--- title: 'Introduction to the npsp Package' author: 'Ruben Fernandez-Casal (ruben.fcasal@udc.es)' date: '`r paste("npsp", packageVersion("npsp"))`' output: html_document: toc: yes toc_depth: 2 rmarkdown::html_vignette: toc: yes toc_depth: 2 vignette: > %\VignetteIndexEntry{Introduction to the npsp Package} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include=FALSE} knitr::opts_chunk$set(cache = FALSE, fig.height=4, fig.width=7, fig.align = 'center', out.width = '100%') # knitr::spin("npsp_intro2.R", knit = FALSE) ``` This vignette (a working draft) tries to illustrate the use of the `npsp` (Nonparametric spatial statistics) package. This package implements nonparametric methods for inference on multidimensional geostatistical processes, avoiding the misspecification problems that may arise when using parametric models. The spatial process can be either stationary or show a non-constant trend. Joint estimation of the trend and the semivariogram can be performed automatically, by using the function `np.fitgeo()`, or by a step-by-step approach. # Introduction We will assume that $\left\{ Y(\mathbf{x}),\mathbf{x}\in D \subset \mathbb{R}^{d}\right\}$ is a spatial process that can be modeled as: $$Y(\mathbf{x})=\mu(\mathbf{x})+\varepsilon(\mathbf{x}),$$ where $\mu(\cdot)$ is the trend function (large-scale variation) and the error term $\varepsilon$ (small-scale variation), is a second order stationary process with zero mean and covariogram $C(\mathbf{u}) = Cov(\varepsilon \left( \mathbf{x}\right) ,\varepsilon \left(\mathbf{x}+\mathbf{u}\right) )$, with $\mathbf{u} \in D$. In this framework, given $n$ observed values $\mathbf{Y}=( Y(\mathbf{x}_{1}),...,Y(\mathbf{x}_{n}))^{t}$, the first step consists in estimate the trend $\mu(\mathbf{x})$ and the semivariogram $\gamma(\mathbf{h}) = C(\mathbf{0}) - C(\mathbf{u})$. There are several `R` packages, such as [`gstat`](https://r-spatial.github.io/gstat/) or [`geoR`](http://www.leg.ufpr.br/geoR/), which implement the traditional geostatistical techniques to approximate these functions. Nevertheless, as these approaches usually assume parametric models, they can present misspecification problems. The aim of the `npsp` package is to provide nonparametric tools for geostatistical modeling and interpolation, under the general spatial model (large + small variation scales), and without assuming any specific form (parametric model) for the trend and the variogram of the process. ```{r } library(npsp) ``` The `precipitation` data set (`sp::SpatialGridDataFrame` class object), supplied with the `npsp` package, will be used in the examples in this work. The data consist of total precipitations (square-root of rainfall inches) during March 2016 recorded over 1053 locations on the continental part of USA. ```{r } library(sp) summary(precipitation) ``` For instance, `spoints()` function may be used to plot the data... ```{r } spoints(precipitation) ``` Function `scattersplot()` may be used to detect the presence of a non-constant trend... ```{r scattersplot, fig.height=9, fig.width=9} scattersplot(precipitation) ``` # Trend estimation The local polynomial trend estimator $\hat{\mu}_{\mathbf{H}}(\mathbf{x})$ (e.g. Fan and Gijbels, 1996), obtained by polynomial smoothing of $\{(\mathbf{x}_{i},Y(\mathbf{x}_{i})):i=1,\ldots,n\}$, is the solution for $\beta_0$ to the least squares minimization problem $$\begin{aligned} \min_{\beta_0 ,\boldsymbol{\beta}_1, \ldots, \boldsymbol{\beta}_p} \sum_{i=1}^{n}\left\{ Y(\mathbf{x}_{i})-\beta_0 -\boldsymbol{\beta}_1^{t}(\mathbf{x}_{i}-\mathbf{x}) - \ldots \right. \nonumber \\ \left. -\boldsymbol{\beta}_p^{t}(\mathbf{x}_{i}-\mathbf{x})^p \right\}^{2} K_{\mathbf{H}}(\mathbf{x}_{i}-\mathbf{x}), \end{aligned}$$ where $\mathbf{H}$ is a $d\times d$ symmetric positive definite matrix; $K$ is a $d$-dimensional kernel and $K_{\mathbf{H}}(\mathbf{u})=\left\vert \mathbf{H}\right\vert ^{-1}K(\mathbf{H}^{-1}\mathbf{u})$. The bandwidth matrix $\mathbf{H}$ controls the shape and size of the local neighborhood used to estimate $\mu(\mathbf{x})$. The local trend estimates can be computed in practice with the `locpol()` function. A full bandwidth matrix and a multiplicative triweight kernel is used to compute the weights. Main calculations are performed in Fortran using the LAPACK library. ```{r } x <- coordinates(precipitation) y <- precipitation$y lp <- locpol(x, y, nbin = c(120, 120), h = diag(c(5, 5))) ``` We can plot the trend estimates with function `simage()` (or `spersp()`). Additionally, in this case the estimates outside of continental part of USA will be masked. ```{r } attr <- attributes(precipitation) border <- attr$border # Mask grid points outside of continental part of USA lp <- mask(lp, window = border) interior <- attr$interior slim <- range(y) col <- jet.colors(256) simage(lp, slim = slim, main = "Trend estimates", col = col, xlab = "Longitude", ylab = "Latitude", asp = 1) plot(border, border = "darkgray", lwd = 2, add = TRUE) plot(interior, border = "lightgray", lwd = 1, add = TRUE) ``` The smoothing procedures in `npsp` use linear binning to discretize the data. For instance, instead of the previous code we can use: ```{r } bin <- binning(x, y, nbin = c(120, 120), window = border) # lp <- locpol(bin, h = diag(c(5, 5))) lp2 <- locpol(bin, h = diag(c(2, 2))) simage(lp2, slim = slim, main = "Trend estimates", col = col, xlab = "Longitude", ylab = "Latitude", asp = 1) plot(border, border = "darkgray", lwd = 2, add = TRUE) plot(interior, border = "lightgray", lwd = 1, add = TRUE) ``` Usually two grids will be considered, a low resolution one to speed up computations during the modeling (e.g. by using the default values), and a higher resolution one to obtain the final results. # Bandwidth selection The trend estimates depends crucially on the bandwidth matrix $H$. A small bandwidth will produce undersmoothed estimates, whereas a big bandwidth will oversmooth the data. Although it may be subjectively choosed by eye, it could be very beneficial to have automatic selection criteria from the data. Traditional bandwidth selectors, such as cross validation (CV) or generalized cross validation (GCV), do not have a good performance for dependent data (Opsomer et al, 2001), since they tend to undersmooth the trend function. The modified cross-validation criteria (MCV), proposed in (Chu and Marron, 1991) for time series, can be adapted for spatial data, by ignoring observations in a neighbourhood $N(i)$ around $\mathbf{x}_{i}$. Note that the ordinary CV approach is a particular case with $N(i)=\left\lbrace \mathbf{x}_{i} \right\rbrace$. In practice, the bandwidth can be selected through the `h.cv()` (or `hcv.data()`) function. ```{r } bin <- binning(x, y, window = border) lp0.h <- h.cv(bin)$h lp0 <- locpol(bin, h = lp0.h, hat.bin = TRUE) ``` An alternative is the corrected generalized cross-validation criterion (CGCV), proposed in Francisco-Fernández and Opsomer (2005), that takes the spatial dependence into account. To use this approach in practice (by setting `objective = "GCV"` and `cov.bin` in `h.cv`), an estimate of the variogram is previously required. # Variogram estimation When the process is assumed stationary, the pilot local linear estimate $\hat\gamma(\mathbf{u})$ is obtained by the linear smoothing of $\left(\mathbf{x}_{i}-\mathbf{x}_{j},(Y(\mathbf{x}_i)- Y(\mathbf{x}_j))^2/2 \right)$. The corresponding bandwidth can be selected, for instance, by minimizing the cross-validation relative squared error. In the general setting with a non constant trend, the natural approach consists in removing the trend and estimating the variogram from the residuals $\mathbf{r}=\mathbf{Y}-\mathbf{SY}$. Nevertheless, the residuals variability may be very different to that of the true errors (see e.g. Cressie, 1993, Section 3.4.3, for the case of the linear trend estimator). As the bias due to the direct use of residuals in variogram estimation may have a significant impact on inference, a similar approach to that described in Fernández-Casal and Francisco-Fernández (2014) could be considered. In practice, the local linear variogram estimates can be computed with functions `np.svar()` and `np.svariso.corr()`. The generic function `h.cv()` may be used to select the corresponding bandwidth. ```{r } svar.bin <- svariso(x, residuals(lp0), nlags = 60, maxlag = 20) svar.h <- h.cv(svar.bin)$h svar.np <- np.svar(svar.bin, h = svar.h) svar.np2 <- np.svariso.corr(lp0, nlags = 60, maxlag = 20, h = svar.h, plot = FALSE) ``` The final variogram estimate is obtained by fitting a "nonparametric" isotropic Shapiro-Botha variogram model (Shapiro and Botha, 1991), to the nonparametric pilot estimate, by using the function `fitsvar.sb.iso()`. ```{r } svm0 <- fitsvar.sb.iso(svar.np, dk = 0) svm1 <- fitsvar.sb.iso(svar.np2, dk = 0) # plot... plot(svm1, main = "Nonparametric bias-corrected semivariogram\nand fitted models", legend = FALSE, xlim = c(0,max(coords(svar.np2))), ylim = c(0,max(svar.np2$biny, na.rm = TRUE))) plot(svm0, lwd = c(1, 1), add = TRUE) plot(svar.np, type = "p", pch = 2, add = TRUE) # abline(h = c(svm1$nugget, svm1$sill), lty = 3) # abline(v = 0, lty = 3) legend("bottomright", legend = c("corrected", 'biased'), lty = c(1, 1), pch = c(1, 2), lwd = c(1, 1)) ``` # Automatic modeling Function `np.fitgeo()` implements an automatic procedure for the joint estimation of the trend and the semivariogram. The algorithm is as follows: 1. Select an initial bandwidth to estimate the trend using the MCV criterion (function `h.cv()`). 2. Compute the local linear trend estimate (function `locpol()`). 3. Obtain a bias-corrected pilot semivariogram estimate from the corresponding residuals (function `np.svariso.corr()`). 4. Fit a valid Shapiro-Botha model to the pilot semivariogram estimates (function `fitsvar.sb.iso()`). 5. Select a new bandwidth matrix for trend estimation with the CGCV criterion, using the fitted variogram model to approximate the data covariance matrix (function `h.cv()` with `objective = "GCV"` and `cov.bin =` fitted variogram model). 6. Repeat steps 2-5 until convergence (in practice, only two iterations are usually needed). ```{r geomod, fig.height=4, fig.width=12} geomod <- np.fitgeo(x, y, nbin = c(30, 30), maxlag = 20, svm.resid = TRUE, window = border) plot(geomod) ``` Setting `corr.svar = TRUE` may be very slow (and memory demanding) when the number of data is large (note also that the bias in the residual variogram decreases when the sample size increases). # Kriging The final trend and variogram estimates could be employed for spatial prediction, by using method `np.kriging()`. ```{r kriging, fig.height=4, fig.width=12} krig.grid <- np.kriging(geomod, ngrid = c(120, 120)) # Plot kriging predictions and kriging standard deviations old.par <- par(mfrow = c(1,2), omd = c(0.0, 0.98, 0.0, 1)) simage(krig.grid, 'kpred', main = 'Kriging predictions', slim = slim, xlab = "Longitude", ylab = "Latitude" , col = col, asp = 1, reset = FALSE) plot(border, border = "darkgray", lwd = 2, add = TRUE) plot(interior, border = "lightgray", lwd = 1, add = TRUE) simage(krig.grid, 'ksd', main = 'Kriging sd', xlab = "Longitude", ylab = "Latitude" , col = hot.colors(256), asp = 1, reset = FALSE) plot(border, border = "darkgray", lwd = 2, add = TRUE) plot(interior, border = "lightgray", lwd = 1, add = TRUE) par(old.par) ``` Currently, only global residual kriging is implemented. Users are encouraged to use `gstat::krige()` (or `krige.cv()`) together with `as.vgm()` for local kriging (see e.g. https://rubenfcasal.github.io/npsp/articles/krigstat.html). # Conclusions and future work Unlike traditional geostatistical methods, the nonparametric procedures avoid problems due to model misspecification. The bias due to the direct use of residuals, may have a significant impact on variogram estimation (and consequently on the selected bandwidth with the CGCV criterion). Therefore, the use of the bias-corrected nonparametric variogram estimator would be recommended. Currently, only isotropic semivariogram estimation is supported, but the intention is to extend this approach to anisotropy in two components (Fernández-Casal et al., 2003), which could be suitable for modeling spatio-temporal dependency. The nonparametric trend and variogram estimates also allow to perform inferences about other characteristics of interest of the process, for instance, by using the bootstrap algorithm described in (Castillo-Páez et al., 2019), which will be implemented in the next version of the package (https://github.com/rubenfcasal/npsp). # Acknowledgments This research has been supported by MINECO grant MTM2017-82724-R, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2020-14 and Centro de Investigación del Sistema universitario de Galicia ED431G 2019/01), all of them through the ERDF. # References Castillo-Páez, S., Fernández-Casal, R., García-Soidán, P. (2019). A nonparametric bootstrap method for spatial data. *Comput. Stat. Data An.*, **137**, 1--15. Chu, C. K. and Marron, J. S. (1991). Comparison of Two Bandwidth Selectors with Dependent Errors. *The Annals of Statistics* **19**, 1906--1918. Cressie, N. (1993). *Statistics for Spatial Data*. Wiley, New York. 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