Package 'coda.base'

Description

The 'alimentation' data set contains the percentage composition of food consumption in 25 European countries during the 1980s. The food categories are:

• 'RM': red meat (pork, veal, beef),

• 'WM': white meat (chicken),

• 'E': eggs,

• 'M': milk,

• 'F': fish,

• 'C': cereals,

• 'S': starch (potatoes),

• 'N': nuts,

• 'FV': fruits and vegetables.

The data set also contains categorical variables indicating whether the country belongs to the North or South/Mediterranean group, and whether it is an Eastern or Western European country.

Usage

alimentation

Format

An object of class data.frame with 25 rows and 13 columns.

Description

Construct the transformation matrix associated with additive log-ratio (alr) coordinates.

Usage

alr_basis(dim, denominator = NULL, numerator = NULL)

Arguments

Value

A matrix defining the alr coordinate system.

References

Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman & Hall, London.

Examples

alr_basis(5)
alr_basis(5, 3)
alr_basis(5, 3, c(1, 5, 2, 4))

Description

The 'arctic_lake' data set records the three-part composition $[sand, silt, clay]$ of 39 sediment samples collected at different water depths in an Arctic lake.

Usage

arctic_lake

Format

An object of class data.frame with 39 rows and 5 columns.

Description

In humans, the main blood group systems are the ABO system, the Rh system, and the MN system. The MN blood system is related to proteins of the red blood cell plasma membrane. Its inheritance pattern is autosomal with codominance, meaning that the heterozygous phenotype is distinct from both homozygous phenotypes.

The three phenotypes are M, N, and MN. Their frequencies vary across populations. Under the Hardy-Weinberg principle, allele and genotype frequencies remain constant across generations in the absence of evolutionary forces, implying that

$\frac{x_{MM} x_{NN}}{x_{MN}^2} = \frac{1}{4}$

where $x_{MM}$ and $x_{NN}$ are the genotype frequencies of the homozygotes and $x_{MN}$ is the genotype frequency of heterozygotes.

Usage

blood_mn

Format

An object of class data.frame with 49 rows and 5 columns.

Description

The 'bmi_activity' data set records the proportion of daily time spent in sleep ('sleep'), sedentary behaviour ('sedent'), light physical activity ('Lpa'), moderate physical activity ('Mpa'), and vigorous physical activity ('Vpa') for 393 children. The standardized body mass index ('zBMI') of each child is also included.

This data set was used in the example of Dumuid et al. (2019) to examine the expected differences in zBMI associated with reallocations of daily time between sleep, sedentary behaviour, and physical activity. Because the original data are confidential, 'bmi_activity' contains simulated data that mimic the main features of the original study.

Usage

bmi_activity

Format

An object of class data.frame with 393 rows and 8 columns.

References

Dumuid, D., Pedisic, Z., Stanford, T. E., Martín-Fernández, J. A., Hron, K., Maher, C., Lewis, L. K., & Olds, T. S. (2019). The Compositional Isotemporal Substitution Model: a Method for Estimating Changes in a Health Outcome for Reallocation of Time between Sleep, Sedentary Behaviour, and Physical Activity. Statistical Methods in Medical Research, 28(3), 846–857.

Description

Construct an ilr basis rotated according to canonical correlations between a compositional response data set and an explanatory data set.

Usage

cc_basis(Y, X)

Arguments

Value

A matrix whose columns define a canonical-correlation-oriented ilr basis.

Description

Construct the default isometric log-ratio basis used in CoDaPack.

Usage

cdp_basis(dim)

Arguments

Value

A matrix with $D$ rows and $D - 1$ columns containing the CoDaPack default ilr basis.

Examples

cdp_basis(5)
cdp_basis(c("a", "b", "c", "d"))

Description

Compute the default binary partition used in CoDaPack's software

Usage

cdp_partition(ncomp)

Arguments

Value

matrix

Examples

cdp_partition(4)

Description

Generic function to calculate the center of a compositional dataset

Usage

center(X, zero.rm = FALSE, na.rm = FALSE)

Arguments

Examples

X = matrix(exp(rnorm(5*100)), nrow=100, ncol=5)
g = rep(c('a','b','c','d'), 25)
center(X)
(by_g <- by(X, g, center))
center(t(simplify2array(by_g)))

Description

Applies the closure operation to a numeric vector, matrix or data frame so that each composition sums to a prescribed constant k.

Usage

closure(X, k = 1)

Arguments

Details

If X is:

• a vector, the returned vector sums to k;

• a matrix or data frame, closure is applied row-wise, and each row sums to the corresponding value of k.

The argument k may be:

• a single positive number, recycled to all rows;

• a numeric vector of length nrow(X), specifying a different closure constant for each row.

For a composition $x = (x_1, \dots, x_D)$ with positive sum, the closure to constant $k$ is

$C(x) = k \frac{x}{\sum_{j=1}^D x_j}.$

This function requires all entries of X to be finite and non-negative, and every row sum (or the vector sum) must be strictly positive.

Value

If X is a vector, a numeric vector of the same length.

If X is a matrix, a numeric matrix with the same dimensions, dimnames, and row-wise sums equal to k.

If X is a data frame, a data frame with the same row and column names, and row-wise sums equal to k.

Examples

closure(c(2, 3, 5))
closure(c(2, 3, 5), k = 100)

X <- matrix(c(1, 1, 2,
              2, 3, 5), nrow = 2, byrow = TRUE)
closure(X)
closure(X, k = c(1, 100))

df <- data.frame(a = c(1, 2), b = c(1, 3), c = c(2, 5))
closure(df, k = 10)

Description

Construct the transformation matrix associated with centered log-ratio (clr) coordinates.

Usage

clr_basis(dim)

Arguments

Details

CLR coordinates are linearly dependent and lie in the $D - 1$ dimensional clr-plane.

Value

A square matrix defining the clr coordinate system.

References

Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman & Hall, London.

Examples

B <- clr_basis(5)
clr_coordinates <- coordinates(c(1, 2, 3, 4, 5), B)
sum(clr_coordinates) < 1e-15

Description

Performs imputation of missing values and/or values below the detection limit in compositional data using an EM algorithm assuming normality on the simplex.

Usage

coda_replacement(
  X,
  DL = NULL,
  dl_prop = 0.65,
  eps = 1e-04,
  parameters = FALSE,
  debug = FALSE,
  maxit = 500
)

Arguments

Value

If 'parameters = FALSE', the imputed object with the same format as 'X' ('matrix' or 'data.frame', preserving data-frame subclasses when possible) and preserving original names. If 'parameters = TRUE', a list with the estimated clr mean, clr covariance, and imputed clr coordinates.

Examples

X <- matrix(c(
  0.00, 0.30, 0.70,
  0.20,   NA, 0.80,
  0.40, 0.60, 0.00,
  0.25, 0.25, 0.50,
  0.10, 0.30, 0.60
), ncol = 3, byrow = TRUE)
colnames(X) <- c("sand", "silt", "clay")

DL <- c(0.05, 0.05, 0.05)

X_imp <- coda_replacement(X, DL = DL, maxit = 20)
X_imp

set.seed(10)
X <- composition(matrix(rnorm(3*10), ncol = 3))
X[sample(c(TRUE, FALSE), 4*10, replace = TRUE, c(1, 3))] <- NA
params <- coda_replacement(X, parameters = TRUE, debug = TRUE)
names(params)
params$clr_mu
composition(params$clr_h)

Description

A minimum set of functions to perform compositional data analysis using the log-ratio approach introduced by John Aitchison (1982) <https://www.jstor.org/stable/2345821>. Main functions have been implemented in c++ for better performance.

Author(s)

Marc Comas-Cufí

See Also

Useful links:

• https://mcomas.net/coda.base/

• https://github.com/mcomas/coda.base

Description

Reconstruct a composition from coordinates with respect to a given basis.

Usage

composition(H, basis = "ilr")

comp(H, basis = "ilr")

Arguments

Value

A composition corresponding to the given coordinates.

See Also

coordinates, ilr_basis, alr_basis, clr_basis, sbp_basis

Description

Compute orthonormal ilr bases adapted to row-wise conditioning patterns.

Usage

conditional_obasis(X, scheme = c("zero", "zero_na"))

Arguments

Details

Each row of 'X' defines one conditioning pattern on the parts of a composition. According to 'scheme', the parts are split into ordered blocks:

• '"zero"': parts equal to '0' and parts with strictly positive values,

• '"zero_na"': missing values ('NA'), zeros, and strictly positive values.

For each row, the function constructs an orthonormal basis of the clr-plane preserving the block structure induced by the selected scheme.

Under 'scheme = "zero"', if a row contains 'nz' zeros, then:

• the first 'nz - 1' coordinates describe the internal log-ratio structure of the zero block,

• the coordinate 'nz' describes the balance between the zero block and the positive block,

• the remaining coordinates describe the internal log-ratio structure of the positive block.

Under 'scheme = "zero_na"', the blocks are ordered as:

• missing values ('NA'),

• zeros,

• strictly positive values.

In this case:

• the first coordinates describe the internal structure of the 'NA' block,

• the next coordinate contrasts the 'NA' block with the positive block,

• the following coordinates describe the internal structure of the zero block,

• the next coordinate contrasts the zero block with the positive block,

• the remaining coordinates describe the internal structure of the positive block.

Value

A three-dimensional array of dimension '(D - 1, D, nrow(X))', where 'D' is the number of parts. Each slice contains one orthonormal ilr basis.

Examples

C <- rbind(
  c(0, 0, 1, 1, 0),
  c(0, 1, 0, 1, 0)
)

conditional_obasis(C)

X <- rbind(
  c(1, NA, 0, 2),
  c(NA, 3, 0, 4),
  c(1, 2, 3, 4)
)

conditional_obasis(X, scheme = "zero_na")

Description

Compute coordinates of a composition or a compositional data set with respect to a given log-ratio basis.

The 'basis' argument can be either:

• a character string identifying a predefined coordinate system, or

• a matrix whose columns define a system of log-contrasts.

The predefined options are:

• '"ilr"': isometric log-ratio coordinates,

• '"olr"': orthonormal log-ratio coordinates,

• '"clr"': centered log-ratio coordinates,

• '"alr"': additive log-ratio coordinates,

• '"pw"': pairwise log-ratios,

• '"pc"': principal component log-ratio coordinates,

• '"pb"': principal balance coordinates,

• '"cdp"': CoDaPack default balances.

Usage

coordinates(X, basis = "ilr")

coord(..., basis = "ilr")

alr_c(X)

clr_c(X)

ilr_c(X)

olr_c(X)

Arguments

Value

Coordinates of 'X' with respect to the given 'basis'. The returned object has the same general type as the input when possible.

See Also

ilr_basis, alr_basis, clr_basis, sbp_basis, composition

Examples

coordinates(1:5)

B <- ilr_basis(5)
coordinates(1:5, B)

X <- rbind(1:5, 2:6)
coordinates(X, "clr")

Description

Compute a distance matrix for compositional data, including the Aitchison distance as an extension of dist.

Usage

dist(x, method = "euclidean", ...)

Arguments

Value

An object of class "dist".

See Also

dist_coda, dist

Examples

X <- exp(matrix(rnorm(10 * 50), ncol = 50, nrow = 10))

(d <- dist_coda(X, method = "aitchison"))
plot(hclust(d))

# In contrast to Euclidean distance
dist(rbind(c(1, 1, 1), c(100, 100, 100)), method = "euc")

# Using Aitchison distance, only relative information is of importance
dist_coda(rbind(c(1, 1, 1), c(100, 100, 100)), method = "ait")

Description

Compute a distance matrix for compositional data using selected CoDa distances.

Usage

dist_coda(x, method = "aitchison", ...)

Arguments

Value

An object of class "dist".

References

Saperas-Riera, J.; Mateu-Figueras, G.; Martín-Fernández, J.A. (2024). Lp-Norm for Compositional Data: Exploring the CoDa L1-Norm in Penalised Regression. Mathematics, 12(9), 1388. doi:10.3390/math12091388.

See Also

dist, dist

Examples

set.seed(1)
X <- exp(matrix(rnorm(10 * 5), ncol = 5, nrow = 10))

dist_coda(X, method = "aitchison")
dist_coda(X, method = "L2")
dist_coda(X, method = "L1")
dist_coda(X, method = "L1-pw")
dist_coda(X, method = "L1-clr")

Description

According to the three-sector theory, employment shifts from the primary sector (raw material extraction), to the secondary sector (industry, energy, and construction), and then to the tertiary sector (services) as economies develop. The 'eurostat_employment' data set contains EUROSTAT data on employment, aggregated for both sexes and all ages, distributed by economic activity in 2008 for 29 EUROSTAT member countries.

A related variable is the logarithm of gross domestic product per person in EUR at current prices ('logGDP'). For exploratory purposes, it is also categorised as a binary variable indicating values above or below the median ('Binary GDP').

The employment composition has 11 parts:

• Primary sector

• Manufacturing

• Energy

• Construction

• Trade repair transport

• Hotels restaurants

• Financial intermediation

• Real estate

• Educ admin defense soc sec

• Health social work

• Other services

Usage

eurostat_employment

Format

An object of class data.frame with 29 rows and 17 columns.

Description

The 'foraminiferals' data set (Aitchison, 1986) is a classical example of paleocological compositional data. It contains the composition of four fossil types (Neogloboquadrina atlantica, Neogloboquadrina pachyderma, Globorotalia obesa, and Globigerinoides triloba) at 30 different depths.

Because the data contain rounded zeros, zero-replacement techniques are typically required before analysis. A natural goal is then to study the association between fossil composition and depth.

Usage

foraminiferals

Format

An object of class data.frame with 30 rows and 5 columns.

Description

Simulate compositional data and optionally introduce structural zeros (interpreted as values below a detection limit) and missing values.

The function first generates a compositional data set 'X0', then creates a modified version 'X' by:

• replacing values below 'dl_par' by zero, if 'zeros = TRUE',

• introducing missing values at random, if 'missings = TRUE'.

A matrix of detection limits 'DL' is also returned. It contains 'dl_par' in the positions that were censored to zero, and '0' elsewhere.

Usage

gen_coda_with_zeros_and_missings(
  n,
  d,
  missings = TRUE,
  zeros = TRUE,
  dl_par = 0.05,
  na_p = 0.15
)

Arguments

Details

Compositions are generated from multivariate normal coordinates and mapped to the simplex through 'composition()'. The eigenvector rotation is included to induce a non-trivial covariance structure in the generated coordinates.

Missing values are introduced completely at random, independently for each cell, with probability 'na_p'.

Value

A list with three components:

X

The generated compositional data set with simulated zeros and/or missing values.

DL

A matrix of detection limits, with 'dl_par' in censored positions and '0' elsewhere.

X0

The original simulated compositional data set before introducing zeros or missing values.

Examples

set.seed(123)
sim <- gen_coda_with_zeros_and_missings(100, 4)

str(sim)
summary(sim$X0)
summary(sim$X)
table(sim$X == 0, useNA = "ifany")

Description

Generic function for the (trimmed) geometric mean.

Usage

gmean(x, zero.rm = FALSE, trim = 0, na.rm = FALSE)

Arguments

See Also

center

Description

The 'house_expend' data set, obtained from Eurostat, records the composition of mean household consumption expenditure across 12 expenditure categories in 27 European Union countries. Some values are rounded zeros.

In addition, the data set contains gross domestic product values for 2005 ('GDP05') and 2014 ('GDP14'). A relevant analysis is the relationship between expenditure compositions and GDP.

Usage

house_expend

Format

An object of class data.frame with 27 rows and 15 columns.

Description

In a sample survey of single persons living alone in rented accommodation, twenty men and twenty women were randomly selected and asked to record their expenditure over one month in the following four mutually exclusive and exhaustive commodity groups:

• 'Hous': housing, including fuel and light,

• 'Food': foodstuffs, including alcohol and tobacco,

• 'Serv': services, including transport and vehicles,

• 'Other': other goods, including clothing, footwear, and durable goods.

Usage

household_budget

Format

An object of class data.frame with 40 rows and 6 columns.

Description

Construct an isometric log-ratio (ilr) basis for a composition with $D$ parts. The ilr basis is an orthonormal basis of the clr-plane and provides $D - 1$ coordinates. The same basis is sometimes referred to as an orthonormal log-ratio (olr) basis.

Usage

ilr_basis(dim, type = "default")

olr_basis(dim, type = "default")

Arguments

Details

For 'type = "default"', the function returns the standard Helmert-type ilr basis. Alternative constructions are available through 'type = "pivot"' and 'type = "cdp"'.

The default basis vectors are:

$h_i = \sqrt{\frac{i}{i+1}} \log \frac{\sqrt[i]{\prod_{j=1}^i x_j}}{x_{i+1}}, \qquad i = 1, \ldots, D - 1$

Value

A matrix with $D$ rows and $D - 1$ columns representing an orthonormal log-ratio basis.

References

Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., & Barceló-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35(3), 279–300.

Examples

ilr_basis(5)
ilr_basis(alimentation[, 1:9])
ilr_basis(c("a", "b", "c", "d"), type = "pivot")

Description

The 'kilauea_iki' data set contains the chemical composition of volcanic rocks sampled from the lava lake at Kilauea Iki (Hawaii). The data represent major oxide concentrations in fractional form.

Usage

kilauea_iki

Format

A data frame with 17 observations and 11 variables:

SiO2

Silicon dioxide

TiO2

Titanium dioxide

Al2O3

Aluminium oxide

Fe2O3

Ferric oxide

FeO

Ferrous oxide

MnO

Manganese oxide

MgO

Magnesium oxide

CaO

Calcium oxide

Na2O

Sodium oxide

K2O

Potassium oxide

P2O5

Phosphorus pentoxide

Details

The variability in oxide concentrations is attributed to magnesian olivine fractionation from a single magmatic mass, as suggested by Richter and Moore (1966).

Source

Richter, D. H., & Moore, J. G. (1966). Petrology of Kilauea Iki lava lake, Hawaii. Geological Survey Professional Paper 537-B.

Description

The 'mammals_milk' data set contains the percentages of five constituents of the milk of 24 mammals: $[W, P, F, L, A]$, where 'W' is water, 'P' is protein, 'F' is fat, 'L' is lactose, and 'A' is ash.

Usage

mammals_milk

Format

An object of class data.frame with 24 rows and 6 columns.

Description

In an attempt to improve the quality of cow milk, milk from thirty cows was assessed before and after a controlled dietary and hormonal regime over eight weeks. A control group of thirty cows kept under the usual regime was also included.

The 'milk_cows' data set provides the complete before/after milk composition data for the sixty cows, with the proportions of protein ('pr'), milk fat ('mf'), carbohydrate ('ch'), calcium ('Ca'), sodium ('Na'), and potassium ('K').

Usage

milk_cows

Format

An object of class tbl_df (inherits from tbl, data.frame) with 116 rows and 10 columns.

Description

The 'montana' data set contains 229 samples of the concentration (in ppm) of five minor elements $[Cr, Cu, Hg, U, V]$ in coal ashes from the Fort Union formation (Montana, USA), in the Powder River Basin.

The five measured elements form a fully observed subcomposition of a much larger chemical composition. Since the data are given in parts per million and all concentrations were measured, a residual component could in principle be added to close the compositions to $10^6$.

Usage

montana

Format

An object of class data.frame with 229 rows and 6 columns.

Description

Construct the system of all pairwise log-ratios between parts.

Usage

pairwise_basis(dim)

Arguments

Value

A matrix, or a sparse matrix for large dimensions, whose columns represent all pairwise log-ratio generators.

Description

The 'parliament2017' data set contains the results of the 2017 Catalan Parliament election aggregated by region.

Usage

parliament2017

Format

A data frame with 42 rows and 9 variables:

com

Region

cs

Votes for the Ciutadans party

jxcat

Votes for the Junts per Catalunya party

erc

Votes for the Esquerra Republicana de Catalunya party

psc

Votes for the Partit Socialista de Catalunya party

catsp

Votes for the Catalunya Sí que es Pot party

cup

Votes for the Candidatura d'Unitat Popular party

pp

Votes for the Partit Popular party

other

Votes for other parties

Source

Idescat, statistics on Catalan Parliament elections.

Description

Builds a single grouped constrained principal balance from the first principal component of the grouped composition.

Usage

partial_pb_constrained(X, lI = NULL, constrained.criterion = "variance")

Arguments

Value

A list with the following elements:

dim

Dimension of the grouped problem, equal to length(lI) - 1.

lI

The input grouping structure.

variance

Variance criterion of the selected grouped balance.

balance_raw

Integer vector in $\{-1,0,1\}$ describing the selected grouped split.

balance

The corresponding one-column balance basis.

constrained.criterion

Criterion used to construct the balance.

Description

Finds the grouped balance with maximum variance among all assignments whose number of active groups is between min_parts and max_parts.

Usage

partial_pb_exact(
  X,
  lI = NULL,
  min_parts = 2,
  max_parts = NULL,
  method = "restricted"
)

Arguments

Details

The search enumerates only supports whose size is between min_parts and max_parts. For each support, signs are generated in binary Gray-code order, fixing the first active group on the left side to avoid evaluating both a balance and its sign reversal.

Value

A list with the following elements:

dim

Dimension of the grouped problem, equal to length(lI) - 1.

lI

The input grouping structure.

variance

Variance criterion of the best grouped balance.

balance_raw

Integer vector in $\{-1,0,1\}$ describing the best grouped split.

balance

The corresponding one-column balance basis.

min_parts

Minimum number of active groups.

max_parts

Maximum number of active groups.

Description

Finds a single grouped balance by tabu search over a partition of selected parts. The search is carried out on groups of parts defined by lI, using configurable neighbourhood moves.

Usage

partial_pb_tabu_search(
  X,
  lI = NULL,
  min_parts = 2,
  max_parts = NULL,
  iter = 100,
  tabu_size = length(lI),
  ini = NULL,
  remove_active = TRUE,
  add_left = TRUE,
  add_right = TRUE,
  flip_side = FALSE,
  swap_zero = FALSE,
  swap_sides = FALSE,
  debug = FALSE,
  constrained.criterion = "variance"
)

Arguments

Details

When ini = NULL, the constrained grouped balance is adjusted greedily so that the initial solution has exactly max_parts active groups.

Value

A list with the selected balance, its variance criterion, the search path, and a neighbourhoods element recording the active neighbourhood types.

Description

Construct a basis of principal balances for a compositional data set.

Usage

pb_basis(
  X,
  method,
  constrained.criterion = "variance",
  cluster.method = "ward.D2",
  ordering = TRUE,
  ...
)

Arguments

Details

Several methods are available:

• '"exact"': exact computation of principal balances,

• '"exact2"': exact computation using incremental Gray-code updates,

• '"constrained"': constrained approximation based on a target criterion,

• '"cluster"': approximation based on hierarchical clustering.

Value

A matrix whose columns are principal balances.

References

Martín-Fernández, J. A., Pawlowsky-Glahn, V., Egozcue, J. J., & Tolosana-Delgado, R. (2018). Advances in Principal Balances for Compositional Data. Mathematical Geosciences, 50, 273–298.

Examples

set.seed(1)
X <- matrix(exp(rnorm(5 * 100)), nrow = 100, ncol = 5)

v1 <- apply(coordinates(X, "pc"), 2, var)
v2 <- apply(coordinates(X, pb_basis(X, method = "exact")), 2, var)
v3 <- apply(coordinates(X, pb_basis(X, method = "constrained")), 2, var)
v4 <- apply(coordinates(X, pb_basis(X, method = "cluster")), 2, var)

barplot(
  rbind(v1, v2, v3, v4),
  beside = TRUE,
  ylim = c(0, 2),
  legend = c(
    "Principal Components",
    "PB (Exact method)",
    "PB (Constrained)",
    "PB (Ward approximation)"
  ),
  names = paste0("Comp.", 1:4),
  args.legend = list(cex = 0.8),
  ylab = "Variance"
)

Description

Builds a sequential binary partition (SBP) by repeatedly applying grouped tabu search to select balances over the current sets of parts. At each step, the best candidate split is retained and the remaining candidate subproblems are explored until an SBP with $D - 1$ balances is obtained.

Usage

pb_tabu_search(X, iter = 100, debug = FALSE)

Arguments

Details

This function provides a heuristic approximation to a principal balance basis. The first balance is searched on the full set of parts, and the subsequent balances are obtained by recursively refining the best currently available split.

All partial searches are initialized with the constrained principal balance of the corresponding grouped composition.

The procedure starts from the trivial grouping where each part forms its own singleton group. After each partial tabu search, up to three candidate subproblems may be generated from the selected solution:

• the split between active and inactive groups,

• the left active branch,

• the right active branch.

All generated candidates are stored, and at each stage the candidate with the largest variance criterion is selected for inclusion in the SBP and for further refinement.

This is a heuristic search strategy and does not guarantee a globally optimal SBP.

Value

An integer matrix representing a sequential binary partition. Rows correspond to the original parts of X and columns correspond to balances. Entries are in $\{-1,0,1\}$. The returned matrix has attribute "max_steps", giving the largest iteration index at which a best partial solution was found among all partial searches performed.

See Also

partial_pb_tabu_search, sbp_basis

Examples

set.seed(1)
X <- matrix(rexp(500), ncol = 5)

SBP <- pb_tabu_search(X, iter = 30)
SBP
attr(SBP, "max_steps")

Description

Construct an ilr basis rotated according to the principal components of the log-ratio coordinates of a compositional data set.

Usage

pc_basis(X)

Arguments

Value

A matrix whose columns define a principal-component-oriented ilr basis.

Description

The perturbation operation combines two compositions by component-wise multiplication and then applies closure to ensure the result remains a valid composition.

Usage

perturbation(X, Y)

Arguments

Details

Perturbation is the analogue of addition in the simplex. Each part of X is multiplied by the corresponding part of Y, and the result is closed with closure so that each composition has constant sum.

Value

An object with the same format as X containing the perturbed compositions, except that vector X with matrix or data.frame Y returns the same rectangular format as Y.

Examples

x <- c(a = 1, b = 2, c = 3)
y <- c(a = 1, b = 1, c = 2)
perturbation(x, y)

X <- rbind(
  c(1, 2, 3),
  c(4, 5, 6)
)
perturbation(X, c(1, 1, 2))
perturbation(c(1, 1, 2), X)

Description

The 'petrafm' data set contains 100 classified volcanic rock samples from Ontario (Canada). The three-part composition is

$[A: Na_2O + K_2O;\ F: FeO + 0.8998\,Fe_2O_3;\ M: MgO]$

Rocks from the calc-alkaline magma series (25 samples) can be distinguished from those of the tholeiitic magma series (75 samples) using an AFM diagram.

Usage

petrafm

Format

An object of class data.frame with 100 rows and 4 columns.

Description

Plot a balance with node labels under horizontal branches

Usage

plot_balance(
  B,
  data = NULL,
  main = "Balance dendrogram",
  summary_fun = NULL,
  cex_node = 0.9,
  offset_node = 0.05,
  ...
)

Arguments

Value

Invisibly returns a data.frame with node coordinates and labels

Examples

X = waste[,5:9]
B = pb_basis(X, method = 'exact')

plot_balance(B)

plot_balance(B, data = X,
             summary_fun = function(x){
               q = quantile(x, probs = c(0.25, 0.5, 0.75))
               sprintf("%0.2f [%0.2f-%0.2f]", q[2], q[1], q[3])
             })

Description

The 'pollen' data set contains 30 fossil pollen samples from three different locations (recorded in variable 'group'). The measured composition is the three-part composition $[pinus, abies, quercus]$.

Usage

pollen

Format

An object of class data.frame with 30 rows and 4 columns.

Description

The 'pottery' data set contains the chemical composition of 45 specimens of Romano-British pottery. The measurements were obtained by atomic absorption spectrophotometry and include nine oxides: Al2O3, Fe2O3, MgO, CaO, Na2O, K2O, TiO2, MnO, and BaO.

The specimens come from five different kiln sites.

Usage

pottery

Format

An object of class data.frame with 45 rows and 11 columns.

Description

The powering operation raises each part of a composition to a scalar exponent and then applies closure to re-normalize the result as a composition.

Usage

powering(X, alpha)

Arguments

Details

Powering is the analogue of scalar multiplication in the simplex. Each part is raised to alpha, and the result is closed with closure. When alpha has one value per row, each composition is powered by its corresponding value. When it has one value per part, each part receives its corresponding exponent. For vector X and vector alpha, each row of the result is X powered by the corresponding element of alpha.

Value

An object with the same format as X containing the powered compositions, except that vector X with vector alpha returns a matrix.

Examples

x <- c(a = 1, b = 2, c = 3)
powering(x, 2)
powering(x, c(1, 2))

X <- rbind(
  c(1, 2, 3),
  c(4, 5, 6)
)
powering(X, c(1, 2))

Description

Simulates a random composition whose coordinate system is constructed from a sequential binary partition induced by a given first balance. The supplied balance is completed to a full orthonormal basis using sbp_basis with fill = TRUE.

Usage

random_composition_with_fixed_pb(principal_balance, n = 100, sd1 = 5)

Arguments

Details

Standard normal latent coordinates are first generated in dimension $D - 1$, where $D$ is the number of parts. Their sample covariance matrix is then diagonalized, and the associated eigenvectors are used to rotate the latent coordinates before mapping them back to the simplex using the basis induced by principal_balance.

This function is mainly intended for examples, simulation studies, and experiments where a specific first balance structure is desired.

Value

A composition matrix with n rows and length(principal_balance) columns.

See Also

sbp_basis, composition

Description

Import data from a codapack workspace

Usage

read_cdp(fname)

Arguments

Description

Construct a balance basis from a sequential binary partition (SBP) or from a more general collection of balances.

Usage

sbp_basis(sbp, data = NULL, fill = FALSE, silent = FALSE)

Arguments

Details

The argument 'sbp' can be specified in two ways:

• as a list of formulas, where each formula defines the numerator and the denominator groups of a balance,

• as a matrix with one column per balance and one row per part. Positive entries indicate parts in the numerator, negative entries indicate parts in the denominator, and zeros indicate unused parts.

Value

A matrix whose columns are balances.

Examples

X <- data.frame(
  a = 1:2, b = 2:3, c = 4:5,
  d = 5:6, e = 10:11, f = 100:101, g = 1:2
)

# Sequential SBP construction
sbp_basis(list(
  b1 = a ~ b + c + d + e + f + g,
  b2 = b ~ c + d + e + f + g,
  b3 = c ~ d + e + f + g,
  b4 = d ~ e + f + g,
  b5 = e ~ f + g,
  b6 = f ~ g
), data = X)

# Chain construction
sbp_basis(list(
  b1 = a ~ b,
  b2 = b1 ~ c,
  b3 = b2 ~ d,
  b4 = b3 ~ e,
  b5 = b4 ~ f,
  b6 = b5 ~ g
), data = X)

# Formula parts can be inferred when data is not supplied
sbp_basis(list(
  b1 = a ~ b,
  b2 = b1 ~ c
))

# Non-orthogonal system of balances
sbp_basis(list(
  b1 = a + b + c ~ e + f + g,
  b2 = d ~ a + b + c,
  b3 = d ~ e + g,
  b4 = a ~ e + b,
  b5 = b ~ f,
  b6 = c ~ g
), data = X)

# Direct construction from a contrast matrix
sbp_basis(cbind(
  c( 1,  1, -1, -1),
  c( 1, -1,  1, -1),
  c( 1, -1, -1,  1)
))

Description

The 'serprot' data set records the percentages of four serum proteins from blood samples of 30 patients. Fourteen patients have one disease and sixteen have another.

The four-part compositions are formed by $[albumin, pre\text{-}albumin, globulin\ A, globulin\ B]$.

Usage

serprot

Format

An object of class data.frame with 36 rows and 7 columns.

Description

The 'statistician_time' data set records the daily time budget of an academic statistician across 20 working days. The six activities are teaching ('T'), consultation ('C'), administration ('A'), research ('R'), other wakeful activities ('O'), and sleep ('S').

These activities may also be grouped into work ('T', 'C', 'A', 'R') and leisure ('O', 'S'). The data allow investigation of the relationship between detailed time-allocation patterns and the broader division between work and leisure.

Usage

statistician_time

Format

An object of class data.frame with 20 rows and 7 columns.

Description

Variation array is returned.

Usage

variation_array(X, include_means = FALSE, ml_covariance = FALSE)

Arguments

Value

variation array matrix

Examples

set.seed(1)
X = matrix(exp(rnorm(5*100)), nrow=100, ncol=5)
variation_array(X)
variation_array(X, include_means = TRUE)

Description

The 'waste' data set studies the relationship between waste composition and floating population in Catalonia. The actual population of a municipality combines census population and floating population (tourists, seasonal visitors, temporary workers, and similar short-term residents), expressed as equivalent full-time residents.

The composition of urban solid waste is classified into five parts:

• 'x1': non-recyclable waste,

• 'x2': glass,

• 'x3': light containers,

• 'x4': paper and cardboard,

• 'x5': biodegradable waste.

Waste generation and composition are influenced by floating population, which makes waste composition a useful predictor of this difficult-to-measure demographic quantity.

Usage

waste

Format

An object of class data.frame with 215 rows and 10 columns.

References

Coenders, G., Martín-Fernández, J. A., & Ferrer-Rosell, B. (2017). When relative and absolute information matter: compositional predictor with a total in generalized linear models. Statistical Modelling, 17(6), 494–512.

Description

The 'weibo_hotels' data set compares the use of Weibo (the Chinese equivalent of Facebook) in hospitality e-marketing between small and medium establishments and larger hotel businesses in China.

The 50 latest posts from the Weibo page of each hotel ($n = 10$) were content-analysed and coded into a four-part composition: $[facilities, food, events, promotions]$. Hotels were also classified by size as large ('L') or small ('S').

Usage

weibo_hotels

Format

An object of class data.frame with 10 rows and 5 columns.