Package 'lazy'

A formula specifying the response and some numeric predictors.

An optional data frame within which to look first for the response, predictors, and weights (the latter will be ignored).

Optional weights for each case (ignored).

An optional specification of a subset of the data to be used.

The action to be taken with missing values in the response or predictors. The default is to stop.

Control parameters: see lazy.control

Control parameters can also be supplied directly.

Parameter controlling the number of neighbors to be used for identifying and validating constant models. conIdPar can assume different forms:

conIdPar=c(idm0,idM0,valM0):

In this case, idm0:idM0 is the range in which the best number of neighbors is searched when identifying the local polynomial models of degree 0 and where valM0 is the maximum number of neighbors used for their validation. This means that the constant models identified with k neighbors, are validated on the first v neighbors, where v=min(k,valM0). If valM0=0, valM0 is set to idMO: see next case for details.

conIdPar=c(idm0,idM0):

Here idm0 and idM0 have the same role as in previous case, and valM0 is by default set to idM0: each model is validated on all the neighbors used in identification.

conIdPar=p:

Here idmO and idMO are obtained according to the following formulas: idm0=3 and idMX=5*p. Recommended choice: p=1. As far as the quantity valM0 is concerned, it gets the default value as in previous case.

conIdPar=NULL:

No constant model is considered.

Parameter controlling the number of neighbors to be used for identifying and validating linear models. linIdPar can assume different forms:

linIdPar=c(idm1,idM1,valM1):

In this case, idm1:idM1 is the range in which the best number of neighbors is searched when identifying the local polynomial models of degree 1 and where valM1 is the maximum number of neighbors used for their validation. This means that the linear models identified with k neighbors, are validated on the first v neighbors, where v=min(k,valM1). If valM1=0, valM1 is set to idM1: see next case for details.

linIdPar=c(idm1,idM1):

Here idm1 and idM1 have the same role as in previous case, and valM1 is by default set to idM1: each model is validated on all the neighbors used in identification.

linIdPar=p:

Here idmO and idMO are obtained according to the following formulas: idm1=3*noPar and idM1=5*p*noPar, where noPar=nx+1 is the number of parameter of the polynomial model of degree 1, and nx is the dimensionality of the input space. Recommended choice: p=1. As far as the quantity valM1 is concerned, it gets the default value as in previous case.

linIdPar=NULL:

No linear model is considered.

Parameter controlling the number of neighbors to be used for identifying and validating quadratic models. quaIdPar can assume different forms:

quaIdPar=c(idm2,idM2,valM2):

In this case, idm2:idM2 is the range in which the best number of neighbors is searched when identifying the local polynomial models of degree 2 and where valM2 is the maximum number of neighbors used for their validation. This means that the quadratic models identified with k neighbors, are validated on the first v neighbors, where v=min(k,valM2). If valM2=0, valM2 is set to idM2: see next case for details.

quaIdPar=c(idm2,idM2):

Here idm2 and idM2 have the same role as in previous case, and valM2 is by default set to idM2: each model is validated on all the neighbors used in identification.

quaIdPar=p:

Here idmO and idMO are obtained according to the following formulas: idm2=3*noPar and idM2=5*p*noPar, where in this case the number of parameters is noPar=(nx+1)*(nx+2)/2, and nx is the dimensionality of the input space. Recommended choice: p=1. As far as the quantity valM2 is concerned, it gets the default value as in previous case.

quaIdPar=NULL:

No quadratic model is considered.

The distance metric: can be manhattan or euclidean.

Vector of n elements. Weights used to evaluate the distance between query point and neighbors.

Parameter controlling the local combination of models. cmbPar can assume different forms:

cmbPar=c(cmb0,cmb1,cmb2):

In this case, cmbX is the number of polynomial models of degree X that will be included in the local combination. Each local model will be therfore a combination of the best cmb0 models of degree 0, the best cmb1 models of degree 1, and the best cmb2 models of degree 2 identified as specified by idPar.

cmbPar=cmb:

Here cmb is the number of models that will be combined, disregarding any constraint on the degree of the models that will be considered. Each local model will be therfore a combination of the best cmb models, identified as specified by id_par.

Initialization of the diagonal elements of the local variance/covariance matrix for Ridge Regression.

Object of class inheriting from lazy.

Data frame (or matrix, vector, etc...) defining of the query points for which a prediction is to be produced.

Logical switch indicating if the function should return the parameters of the local models used to perform each estimation.

Logical switch indicating if the function should return the number of neighbors used to perform each estimation.

Logical switch indicating if the function should return the estimated variance of the prediction suggested by all the models identified for each query point.

Logical switch indicating if the function should return the parameters of all the models identified for each query point.

Logical switch indicating if the function should return the index i of all the samples (X[i,],Y[i]) used to perform each estimation.

Arguments passed to or from other methods.

Vector of q elements, where q is the number of rows in newdata, i.e. the number of query points. The element in position i is the estimate of the value of the unknown function in the query point newdata[i,]. The component h is always returned.

Matrix of z*q elements, where z=z2 i.e., number of parameters of a quadratic model if at least one model of degree 2 was identified (see quaIdPar in lazy.control), otherwise z=z1 i.e., number of parameters of a linear model if at least one model of degree 1 was identified (see linIdPar in lazy.control), or z=1 if only models of degree 0 where considered. In the general case, the elements of the vector t[,j]=c(a0, a1,..., an, a11, a12,..., a22, a23,..., a33, a34,..., ann) are the parameters of the local model used for estimating the function in the jth query point: the cross-terms terms a11,a12,...,ann wil be missing if no quadratic model is identified and the terms a1,...,an, will be missing if no linear model is identified. If, according to cmbPar (see lazy.control), estimations are to be performed by a combination of models, the elements of t[,j] are a weighted average of the parameters of the selected models where the weight of each model is the inverse of the a leave-one-out estimate of the variances of the model itself. REMARK: a translation of the axes is considered which centers all the local models in the respective query point.

Vector of q elements. Selected number of neighbors for each query point. If, according to cmbPar (see lazy.control), a local combination of models is considered, k[j] is the largest value among the number of neighbors used by the selected models for estimating the value in the jth query point.

List of up to 3 components: Each component is a matrix containing an estimate, obtained through a leave-one-out cross-valication, of the variance of local models.

con

Matrix of idM0*q elements, where idM0 is the maximum number of neighbors used to fit local polynomial models of degree 0 (see lazy.control): Estimated variance of all the constant models identified for each query point. If no constant model is identified (see conIdPar and cmbPar in lazy.control) S$con is not returned.

lin

Matrix of idM1*q elements, where idM1 is the maximum number of neighbors used to fit local polynomial models of degree 1 (see lazy.control): Estimated variance of all the linear models identified for each query point. If no linear model is identified (see linIdPar and cmbPar in lazy.control) S$lin is not returned.

qua

Matrix of idM2*q elements, where idM1 is the maximum number of neighbors used to fit local polynomial models of degree 1 (see lazy.control): Estimated variance of all the quadratic models identified for each query point. If no quadratic model is identified (see quaIdPar and cmbPar in lazy.control) S$qua is not returned.

The component S is returned only if S.out=TRUE in the function call.

List of up to 3 components:

con

Array of z0*idM0*q elements, where z0=1 is the number of parameters of a model of degree 0. The element T$con[1,i,j]=a0 is the single parameter of the local model identified on i neighbors of the qth query point.

lin

Array of z1*idM1*q elements where, if n is the dimensionality of the input space, z1=n+1 is the number of parameter of a model of degree 1. The vector T$lin[,i,j]=c(a0,a1,...,an) is the vector of parameters of the local model identified on i neighbors of the qth query point. In particular, a0 is the constant term, a1 is the parameter associated with the first input variable and so on.

qua

Array of z2*idM2*q elements where, if n is the dimensionality of the input space, z2=(n+1)*(n+2)/2 is the number of parameter of a model of degree 2. The vector T$qua[,i,j]=c(a0, a1,..., an, a11, a12,..., a22, a23,..., a33, a34,..., ann) is the vector of parameters of the local quadratic model identified on i neighbors of the qth query point. In particular, a0,...,a1 are the constant and liner parameters as in T$lin, while a11,a12,...,ann are the quadratic ones: a11 is associated with the quadratic term x1^2, a12 with the cross-term x1*x2, and so on.

REMARK: a translation of the axes is considered which centers all the local models in the respective query point. The component T is returned only if T.out=TRUE in the function call.

Matrix of idM*q elements, where idM is the largest of idM0, idM1, and idM2. Contains the index of the neighbors of each query point in newdata. In particular, I[i,j] is the ith nearest neighbor of the qth query point.