Global and local distance-based generalized linear models

Abstract

This paper introduces local distance-based generalized linear models. These models extend (weighted) distance-based linear models first to the generalized linear model framework. Then, a nonparametric version of these models is proposed by means of local fitting. Distances between individuals are the only predictor information needed to fit these models. Therefore, they are applicable, among others, to mixed (qualitative and quantitative) explanatory variables or when the regressor is of functional type. An implementation is provided by the R package dbstats, which also implements other distance-based prediction methods. Supplementary material for this article is available online, which reproduces all the results of this article.

Appendices

Appendix A: The dbstats package

The dbstats package (Boj et al. 2014) for R (R Development Core Team 2015) implements several distance-based prediction methods. The main functions of dbstats are: dblm for DB-LM, ldblm for local DB-LM, dbglm for DB-GLM, ldbglm for local DB-GLM and dbplsr for DB-PLSR.

In Sect. A.1, we describe the usage of function dbglm whereas ldbglm is described in Sect. A.2. Two examples illustrating the usage of these functions from a user perspective have been presented, respectively, in Sects. 3.1 and 4.1. For details of dblm, ldblm and plsr, we refer to Boj et al. (2014).

1.1 A.1 Function dbglm

Function dbglm fits DB-GLM. In this function, distances can be directly provided as: an interdistances matrix (class dist or dissimilarity as in stats package); a squared interdistances matrix (class D2); or an inner-products matrix (class Gram). Classes D2 and Gram have been implemented in the dbstats package. It is also possible to compute distances directly from observed explanatory variables (using a class formula object in the call to dbglm).

The dbstats package does not provide specific methods for computing distances, depending instead on other available functions and packages such as dist in the stats package, daisy in the cluster package (Maechler 2015) or dist in the proxy package (Meyer and Buchta 2015). Utility functions such as as.D2, as.Gram, D2toDist, D2toG, distoD2 and GtoD2 allow the user mutual interconversions (see Boj et al. 2014 for details).

Response and link function are as in the glm function of stats for ordinary GLM.

The usage of dbglm is:

where the argument distance is of class dist or dissimilarity. The same information can be provided replacing distance by an object of class either D2 or Gram.

When calling dbglm using an object of class formula, the first and second arguments in the previous call to dbglm are replaced by other three arguments: formula (of the form ${\varvec{y}}\sim {\varvec{Z}}$), data (a data frame containing the variables in the model: both response ${\varvec{y}}$ and explanatory variables ${\varvec{Z}}$) and metric (that indicates how to compute distances between the rows of ${\varvec{Z}}$; it must be one of the strings “euclidean” (the default), “manhattan” or “gower” to be passed to function daisy of cluster package).

In addition to the response y, the distance matrix distance (or equivalent information), the formula formula, data data and metric, it is worth mentioning the following arguments of dbglm:

• family, weights, offset, mustart are arguments with the same role that they have in the glm function.

• method sets the method to be used in deciding the effective rank. There are five different methods: “AIC”, “BIC”, “GCV” (default), “eff.rank” and “rel.gvar”. See Sect. 3 (before Sect. A.1) for details on these criteria.

• range.eff.rank a vector defining the range of possible values for the effective rank in the dblm iterations to be evaluated when method is “AIC”, “BIC” or “GCV”. It should be restricted between $c(1,n-1)$.

• full.search sets the optimization procedure to be used to minimize the modeling criterion specified in method when “AIC”, “BIC” or “GCV” criteria are specified. See the help of dbstats package Boj et al. (2014) for details.

• rel.gvar relative geometric variability (a real number between 0 and 1; default is 0.95). More details can be found at the end of Sect. 3 (before Sect. A.1).

• eff.rank integer between 1 and $n-1$. If specified its value overrides rel.gvar. When eff.rank = NULL (default), calls to dblm are made with method = “rel.gvar”. More details can be found in Sect. 3 (before Sect. 3.1).

• maxiter, eps1, eps2 are stopping criteria for the iterative algorithm that fits the DB-GLM.

The function returns a list of class dbglm containing the following components:

• Common elements with the output of glm function for R: residuals, fitted.values, family, deviance, aic.model, null.deviance, iter, prior.weights, weights, df.residual, df.null, y, call.

• H hat matrix projector of the last dblm iteration.

• convcrit convergence criterion. One of: “DevStat” (stopping criterion 1: when the relative decrement of deviance in one step is less than eps1), “muStat” (stopping criterion 2: when the relative change of the estimated expected values of the responses in one step is less than eps2), “maxiter” (maximum allowed number of iterations has been exceeded).

• eff.rank, rel.gvar effective rank and relative geometric variability that have been finally used.

• bic.model, gcv.model BIC and GCV criteria of the final DB-GLM.

• dev.resids deviance residuals (the way they are computed depends on the specified family).

• varmu vector of estimated variance of each observation (that depends on the estimated vector of expected values and on the specified family).

1.2 A.2 Function ldbglm

Function ldbglm is a localized version of a DB-GLM. As in the global model dbglm, explanatory information is coded as distances between individuals, that can either be computed from observed explanatory variables or directly provided to function ldbglm as a (possibly squared) interdistances matrix or as a inner-products matrix (a Gram matrix).

Remember that in local DB-GLM there appear two distance functions, $\delta _1$ and $\delta _2$, playing different roles. Accordingly, function ldbglm has two different arguments, dist1 and dist2 of class dist or dissimilarity, where distances $\delta _1$ and $\delta _2$ are specified: dist1 defines the neighborhood delimiting what observations (and with what weight) are used when locally fitting a DB-GLM, whereas dist2 (which may coincide with dist1) is used specifically for fitting this weighted DB-GLM.

The usage of ldbglm is:

In the same way that it was explained in the overview of function dbglm, the predictive information contained in distance matrices dist1 and dist2 can be provided to function ldbglm in three alternative ways: two squared distances matrices (D2.1 and D2.2), two inner-products matrices (G1 and G2), or a formula jointly with a data set and two metrics (formula, data, metric1 and metric2).

The following are other arguments of ldbglm that are specific of this function because they control its local character:

• kind.of.kernel integer number between 1 and 6 which determines the user’s choice of smoothing kernel K (see Eq. 14): (1) Epanechnikov (Default), (2) Biweight, (3) Triweight, (4) Normal, (5) Triangular, (6) Uniform.

• method.h sets the method to be used when choosing the bandwidth h to be used in Eq. (14). There are four different methods, AIC, BIC, GCV (default) and user.h. AIC, BIC and GCV take the bandwidth minimizing the Akaike or Bayesian Information Criterion or the generalized cross-validation, respectively. When method is user.h, the bandwidth is explicitly set by the user through the user.h optional parameter which, in this case, becomes mandatory.

• user.h global bandwidth set by the user. The default value is the first quartile of all the distances d(i, j) in matrix dist1. It applies only if method = "user.h".

• h.range a vector of length 2 giving the range for automatic bandwidth choice. (Default value: quantiles 0.05 and 0.5 of d(i, j) in matrix dist1). It applies when method != "user.h".

• noh number of bandwidth h values within h.range for automatic bandwidth choice. It applies when method != "user.h".

• k.knn minimum number of observations with positive weight in any local fit of a DB-GLM model. A too small value of bandwidth h could originate a neighborhood with only one observation producing a runtime error when trying to fit a local fit of a DB-GLM model. Choosing k.knn > 1 prevents from this problem. By default k.knn = 3.

The function returns a list of class ldbglm containing the following components:

• Common elements with the output of dbglm: residuals, fitted.values, family, weights, y, call.

• dist1, dist2 the distances matrices used to calculate the local weights of the observations and to locally fit the dbglm’s, respectively.

• h.opt the optimal bandwidth h used in the fitting process (if method != user.h).

• S the smoothing matrix in the last iteration of the IRWLS. See Boj et al. (2010) for details on the definition of the smoothing matrix.

Appendix B: Supplementary material

Code for fitting DB-GLM and local DB-GLM is available in the R package dbstats (http://CRAN.R-project.org/package=dbstats). Additional code for reproducing the computations and graphics in the paper are included in the R script ExamplesDB.R. The data files protein.asc and whtspec.asc are provided as supplementary material with the permission of John H. Kalivas, Idaho Satate University.