Package 'lyubishchev'

Classify a Specimen by Multivariate Posterior Probability

Description

Assigns posterior class probabilities to a new specimen using the Edgeworth-Pearson multivariate Gaussian likelihood for each class scatter ellipse. For each class the log-likelihood of the specimen under a multivariate normal with the class mean and covariance is computed, and a softmax over the per-class log-likelihoods yields posterior probabilities.

Usage

classify(specimen, ellipses)

Arguments

specimen	A numeric vector of feature values for a single observation.
ellipses	A named list of scatter ellipses as returned by scatter_ellipse.

Details

The log-likelihood for class $k$ is

$-\tfrac{1}{2}\left(p\log 2\pi + \log|\Sigma_k| + (x-\mu_k)^\top \Sigma_k^{-1} (x-\mu_k)\right)$

where $p$ is the number of features, $\mu_k$ and $\Sigma_k$ are the class mean and covariance, and $x$ is the specimen.

Value

A named list with one element per class. Each element is a list with components:

mahalanobis_distance

Squared Mahalanobis distance from the specimen to the class centroid.

log_likelihood

Multivariate Gaussian log-likelihood of the specimen under the class.

posterior

Posterior probability of the class (softmax over the per-class log-likelihoods). Posteriors sum to 1 across classes.

References

Lubischew, A.A. (1962). On the use of discriminant functions in taxonomy. Biometrics, 18(4), 455-477.

See Also

scatter_ellipse

Examples

ellipses <- scatter_ellipse(iris[, 1:4], iris$Species)
specimen <- c(5.1, 3.5, 1.4, 0.2)
result <- classify(specimen, ellipses)
sapply(result, function(r) r$posterior)

---

Lyubishchev's Divergence Coefficient

Description

Computes Lyubishchev's divergence coefficient $D$ between two groups measured on one or more continuous features. The coefficient summarises the standardised separation between the group means, summed across features:

$D = \sum_j \frac{(M_{1j} - M_{2j})^2}{\sigma_{1j}^2 + \sigma_{2j}^2}$

where $M_{ij}$ and $\sigma_{ij}^2$ are the mean and (sample) variance of feature $j$ in group $i$. Features whose pooled variance is zero are skipped to avoid division by zero.

Usage

divergence_coefficient(a, b)

Arguments

a	A numeric matrix or data frame for the first group, with one row per observation and one column per feature. A numeric vector is treated as a single-feature group.
b	A numeric matrix or data frame for the second group, with the same columns (features) as a.

Details

This is the measure described in Lyubishchev's 1943 manuscript and later published in English by Lubischew (1962). It predates and is more general than the binary-character similarity coefficients of Sokal and Sneath (1963), operating directly on continuous measurements.

Value

A single numeric value, the divergence coefficient $D$. Larger values indicate greater separation between the groups.

References

Lyubishchev, A.A. (1943). Programma obshchey sistematiki [Program of General Systematics]. Manuscript, 22 November 1943.

Lubischew, A.A. (1962). On the use of discriminant functions in taxonomy. Biometrics, 18(4), 455-477.

Examples

setosa <- as.matrix(iris[iris$Species == "setosa", 1:4])
versicolor <- as.matrix(iris[iris$Species == "versicolor", 1:4])
divergence_coefficient(setosa, versicolor)

---

Fit Scatter Ellipses per Class

Description

Fits a covariance ellipse to each class in a labelled multivariate data set. For every class the function computes the centroid (mean vector), the feature covariance matrix and the sample size. These ellipses are the building blocks for transgression and classify.

Usage

scatter_ellipse(X, y)

Arguments

X	A numeric matrix or data frame of observations, with one row per observation and one column per feature.
y	A vector of class labels of length nrow(X). May be a factor, character or numeric vector.

Value

A named list with one element per class. Each element is itself a list with components:

mean

Numeric vector of feature means for the class.

cov

Feature covariance matrix for the class.

n_samples

Integer count of observations in the class.

The names of the list are the class labels (coerced to character).

References

Lubischew, A.A. (1962). On the use of discriminant functions in taxonomy. Biometrics, 18(4), 455-477.

See Also

transgression, classify

Examples

ellipses <- scatter_ellipse(iris[, 1:4], iris$Species)
ellipses[["setosa"]]$mean
ellipses[["setosa"]]$n_samples

---

Detect Overlap (Transgression) Between Two Scatter Ellipses

Description

Tests whether two class scatter ellipses overlap, in Lyubishchev's sense of "transgression" between groups. The centroids are compared using the squared Mahalanobis distance under the pooled covariance of the two classes, and that distance is compared against a chi-squared threshold with degrees of freedom equal to the number of features. When the Mahalanobis distance is below the threshold the groups are deemed to transgress (overlap).

Usage

transgression(ellipses, class_a, class_b, confidence = 0.95)

Arguments

ellipses	A named list of scatter ellipses as returned by scatter_ellipse.
class_a	Name (character) of the first class in ellipses.
class_b	Name (character) of the second class in ellipses.
confidence	Confidence level for the chi-squared threshold, between 0 and 1. Defaults to 0.95.

Value

A list with components:

mahalanobis_distance

Squared Mahalanobis distance between the two centroids under the pooled covariance.

threshold

Chi-squared threshold at the requested confidence with degrees of freedom equal to the number of features.

transgression

Logical; TRUE when the distance is below the threshold (the ellipses overlap).

separation_ratio

Ratio of the Mahalanobis distance to the threshold. Values above 1 indicate well-separated groups.

References

Lubischew, A.A. (1962). On the use of discriminant functions in taxonomy. Biometrics, 18(4), 455-477.

See Also

scatter_ellipse

Examples

ellipses <- scatter_ellipse(iris[, 1:4], iris$Species)
transgression(ellipses, "versicolor", "virginica")

---