Local linear spatial regression

A local linear kernel estimator of the regression function x\mapsto g(x):=E[Y_i|X_i=x], x\in R^d, of a stationary (d+1)-dimensional spatial process {(Y_i,X_i),i\in Z^N} observed over a rectangular domain of the form I_n:={i=(i_1,...,i_N)\in Z^N| 1\leq i_k\leq n_k,k=1,...,N}, n=(n_1,...,n_N)\in Z^N, is proposed and investigated. Under mild regularity assumptions, asymptotic normality of the estimators of g(x) and its derivatives is established. Appropriate choices of the bandwidths are proposed. The spatial process is assumed to satisfy some very general mixing conditions, generalizing classical time-series strong mixing concepts. The size of the rectangular domain I_n is allowed to tend to infinity at different rates depending on the direction in Z^N.Comment: Published at http://dx.doi.org/10.1214/009053604000000850 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local linear spatial quantile regression

Copyright @ 2009 International Statistical Institute / Bernoulli Society for Mathematical Statistics and Probability.Let {(Yi,Xi), i ∈ ZN} be a stationary real-valued (d + 1)-dimensional spatial processes. Denote by x → qp(x), p ∈ (0, 1), x ∈ Rd , the spatial quantile regression function of order p, characterized by P{Yi ≤ qp(x)|Xi = x} = p. Assume that the process has been observed over an N-dimensional rectangular domain of the form In := {i = (i1, . . . , iN) ∈ ZN|1 ≤ ik ≤ nk, k = 1, . . . , N}, with n = (n1, . . . , nN) ∈ ZN. We propose a local linear estimator of qp. That estimator extends to random fields with unspecified and possibly highly complex spatial dependence structure, the quantile regression methods considered in the context of independent samples or time series. Under mild regularity assumptions, we obtain a Bahadur representation for the estimators of qp and its first-order derivatives, from which we establish consistency and asymptotic normality. The spatial process is assumed to satisfy general mixing conditions, generalizing classical time series mixing concepts. The size of the rectangular domain In is allowed to tend to infinity at different rates depending on the direction in ZN (non-isotropic asymptotics). The method provides muchAustralian Research Counci

Spatial Quantile Regression

In a number of applications, a crucial problem consists in describing and analyzing the influence of a vector Xi of covariates on some real-valued response variable Yi. In the present context, where the observations are made over a collection of sites, this study is more difficult, due to the complexity of the possible spatial dependence among the various sites. In this paper, instead of spatial mean regression, we thus consider the spatial quantile regression functions. Quantile regression has been considered in a spatial context. The main aim of this paper is to incorporate quantile regression and spatial econometric modeling. Substantial variation exists across quantiles, suggesting that ordinary regression is insufficient on its own. Quantile estimates of a spatial-lag model show considerable spatial dependence in the different parts of the distribution.W wielu zastosowaniach, podstawowym problemem jest opis i analiza wpływu wektora skorelowanych zmiennych objaśniających X na zmienna objaśnianą Y. W przypadku, gdy obserwacje badanych zmiennych są dodatkowo rozmieszczone przestrzennie, zadanie jest jeszcze trudniejsze, ponieważ mamy dodatkowe zależności, wynikające ze zmienności przestrzennej. W tej pracy, w miejsce przestrzennej regresji wykorzystującej średnią, rozpatrzymy przestrzenna regresję kwantylową. Regresja kwantylowa zostanie omówiona w przestrzennym kontekście. Głównym celem pracy jest wskazanie na możliwości powiązania metodologii regresji kwantylowej i ekonometrycznego modelowania przestrzennego. Dodatkowe zasoby informacji o zmienności otrzymujemy badając kwantyle, wychodząc poza tradycyjny opis klasycznej regresji. Estymacja kwantylowa w modelu przestrzennym uwydatnia zależności przestrzenne dla różnych fragmentów rozważanych rozkładów

Spatial adaptation in heteroscedastic regression: Propagation approach

The paper concerns the problem of pointwise adaptive estimation in regression when the noise is heteroscedastic and incorrectly known. The use of the local approximation method, which includes the local polynomial smoothing as a particular case, leads to a finite family of estimators corresponding to different degrees of smoothing. Data-driven choice of localization degree in this case can be understood as the problem of selection from this family. This task can be performed by a suggested in Katkovnik and Spokoiny (2008) FLL technique based on Lepski's method. An important issue with this type of procedures - the choice of certain tuning parameters - was addressed in Spokoiny and Vial (2009). The authors called their approach to the parameter calibration "propagation". In the present paper the propagation approach is developed and justified for the heteroscedastic case in presence of the noise misspecification. Our analysis shows that the adaptive procedure allows a misspecification of the covariance matrix with a relative error of order 1/log(n), where n is the sample size.Comment: 47 pages. This is the final version of the paper published in at http://dx.doi.org/10.1214/08-EJS180 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Age Progression and Regression with Spatial Attention Modules

Age progression and regression refers to aesthetically render-ing a given face image to present effects of face aging and rejuvenation, respectively. Although numerous studies have been conducted in this topic, there are two major problems: 1) multiple models are usually trained to simulate different age mappings, and 2) the photo-realism of generated face images is heavily influenced by the variation of training images in terms of pose, illumination, and background. To address these issues, in this paper, we propose a framework based on conditional Generative Adversarial Networks (cGANs) to achieve age progression and regression simultaneously. Particularly, since face aging and rejuvenation are largely different in terms of image translation patterns, we model these two processes using two separate generators, each dedicated to one age changing process. In addition, we exploit spatial attention mechanisms to limit image modifications to regions closely related to age changes, so that images with high visual fidelity could be synthesized for in-the-wild cases. Experiments on multiple datasets demonstrate the ability of our model in synthesizing lifelike face images at desired ages with personalized features well preserved, and keeping age-irrelevant regions unchanged

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al., 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for connectome reconstruction problem in very high dimensions. The algorithm approximates the solution by a sequence of rank-one updates which exploit the sparse and positive definite problem structure. This algorithm was described previously (Kressner and Sirkovi\'c, 2015) but never implemented for this connectome problem, leading to a number of challenges. We have had to design judicious stopping criteria and employ efficient solvers for the three main sub-problems of the algorithm, including an efficient GPU implementation that alleviates the main bottleneck for large datasets. The performance of the method is evaluated on three examples: an artificial "toy" dataset and two whole-cortex instances using data from the Allen Mouse Brain Connectivity Atlas. We find that the method is significantly faster than previous methods and that moderate ranks offer good approximation. This speedup allows for the estimation of increasingly large-scale connectomes across taxa as these data become available from tracing experiments. The data and code are available online

A Test Strategy for Spurious Spatial Regression, Spatial Nonstationarity, and Spatial Cointegration

A test strategy consisting of a twofold application of a Lagrange Multiplier test is suggested as a device to reveal spatial nonstationarity and spurious spatial regeression. It is further illustrated how the test strategy can be used as a diagnostic for presence of a spatial cointegrating relationship between two variables. Using Monte Carlo simulations it is shown that the small sample behaviour of the test strategy is as desired in these cases.

Choosing the Right Spatial Weighting Matrix in a Quantile Regression Model

This paper proposes computationally tractable methods for selecting the appropriate spatial weighting matrix in the context of a spatial quantile regression model. This selection is a notoriously difficult problem even in linear spatial models and is even more difficult in a quantile regression setup. The proposal is illustrated by an empirical example and manages to produce tractable models. One important feature of the proposed methodology is that by allowing different degrees and forms of spatial dependence across quantiles it further relaxes the usual quantile restriction attributable to the linear quantile regression. In this way we can obtain a more robust, with regard to potential functional misspecification, model, but nevertheless preserve the parametric rate of convergence and the established inferential apparatus associated with the linear quantile regression approach