# Local distance-based linear model `ldblm` is a localized version of a distance-based linear model. As in the global model `dblm`, explanatory information is coded as distances between individuals. Neighborhood definition for localizing is done by the (semi)metric `dist1` whereas a second (semi)metric `dist2` (which may coincide with `dist1`) is used for distance-based prediction. Both `dist1` and `dist2` can either be computed from observed explanatory variables or directly input as a squared distances matrix or as a `Gram` matrix. The response is a continuous variable as in the ordinary linear model. The model allows for a mixture of continuous and qualitative explanatory variables or, in fact, from more general quantities such as functional data. Notation convention: in distance-based methods we must distinguish **observed explanatory variables** which we denote by Z or z, from **Euclidean coordinates** which we denote by X or x. For explanation on the meaning of both terms see the bibliography references below. ```r ## S3 method for class 'formula' ldblm(formula,data,...,kind.of.kernel=1, metric1="euclidean",metric2=metric1,method.h="GCV",weights, user.h=NULL,h.range=NULL,noh=10,k.knn=3,rel.gvar=0.95,eff.rank=NULL) ## S3 method for class 'dist' ldblm(dist1,dist2=dist1,y,kind.of.kernel=1, method.h="GCV",weights,user.h=quantile(dist1,.25), h.range=quantile(as.matrix(dist1),c(.05,.5)),noh=10, k.knn=3,rel.gvar=0.95,eff.rank=NULL,...) ## S3 method for class 'D2' ldblm(D2.1,D2.2=D2.1,y,kind.of.kernel=1,method.h="GCV", weights,user.h=quantile(D2.1,.25)^.5, h.range=quantile(as.matrix(D2.1),c(.05,.5))^.5,noh=10,k.knn=3, rel.gvar=0.95,eff.rank=NULL,...) ## S3 method for class 'Gram' ldblm(G1,G2=G1,y,kind.of.kernel=1,method.h="GCV", weights,user.h=NULL,h.range=NULL,noh=10,k.knn=3,rel.gvar=0.95, eff.rank=NULL,...) ``` ## Arguments - `formula`: an object of class `formula`. A formula of the form `y~Z`. This argument is a remnant of the `loess` function, kept for compatibility. - `data`: an optional data frame containing the variables in the model (both response and explanatory variables, either the observed ones, Z, or a Euclidean configuration X). - `y`: (required if no formula is given as the principal argument). Response (dependent variable) must be numeric, matrix or data.frame. - `dist1`: a `dist` or `dissimilarity` class object. Distances between observations, used for neighborhood localizing definition. Weights for observations are computed as a decreasing function of their `dist1` distances to the neighborhood center, e.g. a new observation whose reoponse has to be predicted. These weights are then entered to a `dblm`, where distances are evaluated with `dist2`. - `dist2`: a `dist` or `dissimilarity` class object. Distances between observations, used for fitting `dblm`. Default `dist2=dist1`. - `D2.1`: a `D2` class object. Squared distances matrix between individuals. One of the alternative ways of entering distance information to a function. See the Details section in `dblm`. See above `dist1` for explanation of its role in this function. - `D2.2`: a `D2` class object. Squared distances between observations. One of the alternative ways of entering distance information to a function. See the Details section in `dblm`. See above `dist2` for explanation of its role in this function. Default `D2.2=D2.1`. - `G1`: a `Gram` class object. Doubly centered inner product matrix associated with the squared distances matrix `D2.1`. - `G2`: a `Gram` class object. Doubly centered inner product matrix associated with the squared distances matrix `D2.2`. Default `G2=G1` - `kind.of.kernel`: integer number between 1 and 6 which determines the user's choice of smoothing kernel. (1) Epanechnikov (Default), (2) Biweight, (3) Triweight, (4) Normal, (5) Triangular, (6) Uniform. - `metric1`: metric function to be used when computing `dist1` from observed explanatory variables. One of `"euclidean"` (default), `"manhattan"`, or `"gower"`. - `metric2`: metric function to be used when computing `dist2` from observed explanatory variables. One of `"euclidean"` (default), `"manhattan"`, or `"gower"`. - `method.h`: sets the method to be used in deciding the **optimal bandwidth h**. There are five different methods, `AIC`, `BIC`, `OCV`, `GCV` (default) and `user.h`. `OCV` and `GCV` take the optimal bandwidth minimizing a cross-validatory quantity (either `ocv` or `gcv`). `AIC` and `BIC` take the optimal bandwidth minimizing, respectively, the Akaike or Bayesian Information Criterion (see `AIC` for more details). When `method.h` is `user.h`, the bandwidth is explicitly set by the user through the `user.h` optional parameter which, in this case, becomes mandatory. - `weights`: an optional numeric vector of weights to be used in the fitting process. By default all individuals have the same weight. - `user.h`: global bandwidth `user.h`, set by the user, controlling the size of the local neighborhood of Z. Smoothing parameter (Default: 1st quartile of all the distances d(i,j) in `dist1`). Applies only if `method.h="user.h"`. - `h.range`: a vector of length 2 giving the range for automatic bandwidth choice. (Default: quantiles 0.05 and 0.5 of d(i,j) in `dist1`). - `noh`: number of bandwidth `h` values within `h.range` for automatic bandwidth choice (if `method.h!="user.h"`). - `k.knn`: minimum number of observations with positive weight in neighborhood localizing. To avoid runtime errors due to a too small bandwidth originating neighborhoods with only one observation. By default `k.nn=3`. - `rel.gvar`: relative geometric variability (a real number between 0 and 1). In each `dblm` iteration, take the lowest effective rank, with a relative geometric variability higher or equal to `rel.gvar`. Default value (`rel.gvar=0.95`) uses the 95% of the total variability. - `eff.rank`: integer between 1 and the number of observations minus one. Number of Euclidean coordinates used for model fitting in each `dblm` iteration. If specified its value overrides `rel.gvar`. When `eff.rank=NULL` (default), calls to `dblm` are made with `method=rel.gvar`. - ``...``: arguments passed to or from other methods to the low level. ## Details There are two semi-metrics involved in local linear distance-based estimation: `dist1` and `dist2`. Both semi-metrics can coincide. For instance, when `dist1=||xi-xj||` and `dist2=||(xi,xi^2,xi^3)-(xj,xj^2,xj^3)||` the estimator for new observations coincides with fitting a local cubic polynomial regression. The set of bandwidth `h` values checked in automatic bandwidth choice is defined by `h.range` and `noh`, together with `k.knn`. For each `h` in it a local linear model is fitted and the optimal `h` is decided according to the statistic specified in `method.h`. `kind.of.kernel` designates which kernel function is to be used in determining individual weights from `dist1` values. See `density` for more information. ## Returns A list of class `ldblm` containing the following components: - **residuals**: the residuals (response minus fitted values). - **fitted.values**: the fitted mean values. - **h.opt**: the optimal bandwidth h used in the fitting proces (`if method.h!=user.h`). - **S**: the Smoother hat projector. - **weights**: the specified weights. - **y**: the response variable used. - **call**: the matched call. - **dist1**: the distance matrix (object of class `"D2"` or `"dist"`) used to calculate the weights of the observations. - **dist2**: the distance matrix (object of class `"D2"` or `"dist"`) used to fit the `dblm`. ## References Boj E, Caballe, A., Delicado P, Esteve, A., Fortiana J (2016). **Global and local distance-based generalized linear models**. TEST 25, 170-195. Boj E, Delicado P, Fortiana J (2010). **Distance-based local linear regression for functional predictors**. Computational Statistics and Data Analysis 54, 429-437. Boj E, Grane A, Fortiana J, Claramunt MM (2007). **Selection of predictors in distance-based regression**. Communications in Statistics B - Simulation and Computation 36, 87-98. Cuadras CM, Arenas C, Fortiana J (1996). **Some computational aspects of a distance-based model for prediction**. Communications in Statistics B - Simulation and Computation 25, 593-609. Cuadras C, Arenas C (1990). **A distance-based regression model for prediction with mixed data**. Communications in Statistics A - Theory and Methods 19, 2261-2279. Cuadras CM (1989). **Distance analysis in discrimination and classification using both continuous and categorical variables**. In: Y. Dodge (ed.), **Statistical Data Analysis and Inference**. Amsterdam, The Netherlands: North-Holland Publishing Co., pp. 459-473. ## Author(s) Boj, Eva , Caballe, Adria , Delicado, Pedro and Fortiana, Josep ## Note Model fitting is repeated `n` times (`n=` number of observations) for each bandwidth (`noh*n` times). For a `noh` too large or a sample with many observations, the time of this function can be very high. ## See Also `dblm` for distance-based linear models. `ldbglm` for local distance-based generalized linear models. `summary.ldblm` for summary. `plot.ldblm` for plots. `predict.ldblm` for predictions. ## Examples ```r # example to use of the ldblm function n <- 100 p <- 1 k <- 5 Z <- matrix(rnorm(n*p),nrow=n) b1 <- matrix(runif(p)*k,nrow=p) b2 <- matrix(runif(p)*k,nrow=p) b3 <- matrix(runif(p)*k,nrow=p) s <- 1 e <- rnorm(n)*s y <- Z%*%b1 + Z^2%*%b2 +Z^3%*%b3 + e D2 <- as.matrix(dist(Z)^2) class(D2) <- "D2" ldblm1 <- ldblm(y~Z,kind.of.kernel=1,method="GCV",noh=3,k.knn=3) ldblm2 <- ldblm(D2.1=D2,D2.2=D2,y,kind.of.kernel=1,method="user.h",k.knn=3) ```