histogram {histogram} | R Documentation |
histogram with automatic choice of bins
Description
Construction of regular and irregular histograms with different options for choosing the number and widths of the bins. By default, both a regular and an irregular histogram using a data-dependent penalty as described in detail in Rozenholc/Mildenberger/Gather (2009) are constructed. The final estimate is the one with the larger penalized likelihood.
Usage
histogram(y, type = "combined", grid = "data",
breaks = NULL, penalty = "default",
greedy = TRUE, right=TRUE, freq=FALSE, control = list(),
verbose = TRUE, plot = TRUE, ...)
Arguments
y |
a vector of values for which the histogram is desired. |
type |
use |
grid |
if |
breaks |
controls the maximum number of bins allowed in a regular histogram, or the size of
the finest grid in an irregular histogram when |
penalty |
controls which penalty is used. See description of penalties below. |
greedy |
logical; if |
right |
logical; if |
freq |
logical; if |
control |
list of additional control parameters. Meaning and default values depend on settings of |
verbose |
logical; if |
plot |
logical. If |
... |
further arguments and graphical parameters passed to |
Details
The histogram
procedure produces a histogram, i.e. a piecewise constant density estimate
from a univariate real-valued sample stored in a vector y
. Let n
denote the length of y
.
The range of the data is partitioned into D
intervals - called bins - and the density estimate on the i
-th bin is
given by N_i/(n*w_i)
where N_i
is the number of observations in the i
-th bin and w_i
is its width. The histogram thus defined is the maximum likelihood estimate among all densities that are piecewise constant w.r.t. this partition.
The arguments of histogram
given above determine the way the partition is chosen. In a regular histogram, the partition
consists of D
bins of the same widths, and the histogram is determined by the choice of D
. Strategies based on different
criteria can be chosen using the penalty
option. The maximum number of bins can be controlled by either the breaks
argument or the entries g1
, g2
and g3
in the control
argument.
An irregular histogram allows for bins of different widths. In this case, not only the number D
of bins but also the breakpoints
between the bins must be chosen. The set of allowed breakpoints is given by the finest partition selected using the grid
argument.
At the moment a finest regular grid is supported (grid="regular"
) as well as grids with possible breakpoints either equal
to the observations or between the observations (grid="data"
and between
in the control
argument set to
FALSE
or TRUE
, respectively). Setting grid="quantiles"
gives a grid based on regular sample quantiles.
If the breaks
argument is NULL
,
G(n)=g1*n^{g2}*(\log(n))^{g3}
controls the grid in the following way: the smallest
allowed bin width in a "data"
grid is 1/G(n)
times the sample range, while for grid="regular"
and
grid="quantiles"
the finest grid has floor(G(n))
bins. The parameters g1
, g2
and g3
can be changed by modifying the corresponding components in the control
argument. If breaks
is a positive number,
its integer part is used instead of G(n)
. Different strategies for selection of D
and the bin boundaries can be chosen using the penalty
option.
To reduce calculation time for irregular histograms, a subset of the breakpoints of the finest grid can be chosen by starting
from a one-bin histogram and then subsequently finding the split of an existing bin that leads to the largest increase in the
loglikelihood. The full optimization is then performed only over all partitions with endpoints from the subset
thus constructed. This is achieved by setting greedy=TRUE
. To reduce calculation time for regular histograms, the maxbin
parameter in the control
argument gives an upper bound for the number of bins. The default value is 1000.
Using type="combined"
(the default value), both a regular and an irregular histogram are constructed using a penalized likelihood approach
and the one with the larger penalized likelihood is chosen. In this case, the regular histogram is always constructed using the br
penalty.
The penalty
parameter and all other options control the construction of the irregular histogram. penalty
must be equal to "penA"
, "penB"
or "penR"
, since otherwise
comparison of penalized likelihood values would not be meaningful.
Value
an object of class "histogram" which is a list with the same components as in the hist
command.
Penalties
Most settings of penalty
lead to a penalized maximum likelihood histogram. For a sample of size n
and a partition J
that divides the sample range into D
bins, define N_i
as the number of observations in the i
-th bin,
i=1,...,D
and w_i
as the width of the the i
-th bin, i=1,...,D
. In this section,
the index in sums and products is always i=1,\ldots,D
. For any partition J
,
and a fixed sample, the penalized loglikelihood is defined as
\sum N_i * \log(N_i/(n * w_i))-pen(J).
The possible penalties are:
penA
Penalty given in formula (5) in in Rozenholc, Mildenberger and Gather (2009):
pen(J)=c \log {{n-1} \choose {D-1}} + \alpha(D-1) + ck\log(D) + 2\sqrt{c\alpha(D-1)(\log{{n-1} \choose {D-1}} +k \log D) } ,
where the default values are
c=1
,\alpha=0.5
andk=2
. These can be changed using thec
,alpha
andk
components ofcontrol
.penB
Simplified version of formula (5) in Rozenholc, Mildenberger and Gather (2009):
pen(J)=c \log {{n-1} \choose {D-1}} + \alpha(D-1) + \log^{2.5} D,
where the default values are
c=1
and\alpha=1
. These can be changed using thec
andalpha
components ofcontrol
. Default penalty for irregular and combined histograms.penR
Data-dependent penalty as given in formula (6) in Rozenholc, Mildenberger and Gather (2009):
pen(J)=c \log {{n-1} \choose {D-1}} + (\alpha/n) \sum N_i/w_i + \log^{2.5} D,
where the default values are
c=1
and\alpha=0.5
. These can be changed using thec
andalpha
components ofcontrol
.aic
Akaike's Information Criterion (AIC). Defined by
pen(J)=\alpha*D
, where\alpha
is 1 by default and may be changed using thealpha
parameter in thecontrol
argument.bic
Bayesian Information Criterion (BIC). Defined by
pen(J)=\alpha*\log(n)*D
, where\alpha
is 0.5 by default and may be changed using thealpha
parameter in thecontrol
argument.nml
Normalized Maximum Likelihood. Formula is given in Davies, Gather, Nordman, Weinert (2009). Only available for regular histograms.
br
Improved version of AIC for regular histograms as given in Birge and Rozenholc (2006). Defined as
pen(J)=D + \log^{2.5} (D)
. Default penalty for regular histograms, not available for irregular histograms.
Some settings of penalty
do not lead to maximization of a penalized likelihood but optimzation of different measures. These are:
cv
Leave-p-out crossvalidation. Different variants can be chosen by setting the
cvformula
andp
components in thecontrol
argument.cvformula=1
andcvformula=2
are available both for regular and irregular histograms. These are different versions of leave-p-out L2-crossvalidation, where choice of a partition is achieved by minimizing2\sum N_i/w_i - (n+1)\sum N_i^2/(n*w_i),
or
2(n-p) \sum N_i/w_i - (n-p+1) \sum N_i^2/w_i
respectively, see formulas (11) and (12) in Celisse and Robin (2008). Since formula
1
does not depend onp
, if the control parameterp
is set to a value greater than 1cvformula
is set to2
. Kullback-Leibler crossvalidation can be performed by settingcvformula=3
. This is only available ifp=1
andtype="regular"
. The number of bins chosen is the maximizer of\sum N_i \log(N_i-1) + n \log(D),
see remark 2.3 in Hall and Hannan (1988).
sc
Stochastic Complexity criterion, only available for regular histograms. Number of bins is chosen by maximizing
\prod N_i! D^n(D-1)!/(D+n-1)!,
see formula (2.3) in Hall and Hannan (1988).
mdl
Minimum Description Length criterion, only available for regular histograms. Number of bins is chosen by maximizing
\sum (N_i-0.5)\log(N_i-0.5)-(n-0.5D)\log(n-0.5D)+n\log D -0.5D \log n ,
see formula (2.5) in Hall and Hannan (1988).
Control
The control parameter is a list with different components that affect the construction of the histogram. Meaning and default values depend on setting of the other parameters.
alpha
Coefficient of the number of bins in penalties
penA
,penB
,aic
,bic
. Coefficient of the data-driven part in thepenR
penalty.between
logical; if
TRUE
andgrid="data"
, possible bin ends are put between the observations, ifFALSE
(default) they are placed at the observationsc
Controls the weight of the penalty component that corrects for the multiplicity of partitions with the same number of bins in irregular histograms; only used in penalties
penA
,penB
andpenR
.cvformula
determines the type of crossvalidation to be performed. Can take the values 1, 2 and 3. 1 and 2 correspond to different versions of L2 crossvalidation, while
cvformula=3
performs Kullback-Leibler crossvalidation, which is at the moment only available for regular histograms. Note thatcvformula=3
automatically forces every bin to include at least 2 observations. Ifp
is set to a value greater than 1,cvformula=2
is used automatically.g1
The parameters
g1
,g2
andg3
control the maximum number of bins in a regular histogram as well as the bin width and/or number for irregular histograms. DefineG(n)=g1*n^{g2}*(\log(n))^{g3}.
The maximum number of bins allowed in a regular histogram is given by
floor(G(n))
, the finest grid in an irregular histogram withgrid="regular"
is obtained by dividing the sample range intofloor(G(n))
equisized bins, and ifgrid="quantiles"
, the finest grid is obtained by dividing the interval[0,1]
into equisized intervals and using the sample quantiles corresponding to the boundary points. For an irregular histogram withgrid="data"
, a mimimum allowed bin size of1/G(n)
is enforced. This can be disabled by settingg3
toInf
, causing1/G(n)
to be zero. Default settings areg1=1
andg2=1
for all grids. Default values forg3
are-1
forgrid="regular"
andgrid="quantiles"
andInf
forgrid="data"
. Also seemaxbin
.g2
see
g1
.g3
see
g1
.k
Tuning parameter that only has an effect if
penalty="penA"
. Default value is 2.maxbin
Gives an absolute upper bound on the number of bins in order to keep the calculations feasible for large data sets. If the number of bins chosen via
breaks
org1
,g2
andg3
exceedsmaxbin
,maxbin
in used as the maximum number of bins. Only has an effect for regular histograms. Defaults to 1000.p
Controls the number p of data points left out in the crossvalidation. Can take integer values between
1
(default) andn-1
. If a value greater than 1 is chosen,cvformula
is automatically set to 2 since crossvalidation formula 1 does not depend on p and Kullback-Leibler crossvalidation is only supported forp=1
.quanttype
Determines the way the quantiles are calculated if
grid="quantiles"
. Corresponds to thetype
argument inquantile
, whose default7
is also the default here.
Author(s)
Thoralf Mildenberger, Yves Rozenholc, David Zasada.
References
Birg?, L. and Rozenholc, Y. (2006). How many bins should be put in a regular histogram? ESAIM: Probability and Statistics, 10, 24-45.
Celisse, A. and Robin, S. (2008). Nonparametric density estimation by exact leave-p-out cross-validation. Computational Statistics and Data Analysis 52, 2350-2368.
Davies, P. L., Gather, U., Nordman, D. J., and Weinert, H. (2009): A comparison of automatic histogram constructions. ESAIM: Probability and Statistics, 13, 181-196.
Hall, P. and Hannan, E. J. (1988). On stochastic complexity and nonparametric density estimation. Biometrika 75, 705-714.
Rozenholc, Y, Mildenberger, T. and Gather, U. (2009). Combining regular and irregular histograms by penalized likelihood. Discussion Paper 31/2009, SFB 823, TU Dortmund. https://eldorado.tu-dortmund.de/handle/2003/26529
Rozenholc, Y., Mildenberger, T., Gather, U. (2010). Combining regular and irregular histograms by penalized likelihood. Computational Statistics and Data Analysis 54, 3313-3323.
See Also
Examples
## draw a histogram from a standard normal sample
y<-rnorm(100)
histogram(y)
## draw a histogram from a standard exponential sample
y<-rexp(1500)
histogram(y)
## draw a histogram from a normal mixture
n<-sum(sample(c(0,1),1000,replace=TRUE))
y<-c(rnorm(n,mean=5,sd=0.1),rnorm(1000-n))
histogram(y)
## the same using a regular histogram with Kullback-Leibler CV
n<-sum(sample(c(0,1),1000,replace=TRUE))
y<-c(rnorm(n,mean=5,sd=0.1),rnorm(1000-n))
histogram(y,type="regular",penalty="cv",control=list(cvformula=3))