative distribution value, e.g. any point on the top-most surface can be associated with a high cdf value. Counter-monotonicity as the name suggests, the 2 events move in opposite directions. e.g. a game of tennis, with U1 being player1 winning and U2 being player2 winning. we can't have both players winning - this is probability 0. Hence the flat surface for the "triangle" - because you can't have both events occurring in the same direction. As it is monotonic - you would expect the surface of the cdf for anything that interprets as "success of only one" to be like a "flat" slope. (can't think of a better way to explain this one) Independence the 2 variables are independent to one another. The probabilities associated with the materialisation of the 2 events are simply the multiplication of the 2 events. So, in the event that both events are likely to happen (i.e. close to 1), e.g. P(U1) = 0.9 and P(U2) = 0.9, then this lies in the top most surface of the cdf (indicating a high level of probability). Co-monotonicity The 2 variables move in the same direction, for example U2, the price of a burger that moves linearly with U1, the cost of its inputs (if input costs increases, the price of the burger will increase). Even if there was a probability of 0.2 that burger prices will go up, we would say that there is a 0.1 chance that its prices increases if there is a 0.1 probability that the input cost increases (taking the minimum), regardless of how much the increase is (does this make sense?). with Counter and Co-monotonicity, the "surfaces" are usually "flat" slopes because they move in linear fashion (i.e. either move in the same direction or opposite directions). does this help? or is it still unclear?
Have a look at the attached in respect of the comonotinicity or minimum copula. Here there is "perfect co-dependence" and u1=u2 for all values of u1,u2 in [0,1]. This is represented by the 45-degree line in Figure A. The unit mass of probability is spread evenly over this line. Figure B shows the comonotinicity Copula function (as per page 203 in Sweeting). This is a plot of the Copula's cumulative density function. So, the height indicated by the red dot (at u1=0.5 and u2=0.5) is the sum of the probabilities of observing each and every combination of u1 and u2 where both are < or = to 0.5. In other words, it is the sum of the probability mass in Figure B within the red square. This height is 0.5. The height of the blue dot is also 0.5 because the probability mass in Figure B within the blue square is the same as that within the red square. This is why the formula for this copula is min(u1, u2). Figure C is a 2D representation of Figure B. The contours (lines) link points in (u1,u2)-space which have the same height in Figure B. In other words, the contours link points for which the comonotinicity copula takes the same value. You can see that the red and blue dots both lie on the same contour because the value of the copula at both points is the same (=0.5). OK?
Great thread. Follow-up question: could anyone provide a situation in which the Generalized Clayton copula (having both lower AND upper tail dependence) might be appropriate? I'm pretty comfortable (I think) with the other Archimedean copulas, but I'm somewhat struggling with this one. Thanks.
