Type: Package
Title: Bayesian Additive Regression Kernels
Version: 1.0.5
Date: 2024-10-05
Description: Bayesian Additive Regression Kernels (BARK) provides an implementation for non-parametric function estimation using Levy Random Field priors for functions that may be represented as a sum of additive multivariate kernels. Kernels are located at every data point as in Support Vector Machines, however, coefficients may be heavily shrunk to zero under the Cauchy process prior, or even, set to zero. The number of active features is controlled by priors on precision parameters within the kernels, permitting feature selection. For more details see Ouyang, Z (2008) "Bayesian Additive Regression Kernels", Duke University. PhD dissertation, Chapter 3 and Wolpert, R. L, Clyde, M.A, and Tu, C. (2011) "Stochastic Expansions with Continuous Dictionaries Levy Adaptive Regression Kernels, Annals of Statistics Vol (39) pages 1916-1962 <doi:10.1214/11-AOS889>.
License: GPL (≥ 3)
URL: https://www.R-project.org, https://github.com/merliseclyde/bark
BugReports: https://github.com/merliseclyde/bark/issues
Depends: R (≥ 3.5.0)
Suggests: BART, e1071, fdm2id, rmarkdown, knitr, roxygen2, testthat, covr
LazyData: yes
Repository: CRAN
NeedsCompilation: yes
ByteCompile: yes
Encoding: UTF-8
RoxygenNote: 7.3.2
Language: en-US
VignetteBuilder: knitr
Packaged: 2024-10-05 21:16:20 UTC; clyde
Author: Merlise Clyde [aut, cre, ths] (ORCID=0000-0002-3595-1872), Zhi Ouyang [aut], Robert Wolpert [ctb, ths]
Maintainer: Merlise Clyde <clyde@duke.edu>
Date/Publication: 2024-10-05 22:40:28 UTC

bark: Bayesian Additive Regression Trees

Description

Implementation of Bayesian Additive Regression Kernels with Feature Selection for Nonparametric Regression for Gaussian regression and classification for binary Probit models

_PACKAGE

Details

BARK is a Bayesian sum-of-kernels model or because of the Bayesian priors is a Bayesian Additive Regression Kernel model.
For numeric response y, we have y = f(x) + \epsilon, where \epsilon \sim N(0,\sigma^2).
For a binary response y, P(Y=1 | x) = F(f(x)), where F denotes the standard normal cdf (probit link). In both cases, f is the sum of many Gaussian kernel functions. The goal is to have very flexible inference for the unknown function f. bark uses an approximated Cauchy process as the prior distribution for the unknown function f. Feature selection can be achieved through the inference on the scale parameters in the Gaussian kernels. BARK accepts four different types of prior distributions through setting values for selection (TRUE or FALSE), which allows scale parameters for some variables to be set to zero, removing the variables from the kernels selection = TRUE; this enables either soft shrinkage or hard shrinkage for the scale parameters. The input common_lambdas (TRUE or FALSE) specifies whether a common scale parameter should be used for all predictors (TRUE) or if FALSE allows the scale parameters to differ across all variables in the kernel.

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, Chapter 3.

See Also

Other bark functions: bark(), bark-package-deprecated, sim_Friedman1(), sim_Friedman2(), sim_Friedman3(), sim_circle()

Examples


 # Simulate regression example
 # Friedman 2 data set, 200 noisy training, 1000 noise free testing
 # Out of sample MSE in SVM (default RBF): 6500 (sd. 1600)
 # Out of sample MSE in BART (default):    5300 (sd. 1000)
 traindata <- sim_Friedman2(200, sd=125)
 testdata <- sim_Friedman2(1000, sd=0)
 fit.bark.d <- bark(y ~ ., data = data.frame(traindata),
                    testdata = data.frame(testdata), 
                    classification = FALSE,
                    selection = FALSE,
                    common_lambdas = TRUE)
 boxplot(as.data.frame(fit.bark.d$theta.lambda))
 mean((fit.bark.d$yhat.test.mean-testdata$y)^2)
 # Simulate classification example
 # Circle 5 with 2 signals and three noisy dimensions
 # Out of sample erorr rate in SVM (default RBF): 0.110 (sd. 0.02)
 # Out of sample error rate in BART (default):    0.065 (sd. 0.02)
 traindata <- sim_circle(200, dim=5)
 testdata <- sim_circle(1000, dim=5)
 fit.bark.se <- bark(y ~ ., data= data.frame(traindata), 
                     testdata= data.frame(testdata),
                     classification=TRUE, 
                     selection=TRUE,
                     common_lambdas = FALSE)
                   
 boxplot(as.data.frame(fit.bark.se$theta.lambda))
 mean((fit.bark.se$yhat.test.mean>0)!=testdata$y)

Swiss Bank Notes

Description

This data set contains six measurements on 100 genuine and 100 fradulent old Swiss banknotes

Usage

data(banknotes)

Format

a dataframe with the following variables:

Status

the status of the banknote: genuine or counterfeit

Length

Length of bill (mm)

Left

Width of left edge (mm)

Right

Width of right edge (mm)

Bottom

Bottom margin width (mm)

Top

Top margin width (mm)

Diagonal

Length of diagonal (mm)

Source

Flury, B. and Riedwyl, H. (1988). Multivariate Statistics: A practical approach. London: Chapman & Hall, Tables 1.1 and 1.2, pp. 5-8.

Nonparametric Regression using Bayesian Additive Regression Kernels

Description

BARK is a Bayesian sum-of-kernels model.
For numeric response y, we have y = f(x) + \epsilon, where \epsilon \sim N(0,\sigma^2).
For a binary response y, P(Y=1 | x) = F(f(x)), where F denotes the standard normal cdf (probit link).
In both cases, f is the sum of many Gaussian kernel functions. The goal is to have very flexible inference for the unknown function f. BARK uses an approximation to a Cauchy process as the prior distribution for the unknown function f.

Feature selection can be achieved through the inference on the scale parameters in the Gaussian kernels. BARK accepts four different types of prior distributions, e, d, enabling either soft shrinkage or se, sd, enabling hard shrinkage for the scale parameters.

Usage

bark(
  formula,
  data,
  subset,
  na.action = na.omit,
  testdata = NULL,
  selection = TRUE,
  common_lambdas = TRUE,
  classification = FALSE,
  keepevery = 100,
  nburn = 100,
  nkeep = 100,
  printevery = 1000,
  keeptrain = FALSE,
  verbose = FALSE,
  fixed = list(),
  tune = list(lstep = 0.5, frequL = 0.2, dpow = 1, upow = 0, varphistep = 0.5, phistep =
    1),
  theta = list()
)

Arguments

formula

model formula for the model with all predictors, Y ~ X. The X variables will be centered and scaled as part of model fitting.

data

a data frame. Factors will be converted to numerical vectors based on the using 'model.matrix'.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

na.action

a function which indicates what should happen when the data contain NAs. The default is "na.omit".

testdata

Dataframe with test data for out of sample prediction.
Should have same structure as data.

selection

Logical variable indicating whether variable dependent kernel parameters \lambda may be set to zero in the MCMC; default is TRUE.

common_lambdas

Logical variable indicating whether kernel parameters \lambda should be predictor specific or common across predictors; default is TRUE. Note if common_lambdas = TRUE and selection = TRUE this applies just to the non-zero lambda_j.

classification

TRUE/FALSE logical variable, indicating a classification or regression problem.

keepevery

Every keepevery draw is kept to be returned to the user

nburn

Number of MCMC iterations (nburn*keepevery) to be treated as burn in.

nkeep

Number of MCMC iterations kept for the posterior inference.
nkeep*keepevery iterations after the burn in.

printevery

As the MCMC runs, a message is printed every printevery draws.

keeptrain

Logical, whether to keep results for training samples.

verbose

Logical, whether to print out messages

fixed

A list of fixed hyperparameters, using the default values if not specified.
alpha = 1: stable index, must be 1 currently.
eps = 0.5: approximation parameter.
gam = 5: intensity parameter.
la = 1: first argument of the gamma prior on kernel scales.
lb = 2: second argument of the gamma prior on kernel scales.
pbetaa = 1: first argument of the beta prior on plambda.
pbetab = 1: second argument of the beta prior on plambda.
n: number of training samples, automatically generates.
p: number of explanatory variables, automatically generates.
meanJ: the expected number of kernels, automatically generates.

tune

A list of tuning parameters, not expected to change.
lstep: the stepsize of the lognormal random walk on lambda.
frequL: the frequency to update L.
dpow: the power on the death step.
upow: the power on the update step.
varphistep: the stepsize of the lognormal random walk on varphi.
phistep: the stepsize of the lognormal random walk on phi.

theta

A list of the starting values for the parameter theta, use defaults if nothing is given.

Details

BARK is implemented using a Bayesian MCMC method. At each MCMC interaction, we produce a draw from the joint posterior distribution, i.e. a full configuration of regression coefficients, kernel locations and kernel parameters.

Thus, unlike a lot of other modelling methods in R, we do not produce a single model object from which fits and summaries may be extracted. The output consists of values f^*(x) (and \sigma^* in the numeric case) where * denotes a particular draw. The x is either a row from the training data (x.train)

Value

bark returns an object of class 'bark' with a list, including:

call

the matched call

fixed

Fixed hyperparameters

tune

Tuning parameters used

theta.last

The last set of parameters from the posterior draw

theta.nvec

A matrix with nrow(x.train)+1 rows and (nkeep) columns, recording the number of kernels at each training sample

theta.varphi

A matrix with nrow(x.train) +1 rows and (nkeep) columns, recording the precision in the normal gamma prior distribution for the regression coefficients

theta.beta

A matrix with nrow(x.train)+1 rows and (nkeep) columns, recording the regression coefficients

theta.lambda

A matrix with ncol(x.train) rows and (nkeep) columns, recording the kernel scale parameters

thea.phi

The vector of length nkeep, recording the precision in regression Gaussian noise (1 for the classification case)

yhat.train

A matrix with nrow(x.train) rows and (nkeep) columns. Each column corresponds to a draw f^* from the posterior of f and each row corresponds to a row of x.train. The (i,j) value is f^*(x) for the j^{th} kept draw of f and the i^{th} row of x.train.
For classification problems, this is the value of the expectation for the underlying normal random variable.
Burn-in is dropped

yhat.test

Same as yhat.train but now the x's are the rows of the test data; NULL if testdata are not provided

yhat.train.mean

train data fits = row mean of yhat.train

yhat.test.mean

test data fits = row mean of yhat.test

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, page 58.

See Also

Other bark functions: bark-package, bark-package-deprecated, sim_Friedman1(), sim_Friedman2(), sim_Friedman3(), sim_circle()

Examples

##Simulated regression example
# Friedman 2 data set, 200 noisy training, 1000 noise free testing
# Out of sample MSE in SVM (default RBF): 6500 (sd. 1600)
# Out of sample MSE in BART (default):    5300 (sd. 1000)
traindata <- data.frame(sim_Friedman2(200, sd=125))
testdata <- data.frame(sim_Friedman2(1000, sd=0))
# example with a very small number of iterations to illustrate usage
fit.bark.d <- bark(y ~ ., data=traindata, testdata= testdata,
                   nburn=10, nkeep=10, keepevery=10,
                   classification=FALSE, 
                   common_lambdas = FALSE,
                   selection = FALSE)
boxplot(data.frame(fit.bark.d$theta.lambda))
mean((fit.bark.d$yhat.test.mean-testdata$y)^2)

 ##Simulate classification example
 # Circle 5 with 2 signals and three noisy dimensions
 # Out of sample erorr rate in SVM (default RBF): 0.110 (sd. 0.02)
 # Out of sample error rate in BART (default):    0.065 (sd. 0.02)
 traindata <- sim_circle(200, dim=5)
 testdata <- sim_circle(1000, dim=5)
 fit.bark.se <- bark(y ~ ., 
                     data=data.frame(traindata), 
                     testdata= data.frame(testdata), 
                     classification=TRUE,
                     nburn=100, nkeep=200, )
 boxplot(as.data.frame(fit.bark.se$theta.lambda))
 mean((fit.bark.se$yhat.test.mean>0)!=testdata$y)

NonParametric Regression using Bayesian Additive Regression Kernels

Description

BARK is a Bayesian sum-of-kernels model.
For numeric response y, we have y = f(x) + \epsilon, where \epsilon \sim N(0,\sigma^2).
For a binary response y, P(Y=1 | x) = F(f(x)), where F denotes the standard normal cdf (probit link).
In both cases, f is the sum of many Gaussian kernel functions. The goal is to have very flexible inference for the unknown function f. BARK uses an approximation to a Cauchy process as the prior distribution for the unknown function f.

Feature selection can be achieved through the inference on the scale parameters in the Gaussian kernels. BARK accepts four different types of prior distributions, e, d, enabling either soft shrinkage or se, sd, enabling hard shrinkage for the scale parameters.

Arguments

x.train

Explanatory variables for training (in sample) data.
Must be a matrix of doubles, with (as usual) rows corresponding to observations and columns to variables.

y.train

Dependent variable for training (in sample) data.
If y is numeric a continuous response model is fit (normal errors).
If y is a logical (or just has values 0 and 1), then a binary response model with a probit link is fit.

x.test

Explanatory variables for test (out of sample) data.
Should have same structure as x.train.

type

BARK type, e, d, se, or sd, default choice is se.
e: BARK with equal weights.
d: BARK with different weights.
se: BARK with selection and equal weights.
sd: BARK with selection and different weights.

classification

TRUE/FALSE logical variable, indicating a classification or regression problem.

keepevery

Every keepevery draw is kept to be returned to the user

nburn

Number of MCMC iterations (nburn*keepevery) to be treated as burn in.

nkeep

Number of MCMC iterations kept for the posterior inference.
nkeep*keepevery iterations after the burn in.

printevery

As the MCMC runs, a message is printed every printevery draws.

keeptrain

Logical, whether to keep results for training samples.

fixed

A list of fixed hyperparameters, using the default values if not specified.
alpha = 1: stable index, must be 1 currently.
eps = 0.5: approximation parameter.
gam = 5: intensity parameter.
la = 1: first argument of the gamma prior on kernel scales.
lb = 2: second argument of the gamma prior on kernel scales.
pbetaa = 1: first argument of the beta prior on plambda.
pbetab = 1: second argument of the beta prior on plambda.
n: number of training samples, automatically generates.
p: number of explanatory variables, automatically generates.
meanJ: the expected number of kernels, automatically generates.

tune

A list of tuning parameters, not expected to change.
lstep: the stepsize of the lognormal random walk on lambda.
frequL: the frequency to update L.
dpow: the power on the death step.
upow: the power on the update step.
varphistep: the stepsize of the lognormal random walk on varphi.
phistep: the stepsize of the lognormal random walk on phi.

theta

A list of the starting values for the parameter theta, use defaults if nothing is given.

Details

BARK is implemented using a Bayesian MCMC method. At each MCMC interaction, we produce a draw from the joint posterior distribution, i.e. a full configuration of regression coefficients, kernel locations and kernel parameters etc.

Thus, unlike a lot of other modelling methods in R, we do not produce a single model object from which fits and summaries may be extracted. The output consists of values f^*(x) (and \sigma^* in the numeric case) where * denotes a particular draw. The x is either a row from the training data (x.train)

Value

bark returns a list, including:

fixed

Fixed hyperparameters

tune

Tuning parameters used

theta.last

The last set of parameters from the posterior draw

theta.nvec

A matrix with nrow(x.train)+1 rows and (nkeep) columns, recording the number of kernels at each training sample

theta.varphi

A matrix with nrow(x.train) +1 rows and (nkeep) columns, recording the precision in the normal gamma prior distribution for the regression coefficients

theta.beta

A matrix with nrow(x.train)+1 rows and (nkeep) columns, recording the regression coefficients

theta.lambda

A matrix with ncol(x.train) rows and (nkeep) columns, recording the kernel scale parameters

thea.phi

The vector of length nkeep, recording the precision in regression Gaussian noise (1 for the classification case)

yhat.train

A matrix with nrow(x.train) rows and (nkeep) columns. Each column corresponds to a draw f^* from the posterior of f and each row corresponds to a row of x.train. The (i,j) value is f^*(x) for the j^{th} kept draw of f and the i^{th} row of x.train.
For classification problems, this is the value of the expectation for the underlying normal random variable.
Burn-in is dropped

yhat.test

Same as yhat.train but now the x's are the rows of the test data

yhat.train.mean

train data fits = row mean of yhat.train

yhat.test.mean

test data fits = row mean of yhat.test

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, page 58.

See Also

Other bark deprecated functions: bark-package-deprecated, sim.Circle-deprecated, sim.Friedman1-deprecated, sim.Friedman2-deprecated, sim.Friedman3-deprecated

Examples

# Simulate regression example
#  Friedman 2 data set, 200 noisy training, 1000 noise free testing
#  Out of sample MSE in SVM (default RBF): 6500 (sd. 1600)
#  Out of sample MSE in BART (default):    5300 (sd. 1000)
traindata <- sim_Friedman2(200, sd=125)
testdata <- sim_Friedman2(1000, sd=0)
# example with a very small number of iterations to illustrate the method
fit.bark.d <- bark_mat(traindata$x, traindata$y, testdata$x,
                  nburn=10, nkeep=10, keepevery=10,
                  classification=FALSE, type="d")
boxplot(data.frame(fit.bark.d$theta.lambda))
mean((fit.bark.d$yhat.test.mean-testdata$y)^2)

 # Simulate classification example
 #  Circle 5 with 2 signals and three noisy dimensions
 #  Out of sample erorr rate in SVM (default RBF): 0.110 (sd. 0.02)
 #  Out of sample error rate in BART (default):    0.065 (sd. 0.02)
 traindata <- sim_circle(200, dim=5)
 testdata <- sim_circle(1000, dim=5)
 fit.bark.se <- bark_mat(traindata$x, traindata$y, testdata$x, classification=TRUE, type="se")
 boxplot(data.frame(fit.bark.se$theta.lambda))
 mean((fit.bark.se$yhat.test.mean>0)!=testdata$y)

Deprecated functions in package bark.

Description

The functions listed below are deprecated and will be defunct in the near future. When possible, alternative functions with similar functionality are also mentioned. Help pages for deprecated functions are available at help("<function>-deprecated").

Usage

bark_mat(
  x.train,
  y.train,
  x.test = matrix(0, 0, 0),
  type = "se",
  classification = FALSE,
  keepevery = 100,
  nburn = 100,
  nkeep = 100,
  printevery = 1000,
  keeptrain = FALSE,
  fixed = list(),
  tune = list(lstep = 0.5, frequL = 0.2, dpow = 1, upow = 0, varphistep = 0.5, phistep =
    1),
  theta = list()
)

sim.Friedman1(n, sd = 1)

sim.Friedman2(n, sd = 125)

sim.Friedman3(n, sd = 0.1)

sim.Circle(n, dim = 5)

Value

List of deprecated functions

bark_mat

Old version with matrix inputs used for testing;

sim.Friedman1

For sim.Friedman1, use sim_Friedman1.

sim.Friedman2

For sim.Friedman2, use sim_Friedman2.

sim.Friedman3

For sim.Friedman3, use sim_Friedman3.

sim.Circle

For sim.Circle, use sim_circle.

See Also

Other bark deprecated functions: bark-deprecated, sim.Circle-deprecated, sim.Friedman1-deprecated, sim.Friedman2-deprecated, sim.Friedman3-deprecated

Other bark functions: bark(), bark-package, sim_Friedman1(), sim_Friedman2(), sim_Friedman3(), sim_circle()

Simulate Data from Hyper-Sphere for Classification Problems

Description

The classification problem Circle is described in the BARK paper (2008). Inputs are dim independent variables uniformly distributed on the interval [-1,1], only the first 2 out of these dim are actually signals. Outputs are created according to the formula

y = 1(x1^2+x2^2 \le 2/\pi)

Usage

sim.Circle(n, dim=5)

Arguments

n

number of data points to generate

dim

number of dimension of the problem, no less than 2

Value

Returns a list with components

x

input values (independent variables)

y

0/1 output values (dependent variable)

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, Chapter 3.

See Also

Other bark deprecated functions: bark-deprecated, bark-package-deprecated, sim.Friedman1-deprecated, sim.Friedman2-deprecated, sim.Friedman3-deprecated

Examples

## Not run: 
  sim.Circle(n=100, dim = 5)

## End(Not run)

Simulated Regression Problem Friedman 1

Description

The regression problem Friedman 1 as described in Friedman (1991) and Breiman (1996). Inputs are 10 independent variables uniformly distributed on the interval [0,1], only 5 out of these 10 are actually used. Outputs are created according to the formula

y = 10 \sin(\pi x1 x2) + 20 (x3 - 0.5)^2 + 10 x4 + 5 x5 + e

where e is N(0,sd^2).

Usage

sim.Friedman1(n, sd=1)

Arguments

n

number of data points to create

sd

standard deviation of noise, with default value 1

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark deprecated functions: bark-deprecated, bark-package-deprecated, sim.Circle-deprecated, sim.Friedman2-deprecated, sim.Friedman3-deprecated

Examples

## Not run: 
sim.Friedman1(100, sd=1)

## End(Not run)

Simulated Regression Problem Friedman 2

Description

The regression problem Friedman 2 as described in Friedman (1991) and Breiman (1996). Inputs are 4 independent variables uniformly distributed over the ranges

0 \le x1 \le 100

40 \pi \le x2 \le 560 \pi

0 \le x3 \le 1

1 \le x4 \le 11

The outputs are created according to the formula

y = (x1^2 + (x2 x3 - (1/(x2 x4)))^2)^{0.5} + e

where e is N(0,sd^2).

Usage

sim.Friedman2(n, sd=125)

Arguments

n

number of data points to create

sd

Standard deviation of noise. The default value of 125 gives a signal to noise ratio (i.e., the ratio of the standard deviations) of 3:1. Thus, the variance of the function itself (without noise) accounts for 90% of the total variance.

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark deprecated functions: bark-deprecated, bark-package-deprecated, sim.Circle-deprecated, sim.Friedman1-deprecated, sim.Friedman3-deprecated

Examples

## Not run: 
sim.Friedman2(100, sd=125)

## End(Not run)

Simulated Regression Problem Friedman 3

Description

The regression problem Friedman 3 as described in Friedman (1991) and Breiman (1996). Inputs are 4 independent variables uniformly distributed over the ranges

0 \le x1 \le 100

40 \pi \le x2 \le 560 \pi

0 \le x3 \le 1

1 \le x4 \le 11

The outputs are created according to the formula

\mbox{atan}((x2 x3 - (1/(x2 x4)))/x1) + e

where e is N(0,sd^2).

Usage

sim.Friedman3(n, sd=0.1)

Arguments

n

number of data points to create

sd

Standard deviation of noise. The default value of 125 gives a signal to noise ratio (i.e., the ratio of the standard deviations) of 3:1. Thus, the variance of the function itself (without noise) accounts for 90% of the total variance.

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark deprecated functions: bark-deprecated, bark-package-deprecated, sim.Circle-deprecated, sim.Friedman1-deprecated, sim.Friedman2-deprecated

Examples

## Not run: 
sim.Friedman3(n=100, sd=0.1)

## End(Not run)

Simulated Regression Problem Friedman 1

Description

The regression problem Friedman 1 as described in Friedman (1991) and Breiman (1996). Inputs are 10 independent variables uniformly distributed on the interval [0,1], only 5 out of these 10 are actually used. Outputs are created according to the formula

y = 10 \sin(\pi x1 x2) + 20 (x3 - 0.5)^2 + 10 x4 + 5 x5 + e

where e is N(0,sd^2).

Usage

sim_Friedman1(n, sd = 1)

Arguments

n

number of data points to create

sd

standard deviation of noise, with default value 1

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark simulation functions: sim_Friedman2(), sim_Friedman3(), sim_circle()

Other bark functions: bark(), bark-package, bark-package-deprecated, sim_Friedman2(), sim_Friedman3(), sim_circle()

Examples

sim_Friedman1(100, sd=1)

Simulated Regression Problem Friedman 2

Description

The regression problem Friedman 2 as described in Friedman (1991) and Breiman (1996). Inputs are 4 independent variables uniformly distributed over the ranges

0 \le x1 \le 100

40 \pi \le x2 \le 560 \pi

0 \le x3 \le 1

1 \le x4 \le 11

The outputs are created according to the formula

y = (x1^2 + (x2 x3 - (1/(x2 x4)))^2)^{0.5} + e

where e is N(0,sd^2).

Usage

sim_Friedman2(n, sd = 125)

Arguments

n

number of data points to create

sd

Standard deviation of noise. The default value of 125 gives a signal to noise ratio (i.e., the ratio of the standard deviations) of 3:1. Thus, the variance of the function itself (without noise) accounts for 90% of the total variance.

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark simulation functions: sim_Friedman1(), sim_Friedman3(), sim_circle()

Other bark functions: bark(), bark-package, bark-package-deprecated, sim_Friedman1(), sim_Friedman3(), sim_circle()

Examples

sim_Friedman2(100, sd=125)

Simulated Regression Problem Friedman 3

Description

The regression problem Friedman 3 as described in Friedman (1991) and Breiman (1996). Inputs are 4 independent variables uniformly distributed over the ranges

0 \le x1 \le 100

40 \pi \le x2 \le 560 \pi

0 \le x3 \le 1

1 \le x4 \le 11

The outputs are created according to the formula

\mbox{atan}((x2 x3 - (1/(x2 x4)))/x1) + e

where e is N(0,sd^2).

Usage

sim_Friedman3(n, sd = 0.1)

Arguments

n

number of data points to create

sd

Standard deviation of noise. The default value of 125 gives a signal to noise ratio (i.e., the ratio of the standard deviations) of 3:1. Thus, the variance of the function itself (without noise) accounts for 90% of the total variance.

Value

Returns a list with components

x

input values (independent variables)

y

output values (dependent variable)

References

Breiman, Leo (1996) Bagging predictors. Machine Learning 24, pages 123-140.
Friedman, Jerome H. (1991) Multivariate adaptive regression splines. The Annals of Statistics 19 (1), pages 1-67.

See Also

Other bark simulation functions: sim_Friedman1(), sim_Friedman2(), sim_circle()

Other bark functions: bark(), bark-package, bark-package-deprecated, sim_Friedman1(), sim_Friedman2(), sim_circle()

Examples

sim_Friedman3(n=100, sd=0.1)

Simulate Data from Hyper-Sphere for Classification Problems

Description

The classification problem Circle is described in the BARK paper (2008). Inputs are dim independent variables uniformly distributed on the interval [-1,1], only the first 2 out of these dim are actually signals. Outputs are created according to the formula

y = 1(x1^2+x2^2 \le 2/\pi)

Usage

sim_circle(n, dim = 5)

Arguments

n

number of data points to generate

dim

number of dimension of the problem, no less than 2

Value

Returns a list with components

x

input values (independent variables)

y

0/1 output values (dependent variable)

References

Ouyang, Zhi (2008) Bayesian Additive Regression Kernels. Duke University. PhD dissertation, Chapter 3.

See Also

Other bark simulation functions: sim_Friedman1(), sim_Friedman2(), sim_Friedman3()

Other bark functions: bark(), bark-package, bark-package-deprecated, sim_Friedman1(), sim_Friedman2(), sim_Friedman3()

Examples

sim_circle(n=100, dim=5)