dblm {dbstats}

Distance-based linear model

Description

dblm is a variety of linear model where explanatory information is coded as distances between individuals. These distances can either be computed from observed explanatory variables or directly input as a squared distances matrix. The response is a continuous variable as in the ordinary linear model. Since distances can be computed from a mixture of continuous and qualitative explanatory variables or, in fact, from more general quantities, dblm is a proper extension of lm.

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'
dblm(formula,data,...,metric="euclidean",method="OCV",full.search=TRUE,
        weights,rel.gvar=0.95,eff.rank)

## S3 method for class 'dist'
dblm(distance,y,...,method="OCV",full.search=TRUE,
        weights,rel.gvar=0.95,eff.rank)

## S3 method for class 'D2'
dblm(D2,y,...,method="OCV",full.search=TRUE,weights,rel.gvar=0.95,
        eff.rank)
              
## S3 method for class 'Gram'
dblm(G,y,...,method="OCV",full.search=TRUE,weights,rel.gvar=0.95,
        eff.rank)

Arguments

formula

an object of class formula. A formula of the form y~Z. This argument is a remnant of the lm 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.

distance

a dist or dissimilarity class object. See functions dist in the package stats and daisy in the package cluster.

D2

a D2 class object. Squared distances matrix between individuals.

G

a Gram class object. Doubly centered inner product matrix of the squared distances matrix D2.

metric

metric function to be used when computing distances from observed explanatory variables. One of "euclidean" (default), "manhattan", or "gower".

method

sets the method to be used in deciding the effective rank, which is defined as the number of linearly independent Euclidean coordinates used in prediction. There are six different methods: "AIC", "BIC", "OCV" (default), "GCV", "eff.rank" and "rel.gvar". OCV and GCV take the effective rank minimizing a cross-validatory quantity (either ocv or gcv). AIC and BIC take the effective rank minimizing, respectively, the Akaike or Bayesian Information Criterion (see AIC for more details). The optimizacion procedure to be used in the above four methods can be set with the full.search optional parameter.

When method is eff.rank, the effective rank is explicitly set by the user through the eff.rank optional parameter which, in this case, becomes mandatory.

When method is rel.gvar, the fraction of the data geometric variability for model fitting is explicitly set by the user through the rel.gvar optional parameter which, in this case, becomes mandatory.

full.search

sets which optimization procedure will be used to minimize the modelling criterion specified in method. Needs to be specified only if method is "AIC", "BIC", "OCV" or "GCV". If full.search=TRUE, effective rank is set to its global best value, after evaluating the criterion for all possible ranks. Potentially too computationally expensive. If full.search=FALSE, the optimize function is called. Then computation time is shorter, but the result may be found a local minimum.

weights

an optional numeric vector of weights to be used in the fitting process. By default all individuals have the same weight.

rel.gvar

relative geometric variability (real between 0 and 1). Take the lowest effective rank with a relative geometric variability higher or equal to rel.gvar. Default value (rel.gvar=0.95) uses a 95% of the total variability. Applies only rel.gvar if method="rel.gvar".

eff.rank

integer between 1 and the number of observations minus one. Number of Euclidean coordinates used for model fitting. Applies only if method="eff.rank".

...

arguments passed to or from other methods to the low level.

Details

The dblm model uses the distance matrix between individuals to find an appropriate prediction method. There are many ways to compute and calculate this matrix, besides the three included as parameters in this function. Several packages in R also study this problem. In particular dist in the package stats and daisy in the package cluster (the three metrics in dblm call the daisy function).

Another way to enter a distance matrix to the model is through an object of class "D2" (containing the squared distances matrix). An object of class "dist" or "dissimilarity" can easily be transformed into one of class "D2". See disttoD2. Reciprocally, an object of class "D2" can be transformed into one of class "dist". See D2toDist.

S3 method Gram uses the Doubly centered inner product matrix G=XX'. Its also easily to transformed into one of class "D2". See D2toG and GtoD2.

The weights array is adequate when responses for different individuals have different variances. In this case the weights array should be (proportional to) the reciprocal of the variances vector.

When using method method="eff.rank" or method="rel.gvar", a compromise between possible consequences of a bad choice has to be reached. If the rank is too large, the model can be overfitted, possibly leading to an increased prediction error for new cases (even though R2 is higher). On the other hand, a small rank suggests a model inadequacy (R2 is small). The other four methods are less error prone (but still they do not guarantee good predictions).

Value

A list of class dblm containing the following components:

residuals

the residuals (response minus fitted values).

fitted.values

the fitted mean values.

df.residuals

the residual degrees of freedom.

weights

the specified weights.

y

the response used to fit the model.

H

the hat matrix projector.

call

the matched call.

rel.gvar

the relative geometric variabiliy, used to fit the model.

eff.rank

the dimensions chosen to estimate the model.

ocv

the ordinary cross-validation estimate of the prediction error.

gcv

the generalized cross-validation estimate of the prediction error.

aic

the Akaike Value Criterium of the model (only if method="AIC").

bic

the Bayesian Value Criterium of the model (only if method="BIC").

Note

When the Euclidean distance is used the dblm model reduces to the linear model (lm).

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

summary.dblm for summary.
plot.dblm for plots.
predict.dblm for predictions.
ldblm for distance-based local linear models.

Examples

# easy example to illustrate usage of the dblm function
n <- 100
p <- 3
k <- 5

Z <- matrix(rnorm(n*p),nrow=n)
b <- matrix(runif(p)*k,nrow=p)
s <- 1
e <- rnorm(n)*s
y <- Z%*%b + e

D<-dist(Z)

dblm1 <- dblm(D,y)
lm1 <- lm(y~Z)
# the same fitted values with the lm
mean(lm1$fitted.values-dblm1$fitted.values)