Depth-Based Classification and Calculation of Data Depth

Description

The package provides many procedures for calculating the depth of points in an empirical distribution for many notions of data depth. Further it provides implementations for depth-based classification, for multivariate and functional data.

The package implements the DD\alpha-classifier (Lange, Mosler and Mozharovskyi, 2014), a nonparametric procedure for supervised binary classification with q\ge 2 classes. In the training step, the sample is first transformed into a q-dimensional cube of depth vectors, then a linear separation rule in its polynomial extension is constructed with the \alpha-procedure. The classification step involves alternative treatments of 'outsiders'.

Details

Use ddalpha.train to train the DD-classifier and ddalpha.classify to classify with it. Load sample classification problems using getdata. The package contains 50 classification problems built of 33 sets of real data.

The list of the implemented multivariate depths is found in topic depth., for functional depths see depthf.. The depth representations of the multivariate data are obtained with depth.space.. Functions depth.contours and depth.contours.ddalpha build depth contours, and depth.graph builds depth graphs for two-dimensional data. Function draw.ddplot draws DD-plot for the existing DD-classifier, or for pre-calculated depth space.

The package supports user-defined depths and classifiers, see topic Custom Methods. A pre-calculated DD-plot may also be used as data, see topic ddalpha.train.

is.in.convex shows whether an object is no 'outsider', i.e. can be classified by its depth values. Outsiders are alternatively classified by LDA, kNN and maximum Mahalanobis depth as well as by random assignment.

Use compclassf.train and ddalphaf.train to train the functional DD-classifiers and compclassf.classify ddalpha.classify to classify with them. Load sample functional classification problems with dataf.*. The package contains 4 functional data sets and 2 data set generators. The functional data are visualized with plot.functional.

Author(s)

Oleksii Pokotylo, <alexey.pokotylo@gmail.com>

Pavlo Mozharovskyi, <pavlo.mozharovskyi@telecom-paris.fr>

Rainer Dyckerhoff, <rainer.dyckerhoff@statistik.uni-koeln.de>

Stanislav Nagy, <nagy@karlin.mff.cuni.cz>

References

Pokotylo, O., Mozharovskyi, P., Dyckerhoff, R. (2019). Depth and depth-based classification with R-package ddalpha. Journal of Statistical Software 91 1–46.

Lange, T., Mosler, K., and Mozharovskyi, P. (2014). Fast nonparametric classification based on data depth. Statistical Papers 55 49–69.

Lange, T., Mosler, K., and Mozharovskyi, P. (2014). DD\alpha-classification of asymmetric and fat-tailed data. In: Spiliopoulou, M., Schmidt-Thieme, L., Janning, R. (eds), Data Analysis, Machine Learning and Knowledge Discovery, Springer (Berlin), 71–78.

Mosler, K. and Mozharovskyi, P. (2017). Fast DD-classification of functional data. Statistical Papers 58 1055–1089.

Mozharovskyi, P. (2015). Contributions to Depth-based Classification and Computation of the Tukey Depth. Verlag Dr. Kovac (Hamburg).

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics. To appear.

See Also

ddalpha.train, ddalpha.classify,

ddalphaf.train, ddalphaf.classify, compclassf.train, compclassf.classify

depth., depthf., depth.space.,

is.in.convex,

getdata, dataf.*,

plot.ddalpha, plot.ddalphaf, plot.functional, depth.graph, draw.ddplot.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train the DDalpha-classifier
ddalpha <- ddalpha.train(data$train)

# Classify by means of DDalpha-classifier
classes <- ddalpha.classify(ddalpha, data$test[,propertyVars])
cat("Classification error rate:", 
    sum(unlist(classes) != data$test[,classVar])/200, "\n")

# Calculate zonoid depth of top 10 testing objects w.r.t. 1st class
depths.zonoid <- depth.zonoid(data$test[1:10,propertyVars], 
                              data$train[trainIndices,propertyVars])
cat("Zonoid depths:", depths.zonoid, "\n")

# Calculate the random Tukey depth of top 10 testing objects w.r.t. 1st class
depths.halfspace <- depth.halfspace(data$test[1:10,propertyVars], 
                                        data$train[trainIndices,propertyVars])
cat("Random Tukey depths:", depths.halfspace, "\n")

# Calculate depth space with zonoid depth
dspace.zonoid <- depth.space.zonoid(data$train[,propertyVars], c(100, 100))

# Calculate depth space with the exact Tukey depth
dspace.halfspace <- depth.space.halfspace(data$train[,propertyVars], c(100, 100), exact = TRUE)

# Count outsiders
numOutsiders = sum(rowSums(is.in.convex(data$test[,propertyVars], 
                                data$train[,propertyVars], c(100, 100))) == 0)
cat(numOutsiders, "outsiders found in the testing sample.\n")

Fast Computation of the Uniform Metric for Sets of Functional Data

Description

Returns the matrix of C (uniform) distances between two sets of functional data.

Usage

Cmetric(A, B)

Arguments

Details

For two sets of functional data of sizes m and n represented by matrices of their functional values, this function returns the symmetric matrix of size m*n whose entry in the i-th row and j-th column is the approximated C (uniform) distance of the i-th function from the first set, and the j-th function from the second set. This function is utilized in the computation of the h-mode depth.

Value

A symmetric matrix of the distances of the functions of size m*n.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

depthf.hM

dataf2rawfd

Examples

datapop = dataf2rawfd(dataf.population()$dataf,range=c(1950,2015),d=66)
A = datapop[1:20,]
B = datapop[21:50,]
Cmetric(A,B)

Classify using Functional Componentwise Classifier

Description

Classifies data using the functional componentwise classifier.

Usage

compclassf.classify(compclassf, objectsf, subset, ...)

## S3 method for class 'compclassf'
predict(object, objectsf, subset, ...)

Arguments

Value

List containing class labels.

References

Delaigle, A., Hall, P., and Bathia, N. (2012). Componentwise classification and clustering of functional data. Biometrika 99 299–313.

See Also

compclassf.train to train the functional componentwise classifier.

Examples

## Not run: 
## load the Growth dataset
dataf = dataf.growth()

learn = c(head(dataf$dataf, 49), tail(dataf$dataf, 34))
labels =c(head(dataf$labels, 49), tail(dataf$labels, 34)) 
test = tail(head(dataf$dataf, 59), 10)    # elements 50:59. 5 girls, 5 boys

c = compclassf.train (learn, labels, classifier.type = "ddalpha")

classified = compclassf.classify(c, test)

print(unlist(classified))


## End(Not run)

Functional Componentwise Classifier

Description

Trains the functional componentwise classifier

Usage

compclassf.train (dataf, labels, subset,
                  to.equalize = TRUE, 
                  to.reduce = TRUE, 
                  classifier.type = c("ddalpha", "maxdepth", "knnaff", "lda", "qda"), 
                  ...)

Arguments

Details

The finite-dimensional space is directly constructed from the observed values. Delaigle, Hall and Bathia (2012) consider (almost) all sets of discretization points that have a given cardinality.

The usual classifiers are then trained on the constructed finite-dimensional space.

Value

Trained functional componentwise classifier

References

Delaigle, A., Hall, P., and Bathia, N. (2012). Componentwise classification and clustering of functional data. Biometrika 99 299–313.

See Also

compclassf.classify for classification using functional componentwise classifier,

ddalphaf.train to train the functional DD-classifier,

dataf.* for functional data sets included in the package.

Examples

## Not run: 
## load the Growth dataset
dataf = dataf.growth()

learn = c(head(dataf$dataf, 49), tail(dataf$dataf, 34))
labels =c(head(dataf$labels, 49), tail(dataf$labels, 34)) 
test = tail(head(dataf$dataf, 59), 10)    # elements 50:59. 5 girls, 5 boys

c = compclassf.train (learn, labels, classifier.type = "ddalpha")

classified = compclassf.classify(c, test)

print(unlist(classified))


## End(Not run)

Using Custom Depth Functions and Classifiers

Description

To use a custom depth function or classifier one has to implement three functions: parameters validator, learning and calculating functions.

Details

To define a depth function:

.<NAME>_validate

validates parameters passed to ddalpha.train and passes them to the ddalpha object.

IN:
ddalpha	the ddalpha object, containing the data and settings (mandatory)
<custom parameters>	parameters that are passed to the user-defined method
...	other parameters (mandatory)
OUT:
list()	list of output parameters, after the validation is finished, these parameters are stored in the ddalpha object

.<NAME>_learn

trains the depth

IN:
ddalpha	the ddalpha object, containing the data and settings
MODIFIES:
ddalpha	store the calculated statistics in the ddalpha object
depths	calculate the depths of each pattern, e.g.
	for (i in 1:ddalpha$numPatterns) ddalpha$patterns[[i]]$depths = .<NAME>_depths(ddalpha, ddalpha$patterns[[i]]$points)
OUT:
ddalpha	the updated ddalpha object

.<NAME>_depths

calculates the depths

IN:
ddalpha	the ddalpha object, containing the data and settings
objects	the objects for which the depths are calculated
OUT:
depths	the calculated depths for each object (rows), with respect to each class (cols)

Usage: ddalpha.train(data, depth = "<NAME>", <custom parameters>, ...)

⁠ #### Custom depths #### .MyDepth_validate <- function(ddalpha, mydepth.parameter = "value", ...){ print("MyDepth validating") # validate the parameters if (!is.valid(mydepth.parameter)){ warning("Argument \"mydepth.parameter\" not specified correctly. Default value is used") mydepth.parameter = "value" # or stop("Argument \"mydepth.parameter\" not specified correctly.") } # the values from the return list will be stored in the ddalpha object return (list(mydepthpar = mydepth.parameter)) } .MyDepth_learn <- function(ddalpha){ print("MyDepth learning") #1. Calculate any statistics based on data that .MyDepth_depths needs # and store them to the ddalpha object: ddalpha$mydepth.statistic = "some value" #2. Calculate depths for each pattern for (i in 1:ddalpha$numPatterns){ ddalpha$patterns[[i]]$depths = .MyDepth_depths(ddalpha, ddalpha$patterns[[i]]$points) } return(ddalpha) } .MyDepth_depths <- function(ddalpha, objects){ print("MyDepth calculating") depths <- NULL # The depth parameters are accessible in the ddalpha object: mydepth.parameter = ddalpha$mydepth.parameter mydepth.statistic = ddalpha$mydepth.statistic #calculate the depths of the objects w.r.t. each pattern for (i in 1:ddalpha$numPatterns){ depth_wrt_i = #calculate depths of the objects, as vector depths <- cbind(depths, depth_wrt_i) } return (depths) } ddalpha.train(data, depth = "MyDepth", ...) ⁠

To define a classifier:

.<NAME>_validate

validates parameters passed to ddalpha.train and passes them to the ddalpha object

IN:
ddalpha	the ddalpha object, containing the data and settings (mandatory)
<custom parameters>	parameters that are passed to the user-defined method
...	other parameters (mandatory)
OUT:
list()	list of output parameters, after the validation is finished, these parameters are stored in the ddalpha object. In case of a multiclass classifier the validator must return methodSeparatorBinary = F and/or pass aggregation.method = "none" to ddalpha.train

.<NAME>_learn

trains the classifier. Is different for binnary and mylticlass classifiers.

IN:
ddalpha	the ddalpha object, containing the data and settings
index1	(only for binary) index of the first class
index2	(only for binary) index of the second class
depths1	(only for binary) depths of the first class w.r.t. all classes
depths2	(only for binary) depths of the second class w.r.t. all classes
	depths w.r.t. only given classes are received by depths1[,c(index1, index2)]
	for the multiclass classifiers the depths are accessible via ddalpha$patterns[[i]]$depths
OUT:
classifier	the trained classifier object

.<NAME>_classify

classifies the objects

IN:
ddalpha	the ddalpha object, containing the data and global settings
classifier	the previously trained classifier
objects	the objects (depths) that are classified
OUT:
result	a vector with classification results
	(binary) the objects from class "classifier$index1" get positive values
	(multiclass) the objects get the numbers of patterns in ddalpha

Usage: ddalpha.train(data, separator = "<NAME>", ...)

⁠ #### Custom classifiers #### .MyClassifier_validate <- function(ddalpha, my.parameter = "value", ...){ print("MyClassifier validating") # validate the parameters ... # always include methodSeparatorBinary. # TRUE for the binary classifier, FALSE otherwise return(list(methodSeparatorBinary = T, my.parameter = my.parameter )) } # a binary classifier # the package takes care of the voting procedures. Just train it as if there are only two classes .MyClassifier_learn <- function(ddalpha, index1, index2, depths1, depths2){ print("MyClassifier (binary) learning") # The parameters are accessible in the ddalpha object: my.parameter = ddalpha$my.parameter #depths w.r.t. only given classes are received by depths1[,c(index1, index2)] depths2[,c(index1, index2)] # train the classifier classifier <- ... #return the needed values in a list, e.g. return(list( coefficients = classifier$coefficients, ... )) } # a multiclass classifier .MyClassifier_learn <- function(ddalpha){ print("MyClassifier (multiclass) learning") # access the data through the ddalpha object, e.g. for (i in 1:ddalpha$numPatterns){ depth <- ddalpha$patterns[[i]]$depths number <- ddalpha$patterns[[i]]$cardinality ... } # train the classifier classifier <- ... # return the classifier return(classifier) } # the interface of the classify function is equal for binary and multiclass classifiers .MyClassifier_classify <- function(ddalpha, classifier, depths){ print("MyClassifier classifying") # The global parameters are accessible in the ddalpha object: my.parameter = ddalpha$my.parameter # The parameters generated by .MyClassifier_learn are accessible in the classifier object: classifier$coefficients # here are the depths w.r.t. the first class depths[,classifier$index1] # here are the depths w.r.t. the second class depths[,classifier$index2] # fill results in a vector, so that: # (binary) the objects from class "classifier$index1" get positive values # (multiclass) the objects get the numbers of patterns in ddalpha result <- ... return(result) } ddalpha.train(data, separator = "MyClassifier", ...) ⁠

See Also

ddalpha.train

Examples

## Not run: 

#### example:  Euclidean depth ####

#.Euclidean_validate is basically not needed

.Euclidean_learn <- function(ddalpha){
  print("Euclidean depth learning")
  
  #1. Calculate any statistics based on data that .MyDepth_depths needs
  #   and store them to the ddalpha object: 
  for (i in 1:ddalpha$numPatterns){
    ddalpha$patterns[[i]]$center <- colMeans(ddalpha$patterns[[i]]$points)
  }
  
  #2. Calculate depths for each pattern
  for (i in 1:ddalpha$numPatterns){
    ddalpha$patterns[[i]]$depths = .Euclidian_depths(ddalpha, ddalpha$patterns[[i]]$points)
  }
  
  return(ddalpha)
}

.Euclidean_depths <- function(ddalpha, objects){
  print("Euclidean depth calculating")
  depths <- NULL
  
  #calculate the depths of the objects w.r.t. each pattern
  for (i in 1:ddalpha$numPatterns){
    # The depth parameters are accessible in the ddalpha object:
    center = ddalpha$patterns[[i]]$center
    
    depth_wrt_i <- 1/(1 + colSums((t(objects) - center)^2))
    depths <- cbind(depths, depth_wrt_i)
  }
  
  return (depths)
}

#### example:  binary decision tree ####

library(rpart)

.tree_validate  <- function(ddalpha, ...){
  print("tree validating")
  return(list(methodSeparatorBinary = T))
}

# a binary classifier 
# the package takes care of the voting procedures. Just train it as if there are only two classes
.tree_learn <- function(ddalpha, index1, index2, depths1, depths2){
  print("tree learning")
  
  # prepare the data
  data = as.data.frame(cbind( (rbind(depths1, depths2)), 
                              c(rep(1, times = nrow(depths1)), rep(-1, times = nrow(depths1)))))
  names(data) <- paste0("V",seq_len(ncol(data)))
  names(data)[ncol(data)] <- "O"
  # train the classifier
  classifier <- rpart(O~., data = data)
  
  #return the needed values in a list, e.g.
  return(classifier)
}

# the interface of the classify function is equal for binary and multiclass classifiers
.tree_classify <- function(ddalpha, classifier, depths){
  print("tree classifying")
  
  # fill results in a vector, so that the objects from class "classifier$index1" get positive values
  data = as.data.frame(depths)
  names(data) <- paste0("V",seq_len(ncol(data)))
  result <- predict(classifier, as.data.frame(depths), type = "vector")
  
  return(result)
}




#### checking ####

library(ddalpha)
data = getdata("hemophilia")

ddalpha = ddalpha.train(data = data, depth = "Euclidean", separator = "tree")
c = ddalpha.classify(ddalpha, data[,1:2])
errors = sum(unlist(c) != data[,3])/nrow(data)
print(paste("Error rate: ",errors))

# building the depth contours using the custom depth
depth.contours.ddalpha(ddalpha, asp = T, levels = seq(0.5, 1, length.out = 10))


## End(Not run)

Converts data from fdata class to the functional class.

Description

fda.usc contains a handy function fdata that converts varios types of functional data to the fdata class. To use these data in ddalphaf.train it must first be converted with dataf.

The function may be used either to convert a fdata object that contains multiple classes, or to convert multiple fdata objects, each of which contains one class.

Note that fdata$fdata2d = TRUE is not supported.

Usage

dataf(fdatas, labels) 

Arguments

Format

The functional data as a data structure (see dataf.*).

dataf

The functional data as a list of objects. Each object is characterized by two coordinates

args

The arguments vector

vals

The values vector

labels

The classes of the objects

See Also

dataf.* for the functional data format.

ddalphaf.train to train the functional DD\alpha-classifier

compclassf.train to train the functional componentwise classifier

plot.functional for building plots of functional data

Examples

## Not run: 
library(fda.usc)
data(phoneme)

# 1. convert a fdata object that contains multiple classes.
#    labels are defined for each curve
converted = dataf(phoneme$learn, phoneme$classlearn)
plot.functional(converted)

# 2. convert multiple fdata objects, each of which contains one class
#    the same label is applied to all curves of each fdata object
converted = dataf(list(phoneme$learn, phoneme$test), c("1 red", "2 blue"))
converted$name = "Phoneme learn (red) and test (blue)"
plot.functional(converted)

## End(Not run)

Functional Data Sets

Description

The functions generate data sets of functional two-dimensional data of two or more classes.

Usage

# dataf.[name]()

# load the data set by name
# data(list = "name")

# load the data set by name to a variable
# getdata("name")

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates

args

The arguments vector

vals

The values vector

labels

The classes of the objects

Details

More details about the datasets in the topics:

dataf.geneexp

dataf.growth

dataf.medflies

dataf.population

dataf.population2010

dataf.tecator

dataf.tecator

The following datasets provide simulated data:

dataf.sim.1.CFF07

dataf.sim.2.CFF07

See Also

plot.functional for building plots of functional data

Examples

## load the Growth dataset
dataf = dataf.growth()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## Not run: plot.functional(dataf)

Gene Expression Profile Data

Description

A subet of the Drosophila life cycle gene expression data of Arbeitman et al. (2002). The original data set contains 77 gene expression profiles during 58 sequential time points from the embryonic, larval, and pupal periods of the life cycle. The gene expression levels were obtained by a cDNA microarray experiment.

Usage

dataf.geneexp()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates.

args

Time - a numeric vector of time periods

vals

Gene Expression Level - a numeric vector

labels

Biological classifications identified in Arbeitman et al.(2002) (1 = transient early zygotic genes; 2 = muscle-specific genes; 3 = eye-specific genes. )

Source

Chiou, J.-M. and Li, P.-L. Functional clustering and identifying substructures of longitudinal data, J. R. Statist. Soc. B, Volume 69 (2007), 679-699

Arbeitman, M.N., Furlong, E.E.M., Imam,F., Johnson, E., Null,B.H., Baker,B.S., Krasnow, M.A., Scott,M.P., Davis,R.W. and White,K.P. (2002) Gene expression during the life cycle of Drosophila melanogaster. Science, 297, 2270-2274.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the dataset
dataf = dataf.geneexp()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run: 
labels = unlist(dataf$labels)
plot(dataf, 
  xlab="Time", ylab="Gene Expression Level", 
  main=paste0("Gene Expression:  1 red (", sum(labels == 1), "), ", 
            "2 green (", sum(labels == 2), "), ", 
            "3 blue (", sum(labels == 3), ")"),
  colors = c("red", "green", "blue"))

## End(Not run)

Berkeley Growth Study Data

Description

The data set contains the heights of 39 boys and 54 girls from age 1 to 18 and the ages at which they were collected.

Usage

dataf.growth()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates

args

age - a numeric vector of length 31 giving the ages at which the heights were measured

vals

height - a numeric vector of heights in centimeters of 39 boys and 54 girls

labels

The classes of the objects: boy, girl

Details

The ages are not equally spaced.

Source

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York, ch. 6.

Tuddenham, R. D., and Snyder, M. M. (1954) "Physical growth of California boys and girls from birth to age 18", University of California Publications in Child Development, 1, 183-364.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the Growth dataset
dataf = dataf.growth()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run: 
  labels = unlist(dataf$labels)
  plot(dataf, 
    main = paste("Growth: girls red (", sum(labels == "girl"), "),", 
                      " boys blue (", sum(labels == "boy"), ")", sep=""),
    xlab="Year", ylab="Height, cm",
    colors = c("blue", "red")   # in alphabetical order of class labels   
    )

## End(Not run)

Relationship of Age Patterns of Fecundity to Mortality for Female Medflies.

Description

The data set consists of number of eggs laid daily for each of 1000 medflies (Mediterranean fruit flies, Ceratitis capitata) until time of death. Data were obtained in Dr. Carey's laboratory. The main questions are to explore the relationship of age patterns of fecundity to mortality, longevity and lifetime reproduction.

A basic finding was that individual mortality is associated with the time-dynamics of the egg-laying trajectory. An approximate parametric model of the egg laying process was developed and used in Muller et al. (2001). Non-parametric approaches which extend principal component analysis for curve data to the situation when covariates are present have been developed and discussed in Chiou, Muller and Wang (2003) and Chiou et al. (2003).

Usage

dataf.medflies()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates.

args

day - a numeric vector of the days numbers

vals

#eggs - a numeric vector of numbers of eggs laid daily

labels

The classes of the objects: long-lived, short-lived

Source

Carey, J.R., Liedo, P., Muller, H.G., Wang, J.L., Chiou, J.M. (1998). Relationship of age patterns of fecundity to mortality, longevity, and lifetime reproduction in a large cohort of Mediterranean fruit fly females. J. of Gerontology –Biological Sciences 53, 245-251.

Muller, H.G., Carey, J.R., Wu, D., Liedo, P., Vaupel, J.W. (2001). Reproductive potential predicts longevity of female Mediterranean fruit flies. Proceedings of the Royal Society B 268, 445-450.

Chiou, J.M., Muller, H.G., Wang, J.L. (2003). Functional quasi-likelihood regression models with smooth random effects. J. Royal Statist. Soc. B65, 405-423.

Chiou, J.M., Muller, H.G., Wang, J.L., Carey, J.R. (2003). A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies. Statistica Sinica 13, 1119-1133.

Chiou, J.M., Muller, H.G., Wang, J.L. (2004).Functional response models. Statistica Sinica 14,675-693.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the dataset
dataf = dataf.medflies()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run: 
labels = unlist(dataf$labels)
plot(dataf,
  xlab="Day", ylab="# eggs", 
  main=paste("Medflies (training time):\n short-lived red (", sum(labels == "short-lived"), "),", 
                    " long-lived blue (", sum(labels == "long-lived"), ")", sep=""),
  colors = c("blue", "red")   # in alphabetical order of class labels
  )


## End(Not run)

World Historical Population-by-Country Dataset

Description

Historical world population data by countries.

Usage

dataf.population()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates

args

year - a numeric vector of years 1950-2015 (66 years)

vals

population - a numeric vector of the estimated total population in thousands in 233 countries and regions

labels

The geographic region of the country: Africa, Asia, Europe, Latin America, North America, Oceania

identifier

The name of country or region

Details

World population data by a country, area or region as of 1 July of the year indicated. Figures are presented in thousands.

Source

United Nations, Department of Economic and Social Affairs, Population Division, https://esa.un.org/unpd/wpp/Download/Standard/Population/, file Total population - Both sexes

See Also

dataf.population2010

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the Population dataset
dataf = dataf.population()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run:
labels = unlist(dataf$labels)
plot(dataf,
  main = "World population data",
  xlab="Year", ylab="Population (in thousands)"
  )
## End(Not run)

## compute the integrated and infimal depths of the data curves
## with respect to the same set of curves
depthf.fd1(dataf$dataf, dataf$dataf)

World Historical Population-by-Country Dataset (2010 Revision)

Description

Historical world population data by countries.

Usage

dataf.population2010()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates

args

year - a numeric vector of years 1950-2010 (61 years)

vals

population - a numeric vector of the estimated total population in thousands in 233 countries and regions

labels

The geographic region of the country

identifier

The name of country or region

Details

World population data by a country, area or region as of 1 July of the year indicated. Figures are presented in thousands.

Source

United Nations, Department of Economic and Social Affairs, Population Division, https://esa.un.org/unpd/wpp/Download/Standard/Population/, file Total population - Both sexes

See Also

dataf.population

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the Population dataset
dataf = dataf.population2010()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run:
labels = unlist(dataf$labels)
plot(dataf,
  main = "World population data",
  xlab="Year", ylab="Population (in thousands)"
  )
## End(Not run)

## compute the integrated and infimal depths of the data curves
## with respect to the same set of curves
depthf.fd1(dataf$dataf, dataf$dataf)

Model 1 from Cuevas et al. (2007)

Description

Model 1 from Cuevas et al. (2007)

Processes:
X(t) = m_0(t) + e(t), m_0(t) = 30*(1-t)*t^1.2
Y(t) = m_1(t) + e(t), m_1(t) = 30*(1-t)^1.2*t
e(t): Gaussian with mean = 0, cov(X(s), X(t)) = 0.2*exp(-abs(s - t)/0.3)
the processes are discretized at numDiscrets equally distant points on [0, 1]. The functions are smooth and differ in mean only, which makes the classification task rather simple.

Usage

dataf.sim.1.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51, plot = FALSE)

Arguments

Format

A data strusture containing $learn and $test functional data. The functional data are given as data structures.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates.

args

a numeric vector

vals

a numeric vector

labels

The classes of the objects: 0 for X(t), 1 for Y(t)

Source

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 481-496.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the dataset
dataf = dataf.sim.1.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51)
learn = dataf$learn
test = dataf$test

## view the classes
unique(learn$labels)

## access the 5th point of the 2nd object
learn$dataf[[2]]$args[5]
learn$dataf[[2]]$vals[5]

## Not run: 
## plot the data
plot(learn)
plot(test)

## or
dataf = dataf.sim.1.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51, plot = TRUE)

## End(Not run)

Model 2 from Cuevas et al. (2007)

Description

Model 2 from Cuevas et al. (2007)

Processes:
X(t) = m_0(t) + e(t), m_0(t) = 30*(1-t)*t^2 + 0.5*abs(sin(20*pi*t))
Y(t) = an 8-knot spline approximation of X
e(t): Gaussian with mean = 0, cov(X(s), X(t)) = 0.2*exp(-abs(s - t)/0.3)
the processes are discretized at numDiscrets equally distant points on [0, 1].

Usage

dataf.sim.2.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51, plot = FALSE)

Arguments

Format

A data strusture containing $learn and $test functional data. The functional data are given as data structures.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates.

args

a numeric vector

vals

a numeric vector

labels

The classes of the objects: 0 for X(t), 1 for Y(t)

Source

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 481-496.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the dataset
dataf = dataf.sim.2.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51)
learn = dataf$learn
test = dataf$test

## view the classes
unique(learn$labels)

## access the 5th point of the 2nd object
learn$dataf[[2]]$args[5]
learn$dataf[[2]]$vals[5]

## Not run: 
## plot the data
plot(learn)
plot(test)

## or
dataf = dataf.sim.2.CFF07(numTrain = 100, numTest = 50, numDiscrets = 51, plot = TRUE)

## End(Not run)

Functional Data Set Spectrometric Data (Tecator)

Description

This dataset is a part of the original one which can be found at https://www.cmu.edu/dietrich/statistics-datascience/. For each peace of finely chopped meat we observe one spectrometric curve which corresponds to the absorbance measured at 100 wavelengths. The peaces are split according to Ferraty and Vieu (2006) into two classes: with small (<20) and large fat content obtained by an analytical chemical processing.

Usage

dataf.tecator()

Format

The functional data as a data structure.

dataf

The functional data as a list of objects. Each object is characterized by two coordinates.

args

wavelength - a numeric vector of discretization points from 850 to 1050mm

vals

absorbance - a numeric vector of absorbance values

labels

The classes of the objects: "small" (<20) and "large" fat content

Author(s)

Febrero-Bande, M and Oviedo de la Fuente, Manuel

Source

https://www.cmu.edu/dietrich/statistics-datascience/

References

Ferraty, F. and Vieu, P. (2006). Nonparametric functional data analysis: theory and practice. Springer.

See Also

dataf.* for other functional data sets

plot.functional for building plots of functional data

Examples

## load the dataset
dataf = dataf.tecator()

## view the classes
unique(dataf$labels)

## access the 5th point of the 2nd object
dataf$dataf[[2]]$args[5]
dataf$dataf[[2]]$vals[5]

## plot the data
## Not run: 
labels = unlist(dataf$labels)
plot(dataf, 
  xlab="Wavelengths", ylab="Absorbances", 
  main=paste("Tecator: < 20 red (", sum(labels == "small"), "),", 
            " >= 20 blue (", sum(labels == "large"), ")", sep=""),
    colors = c("blue", "red"))

## End(Not run)

Transform a dataf Object to Raw Functional Data

Description

From a (possibly multivariate) functional data object dataf constructs an array of the functional values evaluated at an equi-distant grid of points.

Usage

dataf2rawfd(dataf, range = NULL, d = 101)

Arguments

Value

If the functional data are univariate (scalar-valued), a matrix of size m*d is given, with each row corresponding to one function. If the functional data are k-variate with k>1, an array of size m*d*k of the functional values is given.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

rawfd2dataf

depthf.fd1

depthf.fd2

Examples

## transform a matrix into a functional data set and back
n = 5
d = 21
X = matrix(rnorm(n*d),ncol=d)
R = rawfd2dataf(X,range=c(0,1))
R2 = dataf2rawfd(R,range=c(0,1),d=d)
all.equal(X,R2)

## transform a functional dataset into a raw matrix of functional values
dataf = dataf.population()$dataf
dataf2rawfd(dataf,range=c(1950,2015),d=66)

## transform an array into a multivariate functional data set and back
k = 3
X = array(rnorm(n*d*k),dim=c(n,d,k))
R = rawfd2dataf(X,range=c(-1,1))
dataf2rawfd(R,range=c(-1,1),d=50)

Classify using DD-Classifier

Description

Classifies data using the DD-classifier and a specified outsider treatment.

Usage

ddalpha.classify(ddalpha, objects, subset, outsider.method = NULL, use.convex = NULL)

## S3 method for class 'ddalpha'
predict(object, objects, subset, outsider.method = NULL, use.convex = NULL, ...)

Arguments

Details

Only one outsider treatment can be specified.

See Lange, Mosler and Mozharovskyi (2014) for details and additional information.

Value

List containing class labels, or character string "Ignored" for the outsiders if "Ignore" was specified as the outsider treating method.

References

Dyckerhoff, R., Koshevoy, G., and Mosler, K. (1996). Zonoid data depth: theory and computation. In: Prat A. (ed), COMPSTAT 1996. Proceedings in computational statistics, Physica-Verlag (Heidelberg), 235–240.

Lange, T., Mosler, K., and Mozharovskyi, P. (2014). Fast nonparametric classification based on data depth. Statistical Papers 55 49–69.

Li, J., Cuesta-Albertos, J.A., and Liu, R.Y. (2012). DD-classifier: Nonparametric classification procedure based on DD-plot. Journal of the American Statistical Association 107 737–753.

Mozharovskyi, P. (2015). Contributions to Depth-based Classification and Computation of the Tukey Depth. Verlag Dr. Kovac (Hamburg).

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

Vasil'ev, V.I. (2003). The reduction principle in problems of revealing regularities I. Cybernetics and Systems Analysis 39 686–694.

See Also

ddalpha.train to train the DD-classifier.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train the DDalpha-Classifier (zonoid depth, maximum Mahalanobis depth 
# classifier with defaults as outsider treatment)
ddalpha <- ddalpha.train(data$train, 
                         depth = "zonoid", 
                         outsider.methods = "depth.Mahalanobis")
# Get the classification error rate
classes <- ddalpha.classify(data$test[,propertyVars], ddalpha, 
                            outsider.method = "depth.Mahalanobis")
cat("Classification error rate: ", 
    sum(unlist(classes) != data$test[,classVar])/200, ".\n", sep="")

Test DD-Classifier

Description

Performs a cross-validation procedure over the given data. On each step every numchunks observation is removed from the data, the DD-classifier is trained on these data and tested on the removed observations.

Usage

ddalpha.getErrorRateCV (data, numchunks = 10,  ...)

Arguments

Value

See Also

ddalpha.train to train the DD\alpha-classifier, ddalpha.classify for classification using DD\alpha-classifier, ddalpha.test to test the DD-classifier on particular learning and testing data, ddalpha.getErrorRatePart to perform a benchmark study of the DD-classifier on particular data.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(150, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(150, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
propertyVars <- c(1:2)
classVar <- 3
data <- rbind(cbind(class1, rep(1, 150)), cbind(class2, rep(2, 150)))

# Train 1st DDalpha-classifier (default settings) 
# and get the classification error rate
stat <- ddalpha.getErrorRateCV(data, numchunks = 5)
cat("1. Classification error rate (defaults): ", 
    stat$error, ".\n", sep = "")

# Train 2nd DDalpha-classifier (zonoid depth, maximum Mahalanobis 
# depth classifier with defaults as outsider treatment) 
# and get the classification error rate
stat2 <- ddalpha.getErrorRateCV(data, depth = "zonoid", 
                          outsider.methods = "depth.Mahalanobis")
cat("2. Classification error rate (depth.Mahalanobis): ", 
    stat2$error, ".\n", sep = "")

Test DD-Classifier

Description

Performs a benchmark procedure by partitioning the given data. On each of times steps size observations are removed from the data, the DD-classifier is trained on these data and tested on the removed observations.

Usage

ddalpha.getErrorRatePart(data, size = 0.3, times = 10,  ...)

Arguments

Value

See Also

ddalpha.train to train the DD\alpha-classifier, ddalpha.classify for classification using DD\alpha-classifier, ddalpha.test to test the DD-classifier on particular learning and testing data, ddalpha.getErrorRateCV to get error rate of the DD-classifier on particular data.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 objects
class1 <- mvrnorm(100, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(100, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
propertyVars <- c(1:2)
classVar <- 3
data <- rbind(cbind(class1, rep(1, 100)), cbind(class2, rep(2, 100)))

# Train 1st DDalpha-classifier (default settings) 
# and get the classification error rate
stat <- ddalpha.getErrorRatePart(data, size = 10, times = 10)
cat("1. Classification error rate (defaults): ", 
    stat$error, ".\n", sep = "")

# Train 2nd DDalpha-classifier (zonoid depth, maximum Mahalanobis 
# depth classifier with defaults as outsider treatment) 
# and get the classification error rate
stat2 <- ddalpha.getErrorRatePart(data, depth = "zonoid", 
                                outsider.methods = "depth.Mahalanobis", size = 0.2, times = 10)
cat("2. Classification error rate (depth.Mahalanobis): ", 
    stat2$error, ".\n", sep = "")

Test DD-Classifier

Description

Trains DD-classifier on the learning sequence of the data and tests it on the testing sequence.

Usage

ddalpha.test(learn, test, ...)

Arguments

Value

See Also

ddalpha.train to train the DD-classifier, ddalpha.classify for classification using DD-classifier, ddalpha.getErrorRateCV and ddalpha.getErrorRatePart to get error rate of the DD-classifier on particular data.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train 1st DDalpha-classifier (default settings) 
# and get the classification error rate
stat <- ddalpha.test(data$train, data$test)
cat("1. Classification error rate (defaults): ", 
    stat$error, ".\n", sep = "")

# Train 2nd DDalpha-classifier (zonoid depth, maximum Mahalanobis 
# depth classifier with defaults as outsider treatment) 
# and get the classification error rate
stat2 <- ddalpha.test(data$train, data$test, depth = "zonoid", 
                          outsider.methods = "depth.Mahalanobis")
cat("2. Classification error rate (depth.Mahalanobis): ", 
    stat2$error, ".\n", sep = "")

Train DD-Classifier

Description

Trains the DD-classifier using a training sample according to given parameters. The DD-classifier is a non-parametric procedure that first transforms the training sample into the depth space calculating the depth of each point w.r.t each class (dimension of this space equals the number of classes in the training sample), and then constructs a separating rule in this depth space. If in the classification phase an object does not belong to the convex hull of at least one class (we mention such an object as an 'outsider'), it is mapped into the origin of the depth space and hence cannot be classified in the depth space. For these objects, after 'outsiderness' has been assured, an outsider treatment, i.e. a classification procedure functioning outside convex hulls of the classes is applied; it has to be trained too.

The current realization of the DD-classifier allows for several alternative outsider treatments; they involve different traditional classification methods, see 'Details' and 'Arguments' for parameters needed.

The function allows for classification with q\ge 2 classes, see aggregation.method in 'Arguments'.

Usage

ddalpha.train(formula, data, subset,
              depth = "halfspace", 
              separator = "alpha", 
              outsider.methods = "LDA", 
              outsider.settings = NULL, 
              aggregation.method = "majority",
              pretransform = NULL,
              use.convex = FALSE,     
              seed = 0,
              ...)

Arguments

Details

Depths

For depth="ddplot" the pre-calculated DD-plot shall be passed as data.

To use a custom depth, see topic Custom Methods.

To use an outsider treatment only set depth = NULL.

The following depths are supported:

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

The additional parameters are described in the corresponding topics.

Separators

The separators classify data on the 2-dimensional space of a DD-plot built using the depths.

To use a custom separator, see topic Custom Methods.

alpha

Trains the DD\alpha-classifier (Lange, Mosler and Mozharovskyi, 2014; Mozharovskyi, Mosler and Lange, 2015). The DD\alpha-classifier constructs a linear separating rule in the polynomial extension of the depth space with the \alpha-procedure (Vasil'ev, 2003); maximum degree of the polynomial products is determined via cross-validation (in the depth space).

The additional parameters:

max.degree

Maximum of the range of degrees of the polynomial depth space extension over which the \alpha-procedure is to be cross-validated; can be 1, 2 or 3 (default).

num.chunks

Number of chunks to split data into when cross-validating the \alpha-procedure; should be >0, and smaller than the total number of points in the two smallest classes when aggregation.method = "majority" and smaller than the total number of points in the training sample when aggregation.method = "sequent". The default value is 10.

polynomial

Trains the polynomial DD-classifier (Li, Cuesta-Albertos and Liu, 2012). The DD-classifier constructs a polynomial separating rule in the depth space; the degree of the polynomial is determined via cross-validation (in the depth space).

The additional parameters:

max.degree

Maximum of the range of degrees of the polynomial over which the separator is to be cross-validated; can be in [1:10], the default value is 3.

num.chunks

Number of chunks to split data into when cross-validating the separator; should be >0, and smaller than the total number of points in the two smallest classes when aggregation.method = "majority" and smaller than the total number of points in the training sample when aggregation.method = "sequent". The default value is 10.

knnlm

Trains the k-nearest neighbours classifier in the depth space.

The additional parameters:

knnrange

The maximal number of neighbours for kNN separation. The value is bounded by 2 and n/2.

NULL for the default value 10*(n^{1/q})+1, where n is the number of objects, q is the number of classes.

"MAX" for the maximum value n/2

maxD

The maximum depth separator classifies an object to the class that provides it the largest depth value.

Outsider treatment

An outsider treatment is a supplementary classifier for data that lie outside the convex hulls of all q training classes. Available methods are: Linear Discriminant Analysis (referred to as "LDA"), see lda; k-Nearest-Neighbor Classifier ("kNN"), see knn, knn.cv; Affine-Invariant kNN ("kNNAff"), an affine-invariant version of the kNN, suited only for binary classification (some aggregation is used with multiple classes) and not accounting for ties (at all), but very fast by that; Maximum Mahalanobis Depth Classifier ("depth.Mahalanobis"), the outsider is referred to a class w.r.t. which it has the highest depth value scaled by (approximated) priors; Proportional Randomization ("RandProp"), the outsider is referred to a class randomly with probability equal to it (approximated) prior; Equal Randomization ("RandEqual"), the outsider is referred to a class randomly, chances for each class are equal; Ignoring ("Ignore"), the outsider is not classified, the string "Ignored" is returned instead.

An outsider treatment is specified by a list containing a name and parameters:

name is a character string, name of the outsider treatment to be freely specified; should be unique; is obligatory.

method is a character string, name of the method to use, can be "LDA", "kNN", "kNNAff", "depth.Mahalanobis", "RandProp", "RandEqual" and "Ignore"; is obligatory.

priors is a numerical vector specifying prior probabilities of classes; class portions in the training sample are used by the default. priors is used in methods "LDA", "depth.Mahalanobis" and "RandProp".

knn.k is the number of the nearest neighbors taken into account; can be between 1 and the number of points in the training sample. Set to -1 (the default) to be determined by the leave-one-out cross-validation. knn.k is used in method "kNN".

knn.range is the upper bound on the range over which the leave-one-out cross-validation is performed (the lower bound is 1); can be between 2 and the number of points in the training sample -1. Set to -1 (the default) to be calculated automatically accounting for number of points and dimension. knn.range is used in method "kNN".

knnAff.methodAggregation is a character string specifying the aggregation technique for method "kNNAff"; works in the same way as the function argument aggregation.method. knnAff.methodAggregation is used in method "kNNAff".

knnAff.k is the number of the nearest neighbors taken into account; should be at least 1 and up to the number of points in the training sample when knnAff.methodAggregation = "sequent", and up to the total number of points in the training sample when knnAff.methodAggregation = "majority". Set to -1 (the default) to be determined by the leave-one-out cross-validation. knnAff.k is used in method "kNNAff".

knnAff.range is the upper bound on the range over which the leave-one-out cross-validation is performed (the lower bound is 1); should be >1 and smaller than the total number of points in the two smallest classes when knnAff.methodAggregation = "majority", and >1 and smaller than the total number of points in the training sample when knnAff.methodAggregation = "sequent". Set to -1 to be calculated automatically accounting for number of points and dimension. knnAff.range is used in method "kNNAff".

mah.estimate is a character string specifying which estimates to use when calculating the Mahalanobis depth; can be "moment" or "MCD", determining whether traditional moment or Minimum Covariance Determinant (MCD) (see covMcd) estimates for mean and covariance are used. mah.estimate is used in method "depth.Mahalanobis".

mcd.alpha is the value of the argument alpha for the function covMcd; is used in method "depth.Mahalanobis" when mah.estimate = "MCD".

Value

Trained DD\alpha-classifier containing following - rather informative - fields:

References

Dyckerhoff, R., Koshevoy, G., and Mosler, K. (1996). Zonoid data depth: theory and computation. In: Prat A. (ed), COMPSTAT 1996. Proceedings in computational statistics, Physica-Verlag (Heidelberg), 235–240.

Lange, T., Mosler, K., and Mozharovskyi, P. (2014). Fast nonparametric classification based on data depth. Statistical Papers 55 49–69.

Li, J., Cuesta-Albertos, J.A., and Liu, R.Y. (2012). DD-classifier: Nonparametric classification procedure based on DD-plot. Journal of the American Statistical Association 107 737–753.

Mozharovskyi, P. (2015). Contributions to Depth-based Classification and Computation of the Tukey Depth. Verlag Dr. Kovac (Hamburg).

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

Vasil'ev, V.I. (2003). The reduction principle in problems of revealing regularities I. Cybernetics and Systems Analysis 39 686–694.

See Also

ddalpha.classify for classification using DD-classifier, depth. for calculation of depths, depth.space. for calculation of depth spaces, is.in.convex to check whether a point is not an outsider.

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train 1st DDalpha-classifier (default settings) 
# and get the classification error rate
ddalpha1 <- ddalpha.train(data$train)
classes1 <- ddalpha.classify(ddalpha1, data$test[,propertyVars])
cat("1. Classification error rate (defaults): ", 
    sum(unlist(classes1) != data$test[,classVar])/200, ".\n", sep = "")

# Train 2nd DDalpha-classifier (zonoid depth, maximum Mahalanobis 
# depth classifier with defaults as outsider treatment) 
# and get the classification error rate
ddalpha2 <- ddalpha.train(data$train, depth = "zonoid", 
                          outsider.methods = "depth.Mahalanobis")
classes2 <- ddalpha.classify(ddalpha2, data$test[,propertyVars], 
                               outsider.method = "depth.Mahalanobis")
cat("2. Classification error rate (depth.Mahalanobis): ", 
    sum(unlist(classes2) != data$test[,classVar])/200, ".\n", sep = "")

# Train 3rd DDalpha-classifier (100 random directions for the Tukey depth, 
# adjusted maximum Mahalanobis depth classifier 
# and equal randomization as outsider treatments) 
# and get the classification error rates
treatments <- list(list(name = "mahd1", method = "depth.Mahalanobis", 
                        mah.estimate = "MCD", mcd.alpha = 0.75, priors = c(1, 1)/2), 
                   list(name = "rand1", method = "RandEqual"))
ddalpha3 <- ddalpha.train(data$train, outsider.settings = treatments, 
                          num.direction = 100)
classes31 <- ddalpha.classify(ddalpha3, data$test[,propertyVars], 
                              outsider.method = "mahd1")
classes32 <- ddalpha.classify(ddalpha3, data$test[,propertyVars], 
                              outsider.method = "rand1")
cat("3. Classification error rate (by treatments):\n")
cat("   Error (mahd1): ", 
    sum(unlist(classes31) != data$test[,classVar])/200, ".\n", sep = "")
cat("   Error (rand1): ", 
    sum(unlist(classes32) != data$test[,classVar])/200, ".\n", sep = "")
    
# Train using some weird formula
ddalpha = ddalpha.train(
    I(mpg >= 19.2) ~ log(disp) + I(disp^2) + disp + I(disp * drat),
    data = mtcars, subset = (carb!=1), 
    depth = "Mahalanobis", separator = "alpha")
print(ddalpha) # make sure that the resulting table is what you wanted
CC = ddalpha.classify(ddalpha, mtcars)
sum((mtcars$mpg>=19.2)!= unlist(CC))/nrow(mtcars) # error rate
    
#Use the pre-calculated DD-plot
data = cbind(rbind(mvrnorm(n = 50, mu = c(0,0), Sigma = diag(2)),
                   mvrnorm(n = 50, mu = c(5,10), Sigma = diag(2)),
                   mvrnorm(n = 50, mu = c(10,0), Sigma = diag(2))),
             rep(c(1,2,3), each = 50))
plot(data[,1:2], col = (data[,3]+1))

ddplot = depth.space.Mahalanobis(data = data[,1:2], cardinalities = c(50,50,50))
ddplot = cbind(ddplot, data[,3])
ddalphaD = ddalpha.train(data = ddplot, depth = "ddplot", separator = "alpha")
c = ddalpha.classify(ddalphaD, ddplot[,1:3])
errors = sum(unlist(c) != data[,3])/nrow(data)
print(paste("Error rate: ",errors))

ddalpha = ddalpha.train(data = data, depth = "Mahalanobis", separator = "alpha")
c = ddalpha.classify(ddalpha, data[,1:2])
errors = sum(unlist(c) != data[,3])/nrow(data)
print(paste("Error rate: ",errors))

Classify using Functional DD-Classifier

Description

Classifies data using the functional DD-classifier.

Usage

ddalphaf.classify(ddalphaf, objectsf, subset, ...)

## S3 method for class 'ddalphaf'
predict(object, objectsf, subset, ...)

Arguments

Value

List containing class labels.

References

Mosler, K. and Mozharovskyi, P. (2017). Fast DD-classification of functional data. Statistical Papers 58 1055–1089.

Mozharovskyi, P. (2015). Contributions to Depth-based Classification and Computation of the Tukey Depth. Verlag Dr. Kovac (Hamburg).

See Also

ddalphaf.train to train the functional DD\alpha-classifier.

Examples

## Not run: 
## load the Growth dataset
dataf = dataf.growth()

learn = c(head(dataf$dataf, 49), tail(dataf$dataf, 34))
labels= c(head(dataf$labels, 49), tail(dataf$labels, 34)) 
test  = tail(head(dataf$dataf, 59), 10)    # elements 50:59. 5 girls, 5 boys

c = ddalphaf.train (learn, labels, classifier.type = "ddalpha")

classified = ddalphaf.classify(c, test)

print(unlist(classified))


## End(Not run)

Test Functional DD-Classifier

Description

Performs a cross-validation procedure over the given data. On each step every numchunks observation is removed from the data, the functional DD-classifier is trained on these data and tested on the removed observations.

Usage

ddalphaf.getErrorRateCV (dataf, labels, numchunks = 10, disc.type = c("LS", "comp"),  ...)

Arguments

Value

See Also

ddalphaf.train to train the functional DD\alpha-classifier, ddalphaf.classify for classification using functional DD\alpha-classifier, ddalphaf.test to test the functional DD-classifier on particular learning and testing data, ddalphaf.getErrorRatePart to perform a benchmark study of the functional DD-classifier on particular data.

Examples

# load the fdata
df = dataf.growth()

stat <- ddalphaf.getErrorRateCV(dataf = df$dataf, labels = df$labels, 
                                numchunks = 5,
                                adc.args = list(instance = "avr", 
                                                numFcn = 2, 
                                                numDer = 2))
                                                
cat("Classification error rate: ", stat$errors, ".\n", sep = "")

    

Test Functional DD-Classifier

Description

Performs a benchmark procedure by partitioning the given data. On each of times steps size observations are removed from the data, the functional DD-classifier is trained on these data and tested on the removed observations.

Usage

ddalphaf.getErrorRatePart(dataf, labels, size = 0.3, times = 10, 
                          disc.type = c("LS", "comp"),  ...)

Arguments

Value

See Also

ddalphaf.train to train the functional DD\alpha-classifier, ddalphaf.classify for classification using functional DD\alpha-classifier, ddalphaf.test to test the functional DD-classifier on particular learning and testing data, ddalphaf.getErrorRateCV to get error rate of the functional DD-classifier on particular data.

Examples

# load the fdata
df = dataf.growth()

stat <- ddalphaf.getErrorRatePart(dataf = df$dataf, labels = df$labels, 
                          size = 0.3, times = 5,
                          adc.args = list(instance = "avr", 
                                         numFcn = 2, 
                                         numDer = 2))

cat("Classification error rate: ", stat$errors, ".\n", sep = "")

Test Functional DD-Classifier

Description

Trains functional DD-classifier on the learning sequence of the data and tests it on the testing sequence.

Usage

ddalphaf.test(learn, learnlabels, test, testlabels, disc.type = c("LS", "comp"), ...)

Arguments

Value

See Also

ddalphaf.train to train the functional DD\alpha-classifier, ddalphaf.classify for classification using functonal DD\alpha-classifier, ddalphaf.getErrorRateCV and ddalphaf.getErrorRatePart to get error rate of the functional DD-classifier on particular data.

Examples

# load the fdata
df = dataf.growth()

samp = c(35:70)

ddalphaf.test(learn = df$dataf[-samp], learnlabels = df$labels[-samp], 
              test =  df$dataf[samp],  testlabels =  df$labels[samp], 
              adc.args = list(instance = "avr", 
                              numFcn = 2, 
                              numDer = 2))

Functional DD-Classifier

Description

Trains the functional DD-classifier

Usage

ddalphaf.train(dataf, labels, subset, 
                adc.args = list(instance = "avr", 
                               numFcn = -1, 
                               numDer = -1), 
                classifier.type = c("ddalpha", "maxdepth", "knnaff", "lda", "qda"), 
                cv.complete = FALSE, 
                maxNumIntervals = min(25, ceiling(length(dataf[[1]]$args)/2)),
                seed = 0,
                ...)

Arguments

Details

The functional DD-classifier is fast nonparametric procedure for classifying functional data. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the DD-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification. The resulting classifier is cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust and efficiently computable.

Value

Trained functional DD-classifier

References

Mosler, K. and Mozharovskyi, P. (2017). Fast DD-classification of functional data. Statistical Papers 58 1055–1089.

Mozharovskyi, P. (2015). Contributions to Depth-based Classification and Computation of the Tukey Depth. Verlag Dr. Kovac (Hamburg).

See Also

ddalphaf.classify for classification using functional DD\alpha-classifier, compclassf.train to train the functional componentwise classifier, dataf.* for functional data sets included in the package.

Examples

## Not run: 

## load the Growth dataset
dataf = dataf.growth()

learn = c(head(dataf$dataf, 49), tail(dataf$dataf, 34))
labels= c(head(dataf$labels, 49), tail(dataf$labels, 34)) 
test  = tail(head(dataf$dataf, 59), 10)    # elements 50:59. 5 girls, 5 boys

#cross-validate over the whole variants up to dimension 3
c1 = ddalphaf.train (learn, labels, classifier.type = "ddalpha", maxNumIntervals = 3)

classified1 = ddalphaf.classify(c1, test)

print(unlist(classified1))
print(c1$adc.args) 

# cross-validate over these two variants
c2 = ddalphaf.train (learn, labels, classifier.type = "ddalpha", 
                     adc.args = list(
                       list(instance = "avr", 
                            numFcn = 1, 
                            numDer = 2),
                       list(instance = "avr", 
                            numFcn = 0, 
                            numDer = 2)))

classified2 = ddalphaf.classify(c2, test)

print(unlist(classified2))
print(c2$adc.args) 


## End(Not run)

Calculate Depth

Description

Calculates the depth of points w.r.t. a multivariate data set.

The detailed descriptions are found in the corresponding topics.

Usage

depth.(x, data, notion, ...)

## beta-skeleton depth
# depth.betaSkeleton(x, data, beta = 2, distance = "Lp", Lp.p = 2, 
#                   mah.estimate = "moment", mah.parMcd = 0.75)

## Tukey depth
# depth.halfspace(x, data, exact, method, num.directions = 1000, seed = 0)

## L2-depth
# depth.L2(x, data, mah.estimate = "moment", mah.parMcd = 0.75)

## Mahalanobis depth
# depth.Mahalanobis(x, data, mah.estimate = "moment", mah.parMcd = 0.75)

## projection depth
# depth.projection(x, data, method = "random", num.directions = 1000)

## simplicial depth
# depth.simplicial(x, data, exact = F, k = 0.05, seed = 0)

## simplicial volume depth
# depth.simplicialVolume(x, data, exact = F, k = 0.05, seed = 0)

## spatial depth
# depth.spatial(x, data)

## zonoid depth
# depth.zonoid(x, data)

## potential
# depth.potential (x, data, pretransform = "1Mom", 
#            kernel = "GKernel", kernel.bandwidth = NULL, mah.parMcd = 0.75)

## convex hull peeling depth
# depth.qhpeeling(x, data)

Arguments

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

See Also

depth.betaSkeleton

depth.halfspace

depth.L2

depth.Mahalanobis

depth.projection

depth.simplicial

depth.simplicialVolume

depth.spatial

depth.zonoid

depth.potential

depth.qhpeeling

depth.graph for building the depth surfaces of the two dimensional data.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.(x, data, notion = "zonoid")
cat("Depths: ", depths, "\n")

Calculate Beta-Skeleton Depth

Description

Calculates the beta-skeleton depth of points w.r.t. a multivariate data set.

Usage

depth.betaSkeleton(x, data, beta = 2, distance = "Lp", Lp.p = 2, 
                   mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

Calculates the beta-skeleton depth, see Yang and Modarres (2017). Its particular case, lens depth, see Liu and Modarres (2011), is obtained when beta = 2, distance = "Lp" and Lp.p = 2 (default settings). For tne example of the lens depth, the depth of an observation x is calculated as the portion of lens containing x, with lens being an intersection of two closed balls centered at two sample's points each having radius equal to the distance between these two points.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Liu, Z. and Modarres, R. (2011). Lens data depth and median. Journal of Nonparametric Statistics 23(4) 1063–1074.

Yang, M. and Modarres, R. (2017). \beta-skeleton depth functions and medians. Commmunications in Statistics - Theory and Methods to appear.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.betaSkeleton(x, data)
cat("Depths:", depths, "\n")

Depth Contours

Description

Builds the data depth contours for 2-dimensional data.

Usage

depth.contours(data, depth, 
              main = "", xlab="", ylab = "", 
              drawplot = T, frequency=100, levels = 10,
              col = "red",
              ...)

Arguments

See Also

depth., depth.contours.ddalpha, depth.graph.

Examples

## Not run: 

par(mfrow = c(2,2))
data(hemophilia)

depth.contours(hemophilia[,1:2], depth = "none", main = "data")

for (depth in c("zonoid", "Mahalanobis", "projection", "spatial")){
  depth.contours(hemophilia[,1:2], depth = depth, main = depth)
}

for (depth in c("halfspace", "simplicial", "simplicialVolume")){
  depth.contours(hemophilia[,1:2], depth = depth, main = depth, exact = T)
}



## End(Not run)

Depth Contours

Description

Builds the data depth contours for multiclass 2-dimensional data using the trained classifier. Also accessible from plot.ddalpha.

Usage

depth.contours.ddalpha(ddalpha,
              main = "", xlab="", ylab = "", 
              drawplot = T, frequency=100, levels = 10, drawsep = T, ...)

Arguments

See Also

depth., depth.contours, depth.graph.

Examples

## Not run: 

par(mfrow = c(2,2))
data(hemophilia)

ddalpha = ddalpha.train(hemophilia, depth = "none")
depth.contours.ddalpha(ddalpha, main = "data")

for (depth in c("zonoid", "Mahalanobis", "projection", "spatial")){
  ddalpha = ddalpha.train(hemophilia, depth = depth)
  depth.contours.ddalpha(ddalpha, main = depth)
}

for (depth in c("halfspace", "simplicial", "simplicialVolume")){
  ddalpha = ddalpha.train(hemophilia, depth = depth, exact = T)
  depth.contours.ddalpha(ddalpha, main = depth)
}



## End(Not run)

Depth Graph

Description

Builds the data depth graphs for 2-dimensional data. The graph is built using persp.

Usage

depth.graph(data, 
  depth_f = c("halfspace", "Mahalanobis", "projection", "simplicial", 
              "simplicialVolume", "spatial", "zonoid", "none"), 
  apoint = NULL, 
  main = depth_f,  
  xlim = c(min(data[, 1]), max(data[, 1])), 
  ylim = c(min(data[, 2]), max(data[, 2])), 
  zlim = c(0, max(z)), 
  xnum = 250, 
  ynum = 250, 
  theta=15, phi=60,
  bold = F,
  ...)

Arguments

See Also

depth.

persp

Examples

## Not run: 

par(mfrow = c(2,3), mar = c(0,0,0,0), mai = c(0,0,0.2,0))
data(hemophilia)
depth.graph(hemophilia, "none", xnum = 100, ynum = 100)
depth.graph(hemophilia, "Mahalanobis", xnum = 100, ynum = 100)
depth.graph(hemophilia, "halfspace", xnum = 100, ynum = 100)
depth.graph(hemophilia, "projection", xnum = 100, ynum = 100)
depth.graph(hemophilia, "zonoid", xnum = 100, ynum = 100)
depth.graph(hemophilia, "spatial", xnum = 100, ynum = 100)


## End(Not run)

Calculate Halfspace Depth

Description

Calculates the exact or random Tukey (=halfspace, location) depth (Tukey, 1975) of points w.r.t. a multivariate data set.

Usage

depth.halfspace(x, data, exact, method, num.directions = 1000, seed = 0)

Arguments

Details

For exact=F, if method="Sunif.1D", the Tukey depth is computed approximately using the random Tukey depth method proposed by Cuesta-Albertos and Nieto-Reyes (2008). Here the depth is determined as the minimum univariate Tukey depth of the - on lines in several directions - projected data. The directions are distributed uniformly on the (d-1)-sphere; the same direction set is used for all points.

For exact=T, the Tukey depth is computed exactly as the minimum of the sum of the depths in two orthogonal complementary affine subspaces, which dimensions add to d: one of the subspaces (combinatorial) is the k-dimensional hyperplane through (a point from) x and k points from data, another one is its orthogonal complement (see Dyckerhoff and Mozharovskyi, 2016 for the detailed description of the algorithmic framework). The algorithm then minimizes the depth over all combinations of k points, in which the depth in the orthogonal complements is computed using an exact algorithm. In this case, parameter method specifies the dimensionality k of the combinatorial space. The implemented (reasonable) algorithms (and corresponding names) are: k=1 (or method="recursive"), k=d-2 (or method="plane"), and k=d-1 (or method="line").

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Cuesta-Albertos, J.A. and Nieto-Reyes, A. (2008). The random Tukey depth. Computational Statistics and Data Analysis 52 4979–4988.

Dyckerhoff, R. and Mozharovskyi, P. (2016). Exact computation of the halfspace depth. Computational Statistics and Data Analysis 98 19–30.

Rousseeuw, P.J. and Ruts, I. (1996). Algorithm AS 307: Bivariate location depth. Journal of the Royal Statistical Society. Seriec C (Applied Statistics) 45 516–526.

Tukey, J.W. (1974). Mathematics and the picturing of data. In: Proceeding of the International Congress of Mathematicians, Vancouver, 523–531.

See Also

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 3-dimensional normal distribution
data <- mvrnorm(200, rep(0, 3), 
                matrix(c(1, 0, 0,
                         0, 2, 0, 
                         0, 0, 1),
                nrow = 3))
x <- mvrnorm(10, rep(1, 3), 
             matrix(c(1, 0, 0,
                      0, 1, 0, 
                      0, 0, 1),
             nrow = 3))
              
# default - random Tukey depth
depths <- depth.halfspace(x, data)
cat("Depths: ", depths, "\n")

# default exact method - "recursive"
depths <- depth.halfspace(x, data, exact = TRUE)
cat("Depths: ", depths, "\n")

# method "line"
depths <- depth.halfspace(x, data, method = "line")
cat("Depths: ", depths, "\n")

Calculate L2-Depth

Description

Calculates the L2-depth of points w.r.t. a multivariate data set.

Usage

depth.L2(x, data, mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

Calculates L2-depth (Mosler, 2013). L2-depth is based on the oultyingness distance calculated as the average L2-distance from (a row of) x to each point in data.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Mosler, K. (2013). Depth statistics. In: Becker, C., Fried, R. and Kuhnt, S. (eds), Robustness and Complex Data Structures: Festschrift in Honour of Ursula Gather, Springer-Verlag (Berlin, Heidelberg), 17–34.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.qhpeeling for calculation of convex hull peeling depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.potential for calculation of data potential.

depth.zonoid for calculation of zonoid depth.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.spatial(x, data)
cat("Depths:", depths, "\n")

Calculate Mahalanobis Depth

Description

Calculates the Mahalanobis depth of points w.r.t. a multivariate data set.

Usage

depth.Mahalanobis(x, data, mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

Calculates Mahalanobis depth. Mahalanobis depth is based on an outlyingness measure (Zuo & Serfling, 2000), viz. the Mahalanobis distance between the given point and the center of the data (Mahalanobis, 1936).

Moment estimates may be used i.e. traditional mean and covariance matrix, the corresponding depth may be sensitive to outliers. A more robust depth is obtained with minimum volume ellipsoid (MVE) or minimum covariance determinant (MCD) estimators, see Rousseeuw & Leroy (1987) and Lopuhaa & Rousseeuw (1991).

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Mahalanobis, P. (1936). On the generalized distance in statistics. Proceedings of the National Academy India 12 49–55.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279–294.

Lopuhaa, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. The Annals of Statistics 19 229–248.

Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. John Wiley & Sons (New York).

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461–482.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.Mahalanobis(x, data)
cat("Depths moment: ", depths, "\n")
depths <- depth.Mahalanobis(x, data, mah.estimate = "MCD", mah.parMcd = 0.75)
cat("Depths MCD: ", depths, "\n")

Calculate Potential of the Data

Description

Calculate the potential of the points w.r.t. a multivariate data set. The potential is the kernel-estimated density multiplied by the prior probability of a class. Different from the data depths, a density estimate measures at a given point how much mass is located around it.

Usage

depth.potential (x, data, pretransform = "1Mom", 
                kernel = "GKernel", kernel.bandwidth = NULL, mah.parMcd = 0.75)

Arguments

Details

The potential is the kernel-estimated density multiplied by the prior probability of a class. The kernel bandwidth matrix is decomposed into two parts, one of which describes the form of the data, and the other the width of the kernel. Then the first part is used to transform the data using the moments, while the second is employed as a parameter of the kernel and tuned to achieve the best separation. For details see Pokotylo and Mosler (2015).

Value

Numerical vector of potentials, one for each row in x; or one potential value if x is a numerical vector.

References

Aizerman, M.A., Braverman, E.M., and Rozonoer, L.I. (1970). The Method of Potential Functions in the Theory of Machine Learning. Nauka (Moscow).

Pokotylo, O. and Mosler, K. (2015). Classification with the pot-pot plot. Mimeo.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

Examples

# 3-dimensional normal distribution
data <- mvrnorm(200, rep(0, 3), 
                matrix(c(1, 0, 0,
                         0, 2, 0, 
                         0, 0, 1),
                       nrow = 3))
x <- mvrnorm(10, rep(1, 3), 
             matrix(c(1, 0, 0,
                      0, 1, 0, 
                      0, 0, 1),
                    nrow = 3))

# potential with rule of thumb bandwidth
pot <- depth.potential(x, data)
cat("Potentials: ", pot, "\n")

# potential with bandwidth = 0.1
pot <- depth.potential(x, data, kernel.bandwidth = 0.1)
cat("Potentials: ", pot, "\n")

# potential with robust MCD scaling
pot <- depth.potential(x, data, kernel.bandwidth = 0.1, 
                      pretransform = "NMCD", mah.parMcd = 0.6)
cat("Potentials: ", pot, "\n")

Calculate Projection Depth

Description

Calculates the projection depth of points w.r.t. a multivariate data set.

Usage

  depth.projection(x, data, method = "random", num.directions = 1000, seed = 0)

Arguments

Details

Calculates projection depth. Projection depth, similar to Mahalanobis depth, is based on a measure of outlyingness, used by Stahel (1981) and Donoho (1982), and has been first formulated by Liu (1992). The worst case outlyingness is obtained by maximizing an outlyingness measure over all univariate projections. In practice most often median, and median absolute deviation from the median (MAD), are used as they are robust measures.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

Author(s)

R-codes for the "linearize" method were written by Subhajit Dutta.

References

Donoho, D.L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper. Department of Statistics, Harvard University.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279–294.

Liu, X. and Zuo, Y. (2014). Computing projection depth and its associated estimators. Statistics and Computing 24 51–63.

Stahel, W.A. (1981). Robust estimation: infinitesimal optimality and covariance matrix estimators. Ph.D. thesis (in German). Eidgenossische Technische Hochschule Zurich.

Zuo, Y.J. and Lai, S.Y. (2011). Exact computation of bivariate projection depth and the Stahel-Donoho estimator. Computational Statistics and Data Analysis 55 1173–1179.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

  # 5-dimensional normal distribution
  data <- mvrnorm(100, rep(0, 5), 
                  matrix(c(1, 0, 0, 0, 0, 
                           0, 2, 0, 0, 0, 
                           0, 0, 3, 0, 0, 
                           0, 0, 0, 2, 0, 
                           0, 0, 0, 0, 1),
                         nrow = 5))
  x <- mvrnorm(10, rep(1, 5), 
               matrix(c(1, 0, 0, 0, 0, 
                        0, 1, 0, 0, 0, 
                        0, 0, 1, 0, 0, 
                        0, 0, 0, 1, 0, 
                        0, 0, 0, 0, 1),
                      nrow = 5))
  
  depths <- depth.projection(x, data, method = "random", num.directions = 1000)
  cat("Depths random: ", depths, "\n")
  depths <- depth.projection(x, data, method = "linearize")
  cat("Depths linearize: ", depths, "\n")

Calculate Convex Hull Peeling Depth

Description

Calculates the convex hull peeling depth of points w.r.t. a multivariate data set.

Usage

depth.qhpeeling(x, data)

Arguments

Details

Calculates the convex hull peeling depth (Eddy, 1982; see also Cascos, 2009).

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector. Each depth value equals the number of the convex hulls to be peeled from data so that (the corresponding row of) x is not contained in the convex hull of the rest of the data; the depths are normalized by the number of points in data.

References

Eddy, W.F. (1982). Convex hull peeling. In: Caussinus, H., Ettinger, P. and Tomassone, R. (eds), COMPSTAT 1982. Proceedings in computational statistics, Physica-Verlag (Vienna), 42–47.

Cascos, I. (2009). Data depth: multivariate statistics and geometry. In: Kendall, W.S. and Molchanov, I. (eds) New Perspectives in Stochastic Geometry, Clarendon/Oxford University Press (Oxford).

See Also

depth.halfspace for calculation of the Tukey depth.

depth.L2 for calculation of L2-depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.potential for calculation of data potential.

depth.zonoid for calculation of zonoid depth.

Examples

# Mixture of 3-variate normal distributions
data <- mvrnorm(25, rep(0, 3), diag(3))
x <- rbind(mvrnorm(10, rep(1, 3), diag(3)), data)
depths <- depth.qhpeeling(x, data)
cat("Depths:", depths, "\n")

Fast Depth Computation for Univariate and Bivariate Random Samples

Description

Faster implementation of the halfspace and the simplicial depth. Computes the depth of a whole random sample of a univariate or a bivariate data in one run.

Usage

depth.sample(A, B)

Arguments

Details

The function returns vectors of sample halfspace and simplicial depth values.

Value

Vector of length m of depth halfspace depth values is returned.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

depth.halfspace

depth.simplicial

Examples

n = 100
m = 150
A = matrix(rnorm(2*n),ncol=2)
B = matrix(rnorm(2*m),ncol=2)
depth.sample(A,B)
system.time(D1<-depth.halfspace(A,B))
system.time(D2<-depth.sample(A,B))
max(D1-D2$Half)

A = rnorm(100)
B = rnorm(150)
depth.sample(A,B)
# depth.halfspace(matrix(A,ncol=1),matrix(B,ncol=1))

Calculate Simplicial Depth

Description

Calculates the simplicial depth of points w.r.t. a multivariate data set.

Usage

depth.simplicial(x, data, exact = F, k = 0.05, seed = 0)

Arguments

Details

Calculates simplicial depth. Simplicial depth is counted as a probability that a point lies in a simplex, built on d+1 data points.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association 91 862–872.

Liu, R. Y. (1990). On a notion of data depth based on random simplices. The Annals of Statistics 18 405–414.

Rousseeuw, P.J. and Ruts, I. (1996). Algorithm AS 307: Bivariate location depth. Journal of the Royal Statistical Society. Seriec C (Applied Statistics) 45 516–526.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 3-dimensional normal distribution
data <- mvrnorm(20, rep(0, 3), 
                matrix(c(1, 0, 0,
                         0, 2, 0,
                         0, 0, 1),
                       nrow = 3))
x <- mvrnorm(10, rep(1, 3), 
             matrix(c(1, 0, 0,
                      0, 1, 0,
                      0, 0, 1),
                    nrow = 3))

#exact
depths <- depth.simplicial(x, data, exact = TRUE)
cat("Depths: ", depths, "\n")

#approximative
depths <- depth.simplicial(x, data, exact = FALSE, k = 0.2)
cat("Depths: ", depths, "\n")

Calculate Simplicial Volume Depth

Description

Calculates the simpicial volume depth of points w.r.t. a multivariate data set.

Usage

depth.simplicialVolume(x, data, exact = F, k = 0.05, mah.estimate = "moment", 
                       mah.parMcd = 0.75, seed = 0)

Arguments

Details

Calculates Oja depth (also: Simplicial volume depth). At first the Oja outlyingness function O(x,data) is calculated as the average of the volumes of simplices built on d data points and the measurement point x (Oja, 1983).

Zuo and Serfling (2000) proposed Oja depth based on the Oja outlyingness function as 1/(1 + O(x,data)/S), where S is a square root of the determinant of cov(data), which makes the depth function affine-invariant.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics & Probability Letters 1 327–332.

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461–482.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 3-dimensional normal distribution
data <- mvrnorm(20, rep(0, 3), 
                matrix(c(1, 0, 0,
                         0, 2, 0,
                         0, 0, 1),
                       nrow = 3))
x <- mvrnorm(10, rep(1, 3), 
             matrix(c(1, 0, 0,
                      0, 1, 0,
                      0, 0, 1),
                    nrow = 3))

#exact
depths <- depth.simplicialVolume(x, data, exact = TRUE)
cat("Depths: ", depths, "\n")

#approximative
depths <- depth.simplicialVolume(x, data, exact = FALSE, k = 0.2)
cat("Depths: ", depths, "\n")

Calculate Depth Space using the Given Depth

Description

Calculates the representation of the training classes in depth space.

The detailed descriptions are found in the corresponding topics.

Usage

depth.space.(data, cardinalities, notion, ...)

## Mahalanobis depth
# depth.space.Mahalanobis(data, cardinalities, mah.estimate = "moment", mah.parMcd = 0.75)

## projection depth
# depth.space.projection(data, cardinalities, method = "random", num.directions = 1000)

## Tukey depth
# depth.space.halfspace(data, cardinalities, exact, alg, num.directions = 1000)

## spatial depth
# depth.space.spatial(data, cardinalities)

## zonoid depth
# depth.space.zonoid(data, cardinalities)

# Potential
# depth.space.potential(data, cardinalities, pretransform = "NMom", 
#            kernel = "GKernel", kernel.bandwidth = NULL, mah.parMcd = 0.75)

Arguments

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

See Also

depth.space.Mahalanobis

depth.space.projection

depth.space.halfspace

depth.space.spatial

depth.space.zonoid

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using zonoid depth
depth.space.(data, c(10, 10), notion = "zonoid")

Calculate Depth Space using Halfspace Depth

Description

Calculates the representation of the training classes in depth space using the halfspace depth.

Usage

depth.space.halfspace(data, cardinalities, exact, method, num.directions = 1000, seed = 0)

Arguments

Details

The depth representation is calculated in the same way as in depth.halfspace, see References below for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Cuesta-Albertos, J.A. and Nieto-Reyes, A. (2008). The random Tukey depth. Computational Statistics and Data Analysis 52 4979–4988.

Dyckerhoff, R. and Mozharovskyi, P. (2016). Exact computation of the halfspace depth. Computational Statistics and Data Analysis 98 19–30.

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

Rousseeuw, P.J. and Ruts, I. (1996). Algorithm AS 307: Bivariate location depth. Journal of the Royal Statistical Society. Series C (Applied Statistics) 45 516–526.

Tukey, J.W. (1974). Mathematics and the picturing of data. In: Proceeding of the International Congress of Mathematicians, Vancouver, 523–531.

See Also

ddalpha.train and ddalpha.classify for application, depth.halfspace for calculation of the Tukey depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(1,1), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
plot(data, col = c(rep(1,10), rep(2,10)))
# Get depth space using the random Tukey depth
dhA = depth.space.halfspace(data, c(10, 10))
(dhA)

# Get depth space using default exact method - "recursive"
dhE = depth.space.halfspace(data, c(10, 10), exact = TRUE)
(dhE)

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.halfspace(data[,1:2], cardinalities)

Calculate Depth Space using Mahalanobis Depth

Description

Calculates the representation of the training classes in depth space using Mahalanobis depth.

Usage

depth.space.Mahalanobis(data, cardinalities, mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

The depth representation is calculated in the same way as in depth.Mahalanobis, see 'References' for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Mahalanobis, P. (1936). On the generalized distance in statistics. Proceedings of the National Academy India 12 49–55.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279–294.

Lopuhaa, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. The Annals of Statistics 19 229–248.

Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. John Wiley & Sons (New York).

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461–482.

See Also

ddalpha.train and ddalpha.classify for application, depth.Mahalanobis for calculation of Mahalanobis depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using Mahalanobis depth
depth.space.Mahalanobis(data, c(10, 10))
depth.space.Mahalanobis(data, c(10, 10), mah.estimate = "MCD", mah.parMcd = 0.75)

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.Mahalanobis(data[,1:2], cardinalities)

Calculate Potential Space

Description

Calculates the representation of the training classes in potential space.

Usage

depth.space.potential(data, cardinalities, pretransform = "NMom", 
            kernel = "GKernel", kernel.bandwidth = NULL, mah.parMcd = 0.75)

Arguments

Details

The potential representation is calculated in the same way as in depth.potential, see References below for more information and details.

Value

Matrix of objects, each object (row) is represented via its potentials (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Aizerman, M.A., Braverman, E.M., and Rozonoer, L.I. (1970). The Method of Potential Functions in the Theory of Machine Learning. Nauka (Moscow).

Pokotylo, O. and Mosler, K. (2015). Classification with the pot-pot plot. Mimeo.

See Also

ddalpha.train and ddalpha.classify for application, depth.potential for calculation of the potential.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(50, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(50, c(1,1), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
plot(data, col = c(rep(1,50), rep(2,50)))
# potential with rule of thumb bandwidth
ds = depth.space.potential(data, c(50, 50))
# draw.ddplot(depth.space = ds, cardinalities = c(50, 50))

# potential with bandwidth = 0.5 and joint scaling
ds = depth.space.potential(data, c(50, 50), kernel.bandwidth = 0.5,
                           pretransform = "1Mom")
# draw.ddplot(depth.space = ds, cardinalities = c(50, 50))

# potential with bandwidth = 0.5 and separate scaling
ds = depth.space.potential(data, c(50, 50), kernel.bandwidth = 0.5, 
                           pretransform = "NahMom") # or without pretransform
# draw.ddplot(depth.space = ds, cardinalities = c(50, 50))

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
ds = depth.space.potential(data[,1:2], cardinalities)
# draw.ddplot(depth.space = ds, cardinalities = cardinalities)

Calculate Depth Space using Projection Depth

Description

Calculates the representation of the training classes in depth space using projection depth.

Usage

depth.space.projection(data, cardinalities, 
                       method = "random", num.directions = 1000, seed = 0)

Arguments

Details

The depth representation is calculated in the same way as in depth.projection, see 'References' for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Donoho, D.L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper. Department of Statistics, Harvard University.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279–294.

Liu, X. and Zuo, Y. (2014). Computing projection depth and its associated estimators. Statistics and Computing 24 51–63.

Stahel, W.A. (1981). Robust estimation: infinitesimal optimality and covariance matrix estimators. Ph.D. thesis (in German). Eidgenossische Technische Hochschule Zurich.

Zuo, Y.J. and Lai, S.Y. (2011). Exact computation of bivariate projection depth and the Stahel-Donoho estimator. Computational Statistics and Data Analysis 55 1173–1179.

See Also

ddalpha.train and ddalpha.classify for application, depth.projection for calculation of projection depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using projection depth
depth.space.projection(data, c(10, 10), method = "random", num.directions = 1000)
depth.space.projection(data, c(10, 10), method = "linearize")

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.projection(data[,1:2], cardinalities)

Calculate Depth Space using Simplicial Depth

Description

Calculates the representation of the training classes in depth space using simplicial depth.

Usage

depth.space.simplicial(data, cardinalities, exact = F, k = 0.05, seed = 0)

Arguments

Details

The depth representation is calculated in the same way as in depth.simplicial, see 'References' for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association 91 862–872.

Liu, R. Y. (1990). On a notion of data depth based on random simplices. The Annals of Statistics 18 405–414.

Rousseeuw, P.J. and Ruts, I. (1996). Algorithm AS 307: Bivariate location depth. Journal of the Royal Statistical Society. Seriec C (Applied Statistics) 45 516–526.

See Also

ddalpha.train and ddalpha.classify for application, depth.simplicial for calculation of simplicial depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(1,1), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using simplicial depth
depth.space.simplicial(data, c(10, 10))

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.simplicial(data[,1:2], cardinalities)

Calculate Depth Space using Simplicial Volume Depth

Description

Calculates the representation of the training classes in depth space using simplicial volume depth.

Usage

depth.space.simplicialVolume(data, cardinalities, exact = F, k = 0.05, 
                             mah.estimate = "moment", mah.parMcd = 0.75, seed = 0)

Arguments

Details

The depth representation is calculated in the same way as in depth.simplicialVolume, see References below for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics & Probability Letters 1 327–332.

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461–482.

See Also

ddalpha.train and ddalpha.classify for application, depth.simplicialVolume for calculation of simplicial depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using Oja depth
depth.space.simplicialVolume(data, c(10, 10))

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.simplicialVolume(data[,1:2], cardinalities)

Calculate Depth Space using Spatial Depth

Description

Calculates the representation of the training classes in depth space using spatial depth.

Usage

depth.space.spatial(data, cardinalities, mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

The depth representation is calculated in the same way as in depth.spatial, see 'References' for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the Americal Statistical Association 91 862–872.

Koltchinskii, V.I. (1997). M-estimation, convexity and quantiles. The Annals of Statistics 25 435–477.

Serfling, R. (2006). Depth functions in nonparametric multivariate inference. In: Liu, R., Serfling, R., Souvaine, D. (eds.), Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, American Mathematical Society, 1–16.

Vardi, Y. and Zhang, C.H. (2000). The multivariate L1-median and associated data depth. Proceedings of the National Academy of Sciences, U.S.A. 97 1423–1426.

See Also

ddalpha.train and ddalpha.classify for application, depth.spatial for calculation of spatial depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using spatial depth
depth.space.spatial(data, c(10, 10))

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.spatial(data[,1:2], cardinalities)

Calculate Depth Space using Zonoid Depth

Description

Calculates the representation of the training classes in depth space using zonoid depth.

Usage

depth.space.zonoid(data, cardinalities, seed = 0)

Arguments

Details

The depth representation is calculated in the same way as in depth.zonoid, see 'References' for more information and details.

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

References

Dyckerhoff, R., Koshevoy, G., and Mosler, K. (1996). Zonoid data depth: theory and computation. In: Prat A. (ed), COMPSTAT 1996. Proceedings in computational statistics, Physica-Verlag (Heidelberg), 235–240.

Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate distributions Annals of Statistics 25 1998–2017.

Mosler, K. (2002). Multivariate dispersion, central regions and depth: the lift zonoid approach Springer (New York).

See Also

ddalpha.train and ddalpha.classify for application, depth.zonoid for calculation of zonoid depth.

Examples

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using zonoid depth
depth.space.zonoid(data, c(10, 10))

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.zonoid(data[,1:2], cardinalities)

Calculate Spatial Depth

Description

Calculates the spatial depth of points w.r.t. a multivariate data set.

Usage

depth.spatial(x, data, mah.estimate = "moment", mah.parMcd = 0.75)

Arguments

Details

Calculates spatial depth. Spatial depth (also L1-depth) is a distance-based depth exploiting the idea of spatial quantiles of Chaudhuri (1996) and Koltchinskii (1997), formulated by Vardi & Zhang (2000) and Serfling (2002).

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the Americal Statistical Association 91 862–872.

Koltchinskii, V.I. (1997). M-estimation, convexity and quantiles. The Annals of Statistics 25 435–477.

Serfling, R. (2006). Depth functions in nonparametric multivariate inference. In: Liu, R., Serfling, R., Souvaine, D. (eds.), Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, American Mathematical Society, 1–16.

Vardi, Y. and Zhang, C.H. (2000). The multivariate L1-median and associated data depth. Proceedings of the National Academy of Sciences, U.S.A. 97 1423–1426.

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.spatial(x, data)
cat("Depths: ", depths, "\n")

Calculate Zonoid Depth

Description

Calculates the zonoid depth of points w.r.t. a multivariate data set.

Usage

depth.zonoid(x, data, seed = 0)

Arguments

Details

Calculates zonoid depth (Koshevoy and Mosler, 1997; Mosler, 2002) exactly based on the algorithm of Dyckerhoff, Koshevoy and Mosler (1996), implemented in C++ (and provided) by Rainer Dyckerhoff.

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

References

Dyckerhoff, R., Koshevoy, G., and Mosler, K. (1996). Zonoid data depth: theory and computation. In: Prat A. (ed), COMPSTAT 1996. Proceedings in computational statistics, Physica-Verlag (Heidelberg), 235–240.

Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate distributions Annals of Statistics 25 1998–2017.

Mosler, K. (2002). Multivariate dispersion, central regions and depth: the lift zonoid approach Springer (New York).

See Also

depth.halfspace for calculation of the Tukey depth.

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.potential for calculation of data potential.

Examples

# 5-dimensional normal distribution
data <- mvrnorm(1000, rep(0, 5), 
                matrix(c(1, 0, 0, 0, 0, 
                         0, 2, 0, 0, 0, 
                         0, 0, 3, 0, 0, 
                         0, 0, 0, 2, 0, 
                         0, 0, 0, 0, 1),
                nrow = 5))
x <- mvrnorm(10, rep(1, 5), 
             matrix(c(1, 0, 0, 0, 0, 
                      0, 1, 0, 0, 0, 
                      0, 0, 1, 0, 0, 
                      0, 0, 0, 1, 0, 
                      0, 0, 0, 0, 1),
             nrow = 5))
                
depths <- depth.zonoid(x, data)
cat("Depths: ", depths, "\n")

Calculate Functional Depth

Description

Calculates the depth of functions w.r.t. a functional data set.

The detailed descriptions are found in the corresponding topics.

Usage

depthf.(datafA, datafB, notion, ...)

## Adjusted band depth
# depthf.ABD(datafA, datafB, range = NULL, d = 101, norm = c("C", "L2"), 
# J = 2, K = 1)

## Band depth
# depthf.BD(datafA, datafB, range = NULL, d = 101)

## Univariate integrated and infimal depth
# depthf.fd1(datafA, datafB, range = NULL, d = 101, order = 1, approx = 0)

## Bivariate integrated and infimal depth
# depthf.fd2(datafA, datafB, range = NULL, d = 101)

## h-mode depth
# depthf.hM(datafA, datafB, range = NULL, d = 101, norm = c("C", "L2"),
#  q = 0.2)

## Bivariate h-mode depth
# depthf.hM2(datafA, datafB, range = NULL, d = 101, q = 0.2)

## Half-region depth
# depthf.HR(datafA, datafB, range = NULL, d = 101)

## Univariate random projection depths
# depthf.RP1(datafA, datafB, range = NULL, d = 101, nproj = 50, nproj2 = 5)

# Bivariate random projection depths
# depthf.RP2(datafA, datafB, range = NULL, d = 101, nproj = 51)

Arguments

Value

Numerical vector of depths, one for each function in datafA; or one depth value if datafA is a single function.

See Also

depthf.ABD

depthf.BD

depthf.fd1

depthf.fd2

depthf.hM

depthf.hM2

depthf.HR

depthf.RP1

depthf.RP2

Examples

# real data example
datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
                
depthf.(datafA, datafB, notion = "HR")

dataf2A = derivatives.est(datafA,deriv=c(0,1))
dataf2B = derivatives.est(datafB,deriv=c(0,1))

depthf.(dataf2A, dataf2B, notion = "fd2")

Adjusted Band Depth for Functional Data

Description

The adjusted band depth of functional real-valued data based on either the C (uniform) norm, or on the L^2 norm of functions.

Usage

depthf.ABD(datafA, datafB, range = NULL, d = 101, norm = c("C", "L2"),
  J = 2, K = 1)

Arguments

Details

The function returns the vector of the sample adjusted band depth values. The kernel used in the evaluation is the function K(u) = exp(-u).

Value

A vectors of length m of the adjusted band depths.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Gijbels, I., Nagy, S. (2015). Consistency of non-integrated depths for functional data. Journal of Multivariate Analysis 140, 259–282.

Nagy, S., Gijbels, I. and Hlubinka, D. (2016). Weak convergence of discretely observed functional data with applications. Journal of Multivariate Analysis, 146, 46–62.

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

See Also

depthf.BD

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.ABD(datafA,datafB)
depthf.ABD(datafA,datafB,norm="L2")

Band Depth for Functional Data

Description

The (unadjusted) band depth for functional real-valued data of order J=2.

Usage

depthf.BD(datafA, datafB, range = NULL, d = 101)

Arguments

Details

The function returns the vector of the sample (unadjusted) band depth values.

Value

A vector of length m of the band depth values.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Lopez-Pintado, S. and Romo, J. (2009), On the concept of depth for functional data, J. Amer. Statist. Assoc. 104 (486), 718 - 734.

See Also

depthf.ABD, depthf.fd1

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.BD(datafA,datafB)

Univariate Integrated and Infimal Depth for Functional Data

Description

Usual, and order extended integrated and infimal depths for real-valued functional data based on the halfspace and simplicial depth.

Usage

depthf.fd1(datafA, datafB, range = NULL, d = 101, order = 1, approx = 0)

Arguments

Details

The function returns vectors of sample integrated and infimal depth values.

Value

Four vectors of length m of depth values are returned:

• Simpl_FD the integrated depth based on the simplicial depth,

• Half_FD the integrated depth based on the halfspace depth,

• Simpl_ID the infimal depth based on the simplicial depth,

• Half_ID the infimal depth based on the halfspace depth.

In addition, two vectors of length m of the relative area of smallest depth values is returned:

• Simpl_IA the proportions of points at which the depth Simpl_ID was attained,

• Half_IA the proportions of points at which the depth Half_ID was attained.

The values Simpl_IA and Half_IA are always in the interval [0,1]. They introduce ranking also among functions having the same infimal depth value - if two functions have the same infimal depth, the one with larger infimal area IA is said to be less central. For order=2 and m=1, two additional matrices of pointwise depths are also returned:

• PSD the matrix of size d*d containing the computed pointwise bivariate simplicial depths used for the computation of Simpl_FD and Simpl_ID,

• PHD the matrix of size d*d containing the computed pointwise bivariate halfspace depths used for the computation of Half_FD and Half_ID.

For order=3, only Half_FD and Half_ID are provided.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Nagy, S., Gijbels, I. and Hlubinka, D. (2016). Weak convergence of discretely observed functional data with applications. Journal of Multivariate Analysis, 146, 46–62.

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

See Also

depthf.fd2, infimalRank

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.fd1(datafA,datafB)
depthf.fd1(datafA,datafB,order=2)
depthf.fd1(datafA,datafB,order=3,approx=51)

Bivariate Integrated and Infimal Depth for Functional Data

Description

Integrated and infimal depths of functional bivariate data (that is, data of the form X:[a,b] \to R^2, or X:[a,b] \to R and the derivative of X) based on the bivariate halfspace and simplicial depths.

Usage

depthf.fd2(datafA, datafB, range = NULL, d = 101)

Arguments

Details

The function returns the vectors of sample integrated and infimal depth values.

Value

Four vectors of length m are returned:

• Simpl_FD the integrated depth based on the bivariate simplicial depth,

• Half_FD the integrated depth based on the bivariate halfspace depth,

• Simpl_ID the infimal depth based on the bivariate simplicial depth,

• Half_ID the infimal depth based on the bivariate halfspace depth.

In addition, two vectors of length m of the relative area of smallest depth values is returned:

• Simpl_IA the proportions of points at which the depth Simpl_ID was attained,

• Half_IA the proportions of points at which the depth Half_ID was attained.

The values Simpl_IA and Half_IA are always in the interval [0,1]. They introduce ranking also among functions having the same infimal depth value - if two functions have the same infimal depth, the one with larger infimal area IA is said to be less central.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Hlubinka, D., Gijbels, I., Omelka, M. and Nagy, S. (2015). Integrated data depth for smooth functions and its application in supervised classification. Computational Statistics, 30 (4), 1011–1031.

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

See Also

depthf.fd1, infimalRank

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]

dataf2A = derivatives.est(datafA,deriv=c(0,1))
dataf2B = derivatives.est(datafB,deriv=c(0,1))
depthf.fd2(dataf2A,dataf2B)

h-Mode Depth for Functional Data

Description

The h-mode depth of functional real-valued data.

Usage

depthf.hM(datafA, datafB, range = NULL, d = 101, norm = c("C", "L2"),
  q = 0.2)

Arguments

Details

The function returns the vectors of the sample h-mode depth values. The kernel used in the evaluation is the standard Gaussian kernel, the bandwidth value is chosen as a quantile of the non-zero distances between the random sample curves.

Value

A vector of length m of the h-mode depth values.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 (3), 481–496.

Nagy, S., Gijbels, I. and Hlubinka, D. (2016). Weak convergence of discretely observed functional data with applications. Journal of Multivariate Analysis, 146, 46–62.

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.hM(datafA,datafB)
depthf.hM(datafA,datafB,norm="L2")

Bivariate h-Mode Depth for Functional Data Based on the L^2 Metric

Description

The h-mode depth of functional bivariate data (that is, data of the form X:[a,b] \to R^2, or X:[a,b] \to R and the derivative of X) based on the L^2 metric of functions.

Usage

depthf.hM2(datafA, datafB, range = NULL, d = 101, q = 0.2)

Arguments

Details

The function returns the vectors of sample h-mode depth values. The kernel used in the evaluation is the standard Gaussian kernel, the bandwidth value is chosen as a quantile of the non-zero distances between the random sample curves.

Value

Three vectors of length m of h-mode depth values are returned:

• hM the unscaled h-mode depth,

• hM_norm the h-mode depth hM linearly transformed so that its range is [0,1],

• hM_norm2 the h-mode depth FD linearly transformed by a transformation such that the range of the h-mode depth of B with respect to B is [0,1]. This depth may give negative values.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 (3), 481–496.

See Also

depthf.hM

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]

datafA2 = derivatives.est(datafA,deriv=c(0,1))
datafB2 = derivatives.est(datafB,deriv=c(0,1))

depthf.hM2(datafA2,datafB2)

depthf.hM2(datafA2,datafB2)$hM
# depthf.hM2(cbind(A2[,,1],A2[,,2]),cbind(B2[,,1],B2[,,2]))$hM
# the two expressions above should give the same result

Half-Region Depth for Functional Data

Description

The half-region depth for functional real-valued data.

Usage

depthf.HR(datafA, datafB, range = NULL, d = 101)

Arguments

Details

The function returns the vector of the sample half-region depth values.

Value

A vector of length m of the half-region depth values.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Lopez-Pintado, S. and Romo, J. (2011). A half-region depth for functional data. Computational Statistics & Data Analysis 55 (4), 1679–1695.

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
depthf.HR(datafA,datafB)

Univariate Random Projection Depths for Functional Data

Description

Random projection depth and random functional depth for functional data.

Usage

depthf.RP1(datafA, datafB, range = NULL, d = 101, nproj = 50, nproj2 = 5)

Arguments

Details

The function returns the vectors of sample random projection, and random functional depth values. The random projection depth described in Cuevas et al. (2007) is based on the average univariate depth of one-dimensional projections of functional data. The projections are taken randomly as a sample of standard normal d-dimensional random variables, where d stands for the dimensionality of the discretized functional data.

The random functional depth (also called random Tukey depth, or random halfspace depth) is described in Cuesta-Albertos and Nieto-Reyes (2008). The functional data are projected into the real line in random directions as for the random projection depths. Afterwards, an approximation of the halfspace (Tukey) depth based on this limited number of univariate projections is assessed.

Value

Three vectors of depth values of length m are returned:

• Simpl_FD the random projection depth based on the univariate simplicial depth,

• Half_FD the random projection depth based on the univariate halfspace depth,

• RHalf_FD the random halfspace depth.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions, Computational Statistics 22 (3), 481–496.

Cuesta-Albertos, J.A. and Nieto-Reyes, A. (2008). The random Tukey depth. Computational Statistics & Data Analysis 52 (11), 4979–4988.

See Also

depthf.RP2

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]

depthf.RP1(datafA,datafB)

Bivariate Random Projection Depths for Functional Data

Description

Double random projection depths of functional bivariate data (that is, data of the form X:[a,b] \to R^2, or X:[a,b] \to R and the derivative of X).

Usage

depthf.RP2(datafA, datafB, range = NULL, d = 101, nproj = 51)

Arguments

Details

The function returns the vectors of sample double random projection depth values. The double random projection depths are described in Cuevas et al. (2007). They are of two types: RP2 type, and RPD type. Both types of depths are based on bivariate projections of the bivariate functional data. These projections are taken randomly as a sample of standard normal d-dimensional random variables, where d stands for the dimensionality of the internally represented discretized functional data. For RP2 type depths, the average bivariate depth of the projected quantities is assessed. For RPD type depths, further univariate projections of these bivariate projected quantities are evaluated, and based on these final univariate quantities, the average univariate depth is computed.

Value

Five vectors of length m are returned:

• Simpl_FD the double random projection depth RP2 based on the bivariate simplicial depth,

• Half_FD the double random projection depth RP2 based on the bivariate halfspace depth,

• hM_FD the double random projection depth RP2 based on the bivariate h-mode depth,

• Simpl_DD the double random projection depth RPD based on the univariate simplicial depth,

• Half_DD the random projection depth RPD based on the univariate halfspace depth,

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22 (3), 481–496.

See Also

depthf.RP1

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]

dataf2A = derivatives.est(datafA,deriv=c(0,1))
dataf2B = derivatives.est(datafB,deriv=c(0,1))
depthf.RP2(dataf2A,dataf2B)

Calculate Simplicial Band Depth

Description

Calculate the simplicial band depth defined by Lopez-Pintado, Sun, Lin, Genton (2014).

Usage

depthf.simplicialBand(objectsf, dataf, modified = TRUE, J = NULL, 
                      range = NULL, d = 101)

Arguments

Value

A vector of depths of each of objectsf w.r.t. dataf.

References

Lopez-Pintado, Sun, Lin, Genton (2014). Simplicial band depth for multivariate data. Advances in Data Analysis and Classification 8(3) 321–338.

See Also

depthf.BD, depthf.ABD.

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]

dataf2A = derivatives.est(datafA,deriv=c(0,1))
dataf2B = derivatives.est(datafB,deriv=c(0,1))
depthf.simplicialBand(dataf2A,dataf2B)

Estimation of the First Two Derivatives for Functional Data

Description

Returns the estimated values of derivatives of functional data.

Usage

derivatives.est(dataf, range = NULL, d = 101, spar = NULL, deriv = c(0,1))

Arguments

Details

If the input dataf is a functional random sample of size m, the function returns a dataf object of nd-dimensional functional data, where in the elements of the vector-valued functional data represent the estimated values of the derivatives of dataf. All derivatives are evaluated at an equi-distant grid of d points in the domain given by range. nd here stands for 1, 2 or 3, depending on how many derivatives of dataf are requested to be computed. For the estimation, functions D1ss and D2ss from the package sfsmisc are utilized.

Value

A multivariate dataf object of the functional values and / or the derivatives of dataf. The dimensionality of the vector-valued functional data is nd. The arguments of the data are all equal to an equi-distant grid of d points in the domain given by range. nd is the demanded number of derivatives at the output, i.e. the length of the vector deriv.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

D1ss in package sfsmisc

D2ss in package sfsmisc

Examples

dataf = dataf.population()$dataf
derivatives.est(dataf,deriv=c(0,1,2))

Depth-Based kNN

Description

The implementation of the affine-invariant depth-based kNN of Paindaveine and Van Bever (2015).

Usage

dknn.classify(objects, data, k, depth = "halfspace", seed = 0)

Arguments

Value

List containing class labels, or character string "Ignored" for the outsiders if "Ignore" was specified as the outsider treating method.

References

Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli 21 62–82.

See Also

dknn.train to train the Dknn-classifier.

dknn.classify.trained to classify with the Dknn-classifier.

ddalpha.train to train the DD\alpha-classifier.

ddalpha.getErrorRateCV and ddalpha.getErrorRatePart to get error rate of the Dknn-classifier on particular data (set separator = "Dknn").

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train the classifier
# and get the classification error rate
cls <- dknn.train(data$train, kMax = 20, depth = "Mahalanobis")
cls$k
classes1 <- dknn.classify.trained(data$test[,propertyVars], cls)
cat("Classification error rate: ", 
    sum(unlist(classes1) != data$test[,classVar])/200)

# Classify the new data based on the old ones in one step
classes2 <- dknn.classify(data$test[,propertyVars], data$train, k = cls$k, depth = "Mahalanobis")
cat("Classification error rate: ", 
    sum(unlist(classes2) != data$test[,classVar])/200)

Depth-Based kNN

Description

The implementation of the affine-invariant depth-based kNN of Paindaveine and Van Bever (2015).

Usage

dknn.classify.trained(objects, dknn)

Arguments

Value

List containing class labels, or character string "Ignored" for the outsiders if "Ignore" was specified as the outsider treating method.

References

Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli 21 62–82.

See Also

dknn.train to train the Dknn-classifier.

dknn.classify to classify with the Dknn-classifier.

ddalpha.train to train the DD\alpha-classifier.

ddalpha.getErrorRateCV and ddalpha.getErrorRatePart to get error rate of the Dknn-classifier on particular data (set separator = "Dknn").

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train the classifier
# and get the classification error rate
cls <- dknn.train(data$train, kMax = 20, depth = "Mahalanobis")
cls$k
classes1 <- dknn.classify.trained(data$test[,propertyVars], cls)
cat("Classification error rate: ", 
    sum(unlist(classes1) != data$test[,classVar])/200)

# Classify the new data based on the old ones in one step
classes2 <- dknn.classify(data$test[,propertyVars], data$train, k = cls$k, depth = "Mahalanobis")
cat("Classification error rate: ", 
    sum(unlist(classes2) != data$test[,classVar])/200)

Depth-Based kNN

Description

The implementation of the affine-invariant depht-based kNN of Paindaveine and Van Bever (2015).

Usage

dknn.train(data, kMax = -1, depth = "halfspace", seed = 0)

Arguments

Value

The returned object contains technical information for classification, including the found optimal value k.

References

Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli 21 62–82.

See Also

dknn.classify and dknn.classify.trained to classify with the Dknn-classifier.

ddalpha.train to train the DD\alpha-classifier.

ddalpha.getErrorRateCV and ddalpha.getErrorRatePart to get error rate of the Dknn-classifier on particular data (set separator = "Dknn").

Examples

# Generate a bivariate normal location-shift classification task
# containing 200 training objects and 200 to test with
class1 <- mvrnorm(200, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(200, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:100)
testIndices <- c(101:200)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 100)), 
                   cbind(class2[trainIndices,], rep(2, 100)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 100)), 
                  cbind(class2[testIndices,], rep(2, 100)))
data <- list(train = trainData, test = testData)

# Train the classifier
# and get the classification error rate
cls <- dknn.train(data$train, kMax = 20, depth = "Mahalanobis")
cls$k
classes1 <- dknn.classify.trained(data$test[,propertyVars], cls)
cat("Classification error rate: ", 
    sum(unlist(classes1) != data$test[,classVar])/200)

# Classify the new data based on the old ones in one step
classes2 <- dknn.classify(data$test[,propertyVars], data$train, k = cls$k, depth = "Mahalanobis")
cat("Classification error rate: ", 
    sum(unlist(classes2) != data$test[,classVar])/200)

Draw DD-Plot

Description

The function draws the DD-plot either of the existing DD\alpha-classifier of the depth space. Also accessible from plot.ddalpha.

Usage

draw.ddplot(ddalpha, depth.space, cardinalities, 
            main = "DD plot", xlab = "C1", ylab = "C2", xlim, ylim,
            classes = c(1, 2), colors = c("red", "blue", "green"), drawsep = T)

Arguments

See Also

ddalpha.train

depth.space.

Examples

  data = getdata("kidney")
  
  #1. using the existing ddalpha classifier
  ddalpha = ddalpha.train(data, depth = "spatial")
  draw.ddplot(ddalpha, main = "DD-plot")
  
  #2. using depth.space.
  # Sort the data w.r.t. classes
  data = rbind(data[data$C == 1,], data[data$C == 2,])
  cardinalities = c(sum(data$C == 1), sum(data$C == 2))
  
  dspace = depth.space.spatial(data[,-6], cardinalities = cardinalities)
  draw.ddplot(depth.space = dspace, cardinalities = cardinalities, 
              main = "DD-plot", xlab = 1, ylab = 2)

Fast Kernel Smoothing

Description

Produces a kernel smoothed version of a function based on the vectors given in the input. Bandwidth is selected using cross-validation.

Usage

FKS(dataf, Tout, kernel = c("uniform", "triangular", "Epanechnikov",
  "biweight", "triweight", "Gaussian"), m = 51, K = 20)

Arguments

Details

A vector of the same length as Tout corresponding to the values of the function produced using kernel smoothing, is provided. Bandwidth is selected using the K-fold cross-validation of randomly shuffled input values.

Value

A dataf object corresponding to Tout of smoothed functional values.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

Examples

d = 10
T = sort(runif(d))
X = T^2+ rnorm(d,sd=.1)
Tout = seq(0,1,length=101)

plot(T,X)
dataf = list(list(args=T,vals=X))
data.sm = FKS(dataf,Tout,kernel="Epan")
lines(data.sm[[1]]$args,data.sm[[1]]$vals,col=2)

datafs = structure(list(dataf=dataf,labels=1:length(dataf)),class="functional")
plot(datafs)
points(T,X)
data.sms = structure(list(dataf=data.sm,labels=1:length(data.sm)),class="functional")
plot(data.sms)

n = 6
dataf = list()
for(i in 1:n) dataf[[i]] = list(args = T<-sort(runif(d)), vals = T^2 + rnorm(d,sd=.1))
data.sm = FKS(dataf,Tout,kernel="triweight")
data.sms = structure(list(dataf=data.sm,labels=1:length(data.sm)),class="functional")
plot(data.sms)

Data for Classification

Description

50 multivariate data sets for binary classification. For more details refer https://wisostat.uni-koeln.de/de/forschung/software-und-daten/data-for-classification/

The getdata function gets the data set from the package, and returns it. The dataset itself does not appear in the global environment and the existing variables with the same name remain unchanged.

Usage

# load the data set
# data(name)

# load the data set by name
# data(list = "name")

# load the data set by name to a variable
# getdata("name")

Arguments

Format

A data frame with n observations on the d variables. The last d+1 column is the class label.

x[,1:d]

numeric values

x[,d+1]

the numeric class label (0 or 1) or (1 or 2)

Details

The package contains data sets used in the joint project of the University of Cologne and the Hochschule Merseburg "Classifying real-world data with the DDalpha-procedure". Comprehensive description of the methodology, and experimental settings and results of the study are presented in the work:

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

For a more complete explanation of the technique and further experiments see: Lange, T., Mosler, K., and Mozharovskyi, P. (2014). Fast nonparametric classification based on data depth. Statistical Papers 55 49–69.

50 binary classification tasks have been obtained from partitioning 33 freely accessible data sets. Multiclass problems were reasonably split into binary classification problems, some of the data set were slightly processed by removing objects or attributes and selecting prevailing classes. Each data set is provided with a (short) description and brief descriptive statistics. The name reflects the origination of the data. A letter after the name is a property filter, letters (also their combinations) in brackets separated by "vs" are the classes opposed. The letters (combinations or words) stand for labels of classes (names of properties) and are intuitive. Each description contains a link to the original data.

The data have been collected as open source data in January 2013. Owners of the package decline any responsibility regarding their correctness or consequences of their usage. If you publish material based on these data, please quote the original source. Special requests regarding citations are found on data set's web page.

Note

List of the datasets:

baby
banknoten
biomed
bloodtransfusion
breast_cancer_wisconsin
bupa
chemdiab_1vs2
chemdiab_1vs3
chemdiab_2vs3
cloud
crabB_MvsF
crabF_BvsO
crabM_BvsO
crabO_MvsF
crab_BvsO
crab_MvsF
cricket_CvsP
diabetes
ecoli_cpvsim
ecoli_cpvspp
ecoli_imvspp
gemsen_MvsF
glass
groessen_MvsF
haberman
heart
hemophilia
indian_liver_patient_1vs2
indian_liver_patient_FvsM
iris_setosavsversicolor
iris_setosavsvirginica
iris_versicolorvsvirginica
irish_ed_MvsF
kidney
pima
plasma_retinol_MvsF
segmentation
socmob_IvsNI
socmob_WvsB
tae
tennis_MvsF
tips_DvsN
tips_MvsF
uscrime_SvsN
vertebral_column
veteran_lung_cancer
vowel_MvsF
wine_1vs2
wine_1vs3
wine_2vs3

Also functional data sets can be loaded:

geneexp
growth
medflies
population
population2010
tecator

References

Lange, T., Mosler, K., and Mozharovskyi, P. (2014). Fast nonparametric classification based on data depth. Statistical Papers 55 49–69.

Mozharovskyi, P., Mosler, K., and Lange, T. (2015). Classifying real-world data with the DD\alpha-procedure. Advances in Data Analysis and Classification 9 287–314.

The general list of sources consists of:

UCI Machine Learning Repository https://archive.ics.uci.edu/ml/
R-packages https://CRAN.R-project.org/
https://www.cmu.edu/dietrich/statistics-datascience/
https://stat.ethz.ch/Teaching/Datasets/
https://www.stats.ox.ac.uk/pub/PRNN/

See Also

utils:data

Examples

# load a dataset using data()
data(hemophilia)
data(list = "hemophilia")
rm(hemophilia)

# load data set using getdata()
hemophilia = "This is some existing object called 'hemophilia'. It remains unchanged"
d = getdata("hemophilia")
head(d)
print(hemophilia)

#get the list of all data sets
names = data(package = "ddalpha")$results[,3]

Adjusted Ranking of Functional Data Based on the Infimal Depth

Description

Returns a vector of adjusted depth-based ranks for infimal depth for functional data.

Usage

infimalRank(ID, IA, ties.method = "max")

Arguments

Details

Infimal depths for functional data tend to give to many functional observations the same value of depth. Using this function, the data whose depth is the same is ranked according to the infimal area indicator. This indicator is provided in functions depthf.fd1 along the value of the infimal depth.

Value

A vector of length n. Low depth values mean high ranks, i.e. potential outlyingness. If some of the infimal depths are identical, the ranking of these functions is made according to the values of the infimal area. There, higher infimal area index means higher rank, i.e. non-centrality.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

Examples

datafA = dataf.population()$dataf[1:20]
datafB = dataf.population()$dataf[21:50]
D = depthf.fd1(datafA,datafB)
infimalRank(D$Half_ID,D$Half_IA) 

ID = c(0,1,0,0,0,1,1)
IA = c(2,3,1,0,2,4,1)
infimalRank(ID,IA)

Check Outsiderness

Description

Checks the belonging to at least one of class convex hulls of the training sample.

Usage

is.in.convex(x, data, cardinalities, seed = 0)

Arguments

Details

Checks are conducted w.r.t. each separate class in data using the simplex algorithm, taken from the C++ implementation of the zonoid depth calculation by Rainer Dyckerhoff.

Value

Matrix of number of objects rows and number of classes columns, containing 1 if an object belongs to the convex hull of the corresponding class, and 0 otherwise.

Author(s)

Implementation of the simplex algorithm is taken from the algorithm for computation of zonoid depth (Dyckerhoff, Koshevoy and Mosler, 1996) that has been implemented in C++ by Rainer Dyckerhoff.

References

Dyckerhoff, R., Koshevoy, G., and Mosler, K. (1996). Zonoid data depth: theory and computation. In: Prat A. (ed), COMPSTAT 1996. Proceedings in computational statistics, Physica-Verlag (Heidelberg), 235–240.

See Also

ddalpha.train and ddalpha.classify for application.

Examples

# Generate a bivariate normal location-shift classification task
# containing 400 training objects and 1000 to test with
class1 <- mvrnorm(700, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(700, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
trainIndices <- c(1:200)
testIndices <- c(201:700)
propertyVars <- c(1:2)
classVar <- 3
trainData <- rbind(cbind(class1[trainIndices,], rep(1, 200)), 
                   cbind(class2[trainIndices,], rep(2, 200)))
testData <- rbind(cbind(class1[testIndices,], rep(1, 500)), 
                  cbind(class2[testIndices,], rep(2, 500)))
data <- list(train = trainData, test = testData)

# Count outsiders
numOutsiders = sum(rowSums(is.in.convex(data$test[,propertyVars], 
                                data$train[,propertyVars], c(200, 200))) == 0)
cat(numOutsiders, "outsiders found in the testing sample.\n")

Fast Computation of the L^2 Metric for Sets of Functional Data

Description

Returns the matrix of L^2 distances between two sets of functional data.

Usage

L2metric(A, B)

Arguments

Details

For two sets of functional data of sizes m and n represented by matrices of their functional values on the common domain {1,...,d}, this function returns the symmetric matrix of size m*n whose entry in the i-th row and j-th column is the approximated L^2 distance of the i-th function from the first set, and the j-th function from the second set. This function is utilized in the computation of the h-mode depth.

Value

A symmetric matrix of the distances of the functions of size m*n.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

depthf.hM

dataf2rawfd

Examples

datapop = dataf2rawfd(dataf.population()$dataf,range=c(1950,2015),d=66)
A = datapop[1:20,]
B = datapop[21:50,]
L2metric(A,B)

Plots for the "ddalpha" Class

Description

depth.contours.ddalpha – builds the data depth contours for multiclass 2-dimensional data using the trained classifier. draw.ddplot – draws the DD-plot of the existing DD\alpha-classifier.

Usage

## S3 method for class 'ddalpha'
plot(x, type = c("ddplot", "depth.contours"), ...)

Arguments

See Also

depth.

depth.contours

depth.graph

Examples

## Not run: 


par(mfrow = c(2,2))
data(hemophilia)

ddalpha = ddalpha.train(hemophilia, depth = "none")
plot(ddalpha, type = "depth.contours", main = "data")
plot(ddalpha, type = "ddplot", main = "data", drawsep = F)

for (depth in c("zonoid", "Mahalanobis", "projection", "spatial")){
  ddalpha = ddalpha.train(hemophilia, depth = depth)
  plot(ddalpha, type = "depth.contours", main = depth, drawsep = T)
  plot(ddalpha, type = "ddplot", main = depth)
}


## End(Not run)

Plots for the "ddalphaf" Class

Description

plot.functional – plots the functional data used by classifier

depth.contours.ddalpha – builds the data depth contours for multiclass 2-dimensional data using the trained classifier. draw.ddplot – draws the DD-plot of the existing DD\alpha-classifier.

Usage

## S3 method for class 'ddalphaf'
plot(x, type = c("functional.data", "ddplot", "depth.contours"), ...)

Arguments

See Also

depth.

depth.contours

depth.graph

Examples

## Not run: 

dataf = dataf.growth()
ddalphaf = ddalphaf.train (dataf$dataf, dataf$labels, 
                            classifier.type = "ddalpha", maxNumIntervals = 2)

# plot the functional data
plot(ddalphaf)

# plot depth contours and separation in the transformed space 
# (possible only if maxNumIntervals = 2)
plot(ddalphaf, type = "depth.contours")

# plot the DD-plot
plot(ddalphaf, type = "ddplot")


## End(Not run)

Plot functions for the Functional Data

Description

Plots the functional data given in the form which is described in the topic dataf.*.

Usage

## S3 method for class 'functional'
plot(x, 
      main = "Functional data", xlab = "args", ylab = "vals", 
      colors = c("red", "blue", "green", "black", "orange", "pink"), ...)
      
## S3 method for class 'functional'
lines(x, 
      colors = c("red", "blue", "green", "black", "orange", "pink"), ...)
      
## S3 method for class 'functional'
points(x, 
      colors = c("red", "blue", "green", "black", "orange", "pink"), ...)

Arguments

See Also

dataf.* for functional data description

Examples

## Not run: 
  ## load the Growth dataset
  dataf = dataf.growth()
  
  labels = unlist(dataf$labels)
  plot(dataf, 
       main = paste("Growth: girls red (", sum(labels == "girl"), "),", 
                    " boys blue (", sum(labels == "boy"), ")", sep=""),
       xlab="Year", ylab="Height, cm",
       colors = c("blue", "red")   # in alphabetical order of class labels   
  )
  
  # plot an observation as a line
  observation = structure(list(dataf = list(dataf$dataf[[1]])), class = "functional")
  lines(observation, colors = "green", lwd = 3)
  
  # plot hight at the age of 14 
  indexAge14 = which(observation$dataf[[1]]$args == 14)
  hightAge14 = observation$dataf[[1]]$vals[indexAge14]
  atAge14 = structure(list(
                      dataf = list(dataf = list(args = 14, vals = hightAge14))
                      ), class = "functional")
  points(atAge14, colors = "yellow", pch = 18)

## End(Not run)

Transform Raw Functional Data to a dataf Object

Description

Constructs a (possibly multivariate) functional data object given by an array of its functional values evaluated at an equi-distant grid of points, and transforms it into a dataf object more suitable for work in the ddalpha package.

Usage

rawfd2dataf(X, range)

Arguments

Value

A (possibly multivariate) dataf object corresponding to the functional data X evaluated at an equi-distant grid of points.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

See Also

dataf2rawfd

depthf.fd1

depthf.fd2

Examples

## transform a matrix into a functional data set
n = 5
d = 21
X = matrix(rnorm(n*d),ncol=d)
rawfd2dataf(X,range=c(0,1))

## transform an array into a multivariate functional data set
k = 3
X = array(rnorm(n*d*k),dim=c(n,d,k))
rawfd2dataf(X,range=c(-1,1))

Reset Graphical Parameters

Description

The function returns the default graphical parameters for par.

Usage

resetPar()

Details

The returned parameters are used as input parameters for par.

Value

The list of graphical parameters described in the 'Graphical Parameters' section of par.

Examples

par(mfrow = c(1,2), mar = c(2,2,2,2))
plot(sin, -pi, 2*pi)
plot(cos, -pi, 2*pi)

par(resetPar())
plot(sin, -pi, 2*pi)
plot(cos, -pi, 2*pi)

Diagnostic Plot for First and Second Order Integrated and Infimal Depths

Description

Produce the diagnostic plot based on the fist or second order extended integrated / infimal depths.

Usage

shape.fd.analysis(datafA, datafB, range = NULL, d = 101, order = 1,
  method = c("halfspace", "simplicial"), approx = 0, title = "",
  nfun = 10, plot = TRUE)

Arguments

Details

Plots a diagnostic plot of pointwise univariate (or bivariate) depths for all possible points (or couples of points) from the domain of the functional data. From such a plot it is possible to infer into the first order (or second order) properties of a single function x with respect to the given set of functional data. For order=1, the integral of the displayed function is the integrated depth of x, the smallest value of the function is the infimal depth of x. For order=2, the bivariate integral of the displayed surface gives the second order extended integrated depth of x, the infimum of this bivariate function gives the second order infimal depth of x. For details see Nagy et al. (2016) and depthf.fd1.

Value

For order=1 two depth values, and two vectors of pointwise depths:

• Simpl_FD the first order integrated depth based on the simplicial depth,

• Half_FD the first order integrated depth based on the halfspace depth,

• Simpl_ID the first order infimal depth based on the simplicial depth,

• Half_ID the first order infimal depth based on the halfspace depth,

• PSD the vector of length d containing the computed pointwise univariate simplicial depths used for the computation of Simpl_FD and Simpl_ID,

• PHD the vector of length d containing the computed pointwise univariate halfspace depths used for the computation of Half_FD and Half_ID.

In addition, the first order integrated / infimal depth diagnostic plot of the function A with respect to the random sample given by the functions corresponding to the rows of the matrix B is produced.

For order=2 four depth values, and two matrices of pointwise depths:

• Simpl_FD the second order integrated depth based on the simplicial depth,

• Half_FD the second order integrated depth based on the halfspace depth,

• Simpl_ID the second order infimal depth based on the simplicial depth,

• Half_ID the second order infimal depth based on the halfspace depth,

• PSD the matrix of size d*d containing the computed pointwise bivariate simplicial depths used for the computation of Simpl_FD and Simpl_ID,

• PHD the matrix of size d*d containing the computed pointwise bivariate halfspace depths used for the computation of Half_FD and Half_ID.

In addition, the second order integrated / infimal depth diagnostic plot of the function A with respect to the random sample given by the functions corresponding to the rows of the matrix B is produced.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

See Also

depthf.fd1

Examples

datafA = dataf.population()$dataf[1]
dataf = dataf.population()$dataf[2:20]
shape.fd.analysis(datafA,dataf,order=1)
shape.fd.analysis(datafA,dataf,order=2,approx=0)

Functional Depth-Based Shape Outlier Detection

Description

Detects functional outliers of first three orders, based on the order extended integrated depth for functional data.

Usage

shape.fd.outliers(dataf, range = NULL, d = 101, q = 0.05,
  method = c("halfspace", "simplicial"), approx = 100, print = FALSE,
  plotpairs = FALSE, max.order = 3, exclude.out = TRUE,
  output = c("matrix", "list"), identifiers = NULL)

Arguments

Details

Using the procedure described in Nagy et al. (2016), the function uses the order extended integrated depths for functions, see depthf.fd1 and shape.fd.analysis, to perform informal functional shape outlier detection. Outliers of the first order (horizontal shift outliers) are found as the functions with q % of smallest (first order) integrated depth values. Second and third order outliers (shape outliers) are found using the extension of the boxplot method for depths as described in the paper Nagy et al. (2016).

Value

A matrix of logical values of size n*4, where n is the sample size. In the first three rows indicators of outlyingness of the corresponding functions for orders 1, 2 and 3 are given, in the fourth row the indicator of outlyingness with respect to the comparison of the first, and third order depths is given. That is, the fist row corresponds to the first order outliers, the second row to the second order outliers, and the last two rows formally to the third order outliers. Please consult Nagy et al. (2016) to interpret the notion of shape outlyingness.

Author(s)

Stanislav Nagy, nagy@karlin.mff.cuni.cz

References

Nagy, S., Gijbels, I. and Hlubinka, D. (2017). Depth-based recognition of shape outlying functions. Journal of Computational and Graphical Statistics, 26 (4), 883–893.

See Also

depthf.fd1, shape.fd.analysis

Examples

n = 30
dataf = dataf.population()$dataf[1:n]
shape.fd.outliers(dataf,print=TRUE,plotpairs=TRUE,
identifiers=unlist(dataf.population()$identifier)[1:n])