Excess Mass Calculation and Plots

Description

Implementation of a function which calculates the empirical excess mass for a given \lambda and given maximal number of modes (excessm). Offering powerful plot features to visualize empirical excess mass (exmsilhouette). This includes the possibility of drawing several plots (with different maximal number of modes / cut off values) in a single graph. Furthermore, plotting the empirical excess mass against lambda is implemented (exmplot).

Details

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

Muller, D. W., 12.1992, "The Excess Mass Approach in Statistics", Beitraege zur Statistik – StatLab Heidelberg, http://archiv.ub.uni-heidelberg.de/volltextserver/21357/

Examples

library(MASS)
attach(geyser)
excessm(duration, lambda=0.2)

x <- rnorm(1000)
exmsilhouette(x, M=2, CutOff=0.5)

mexmsilhouette(duration, CutOff=c(1,2), steps=60)

Excess Mass Function

Description

Algorithm which calculates the empirical excess mass for a given \lambda and given maximal number of modes.

Usage

excessm(x, lambda, M = 1, UpToM = FALSE)

Arguments

Value

Note

Please note that an allowance for M modes does not necessarily result in M \lambda-clusters. Hence, the number of intervals returned can be smaller than M. In this case a warning will be displayed. The vector excess_mass does have less than M entries.

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

See Also

exmplot, exmsilhouette, mexmsilhouette

Examples

library(MASS)
attach(geyser)

##calculating excess mass for duration of 'Old Faithful Geyser' for lambda=0.2 allowing for one mode
excessm(duration, lambda=0.2)

##same as above, but allowing for up to three modes
excessm(duration, lambda=0.2, M=3) 

#returns the intervals for modes 1,2 and 3
excessm(duration, lambda=0.2, M=3, UpToM=TRUE) 

Excess Mass Lambda Plot

Description

Produces an excess mass lambda plot and calculates the maximal excess mass difference achieved by allowing for an additional mode.

Usage

exmplot(xdata, M=1, CutOff=1, steps=50, Lambda=NULL)

Arguments

Details

CutOff should not be set too small or too large, as this results in meaningless graphs. The excess mass for several \lambdas can be calculated by specifying the Lambda.

Value

An Excess Mass Lambda plot is produced. The lines in the plot are sorted by the maximal number of modes from left to right, due to the monotonicity of the excess mass in M.

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

See Also

excessm, exmsilhouette, mexmsilhouette

Examples

library(MASS)
attach(geyser)

##calculating the maximal excess mass difference for duration of 'Old Faithful Geyser' for M=3
exmplot(duration, M=3)

##Plotting the excess mass against lambda for modes 1-5, 
##increase CutOff value, double the number of steps
exmplot(duration, M=5, CutOff=1.2, steps=100)

##Specifying Lambda
Lambda=seq(.0,0.5,0.005)
exmplot(duration, M=7, Lambda=Lambda)

(Single) Excess Mass Silhouette

Description

Produces an excess mass plot and the corresponding numerical values if required.

Usage

exmsilhouette(xdata, M = 1, CutOff = 1,steps = 50,rug = TRUE,
Lambda = NULL,col = FALSE,rdata = FALSE,label = TRUE)

Arguments

Details

CutOff should not be set too small or too large, as this results in meaningless graphs. The excess mass for several \lambdas can be calculated by specifying the Lambda.

Value

A plot is always produced. By setting rdata=TRUE numerical results are returned in form of a two-dimensional list. The first argument specifies \lambda. This means that if steps=50 with 1 (50) you get access to the numerical results for the smallest (largest) \lambda.

In case no \lambda-vector is used for each \lambda, the following information is displayed:

The last two components are presented in the way known from the excess mass function. In case Lambda was set manually the value of \lambda is not returned, as it is known.

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

See Also

excessm, mexmsilhouette, exmplot

Examples

library(MASS)
attach(geyser)

##Plot allowing for up to two modes and reduced CutOff value
exmsilhouette(duration, M=2, CutOff=1.25)

##Plot with twice the default number of steps, omitting rug plot, 
##colorizing the graph and asking for numerical output
res <- exmsilhouette(duration, M=2, CutOff=1.25, steps=100, rug=FALSE, col=TRUE, rdata=TRUE)

##Specifying Lambda and requesting numerical output
L=seq(.01,0.25,0.005)
res <- exmsilhouette(duration, M=3, Lambda=L, col=TRUE, rdata=TRUE)

(Multiple) Excess Mass Silhouette

Description

Produces a graph with several excess mass plots allowing for different maximal numbers of modes/ cut off values.

Usage

mexmsilhouette(xdata, M = 1:3, CutOff = c(1,2,5), steps = 30,
Lambda = NULL, col = FALSE, rug = TRUE, rdata = FALSE)

Arguments

Details

CutOff should not be set too small or too large, as this results in meaningless graphs.

Value

Always a graph with multiple plots is produced. Each column contains another maximal number of modes and each row another CutOff factor.

Setting rdata=TRUE numerical results are returned in form of a list. If the number of modes and the CutOff parameter contain just one element, the output of "mexmsilhouette" and "exmplot" are equal.

Otherwise we can distinguish between two cases. First Lambda is not specified, hence the list is four-dimensional. The first element determines the CutOff value of the data by using the sorted CutOff vector (using the plot, this means the row in which the graph is shown). The second element specifies the maximal number of modes by using the sorted mode vector (again using the plot, this means the column of the plot). The third element selects the \lambda of the graph. For each plot and each \lambda, the following information is stored: the value of \lambda, the \lambda-clusters and the excess mass vector. Using the default setting [2,2,5,2] shows the \lambda-clusters of the fifth smallest \lambda of the CutOff=2-M=2-plot.

If Lambda is declared manually, the list is three-dimensional. Hence, the first argument denotes the maximal number of modes (the column of the graph). The second argument indicates the \lambda by the position held by it in the Lambda vector. As in "exmplot" only two information are shown. The \lambda-clusters ([,,1]) and the vector of excess mass ([,,2]), as the value of \lambda is known.

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

See Also

excessm, exmplot, exmsilhouette

Examples

library(MASS)
attach(geyser)

##calculating excess mass plots for duration of 'Old Faithful Geyser',
##specifying CutOff and number of steps manually
mexmsilhouette(duration, CutOff=c(1,2), steps=60)

##Allowing for three different maximal number of modes 
##and CutOff factors as well as color. 
##The rug plot is omitted and numerical data is requested.
res=mexmsilhouette(duration, M=c(2,3,7), CutOff=c(0.8,1,2), col=TRUE, rug=FALSE, rdata=TRUE)

##Lambda is specified, color is set to true, numerical data is requested
L=seq(.01,.25,0.005)
res=mexmsilhouette(duration, M=c(2,3,4), Lambda=L, col=TRUE, rdata=TRUE)

Searching For A Maximal Lambda

Description

Function which gives a rough approximation of maximal \lambda.

Usage

searchMaxLambda(x, limcount = 4, step = 1.05, trylambda = 0.01)

Arguments

Details

Excess mass is calculated for trylambda. In case the resulting excess mass is larger (smaller) than the cut off value, trylambda is set as trylambda * step (respectively trylambda / step) and excess mass is calculated again until it is smaller (larger) than the cut off value. The corresponding trylambda is returned.

The approximation is done allowing only for one \lambda-cluster, as scans including more \lambda-clusters have high computational costs due to the recursive structure of the algorithm.

Value

trylambda calculated as described in Details.

Author(s)

Marc-Daniel Mildenberger mildenberger.stat@web.de, based on earlier code from Dr. Guenther Sawitzki gs@statlab.uni-heidelberg.de

References

Muller, D. W. and Sawitzki, G., 09.1991, "Excess Mass Estimates and Tests for Multimodality", Journal of the American Statistical Association , Vol. 86, No. 415, pp. 738–746, http://www.jstor.org/stable/2290406

See Also

excessm, exmplot, exmsilhouette, mexmsilhouette

Examples

library(MASS)
attach(geyser)

#Calculating Lambda using standard settings
searchMaxLambda(duration)

#Calculating Lambda, reducing cut off value and step. Setting another initial lambda
searchMaxLambda(duration, limcount = 5, step = 1.01, trylambda = 1)