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.

Virtual Reality as a Support Tool for Ergonomic-Style Convergence

The competitive industrial context compels companies to speed-up every new product design. In order to keep designing products that meet the needs of the end user, a human centered concurrent product design methodology has been proposed. Its setting up is complicated by the difficulties of collaboration between experts involved inthe design process. In order to ease this collaboration, we propose the use of virtual reality as an intermediate design representation in the form of light and specialized immersive convergence support applications. In this paper, we present the As Soon As Possible (ASAP) methodology making possible the development of these tools while ensuring their usefulness and usability. The relevance oft his approach is validated by an industrial use case through the design of an ergonomic-style convergence support tool.

‘Continue or terminate?’ Determinants of long-term volunteering in sports clubs

This study analyzed the determinants underlying sports club volunteers' tendencies to continue or terminate their long-term commitment to volunteering in order to help sports clubs improve their volunteer management. Their risk of terminating was viewed in terms of subjective expectations and evaluations (satisfaction) regarding club-related working conditions and normative commitments (solidarity) to the sports club. These relationships were tested empirically with an online questionnaire of 441 sports club volunteers in a selection of 45 Swiss sports clubs. Results showed that the constructs orientation toward collective solidarity and volunteer job satisfaction correlated positively with long-term volunteering commitment. The effect of the former was stronger than that of the latter. Volunteers with a higher orientation toward collective solidarity were unlikely to terminate their voluntary engagement in their club. The discussion presents recommendations to help clubs retain volunteers.