# prWarp: Homo example

#### Anne Le Maitre, Philipp Mitteroecker, Silvester Bartsch, Corinna Erkinger, Nicole D. S. Grunstra, Fred L. Bookstein

#### 2024-03-20

This document corresponds to the worked example of the paper Morphometric variation at different spatial scales: coordination and compensation in the emergence of organismal form. (Mitteroecker et al., 2020a). The dataset is a set of 87 2D landmarks on the midsagittal plane of the skull in 24 adult modern humans, which are also accessible via the DRYAD repository (Mitteroecker et al., 2020b). The example below describes how to decompose shape variation into partial warps, in order to obtain bending energies, principal warps, partial warp scores, and the non-affine component of shape variation for 2D landmark configurations. In also describes how to compute Mardia-Dryden distributions and self-similar distributions of landmarks. For further details about results interpretation, please read the associated paper.

## Data preparation

Install and load the packagein R.

    library("prWarp")

Load the dataset `HomoMidSag`. This dataset comprises the coordinates of 87 two-dimensional landmarks for the 24 specimens as a data frame. Semilandmarks are already slid, but not superimposed.

    data("HomoMidSag")
    k <- dim(HomoMidSag)[2] / 2  # number of landmarks
    n_spec <- dim(HomoMidSag)[1]  # number of specimens
    homo_ar <- geomorph::arrayspecs(HomoMidSag, k, 2)  # create an array
    dimnames(homo_ar)[[1]] <- 1:k
    dimnames(homo_ar)[[2]] <- c("X", "Y")

Superimpose landmarks using the generalized Procrustes analysis (GPA).

    homo_gpa <- Morpho::procSym(homo_ar)

    ## performing Procrustes Fit

    ## in...  0.05659914 secs

    ## Operation completed in 0.0921368598937988 secs

    m_overall <- homo_gpa$rotated  # Procrustes coordinates
    m_mshape <- homo_gpa$mshape  # average shape

Visualization of the mean shape

    plot(m_mshape, asp = 1, main = "Average shape", xlab = "X", ylab = "Y")

## Partial warp decomposition

Decompose the 87 Procrustes aligned landmarks into partial warps. The reference matrix is generally the average landmark configuration. Note that the function `create.pw.be` returns principal warps, partial warp scores, partial warp variation and associated bending energies, but also the non-affine componennt of shape variation.

    homo_be_pw <- create.pw.be(m_overall, m_mshape)

Plot the partial warp variance as a function of the log of the inverse bending energy. The first pairs of partial warps correspond to small-scale shape variation whereas the last pairs correspond to the large-scale shape variation.

    # Computation of log BE^-1 for the (k-3) partial warps
    logInvBE <- log((homo_be_pw$bendingEnergy)^(-1))
    # Computation of log PW variance for the (k-3) partial warps
    logPWvar <- log(homo_be_pw$variancePW)
    # Linear regression of the log PW variance on the log BE^-1
    mod <- lm(logPWvar ~ logInvBE)
    # Plot log PW variance on log BE^-1 with regression line
    plot(logInvBE, logPWvar, col = "white", asp = 1, main = "PW variance against inverse BE", sub = paste("slope =", round(mod$coefficients[2], 2)), xlab = "log 1/BE", ylab = "log PW variance")
    text(logInvBE, logPWvar, labels = names(logPWvar), cex = 0.5)
    abline(mod, col = "blue")

Visualize the non-affine component of shape variation.

    # Compute the trace of t(Xnonaf) %*% Xnonaf
    tr_nonaf <- sum(diag(t(homo_be_pw$Xnonaf) %*% homo_be_pw$Xnonaf))
    # Convert matrix into a 3D array
    Anonaf <- xxyy.to.array(homo_be_pw$Xnonaf, p = k, k = 2)
    # Plot the non-affine shape variation around the mean
    geomorph::plotAllSpecimens(Anonaf, plot_param = list(pt.cex = 0.3, mean.cex = 0.8, mean.col = "red"))

## Self-similar and Mardia-Dryden distributions

Compute a self-similar distribution around the average landamark configuration.

    # Compute the self-similar distribution
    Xdefl <- ssim.distri(m_mshape, n = n_spec, sd = 0.05, f = 1)
    # Compute the trace of t(Xdefl) %*% Xdefl
    tr_defl <- sum(diag(t(Xdefl) %*% Xdefl))
    # Convert matrix into a 3D array
    Adefl <- xxyy.to.array(Xdefl, p = k, k = 2)
    # Plot the self-similar distribution
    geomorph::plotAllSpecimens(Adefl, plot_param = list(pt.cex = 0.3, mean.cex = 0.8, mean.col = "red"))

Compute a Mardia-Dryden distribution around the average landamark configuration.

    # Compute the Mardia-Dryden distribution
    Xmd <- md.distri(m_mshape, n = n_spec, sd = 0.005)
    # Convert matrix into a 3D array
    Amd <- xxyy.to.array(Xmd, p = k, k = 2)
    # Plot the Mardia-Dryden distribution
    geomorph::plotAllSpecimens(Amd, plot_param = list(pt.cex = 0.3, mean.cex = 0.8, mean.col = "red"))

## References

Bartsch S (2019). The ontogeny of hominid cranial form: A geometric morphometric analysis of coordinated and compensatory processes. Master’s thesis, University of Vienna.

Bookstein FL (1989). Principal Warps: Thin-plate splines and the decomposition of deformations. IEEE Transactions on pattern analysis and machine intelligence, 11(6): 567–585. https://ieeexplore.ieee.org/abstract/document/24792

Mitteroecker P et al. (2020a). Morphometric variation at different spatial scales: coordination and compensation in the emergence of organismal form. Systematic Biology, 69(5): 913–926. https://doi.org/10.1093/sysbio/syaa007

Mitteroecker P et al. (2020b). Data form: Morphometric variation at different spatial scales: coordination and compensation in the emergence of organismal form.Dryad Digital Repository. https://doi.org/10.5061/dryad.j6q573n8s