Functional principal components analysis via penalized rank one approximation

Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alternative approach to FPCA using penalized rank one approximation to the data matrix. Our contributions are four-fold: (1) by considering invariance under scale transformation of the measurements, the new formulation sheds light on how regularization should be performed for FPCA and suggests an efficient power algorithm for computation; (2) it naturally incorporates spline smoothing of discretized functional data; (3) the connection with smoothing splines also facilitates construction of cross-validation or generalized cross-validation criteria for smoothing parameter selection that allows efficient computation; (4) different smoothing parameters are permitted for different FPCs. The methodology is illustrated with a real data example and a simulation.Comment: Published in at http://dx.doi.org/10.1214/08-EJS218 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust regularized singular value decomposition with application to mortality data

We develop a robust regularized singular value decomposition (RobRSVD) method for analyzing two-way functional data. The research is motivated by the application of modeling human mortality as a smooth two-way function of age group and year. The RobRSVD is formulated as a penalized loss minimization problem where a robust loss function is used to measure the reconstruction error of a low-rank matrix approximation of the data, and an appropriately defined two-way roughness penalty function is used to ensure smoothness along each of the two functional domains. By viewing the minimization problem as two conditional regularized robust regressions, we develop a fast iterative reweighted least squares algorithm to implement the method. Our implementation naturally incorporates missing values. Furthermore, our formulation allows rigorous derivation of leave-one-row/column-out cross-validation and generalized cross-validation criteria, which enable computationally efficient data-driven penalty parameter selection. The advantages of the new robust method over nonrobust ones are shown via extensive simulation studies and the mortality rate application.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS649 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Functional dynamic factor models with application to yield curve forecasting

Accurate forecasting of zero coupon bond yields for a continuum of maturities is paramount to bond portfolio management and derivative security pricing. Yet a universal model for yield curve forecasting has been elusive, and prior attempts often resulted in a trade-off between goodness of fit and consistency with economic theory. To address this, herein we propose a novel formulation which connects the dynamic factor model (DFM) framework with concepts from functional data analysis: a DFM with functional factor loading curves. This results in a model capable of forecasting functional time series. Further, in the yield curve context we show that the model retains economic interpretation. Model estimation is achieved through an expectation-maximization algorithm, where the time series parameters and factor loading curves are simultaneously estimated in a single step. Efficient computing is implemented and a data-driven smoothing parameter is nicely incorporated. We show that our model performs very well on forecasting actual yield data compared with existing approaches, especially in regard to profit-based assessment for an innovative trading exercise. We further illustrate the viability of our model to applications outside of yield forecasting.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS551 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Existence of positive solution for second-order impulsive boundary value problems on infinity intervals

We deal with the existence of positive solutions to impulsive second-order differential equations subject to some boundary conditions on the semi-infinity interval

New comparison results for impulsive functional differential equations

AbstractComparison principles play an important role in the qualitative and quantitative study of differential equations. In this paper, we establish new maximum principles for impulsive functional differential equations

A two-way regularization method for MEG source reconstruction

The MEG inverse problem refers to the reconstruction of the neural activity of the brain from magnetoencephalography (MEG) measurements. We propose a two-way regularization (TWR) method to solve the MEG inverse problem under the assumptions that only a small number of locations in space are responsible for the measured signals (focality), and each source time course is smooth in time (smoothness). The focality and smoothness of the reconstructed signals are ensured respectively by imposing a sparsity-inducing penalty and a roughness penalty in the data fitting criterion. A two-stage algorithm is developed for fast computation, where a raw estimate of the source time course is obtained in the first stage and then refined in the second stage by the two-way regularization. The proposed method is shown to be effective on both synthetic and real-world examples.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS531 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Oscillation for solutions of nonlinear neutral differential equations with impulses

AbstractThis paper is concerned with nonlinear neutral differential equations with impulses of the form Some oscillation criteria for solutions of this equation are established. An interesting example is also given, which illustrates that impulses play an important role in giving rise to the oscillation of equations

Oscillation criteria for first-order impulsive differential equations with positive and negative coefficients

AbstractSome sufficient conditions are obtained for oscillation of all solutions of the first-order impulsive differential equation with positive and negative coefficients[x(t)-R(t)x(t-r)]′+P(t)x(t-τ)-Q(t)x(t-σ)=0,τ⩾σ>0,t⩾t0,x(tk+)=Ik(x(tk)),k=1,2,….Our results improve the known results in the literature

Oscillation of solutions of impulsive neutral difference equations with continuous variable

We obtain sufficient conditions for oscillation of all solutions of the neutral impulsive difference equation with continuous variable Δτ(y(t)+p(t)y(t−mτ))+Q(t)y(t−lτ)=0, t≥t0−τ, t≠tk, y(tk+τ)−y(tk)=bky(tk), k∈ℕ(1), where Δτ denotes the forward difference operator, that is, Δτz(t)=z(t+τ)−z(t), p(t)∈C([t0−τ,∞),ℝ), Q(t)∈C([t0−τ,∞),(0,∞)), m, l are positive integers, τ>0 and bk are constants, 0≤t0<t1<t2<⋯<tk<⋯ with limk→∞tk=∞