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.

Percutaneous coronary interventions in Europe: prevalence, numerical estimates, and projections based on data up to 2004

AIMS: A registry mandated by the European Society of Cardiology collects data on trends in interventional cardiology within Europe. Special interest focuses on relative increases and ratios in new techniques and their distributions across Europe. We report the data through 2004 and give an overview of the development of coronary interventions since the first data collection in 1992. METHODS AND RESULTS: Questionnaires were distributed yearly to delegates of all national societies of cardiology represented in the European Society of Cardiology. The goal was to collect the case numbers of all local institutions and operators. The overall numbers of coronary angiographies increased from 1992 to 2004 from 684 000 to 2 238 000 (from 1250 to 3930 per million inhabitants). The respective numbers for percutaneous coronary interventions (PCIs) and coronary stenting procedures increased from 184 000 to 885 000 (from 335 to 1550) and from 3000 to 770 000 (from 5 to 1350), respectively. Germany was the most active country with 712 000 angiographies (8600), 249 000 angioplasties (3000), and 200 000 stenting procedures (2400) in 2004. The indication has shifted towards acute coronary syndromes, as demonstrated by rising rates of interventions for acute myocardial infarction over the last decade. The procedures are more readily performed and perceived safer, as shown by increasing rate of "ad hoc" PCIs and decreasing need for emergency coronary artery bypass grafting (CABG). In 2004, the use of drug-eluting stents continued to rise. However, an enormous variability is reported with the highest rate in Switzerland (70%). If the rate of progression remains constant until 2010 the projected number of coronary angiographies will be over three million, and the number of PCIs about 1.5 million with a stenting rate of almost 100%. CONCLUSION: Interventional cardiology in Europe is ever expanding. New coronary revascularization procedures, alternative or complementary to balloon angioplasty, have come and gone. Only stenting has stood the test of time and matured to the default technique. Facilitated access to PCI, more complete and earlier detection of coronary artery disease promise continued growth of the procedure despite the uncontested success of prevention.

Numerical and granulometric approaches to geophysical granular flows

Volcanic ash clouds can be fed by an upward-directed eruption column (Plinian column) or by elutriation from extensive pyroclastic-flows (coignimbrite cloud). For large-scale eruptions, there is considerable uncertainty about which mechanism is dominant. Here we analyze in a novel way a comprehensive grainsize database for pyroclastic deposits. We demonstrate that the Mount Pinatubo climactic eruption deposits were substantially derived from coignimbrite clouds, and not only by a Plinian cloud as generally thought. Coignimbrite ash-fall deposits are much richer in breathable <10 m ash (5–25 wt%) than pure Plinian ash at most distances from the source volcano. We also show that coignimbrite ash clouds, as at Pinatubo, are expected to be more water rich than Plinian clouds, leading to removal of more HCl prior to stratospheric injection, thereby reducing their atmospheric impact.

Pregnancy induces numerical and functional changes of CD4+CD25 high regulatory T cells in patients with rheumatoid arthritis

OBJECTIVE: In a prospective study we investigated whether numerical and functional changes of CD4+CD25(high) regulatory T cells (Treg) were associated with changes of disease activity observed during pregnancy and post partum in patients with rheumatoid arthritis (RA). METHODS: The frequency of CD4+CD25(high) T cells was determined by flow cytometry in 12 patients with RA and 14 healthy women during and after pregnancy. Fluorescence-activated cell sorting (FACS) was used to sort CD4+CD25(high) T cells and CD4+CD25- T cells were stimulated with anti-CD3 and anti-CD28 monoclonal antibodies alone or in co-culture to investigate proliferation and cytokine secretion. RESULTS: Frequencies of CD4+CD25(high) Treg were significantly higher in the third trimester compared to 8 weeks post partum in patients and controls. Numbers of CD4+CD25(high) Treg inversely correlated with disease activity in the third trimester and post partum. In co-culture experiments significantly higher amounts of IL10 and lowered levels of tumour necrosis factor (TNF)alpha and interferon (IFN)gamma were found in supernatants of the third trimester compared to postpartum samples. These findings were independent from health or disease in pregnancy, however postpartum TNFalpha and IFN gamma levels were higher in patients with disease flares. CONCLUSION: The amelioration of disease activity in the third trimester corresponded to the increased number of Treg that induced a pronounced anti-inflammatory cytokine milieu. The pregnancy related quantitative and qualitative changes of Treg suggest a beneficial effect of Treg on disease activity.

Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.