Effects of experimental freezing on soil nitrogen dynamics in soils from a net nitrification gradient in a nitrogen-saturated hardwood forest ecosystem

This study examined effects of soil freezing on N dynamics in soil along an N processing gradient within a mixed hardwood dominated watershed at Fernow Experimental Forest, West Virginia. Sites were designated as LN (low rates of N processing), ML (moderately low), MH (moderately high), and HN (high). Soils underwent three 7-day freezing treatments (0, –20, or –80 °C) in the laboratory. Responses varied between temperature treatments and along the gradient. Initial effects differed among freezing treatments for net N mineralization, but not nitrification, in soils across the gradient, generally maintained at LN < ML ≤ MH < HN for all treatments. Net N mineralization potential was higher following freezing at –20 and –80 °C than control; all were higher than at 0 °C. Net nitrification potential exhibited similar patterns. LN was an exception, with net nitrification low regardless of treatment. Freezing response of N mineralization differed greatly from that of nitrification, suggesting that soil freezing may decouple two processes of the soil N cycle that are otherwise tightly linked at our site. Results also suggest that soil freezing at temperatures commonly experienced at this site can further increase net nitrification in soils already exhibiting high nitrification from N saturation.

Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding.