Package 'bestglm'

Title:	Best Subset GLM and Regression Utilities
Description:	Best subset glm using information criteria or cross-validation, carried by using 'leaps' algorithm (Furnival and Wilson, 1974) <doi:10.2307/1267601> or complete enumeration (Morgan and Tatar, 1972) <doi:10.1080/00401706.1972.10488918>. Implements PCR and PLS using AIC/BIC. Implements one-standard deviation rule for use with the 'caret' package.
Authors:	A.I. McLeod, Changjiang Xu and Yuanhao Lai
Maintainer:	Yuanhao Lai <[email protected]>
License:	GPL (>= 2)
Version:	0.37.3
Built:	2024-09-19 06:38:07 UTC
Source:	CRAN

Help Index

bestglm: Best Subset GLM
Daily ozone pollution with meteorlogical and date inputs
Binary representation of non-negative integer
Best Subset GLM using Information Criterion or Cross-Validation
Cross-validation using delete-d method.
Adjusted K-fold Cross-Validation
K-fold Cross-Validation
Detroit homicide data for 1961-73 used in the book Subset Regression by A.J. Miller
Scaled Variables Dependency Plots: Output vs Inputs
Forest fires in Montesinho natural park. Standardized inputs.
Fitted values in PCR and PLS.
Multipanel Display and Table Glmnet CV Output.
Glmnet Prediction Using CVAV.
Predictions on Test Data with Grpreg
Simulated Linear Regression (Train) with Nine Highly Correlated Inputs
Iowa School Test
Leave-one-out cross-validation
Hospital manpower data
Pollution dataset from McDonald and Schwing (1973)
Forest fires in Montesinho natural park
Nearest Neighbour Regression Prediction
Utility function. Implements the 1-sd rule.
Principal Component and Partial Least Squares Regression
Diagnostic plots for PCR and PLS
Plot Regularization Path and One Standard Deviation Rule
Predict Method for Pcreg.
Print method for 'bestglm' object
Print method for 'pcreg' object
Residuals Fitted PCR or PLS
Abrasion loss for various hardness and tensile strength
South African Hearth Disease Data
Simulated Regression Data
Sphere Data Matrix
summary of 'bestglm' object
Summary Method for Pcreg.
Partition Dataframe into Train/Test Samples
Variance Inflation Factor for a Design Matrix
Nuclear plant data. Quantitative inputs logged and standardized.
Prostate cancer data. Standardized.

bestglm: Best Subset GLM

Description

Provides new information criterion BICq as well as AIC, BIC and EBIC for selecting the best model. Additionally, various CV algorithms are also provided.

Details

Package:	bestglm
Type:	Package
Version:	0.33
Date:	2011-11-03
License:	GLP 2.0 or greater
LazyData:	yes
LazyLoad:	yes

bestglm is the main function. All other functions are utility functions and are not normally invoked.

Many examples are provided in the vignettes accompanying this package. The vignettes are produced using the R package Sweave and so R scripts can easily be extracted.

The R package xtable is needed for the vignette in SimExperimentBICq.Rnw.

Author(s)

A.I. McLeod and Changjiang Xu

References

Xu, C. and McLeod, A.I. (2009). Bayesian Information Criterion with Bernouilli Prior.

Examples

## Not run: 
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
#Best subset using AIC
bestglm(train, IC="AIC")
#Best subset using BIC
bestglm(train, IC="BIC")
#Best subset using EBIC
bestglm(train, IC="BICg")
#Best subset using BICg with g=0.5 (tuning parameter)
bestglm(train, IC="BICg", t=0.5)
#Best subset using BICq. Note BICq with q=0.25 is default.
bestglm(train, IC="BICq")
#Best subset using BICq with q=0.5 (equivalent to BIC)
bestglm(train, IC="BICq", t=0.5)
#Remark: set seed since CV depends on it
set.seed(123321123)
bestglm(train, IC="CV", t=10)
#using HTF method
bestglm(train, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1))
#Best subset, logistic regression
data(SAheart)
bestglm(SAheart, IC="BIC", family=binomial)
#Best subset, factor variables with more than 2 levels
data(AirQuality)
#subset
bestglm(AirQuality, IC="BICq")

## End(Not run)
## Not run: 
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
#Best subset using AIC
bestglm(train, IC="AIC")
#Best subset using BIC
bestglm(train, IC="BIC")
#Best subset using EBIC
bestglm(train, IC="BICg")
#Best subset using BICg with g=0.5 (tuning parameter)
bestglm(train, IC="BICg", t=0.5)
#Best subset using BICq. Note BICq with q=0.25 is default.
bestglm(train, IC="BICq")
#Best subset using BICq with q=0.5 (equivalent to BIC)
bestglm(train, IC="BICq", t=0.5)
#Remark: set seed since CV depends on it
set.seed(123321123)
bestglm(train, IC="CV", t=10)
#using HTF method
bestglm(train, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1))
#Best subset, logistic regression
data(SAheart)
bestglm(SAheart, IC="BIC", family=binomial)
#Best subset, factor variables with more than 2 levels
data(AirQuality)
#subset
bestglm(AirQuality, IC="BICq")

## End(Not run)

Daily ozone pollution with meteorlogical and date inputs

Description

This dataset was derived from the R built-in dataset 'airquality' by adding date information and deleting all missing values. This dataset is referred to as 'environmental' in Cleveland (1993).

Usage

data(AirQuality)data(AirQuality)

Format

A data frame with 111 observations on the following 6 variables.

Solar.R: input, a numeric vector
Wind: input, a numeric vector
Temp: input, a numeric vector
month: input, a factor with levels May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr
weekday: input, a factor with levels Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Ozone: output, a numeric vector

Details

Cleveland (1993, Chapter 5) presents an insightful analysis using co-plots and the scatterplot matrix. Several interesting interactions are noted. For a fixed 'Wind‘, the effect of ’Solar.R' changes as 'Temp' increases. And for a fixed 'Temp', as 'Wind' decreases, the effect of 'Solar.R' is less.

Source

airquality

References

Cleveland, W.S. (1993). Visualizing Data.

Examples

data(AirQuality)
#Example 1. Find best model
bestglm(AirQuality, IC="BIC")

data(AirQuality)
#Example 1. Find best model
bestglm(AirQuality, IC="BIC")

Binary representation of non-negative integer

Description

A non-negative integer is represented as a binary number. The digits, 0 or 1, of this number are returned in a vector.

Usage

to.binary(n, k = ceiling(logb(n+1,base=2)))
to.binary(n, k = ceiling(logb(n+1,base=2)))

Arguments

`n`	a non-negative integers
`k`	number of digits to be returned.

Value

A vector of length k. The first element is the least significant digit.

Author(s)

A.I. McLeod

Examples

to.binary(63)
to.binary(64)
#sometimes we want to pad result with 'leading' 0's
to.binary(63, k=20)
to.binary(64, k=20)
to.binary(63)
to.binary(64)
#sometimes we want to pad result with 'leading' 0's
to.binary(63, k=20)
to.binary(64, k=20)

Best Subset GLM using Information Criterion or Cross-Validation

Description

Best subset selection using 'leaps' algorithm (Furnival and Wilson, 1974) or complete enumeration (Morgan and Tatar, 1972). Complete enumeration is used for the non-Gaussian and for the case where the input matrix contains factor variables with more than 2 levels. The best fit may be found using the information criterion IC: AIC, BIC, EBIC, or BICq. Alternatively, with IC=‘CV’ various types of cross-validation may be used.

Usage

bestglm(Xy, family = gaussian, IC = "BIC", t = "default", 
 CVArgs = "default", qLevel = 0.99, TopModels = 5, 
 method = "exhaustive", intercept = TRUE, weights = NULL, 
 nvmax = "default", RequireFullEnumerationQ = FALSE, ...)
bestglm(Xy, family = gaussian, IC = "BIC", t = "default", 
 CVArgs = "default", qLevel = 0.99, TopModels = 5, 
 method = "exhaustive", intercept = TRUE, weights = NULL, 
 nvmax = "default", RequireFullEnumerationQ = FALSE, ...)

Arguments

`Xy`	Dataframe containing the design matrix X and the output variable y. All columns must be named.
`family`	One of the glm distribution functions. The glm function is not used in the Gaussian case. Instead for efficiency either 'leaps' is used or when factor variables are present with more than 2 levels, 'lm' may be used.
`IC`	Information criteria to use: "AIC", "BIC", "BICg", "BICq", "LOOCV", "CV".
`t`	adjustable parameter for BICg, BICq or CV. For BICg, default is g=t=1. For BICq, default is q=t=0.25. For CV, default the delete-d method with d=ceil(n(1-1/(log n - 1))) and REP=t=1000. The default value of the parameter may be changed by changing t.
`CVArgs`	Used when IC is set to 'CV'. The default is use the delete-d algorithm with d=ceil(n(1-1/(log n - 1))) and t=100 repetitions. Note that the number of repetitions can be changed using t. More generally, CVArgs is a list with 3 named components: Method, K, REP, where Method is one of \"HTF\", \"DH\", \"d\" corresponding to using the functions CVHTM (Hastie et al., 2009, K-fold CV), CVDH (adjusted K-fold CV, Davison and Hartigan, 1997) and CVd (delete-d CV with random subsamples, Shao, 1997).
`qLevel`	the alpha level for determining interval for best q. Larger alpha's result in larger intervals.
`TopModels`	Finds the best `TopModels` models.
`method`	Method used in leaps algorithm for searching for the best subset.
`intercept`	Default TRUE means the intercept term is always included. If set to FALSE, no intercept term is included. If you want only include the intercept term when it is signficant then set IncludeInterceptQ=FALSE and include a column of 1's in the design matrix.
`weights`	weights
`nvmax`	maximum number of independent variables allowed. By default, all variables
`RequireFullEnumerationQ`	Use exhaustive search algorithm instead of 'leaps'
`...`	Optional arguments which are passed to `lm` or `glm`

Details

In the Gaussian case, the loglikelihood may be written $logL = -(n/2) log (RSS/n)$ , where RSS is the residual sum-of-squares and n is the number of observations. When the function 'glm' is used, the log-likelihood, logL, is obtained using 'logLik'. The penalty for EBIC and BICq depends on the tuning parameter argument, t. The argument t also controls the number of replications used when the delete-d CV is used as default. In this case, the parameter d is chosen using the formula recommended by Shao (1997). See CVd for more details.

In the binomial GLM, nonlogistic, case the last two columns of Xy are the counts of 'success' and 'failures'.

Cross-validation may also be used to select the best subset. When cross-validation is used, the best models of size k according to the log-likelihood are compared for k=0,1,...,p, where p is the number of inputs. Cross-validation is not available when there are categorical variables since in this case it is likely that the training sample may not contain all levels and in this case we can't predict the response in the validation sample. In the case of GLM, the \"DH\" method for CV is not available.

Usually it is a good idea to keep the intercept term even if it is not significant. See discussion in vignette.

Cross-validation is not available for models with no intercept term or when force.in is non-null or when nvmax is set to less than the full number of independent variables.

Please see the package vignette for more details and examples.

Value

A list with class attribute 'bestglm' and named components:

`BestModel`	An lm-object representing the best fitted regression.
`Title`	A brief title describing the algorithm used: CV(K=K), CVadj(K=K), CVd(d=K). The range of q for an equivalent BICq model is given.
`Subsets`	The best subsets of size, k=0,1,...,p are indicated as well the value of the log-likelihood and information criterion for each best subset. In the case of categorical variables with more than 2 levels, the degrees of freedom are also shown.
`qTable`	Table showing range of q for choosing each possible subset size. Assuming intercept=TRUE, k=1 corresponds to model with only an intercept term and k=p+1, where p is the number of input variables, corresponds to including all variables.
`Bestq`	Optimal q
`ModelReport`	A list with components: NullModel, LEAPSQ, glmQ, gaussianQ, NumDF, CategoricalQ, Bestk.
`BestModels`	Variables in the `TopModels` best list

Methods function 'print.bestglm' and 'summary.bestglm' are provided.

Author(s)

C. Xu and A.I. McLeod

References

Xu, C. and McLeod, A.I. (2009). Bayesian Information Criterion with Bernouilli Prior.

Chen, J. and Chen, Z. (2008). Extended Bayesian Information Criteria for Model Selection with Large Model Space. Biometrika 2008 95: 759-771.

Furnival, G.M. and Wilson, R. W. (1974). Regressions by Leaps and Bounds. Technometrics, 16, 499–511.

Morgan, J. A. and Tatar, J. F. (1972). Calculation of the Residual Sum of Squares for All Possible Regressions. Technometrics 14, 317-325.

Miller, A. J. (2002), Subset Selection in Regression, 2nd Ed. London, Chapman and Hall.

Shao, Jun (1997). An Asymptotic Theory for Linear Model Selection. Statistica Sinica 7, 221-264.

Examples

#Example 1. 
#White noise test.
set.seed(123321123)
p<-25   #number of inputs
n<-100  #number of observations
X<-matrix(rnorm(n*p), ncol=p)
y<-rnorm(n)
Xy<-as.data.frame(cbind(X,y))
names(Xy)<-c(paste("X",1:p,sep=""),"y")
bestAIC <- bestglm(Xy, IC="AIC")
bestBIC <- bestglm(Xy, IC="BIC")
bestEBIC <- bestglm(Xy, IC="BICg")
bestBICq <- bestglm(Xy, IC="BICq")
NAIC <- length(coef(bestAIC$BestModel))-1
NBIC <- length(coef(bestBIC$BestModel))-1
NEBIC <- length(coef(bestEBIC$BestModel))-1
NBICq <- length(coef(bestBICq$BestModel))-1
ans<-c(NAIC, NBIC, NEBIC, NBICq)
names(ans)<-c("AIC", "BIC", "BICg", "BICq")
ans
# AIC  BIC EBIC BICq 
#   3    1    0    0 

#Example 2. bestglm with BICq
#Find best model. Default is BICq with q=0.25
data(znuclear) #standardized data. 
#Rest of examples assume this dataset is loaded.
out<-bestglm(znuclear, IC="BICq")
out
#The optimal range for q
out$Bestq
#The possible models that can be chosen
out$qTable
#The best models for each subset size
out$Subsets
#The overall best models
out$BestModels
#
#Example 3. Normal probability plot, residuals, best model
ans<-bestglm(znuclear, IC="BICq")
e<-resid(ans$BestModel)
qqnorm(e, ylab="residuals, best model")
#
#To save time, none of the remaining examples are run
## Not run: 
#Example 4. bestglm, using EBIC, g=1
bestglm(znuclear, IC="BICg")
#EBIC with g=0.5
bestglm(znuclear, IC="BICg", t=0.5)
#
#Example 5. bestglm, CV
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
#the default CV method takes too long, set t=10 to do only
# 10 replications instead of the recommended 1000
bestglm(train, IC="CV", t=10)
bestglm(train, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1))
#Compare with DH Algorithm. Normally set REP=100 is recommended.
bestglm(train, IC="CV", CVArgs=list(Method="DH", K=10, REP=1))
#Compare LOOCV
bestglm(train, IC="LOOCV")
#
#Example 6. Optimal q for manpower dataset
data(manpower)
out<-bestglm(manpower)
out$Bestq
#
#Example 7. Factors with more than 2 levels
data(AirQuality)
bestglm(AirQuality)
#
#Example 8. Logistic regression
data(SAheart)
bestglm(SAheart, IC="BIC", family=binomial)
#BIC agrees with backward stepwise approach
out<-glm(chd~., data=SAheart, family=binomial)
step(out, k=log(nrow(SAheart)))
#but BICq with q=0.25
bestglm(SAheart, IC="BICq", t=0.25, family=binomial)
#
#Cross-validation with glm
#make reproducible results
set.seed(33997711)
#takes about 15 seconds and selects 5 variables
bestglm(SAheart, IC="CV", family=binomial)
#about 6 seconds and selects 2 variables
bestglm(SAheart, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1), family=binomial)
#Will produce an error -- NA
\dontrun{bestglm(SAheart, IC="CV", CVArgs=list(Method="DH", K=10, REP=1), family=binomial)}
\dontrun{bestglm(SAheart, IC="LOOCV", family=binomial)}
#
#Example 9. Model with no intercept term
X<-matrix(rnorm(200*3), ncol=3)
b<-c(0, 1.5, 0)
y<-X%*%b + rnorm(40)
Xy<-data.frame(as.matrix.data.frame(X), y=y)
bestglm(Xy, intercept=FALSE) 

## End(Not run)

#Example 1. 
#White noise test.
set.seed(123321123)
p<-25   #number of inputs
n<-100  #number of observations
X<-matrix(rnorm(n*p), ncol=p)
y<-rnorm(n)
Xy<-as.data.frame(cbind(X,y))
names(Xy)<-c(paste("X",1:p,sep=""),"y")
bestAIC <- bestglm(Xy, IC="AIC")
bestBIC <- bestglm(Xy, IC="BIC")
bestEBIC <- bestglm(Xy, IC="BICg")
bestBICq <- bestglm(Xy, IC="BICq")
NAIC <- length(coef(bestAIC$BestModel))-1
NBIC <- length(coef(bestBIC$BestModel))-1
NEBIC <- length(coef(bestEBIC$BestModel))-1
NBICq <- length(coef(bestBICq$BestModel))-1
ans<-c(NAIC, NBIC, NEBIC, NBICq)
names(ans)<-c("AIC", "BIC", "BICg", "BICq")
ans
# AIC  BIC EBIC BICq 
#   3    1    0    0 

#Example 2. bestglm with BICq
#Find best model. Default is BICq with q=0.25
data(znuclear) #standardized data. 
#Rest of examples assume this dataset is loaded.
out<-bestglm(znuclear, IC="BICq")
out
#The optimal range for q
out$Bestq
#The possible models that can be chosen
out$qTable
#The best models for each subset size
out$Subsets
#The overall best models
out$BestModels
#
#Example 3. Normal probability plot, residuals, best model
ans<-bestglm(znuclear, IC="BICq")
e<-resid(ans$BestModel)
qqnorm(e, ylab="residuals, best model")
#
#To save time, none of the remaining examples are run
## Not run: 
#Example 4. bestglm, using EBIC, g=1
bestglm(znuclear, IC="BICg")
#EBIC with g=0.5
bestglm(znuclear, IC="BICg", t=0.5)
#
#Example 5. bestglm, CV
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
#the default CV method takes too long, set t=10 to do only
# 10 replications instead of the recommended 1000
bestglm(train, IC="CV", t=10)
bestglm(train, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1))
#Compare with DH Algorithm. Normally set REP=100 is recommended.
bestglm(train, IC="CV", CVArgs=list(Method="DH", K=10, REP=1))
#Compare LOOCV
bestglm(train, IC="LOOCV")
#
#Example 6. Optimal q for manpower dataset
data(manpower)
out<-bestglm(manpower)
out$Bestq
#
#Example 7. Factors with more than 2 levels
data(AirQuality)
bestglm(AirQuality)
#
#Example 8. Logistic regression
data(SAheart)
bestglm(SAheart, IC="BIC", family=binomial)
#BIC agrees with backward stepwise approach
out<-glm(chd~., data=SAheart, family=binomial)
step(out, k=log(nrow(SAheart)))
#but BICq with q=0.25
bestglm(SAheart, IC="BICq", t=0.25, family=binomial)
#
#Cross-validation with glm
#make reproducible results
set.seed(33997711)
#takes about 15 seconds and selects 5 variables
bestglm(SAheart, IC="CV", family=binomial)
#about 6 seconds and selects 2 variables
bestglm(SAheart, IC="CV", CVArgs=list(Method="HTF", K=10, REP=1), family=binomial)
#Will produce an error -- NA
\dontrun{bestglm(SAheart, IC="CV", CVArgs=list(Method="DH", K=10, REP=1), family=binomial)}
\dontrun{bestglm(SAheart, IC="LOOCV", family=binomial)}
#
#Example 9. Model with no intercept term
X<-matrix(rnorm(200*3), ncol=3)
b<-c(0, 1.5, 0)
y<-X%*%b + rnorm(40)
Xy<-data.frame(as.matrix.data.frame(X), y=y)
bestglm(Xy, intercept=FALSE) 

## End(Not run)

Cross-validation using delete-d method.

Description

The delete-d method for cross-validation uses a random sample of d observations as the validation sample. This is repeated many times.

Usage

CVd(X, y, d = ceiling(n * (1 - 1/(log(n) - 1))), REP = 100, family = gaussian, ...)
CVd(X, y, d = ceiling(n * (1 - 1/(log(n) - 1))), REP = 100, family = gaussian, ...)

Arguments

`X`	training inputs
`y`	training output
`d`	size of validation sample
`REP`	number of replications
`family`	glm family
`...`	optional arguments passed to `glm` or `lm`

Details

Shao (1993, 1997) suggested the delete-d algorithm implemented in this function. In this algorithm, a random sample of d observations are taken as the validation sample. This random sampling is repeated REP times. Shao (1997, p.234, eqn. 4.5 and p.236) suggests $d= n(1-1/(log n - 1))$ , This is obtained by taking $\lambda_n = log n$ on page 236 (Shao, 1997). As shown in the table Shao's recommended choice of the d parameter corresponds to validation samples that are typically much larger that used in 10-fold or 5-fold cross-validation. LOOCV corresponds to d=1 only!

n	d	K=10	K=5
50	33	5	10
100	73	10	20
200	154	20	40
500	405	50	100
1000	831	100	200

Value

Vector of two components comprising the cross-validation MSE and its sd based on the MSE in each validation sample.

Author(s)

A.I. McLeod and C. Xu

References

Shao, Jun (1993). Linear Model Selection by Cross-Validation. Journal of the American Statistical Assocation 88, 486-494.

Shao, Jun (1997). An Asymptotic Theory for Linear Model Selection. Statistica Sinica 7, 221-264.

Examples

#Example 1. delete-d method
#For the training set, n=67. So 10-fold CV is like using delete-d
#with d=7, approximately.
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
set.seed(123321123)
CVd(X, y, d=7, REP=10)
#should set to 1000. Used 10 to save time in example.
#Example 1. delete-d method
#For the training set, n=67. So 10-fold CV is like using delete-d
#with d=7, approximately.
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
set.seed(123321123)
CVd(X, y, d=7, REP=10)
#should set to 1000. Used 10 to save time in example.

Adjusted K-fold Cross-Validation

Description

An adjustment to K-fold cross-validation is made to reduce bias.

Usage

CVDH(X, y, K = 10, REP = 1)
CVDH(X, y, K = 10, REP = 1)

Arguments

`X`	training inputs
`y`	training output
`K`	size of validation sample
`REP`	number of replications

Details

Algorithm 6.5 (Davison and Hinkley, p.295) is implemented.

Value

Vector of two components comprising the cross-validation MSE and its sd based on the MSE in each validation sample.

Author(s)

A.I. McLeod and C. Xu

References

Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and their Application. Cambridge University Press.

Examples

#Example 1. Variability in 10-fold CV with Davison-Hartigan Algorithm.
#Plot the CVs obtained by using 10-fold CV on the best subset
#model of size 2 for the prostate data. We assume the best model is
#the model with the first two inputs and then we compute the CV's
#using 10-fold CV, 100 times. The result is summarized by a boxplot as well 
#as the sd.
NUMSIM<-10
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
cvs<-numeric(NUMSIM)
set.seed(123321123)
for (isim in 1:NUMSIM)
    cvs[isim]<-CVDH(X,y,K=10,REP=1)[1]
summary(cvs)
#Example 1. Variability in 10-fold CV with Davison-Hartigan Algorithm.
#Plot the CVs obtained by using 10-fold CV on the best subset
#model of size 2 for the prostate data. We assume the best model is
#the model with the first two inputs and then we compute the CV's
#using 10-fold CV, 100 times. The result is summarized by a boxplot as well 
#as the sd.
NUMSIM<-10
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
cvs<-numeric(NUMSIM)
set.seed(123321123)
for (isim in 1:NUMSIM)
    cvs[isim]<-CVDH(X,y,K=10,REP=1)[1]
summary(cvs)

K-fold Cross-Validation

Description

K-fold cross-validation.

Usage

CVHTF(X, y, K = 10, REP = 1, family = gaussian, ...)
CVHTF(X, y, K = 10, REP = 1, family = gaussian, ...)

Arguments

`X`	training inputs
`y`	training output
`K`	size of validation sample
`REP`	number of replications
`family`	glm family
`...`	optional arguments passed to `glm` or `lm`

Details

HTF (2009) describe K-fold cross-validation. The observations are partitioned into K non-overlapping subsets of approximately equal size. Each subset is used as the validation sample while the remaining K-1 subsets are used as training data. When $K=n$ , where n is the number of observations the algorithm is equivalent to leave-one-out CV. Normally $K=10$ or $K=5$ are used. When $K<n-1$ , their are may be many possible partitions and so the results of K-fold CV may vary somewhat depending on the partitions used. In our implementation, random partitions are used and we allow for many replications. Note that in the Shao's delete-d method, random samples are used to select the valiation data whereas in this method the whole partition is selected as random. This is acomplished using, fold <- sample(rep(1:K,length=n)). Then fold indicates each validation sample in the partition.

Value

Vector of two components comprising the cross-validation MSE and its sd based on the MSE in each validation sample.

Author(s)

A.I. McLeod and C. Xu

References

Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning. 2nd Ed. Springer-Verlag.

Examples

#Example 1. 10-fold CV
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
CVHTF(X,y,K=10,REP=1)[1]
#Example 1. 10-fold CV
data(zprostate)
train<-(zprostate[zprostate[,10],])[,-10]
X<-train[,1:2]
y<-train[,9]
CVHTF(X,y,K=10,REP=1)[1]

Detroit homicide data for 1961-73 used in the book Subset Regression by A.J. Miller

Description

For convenience we have labelled the input variables 1 through 11 to be consistent with the notation used in Miller (2002). Only the first 11 variables were used in Miller's analyses. The best fitting subset regression with these 11 variables, uses only 3 inputs and has a residual sum of squares of 6.77 while using forward selection produces a best fit with 3 inputs with residual sum of squares 21.19. Backward selection and stagewise methods produce similar results. It is remarkable that there is such a big difference. Note that the usual forward and backward selection algorithms may fail since the linear regression using 11 variables gives essentially a perfect fit.

Usage

data(Detroit)data(Detroit)

Format

A data frame with 13 observations on the following 14 variables.

FTP.1: Full-time police per 100,000 population
UEMP.2: Percent unemployed in the population
MAN.3: Number of manufacturing workers in thousands
LIC.4: Number of handgun licences per 100,000 population
GR.5: Number of handgun registrations per 100,000 population
CLEAR.6: Percent homicides cleared by arrests
WM.7: Number of white males in the population
NMAN.8: Number of non-manufacturing workers in thousands
GOV.9: Number of government workers in thousands
HE.10: Average hourly earnings
WE.11: Average weekly earnings
ACC: Death rate in accidents per 100,000 population
ASR: Number of assaults per 100,000 population
HOM: Number of homicides per 100,000 of population

Details

The data were orginally collected and discussed by Fisher (1976) but the complete dataset first appeared in Gunst and Mason (1980, Appendix A). Miller (2002) discusses this dataset throughout his book. The data were obtained from StatLib.

Source

http://lib.stat.cmu.edu/datasets/detroit

References

Fisher, J.C. (1976). Homicide in Detroit: The Role of Firearms. Criminology, vol.14, 387-400.

Gunst, R.F. and Mason, R.L. (1980). Regression analysis and its application: A data-oriented approach. Marcel Dekker.

Miller, A. J. (2002). Subset Selection in Regression. 2nd Ed. Chapman & Hall/CRC. Boca Raton.

Examples

#Detroit data example
data(Detroit)
#As in Miller (2002) columns 1-11 are used as inputs
p<-11
#For possible comparison with other algorithms such as LARS
#  it is preferable to work with the scaled inputs.
#From Miller (2002, Table 3.14), we see that the
#best six inputs are: 1, 2, 4, 6, 7, 11
X<-as.data.frame(scale(Detroit[,c(1,2,4,6,7,11)]))
y<-Detroit[,ncol(Detroit)]
Xy<-cbind(X,HOM=y)
#Use backward stepwise regression with BIC selects full model
out <- lm(HOM~., data=Xy)
step(out, k=log(nrow(Xy)))
#
#Same story with exhaustive search algorithm
out<-bestglm(Xy, IC="BIC")
out
#But many coefficients have p-values that are quite large considering
#  the selection bias. Note: 1, 6 and 7 are all about 5% only.
#We can use BICq to reduce the number of variables.
#The qTable let's choose q for other possible models,
out$qTable
#This suggest we try q=0.05 or q=0.0005 
bestglm(Xy,IC="BICq", t=0.05)
bestglm(Xy,IC="BICq", t=0.00005)
#It is interesting that the subset model of size 2 is not a subset
# itself of the size 3 model. These results agree with 
#Miller (2002, Table 3.14).
#
#Using delete-d CV with d=4 suggests variables 2,4,6,11
set.seed(1233211)
bestglm(Xy, IC="CV", CVArgs=list(Method="d", K=4, REP=50))
#Detroit data example
data(Detroit)
#As in Miller (2002) columns 1-11 are used as inputs
p<-11
#For possible comparison with other algorithms such as LARS
#  it is preferable to work with the scaled inputs.
#From Miller (2002, Table 3.14), we see that the
#best six inputs are: 1, 2, 4, 6, 7, 11
X<-as.data.frame(scale(Detroit[,c(1,2,4,6,7,11)]))
y<-Detroit[,ncol(Detroit)]
Xy<-cbind(X,HOM=y)
#Use backward stepwise regression with BIC selects full model
out <- lm(HOM~., data=Xy)
step(out, k=log(nrow(Xy)))
#
#Same story with exhaustive search algorithm
out<-bestglm(Xy, IC="BIC")
out
#But many coefficients have p-values that are quite large considering
#  the selection bias. Note: 1, 6 and 7 are all about 5% only.
#We can use BICq to reduce the number of variables.
#The qTable let's choose q for other possible models,
out$qTable
#This suggest we try q=0.05 or q=0.0005 
bestglm(Xy,IC="BICq", t=0.05)
bestglm(Xy,IC="BICq", t=0.00005)
#It is interesting that the subset model of size 2 is not a subset
# itself of the size 3 model. These results agree with 
#Miller (2002, Table 3.14).
#
#Using delete-d CV with d=4 suggests variables 2,4,6,11
set.seed(1233211)
bestglm(Xy, IC="CV", CVArgs=list(Method="d", K=4, REP=50))

Scaled Variables Dependency Plots: Output vs Inputs

Description

A lattice grid plot is produced for the output vs. each input. The variables are scaled to have mean zero and variance one.

Usage

dgrid(XyDF, span=0.8)
dgrid(XyDF, span=0.8)

Arguments

`XyDF`	Must be a dataframe with the last column corresponding to the output
`span`	smoothing parameter for loess

Value

a lattice plot

Author(s)

A. I. McLeod

Examples

data(mcdonald)
dgrid(mcdonald)
data(mcdonald)
dgrid(mcdonald)

Forest fires in Montesinho natural park. Standardized inputs.

Description

The forest fire data were collected during January 2000 to December 2003 for fires in the Montesinho natural park located in the northeast region of Portugal. The response variable of interest was area burned in ha. When the area burned as less than one-tenth of a hectare, the response variable as set to zero. In all there were 517 fires and 247 of them recorded as zero. The region was divided into a 10-by-10 grid with coordinates X and Y running from 1 to 9. The categorical variable xyarea indicates the region in this grid for the fire.

Usage

data(Fires)data(Fires)

Format

A data frame with 517 observations on the following 12 variables. All quantitative variables have been standardized.

xyarea: a factor with 36 levels
month: an ordered factor with 12 levels
day: an ordered factor with 7 levels
FFMC: fine fuel moisture code
DMC: Duff moisture code
DC: drought code
ISI: initial spread index
temp: average ambient temperature
RH: a numeric vector
wind: wind speed
rain: rainfall
lburned: log(x+1), x is burned area with x=0 for small fires

Details

The original data may be found at the website below as well as an analysis. The quantitative variables in this dataset have been standardized. For convenience, the original data is provided in MontesinhoFires.

Source

http://archive.ics.uci.edu/ml/datasets/Forest+Fires

References

P. Cortez and A. Morais, 2007. A Data Mining Approach to Predict Forest Fires using Meteorological Data. In J. Neves, M. F. Santos and J. Machado Eds., New Trends in Artificial Intelligence, Proceedings of the 13th EPIA 2007 - Portuguese Conference on Artificial Intelligence, December, Guimaraes, Portugal, pp. 512-523, 2007.

Examples

data(Fires)
names(Fires)
#ANOVA for xyarea is significant at 1.1%.
summary(aov(lburned~xyarea, data=Fires))
data(Fires)
names(Fires)
#ANOVA for xyarea is significant at 1.1%.
summary(aov(lburned~xyarea, data=Fires))

Fitted values in PCR and PLS.

Description

The fitted values are returned given the output from pcreg.

Usage

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

Arguments

`object`	`object` output
`...`	additional parameters

Details

Method function for pcreg.

Value

residuals

Author(s)

A. I. McLeod

Examples

fitted(pcreg(mcdonald, scale=TRUE))
fitted(pcreg(mcdonald, scale=TRUE))

Multipanel Display and Table Glmnet CV Output.

Description

Four panels.

Usage

glmnetGridTable(XyList, alpha = 0, nfolds=10, family = "gaussian")
glmnetGridTable(XyList, alpha = 0, nfolds=10, family = "gaussian")

Arguments

`XyList`	input
`alpha`	elastic net parameter
`nfolds`	Number of folds, K, in regularized K-fold CV, must be >3 and <=10.
`family`	distribution

Details

tba

Value

plot produced by side-effect. Table.

Note

Set random seed beforehand if you want reproducibility.

Author(s)

A. I. McLeod

Examples

set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)
set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)

Glmnet Prediction Using CVAV.

Description

Predict by averaging the predictions from cv.glmnet().

Usage

glmnetPredict(XyList, NREP = 15, alpha = 0, nfolds=10, 
 family = c("gaussian", "binomial", "poisson", "multinomial"))
glmnetPredict(XyList, NREP = 15, alpha = 0, nfolds=10, 
 family = c("gaussian", "binomial", "poisson", "multinomial"))

Arguments

`XyList`	list with components XyTr, XTr, yTr, XTe.
`NREP`	number of replications to use in average
`alpha`	elastic net parameter
`nfolds`	Number of folds, K, in regularized K-fold CV, must be >3 and <=10.
`family`	model

Value

vector with predictions

Author(s)

A. I. McLeod

Examples

set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)
yh <- glmnetPredict(out, NREP=5)
sqrt(mean((out$yTe - yh)^2))
set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)
yh <- glmnetPredict(out, NREP=5)
sqrt(mean((out$yTe - yh)^2))

Predictions on Test Data with Grpreg

Description

A dataframe is partitioned randomly into training and test samples. The function grpreg::grpreg() is used to fit the training data using Lasso, SCAD and MCP penalty functions. The BIC criterion is used to selecting the penalty parameter lambda.

Usage

grpregPredict(Xy, trainFrac = 2/3, XyList=NULL)
grpregPredict(Xy, trainFrac = 2/3, XyList=NULL)

Arguments

`Xy`	a dataframe that may contain factor variables
`trainFrac`	the fraction of data to be used for training
`XyList`	instead of supplying Xy you can provide XyList.

Value

vector of RMSEs

Examples

grpregPredict(mcdonald)
grpregPredict(mcdonald)

Simulated Linear Regression (Train) with Nine Highly Correlated Inputs

Description

The script that generated this data is given below.

Usage

data("hivif")data("hivif")

Format

A data frame with 1000 observations on the following 10 variables.

x1: a numeric vector
x2: a numeric vector
x3: a numeric vector
x4: a numeric vector
x5: a numeric vector
x6: a numeric vector
x7: a numeric vector
x8: a numeric vector
x9: a numeric vector
y: a numeric vector

Examples

#Simple example
data(hivif)
lm(y ~ ., data=hivif)
#
#This example shows how the original data was simulated and
#how additional test data may be simulated.
## Not run: 
 set.seed(778851) #needed for original training data
 n <- 100
 p <- 9 #9 covariates plus intercept
 sig <- toeplitz(0.9^(0:(p-1)))
 X <- MASS::mvrnorm(n=n, rep(0, p), Sigma=sig)
 colnames(X) <- paste0("x", 1:p)
 b <- c(0,-0.3,0,0,-0.3,0,0,0.3,0.3) #
 names(b) <- paste0("x", 1:p)
 y <- 1 +  X
 Xy <- cbind(as.data.frame.matrix(X), y=y) #=hivif
#Test data
 nTe <- 10^3
 XTe <- MASS::mvrnorm(n=nTe, rep(0, p), Sigma=sig)
 colnames(XTe) <- paste0("x", 1:p)
  yTe <- 1 +  XTe
 XyTe <- cbind(as.data.frame.matrix(XTe), y=yTe) #test data
 ans <- lm(y ~ ., data=Xy) #fit training data
 mean((XyTe$y - predict(ans, newdata=XyTe))^2) #MSE on test data
 
## End(Not run)
#Simple example
data(hivif)
lm(y ~ ., data=hivif)
#
#This example shows how the original data was simulated and
#how additional test data may be simulated.
## Not run: 
 set.seed(778851) #needed for original training data
 n <- 100
 p <- 9 #9 covariates plus intercept
 sig <- toeplitz(0.9^(0:(p-1)))
 X <- MASS::mvrnorm(n=n, rep(0, p), Sigma=sig)
 colnames(X) <- paste0("x", 1:p)
 b <- c(0,-0.3,0,0,-0.3,0,0,0.3,0.3) #
 names(b) <- paste0("x", 1:p)
 y <- 1 +  X
 Xy <- cbind(as.data.frame.matrix(X), y=y) #=hivif
#Test data
 nTe <- 10^3
 XTe <- MASS::mvrnorm(n=nTe, rep(0, p), Sigma=sig)
 colnames(XTe) <- paste0("x", 1:p)
  yTe <- 1 +  XTe
 XyTe <- cbind(as.data.frame.matrix(XTe), y=yTe) #test data
 ans <- lm(y ~ ., data=Xy) #fit training data
 mean((XyTe$y - predict(ans, newdata=XyTe))^2) #MSE on test data
 
## End(Not run)

Iowa School Test

Description

Dataset on poverty and academic performance.

Usage

data("Iowa")data("Iowa")

Format

A data frame with 133 observations on the following 3 variables.

City: a factor with 6 levels Cedar Rapids Davenport Des Moines Iowa City Sioux City Waterloo
Poverty: percentage subsidized
Test: achievement test score

Details

There are n=133 average test scores for schools in the K=6 largest cities. The test score offers a standardized measure of academic achievement. The purpose of the study is to investigate if there is a relationship between academic achievement, as measured by the test, and poverty. It is expected that students from economically disadvantaged backgrounds will do less well. Data on the average income in the school district was not available so a proxy variable for poverty was used. The percentage of students who received subsidized meals was available so this was used as the "Poverty" variable.

Source

Abraham and Ledholter, Introduction to Regression, Wiley.

Examples

data(Iowa)
table(Iowa$City)
data(Iowa)
table(Iowa$City)

Leave-one-out cross-validation

Description

An observation is removed and the model is fit the the remaining data and this fit used to predict the value of the deleted observation. This is repeated, n times, for each of the n observations and the mean square error is computed.

Usage

LOOCV(X, y)
LOOCV(X, y)

Arguments

`X`	training inputs
`y`	training output

Details

LOOCV for linear regression is exactly equivalent to the PRESS method suggested by Allen (1971) who also provided an efficient algorithm.

Value

Vector of two components comprising the cross-validation MSE and its sd based on the MSE in each validation sample.

Author(s)

A.I. McLeod and C. Xu

References

Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning. 2nd Ed.

Allen, D.M. (1971). Mean Square Error of Prediction as a Criterion for Selecting Variables. Technometrics, 13, 469 -475.

Examples

#Example. Compare LOO CV with K-fold CV.
#Find CV MSE's for LOOCV and compare with K=5, 10, 20, 40, 50, 60
#Takes about 30 sec
## Not run: 
 data(zprostate)
 train<-(zprostate[zprostate[,10],])[,-10]
 X<-train[,1:2]
 y<-train[,9]
 CVLOO<-LOOCV(X,y)
 KS<-c(5,10,20,40,50,60)
 nKS<-length(KS)
 cvs<-numeric(nKS)
 set.seed(1233211231)
 for (iK in 1:nKS)
    cvs[iK]<-CVDH(X,y,K=KS[iK],REP=10)[1]
 boxplot(cvs)
 abline(h=CVLOO, lwd=3, col="red")
 title(sub="Boxplot of CV's with K=5,10,20,40,50,60 and LOO CV in red")
 
## End(Not run)
#Example. Compare LOO CV with K-fold CV.
#Find CV MSE's for LOOCV and compare with K=5, 10, 20, 40, 50, 60
#Takes about 30 sec
## Not run: 
 data(zprostate)
 train<-(zprostate[zprostate[,10],])[,-10]
 X<-train[,1:2]
 y<-train[,9]
 CVLOO<-LOOCV(X,y)
 KS<-c(5,10,20,40,50,60)
 nKS<-length(KS)
 cvs<-numeric(nKS)
 set.seed(1233211231)
 for (iK in 1:nKS)
    cvs[iK]<-CVDH(X,y,K=KS[iK],REP=10)[1]
 boxplot(cvs)
 abline(h=CVLOO, lwd=3, col="red")
 title(sub="Boxplot of CV's with K=5,10,20,40,50,60 and LOO CV in red")
 
## End(Not run)

Hospital manpower data

Description

The goal of this study is to predict the manpower requirement as given in the output variable Hours given the five other input variables. Data is from Table 3.8 of Myers (1990). See also Examples 3.8, 4.5, 8.8.

Usage

data(manpower)data(manpower)

Format

A data frame with 17 observations. The output variable is Hours and the inputs are Load, Xray, BedDays, AreaPop and Stay. The site 1 through 17 is indicated by the row name.

Load: a numeric vector
Xray: a numeric vector
BedDays: a numeric vector
AreaPop: a numeric vector
Stay: a numeric vector
Hours: a numeric vector

Details

This data illustrates the multicollinearity problem and the use of VIF to identify it. It provides an illustrative example for ridge regression and more modern methods such as lasso and lars.

Source

Myers (1990) indicates the source was "Procedures and Analysis for Staffing Standards Development: Data/Regression Analysis Handbook", Navy Manpower and Material Analysis Center, San Diego, 1979.

References

Myers, R. (1990). Classical and Modern Regression with Applications. The Duxbury Advanced Series in Statistics and Decision Sciences. Boston: PWS-KENT Publishing Company.

Examples

data(manpower)
data(manpower)

Pollution dataset from McDonald and Schwing (1973)

Description

Regression data used to illustrate ridge regression

Usage

data("mcdonald")data("mcdonald")

Format

A data frame with 60 observations on the following 16 variables.

PREC: Average annual precipitation in inches
JANT: Average January temperature in degrees F
JULT: Same for July
OVR65: Percent of 1960 SMSA population aged 65 or older
POPN: Average household size
EDUC: Median school years completed by those over 22
HOUS: Percent of housing units which are sound & with all facilities
DENS: Population per sq. mile in urbanized areas, 1960
NONW: Percent non-white population in urbanized areas, 1960
WWDRK: Percent employed in white collar occupations
POOR: Percent of families with income < $3000
HC: Relative hydrocarbon pollution potential
NOX: Same for nitric oxides
SOx: Same for sulphur dioxide
HUMID: Annual average percent relative humidity at 1pm
MORT: Total age-adjusted mortality rate per 100,000

Details

Ridge regression example

Source

Gary C. McDonald and Richard C. Schwing (1973), Instabilities of Regression Estimates Relating Air Pollution to Mortality, Technometrics 15/3, 463-481.

Examples

data(mcdonald)
vifx(mcdonald[, -ncol(mcdonald)])
data(mcdonald)
vifx(mcdonald[, -ncol(mcdonald)])

Forest fires in Montesinho natural park

Description

Usage

data(MontesinhoFires)data(MontesinhoFires)

Format

A data frame with 517 observations on the following 13 variables.

X: X coordinate for region, 0-10
Y: X coordinate for region, 0-10
month: an ordered factor with 12 levels
day: an ordered factor with 7 levels
FFMC: fine fuel moisture code
DMC: Duff moisture code
DC: drought code
ISI: initial spread index
temp: average ambient temperature
RH: a numeric vector
wind: wind speed
rain: rainfall
burned: area burned in hectares

Details

This is the original data taken from the website below.

Source

http://archive.ics.uci.edu/ml/datasets/Forest+Fires

References

Examples

data(MontesinhoFires)
names(MontesinhoFires)
data(Fires)
names(Fires)
#Anova for month
summary(aov(burned~month, data=MontesinhoFires))
data(MontesinhoFires)
names(MontesinhoFires)
data(Fires)
names(Fires)
#Anova for month
summary(aov(burned~month, data=MontesinhoFires))

Nearest Neighbour Regression Prediction

Description

Given training/test data in the predictions on the test data computed. L1, L2 and correlation distances may be used. The data is sphered prior to making the NN predictions.

Usage

NNPredict(XyList, dist = c("L2", "COR", "L1"))
NNPredict(XyList, dist = c("L2", "COR", "L1"))

Arguments

`XyList`	list with six elements
`dist`	distance used

Value

vector of predictions

Author(s)

A. I. McLeod

Examples

AQ <- airquality[complete.cases(airquality),c(2,3,4,1)]
XyList <- trainTestPartition(AQ)
NNPredict(XyList) 
AQ <- airquality[complete.cases(airquality),c(2,3,4,1)]
XyList <- trainTestPartition(AQ)
NNPredict(XyList)

Utility function. Implements the 1-sd rule.

Description

The CV and its standard devation are provided for a range of models ordered by the number of parameters estimated.

Usage

oneSDRule(CVout)
oneSDRule(CVout)

Arguments

CVout

A matrix with two columns. First column is the CV and second, its sd. Row ordering is from fewest parameter to most.

Value

The row corresponding to the best model.

Author(s)

A.I. McLeod and C. Xu

References

Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning. 2nd Ed. Springer-Verlag.

Examples

CV<-c(1.4637799,0.7036285,0.6242480,0.6069406,0.6006877,0.6005472,0.5707958,
      0.5907897,0.5895489)
CVsd<-c(0.24878992,0.14160499,0.08714908,0.11376041,0.08522291,
 0.11897327,0.07960879,0.09235052,0.12860983)
CVout <- matrix(c(CV,CVsd), ncol=2)
oneSDRule(CVout)
CV<-c(1.4637799,0.7036285,0.6242480,0.6069406,0.6006877,0.6005472,0.5707958,
      0.5907897,0.5895489)
CVsd<-c(0.24878992,0.14160499,0.08714908,0.11376041,0.08522291,
 0.11897327,0.07960879,0.09235052,0.12860983)
CVout <- matrix(c(CV,CVsd), ncol=2)
oneSDRule(CVout)

Principal Component and Partial Least Squares Regression

Description

Regression using the principal components or latent variables as inputs. The best model is selected using components 1, 2, ..., r, where r, the number of components to use is determined by the AIC or BIC.

Usage

pcreg(Xy, scale = TRUE, method = c("PC", "LV"), ic = c("BIC", "AIC"))
pcreg(Xy, scale = TRUE, method = c("PC", "LV"), ic = c("BIC", "AIC"))

Arguments

`Xy`	dataframe with variable names in columns
`scale`	Whether or not to scale. Default is TRUE.
`method`	either principal components, "PC", or partial least squares latent variables, "LV"
`ic`	"BIC" or "AIC"

Value

An S3 class list "pcreg" with components

`lmfit`	lm model
`PLSFit`	column sd
`Z`	matrix of principal components or latent vector
`method`	'pcr' or 'pls'

Author(s)

A. I. McLeod

Examples

pcreg(mcdonald, scale=TRUE, method="PC")
pcreg(mcdonald, scale=TRUE, method="LV")
pcreg(mcdonald, scale=TRUE, method="PC")
pcreg(mcdonald, scale=TRUE, method="LV")

Diagnostic plots for PCR and PLS

Description

Diagnostic plots available with lm-objects are provided.

Usage

## S3 method for class 'pcreg'
plot(x, ...)
## S3 method for class 'pcreg'
plot(x, ...)

Arguments

`x`	`x` output from pcreg(). It has S3 class 'pcreg'.
`...`	additional parameters

Details

See plot method for S3 class 'lm'.

Value

Nothing. The plot is produced.

Author(s)

A. I. McLeod

Examples

ans <- pcreg(mcdonald, scale=TRUE)
plot(ans)
ans <- pcreg(mcdonald, scale=TRUE)
plot(ans)

Plot Regularization Path and One Standard Deviation Rule

Description

Takes input either matrix with 2 columns or output from caret::train() and produces a plot showing the best model selected using the 1 SD rule.

Usage

plot1SDRule(ans, main = "", sub = "", xlab = "df", ylab = "EPE")
plot1SDRule(ans, main = "", sub = "", xlab = "df", ylab = "EPE")

Arguments

`ans`	matrix or output from train
`main`	optional plot title
`sub`	optional plot subtitle
`xlab`	optional x-axis label
`ylab`	optional y-axis label

Value

tuning parameter value for best model

Author(s)

A. I. McLeod

References

Hastie, Tibsharani and Friedman, "Elements of Statistical Learning".

Examples

CV<-c(1.4637799,0.7036285,0.6242480,0.6069406,0.6006877,0.6005472,0.5707958,
      0.5907897,0.5895489)
CVsd<-c(0.24878992,0.14160499,0.08714908,0.11376041,0.08522291,
 0.11897327,0.07960879,0.09235052,0.12860983)
CVout <- matrix(c(CV,CVsd), ncol=2)
oneSDRule(CVout)
CV<-c(1.4637799,0.7036285,0.6242480,0.6069406,0.6006877,0.6005472,0.5707958,
      0.5907897,0.5895489)
CVsd<-c(0.24878992,0.14160499,0.08714908,0.11376041,0.08522291,
 0.11897327,0.07960879,0.09235052,0.12860983)
CVout <- matrix(c(CV,CVsd), ncol=2)
oneSDRule(CVout)

Predict Method for Pcreg.

Description

Prediction for models fit using pcreg().

Usage

## S3 method for class 'pcreg'
predict(object, newdata, ...)
## S3 method for class 'pcreg'
predict(object, newdata, ...)

Arguments

`object`	the S3 class object produced as output from the function pcreg()
`newdata`	dataframe with new data and with same column names as used in the original argument to pcreg.
`...`	additional arguments

Details

The prediction method, predict.mvr(), which is available in the pls package is used. We take advantage of this since it avoids fussing with scaling issues since it is automatically handled for us by predict.mvr()

Value

the predicted values

Author(s)

A. I. McLeod

Examples

XyList <- trainTestPartition(mcdonald)
XyTr <- XyList$XyTr
XyTe <- XyList$XyTe
ans <- pcreg(XyTr, scale=TRUE)
predict(ans, newdata=XyTe)
XyList <- trainTestPartition(mcdonald)
XyTr <- XyList$XyTr
XyTe <- XyList$XyTe
ans <- pcreg(XyTr, scale=TRUE)
predict(ans, newdata=XyTe)

Print method for 'bestglm' object

Description

A brief description of the best fit is given.

Usage

## S3 method for class 'bestglm'
print(x, ...)
## S3 method for class 'bestglm'
print(x, ...)

Arguments

`x`	Output from the bestglm function
`...`	optional arguments

Value

No value. Output to terminal only.

Author(s)

A.I. McLeod and C. Xu

Examples

data(znuclear)
bestglm(znuclear)
data(znuclear)
bestglm(znuclear)

Print method for 'pcreg' object

Description

A brief description of the best fit is given.

Usage

## S3 method for class 'pcreg'
print(x, ...)
## S3 method for class 'pcreg'
print(x, ...)

Arguments

`x`	Output from the pcreg function
`...`	optional arguments

Value

No value. Output to terminal only.

Author(s)

A.I. McLeod and C. Xu

Examples

pcreg(znuclear, scale=TRUE)
pcreg(znuclear, scale=TRUE)

Residuals Fitted PCR or PLS

Description

The residuals from a model fitted using pcreg are returned.

Usage

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

Arguments

`object`	`object` output
`...`	additional parameters

Details

Method function for pcreg.

Value

residuals

Author(s)

A. I. McLeod

Examples

resid(pcreg(mcdonald, scale=TRUE))
resid(pcreg(mcdonald, scale=TRUE))

Abrasion loss for various hardness and tensile strength

Description

The data come from an experiment to investigate how the resistance of rubber to abrasion is affected by the hardness of the rubber and its tensile strength.

Usage

data(rubber)data(rubber)

Format

A data frame with 30 observations on the following 3 variables.

hardness: hardness in degree Shore
tensile.strength: tensile strength in kg per square meter
abrasion.loss: abrasion loss in gram per hour
ts.low: tensile strength minus the breakpoint 180 km/m^2
ts.high: tensile strength minus the breakpoint 180 km/m^2

Source

Hand, D.J., Daly, F., Lunn, A.D., McConway, K.J. and Ostrowski, E. (1993). A Handbook of Small Datasets. Chapman and Hall.

References

Cleveland, W. S. (1993). Visualizing data. Hobart Press, Summit: New Jersey.

Davies, O.L. and Goldsmith, P.L.(1972) Statistical methods in Research and Production.

Examples

data(rubber)
ans <- lm(abrasion.loss~hardness+tensile.strength, data=rubber)
data(rubber)
ans <- lm(abrasion.loss~hardness+tensile.strength, data=rubber)

South African Hearth Disease Data

Description

A retrospective sample of males in a heart-disease high-risk region of the Western Cape, South Africa.

Usage

data(SAheart)data(SAheart)

Format

A data frame with 462 observations on the following 10 variables.

sbp: systolic blood pressure
tobacco: cumulative tobacco (kg)
ldl: low density lipoprotein cholesterol
adiposity: a numeric vector
famhist: family history of heart disease, a factor with levels Absent Present
typea: type-A behavior
obesity: a numeric vector
alcohol: current alcohol consumption
age: age at onset
chd: response, coronary heart disease

Details

A retrospective sample of males in a heart-disease high-risk region of the Western Cape, South Africa. There are roughly two controls per case of CHD. Many of the CHD positive men have undergone blood pressure reduction treatment and other programs to reduce their risk factors after their CHD event. In some cases the measurements were made after these treatments. These data are taken from a larger dataset, described in Rousseauw et al, 1983, South African Medical Journal.

Source

Rousseauw, J., du Plessis, J., Benade, A., Jordaan, P., Kotze, J. and Ferreira, J. (1983). Coronary risk factor screening in three rural communities, South African Medical Journal 64: 430–436.

Examples

data(SAheart)
str(SAheart) 
summary(SAheart)
data(SAheart)
str(SAheart) 
summary(SAheart)

Simulated Regression Data

Description

Data a simulation study reported by Shao (1993, Table 1). The linear regression model Shao (1993, Table 2) reported 4 simulation experiments using 4 different values for the regression coefficients:

$y = 2 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + e,$

where $e$ is an independent normal error with unit variance.

The four regression coefficients for the four experiments are shown in the table below,

Experiment	$\beta_2$	$\beta_3$	$\beta_4$	$\beta_5$
1	0	0	4	0
2	0	0	4	8
3	9	0	4	8
4	9	6	4	8

The table below summarizes the probability of correct model selection in the experiment reported by Shao (1993, Table 2). Three model selection methods are compared: LOOCV (leave-one-out CV), CV(d=25) or the delete-d method with d=25 and APCV which is a very efficient computation CV method but specialized to the case of linear regression.

Experiment	LOOCV	CV(d=25)	APCV
1	0.484	0.934	0.501
2	0.641	0.947	0.651
3	0.801	0.965	0.818
4	0.985	0.948	0.999

The CV(d=25) outperforms LOOCV in all cases and it also outforms APCV by a large margin in Experiments 1, 2 and 3 but in case 4 APCV is slightly better.

Usage

data(Shao)data(Shao)

Format

A data frame with 40 observations on the following 4 inputs.

x2: a numeric vector
x3: a numeric vector
x4: a numeric vector
x5: a numeric vector

Source

Shao, Jun (1993). Linear Model Selection by Cross-Validation. Journal of the American Statistical Assocation 88, 486-494.

Examples

#In this example BICq(q=0.25) selects the correct model but BIC does not
data(Shao)
X<-as.matrix.data.frame(Shao)
b<-c(0,0,4,0)
set.seed(123321123)
#Note: matrix multiplication must be escaped in Rd file
y<-X%*%b+rnorm(40)
Xy<-data.frame(Shao, y=y)
bestglm(Xy)
bestglm(Xy, IC="BICq")

#In this example BICq(q=0.25) selects the correct model but BIC does not
data(Shao)
X<-as.matrix.data.frame(Shao)
b<-c(0,0,4,0)
set.seed(123321123)
#Note: matrix multiplication must be escaped in Rd file
y<-X%*%b+rnorm(40)
Xy<-data.frame(Shao, y=y)
bestglm(Xy)
bestglm(Xy, IC="BICq")

Sphere Data Matrix

Description

The data matrix is scaled and sphered so it is orthonormal. The Cholesky decomposition is used.

Usage

sphereX(X)
sphereX(X)

Arguments

`X`	`X` rectangular data matrix

Value

sphered matrix

Author(s)

A. I. McLeod

Examples

data(longley)
longley.x <- data.matrix(longley[, 1:6])
sphereX(longley.x)
data(longley)
longley.x <- data.matrix(longley[, 1:6])
sphereX(longley.x)

summary of 'bestglm' object

Description

An analysis of deviance and a likelihood-ratio test with p-value. The p-value is greatly exagerated due to selection.

Usage

## S3 method for class 'bestglm'
summary(object, SubsetsQ=FALSE, ...)
## S3 method for class 'bestglm'
summary(object, SubsetsQ=FALSE, ...)

Arguments

`object`	Output from the bestglm function
`SubsetsQ`	List best subsets of each size
`...`	optional arguments

Value

No value. Output to terminal only.

Author(s)

A.I. McLeod and C. Xu

Examples

data(znuclear)
summary(bestglm(znuclear))
#
#find statistical signficance of overall regression
data(Fires)
summary(bestglm(Fires, IC="BICq", t=1))
data(znuclear)
summary(bestglm(znuclear))
#
#find statistical signficance of overall regression
data(Fires)
summary(bestglm(Fires, IC="BICq", t=1))

Summary Method for Pcreg.

Description

The summary is based on the summary method for S3 class 'lm'.

Usage

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

Arguments

`object`	`object` output
`...`	additional parameters

Details

Method function for pcreg.

Value

residuals

Note

The standard errors and p-values are wrong due to selection bias.

Author(s)

A. I. McLeod

Examples

resid(pcreg(mcdonald, scale=TRUE))
resid(pcreg(mcdonald, scale=TRUE))

Partition Dataframe into Train/Test Samples

Description

Dataframe used to create training and test datasets using specified fraction for the training sample. The data matrix must be comprised of continuous variables only (no factors).

Usage

trainTestPartition(Xy, trainFrac = 2/3)
trainTestPartition(Xy, trainFrac = 2/3)

Arguments

`Xy`	Dataframe with column names, last column is the response variable and others are the regression input variables. The data matrix must be comprised of continuous variables only (no factors).
`trainFrac`	Fraction to be used for the training sample.

Value

A list with components

`XyTr`	Training dataframe.
`XTr`	Matrix, input training variables.
`yTr`	Vector, output training variable.
`XyTe`	Training dataframe.
`XTe`	Matrix, input test variables.
`yTe`	Vector, output test variable.
`XyTr`	Training dataframe.
`XyTr`	Training dataframe.
`XyTr`	Training dataframe.

Author(s)

A. I. McLeod

Examples

set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)
set.seed(7733551)
out <- trainTestPartition(mcdonald)
round(glmnetGridTable(out),4)

Variance Inflation Factor for a Design Matrix

Description

Barplot of the VIF is produced

Usage

vifx(X)
vifx(X)

Arguments

`X`	A design matrix

Details

The VIF are the diagonal elements in the inverse $t(X*) X*$ , where X* is the rescaled design matrix.

Value

vector with VIF's

Author(s)

A. I. McLeod

References

Marquardt, D. W. (1970). Generalized Inverses, Ridge Regression, Biased Linear Estimation, and Nonlinear Estimation. Technometrics 12(3), 591-612.

Examples

data(mcdonald)
vifx(mcdonald[, -ncol(mcdonald)])
data(mcdonald)
vifx(mcdonald[, -ncol(mcdonald)])

Nuclear plant data. Quantitative inputs logged and standardized.

Description

Data on 32 nuclear power plants. The response variable is cost and there are ten covariates.

Usage

data(znuclear)data(znuclear)

Format

A data frame with 32 observations on the following 12 variables. All quantitative variables, except date, have been logged and standardized to have mean 0 and variance 1.

date: Quantitative covariate. The date on which the construction permit was issued. The data are measured in years since January 1 1990 to the nearest month.
T1: Quantitative covariate. The time between application for and issue of the construction permit.
T2: Quantitative covariate. The time between issue of operating license and construction permit.
capacity: Quantitative covariate. The net capacity of the power plant (MWe).
PR: Binary covariate. Value 1, indicates the prior existence of a LWR plant at the same site.
NE: Binary covariate, located in North-East USA
CT: Binary covariate, presence of cooling tower
BW: Binary covariate, where 1 indicates that the nuclear steam supply system was manufactured by Babcock-Wilcox.
N: Quantitative covariate. The cumulative number of power plants constructed by each architect-engineer.
PT: Binary covariate, partial turnkey guarantee.
cost: Outcome. The capital cost of construction in millions of dollars adjusted to 1976 base.

Details

Davison (2003) explores fitting models to this data using forward and backward stepwise regression. In this modelling logs of quantiative variablesare used. We have also standardized this data to facilitate comparison with other techniques such as LARS and principal component regression.

Davison and Hinkley (1997, Example 6.8, 6.10, 6.12) use this data in a series of examples. Example 6.8: estimation of prediction error. Example 6.10: prediction error using cross-validation and bootstrapping. Example 6.12: subset model selection using cross-validation.

Source

Obtained from the CRAN package boot.

References

Davison, A. C. (2003). Statistical Models. Cambridge: Cambridge University Press.

Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and their Application. Cambridge University Press.

Examples

data(znuclear)
bestglm(znuclear, IC="BICq")
data(znuclear)
bestglm(znuclear, IC="BICq")

Prostate cancer data. Standardized.

Description

Data with 8 inputs and one output used to illustrate the prediction problem and regression in the textbook of Hastie, Tibshirani and Freedman (2009).

Usage

data(zprostate)data(zprostate)

Format

A data frame with 97 observations, 9 inputs and 1 output. All input variables have been standardized.

lcavol: log-cancer volume
lweight: log prostate weight
age: age in years
lbph: log benign prostatic hyperplasia
svi: seminal vesicle invasion
lcp: log of capsular penetration
gleason: Gleason score
pgg45: percent of Gleascores 4/5
lpsa: Outcome. Log of PSA
train: TRUE or FALSE

Details

A study of 97 men with prostate cancer examined the correlation between PSA (prostate specific antigen) and a number of clinical measurements: lcavol, lweight, lbph, svi, lcp, gleason, pgg45

References

Hastie, Tibshirani & Friedman. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd Ed. Springer.

Examples

#Prostate data. Table 3.3 HTF.
data(zprostate)
#full dataset
trainQ<-zprostate[,10]
train <-zprostate[trainQ,-10]
test <-zprostate[!trainQ,-10]
ans<-lm(lpsa~., data=train)
sig<-summary(ans)$sigma
yHat<-predict(ans, newdata=test)
yTest<-zprostate$lpsa[!trainQ]
TE<-mean((yTest-yHat)^2)
#subset
ansSub<-bestglm(train, IC="BICq")$BestModel
sigSub<-summary(ansSub)$sigma
yHatSub<-predict(ansSub, newdata=test)
TESub<-mean((yTest-yHatSub)^2)
m<-matrix(c(TE,sig,TESub,sigSub), ncol=2)
dimnames(m)<-list(c("TestErr","Sd"),c("LS","Best"))
m

#Prostate data. Table 3.3 HTF.
data(zprostate)
#full dataset
trainQ<-zprostate[,10]
train <-zprostate[trainQ,-10]
test <-zprostate[!trainQ,-10]
ans<-lm(lpsa~., data=train)
sig<-summary(ans)$sigma
yHat<-predict(ans, newdata=test)
yTest<-zprostate$lpsa[!trainQ]
TE<-mean((yTest-yHat)^2)
#subset
ansSub<-bestglm(train, IC="BICq")$BestModel
sigSub<-summary(ansSub)$sigma
yHatSub<-predict(ansSub, newdata=test)
TESub<-mean((yTest-yHatSub)^2)
m<-matrix(c(TE,sig,TESub,sigSub), ncol=2)
dimnames(m)<-list(c("TestErr","Sd"),c("LS","Best"))
m

Package 'bestglm'

Help Index

bestglm: Best Subset GLM

Description

Details

Author(s)

References

See Also

Examples

Daily ozone pollution with meteorlogical and date inputs

Description

Usage

Format

Details

Source

References

Examples

Binary representation of non-negative integer

Description

Usage

Arguments

Value

Author(s)

Examples

Best Subset GLM using Information Criterion or Cross-Validation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Cross-validation using delete-d method.

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Adjusted K-fold Cross-Validation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

K-fold Cross-Validation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Detroit homicide data for 1961-73 used in the book Subset Regression by A.J. Miller

Description

Usage

Format

Details

Source

References

Examples

Scaled Variables Dependency Plots: Output vs Inputs

Description

Usage

Arguments

Value

Author(s)

See Also

Examples