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This is a practical guide to P-splines, a simple, flexible and powerful tool for smoothing. P-splines combine regression on B-splines with simple, discrete, roughness penalties. They were introduced by the authors in 1996 and have been used in many diverse applications. The regression basis makes it straightforward to handle non-normal data, like in generalized linear models. The authors demonstrate optimal smoothing, using mixed model technology and Bayesian estimation, in addition to classical tools like cross-validation and AIC, covering theory and applications with code in R. Going far beyond simple smoothing, they also show how to use P-splines for regression on signals, varying-coefficient models, quantile and expectile smoothing, and composite links for grouped data. Penalties are the crucial elements of P-splines; with proper modifications they can handle periodic and circular data as well as shape constraints. Combining penalties with tensor products of B-splines extends these attractive properties to multiple dimensions. An appendix offers a systematic comparison to other smoothers.
Google’s Legal Department addresses cutting-edge issues that run from driverless cars to green-energy power cables for the Eastern Seaboard and legal hot spots from China to Turkey. Our legal department today consists of more than 900 legal team members, a significant growth from the one lawyer that made up the legal department in 2001. The unique culture of Google itself has inspired the legal department to innovate in ways that are more progressive than most companies of a similar size. The Google Legal Team supports the vision of the company’s engineers who are trying to create new technologies that will have an international impact on the lives of people. Accordingly, our legal team focuses its support on the interests of the users of the company’s technology and defends Google so that it can continue to focus on the company vision.