An empirical study of PLAD regression using the bootstrap

Partial LAD regression uses the L ₁ norm associated with least absolute deviations (LAD) regression while retaining the same algorithmic structure of univariate partial least squares (PLS) regression. We use the bootstrap in order to assess the partial LAD regression model performance and to make comparisons to PLS regression. We use a variety of examples coming from NIR experiments as well as two sets of experimental data.     

Selection of smoothing parameters in<Emphasis Type="Italic">B</Emphasis>-spline nonparametric regression models using information criteria

We consider the use ofB-spline nonparametric regression models estimated by the maximum penalized likelihood method for extracting information from data with complex nonlinear structure. Crucial points inB-spline smoothing are the choices of a smoothing parameter and the number of basis functions, for which several selectors have been proposed based on cross-validation and Akaike information criterion known as AIC. It might be however noticed that AIC is a criterion for evaluating models estimated by the maximum likelihood method, and it was derived under the assumption that the ture distribution belongs to the specified parametric model. In this paper we derive information criteria for evaluatingB-spline nonparametric regression models estimated by the maximum penalized likelihood method in the context of generalized linear models under model misspecification. We use Monte Carlo experiments and real data examples to examine the properties of our criteria including various selectors proposed previously.     

A partially penalized P1/CR immersed finite element method for planar elasticity interface problems

We propose a partially penalized P₁/CR immersed finite element (IFE) method with midpoint values on edges as degrees of freedom for CR elements to solve planar elasticity interface problems. Optimal approximation errors in L² norm and H¹ semi‐norm are obtained for the P₁/CR IFE spaces. Moreover, by adding some stabilization terms on the edges of interface elements, we derive an optimal error estimate for the P₁/CR IFE method. Our method differs from the method with average values on edges as degrees of freedom for P₁/CR elements in Qin et al.'s study, where no approximation theoretical result was presented. Numerical examples confirm our theoretical results.     

Robust penalized regression spline fitting with application to additive mixed modeling

An increasingly popular method for smoothing noisy data is penalized regression spline fitting. In this paper a new procedure is proposed for fitting robust penalized regression splines. This procedure is computationally fast, straightforward to implement, and can be paired with any smoothing parameter selection method. In addition, it can also be extended to other settings, such as additive mixed modeling. Both simulated and real data examples are used to illustrate the effectiveness of the procedure.     

Upper bounds for the L 1-risk of the minimum L 1-distance regression estimator

A new estimator of a regression function is introduced via minimizing the L ₁-distance between some empirical function and its theoretical counterpart plus penalty for the roughness. The L ₁-risk of the estimator is bounded from above for every sample size no matter what the dependence structure of the observed random variables is. In the case of independent errors of measurement with a common variance the estimator is shown to achieve the optimal L ₁-rate of convergence within the class of m-times differentiable functions with bounded derivatives.     

Penalized likelihood regression for generalized linear models with non-quadratic penalties

One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression for exponential families. Most of the smoothing methods employ quadratic penalties, leading to linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Uniform bounds from bounded Lp-functionals in reaction-diffusion equations

To prove global existence of classical or mild solutions of reaction-diffusion equations, a priori bounds in the uniform norm are needed. But for interesting examples, often one can only derive bounds for some L_p-norms. Using the structure of the reaction term they can be used to obtain uniform bounds. Two propositions are stated which give conditions for this procedure. The proofs use the smoothing properties of analytic semigroups and the multiplicative Gagliardo-Nirenberg inequality. To illustrate the method, we prove global existence of solutions for the Brusselator and a Volterra-Lotka system with one diffusing and one sedentary species.     

Genetic algorithms for the selection of smoothing parameters in additive models

Summary  Additive models of the type y=f ₁(x ₁)+...+f _p(x _p)+ε where f _j , j=1,..,p, have unspecified functional form, are flexible statistical regression models which can be used to characterize nonlinear regression effects. One way of fitting additive models is the expansion in B-splines combined with penalization which prevents overfitting. The performance of this penalized B-spline (called P-spline) approach strongly depends on the choice of the amount of smoothing used for components f _j . In particular for higher dimensional settings this is a computationaly demanding task. In this paper we treat the problem of choosing the smoothing parameters for P-splines by genetic algorithms. In several simulation studies this approach is compared to various alternative methods of fitting additive models. In particular functions with different spatial variability are considered and the effect of constant respectively local adaptive smoothing parameters is evaluated.     

Strong Stability and Non-smooth Data Error Estimates for Discretizations of Linear Parabolic Problems

We study smoothing properties of discretizations of a linear parabolic initial boundary value problem with a possibly non-selfadjoint elliptic operator. The solution at time t > 0 of this problem, as well as its time derivatives, are in L _r for initial values in L _s even when r > s. We show that similar strong stability results hold for discrete solutions obtained by discretizing in space by linear finite elements and in time by a class of A()-stable implicit rational multistep methods (including single step methods as a special case) with good smoothing properties, as well as for certain combinations of single step methods. Most of our results are derived from the corresponding L ₂-bounds, shown by semigroup techniques, together with a discrete Gagliardo-Nirenberg inequality, and generalize previously known estimates with respect to admissible problems and time discretization methods. Our techniques make it possible to obtain, e.g., supremum norm error estimates for initial data which are only required to be in L ₁.     

Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

A Geometric Programming Framework for Univariate Cubic L 1 Smoothing Splines

Univariate cubic L ₁ smoothing splines are capable of providing shape-preserving C ¹-smooth approximation of multi-scale data. The minimization principle for univariate cubic L ₁ smoothing splines results in a nondifferentiable convex optimization problem that, for theoretical treatment and algorithm design, can be formulated as a generalized geometric program. In this framework, a geometric dual with a linear objective function over a convex feasible domain is derived, and a linear system for dual to primal conversion is established. Numerical examples are given to illustrate this approach. Sensitivity analysis for data with uncertainty is presented. This work is supported by research grant #DAAG55-98-D-0003 of the Army Research Office, USA.     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

On the p version of the finite element method in the presence of numerical integration

We analyze the error in the p version of the finite element method when the effect of the quadrature error is taken into account. We extend some results by Banerjee and Suri [Math. Comput. 59 , 1–20 (1992)] on the H¹-norm error to the case of the error in the L₂ norm. We investigate three sources of quadrature error that can occur: the error due to the numerical integration of the right-hand side, that due to nonconstant coefficients, and that due to the presence of mapped elements. Presented are various theoretical and computational examples regarding the sharpness of our results. In addition, we make a note on the use of numerical quadrature in conjunction with p-adaptive procedures and on the necessity of overintegration in the h version with linear elements, when the L₂ norm is of interest. © 1993 John Wiley & Sons, Inc.     

A Path Algorithm for the Fused Lasso Signal Approximator

The Lasso is a very well-known penalized regression model, which adds an L₁ penalty with parameter λ₁ on the coefficients to the squared error loss function. The Fused Lasso extends this model by also putting an L₁ penalty with parameter λ₂ on the difference of neighboring coefficients, assuming there is a natural ordering. In this article, we develop a path algorithm for solving the Fused Lasso Signal Approximator that computes the solutions for all values of λ₁ and λ₂. We also present an approximate algorithm that has considerable speed advantages for a moderate trade-off in accuracy. In the Online Supplement for this article, we provide proofs and further details for the methods developed in the article.     

Iterated line graphs are maximally ordered

A graph G is k‐ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. We prove that if G is a connected graph distinct from a path, then there is a number t_G such that for every t ≥ t_G the t‐iterated line graph of G, L^t (G), is (δ(L^t (G)) + 1)‐ordered. Since there is no graph H which is (δ(H)+2)‐ordered, the result is best possible. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 171–180, 2006     

On an anti‐Ramsey problem of Burr,Erdős,Graham, and T. Sós

Given a graph L, in this article we investigate the anti‐Ramsey number χ_S(n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G(n,e) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interesting results and problems, Burr, Erd?s, Graham, and T. Sós asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that In this article, we prove a slightly weaker statement, namely we show that the statement is true if L is a connected bipartite graph, which is not a complete bipartite graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 147–156, 2006     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

On constrained optimization problems with nonsmooth cost functionals

In this paper we present a computational procedure for minimizing a class ofL ₁-functionals subject to conventional as well as functional constraints. The computational procedure is based on the idea of enforced smoothing together with a method of converting the functional constraints into conventional equality constraints. For illustration, two examples are solved using the proposed procedure.