# Least Squares Euclidean Multidimensional Scaling

*Started October 02, 2020. Last update April 13, 2021*

# Note

This book will be expanded/updated frequently. The directory deleeuwpdx.net/pubfolders/stress has a pdf version, the bib file, the complete Rmd file with the codechunks, and the R and C source code. All suggestions for improvement of text or code are welcome, and some would be really benificial. For example, I only use base R graphics, nothing more fancy, because base graphics is all I know.

All text and code are in the public domain and can be copied, modified, and used by anybody in any way they want. Attribution will be appreciated, but is not required. For completeness we include a slighty modified version of the Unlicense as appendix C.

I number and label *all* displayed equations. Equations are displayed, instead of inlined, if and only if

- they are important, or
- they are referred to elsewhere in the text, or
- if not displaying them messes up the line spacing.

All code chunks in the text are also named. Theorems, lemmas, and so on are named and numbered, using the captioners package (Letaw (2015)). All chapters, sections, subsections, are also named and numbered using bookdown/Rmarkdown.

I have been somewhat hesitant to use lemmas, theorems, and corollaries in this book. I am not a mathematician, and, given the level of almost all of the text, their use seems somewhat pretentious. But ultimately they do provide a useful, maybe even indispensable, organizational tool. As a compromise I use *result* for those results that are too trivial to be elevated to theorem status. These results often do not even need a proof. If there is a proof of a lemma, theorem, corollary, or result, it ends with a \(\blacksquare\).

Another idiosyncracy: if a line in multiline displayed equation ends with “=,” then the next line begins with “=.” If it ends with “+,” then the next line begin with “+,” and if it ends with “-” the next line begins with “+” as well. I’ll try to avoid ending a line with “+” or “-,” especially with “-,” but if it happens you are warned. A silly example is \[\begin{align} &(x+y)^2-\\ &+4x=\\ &=x^2+y^2-2x=\\ &=(x-y)^2. \end{align}\]

Just as an aside: if I refer to something that has been mentioned “above” I mean something that comes earlier in the book and “below” refers to anything that comes later. This always confuses me, so I had to write it down.

The dilemma of whether to use “we” or “I” throughout the book is solved in the usual way. If I feel that a result is the work of a group (me, my co-workers, and the giants on whose shoulders we stand) then I use “we.” If it’s an individual decision, or something personal, then I use “I.” The default is “we,” as it always should be in scientific writing.

Chapter 19 has some of the necessary mathematical background material, both notation and results, sometimes with specific eleborations that seem useful for the book. The other chapters have references to chapter 19 when additional background may be needed.

There is an appendix A with code, and an appendix B with data sets. These contain brief descriptions and links to the supplementary materials, which are the actual code and data.

I will use this note to thank Rstudio, in particular J. J. Allaire and Yihui Xi, for their contributions to the R universe, and for their effective promotion of open source and open access. Not too long ago I was an ardent LaTeX user, firmly convinced I would never use anything else again in my lifetime. As I was convinced, before that, that I would never use anything besides, in that order, FORTRAN, PL/I, APL, and (X)Lisp. But I lived too long. And then R, Rstudio, (R)Markdown, bookdown, and blogdown came along.