ldblm {dbstats}

Local distance-based linear model

Description

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.

Usage

## 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.

Value

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.

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.

Author(s)

Boj, Eva <[e-mail]>, Caballe, Adria <[e-mail]>, Delicado, Pedro <[e-mail]> and Fortiana, Josep <[e-mail]>

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.

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

# 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)