| Type: | Package |
| Title: | 28 Benchmark Densities from Berlinet/Devroye (1994) |
| Version: | 1.0.8 |
| Date: | 2023-08-31 |
| Maintainer: | Thoralf Mildenberger <mild@zhaw.ch> |
| Description: | Full implementation of the 28 distributions introduced as benchmarks for nonparametric density estimation by Berlinet and Devroye (1994) https://hal.science/hal-03659919. Includes densities, cdfs, quantile functions and generators for samples as well as additional information on features of the densities. Also contains the 4 histogram densities used in Rozenholc/Mildenberger/Gather (2010) <doi:10.1016/j.csda.2010.04.021>. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| URL: | https://github.com/thmild/benchden |
| NeedsCompilation: | no |
| Packaged: | 2023-08-31 08:17:15 UTC; thoralf |
| Author: | Thoralf Mildenberger
 [aut, cre],
Henrike Weinert
[aut],
Sebastian Tiemeyer [aut] |
| Repository: | CRAN |
| Date/Publication: | 2023-08-31 08:50:05 UTC |
Some properties of 28 benchmark densities
Description
Names and points of nonsmoothness for the 28 distributions from Berlinet/Devroye (1994).
Usage
bberdev(dnum = 1)
nberdev(dnum = 1)
Arguments
dnum |
number of distribution as in Berlinet/Devroye (1994), Section 3.2. |
Details
These functions implement the 28 distributions from Berlinet and Devroye (1994), Section 3.2, which are:
dnum == 1 "uniform" on [0,1] as in stats-package
dnum == 2 "exponential" as in stats-package
dnum == 3 "Maxwell"
dnum == 4 "double exponential"
dnum == 5 "logistic" as in stats-package
dnum == 6 "Cauchy" as in stats-package
dnum == 7 "extreme value"
dnum == 8 "infinite peak"
dnum == 9 "Pareto"
dnum == 10 "symmetric Pareto"
dnum == 11 "normal" as in stats-package
dnum == 12 "lognormal"
dnum == 13 "uniform scale mixture"
dnum == 14 "Matterhorn"
dnum == 15 "logarithmic peak"
dnum == 16 "isosceles triangle"
dnum == 17 "beta 2,2" as in stats-package
dnum == 18 "chi-square 1" as in stats-package
dnum == 19 "normal cubed"
dnum == 20 "inverse exponential"
dnum == 21 "Marronite"
dnum == 22 "skewed bimodal"
dnum == 23 "claw"
dnum == 24 "smooth comb"
dnum == 25 "caliper"
dnum == 26 "trimodal uniform"
dnum == 27 "sawtooth"
dnum == 28 "bilogarithmic peak"
Value
nberdev |
gives the name of the distribution (the same as |
bberdev |
Since evaluation of loss functions
in nonparametric density estimation often requires numerical
integration, |
Author(s)
Thoralf Mildenberger, Henrike Weinert and Sebastian Tiemeyer
References
A. Berlinet and L. Devroye, "A comparison of kernel density estimates", Publications de l'Institut de Statistique de l'Universite de Paris, vol. 38(3), pp. 3-59, 1994. https://hal.science/hal-03659919
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Examples
# name of "Claw"-distribution
nberdev(dnum=23)
Some Properties of 28 benchmark densities
Description
Name, position of modes, support and points of nonsmoothness for the 28 distributions from Berlinet/Devroye (1994).
Usage
berdev(dnum = 1)
Arguments
dnum |
number of distribution as in Berlinet/Devroye (1994), Section 3.2. |
Details
These functions implement the 28 distributions from Berlinet and Devroye (1994), Section 3.2, which are:
dnum == 1 "uniform" on [0,1] as in stats-package
dnum == 2 "exponential" as in stats-package
dnum == 3 "Maxwell"
dnum == 4 "double exponential"
dnum == 5 "logistic" as in stats-package
dnum == 6 "Cauchy" as in stats-package
dnum == 7 "extreme value"
dnum == 8 "infinite peak"
dnum == 9 "Pareto"
dnum == 10 "symmetric Pareto"
dnum == 11 "normal" as in stats-package
dnum == 12 "lognormal"
dnum == 13 "uniform scale mixture"
dnum == 14 "Matterhorn"
dnum == 15 "logarithmic peak"
dnum == 16 "isosceles triangle"
dnum == 17 "beta 2,2" as in stats-package
dnum == 18 "chi-square 1" as in stats-package
dnum == 19 "normal cubed"
dnum == 20 "inverse exponential"
dnum == 21 "Marronite"
dnum == 22 "skewed bimodal"
dnum == 23 "claw"
dnum == 24 "smooth comb"
dnum == 25 "caliper"
dnum == 26 "trimodal uniform"
dnum == 27 "sawtooth"
dnum == 28 "bilogarithmic peak"
Value
berdev returns a list with components
name |
gives the name of the distribution, |
peaks |
gives a vector of the positions of peaks or modes of the density, and |
support |
gives a matrix as follows: in each row an interval is defined (with the first column giving the left and the second column the right end of the interval). Together the intervals give the support of the distribution (for most distributions only one interval). |
breaks |
Since evaluation of loss functions
in nonparametric density estimation often requires numerical
integration, |
Author(s)
Thoralf Mildenberger, Henrike Weinert and Sebastian Tiemeyer
References
A. Berlinet and L. Devroye, "A comparison of kernel density estimates", Publications de l'Institut de Statistique de l'Universite de Paris, vol. 38(3), pp. 3-59, 1994. https://hal.science/hal-03659919
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Examples
# position of peaks of "Claw"-distribution
berdev(dnum=23)$peaks
# support of the "Trimodal uniform"
berdev(dnum=26)$support
Some properties of 4 histogram benchmark densities
Description
Names and breakpoints for the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010).
Usage
bhisto(dnum = 1)
nhisto(dnum = 1)
Arguments
dnum |
number of distribution. |
Details
These functions implement the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010). Defined as the following mixtures of uniform distributions:
dnum == 1 5 bin regular histogram:
0.15*U[0,0.2] + 0.35*U(0.2,0.4] + 0.2*U(0.4,0.6] +0.1*U(0.6,0.8]+ 0.2*U(0.8,1.0]
dnum == 2 5 bin irregular histogram:
0.15*U[0,0.13] + 0,35*U(0.13,0.34] + 0.2*U(0.34,0.61]
+0.1*U(0.61,0.65] + 0.2*U(0.65,1.0]
dnum == 3 10 bin regular histogram:
0.01*U[0,0.1] + 0.18*U(0.1,0.2] + 0.16*U(0.2,0.3]
+0.07*U(0.3,0.4] + 0.06*U(0.4,0.5] + 0.01*U(0.5,0.6]
+0.06*U(0.6,0.7] + 0.37*U(0.7,0.8] + 0.06*U(0.8,0.9]
+0.02*U(0.9,1.0]
dnum == 4 10 bin irregular histogram:
0.01*U[0,0.02] + 0.18*U(0.02,0.07] + 0.16*U(0.07,0.14]
+0.07*U(0.14,0.44] + 0.06*U(0.44,0.53] + 0.01*U(0.53,0.56]
+0.06*U(0.56,0.67] + 0.37*U(0.67,0.77] + 0.06*U(0.77,0.91]
+0.02*U(0.91,1.0]
where U[a,b] denotes the uniform distribution on [a,b].
Value
nhisto |
gives the name of the distribution (the same as |
bhisto |
gives the vector of break points (the same as |
Author(s)
Thoralf Mildenberger
References
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Y. Rozenholc, T. Mildenberger and U. Gather (2010), "Combining Regular and Irregular Histograms by Penalized Likelihood", Computational Statistics and Data Analysis, 54, 3313-3323. doi:10.1016/j.csda.2010.04.021 Earlier version including explicit definition of the densities: doi:10.17877/DE290R-15901
Examples
# name string of 5 bin regular histogram
nhisto(dnum=1)
28 benchmark densities from Berlinet/Devroye (1994)
Description
Density, distribution function, quantile function and random variate generation for the 28 distributions from Berlinet/Devroye (1994).
Usage
dberdev(x,dnum = 1)
pberdev(q,dnum = 1)
qberdev(p,dnum = 1)
rberdev(n,dnum = 1)
Arguments
dnum |
number of distribution as in Berlinet/Devroye (1994), Section 3.2. |
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
Details
These functions implement the 28 distributions from Berlinet and Devroye (1994), Section 3.2, which are:
dnum == 1 "uniform" on [0,1] as in stats-package
dnum == 2 "exponential" as in stats-package
dnum == 3 "Maxwell"
dnum == 4 "double exponential"
dnum == 5 "logistic" as in stats-package
dnum == 6 "Cauchy" as in stats-package
dnum == 7 "extreme value"
dnum == 8 "infinite peak"
dnum == 9 "Pareto"
dnum == 10 "symmetric Pareto"
dnum == 11 "normal" as in stats-package
dnum == 12 "lognormal"
dnum == 13 "uniform scale mixture"
dnum == 14 "Matterhorn"
dnum == 15 "logarithmic peak"
dnum == 16 "isosceles triangle"
dnum == 17 "beta 2,2" as in stats-package
dnum == 18 "chi-square 1" as in stats-package
dnum == 19 "normal cubed"
dnum == 20 "inverse exponential"
dnum == 21 "Marronite"
dnum == 22 "skewed bimodal"
dnum == 23 "claw"
dnum == 24 "smooth comb"
dnum == 25 "caliper"
dnum == 26 "trimodal uniform"
dnum == 27 "sawtooth"
dnum == 28 "bilogarithmic peak"
Value
dberdev |
gives the density, |
pberdev |
gives the distribution function, |
qberdev |
gives the quantile function, and |
rberdev |
generates random deviates. |
Acknowledgement
The authors thank Luc Devroye for providing his original implementation for testing purposes.
Author(s)
Thoralf Mildenberger, Henrike Weinert and Sebastian Tiemeyer
References
A. Berlinet and L. Devroye, "A comparison of kernel density estimates," Publications de l'Institut de Statistique de l'Universite de Paris, vol. 38(3), pp. 3-59, 1994. https://hal.science/hal-03659919
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Examples
# histogram and true density of "Claw"-distribution
hist(rberdev(1000,dnum=23),breaks=100, main = " ",freq=FALSE)
lines(seq(-3,3,0.01),dberdev(seq(-3,3,0.01),dnum=23),col="blue",lwd=2)
title(paste(nberdev(dnum=23)))
# plot cdf of simulated data and the df of "Matterhorn"-distribution
plot.stepfun(rberdev(100,dnum=14),do.points=TRUE,main="")
lines(seq(-1,1,0.001),pberdev(seq(-1,1,0.001),dnum=14),col="blue")
title(paste(nberdev(dnum=14)))
# plot quantiles of "smooth comb"-distribution
plot(qberdev(seq(0,1,0.01),dnum=24),t="l")
title(paste(nberdev(dnum=24)))
4 histogram benchmark densities
Description
Density, distribution function, quantile function and random variate generation for the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010).
Usage
dhisto(x,dnum = 1)
phisto(q,dnum = 1)
qhisto(p,dnum = 1)
rhisto(n,dnum = 1)
Arguments
dnum |
number of distribution as in Rozenholc/Mildenberger/Gather (2010) |
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
Details
These functions implement the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010). Defined as the following mixtures of uniform distributions:
dnum == 1 5 bin regular histogram:
0.15*U[0,0.2] + 0.35*U(0.2,0.4] + 0.2*U(0.4,0.6] +0.1*U(0.6,0.8]+ 0.2*U(0.8,1.0]
dnum == 2 5 bin irregular histogram:
0.15*U[0,0.13] + 0,35*U(0.13,0.34] + 0.2*U(0.34,0.61]
+0.1*U(0.61,0.65] + 0.2*U(0.65,1.0]
dnum == 3 10 bin regular histogram:
0.01*U[0,0.1] + 0.18*U(0.1,0.2] + 0.16*U(0.2,0.3]
+0.07*U(0.3,0.4] + 0.06*U(0.4,0.5] + 0.01*U(0.5,0.6]
+0.06*U(0.6,0.7] + 0.37*U(0.7,0.8] + 0.06*U(0.8,0.9]
+0.02*U(0.9,1.0]
dnum == 4 10 bin irregular histogram:
0.01*U[0,0.02] + 0.18*U(0.02,0.07] + 0.16*U(0.07,0.14]
+0.07*U(0.14,0.44] + 0.06*U(0.44,0.53] + 0.01*U(0.53,0.56]
+0.06*U(0.56,0.67] + 0.37*U(0.67,0.77] + 0.06*U(0.77,0.91]
+0.02*U(0.91,1.0]
where U[a,b] denotes the uniform distribution on [a,b].
Value
dhisto |
gives the density, |
phisto |
gives the distribution function, |
qhisto |
gives the quantile function, and |
rhisto |
generates random deviates. |
Author(s)
Thoralf Mildenberger
References
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Y. Rozenholc, T. Mildenberger and U. Gather (2010), "Combining Regular and Irregular Histograms by Penalized Likelihood", Computational Statistics and Data Analysis, 54, 3313-3323. doi:10.1016/j.csda.2010.04.021 Earlier version including explicit definition of the densities: doi:10.17877/DE290R-15901
Examples
# histogram and true density of "5 bin irregular"-distribution
hist(rhisto(2000,dnum=2),breaks=250, main = " ",freq=FALSE)
lines(seq(0,1,0.01),dhisto(seq(0,1,0.01),dnum=2),col="blue",lwd=1)
title(paste("sample from",nhisto(dnum=2),"density"))
Some properties of 4 histogram benchmark densities
Description
Name, position of modes, support and break points for the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010).
Usage
histo(dnum = 1)
Arguments
dnum |
number of distribution. |
Details
These functions implement the 4 histogram benchmark distributions from Rozenholc/Mildenberger/Gather (2010). Defined as the following mixtures of uniform distributions:
dnum == 1 5 bin regular histogram:
0.15*U[0,0.2] + 0.35*U(0.2,0.4] + 0.2*U(0.4,0.6] +0.1*U(0.6,0.8]+ 0.2*U(0.8,1.0]
dnum == 2 5 bin irregular histogram:
0.15*U[0,0.13] + 0,35*U(0.13,0.34] + 0.2*U(0.34,0.61]
+0.1*U(0.61,0.65] + 0.2*U(0.65,1.0]
dnum == 3 10 bin regular histogram:
0.01*U[0,0.1] + 0.18*U(0.1,0.2] + 0.16*U(0.2,0.3]
+0.07*U(0.3,0.4] + 0.06*U(0.4,0.5] + 0.01*U(0.5,0.6]
+0.06*U(0.6,0.7] + 0.37*U(0.7,0.8] + 0.06*U(0.8,0.9]
+0.02*U(0.9,1.0]
dnum == 4 10 bin irregular histogram:
0.01*U[0,0.02] + 0.18*U(0.02,0.07] + 0.16*U(0.07,0.14]
+0.07*U(0.14,0.44] + 0.06*U(0.44,0.53] + 0.01*U(0.53,0.56]
+0.06*U(0.56,0.67] + 0.37*U(0.67,0.77] + 0.06*U(0.77,0.91]
+0.02*U(0.91,1.0]
where U[a,b] denotes the uniform distribution on [a,b].
Value
histo returns a list with the following components:
name |
gives the name of the distribution. |
peaks |
gives a vector of the positions of peaks of the density, defined here as mid points of maximal intervals. |
support |
gives a matrix with one row with the endpoints of the support,
which is |
breaks |
gives the vector of break points. |
Author(s)
Thoralf Mildenberger
References
T. Mildenberger and H. Weinert, "The benchden Package: Benchmark Densities for Nonparametric Density Estimation", Journal of Statistical Software, vol. 46(14), 1-14, 2012. https://www.jstatsoft.org/v46/i14/
Y. Rozenholc, T. Mildenberger and U. Gather (2010), "Combining Regular and Irregular Histograms by Penalized Likelihood", Computational Statistics and Data Analysis, 54, 3313-3323. doi:10.1016/j.csda.2010.04.021 Earlier version including explicit definition of the densities: doi:10.17877/DE290R-15901
Examples
# position of peaks of the 5 bin irregular histogram density
histo(dnum=2)$peaks
# support of the 10 bin regular histogram density
histo(dnum=3)$support