Abstract

This paper is a non-technical and mostly graphical introduction toThe data for this paper are 5888 pot sherds excavated in 1971 and 1975 from several sites in the vicinity of Durant Bend, Dalla County, Alabama. Each sherd was labeled by site and by the depth level in the excavation. In addition each sherd was classified usng three binary variables: design (check stamped vs plain), paste color (dark vs light), and thickness (thick vs thin). For the details we refer to Nance (1976) and Nance and De Leeuw (2018). In the following table the sherds are aggregated over sites/depths.

```
## CS Plain Dark Light Thin Thick
## Ds73S1 118 555 425 248 157 516
## Ds73S2 171 302 304 169 126 347
## Ds73S3 83 156 163 76 67 172
## Ds73S4 36 51 59 28 24 63
## Ds73S5+ 25 50 46 29 19 56
## Ds73NAM 203 964 656 511 183 984
## Ds73NUM 196 389 342 243 100 485
## Ds73NLM 164 199 229 134 64 299
## Ds73NBM 74 85 102 57 27 132
## Ds791 11 292 170 133 125 178
## Ds792 17 247 163 101 116 148
## Ds793 24 163 100 87 62 125
## Ds794+ 3 39 25 17 18 24
## Au1131 7 34 30 11 30 11
## Au1132 14 40 38 16 32 22
## Au1133+ 15 18 25 8 22 11
## Ds98PZ 20 219 166 73 183 56
## DBUCC 11 110 36 85 85 36
## DBLCC 9 45 32 22 47 7
## DBBCC1 13 20 19 14 29 4
## DBBCC2 28 34 45 17 49 13
## DBBCC3 90 79 132 37 147 22
## DBBCC4 83 67 106 44 111 39
## DBBCC5+ 66 45 81 30 81 30
## DS971 1 75 60 16 49 27
## DS972 7 67 47 27 51 23
## DS973+ 9 45 43 11 38 16
```

The technique we will use to analyse the Durent Bend data is Homogeneity Analysis (Gifi (1990)), which is more widely known as *Multiple Correspondence Analysis* (Greenacre (1984), Greenacre and Blasius (2006)). We give a graphical introduction to Homogeneity Analysis, without using formulas.

Suppose we have \(m\) categorical *variables*, and that variable \(j\) has \(k_j\) *categories* \((j=1,\cdots,m)\). The \(m\) variables *categorize* or *measure* \(n\) *objects*. Variable \(j\) partitions the set of \(n\) objects into \(k_j\) subsets, one subset for each category. Before we get to the analysis of the Durant Bend sherds, we will illustrate the main concepts of our approach with a small example in which three variables partition ten objects. The first two variables have three categories, the last one has two categories.

```
## first second third
## 01 a p u
## 02 b q v
## 03 a r v
## 04 a p u
## 05 b p v
## 06 c p v
## 07 a p u
## 08 a p v
## 09 c p v
## 10 a p v
```

In Homogeneity Analysis we aim to make a *joint plot* of the objects and the categories of the variables. Joint plots are also known as *biplots* (Gower and Hand (1996)). In a joint plot both objects and categories are represented as points in a low-dimensional space, usually the plane, in such a way that the relations in the data are represented as precisely as possible in the plot. We will specify what we mean by “as precisely as possible” below. Homogeneity Analysis is defined by defining a measure of the loss of information in a certain way, and then choosing the representation that minimizes that loss.

The \(n\) points in the plan, or more generally in \(p\)-dimensional space, representing the objects (sherds) are collected in a matrix of *object scores*. The \(k_j\) points representing the categories of variable \(j\) are in a matrix of *category quantifications* for variabe \(j\). For each variable we can make a *graph plot* in which each of the \(n\) object scores is connected by a straight line to the quantification of the category that this object falls in. Thus there is one line departing from each object point, while the number of lines arriving at a category point is equal to the number of objects in the category. One graph plot has \(n\) lines, all graph plots together have \(n\times m\) lines.

If all these lines have length zero, then all objects coincide with “their” categories for that variable, and thus we have reproduced the data exactly. If there is more than one variable, however, we cannot expect to have such a perfect representation, because objects which are together in a category for one variable may not be together for another variable.

Homogeneity Analysis is defined as the technique that produces a joint plot of objects and category quantification in such a way that the total length of all \(n\times m\) lines in the \(m\) graph plots is a small as possible. Some qualifications are needed, however. We actually minimizes the sum of the squared length of the lines, for the same reason that we use the squares of the residuals in a regression analysis. It simplifies the mathematics and the computation to use squared distances. Secondly, we could trivially gain our objective of minimizing line length by collapsing all object scores and category quantifications in a single point, which makes our loss function equal to zero, but is useless in representing or reproducing the data. Thus we need some form of *normalization* to prevent this trivial solution from happening. In Homogeneity Analysis we require the columns of the object score matrix add up to zero, have sum of squares equal to one, and are uncorrelated.

Let’s illustrate this with our small example. We start with a completely aribitrary *initial configuration*. The ten objects are placed at equal distances on a circle, and the categories for each of the variables are on the horizontal axis. This leads to the first three graph plots, which we have superimposed to get the fourth plot with \(n\times m = 30\) lines.

For this arbitrary initial configuration the loss, i.e. the sum of squares of the line lengths, or the sum of the squred distances between objects and the categories they fall in, is equal to 8.1081098.

In Homogeneity Analysis we minimize loss by what is known as *reciprocal averaging* or *alternating least squares*. We alternate two substeps. The first substep improves the category quantifications for a given set of object scores, the second substep improves and normalizes the object scores for a given set of category quantifications, namely those we have just computed in the first substep. Taken together these two substeps are an *iteration*. So each iteration starts with object scores and category quantifications and uses its two substeps to improve both. Each of the two substeps decreases the loss functions, i.e. the total squared length of the lines in the graph plots.

and the loss has decreased to 5.2016654. Note that we have not improved the object scores yet, so they are still their initial configuration, equally spaced on a circle. Also note, in variable 2 for instance, that category quantifications coincide with object scores, and thus contribute zero to the loss, if the object is the only observation in the category. In addition, because category quantifications are averages of objects points, they are in the convex hull of the object points, which means in this figure that they are within the circle. Averaging objects points makes the category quantifications move closer to the origin.

The second substep improves the object scores, while keeping the category quantifications in the locations we have just computed in the first substep. The second substaep has itself two substeps, say \(2A\) and \(2B\). In the substep \(2A\) the score of an object for given category quantifications is computed as the average or centroid of the \(m\) category quantifications the object is in.The loss function is down all the way to 0.4852424. This is not a proper loss value, however, because the object scores are no longer centered, standardized, and uncorrelated, and that was a Homogeneity Analysis requirement. Substep 1 shrinks the object scores towards the origin by averaging, substep 2A takes the resulting category quantifications and shrinks them more by even more averaging. Thus in substep \(2B\) we have to renormalize the object scores such that they are centered, standardized, and uncorrelated. This gives