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G. Wahba 485matrix is a rank p projection operator. The degrees of freedom for signal is animportant concept in linear and nonlinear nonparametric regression, and itwas a mistake to hide it inconspicuously in Wahba (1983). Later Brad Efron(2004) gave an alternative definition of degrees of freedom for signal. The definitionin Wahba (1983) depends only on the data; Efron’s is essentially anexpected value. Note that in (41.1),trace{A(λ)} =n∑i=1∂ŷ i∂y i,where ŷ i is the predicted value of y i . This definition can reasonably be appliedto a problem with a nonlinear forward operator (that is, that maps dataonto the predicted data) when the derivatives exist, and the randomized tracemethod is reasonable for estimating the degrees of freedom for signal, althoughcare should be taken concerning the size of δ. Even when the derivatives don’texist the randomized trace can be a reasonable way of getting at the degreesof freedom for signal; see, e.g., Wahba et al. (1995).41.1.5 Yuedong Wang, Chong Gu and smoothing splineANOVASometime in the late 80s or early 90s I heard Graham Wilkinson expoundon ANOVA (Analysis of Variance), where data was given on a regular d-dimensional grid, viz.y ijk , t ijk , i =1,...,I, j =1,...,J, k =1,...,K,for d = 3 and so forth. That is, the domain is the Cartesian product ofseveral one-dimensional grids. Graham was expounding on how fitting a modelfrom observations on such a domain could be described as set of orthogonalprojections based on averaging operators, resulting in main effects, two factorinteractions, etc. “Ah-ha” I thought, we should be able to do exactly samething and more where the domain is the Cartesian product T = T 1 ⊗···⊗T dof d arbitrary domains. We want to fit functions on T , with main effects(functions of one variable), two factor interactions (functions of two variables),and possibly more terms given scattered observations, and we just need todefine averaging operators for each T α .Brainstorming with Yuedong Wang and Chong Gu fleshed out the results.Let H α ,α =1,...,d be d RKHSs with domains T α , each H α containing theconstant functions. H = H 1 ⊗···⊗H d is an RKHS with domain T . For eachα =1,...,d,constructaprobabilitymeasuredµ α on T α ,withthepropertythat the symbol (E α f)(t), the averaging operator, defined by∫(E α f)(t) = f(t 1 ,...,t d )dµ α (t α ),T ( α)