# Procrustean Similarity Transformations Solves the Procrustean problem of fitting one (MDS) configuration (testee) to another (target) MDS configuration. ```r Procrustes(X, Y) ## S3 method for class 'procr' plot(x, plot.type = "jointplot", plot.dim = c(1,2), main, xlab, ylab, xlim, ylim, asp = 1, pch = 20, col.X = "cadetblue", col.Y = "gray", col.Yhat = "coral1", label.conf = list(label = TRUE, pos = 3, cex = 0.8), arrows = TRUE, length = 0.10, legend = list(plot = TRUE, labels = c("Target", "Testee"), pos = "bottomright"), ...) ``` ## Arguments - `X`: Target configuration - `Y`: Testee configuration - `x`: Object of class `procr` - `plot.type`: Either `"jointplot"` or `"transplot"` - `plot.dim`: Vector with dimensions to be plotted. - `main`: Plot title. - `xlab`: Label of x-axis. - `ylab`: Label of y-axis. - `xlim`: Scale x-axis. - `ylim`: Scale y-axis. - `pch`: Plot symbol. - `asp`: Aspect ratio. - `col.X`: Color target configuration. - `col.Y`: Color testee configuration. - `col.Yhat`: Color transformed configuration. - `label.conf`: List with arguments for plotting the labels of the configurations in a configuration plot (logical value whether to plot labels or not, label position, label color). - `length`: length of the edges of the arrow head (in inches). - `arrows`: For `"transplot"` only, whether arrows should be plotted or not. - `legend`: List with arguments for plotting the legend. - `...`: Additional plot arguments. ## Details `Y` is going to be modified by finding an optimal dilation factor, an optimal translation and rotation for `Y` such that it is a similar as possible to `X`. `X` remains untouched. ## Returns Returns an object of class `procr` with: - **X**: Input target configuration - **Y**: Input testee configuration - **Yhat**: Procrustes transformed (fitted) configuration - **translation**: Translation vector - **dilation**: Dilation factor - **rotation**: Rotation-reflection matrix - **confdistX**: Configuration distances X - **confdistY**: Configuration distances Y - **confdistYhat**: Configuration distances of fitted configuration - **congcoef**: Congruence coefficient - **aliencoef**: Alienation coefficient - **r**: Correlation coefficient between X and Y-hat (shrinkage ratio) - **pairdist**: Pairwise object distances (sorted) ## References Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling and Unfolding (2nd ed.). Springer. Mair, P., Groenen, P. J. F., De Leeuw, J. (2022). More on multidimensional scaling in R: smacof version 2, Journal of Statistical Software, 102(10), 1-47. tools:::Rd_expr_doi("10.18637/jss.v102.i10") Borg, I. and Mair, P. (2022). A note on Procrustean fittings of noisy configurations. Austrian Journal of Statistics, 51, 1-9. tools:::Rd_expr_doi("10.17713/ajs.v51i4.1423") ## Examples ```r ## artificial example: X <- matrix(c(1, -1, -1, 1, 2, 2, -2, -2), ncol = 2) Y <- matrix(c(0.07, 0.93, 1.93, 1.07, 2.62, 3.12, 1.38, 0.88), ncol = 2) op <- par(mfrow = c(1,2)) plot(X[,1], X[,2], xlim = c(-3, 3), ylim = c(-2, 3.5), asp = 1, xlab = "", ylab = "") rect(-1, -2, 1, 2) points(Y[,1], Y[,2], xlim = c(-3, 3), col = "gray") polygon(Y[,1], Y[,2], border = "gray") fitp <- Procrustes(X, Y) plot(fitp$Yhat[,1], fitp$Yhat[,2], col = "red", xlim = c(-3, 3), ylim = c(-2, 3.5), asp = 1, xlab = "", ylab = "") polygon(fitp$Yhat[,1], fitp$Yhat[,2], border = "red") par(op) ## MDS example: eastD <- sim2diss(EW_eng$east) attr(eastD, "Labels") <- abbreviate(attr(eastD, "Labels")) fit.east <- mds(eastD, type = "ordinal") westD <- sim2diss(EW_eng$west) attr(westD, "Labels") <- abbreviate(attr(westD, "Labels")) fit.west <- mds(westD, type = "ordinal", init = torgerson(eastD)) fit.proc <- Procrustes(fit.east$conf, fit.west$conf) fit.proc ## Configuration plots; Procrustes plots. plot(fit.east, main = "MDS East Germany") ## MDS plot East Germany plot(fit.west, main = "MDS West Germany") ## MDS plot West Germany ## Procrustes configurations (X and Yhat) plot(fit.proc, ylim = c(-1, 1), col.X = "cadetblue", col.Yhat = "brown", pch = 19, legend = list(pos = "topleft", labels = c("East Germany", "West Germany"))) ## Procrustes transformations (Y and Yhat) plot(fit.proc, plot.type = "transplot", length = 0.05, ylim = c(-1,1), legend = list(pos = "bottomright", labels = c("West Germany (untransformed)", "West Germany (transformed)"))) ```

Procrustes function