Package 'DistributionalDDA'

Description

The function does not normalize the density.

Usage

.DensityFromHist(
  weights,
  breaks,
  background = 0,
  dmu = function(x) rep(1, numberOfCases(x))
)

Arguments

Value

a function that can be evaluated on the support of the underlying measure (given by dmu) and provides the a density at every point.

See Also

Other .DensitFromHist: .LogDensityFromHist()

Examples

f = .DensityFromHist(1:5,0:5)
x = seq(-1,6,by=0.01)
plot(x,f(x),type="b")

Description

The function does not normalize the density.

Usage

.LogDensityFromHist(
  weights,
  breaks,
  background = mean(weights)/(length(weights) * 10),
  dmu = function(x) rep(1, numberOfCases(x))
)

Arguments

Value

a function that can be evaluated on the support of the underlying measure (given by dmu) and provides the a density at every point.

See Also

Other .DensitFromHist: .DensityFromHist()

Examples

f = .LogDensityFromHist(1:5,0:5)
x = seq(-1,6,by=0.01)
plot(x,f(x),type="b")

Description

A subsetting function selecting only a part of the histograms

Usage

## S3 method for class 'BDataHistograms'
x[i, j, ...]

Arguments

Value

A BDataHistograms object with the subset of histograms described by the basis

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(4,hist(Filter(inRange,rnorm(1000)),breaks=breaks,plot=FALSE)$counts))
hists = BDataFrequencyHistograms(X,breaks=breaks,basis=B2N01Basis5)
hists[1:2, ]

Description

The operator acts on all elements of a list and returns a list of the results. The right hand side can be a name in which case the '$' operator is used to e

Usage

X %.% expr

X %..% expr

Arguments

Value

a list, with results of the operations on the individual list elements.

Functions

• X %..% expr: same as '

Examples

data(DTD)
DTD$data %.% shape
DTD$data %.% summary(.$shape)
DTD$data %..% summary(.$shape)

Description

BData objects are considered a elements of a BayesSpace and operated accordingly to Bayes space rules, using the functions [Add], [Sub], [Mul], [Div]. For operating with a scalar, please use the BData object first.

Usage

## S3 method for class 'BData'
x + y

## S3 method for class 'BData'
x - y

## S3 method for class 'BData'
x * y

## S3 method for class 'BData'
x / y

Arguments

Value

A BData object of the same class as 'x', resulting from the corresponding Bayes-space operation (addition, subtraction, scalar multiplication, or scalar division).

Functions

• - : Bayes space subtraction divides densities

• * : Bayes space multiplication works only with scalars

• / : Bayes space division only works with scalars and leads to do the y-th root

Examples

X <- BDataSplines(t(1:5),B2Unif01Basis5)
X+X

Description

This additional is also able to add functions and BayesSpace objects. The sum of two functions is a function that results in the sum of the two results. The sum of two BayesSpace objects (in BData), is given by the BayesSpace operations, which is given by the product of the densities with respect to the reference measure.

Usage

Add(x, y)

## S3 method for class ''function''
Add(x, y)

## S3 method for class 'list'
Add(x, y)

## S3 method for class 'numeric'
Add(x, y)

## S3 method for class ''function''
Add.function(x, y)

## S3 method for class 'list'
Add.function(x, y)

## S3 method for class 'numeric'
Add.function(x, y)

## S3 method for class ''function''
Add.numeric(x, y)

## S3 method for class 'numeric'
Add.numeric(x, y)

## S3 method for class 'numeric'
Add.list(x, y)

## S3 method for class 'list'
Add.numeric(x, y)

## S3 method for class 'list'
Add.list(x, y)

## S3 method for class ''function''
Add.list(x, y)

## S3 method for class 'BDataDensities'
Add(x, y)

## S3 method for class 'BDataLogDensities'
Add(x, y)

## S3 method for class 'BDataSplines'
Add(x, y)

## S3 method for class 'BDataHistograms'
Add(x, y)

## S3 method for class 'BDataDensities'
Add.BDataDensities(x, y)

## S3 method for class 'BDataLogDensities'
Add.BDataLogDensities(x, y)

## Default S3 method:
Add.BDataLogDensities(x, y)

## Default S3 method:
Add.BDataDensities(x, y)

## S3 method for class 'BDataSplines'
Add.BDataSplines(x, y)

## Default S3 method:
Add.BDataSplines(x, y)

## S3 method for class 'BDataHistograms'
Add.BDataHistograms(x, y)

## S3 method for class 'BDataDensities'
Add(x, y)

## S3 method for class 'BDataSplines'
Add(x, y)

## S3 method for class 'BDataLogDensities'
Add(x, y)

## S3 method for class 'BDataPDFs'
Add(x, y)

## S3 method for class 'BDataLogPDFs'
Add(x, y)

Arguments

Value

A BData object of the same subclass as the first argument (the Bayes-space sum / difference / scalar multiple / quotient).

Methods (by class)

• Add(`function`): method for functions

• Add(list): method for lists, iterating over them

• Add(numeric): method for numerics, considering them as constant functions

• Add(BDataDensities): method for BDataDensities (effectively multiplying the densities as addition in the BayesSpace)

• Add(BDataLogDensities): method for BDataLogDensities

• Add(BDataSplines): method for BDataSplines

• Add(BDataHistograms): method for BDataHistograms

• Add(BDataDensities): BayesSpaceAddition for BDataDensities

• Add(BDataSplines): BayesSpaceAddition for BDataSplines

• Add(BDataLogDensities): BayesSpaceAddition for BDataLogDensities

• Add(BDataPDFs): BayesSpaceAddition for BDataPDFs

• Add(BDataLogPDFs): BayesSpaceAddition for BDataLogPDFs

Functions

• Add.function(`function`): method for adding two functions, doing it by creating a function adding their values.

• Add.function(list): method adding a function to a list, creating a list with entries for each list

• Add.function(numeric): method adding a constant to a function

• Add.numeric(`function`): method for adding a constant to a function

• Add.numeric(numeric): method for two numerics, simply adding the values

• Add.list(numeric): method for adding a numeric to every value of the list

• Add.numeric(list): method for adding a numeric to every value of the list

• Add.list(list): method for vectorized addition of list elements

• Add.list(`function`): method for adding a function to every element of a list

• Add.BDataDensities(BDataDensities): method for two BDataDensities

• Add.BDataLogDensities(BDataLogDensities): method for adding two BDataLogDensities

• Add.BDataLogDensities(default): method converting the second argument to a BDataLogDensity and then adding

• Add.BDataDensities(default): method converting the second argument to a BDataDensity object and then adding.

• Add.BDataSplines(BDataSplines): method for two BDataSplines

• Add.BDataSplines(default): method for BDataSplines default

• Add.BDataHistograms(BDataHistograms): method for BDataHistograms

See Also

Other Add: Div(), Mul(), Sub()

Examples

Add(sin,cos)
Add(list(sin,cos),sin)
Add(cos,list(sin,cos))
Add(list(sin,cos),list(sin,cos))
Add(list(sin,cos),10)

Description

Sum together the absolute frequencies of two representations with this information

Usage

addFrequency(x, y, ...)

## S3 method for class 'BDataFrequencyHistograms'
addFrequency.BDataFrequencyHistograms(x, y, ...)

## S3 method for class 'BDataUnivariateSamples'
addFrequency.BDataUnivariateSamples(x, y, ...)

## S3 method for class 'BDataUnivariateSamples'
addFrequency.BDataFrequencyHistograms(
  x,
  y,
  .baseSpaceBasis = getBayesSpaceBasis(x),
  ...
)

## S3 method for class 'BDataFrequencyHistograms'
addFrequency.BDataUnivariateSamples(
  x,
  y,
  .baseSpaceBasis = getBayesSpaceBasis(x),
  ...
)

Arguments

Value

An object of the least informative kind from the two pooled together, which contains the added absolute frequencies of both input objects. The Bayes Space basis is controlled by the first argument.

Functions

• addFrequency.BDataFrequencyHistograms(BDataFrequencyHistograms): method for two "BDataFrequencyHistograms" object. Returns a "BDataFrequencyHistograms" where the frequencies are summed (if breaks are equivalent), or where they are distributed uniformly along the interval bins and rebinned by summing with a break set resulting from the combination of both input bin sets.

• addFrequency.BDataUnivariateSamples(BDataUnivariateSamples): method for two "BDataUnivariateSamples" object. Returns a "BDataUnivariateSamples" where the observations are pooled together (in the order they are given), pair by pair. It can recycle a "BDataUnivariateSamples" of one single population with a multi-population "BDataUnivariateSamples", but will not work with any other combination of "BDataUnivariateSamples" of different lengths.

• addFrequency.BDataFrequencyHistograms(BDataUnivariateSamples): method for a "BDataFrequencyHistograms" and a "BDataUnivariateSamples" objects; classifies the data sets from "BDataUnivariateSamples" into the bins of "BDataFrequencyHistograms". As with other methods of this function, it only works for 'x' and 'y' of the same length, or else either 'x' or 'y' must have length 1.

• addFrequency.BDataUnivariateSamples(BDataFrequencyHistograms): method for a "BDataUnivariateSamples" and a "BDataFrequencyHistograms" objects; same as in the reverse order

Examples

xh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
yh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xh,yh)
xh1 = list(runif(300)) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xh,xh1)
xh1 = list(runif(300)) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5, breaks=c(0:4)/4)
addFrequency(xh,xh1)
xs = lapply(rep(200,4), runif) |> BDataSamples(B2Unif01Basis5)
ys = lapply(rep(10,4), runif) |> BDataSamples(B2Unif01Basis5)
addFrequency(xs,ys) |> summary()
x1 = BDataSamples(list(runif(100)), B2Unif01Basis5)
addFrequency(xs, x1)  |> summary()
xs = lapply(rep(200,4), runif) |> BDataSamples(B2Unif01Basis5)
yh = lapply(rep(1000,4), runif) |> as.BDataFrequencyHistograms(basis=B2Unif01Basis5)
addFrequency(xs,yh)

Description

Age distributions for 57 communes in Upper Austria (Oberoesterreich), separated by sex. Each observation represents the age distribution of either the male or female population in a commune, given as proportions across 19 five-year age groups.

Usage

age

Format

A list with three components:

data

A data frame with 114 rows (57 communes x 2 sexes) and 22 columns:

commune

Character. Four-digit commune code.

municipality

Character. Name of the municipality (e.g., "Linz", "Steyr"). German umlauts have been transliterated to ASCII (ue, oe, ae, ss).

sex

Factor with levels "male" and "female".

AGE_2, AGE_7, ..., AGE_92

Numeric. Proportions for each age group. Column names indicate the midpoint of the 5-year age interval (e.g., AGE_2 = ages 0-4, AGE_7 = ages 5-9, ..., AGE_92 = ages 90+).

ages

Numeric vector of length 19. The midpoint ages: 2, 7, 12, ..., 92.

distributions

A 114 x 19 matrix of age proportions. Row names are formatted as "municipality_sex" (e.g., "Linz_male"). Each row sums to 1.

Details

The data originates from the Austrian Register Census 2021 (Registerzaehlung 2021), which provides population counts by gender, age, and citizenship at the municipality level. The original counts have been converted to proportions (probability distributions) for distributional data analysis.

This dataset is useful for demonstrating PCA on distributional data, as it contains real-world probability distributions that vary in shape (e.g., rural vs urban communes have different age structures).

License

Creative Commons Attribution 4.0 International (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/

Note

Municipality names have been transliterated from German to ASCII for cross-platform compatibility, following the convention ae / oe / ue / ss for the umlauts and sharp-s (so e.g. the German "Altmuenster" and "Schoenegg" are stored with ue / oe). For an unambiguous key independent of spelling, use the four-digit codes in the 'commune' column.

Source

Data from the Austrian Register Census 2021 (Registerzaehlung 2021), provided by the State of Upper Austria (Land Oberoesterreich) via the Open Data Portal Austria.

References

Land Oberoesterreich (2021). Bevoelkerung nach Geschlecht und Alter. Open Data Oesterreich. Old broken link: https://www.data.gv.at/katalog/dataset/land-ooe_bevolkerung-geschlecht-und-alter Current dataset: https://www.data.gv.at/katalog/datasets/cd33f19a-7bcc-4454-987c-9f2d76542872

Examples

data("age")

# Examine the structure
str(age)
head(age$data)

# The age midpoints
age$ages

# Plot a few age distributions
matplot(age$ages, t(age$distributions[1:6, ]), type = "l",
        xlab = "Age", ylab = "Proportion",
        main = "Age distributions for first 6 observations")

# Convert to BData for distributional analysis
# First, create a basis for age distributions (support 0-100)
rmAge <- ReferenceMeasureUniform(0, 100)
B2Age <- BayesSpace(rmAge)
basisAge <- BayesSpaceBasis(B2Age, nbasis = 7)

# Create histograms from the proportional data
breaks <- c(5*(0:18), 100)
ageHist <- BDataDensityHistograms(
    age$distributions, breaks = breaks, basis = basisAge
    )

# Convert to splines: histograms -> densities -> splines
ageDensities <- as.BDataDensities(ageHist)
ageSplines <- as.BDataSplines(ageDensities)
pcaAge <- princomp(ageSplines)
summary(pcaAge)

# Visualize PCA results
plot(pcaAge, type = "screeplot")
plot(pcaAge, type = "scores")

# Score plot with municipality labels
plot(pcaAge, type = "scores", labels = age$data$municipality, cex = 0.6)

# Color by sex
sexColors <- ifelse(age$data$sex == "male", "blue", "red")
plot(pcaAge, type = "scores", col = sexColors)
legend("topright", c("Male", "Female"), col = c("blue", "red"), pch = 19)

Description

Finds a BayesSpaceBasis representation of the object

Usage

as.BayesSpaceBasis(x, ...)

## S3 method for class 'BayesSpaceBasis'
as.BayesSpaceBasis(x, ...)

## S3 method for class 'BayesSpace'
as.BayesSpaceBasis(x, ...)

Arguments

Value

a [BayesSpaceBasis] object

Methods (by class)

• as.BayesSpaceBasis(BayesSpaceBasis): returns x

• as.BayesSpaceBasis(BayesSpace): Calls 'BayesSpaceBasis(x,...)' to create an appropriate basis

See Also

[BayesSpaceBasis]

Examples

as.BayesSpaceBasis(B2Unif01Basis5)
as.BayesSpaceBasis(B2Unif01)

Description

This represents BayesSpace data by a list of density functions, i.e. one density (with respect to the reference measure) per observation. Use this representation to inspect or plot the distributions as densities; for computation (PCA, distances) convert further to splines with [as.BDataSplines].

Usage

as.BDataDensities(x, ...)

## S3 method for class 'BDataDensities'
as.BDataDensities(x, ...)

## S3 method for class 'BDataSplines'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataHistograms'
as.BDataDensities(
  x,
  ...,
  background = 0,
  basis = getBayesSpaceBasis(x),
  dmu = getPDF(getReferenceMeasure(basis))
)

## S3 method for class 'BDataSamples'
as.BDataDensities(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

Arguments

Value

A BDataDensities object (densities w.r.t. the reference measure).

Methods (by class)

• as.BDataDensities(BDataDensities): method for the class itself das nothing

• as.BDataDensities(BDataSplines): constructs the projected functions

• as.BDataDensities(BDataLogDensities): calls [BDataDensities.BDataLogDensities]

• as.BDataDensities(BDataPDFs): divides PDFs by PDF of reference measure

• as.BDataDensities(BDataLogPDFs): divides PDFs by PDF of reference measure

• as.BDataDensities(BDataHistograms): method for histograms (TODO: Uses a very crude heuristic to estimate the log density)

• as.BDataDensities(BDataSamples): constructs a density from the samples

See Also

[.DensityFromHist]

Other BayesSpaceConverters: as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X = BDataLogDensities(list(sin,cos),basis=B2Unif01Basis5)
as.BDataDensities(X)

Description

Density histograms do not require count information, but provide density information about the relative density in the bars.

Usage

as.BDataDensityHistograms(X, ...)

## S3 method for class 'BDataUnivariateSamples'
as.BDataDensityHistograms(
  X,
  ...,
  breaks = "Sturges",
  basis = getBayesSpaceBasis(X)
)

Arguments

Value

a BDataDensityHistograms object

Methods (by class)

• as.BDataDensityHistograms(BDataUnivariateSamples): method for samples computing the histogram

See Also

[as.BDataDensityHistograms.list]

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Other BayesSpaceConverters: as.BDataDensities(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

s = BDataSamples(list(rnorm(300), rnorm(300,1)), basis=B2N01Basis5)
as.BDataDensityHistograms(s)

Description

Constructor for a distributional data set of class "BDataDensityHistograms" out of a list of either precalculated histograms or of numeric vectors.

Usage

## S3 method for class 'list'
as.BDataDensityHistograms(X, breaks, basis, ...)

Arguments

Value

A distributional data container of class ['BDataDensityHistograms'].

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

x = lapply(rep(1000,4), runif)
(bddh = as.BDataDensityHistograms(x, breaks="Sturges", basis=B2Unif01Basis5) )
(bdfh = as.BDataFrequencyHistograms(x, breaks="Sturges", basis=B2Unif01Basis5) )
as.BDataDensityHistograms(bdfh)

Description

Only samples and BDataFrequencyHistograms themselfs can be effectively converted to histograms, because all other representations lack information on the certainty

Usage

as.BDataFrequencyHistograms(X, ...)

## S3 method for class 'BDataUnivariateSamples'
as.BDataFrequencyHistograms(
  X,
  breaks = "Sturges",
  ...,
  basis = getBayesSpaceBasis(X)
)

Arguments

Value

a BDataFrequencyHistograms object representing the same dataset

Methods (by class)

• as.BDataFrequencyHistograms(BDataUnivariateSamples): method for BDataUnivariateSamples

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

Xsamples = BDataSamples(lapply(1:10,function(i) runif(100)),basis=B2Unif01Basis5)
Xhist    = as.BDataFrequencyHistograms(Xsamples)

Description

Constructor for a distributional data set of class "BDataFrequencyHistograms" out of a list of either precalculated histograms or of numeric vectors.

Usage

## S3 method for class 'list'
as.BDataFrequencyHistograms(X, breaks, basis, ...)

Arguments

Value

A distributional data container of class ['BDataFrequencyHistograms'].

See Also

Other BDataHistograms: BDataFrequencyHistograms(), as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

x = lapply(rep(1000,4), runif)
as.BDataFrequencyHistograms(x, breaks="Sturges",basis=B2Unif01Basis5)

Description

The function tries to guess whether its datasets, distributions, Histograms,...

Usage

as.BDataHistograms(X, ...)

## S3 method for class 'list'
as.BDataHistograms(
  X,
  ...,
  breaks = NULL,
  basis = NULL,
  nsim = NULL,
  seed = NULL
)

## S3 method for class 'BDataHistograms'
as.BDataHistograms(
  X,
  ...,
  breaks = NULL,
  basis = NULL,
  nsim = NULL,
  seed = NULL
)

## S3 method for class 'histogram'
as.BDataHistograms(X, ..., breaks = NULL, basis, nsim = NULL, seed = NULL)

## S3 method for class 'BDataFrequencyHistograms'
as.BDataHistograms(X, ...)

## S3 method for class 'BDataDensityHistograms'
as.BDataHistograms(X, ...)

Arguments

Value

either a BDataFrequencyHistograms (if counts are availalbe) , or a BDataDensityHistograms object if only densities are known

Methods (by class)

• as.BDataHistograms(list): method for lists of possibly heterogenious data

• as.BDataHistograms(BDataHistograms): checks breaks if given

• as.BDataHistograms(histogram): checks breaks if given

• as.BDataHistograms(BDataFrequencyHistograms): identity converter

• as.BDataHistograms(BDataDensityHistograms): identity converter

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

myBreaks = seq(0,1,by=0.2)
as.BDataHistograms(replicate(10,runif(100),simplify=FALSE),
                   breaks=myBreaks,basis=B2Unif01Basis5)

Description

This represents BayesSpace data by a list of log-density functions (the logarithm of the density with respect to the reference measure). Log-densities are the natural representation for Bayes-space operations, where adding distributions corresponds to adding their log-densities.

Usage

as.BDataLogDensities(x, ...)

## S3 method for class 'BDataDensities'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSplines'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataLogDensities(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

## S3 method for class 'list'
as.BDataLogDensities(x, ..., basis)

## S3 method for class 'BDataHistograms'
as.BDataLogDensities(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

A BDataLogDensities object.

Methods (by class)

• as.BDataLogDensities(BDataDensities): method for BDataDensities

• as.BDataLogDensities(BDataLogPDFs): method for BDataLogPDFs

• as.BDataLogDensities(BDataPDFs): method for BDataPDFs

• as.BDataLogDensities(BDataSplines): constructs the projected functions

• as.BDataLogDensities(BDataLogDensities): calls [BDataLogDensities.BDataLogDensities]

• as.BDataLogDensities(BDataSamples): constructs a density from the samples

• as.BDataLogDensities(list): constructs a density from the samples

• as.BDataLogDensities(BDataHistograms): constructs a density from the samples (TODO: Uses a very crude heuristic to estimate the log density)

See Also

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

as.BDataLogDensities(list(sin,cos),basis=B2N01)

Description

A probability density function or Lebesgue density of a measure can only be computed for Lebesgue continuous distributions. As a dataset of BayesSpace elements (BData) they can be encoded as BDataLogPDFs objects. This function converts other representations to this representation.

Usage

as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataDensities'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSplines'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataLogPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataLogPDFs(x, ..., basis)

## S3 method for class 'BDataDensityHistograms'
as.BDataLogPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataFrequencyHistograms'
as.BDataLogPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataLogPDFs(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

Arguments

Value

a BDataLogPDFs object

Methods (by class)

• as.BDataLogPDFs(BDataDensities): method for BDataDensities multiplies with reference measure density

• as.BDataLogPDFs(BDataLogDensities): method for BDataLogDensities adds the log density of the reference measure

• as.BDataLogPDFs(BDataSplines): method for BDataSplines

• as.BDataLogPDFs(BDataPDFs): method for BDataLogPDFs exponentiates and multiplies with reference measure density

• as.BDataLogPDFs(BDataLogPDFs): method for BDataLogPDFs possibly changes the basis

• as.BDataLogPDFs(BDataDensityHistograms): method for BDataDensityHistograms interprets histogram as density.

• as.BDataLogPDFs(BDataFrequencyHistograms): method for BDataFrequencyHistograms

• as.BDataLogPDFs(BDataSamples): method for BDataSamples

See Also

[BDataLogPDFs], [as.BDataPDFs]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X <- BDataDensities(list(\(x)(1),\(x)(x)),basis=B2N01Basis5)
Xlpdfs = as.BDataLogPDFs(X)
Xlpdfs
X1 = as.BDataLogPDFs(as.BDataLogDensities(X))
X2 = as.BDataLogPDFs(as.BDataLogPDFs(X))
X3 = as.BDataLogPDFs(Xlpdfs)
set.seed(47)
Xs = BDataSamples(replicate(7,runif(200),simplify=FALSE),B2Unif01Basis5)
X4 = as.BDataLogPDFs(Xs) # TODO
X5 = as.BDataLogPDFs(as.BDataFrequencyHistograms(Xs,breaks=seq(0,1,by=0.1)))
X6 = as.BDataLogPDFs(as.BDataDensityHistograms(Xs,breaks=seq(0,1,by=0.1)^2))
# plot(Xlpdfs)

Description

A probability density function or Lebesgue density of a measure can only be computed for Lebesgue continuous distributions. As a dataset of BayesSpace elements (BData) they can be encoded as BDataPDFs objects. This function converts other representations to this representation.

Usage

as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataDensities'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogDensities'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataLogPDFs'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataPDFs'
as.BDataPDFs(x, ..., basis)

## S3 method for class 'BDataDensityHistograms'
as.BDataPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataFrequencyHistograms'
as.BDataPDFs(x, ..., background = 0, basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataPDFs(
  x,
  ...,
  method = c("Fisher", "kde"),
  basis = getBayesSpaceBasis(x)
)

## S3 method for class 'BDataSplines'
as.BDataPDFs(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

a BDataPDFs object

Methods (by class)

• as.BDataPDFs(BDataDensities): method for BDataDensities multiplies with reference measure density

• as.BDataPDFs(BDataLogDensities): method for BDataLogDensities exponentiates and multiplies with reference measure density

• as.BDataPDFs(BDataLogPDFs): method for BDataLogPDFs exponentiates and multiplies with reference measure density

• as.BDataPDFs(BDataPDFs): method for BDataPDFs possibly changes the basis

• as.BDataPDFs(BDataDensityHistograms): method for BDataDensityHistograms interprets histogram as density.

• as.BDataPDFs(BDataFrequencyHistograms): method for BDataFrequencyHistograms

• as.BDataPDFs(BDataSamples): method for BDataSamples

• as.BDataPDFs(BDataSplines): method for BDataSplines

See Also

[BDataPDFs]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataSamples(), as.BDataSplines(), idt.BData(), toHistogram()

Examples

X <- BDataDensities(list(\(x)(1),\(x)(x)),basis=B2N01Basis5)
Xpdfs = as.BDataPDFs(X)
Xpdfs
X1 = as.BDataPDFs(as.BDataLogDensities(X))
X2 = as.BDataPDFs(as.BDataLogPDFs(X))
X3 = as.BDataPDFs(Xpdfs)
set.seed(47)
Xs = BDataSamples(replicate(7,runif(200),simplify=FALSE),B2Unif01Basis5)
# X4 = as.BDataPDFs(Xs) # TODO
X5 = as.BDataPDFs(as.BDataFrequencyHistograms(Xs,breaks=seq(0,1,by=0.1)))
X6 = as.BDataPDFs(as.BDataDensityHistograms(Xs,breaks=seq(0,1,by=0.1)^2))
# plot(Xpdfs)

Description

Convert any representation to "BDataSamples", most commonly by simulating the data by means of [simulate.BData()].

Usage

as.BDataSamples(x, ...)

## S3 method for class 'BData'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataSamples'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

## S3 method for class 'BDataUnivariateSamples'
as.BDataSamples(x, ..., basis = getBayesSpaceBasis(x))

Arguments

Value

a BDataSamples object

Methods (by class)

• as.BDataSamples(BData): for any "BData" object

• as.BDataSamples(BDataSamples): for "BDataSamples" object, as identity or to change the basis

• as.BDataSamples(BDataUnivariateSamples): for "BDataUnivariateSamples" object, as identity or to change the basis

See Also

[simulate.BData]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSplines(), idt.BData(), toHistogram()

Description

Returns a dataset of densities represented by zero-integral B-spline (ZB-spline) coefficients in a [BDataSplines] object. The spline representation is the compact form used for computation (PCA, distances, inner products). The conversion relies on a [BayesSpaceBasis] that is either taken from the object or supplied via 'basis'.

Usage

as.BDataSplines(X, ...)

## S3 method for class 'BDataLogDensities'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataLogPDFs'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataPDFs'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataDensities'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataSplines'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataSamples'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BDataGeneral'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class ''function''
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'histogram'
as.BDataSplines(X, ..., basis = getBayesSpaceBasis(X))

## S3 method for class 'BayesSpaceBasis'
as.BDataSplines(X, ...)

Arguments

Value

A distributional data container of class ['BDataSplines'].

Methods (by class)

• as.BDataSplines(BDataLogDensities): method for BDataLogDensities

• as.BDataSplines(BDataLogPDFs): method for BDataLogPDFs

• as.BDataSplines(BDataPDFs): method for BDataPDFs

• as.BDataSplines(BDataDensities): method for BDataDensities

• as.BDataSplines(BDataSplines): method for BDataSplines, eventually reprojects if basis given

• as.BDataSplines(BDataSamples): method for BDataSamples

• as.BDataSplines(BDataGeneral): method for BDataGeneral

• as.BDataSplines(`function`): method for function interpreted as density with respect to the reference measure.

• as.BDataSplines(histogram): method for function interpreted as density with respect to the reference measure. (TODO: Uses a very crude heuristic to estimate the log density)

• as.BDataSplines(BayesSpaceBasis): the distributions corresponding to the basis elements.

Note

When converting from densities, log-densities or PDFs, the spline coefficients are the L2 projection onto the (non-orthonormal) ZB-spline basis: they solve the normal equations 'S c = <f, zeta>' with the Gram matrix 'S = getMetricTensor(basis)' (see [getMetricTensor]). The inner products '<f, zeta>' are estimated by Monte-Carlo integration over the reference-measure sample, so the coefficients carry a small sampling error that decreases as the sample size grows.

See Also

[as.BDataDensities], [BayesSpaceBasis], [getMetricTensor]

Other BayesSpaceConverters: as.BDataDensities(), as.BDataDensityHistograms(), as.BDataFrequencyHistograms(), as.BDataHistograms(), as.BDataLogDensities(), as.BDataLogPDFs(), as.BDataPDFs(), as.BDataSamples(), idt.BData(), toHistogram()

Examples

X = BDataDensities(list(\(x) x, \(x) x^2),basis=B2Unif01Basis5)
as.BDataSplines(X)
as.BDataSplines(as.BDataSplines(X))
as.BDataSplines(as.BDataLogDensities(X))
as.BDataSplines(BDataSamples(replicate(10,runif(100),simplify=FALSE),basis=B2Unif01Basis5))

Description

A distribution of distributions based a distribution family with random hyperparameters.

Usage

B2DistHyper(
  density,
  basis,
  paramSimulator,
  ...,
  simulator = NULL,
  .hyperparameters = list(...)
)

Arguments

Value

A B2DistHyper object encoding the distribution.

Examples

D = B2DistHyper(dbeta,B2Unif01Basis5,function() rexp(2),simulator=rbeta)
simulate(D,4,seed=47)

Description

A Gaussian Process distribution for probabilities

Simulate a Gaussian Distribution on B2

Usage

B2GaussianDistribution(
  mean = BDataLogDensities(list(function(x) rep(0, length(x))), basis = B2Unif01Basis7),
  var = varKernel(function(x, y) exp(-abs(outer(x, y, "-"))), basis =
    getBayesSpaceBasis(mean)),
  nEval = 513
)

## S3 method for class 'B2GaussianDistribution'
simulate(object, nsim = 1, seed = NULL, ..., nEval = object$nEval)

Arguments

Value

B2GaussianDistribution: a B2GaussianDistribution object which can be simulated with, simulate: BDataLogDensities-Object with nsim cases with the same BayesSpace and BayesSpaceBasis as the mean.

Methods (by generic)

• simulate(B2GaussianDistribution): Simulaton method

See Also

[

[stats::simulate]

Examples

m = BDataPDFs(list(dunif),basis=B2Unif01Basis7)
G1 = B2GaussianDistribution(m,function(x,y) exp(-(4*abs(x-y)^1.7)))
plot(simulate(G1,10,seed=47),col=2:11)

GD = B2GaussianDistribution(mean=BDataPDFs(\(x)x,basis=B2Unif01Basis5),
                            var=\(x,y) exp(-abs(x-y)))
plot(simulate(GD,5))
GD = B2GaussianDistribution(mean=BDataPDFs(\(x) dnorm(x),basis=B2N01Basis5),
                            var=\(x,y) exp(-abs(x-y)))
plot(as.BDataPDFs(simulate(GD,5)))

Description

B2N01 is the Bayes space B2(N(0,1)), and B2N01BasisN its B-Spline basis with N elements of order 2

Usage

B2N01

B2N01Basis5

B2N01Basis7

B2N01Basis20

Format

An object of class BayesSpace of length 1.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

An object of class BayesSpaceBasis of length 5.

Functions

• B2N01Basis5: a basis for B2N01 with 5 elements

• B2N01Basis7: a basis for B2N01 with 7 elements

• B2N01Basis20: a basis for B2N01 with 20 elements

See Also

[BayesSpaceBasis]

Other ReferenceMeasures: ReferenceMeasureDiscrete(), ReferenceMeasureTransformed(), ReferenceMeasureUniform(), StdReferenceMeasureUnif01, expectation(), getSample(), isInDomain()

Examples

getReferenceMeasure(B2N01)
getMetricTensor(B2N01Basis5)

Description

The basis is represented by log-densities. The function is vectorized, i.e. multiple x values may be provided.

Usage

basisEval(basis, x, ...)

## S3 method for class 'BayesSpaceBasis'
basisEval(basis, x, ...)

## S3 method for class 'basisfd'
basisEval(basis, x, ...)

## S3 method for class 'BayesSpaceBasis'
as.function(x, ...)

Arguments

Value

a matrix with the evaluation for one each of the x in a row.

Methods (by class)

• basisEval(BayesSpaceBasis): the standard method for BayesSpaceBasis

• basisEval(basisfd): a method implementation for basisfd

Functions

• as.function(BayesSpaceBasis): the basis can also be converted to a function or called directly

Examples

x = seq(0,1,by=0.1)
basisEval(B2Unif01Basis5,x)

Description

A Bayes Space is a Hilbert space of distribution and likelihoods having a non zero density with respect to the reference measure (which needs) to be a distribution itself.

Usage

BayesSpace(referenceMeasure = ReferenceMeasureUniform(0, 1))

Arguments

Value

An object of class '"BayesSpace"': a list holding the reference measure.

See Also

[ReferenceMeasure]

Other BayesSpace: L2Space()

Examples

bs = BayesSpace(ReferenceMeasureUniform(0,10))
bs
X = BDataSamples(replicate(10,runif(100,0,10)),basis=BayesSpaceBasis(bs))

Description

Creates the finite-dimensional basis in which distributional data are represented for computation. Besides the functional spline basis and its [BayesSpace], the returned object stores the transformation matrix 'A' and the Gram matrix 'S' (the inner products of the basis elements).

Usage

BayesSpaceBasis(basis, ...)

## S3 method for class 'basisfd'
BayesSpaceBasis(
  basis,
  bayesSpace = BayesSpace(ReferenceMeasureUniform(basis$rangeval[1], basis$rangeval[2])),
  A = computeL22BayesSpacesBasisMatrix(basis),
  S = A %*% fda::inprod(basis, basis) %*% t(A)/diff(basis$rangeval),
  orthonormalize = FALSE,
  ...
)

## S3 method for class 'fd'
BayesSpaceBasis(basis, ...)

## S3 method for class 'BayesSpace'
BayesSpaceBasis(basis, ...)

Arguments

Details

The default basis (constructed from a reference measure or an 'fda' B-spline basis) consists of **zero-integral B-splines (ZB-splines)**: spline functions that integrate to zero, which is required to represent centred-log-ratio (clr) transformed densities. The object stores

'basis'

a function evaluating the underlying (B-spline) basis,

'A'

the transformation matrix mapping those values to the ZB-spline basis, so the basis functions are 'zeta = A %*% B',

'S'

the Gram (metric) matrix 'S[i,j] = <zeta_i, zeta_j>',

'bayesSpace', 'centered'

the space and a centring flag.

ZB-splines are **not orthonormal**, so in general 'S' is not the identity; inner products, norms, variances and PCA all use 'S' (see [getMetricTensor], [princomp.BData]). Pass an explicit 'S' only if you have a better value than the internally computed one.

Value

An object of class '"BayesSpaceBasis"': a list with components 'basis', 'bayesSpace', 'A', 'S' and 'centered'.

Methods (by class)

• BayesSpaceBasis(basisfd): Construct from an basisfd object from functional data

• BayesSpaceBasis(fd): Construct from an fd object from functional data

• BayesSpaceBasis(BayesSpace): Calls BayesSpaceBasis on the Reference measure

See Also

[getMetricTensor], [BayesSpace], [as.BDataSplines], [princomp.BData]

[fda::basisfd]

[fda::fd]

Examples

basis <- BayesSpaceBasis(B2Unif01, nbasis = 7)
getMetricTensor(basis)        # Gram matrix S (not the identity for ZB-splines)

Description

Create a BayesSpaceBasis for a Cartesian Product of Bayes Spaces

Usage

## S3 method for class 'BayesSpaceCartesianProduct'
BayesSpaceBasis(basis, ..., useBases = lapply(basis, BayesSpaceBasis, ...))

## S3 method for class 'ReferenceMeasureCartesianProduct'
BayesSpaceBasis(
  basis,
  ...,
  useBases = lapply(bayesSpace, BayesSpaceBasis, ...),
  bayesSpace = BayesSpaceCartesianProduct(lapply(basis, BayesSpace))
)

Arguments

Value

A BayesSpaceBasisCartesianProduct of the bases in useBases, for the BayesSpace basis.

Functions

• BayesSpaceBasis(ReferenceMeasureCartesianProduct): creates the BayesSpaceCartesian Product for this reference measure and the corresponding product basis.

See Also

BayesSpaceBasis

Other Cartesian: BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Examples

BayesSpaceBasis(BayesSpaceCartesianProduct(
    A=BayesSpace(StdReferenceMeasureUnif01),
    B=BayesSpace(ReferenceMeasureDiscrete(c("A","B","C"),c(0.2,0.3,0.5))),
    C=BayesSpace(StdReferenceMeasureN01)
    )
)

Description

The function needs a vectorized function mapping the support of the Bayes space to real densities.

Usage

## S3 method for class ''function''
BayesSpaceBasis(
  basis,
  bayesSpace = B2Unif01,
  A = diag(rep(1, ncol(basis(getSample(bayesSpace))))),
  S = A %*% scalarProductOfLogDensities(basis, space = bayesSpace) %*% t(A),
  orthonormalize = FALSE,
  ...,
  centered = FALSE
)

Arguments

Details

A and S only need to be provided if there is a better result availabel than from the internal computation of the Bayes space

Value

A BayesSpaceBasis

See Also

BayesSpaceBasis

Examples

BayesSpaceBasis(function(x) outer(x,1:5,function(a,b) cos(a*b*pi)))
BayesSpaceBasis(function(x) outer(x,1:5,function(a,b) cos(a*b*pi)),orthonormalize=TRUE)

Description

Coded Basis for ReferenceMeasureDiscrete

Usage

## S3 method for class 'ReferenceMeasureDiscrete'
BayesSpaceBasis(
  basis,
  bayesSpace = BayesSpace(basis),
  orthonormalize = FALSE,
  ...
)

Arguments

Value

A BayesSpaceBasis for that discrete reference measure. If a nonstandard A is used it might be nonorthogonal.

Examples

X=c("A","B","C")
rmd = ReferenceMeasureDiscrete(X)
basis = BayesSpaceBasis(rmd)
as.function(basis)(X)
weightedStats(rmd,as.function(basis),function(x) rep(1,length(x)))

Description

The basis is a transformation of the Basis that would be computed for the underlying untransformed BayesSpace.

Usage

## S3 method for class 'ReferenceMeasureTransformed'
BayesSpaceBasis(
  basis,
  bayesSpace = getBayesSpace(basis),
  orthonormalize = FALSE,
  ...
)

Arguments

Value

an object of class '"BayesSpaceBasis"', transforming the basis of the underlying untransformed BayesSpace.

Examples

rmt <- ReferenceMeasureTransformed("pnorm", mean=2, sd=2)
basis <- BayesSpaceBasis(rmt)
basis

Description

create the 'BayesSpaceBasis' object for 'ReferenceMeasureUniform'

Usage

## S3 method for class 'ReferenceMeasureUniform'
BayesSpaceBasis(
  basis,
  bayesSpace = getBayesSpace(basis),
  orthonormalize = FALSE,
  nbasis = 7,
  order = 4,
  breaks = seq(basis$a, basis$b, length.out = (nbasis - order + 3)),
  ...
)

Arguments

Value

object of class 'BayesSpaceBasis'

Examples

basis <- BayesSpaceBasis(ReferenceMeasureUniform(0,1))
basis
#basis(seq(0,1,length.out=10))

Description

Computes the product basis of BayesSpaceBasis for the product BayesSpace

Usage

BayesSpaceBasisCartesianProduct(
  ...,
  .LIST = .listCatcher(...),
  bayesSpace = BayesSpaceCartesianProduct(.LIST = lapply(.LIST, getBayesSpace))
)

Arguments

Value

A Bayes Space Basis for the product space as defined the DDA Book.

See Also

[BayesSpaceBasis]

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Description

The Cartesian Product of Bayes Spaces

Usage

BayesSpaceCartesianProduct(...)

L2SpaceCartesianProduct(...)

Arguments

Value

the Product space of the BayesSpaces, a BayesSpace itself.

Functions

• L2SpaceCartesianProduct(): Product of L2 spaces

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), cartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Description

'BData' is the base class for the data containers in DistributionalDDA. A 'BData' object holds a data set of distributions together with the [BayesSpaceBasis] in which they are represented. Several subclasses store the same information in different forms; pick the one that matches how your data arrive, then convert to splines for computation.

Details

# Which representation should I use?

Raw samples

[BDataSamples] – one numeric sample per distribution.

Binned data

[BDataFrequencyHistograms] (counts) or [BDataDensityHistograms] (densities).

Density functions w.r.t. the Lebesgue measure

[BDataPDFs] / [BDataLogPDFs].

Density functions w.r.t. the reference measure

[BDataDensities] / [BDataLogDensities].

Spline coefficients

[BDataSplines] – the compact representation used for computation (PCA, distances, inner products).

# Conversion pipeline The 'as.BData*' converters move between representations. The usual flow is

  Samples / Histograms  ->  Densities  ->  Splines  ->  PDFs 

Converting towards densities/splines requires a basis and is an approximation (the data are projected onto the finite-dimensional spline space); extracting samples or re-expressing a representation as itself is lossless. For most analyses, convert to [BDataSplines] with [as.BDataSplines].

See Also

[BayesSpace], [BayesSpaceBasis], [ReferenceMeasure], [as.BDataSplines], [as.BDataDensities]

Description

The dataset is represented by a list of functions, associated with a Bayes space. The basis is used for functions relying on projections, e.g. when the data is converted to a BDataSpline object.

Usage

BDataDensities(f, basis, ...)

BDataPDFs(f, basis, ...)

## S3 method for class 'list'
BDataLogDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataLogPDFs(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class ''function''
BDataDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class ''function''
BDataPDFs(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataDensities(
  f,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'list'
BDataPDFs(f, basis, ..., close = FALSE, closed = FALSE, centered = FALSE)

Arguments

Details

BDataDensities represent densities (aka as Radon-Nikodym derivatives) with repect to the reference measure of the BayesSpace.

BDataPDFs for (Probablity Density function) represents densities with respect to the Lebesgue measure (Haar measure) of the support. I.e. the measure that is invariant under shift or permutation of the space. BDataPDFs are only reasonable if the reference measure is absolutely continuous, i.e. itself has a finite Lebesgue density.

For the special case of uniform reference measures BDataDensities and BDataPDFs of the same function, represent the same measure.

Value

An object of class 'c("BDataDensities","BData")' representing the densities with respect to the reference measure.

Methods (by class)

• BDataDensities(`function`): Method for a single function

• BDataDensities(list): Method for a list of functions

Functions

• BDataPDFs(): Object representing Lebesgue densities

• BDataLogDensities(list): Method for list of functions interpreted as log densities.

• BDataLogPDFs(list): Method for list of functions interpreted as log probability density functions (PDFs).

• BDataPDFs(`function`): Method for a single function

• BDataPDFs(list): Method for a list of functions

See Also

[BayesSpaceBasis]

Other BData: BDataDensityHistograms(), BDataLogDensities()

Examples

BDataDensities(list(function(x) x, function(x) 1-x),B2Unif01Basis5)

Description

Simple S3 constructor of a data container for distributional data where each distribution is represented as a density histogram.

Usage

BDataDensityHistograms(matrixDensities, breaks, basis)

Arguments

Value

An object of class 'c("BDataDensityHistograms","BDataHistograms","BData")', with the matrix densities as main value and 'breaks' and 'basis' as attributes.

See Also

Other BData: BDataDensities(), BDataLogDensities()

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(10,hist(Filter(inRange,rnorm(1000)),breaks=breaks,plot=FALSE)$density))
hists = BDataDensityHistograms(X,breaks=breaks,basis=B2N01Basis5)
plot(hists)

Description

Simple S3 constructor of a data container for distributional data where each distribution is represented as a (absolute) frequency histogram.

Usage

BDataFrequencyHistograms(matrixCounts, breaks, basis)

Arguments

Value

An object of class 'c("BDataFrequencyHistograms","BDataHistograms","BData")', with the matrix counts as main value and 'breaks' and 'basis' as attributres.

See Also

Other BDataHistograms: as.BDataDensityHistograms(), as.BDataDensityHistograms.list(), as.BDataFrequencyHistograms.list(), getBreaks(), plot.BDataHistograms()

Examples

set.seed(47)
breaks = seq(-3,3,by=1)
inRange = function(x) -3<x & x<3
X = t(replicate(10,hist(Filter(inRange,rnorm(1000)),breaks=breaks)$counts))
hists = BDataFrequencyHistograms(X,breaks=breaks,basis=B2N01Basis5)
plot(hists)

Description

A BDataGeneral object is essentially a list of different representations of BayesSpace elements of the same BayesSpace. They are treated individually as long as possible.

Usage

BDataGeneral(
  data,
  basis,
  ...,
  close = FALSE,
  closed = isClosed(data),
  centered = isCentered(data)
)

Arguments

Value

An object of class 'c("BDataGeneral","BData")': a list bundling the several distributional representations for a common Bayes space.

Examples

X = BDataGeneral(list(sin,hist(runif(100)),BDataSamples(list(runif(100)),basis=B2Unif01Basis7)),
             basis=B2Unif01Basis5)
Xs= as.BDataSplines(X)

Description

The dataset is represented by a list of functions, associated with a Bayes space. The basis is used for functions relying on projections, e.g. when the data is converted to a BDataSpline object

Usage

BDataLogDensities(f, basis, ...)

BDataLogPDFs(f, basis, ...)

## S3 method for class 'BDataLogDensities'
BDataDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataLogDensities'
BDataDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataDensities'
BDataLogDensities(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataLogPDFs'
BDataPDFs(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

## S3 method for class 'BDataPDFs'
BDataLogPDFs(
  f,
  basis = getBayesSpaceBasis(f),
  ...,
  close = FALSE,
  closed = isClosed(f),
  centered = isCentered(f)
)

Arguments

Value

An object of class 'c("BDataLogDensities","BData")' representing the log-densities with respect to the reference measure.

Methods (by class)

• BDataLogDensities(BDataDensities): conversion method

Methods (by generic)

• BDataDensities(BDataLogDensities): Method for a list of functions

• BDataDensities(BDataLogDensities): conversion method

Functions

• BDataLogPDFs(): Object representing Lebesgue densities

• BDataPDFs(BDataLogPDFs): conversion method

• BDataLogPDFs(BDataPDFs): conversion method

See Also

[BayesSpaceBasis]

Other BData: BDataDensities(), BDataDensityHistograms()

Examples

BDataDensities(list(function(x) x, function(x) 1-x),B2Unif01Basis5)

Description

Constructs a BData object where each distribution is represented as a sample. Such a sample can be a vector of values (for univariate cases) or some data structure convertible to a "data.frame" (for multivariate cases).

Usage

BDataSamples(datalist, basis)

## S3 method for class 'BDataUnivariateSamples'
summary(object, ...)

## S3 method for class 'BDataSamples'
summary(object, ...)

## S3 method for class 'BDataUnivariateSamples'
x[...]

## S3 method for class 'BDataSamples'
x[i, j]

Arguments

Value

In multivariate cases, the outcome is an object of class 'c("BDataSamples","BData")' encoding a list of data.frames of the same number of columns. In univariate cases, the outcome is an object of class 'c("BDataUnivariateSamples","BDataSamples","BData")' encoding a list of vectors of samples of each individual distribution.

Methods (by generic)

• summary(BDataSamples): summary method for multivariate cases

• [: subsetting method for multivariate samples

Functions

• summary(BDataUnivariateSamples): summary method for univariate cases

• [: subsetting method for univariate samples

Examples

x = lapply(rep(1000,4), runif) |> BDataSamples(basis=B2Unif01Basis5)
summary(x)

Description

create a 'BDataSplines' object: a set of distributions represented as splines; possible representation of individual distribution through a vector of coefficients.

Usage

BDataSplines(matrixCoefs, basis)

## S3 method for class 'BDataSplines'
x[i, j, ...]

Arguments

Value

object of class 'BDataSplines' describing a set of distributions represented as splines

Methods (by generic)

• [: subsetting method, keeps structure if only i is given.

See Also

BData

Other BDataSplines: plot.BDataSplines()

Examples

coefs=matrix(1:15,nrow=3)
bssplines=BDataSplines(coefs,B2Unif01Basis5)
bssplines

Description

Creates a biplot that simultaneously displays the principal component scores (observations) and loadings (variables/coefficients) in the same plot. This visualization helps understand relationships between observations and the underlying coefficient structure.

Usage

## S3 method for class 'princomp.BData'
biplot(
  x,
  choices = 1:2,
  scale = 0,
  pc.biplot = FALSE,
  main = NULL,
  col = c("black", "red"),
  cex = c(0.8, 0.8),
  xlabs = NULL,
  ylabs = NULL,
  expand = 1,
  arrow.len = 0.1,
  ...
)

Arguments

Details

## Biplot Interpretation

In a biplot:

• Observations (numbers/text) represent individual distributions. Observations close together have similar distribution shapes.

• Loading arrows (red arrows labeled C1, C2, etc.) represent spline basis coefficients. The direction and length indicate how each coefficient contributes to the principal components.

• Projecting an observation onto a loading arrow approximates that observation's value on that coefficient (relative to the mean).

• Arrow angles: Arrows pointing in similar directions indicate positively correlated coefficients; opposite directions indicate negative correlation; perpendicular arrows indicate uncorrelated coefficients.

## Note on Coefficient Interpretation

Unlike PCA on raw data where variables have direct meaning, the "variables" here are spline basis coefficients. These are abstract and their individual interpretation may not be intuitive. For understanding how distributions vary, the plot.princomp.BData function with 'type="eigenfunctions"' provides a more interpretable visualization showing how distribution shapes change along each PC direction.

## Why Biplot Works for ZB-Splines

The biplot is meaningful for the default ZB-spline basis because ZB-splines have **local support**: each basis function is non-zero only in a specific region of the domain. This means coefficient i primarily reflects the behavior of the distribution near knot region i. When a loading arrow for coefficient Ci points strongly in a PC direction, it indicates that variation in that region of the domain is associated with that principal component.

**Important:** If orthonormal splines were used instead, the biplot would lose this interpretability because orthonormal basis functions have global support (they are non-zero across the entire domain), so each coefficient would mix information from everywhere rather than representing a local region.

Value

Invisibly returns a list containing the scaled 'scores' and 'loadings' matrices as plotted.

Author(s)

Matthias Templ

References

Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453-467.

See Also

princomp.BData for performing the PCA.

plot.princomp.BData for other visualization options, especially 'type="loadings"' for functional interpretation.

biplot for the generic biplot function.

Other PCA: plot.princomp.BData(), plot.princomp.BData.grouped(), plot.princomp.BData.multivariate(), predict.princomp.BData(), princomp.BData(), screeplot.princomp.BData(), summary.princomp.BData()

Examples

## Create example PCA result
meanDist <- BDataPDFs(list(dunif), basis = B2Unif01Basis7)
gaussProc <- B2GaussianDistribution(meanDist, function(x,y) exp(-4*abs(x-y)^1.7))
pcaResult <- princomp(as.BDataSplines(simulate(gaussProc, 25, seed = 42)))

## Basic biplot
biplot(pcaResult)

## Biplot of PC2 vs PC3
biplot(pcaResult, choices = c(2, 3))

## Correlation biplot (scale=1)
biplot(pcaResult, scale = 1)

## Customized biplot
biplot(pcaResult,
       col = c("blue", "darkgreen"),
       cex = c(1, 0.7),
       expand = 1.5,
       main = "PCA Biplot of Distributional Data")

Description

Currently this function used the [fda::fbplot] function on the evaluated functions in compbination with [depthRank] as depth measure.

Usage

## S3 method for class 'BData'
boxplot(
  x,
  ...,
  depth,
  depthType = c("depthquantile", "spatialdepth", "meandistance", "median"),
  support = getSupport(x),
  SD = supportDiscretization(support, nEval),
  nEval = 513,
  xlim = getSupport1DPlottingLimits(support)
)

## S3 method for class 'BDataSplines'
boxplot(x, ...)

## S3 method for class 'BDataSamples'
boxplot(x, ...)

Arguments

Value

'NULL' invisibly, called for the side effect of producing a functional boxplot.

Functions

• boxplot(BDataSplines): method for BDataSplines uses classical boxplot for the coefficients.

• boxplot(BDataSamples): method for BDataSamples uses classical boxplot for the the datasets.

See Also

fda::fbplot

Examples

X = BDataSplines(matrix(rnorm(10*5),ncol=5),basis=B2Unif01Basis5)
boxplot(X)
boxplot(clr(X),depthType="meandistance")
boxplot(clr(X))

Description

Compute the cartesian product

Usage

cartesianProduct(...)

## S3 method for class 'Support'
cartesianProduct(...)

## S3 method for class 'ReferenceMeasure'
cartesianProduct(...)

## S3 method for class 'BayesSpace'
cartesianProduct(...)

## S3 method for class 'BayesSpaceBasis'
cartesianProduct(...)

Arguments

Value

the Cartesian product of the objects given. Check [SupportCartesianProduct()] for details about this structure.

The cartesian product of a space (like a vector space or support) is the space of tuples, where each tuple element comes from the space of the same position or name. The cartesian product of measures is the product measure, which gives a cartesian product of sets the product of the measures. The cartesian product of functions(like basis) is a function on the cartesian product of supports and gives the product of the function values for each of the components.

Methods (by class)

• cartesianProduct(Support): method for Supports

• cartesianProduct(ReferenceMeasure): method for ReferenceMeasure

• cartesianProduct(BayesSpace): method for BayesSpace

• cartesianProduct(BayesSpaceBasis): method for BayesSpaceBasis

See Also

Other Cartesian: BayesSpaceBasis.BayesSpaceCartesianProduct(), BayesSpaceBasisCartesianProduct(), BayesSpaceCartesianProduct(), ReferenceMeasureCartesianProduct(), SupportCartesianProduct(), getBayesSpace.ReferenceMeasureCartesianProduct(), getCartesianFactors(), subBasisIndices()

Description

Ideally this should always be called on the original samples.

Usage

caseE(x, X, ...)

## S3 method for class 'BDataSamples'
caseE(x, X, ...)

## S3 method for class 'BData'
caseE(x, X, ...)

Arguments

Value

a vector, matrix or with the Expected values for each element of x. if the expected value is multivariate the last dimension iterates over the cases in x.

Methods (by class)

• caseE(BDataSamples): method for BDataSamples simply uses means

• caseE(BData): method for distributions

See Also

Other caseStatistics: caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseE(X,X=function(x) x,probs=c(0.25,0.75))
caseE(X,X=function(x) x^2,probs=0.5)

Description

Density log ratios independent of the reference measure and any other values.

Usage

caseLogRatios(x, numerator, denominator, ...)

Arguments

Value

a dataset of the same number of cases as x of the corresponding log rations

See Also

Other caseStatistics: caseE(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X  = BDataPDFs(DTD$shapePDF,basis=B2Unif01Basis5)
caseLogRatios(X,0.8,0.9)

Description

Ideally this should always be called on the original samples.

Usage

caseMeans(x, ...)

Arguments

Value

a vector or a matrix (with different cases in columns) of the means of the distributions in the dataset. For multivariate (vector-valued) means, the last dimension iterates over the elements of x.

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseQuantiles(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseMeans(X)
X = BDataSamples(DTD$data,basis=B2Unif01Basis5)
caseMeans(X)

Description

Ideally this should always be called on the original samples.

Usage

caseQuantiles(X, ...)

## S3 method for class 'BDataSamples'
caseQuantiles(X, ...)

## S3 method for class 'BData'
caseQuantiles(X, ...)

Arguments

Value

a vector or a matrix (with different probabilities in colums) of quantiles of the distributions in the dataset.

Methods (by class)

• caseQuantiles(BDataSamples): method for BDataSamples, simply calles [stats::quantile]

• caseQuantiles(BData): method for BData not yet implemented

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseSds(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseQuantiles(X,probs=c(0.25,0.75))
caseQuantiles(X,probs=0.5)

Description

Ideally this should always be called on the original samples.

Usage

caseSds(x, ...)

Arguments

Value

a vector the standard deviations of the distributions in the dataset.

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseVars(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseSds(X)

Description

Ideally this should always be called on the original samples.

Usage

caseVars(x, ..., X = function(x) x)

## S3 method for class 'BDataSamples'
caseVars(x, ..., X = function(x) x)

## S3 method for class 'BData'
caseVars(x, ..., X = function(x) x)

Arguments

Value

A vector or tensor of variances. In case of multiple dimensional means, the last dimension iterates over the elements of x.

Methods (by class)

• caseVars(BDataSamples): method for samples

• caseVars(BData): method for distributions

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), centralityQuantile(), distanceToCenter(), median.BData(), spatialDepth()

Examples

data(DTD)
X = BDataSamples(DTD$data %.% shape,basis=B2Unif01Basis5)
caseVars(X)
X = BDataSamples(DTD$data,basis=B2Unif01Basis5)
caseVars(X)

Description

The function tries to keep the (open) support, by using a one sided derivative at limits of the support, while using a two sided in the middle of the support.

Usage

cdf2density(cdf, ..., .args = list(...), eps = 1e-07, cdf2 = NULL)

Arguments

Value

a vectorized function giving the density obtained by numerically differentiating 'cdf' (a one-sided difference at the boundaries of the support, a two-sided difference in the interior). If 'cdf2' is supplied the result is the density relative to that reference measure instead of the Lebesgue density.

Examples

dUnif = cdf2density(punif,min=-0.2,0.4)
x=seq(-1,2,by=0.01)
plot(x,dUnif(x))

Description

This function essentially tries to infere the support from a cdf.

Usage

cdf2Support(cdf, alpha = 0.01, eps = 1e-07, digits = 5)

Arguments

Value

a 'Support' object describing the support inferred from the 'cdf'. Its 'a' and 'b' elements give the (possibly infinite) lower and upper support boundaries, its 'lim' element the effective lower and upper boundaries (a numeric vector of length two, cut at probability 'alpha/2' and '1 - alpha/2' when the support is unbounded), and 'X' any detected points of positive probability mass.

See Also

Other Support: Support(), pdf2Support(), plotFuncs(), sample2Support()

Examples

cdf2Support(pnorm)
cdf2Support(punif)
cdf2Support(
  function(x){
  ifelse(x< -pi,0,((pmax(x,0)+pi)/pi)*0.5) +
      ifelse(x<0,0,0.25+(pmax(x,pi)/pi)*0.25)
  }
)

Description

This centers an representation in a way that a direct converstion to a log Density has integral 0.

Usage

cen(X, ..., center = TRUE)

## S3 method for class ''function''
cen(X, measure = getReferenceMeasure(X), ..., center)

## S3 method for class 'BDataDensities'
cen(X, ..., center)

## S3 method for class 'BDataPDFs'
cen(X, ..., center)

## S3 method for class 'BDataLogDensities'
cen(X, ..., center)

## S3 method for class 'BDataLogPDFs'
cen(X, ..., center)

## S3 method for class 'BayesSpaceBasis'
cen(X, ..., center)

Arguments

Details

Centring subtracts the integral mean (with respect to the reference measure) so that the result integrates to zero; equivalently it makes the clr representative of the distribution centred.

Value

and object of the same type as the incoming, but closed.

Methods (by class)

• cen(`function`): method for functions

• cen(BDataDensities): method for BDataDensities

• cen(BDataPDFs): method for BDataPDFs

• cen(BDataLogDensities): method for BDataLogDensities

• cen(BDataLogPDFs): method for BDataLogPDFs

• cen(BayesSpaceBasis): method for BayesSpaceBasis returns the centered logDensities

See Also

[clo]

Examples

cen(function(x) x^2,measure=ReferenceMeasureUniform(0,1))

Description

A centering constant is a real value, by which needs to be substracted from a function (of its log in case of densities) get an integral of 0.

Usage

centeringConstant(f, ...)

## S3 method for class ''function''
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BayesSpaceBasis'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataDensities'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogDensities'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogPDFs'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataPDFs'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataSplines'
centeringConstant(f, measure = getReferenceMeasure(f), ...)

Arguments

Value

a (vector) of numeric centering constants. Might contain Inf in case of infinite measures and 0 in case of functions that are constant zeros

Methods (by class)

• centeringConstant(`function`): method for function

• centeringConstant(BayesSpaceBasis): method for BayesSpaceBasis

• centeringConstant(BDataDensities): method for BDataDensities

• centeringConstant(BDataLogDensities): method for BDataLogDensities

• centeringConstant(BDataLogPDFs): method for BDataLogPDFs

• centeringConstant(BDataPDFs): method for BDataPDFs

• centeringConstant(BDataSplines): method for BDataSplines

Examples

# centering constant of f(x) = x^2 on Uniform(0,1)
centeringConstant(function(x) x^2, measure = ReferenceMeasureUniform(0,1))

Description

Computes quantile of distanceToCenter of data in a dataset, 0 is the median, 1 is the farthest point

Usage

centralityQuantile(x, ...)

## Default S3 method:
centralityQuantile(
  x,
  ...,
  d = distanceToCenter(x, ..., distType = distType),
  distType = c("meandistance", "median", "spatialdepth"),
  ties.method = "first"
)

centralityRank(x, ...)

## Default S3 method:
centralityRank(
  x,
  ...,
  d = distanceToCenter(x, ..., distType = distType),
  distType = c("meandistance", "median", "spatialdepth"),
  ties.method = "first"
)

Arguments

Value

A numeric vector in [0,1], 0 for the most central point and 1 for the most peripheral.

Methods (by class)

• centralityQuantile(default): default method

Functions

• centralityRank(): returns the rank rather than the quantile: an integer vector of ranks, 1 for the most central point up to the number of observations for the most peripheral.

• centralityRank(default): default method returning ranks

See Also

Other caseStatistics: caseE(), caseLogRatios(), caseMeans(), caseQuantiles(), caseSds(), caseVars(), distanceToCenter(), median.BData(), spatialDepth()

Examples

x = rnorm(100)
plot(x,centralityQuantile(x))
plot(x,centralityRank(x))

Description

This is a generic version of clo overwriting the one in compositions

Usage

clo(X, ..., close = TRUE)

## Default S3 method:
clo(X, ..., total = 1, close)

## S3 method for class ''function''
clo(X, measure = getReferenceMeasure(X), ..., total = 1, close)

## S3 method for class 'BDataDensities'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataPDFs'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataLogDensities'
clo(X, ..., total = 1, close)

## S3 method for class 'BDataLogPDFs'
clo(X, ..., total = 1, close)

## S3 method for class 'BayesSpaceBasis'
clo(X, ..., total = 1)

Arguments

Details

Closure rescales a (positive) function so that it integrates to 'total' (one by default) with respect to the reference measure, i.e. it projects the function to a proper density and thereby picks the representative of its Bayes-space class.

Value

and object of the same type as the incoming, but closed.

Methods (by class)

• clo(default): the default is [compositions::clo]

• clo(`function`): method for functions

• clo(BDataDensities): method for BDataDensities

• clo(BDataPDFs): method for BDataPDFs

• clo(BDataLogDensities): method for BDataLogDensities

• clo(BDataLogPDFs): method for BDataLogPDFs

• clo(BayesSpaceBasis): method for BayesSpaceBasis returns the closed logDensities

See Also

[compositions::clo] for parameters of the default, [cen]

Examples

clo(function(x) x^2,measure=ReferenceMeasureUniform(0,1))

Description

Closure functions to enforce a distributional representation to integrate to one, if needed.

Usage

## S3 method for class 'BDataHistograms'
clo(X, ...)

Arguments

Value

A fitting object with elements now integrating to one, depending on the specific method.

Examples

# close frequency histograms to density histograms integrating to one
counts = rbind(c(10,20,30,40), c(25,25,25,25))
fh = BDataFrequencyHistograms(counts, breaks = seq(0,1,by=0.25),
                              basis = B2Unif01Basis5)
clo(fh)

Description

A closing constant is a real value, by which the function (or its Exp in case of log functions) needs to be divided to get the representative with measure 1.

Usage

closingConstant(f, ...)

## S3 method for class ''function''
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BayesSpaceBasis'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataDensities'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogDensities'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataLogPDFs'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataPDFs'
closingConstant(f, measure = getReferenceMeasure(f), ...)

## S3 method for class 'BDataSplines'
closingConstant(f, measure = getReferenceMeasure(f), ...)

Arguments

Value

a (vector) of numeric closing constants. Might contain Inf in case of infinite measures and 0 in case of functions that are constant zeros

Methods (by class)

• closingConstant(`function`): method for function

• closingConstant(BayesSpaceBasis): method for BayesSpaceBasis

• closingConstant(BDataDensities): method for BDataDensities

• closingConstant(BDataLogDensities): method for BDataLogDensities

• closingConstant(BDataLogPDFs): method for BDataLogPDFs

• closingConstant(BDataPDFs): method for BDataPDFs

• closingConstant(BDataSplines): method for BDataSplines

Examples

# closing constant of f(x) = x^2 on Uniform(0,1)
closingConstant(function(x) x^2, measure = ReferenceMeasureUniform(0,1))

Description

This is a generic version of clr overwriting the one in compositions

Usage

clr(x, ...)

## Default S3 method:
clr(x, ...)

## S3 method for class 'BData'
clr(x, ...)

Arguments

Details

The centred log-ratio map sends a distribution to its log-density centred to integrate to zero, i.e. to its coordinates in the L2 space underlying the Bayes space. It is an isometry from the Bayes space onto the centred subspace of that L2 space, so Bayes-space distances and inner products become the ordinary L2 ones.

Value

the centred log-ratio (clr) representation of 'x'. For 'BData' objects this is the centred log-density representation (an object of class 'BDataLogDensities' whose log-density integrates to zero); the default method returns the clr transform computed by [compositions::clr].

Methods (by class)

• clr(default): the default is [compositions::clr]

• clr(BData): returns a centered representation as BDataLogDensities

Examples

X1 = BDataPDFs(basis=B2Unif01Basis5,
   list(
    \(x)(dbeta(x,1,2)),
    \(x)(dbeta(x,1,1)),
    \(x)(dbeta(x,1.5,0.7))
   ) )
plot(X1,main="Beta Densities on Unif(0,1)")
plot(clr(X1),main="CLRs of Beta Densities on Unif(0,1)")

Description

This computes a (n-1)x1 matrix mapping the basis results of the L2 basis to L^2_0 basis for a subspace of the Log-Space of a BayesSpace.

Usage

computeL22BayesSpacesBasisMatrix(x, ...)

Arguments

Value

the transformation matrix 'A' mapping the L2 (B-spline) basis to the ZB-spline basis of the Bayes space. For a B-spline basis with 'nbasis' functions this is an '(nbasis - 1) x nbasis' matrix, i.e. one row per ZB-spline basis function.

Description

create the transform matrix A from a B-spline basis (fda::create.bspline.basis())

Usage

## S3 method for class 'basisfd'
computeL22BayesSpacesBasisMatrix(x, ...)

Arguments

Value

matrix

Examples

computeL22BayesSpacesBasisMatrix(
         fda::create.bspline.basis(rangeval = c(0,1),
                                  nbasis=9,
                                  norder=4,
                                  breaks=seq(0,1,length.out=7)
                                  )
                                )
# a cubic B-spline basis with 8 functions yields a 7 x 8 matrix
A = computeL22BayesSpacesBasisMatrix(
        fda::create.bspline.basis(c(0,1), nbasis=8, norder=4)
    )
dim(A)

Description

computes the limit of a function

Usage

computeLimit(f, x, d = 1)

Arguments

Value

the limit it will still return

Examples

computeLimit(\(x)(2*(x-pi)/(x-pi)),x=pi)
np = computeLimit(\(x)(pi*(x-pi)/(x-pi)),x=pi)
np-pi

Description

The implementation depends on the measure

Usage

CovarianceOperator(measure, X, Y, ...)

## S3 method for class 'ReferenceMeasure'
CovarianceOperator(
  measure,
  X,
  Y,
  ...,
  method = c("default", "sample", "MonteCarlo", "integration"),
  tol = 0.01,
  close = FALSE
)

## S3 method for class 'ReferenceMeasureDiscrete'
CovarianceOperator(measure, X, Y, ...)

Arguments

Value

the scalar products (covariances) of 'X' and 'Y' with respect to the reference measure. For the default ReferenceMeasure method these are obtained from [stats::cov()] applied to ‘X' and 'Y' evaluated on the measure’s sample. The result is a numeric scalar, vector, or matrix depending on the number of columns returned by 'X' and 'Y' (one entry per pair of output components); when 'X' or 'Y' is a list, the results are combined columnwise.

Methods (by class)

• CovarianceOperator(ReferenceMeasure): method for ReferenceMeasures using a MonteCarlo integration based on the sample provided by getSample(sample)

• CovarianceOperator(ReferenceMeasureDiscrete): method for ReferenceMeasuresDiscrete explicit computation

Examples

CovarianceOperator(StdReferenceMeasureUnif01,
                   function(x) cbind(x,-x),
                   function(x) cbind(x,-x,2*x+1)
)

Description

A simple fast tool for defining a discretized function defined continuously on [a,b] and discrete values in X, with equidistant data values y in between. The resulting function is vectorized and interpolates linearily with errors or NA outside the region. It handles infinite supports, but not infinite results on the continuous region.

Usage

DiscretizedFunction(y, ...)

## S3 method for class 'numeric'
DiscretizedFunction(y, ..., support = Support(...), transferCdf = NULL)

## S3 method for class ''function''
DiscretizedFunction(
  y,
  a = 0,
  b = a,
  n = 501,
  X = c(),
  transferCdf = NULL,
  transferQf = cdf2qf(transferCdf, ...),
  ...
)

## S3 method for class 'Discretizer'
DiscretizedFunction(y, Y, ...)

## S3 method for class 'ReferenceMeasureTransformed'
DiscretizedFunction(y, f, n = 513, support = getSupport(y), ...)

## S3 method for class 'ReferenceMeasureUniform'
DiscretizedFunction(y, f, n = 513, support = getSupport(y), ...)