Parameter estimation of ODE's via nonparametric estimators

Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.Comment: Published in at http://dx.doi.org/10.1214/07-EJS132 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Appraisal Using Generalized Additive Models

Many of the results from real estate empirical studies depend upon using a correct functional form for their validity. Unfortunately, common parametric statistical tools cannot easily control for the possibility of misspecification. Recently, semiparametric estimators such as generalized additive models (GAMs) have arisen which can automatically control for additive (in price) or multiplicative (in ln(price)) nonlinear relations among the independent and dependent variables. As the paper shows, GAMs can empirically outperform naive parametric and polynomial models in ex-sample predictive behavior. Moreover, GAMs have well-developed statistical properties and can suggest useful transformations in parametric settings.

Feature Extraction in Signal Regression: A Boosting Technique for Functional Data Regression

Main objectives of feature extraction in signal regression are the improvement of accuracy of prediction on future data and identification of relevant parts of the signal. A feature extraction procedure is proposed that uses boosting techniques to select the relevant parts of the signal. The proposed blockwise boosting procedure simultaneously selects intervals in the signal’s domain and estimates the effect on the response. The blocks that are defined explicitly use the underlying metric of the signal. It is demonstrated in simulation studies and for real-world data that the proposed approach competes well with procedures like PLS, P-spline signal regression and functional data regression. The paper is a preprint of an article published in the Journal of Computational and Graphical Statistics. Please use the journal version for citation

Aggregated functional data model for Near-Infrared Spectroscopy calibration and prediction

Calibration and prediction for NIR spectroscopy data are performed based on a functional interpretation of the Beer-Lambert formula. Considering that, for each chemical sample, the resulting spectrum is a continuous curve obtained as the summation of overlapped absorption spectra from each analyte plus a Gaussian error, we assume that each individual spectrum can be expanded as a linear combination of B-splines basis. Calibration is then performed using two procedures for estimating the individual analytes curves: basis smoothing and smoothing splines. Prediction is done by minimizing the square error of prediction. To assess the variance of the predicted values, we use a leave-one-out jackknife technique. Departures from the standard error models are discussed through a simulation study, in particular, how correlated errors impact on the calibration step and consequently on the analytes' concentration prediction. Finally, the performance of our methodology is demonstrated through the analysis of two publicly available datasets.Comment: 27 pages, 7 figures, 7 table

Improving Estimation in Functional Linear Regression with Points of Impact: Insights into Google AdWords

The functional linear regression model with points of impact is a recent augmentation of the classical functional linear model with many practically important applications. In this work, however, we demonstrate that the existing data-driven procedure for estimating the parameters of this regression model can be very instable and inaccurate. The tendency to omit relevant points of impact is a particularly problematic aspect resulting in omitted-variable biases. We explain the theoretical reason for this problem and propose a new sequential estimation algorithm that leads to significantly improved estimation results. Our estimation algorithm is compared with the existing estimation procedure using an in-depth simulation study. The applicability is demonstrated using data from Google AdWords, today's most important platform for online advertisements. The \textsf{R}-package \texttt{FunRegPoI} and additional \textsf{R}-codes are provided in the online supplementary material

Sparse Model Identification and Learning for Ultra-high-dimensional Additive Partially Linear Models

The additive partially linear model (APLM) combines the flexibility of nonparametric regression with the parsimony of regression models, and has been widely used as a popular tool in multivariate nonparametric regression to alleviate the "curse of dimensionality". A natural question raised in practice is the choice of structure in the nonparametric part, that is, whether the continuous covariates enter into the model in linear or nonparametric form. In this paper, we present a comprehensive framework for simultaneous sparse model identification and learning for ultra-high-dimensional APLMs where both the linear and nonparametric components are possibly larger than the sample size. We propose a fast and efficient two-stage procedure. In the first stage, we decompose the nonparametric functions into a linear part and a nonlinear part. The nonlinear functions are approximated by constant spline bases, and a triple penalization procedure is proposed to select nonzero components using adaptive group LASSO. In the second stage, we refit data with selected covariates using higher order polynomial splines, and apply spline-backfitted local-linear smoothing to obtain asymptotic normality for the estimators. The procedure is shown to be consistent for model structure identification. It can identify zero, linear, and nonlinear components correctly and efficiently. Inference can be made on both linear coefficients and nonparametric functions. We conduct simulation studies to evaluate the performance of the method and apply the proposed method to a dataset on the Shoot Apical Meristem (SAM) of maize genotypes for illustration