Non-stationary covariance function modelling in 2D least-squares collocation

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the ob-served data. However, the assumption that the spatial dependence is constant through-out the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage fordealing with non-stationarity in geodetic data. We then compared stationary and non-stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC

Long swings in the Canadian dollar

This paper uses daily, monthly, and quarterly observations for the Canadian dollar - US dollar nominal exchange rate over the recent flexible exchange rate period (from 2 January 1973 to 11 June 2004), and a new statistical model of exchange rate dynamics, developed by Engel and Hamilton to test the null hypothesis that the value of the Canadian dollar is characterized by long swings (i.e. it moves in one direction for long periods of time). Results indicate that only with quarterly data does the segmented trends model outperfom the random walk model. In fact, the performance of the segmented trends model declines as the frequency of the data increases, suggesting that at higher frequencies the segmented trends model has a more difficult time in distinguishing trends.

Chaos in a long-term experiment with a plankton community

Mathematical models predict that species interactions such as competition and predation can generate chaos1, 2, 3, 4, 5, 6, 7, 8. However, experimental demonstrations of chaos in ecology are scarce, and have been limited to simple laboratory systems with a short duration and artificial species combinations9, 10, 11, 12. Here, we present the first experimental demonstration of chaos in a long-term experiment with a complex food web. Our food web was isolated from the Baltic Sea, and consisted of bacteria, several phytoplankton species, herbivorous and predatory zooplankton species, and detritivores. The food web was cultured in a laboratory mesocosm, and sampled twice a week for more than 2,300 days. Despite constant external conditions, the species abundances showed striking fluctuations over several orders of magnitude. These fluctuations displayed a variety of different periodicities, which could be attributed to different species interactions in the food web. The population dynamics were characterized by positive Lyapunov exponents of similar magnitude for each species. Predictability was limited to a time horizon of 15¿30 days, only slightly longer than the local weather forecast. Hence, our results demonstrate that species interactions in food webs can generate chaos. This implies that stability is not required for the persistence of complex food webs, and that the long-term prediction of species abundances can be fundamentally impossible