Regression Methods for Interval-Valued Variables

Description

Contains some important regression methods for interval-valued variables. For each method, it is available the fitted values, residuals and some goodness-of-fit measures.

Details

Some available functions: cm, MinMax, crm, ccrm, bivar

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br , Claudio A. V. de Souza Filho and Pedro R. D. Marinho

Maintainer: Eufrasio de A. Lima Neto eufrasio@de.ufpb.br

Acknowledgments: The authors would like to thank CNPq (Brazilian Agency) for their financial support.

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

Lima Neto, E.A. and De Carvalho, F.A.T. (2010). Constrained linear regression models for symbolic interval-valued variables. Computational Statistics and Data Analysis, 54, 333–347.

Lima Neto, E. A., Cordeiro, G. and De Carvalho, F.A.T. (2011). Bivariate symbolic regression models for interval-valued variables. Journal of Statistical Computation and Simulation (Print), 81, 1727–1744.

Cardiological Interval Data Set (Centre and Range)

Description

A real interval-valued data set represented in terms of the centre and the range of the intervals.

Usage

data("Cardiological.CR")

Format

A data frame containing the following variables:.

PulseC

The midpoint of the response interval-valued variable Pulse

SystC

The midpoint of the explanatory interval-valued variable Systolic Pressure

DiastC

The midpoint of the explanatory interval-valued variable Diastolic Pressure

PulseR

The range of the response interval-valued variable Pulse

SystR

The range of the explanatory interval-valued variable Systolic Pressure

DiastR

The range of the explanatory interval-valued variable Diastolic Pressure

Details

This data set concerns the record of the pulse rate (Y), systolic blood pressure (X1) and diastolic blood pressure (X2) from 11 patients.

Source

Billard and Diday (2000)

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

See Also

crm

Examples

data("Cardiological.CR", package = "iRegression")

crm1 <- crm("PulseC~SystC+DiastC","PulseR~SystR+DiastR",data=Cardiological.CR)
summary(crm1)

Cardiological Interval Data Set

Description

A real interval-valued data set.

Usage

data("Cardiological.CR")

Format

A data frame containing following variables:

PulseMin

Lower bound of the response interval-valued variable Pulse

SystMin

Lower bound of the explanatory interval-valued variable Systolic Pressure

DiastMin

Lower bound of the explanatory interval-valued variable Diastolic Pressure

PulseMax

Upper bound of the response interval-valued variable Pulse

SystMax

Upper bound of the explanatory interval-valued variable Systolic Pressure

DiastMax

Upper bound of the explanatory interval-valued variable Diastolic Pressure

Details

This data set concerns the record of the pulse rate (Y), systolic blood pressure (X1) and diastolic blood pressure (X2) from 11 patients.

Source

Billard and Diday (2000)

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

See Also

cm, MinMax

Examples

data("Cardiological.MinMax", package = "iRegression")

cm1 <- cm(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
summary(cm1)

##
data("Cardiological.MinMax", package = "iRegression")

MinMax1 <- MinMax(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
summary(MinMax1)

MinMax Method

Description

MinMax is used to fit a linear regression model to symbolic interval-valued variables based on the MinMax method (Lima Neto and De Carvalho, 2008).

Usage

MinMax(formula1, formula2, data, ...)

Arguments

Details

The Min-Max Method suggests to estimate the lower and upper bounds of the intervals using different vectors of parameters. This is equivalent to supposing independence between the values of lower and upper bounds of the intervals. The MinMax Method fits two independent linear regression models on the lower and upper bounds of the intervals, respectively, and minimizes the error of the lower bounds plus the error of the upper bounds.

Value

MinMax returns an object of class "MinMax" including at least the following elements:

Note

formula1 must contain the lower limit of the symbolic interval-valued variables. formula2 contain the upper limit of the symbolic interval-valued variables.

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br , Claudio A. V. de Souza Filho and Pedro R. D. Marinho

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

summary.MinMax, coef.MinMax, fitted.MinMax, residuals.MinMax, formula

Examples

data("Cardiological.MinMax", package = "iRegression") ## see Billard, L. and Diday, E. (2000)
ex.MinMax <- MinMax(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
ex.MinMax

Bivariate Symbolic Regression Method

Description

This function fits an bivariate regression model for interval-valued variables, based on bivariate exponential family of distributions, and return the fitted values, the residuals, rho, phi and the goodness-of-fit measure deviance

Usage

bivar(formula1, lig1, formula2, lig2, data, ...)

Arguments

Details

This function fits an bivariate regression model for interval-valued variables considering the bivariate Gaussian distribution in the random component Y = [Y1, Y2]. It is possible consider any pair of interval features for the bivariate random vector Y. For example, the lower and upper interval bounds or the midpoint and the range of intervals, respectively. It also possible to choice different link functions (identity, inverse or log) to connect the random variables Y1 and Y2 with the respective linear predictors.

Value

bivar returns an object of class "bivar" including at least the following elements:

Note

lig1 and lig2 must be "identity", "inverse" or "log" for identity, inverse or logarithmic link functions, respectively.

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br , Claudio A. V. de Souza Filho and Pedro R. D. Marinho

References

Lima Neto, E. A., Cordeiro, G. and De Carvalho, F.A.T. (2011). Bivariate symbolic regression models for interval-valued variables. Journal of Statistical Computation and Simulation (Print), 81, 1727–1744.

See Also

summary.bivar, coef.bivar, fitted.bivar, residuals.bivar, formula

Examples

data("soccer.bivar", package = "iRegression")
ex.bivar <- bivar("yMin~t1Min+t2Min", "identity", "yMax~t1Max+t2Max", "identity", data=soccer.bivar)
ex.bivar

Constrained Centre and Range Method

Description

ccrm is used to fit a linear regression model to symbolic interval-valued variables based on the inequality constraints over the range variables (Lima Neto and De Carvalho, 2010).

Usage

ccrm(formula1, formula2, data, ...)

Arguments

Details

The Constrained Centre and Range method (CCRM) was proposed by Lima Neto and De Carvalho (2010) and fits two independent linear regression models on the midpoint and range of the intervals. In the Constrained Centre and Range Method, the estimative of the parameters of the range's model is based on inequality constraints. There is no constraints over the parameters estimates for the midpoint regression equation. The aim is to guarantee mathematical coherence between the predicted values of the lower and upper bounds of the response interval-valued variable Y, i.e., yL < yU.

Value

ccrm returns an object of class "ccrm" including at least the following elements:

Note

formula1 must contain the midpoint of the symbolic interval-valued variables. formula2 contain the range (upper limit minus lower limit) of the symbolic interval-valued variables.

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br , Claudio A. V. de Souza Filho and Pedro R. D. Marinho

References

Lima Neto, E.A. and De Carvalho, F.A.T. (2010). Constrained linear regression models for symbolic interval-valued variables. Computational Statistics and Data Analysis, 54, 333–347.

See Also

summary.ccrm, coef.ccrm, fitted.ccrm, residuals.ccrm, formula

Examples

data("Cardiological.CR", package = "iRegression")
ex.ccrm <- ccrm("PulseC~SystC+DiastC","PulseR~SystR+DiastR",data=Cardiological.CR)
ex.ccrm

Centre Method

Description

cm is used to fit a linear regression model to symbolic interval-valued variables based on the centre method (Billard and Diday, 2000).

Usage

cm(formula1, formula2, data, ...)

Arguments

Details

Billard and Diday (2000) presented the first approach to fitting a linear regression model to symbolic interval data sets from a SDA of view. Their approach consists on fitting a linear regression model to the mid-points of the interval values assumed by the symbolic interval variables in the learning set and applies this model to the lower and upper bounds of the interval values of the independent symbolic interval variables to be predicted, respectively, the lower and upper bounds of the interval value of the dependent variable. The Centre Method is based on the minimization of the midpoint error. The lower and upper bounds of the dependent variable are predicted, respectively, from the lower and upper bounds of the independent variable using the same vector of parameters beta.

Value

cm returns an object of class "cm" including at least the following elements:

Note

formula1 must contain the lower limit of the symbolic interval-valued variables. formula2 contain the upper limit of the symbolic interval-valued variables.

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br, Claudio A. V. de Souza Filho and Pedro R. D. Marinho

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

summary.cm, coef, fitted.cm, residuals.cm, formula

Examples

data("Cardiological.MinMax", package = "iRegression") ## see Billard and Diday (2000)
ex.cm <- cm(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
ex.cm

Extract Coefficients for the MinMax Method

Description

Returns the coefficients from an object class MinMax.

Usage

## S3 method for class 'MinMax'
coef(object, ...)

Arguments

Value

Coefficients extracted from an object class MinMax.

See Also

MinMax

Extract the Coefficients for the Bivariate Symbolic Regression Method

Description

Returns the coefficients from an object class bivar.

Usage

## S3 method for class 'bivar'
coef(object, ...)

Arguments

Value

Coefficients extracted from an object class bivar.

See Also

bivar

Extract the Coefficients for the Constrained Centre and Range Method

Description

Returns the coefficients from an object class ccrm.

Usage

## S3 method for class 'ccrm'
coef(object, ...)

Arguments

Value

Coefficients extracted from an object class object.

See Also

ccrm

Extract the Coefficients for the Centre and Range Method

Description

Returns the coefficients from an object class crm.

Usage

## S3 method for class 'crm'
coef(object, ...)

Arguments

Value

Coefficients extracted from an object class object.

See Also

crm

Centre and Range Method

Description

crm is used to fit a linear regression model to symbolic interval-valued variables based on the Centre and Range method (Lima Neto and De Carvalho, 2008).

Usage

crm(formula1, formula2, data, ...)

Arguments

Details

In the Center Method, the estimate of the parameters beta is based only on the midpoint of the intervals. However, the Centre and Range Method proposed by Lima Neto and De Carvalho (2008) consider suitable to include both the information given by the center and by the range of an interval-valued variable on a linear regression model to improve the model prediction performance. The Centre and Range Method fits two independent linear regression models on the midpoint and range of the intervals, respectively, and minimizes the error of the midpoint plus the error of the range.

Value

cm returns an object of class "crm" including at least the following elements:

Note

formula1 must contain the midpoint of the symbolic interval-valued variables. formula2 contain the range (upper limit minus lower limit) of the symbolic interval-valued variables.

Author(s)

Eufrasio de A. Lima Neto eufrasio@de.ufpb.br , Claudio A. V. de Souza Filho and Pedro R. D. Marinho

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

summary.crm, coef.crm, fitted.crm, residuals.crm, formula

Examples

data("Cardiological.CR", package = "iRegression")
ex.crm <- crm("PulseC~SystC+DiastC","PulseR~SystR+DiastR",data=Cardiological.CR)
ex.crm

Extract MinMax Method Fitted Values

Description

Returns the fitted values from an object class MinMax.

Usage

## S3 method for class 'MinMax'
fitted(object, ...)

Arguments

Value

Fitted values extracted from the object class MinMax.

See Also

MinMax

Extract Bivariate Symbolic Regression Method Fitted Values

Description

Returns the fitted values from an object class bivar.

Usage

## S3 method for class 'bivar'
fitted(object, ...)

Arguments

Value

Fitted values extracted from the object class bivar.

See Also

bivar

Extract Constrained Centre and Range Method Fitted Values

Description

Returns the fitted values from an object class ccrm.

Usage

## S3 method for class 'ccrm'
fitted(object, ...)

Arguments

Value

Fitted values extracted from the object class object.

See Also

ccrm

Extract Centre Method Fitted Values

Description

Returns the fitted values from an object class cm.

Usage

## S3 method for class 'cm'
fitted(object, ...)

Arguments

Value

Fitted values extracted from an object class cm.

See Also

cm

Extract Centre and Range Method Fitted Values

Description

Returns the fitted values from an object class crm.

Usage

## S3 method for class 'crm'
fitted(object, ...)

Arguments

Value

Fitted values extracted from the object class object.

See Also

crm

Print Values for various iRegression methods

Description

print prints its argument.

Usage

## S3 method for class 'cm'
print(x, ...)
## S3 method for class 'crm'
print(x, ...)
## S3 method for class 'ccrm'
print(x, ...)
## S3 method for class 'MinMax'
print(x, ...)
## S3 method for class 'bivar'
print(x, ...)
## S3 method for class 'summary.cm'
print(x, ...)
## S3 method for class 'summary.crm'
print(x, ...)
## S3 method for class 'summary.ccrm'
print(x, ...)
## S3 method for class 'summary.MinMax'
print(x, ...)
## S3 method for class 'summary.bivar'
print(x, ...)
## S3 method for class 'coef.crm'
print(x, ...)
## S3 method for class 'coef.ccrm'
print(x, ...)
## S3 method for class 'coef.MinMax'
print(x, ...)
## S3 method for class 'coef.bivar'
print(x, ...)

Arguments

See Also

print

Extract MinMax Method Residuals

Description

Returns the residuals from an object class MinMax.

Usage

## S3 method for class 'MinMax'
residuals(object, ...)

Arguments

Value

Residuals extracted from the object class MinMax.

See Also

MinMax

Extract Bivariate Symbolic Regression Method Residuals

Description

Returns the residuals from an object class bivar.

Usage

## S3 method for class 'bivar'
residuals(object, ...)

Arguments

Value

Residuals extracted from the object class bivar.

See Also

bivar

Extract Constrained Centre and Range Method Residuals

Description

Returns the residuals from an object class ccrm.

Usage

## S3 method for class 'ccrm'
residuals(object, ...)

Arguments

Value

Residuals extracted from the object class ccrm.

See Also

ccrm

Extract Centre Method Residuals

Description

Returns the residuals from an object class cm.

Usage

## S3 method for class 'cm'
residuals(object, ...)

Arguments

Value

Residuals extracted from the object class cm.

See Also

cm

Extract Centre and Range Method Residuals

Description

Returns the residuals from an object class crm.

Usage

## S3 method for class 'crm'
residuals(object, ...)

Arguments

Value

Residuals extracted from the object class crm.

See Also

crm

Soccer Interval Data Set

Description

A real interval-valued data set.

Usage

data("soccer.bivar")

Format

A data frame containing following variables:

yMin

Minimum of the response variable Y (weight)

t1Min

Minimum of the explanatory variable T1 (height)

t2Min

Minimum of the explanatory variable T2 (age)

yMax

Maximum of the response variable Y (weight)

t1Max

Maximum of the explanatory variable T1 (height)

t2Max

Maximum of the explanatory variable T2 (age)

Details

This data set concerns the record of the Weight (Y), Height (T1) and Age (T2) from 20 soccer teams of the premiere French championship.

Source

Lima Neto et. al. (2011)

References

Lima Neto, E. A., Cordeiro, G. and De Carvalho, F.A.T. (2011). Bivariate symbolic regression models for interval-valued variables. Journal of Statistical Computation and Simulation (Print), 81, 1727–1744.

See Also

cm, MinMax, bivar

Examples

data("soccer.bivar", package = "iRegression")

bivar1 <- bivar(yMin~t1Min+t2Min, "identity", yMax~t1Max+t2Max, "identity", data=soccer.bivar)
summary(bivar1)

Summarizing MinMax Method Fits

Description

summary method for class MinMax.

Usage

## S3 method for class 'MinMax'
summary(object, ...)

Arguments

.

Value

The function summary.MinMax returns the following elements, given an object of the class "MinMax",

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

MinMax

Examples

##-- Continuing the  MinMax() example:
data("Cardiological.MinMax", package = "iRegression")
ex.MinMax <- MinMax(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
ex.sum <- summary(ex.MinMax)
ex.sum

Summarizing Bivariate Symbolic Regression Method Fits

Description

summary method for class bivar.

Usage

## S3 method for class 'bivar'
summary(object, ...)

Arguments

.

Value

The function summary.bivar returns the following elements, given an object of the class "bivar",

References

Lima Neto, E. A., Cordeiro, G. and De Carvalho, F.A.T. (2011). Bivariate symbolic regression models for interval-valued variables. Journal of Statistical Computation and Simulation (Print), 81, 1727–1744.

See Also

bivar

Examples

##-- Continuing the  bivar() example:
data("soccer.bivar", package = "iRegression")
ex.bivar <- bivar(yMin~t1Min+t2Min, "identity", yMax~t1Max+t2Max, "identity", data=soccer.bivar)
ex.sum <- summary(ex.bivar)
ex.sum

Summarizing Constrained Centre and Range Method Fits

Description

summary method for class ccrm.

Usage

## S3 method for class 'ccrm'
summary(object, ...)

Arguments

Value

The function summary.ccrm returns the following elements, given an object of the class "ccrm",

References

Lima Neto, E.A. and De Carvalho, F.A.T. (2010). Constrained linear regression models for symbolic interval-valued variables. Computational Statistics and Data Analysis, 54, 333–347.

See Also

ccrm

Examples

##-- Continuing the  ccrm() example:
data("Cardiological.CR", package = "iRegression")
ex.ccrm <- ccrm(PulseC~SystC+DiastC,PulseR~SystR+DiastR,data=Cardiological.CR)
ex.sum <- summary(ex.ccrm)
ex.sum

Summarizing Centre Method Fits

Description

summary method for class cm.

Usage

## S3 method for class 'cm'
summary(object, ...)

Arguments

Value

The function summary.cm returns the following elements, given an object of the class "cm",

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

cm

Examples

##-- Continuing the  cm() example:
data("Cardiological.MinMax", package = "iRegression")
ex.cm <- cm(PulseMin~SystMin+DiastMin,PulseMax~SystMax+DiastMax,data=Cardiological.MinMax)
ex.sum <- summary(ex.cm)
ex.sum

Summarizing Centre and Range Method Fits

Description

summary method for class crm.

Usage

## S3 method for class 'crm'
summary(object, ...)

Arguments

Value

The function summary.crm returns the following elements, given an object of the class "crm",

References

Billard, L. and Diday, E. (2000) Regression analysis for interval-valued data. Data Analysis, Classification and Related Methods: Proceedings of the Seventh Conference of the International Federation of Classification Societies, Springer-Verlag, pp. 369-374.

Lima Neto, E.A. and De Carvalho, F.A.T. (2008). Centre and range method to fitting a linear regression model on symbolic interval data. Computational Statistics and Data Analysis, 52, 1500–1515.

See Also

crm

Examples

##-- Continuing the  crm() example:
data("Cardiological.CR", package = "iRegression")
ex.crm <- crm(PulseC~SystC+DiastC,PulseR~SystR+DiastR,data=Cardiological.CR)
ex.sum <- summary(ex.crm)
ex.sum