Given a stationary state-space model that relates a sequence of hidden states and corresponding measurements or observations, Bayesian filtering provides a principled statistical framework for inferring the posterior distribution of the current state given all measurements up to the present time. For example, the Apollo lunar module implemented a Kalman filter to infer its location from a sequence of earth-based radar measurements and land safely on the moon. To perform Bayesian filtering, we require a measurement model that describes the conditional distribution of each observation given state. The Kalman filter takes this measurement model to be linear, Gaussian. Here we show how a nonlinear, Gaussian approximation to the distribution of state given observation can be used in conjunction with Bayes’ rule to build a nonlinear, non-Gaussian measurement model. The resulting approach, called the Discriminative Kalman Filter (DKF), retains fast closed-form updates for the posterior. We argue there are many cases where the distribution of state given measurement is better-approximated as Gaussian, especially when the dimensionality of measurements far exceeds that of states and the Bernstein—von Mises theorem applies. Online neural decoding for brain-computer interfaces provides a motivating example, where filtering incorporates increasingly detailed measurements of neural activity to provide users control over external devices. Within the BrainGate2 clinical trial, the DKF successfully enabled three volunteers with quadriplegia to control an on-screen cursor in real-time using mental imagery alone. Participant “T9” used the DKF to type out messages on a tablet PC. Nonstationarities, or changes to the statistical relationship between states and measurements that occur after model training, pose a significant challenge to effective filtering. In brain-computer interfaces, one common type of nonstationarity results from wonkiness or dropout of a single neuron. We show how a robust measurement model can be used within the DKF framework to effectively ignore large changes in the behavior of a single neuron. At BrainGate2, a successful online human neural decoding experiment validated this approach against the commonly-used Kalman filter.
Consistently lauded for its comprehensiveness and full-color color presentation, the latest edition of Rheumatology by Marc C. Hochberg, MD, MPH et al. continues the tradition of excellence of previous editions. Designed to meet the needs of the practicing clinician, it provides extensive, authoritative coverage of rheumatic disease from basic scientific principles to practical points of clinical management in a lucid, logical, user-friendly manner. Find the critical answers you need quickly and easily thanks to a consistent, highly user-friendly format covering all major disorders of the musculoskeletal system in complete, self-contained chapters. Get trusted perspectives and insights from chapters co-authored by internationally renowned leaders in the field, 25% of whom are new to this edition. Track disease progression and treat patients more effectively with the most current information, including 22 new chapters on genetic findings, imaging outcomes, and cell and biologic therapies as well as rheumatoid arthritis and SLE. Incorporate the latest findings about pathogenesis of disease; imaging outcomes for specific diseases like RA, osteoarthritis, and spondyloarthropathies; cell and biologic therapies; and other timely topics.
In this essential primer, mathematician Michael Frame, a close collaborator with Benoit Mandelbrot, the founder of fractal geometry, and poet Amelia Urry explore the amazing world of fractals as they appear in nature, art, medicine, and technology
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