Estimate true group variance

Description

Estimation of true group variance incorporating observation wise variances. The function uses the data from x and the individual variances for each observation, for example derived from uncertainties, to calculate a 'true' group variance. The variance of the matrix is corrected for the sum of the individual variances of the data set, which is normalized to the number of rows of the matrix.

Usage

calc_estimate_true_var(x, ...)

## Default S3 method:
calc_estimate_true_var(x, individual_var, force_pos_def = T, ...)

## S3 method for class 'rmult'
calc_estimate_true_var(x, individual_var, force_pos_def = T, ...)

Arguments

Value

matrix of corrected group variance

Methods (by class)

• default: for class matrix or data.frame

• rmult: for class rmult

Author(s)

Solveig Pospiech, K. Gerald v.d. Boogaart

Simulated observation data

Description

A data set of 200 simulated observations with two observed variables and two groups, non-compositional

Usage

dataobs

Format

A data frame with 200 rows and 3 columns

Var1

simulated observed variable

Var2

simulated observed variable

Group

Factor with levels 'Group 1' and 'Group 2'

Simulated observation of compositional data

Description

A data set of 200 simulated observations with three observed variables and two groups, compositional

Usage

dataobs_coda

Format

A data frame with 200 rows and 4 columns

Var1

simulated observed variable, compositional

Var2

simulated observed variable, compositional

Var3

simulated observed variable, compositional

Group

Factor with levels 'Group 1' and 'Group 2'

Simulated true data

Description

A data set of 200 simulated 'true' data, from which the observations are deduced, with two observed variables and two groups, non-compositional

Usage

datatrue

Format

A data frame with 200 rows and 3 columns

Var1

simulated variable

Var2

simulated variable

Group

Factor with levels 'Group 1' and 'Group 2'

Simulated true compositional data

Description

A data set of 200 simulated 'true' data, from which the observations are deduced, with three observed variables and two groups, compositional

Usage

datatrue_coda

Format

A data frame with 200 rows and 4 columns

Var1

simulated variable, compositional

Var2

simulated variable, compositional

Var3

simulated variable, compositional

Group

Factor with levels 'Group 1' and 'Group 2'

Force positive definiteness

Description

Function to force positive definiteness on a matrix.

Usage

force_posdef(x, verbose = T)

Arguments

Value

positive definite matrix

Author(s)

Solveig Pospiech

Generalized mean

Description

Calculates the generalized mean of a data set by using a given group variance and individual, observation-wise variances for each observation of the data set

Usage

generalized_mean(x, ...)

## Default S3 method:
generalized_mean(
  x,
  var,
  individual_var = matrix(0, nrow = nrow(x), ncol = ncol(x)),
  ...
)

## S3 method for class 'rmult'
generalized_mean(
  x,
  var,
  individual_var = matrix(0, nrow = nrow(x), ncol = ncol(x)^2),
  ...
)

Arguments

Value

vector of lenght of ncol(x) of generalized means

Methods (by class)

• default: for class matrix or data.frame

• rmult: for class rmult of package 'compositions'

Author(s)

Solveig Pospiech, K. Gerald v.d. Boogaart

predict.vqda

Description

Classify multivariate observations in conjunction with qda() or lda() of class 'vqda' or 'vlda'.

Usage

## S3 method for class 'vqda'
predict(object, newdata, newerror, prior = object$prior, ...)

## S3 method for class 'vlda'
predict(object, newdata, newerror, prior = object$prior, ...)

Arguments

Value

list containing the following components: class factor containing the predicted group likelihood matrix of dimension 'number of samples' x 'number of groups', containing the likelihood for each sample to belong to one of the groups grouping original grouping of the samples, copied from the input object

list containing the following components: class factor containing the predicted group likelihood matrix of dimension 'number of samples' x 'number of groups', containing the likelihood for each sample to belong to one of the groups grouping original grouping of the samples, copied from the input object

Methods (by class)

• vqda: predict() for class 'vqda'

• vlda: predict() for class 'vlda'

Author(s)

Solveig Pospiech, package 'MASS'

Simulated observation uncertainties

Description

A data set of 200 simulated uncertainties with two variables and two groups, non-compositional

Usage

uncertainties

Format

A data frame with 200 rows and 3 columns

Var1

simulated observed variable

Var2

simulated observed variable

Group

Factor with levels 'Group 1' and 'Group 2'

Simulated observation uncertainties of compositional data

Description

A data set of 200 simulated uncertainties with three variables and two groups, compositional

Usage

uncertainties_coda

Format

A data frame with 200 rows and 4 columns

Var1

simulated observed variable, compositional

Var2

simulated observed variable, compositional

Var3

simulated observed variable, compositional

Group

Factor with levels 'Group 1' and 'Group 2'

Weighted Linear Discriminant Analysis

Description

Extension of the qda() of package 'MASS' (not the lda() function) to calculate a LDA incorporating individual, cell-wise uncertainties, e.g. if the uncertainties are expressed as individual variances for each measurand.

Usage

vlda(x, uncertainties, grouping, prior)

Arguments

Details

Uncertainties can be considered in a statistical analysis either by each measured variable, by each observation or by using the individual, cell-wise uncertainties. There are several methods for incorporating variable-wise or observation-wise uncertainties into a QDA, most of them using the uncertainties as weights for the variables or observations of the data set. The term 'cell-wise uncertainties' describe a data set of $d$ analysed variables where each observation has an individual uncertainty for each of the $d$ variables conforming it. Hence, a data set of $n \times d$ data values has associated a data set of $n \times d$ individual uncertainties. Instead of weighting the columns or rows of the data set, the vlda() function uses uncertainties to recalculate better estimates of the group variances and group means. It is internally very similar to the vqda function, but with an averaged group variance for all groups. If the presence of uncertainties is not accounted for, the decision rules are based on the group variances calculated by the given data set. But this observed group variance might deviate notably from the group variance, which can be estimated including the uncertainties. This methodological framework does not only allow to incorporate cell-wise uncertainties, but also would largely be valid if the information about the co-dependency between uncertainties within each observation would be reported.

Value

object of class 'vlda' containing the following components: prior the prior probabilities used. counts counts per group. means the group means. generalizedMeans the group means calculated by the function generalized_mean groupVarCorrected the group variances calculated by the function calc_estimate_true_var lev the levels of the grouping factor. grouping the factor specifying the class for each observation.

Author(s)

Solveig Pospiech, package 'MASS'

References

Pospiech, S., R. Tolosana-Delgado and K.G. van den Boogaart (2020) Discriminant Analysis for Compositional Data Incorporating Cell-Wise Uncertainties, Mathematical Geosciences

Examples

# for non-compositional data:
data("dataobs")
data("uncertainties")
mylda = vlda(x = dataobs[, 1:2], uncertainties = uncertainties[, 1:2], grouping = dataobs$Group)
mypred = predict(mylda, newdata = dataobs[, 1:2], newerror = uncertainties[, 1:2])
forplot = cbind(dataobs, LG1 = mypred$posterior[,1])
if (require("ggplot2")) {
  scatter_plot = ggplot(data = forplot, aes(x = Var1, y = Var2)) +
    geom_point(aes(shape = Group, color = LG1))
  if (require("ggthemes")) {
    scatter_plot = scatter_plot +
        scale_color_gradientn(colours = colorblind_pal()(5))
  }
  scatter_plot
}

# for compositional data
data("dataobs_coda")
data("uncertainties_coda")
require(compositions)
# generate ilr-transformation (from package 'compositions')
data_ilr = ilr(dataobs_coda[, 1:3])
uncert_ilr = t(simplify2array(apply(uncertainties_coda[, 1:3],1,
                       function(Delta) clrvar2ilr(diag(Delta)))))
uncert_ilr = compositions::rmult(uncert_ilr) # change class into rmult from package 'compositions'
mylda_coda = vlda(x = data_ilr, uncertainties = uncert_ilr, grouping = dataobs_coda$Group)
mypred_coda = predict(mylda_coda, newdata = data_ilr, newerror = uncert_ilr)
forplot_coda = cbind(dataobs_coda, LG1 = mypred_coda$posterior[,1])
# if 'ggtern' is installed, you can plot via ggtern:
# if (require("ggtern")) {
#   ternary_plot = ggtern(data = forplot_coda, aes(x = Var1, y = Var2, z = Var3)) +
#     geom_point(aes(shape = Group, color = LG1))
#   if (require("ggthemes")) {
#     ternary_plot = ternary_plot +
#         scale_color_gradientn(colours = colorblind_pal()(5))
#   }
#   ternary_plot
# }

Weighted Quadratic Discriminant Analysis

Description

Extension of the qda() of package 'MASS' to calculate a QDA incorporating individual, cell-wise uncertainties, e.g. if the uncertainties are expressed as individual variances for each measurand.

Usage

vqda(x, uncertainties, grouping, prior)

Arguments

Details

Uncertainties can be considered in a statistical analysis either by each measured variable, by each observation or by using the individual, cell-wise uncertainties. There are several methods for incorporating variable-wise or observation-wise uncertainties into a QDA, most of them using the uncertainties as weights for the variables or observations of the data set. The term 'cell-wise uncertainties' describe a data set of $d$ analysed variables where each observation has an individual uncertainty for each of the $d$ variables conforming it. Hence, a data set of $n \times d$ data values has associated a data set of $n \times d$ individual uncertainties. Instead of weighting the columns or rows of the data set, the vqda() function uses uncertainties to recalculate better estimates of the group variances and group means. If the presence of uncertainties is not accounted for, the decision rules are based on the group variances calculated by the given data set. But this observed group variance might deviate notably from the group variance, which can be estimated including the uncertainties. This methodological framework does not only allow to incorporate cell-wise uncertainties, but also would largely be valid if the information about the co-dependency between uncertainties within each observation would be reported.

Value

object of class 'vqda' containing the following components: prior the prior probabilities used. counts counts per group. means the group means. generalizedMeans the group means calculated by the function generalized_mean groupVarCorrected the group variances calculated by the function calc_estimate_true_var lev the levels of the grouping factor. grouping the factor specifying the class for each observation.

Author(s)

Solveig Pospiech, package 'MASS'

References

Pospiech, S., R. Tolosana-Delgado and K.G. van den Boogaart (2020) Discriminant Analysis for Compositional Data Incorporating Cell-Wise Uncertainties, Mathematical Geosciences

Examples

# for non-compositional data:
data("dataobs")
data("uncertainties")
myqda = vqda(x = dataobs[, 1:2], uncertainties = uncertainties[, 1:2], grouping = dataobs$Group)
mypred = predict(myqda, newdata = dataobs[, 1:2], newerror = uncertainties[, 1:2])
forplot = cbind(dataobs, LG1 = mypred$posterior[,1])
if (require("ggplot2")) {
  scatter_plot = ggplot(data = forplot, aes(x = Var1, y = Var2)) +
    geom_point(aes(shape = Group, color = LG1))
  if (require("ggthemes")) {
    scatter_plot = scatter_plot +
        scale_color_gradientn(colours = colorblind_pal()(5))
  }
  scatter_plot
}

# for compositional data
data("dataobs_coda")
data("uncertainties_coda")
require(compositions)
# generate ilr-transformation (from package 'compositions')
data_ilr = ilr(dataobs_coda[, 1:3])
uncert_ilr = t(simplify2array(apply(uncertainties_coda[, 1:3],1,
                       function(Delta) clrvar2ilr(diag(Delta)))))
uncert_ilr = compositions::rmult(uncert_ilr) # change class into rmult from package 'compositions'
myqda_coda = vqda(x = data_ilr, uncertainties = uncert_ilr, grouping = dataobs_coda$Group)
mypred_coda = predict(myqda_coda, newdata = data_ilr, newerror = uncert_ilr)
forplot_coda = cbind(dataobs_coda, LG1 = mypred_coda$posterior[,1])
# if 'ggtern' is installed, you can plot via ggtern:
# if (require("ggtern")) {
#   ternary_plot = ggtern(data = forplot_coda, aes(x = Var1, y = Var2, z = Var3)) +
#     geom_point(aes(shape = Group, color = LG1))
#   if (require("ggthemes")) {
#     ternary_plot = ternary_plot +
#         scale_color_gradientn(colours = colorblind_pal()(5))
#   }
#   ternary_plot
# }