A Variance-Minimizing Filter for Large-Scale Applications

A truly variance-minimizing filter is introduced and its per for mance is demonstrated with the Korteweg– DeV ries (KdV) equation and with a multilayer quasigeostrophic model of the ocean area around South Africa. It is recalled that Kalman-like filters are not variance minimizing for nonlinear model dynamics and that four - dimensional variational data assimilation (4DV AR)-like methods relying on per fect model dynamics have dif- ficulty with providing error estimates. The new method does not have these drawbacks. In fact, it combines advantages from both methods in that it does provide error estimates while automatically having balanced states after analysis, without extra computations. It is based on ensemble or Monte Carlo integrations to simulate the probability density of the model evolution. When obser vations are available, the so-called importance resampling algorithm is applied. From Bayes’ s theorem it follows that each ensemble member receives a new weight dependent on its ‘ ‘distance’ ’ t o the obser vations. Because the weights are strongly var ying, a resampling of the ensemble is necessar y. This resampling is done such that members with high weights are duplicated according to their weights, while low-weight members are largely ignored. In passing, it is noted that data assimilation is not an inverse problem by nature, although it can be for mulated that way . Also, it is shown that the posterior variance can be larger than the prior if the usual Gaussian framework is set aside. However , i n the examples presented here, the entropy of the probability densities is decreasing. The application to the ocean area around South Africa, gover ned by strongly nonlinear dynamics, shows that the method is working satisfactorily . The strong and weak points of the method are discussed and possible improvements are proposed.

Variance in male reproductive success and sexual size dimorphism in pinnipeds: testing an assumption of sexual selection theory

The theory of evolution by sexual selection for sexual size dimorphism (SSD) postulates that SSD primarily reflects the adaptation of males and females to their different reproductive roles. For example, competition among males for access to females increases male body size because larger males are better able to maintain dominant status than smaller males. Larger dominant males sire most offspring while smaller subordinate males are unsuccessful, leading to skew in reproductive success. Therefore, species with male-biased SSD are predicted to have greater variance in male reproductive success than those in which both sexes are similar in size. We tested this prediction among the Pinnipedia, a mammalian group with a great variation in SSD. From a literature review, we identified genetic estimates of male reproductive success for 10 pinniped taxa (eight unique species and two subspecies of a ninth species) that range from seals with similarly sized males and females to species in which males are more than four times as large as females. We found no support for a positive relationship between variance in reproductive success and SSD among pinnipeds after excluding the elephant seals Mirounga leonina and Mirounga angustirostris, which we discuss as distinctive cases. Several explanations for these results are presented, including the revival of one of Darwin's original ideas. Darwin proposed that natural selection may explain SSD based on differences in energetic requirements between sexes and the potential for sexual niche segregation. Males may develop larger bodies to exploit resources that remain unavailable to females due to the energetic constraints imposed on female mammals by gestation and lactation. The importance of this alternative explanation remains to be tested.

Jump and variance risk premia in the S&P 500

We analyze the risk premia embedded in the S&P 500 spot index and option markets. We use a long time-series of spot prices and a large panel of option prices to jointly estimate the diffusive stock risk premium, the price jump risk premium, the diffusive variance risk premium and the variance jump risk premium. The risk premia are statistically and economically significant and move over time. Investigating the economic drivers of the risk premia, we are able to explain up to 63 % of these variations.