# A data-driven approach to identifying development stages in diachronic corpus linguistics

In a previous post, I showcased the development of the split infinitive ^{1} I wanted to check whether the split infinitive had spiked after the airing of the original Star Trek series in the late 1960s (to find out whether that was indeed the case, I invite you to read the post!). In other words, I had a theoretical time-partition in mind (before and after 1967), and I wanted to check whether it had any empirical relevance. Another strategy consists in letting the data decide what time-partitions are empirically relevant from the start.

The corpora that diachronic linguists work with are pre-partitioned into years or decades. The stacked barplot below compares the distributions of the split infinitive (blue) and the unsplit infinitive (orange) across the 20 decades spanned by the Corpus of Contemporary American English (1810s-2000s).

The barplot does a good job at showing that the unsplit infinitive is more frequent overall than its unsplit counterpart. Also apparent in the plot is that the popularity of the split infinitive increases steadily from the 1940s onwards, whereas the reverse trend is observed for the unsplit infinitive over the same period.

The question is whether the plot can help us identify stages in the development of the split infinitive. To some extent, it can. In the first stage, which spans from the 1810s to the 1840s, the split infinitive is hardly used at all. A second stage spanning from the 1840s to the 1900s (with the exception of the 1890s) shows a pattern of moderate increase. A third stage between the 1900s and the 1940s corresponds to a period of decrease. Finally a fourth stage of dramatic increase is observed between the 1940s and the 2000s. With respect to the unsplit infinitive, we observe an increase from the 1810s until the 1920s and a slight decrease after that (notwithstanding periods of ups and downs at regular intervals).

A more elaborate method is proposed by Gries & Hilpert (2008): Variability-based Neighbor Clustering (VNC). It is similar to hierarchical cluster analysis (HCA, see this post), i.e. a method that displays a hierarchy of clusters, typically in the form of dendrograms with branches and leaves.^{2} The figure below exemplifies what a typical dendrogram looks like.

The dendrogram is based on the frequencies of the 1600 most frequent types of the split infinitive found in COHA. HCA takes as input a contingency table which is then converted into a distance matrix with a distance metric (here Euclidean). Next, an amalgamation rule is applied. It specifies how the elements in the matrix are clustered. The decades that are the most similar are amalgamated first. The plot is complete when all clusters have been joined.

The clusters make sense. We see that the decades 1990s and 2000s belong to the same cluster. This is hardly surprising insofar as these two decades correspond to a period of dramatic increase for the split infinitive.

Here is the R code used to generate the dendrogram.

`# loading the data`

`rm(list=ls(all=TRUE)) # clear R's memory`

`data <- read.table("https://www.nakala.fr/nakala/data/11280/51e80f33", header=T, sep="\t", row.names=1) # load the data`

`head(data) # inspect the contingency table`

`names(data) <- gsub("X", "", names(data)) # customize the header`

`# running HCA`

`method <- "euclidean" # select the distance metric`

`dist.object <- dist(t(data), method=method) # create the distance object`

`dist.matrix <- as.matrix(dist.object) # convert into a dissimilarity matrix`

`clusters <- hclust(dist.object, method="ward.D") # amalgamate the cluster`

`plot(clusters, sub=paste("(", method, "Ward)", sep=" ")) # plot the dendrogram`

VNC proceeds likewise except that it does justice to the chronological linearity of linguistic developments by not separating adjacent decades. When VNC is run on the basis of a single string of frequency values, it uses the standard deviation as a similarity measure and averaging as amalgamation rule.

We apply VNC to the frequency development of the split infinitive. The script proceeds as follows. It takes as input the sequence of frequencies shown in the frequency table below.

For each pair of adjacent decades, the algorithm determines the standard deviation.^{3} For example, for the decades 1830 (frequency = 10) and 1840 (frequency = 25), the standard deviation is 10.61 (see leftmost column in the table below). In R, the standard deviation is calculated with the `sd()`

function.

`round(sd(c(10,25)), 2)`

`[1] 10.61`

In the first iteration (leftmost column in the above table), VNC finds that the two decades with the smallest standard deviation are 1810 and 1820. These two are therefore merged first and assigned a mean frequency of \frac{4+4}{2}=4. This new value, combined with the others (the column “freqs” in Iteration 2) serves as the basis for the second VNC iteration. This time, the closest neighbors are [1810-1820] and 1830 (standard deviation = 3.46).

`round(sd(c(4,4,10)), 2)`

[1] 3.46

These two periods are merged and a mean value is computed for the third iteration.

As VNC proceeds through the iterations (five of which are displayed in the above table), the time groupings become larger until all 20 decades are merged into a single large cluster. This happens when the last decade (2000) is merged with the last-but-one cluster.

The script displays the clusters in the form of a dendrogram.

The dendrogram shows which periods have been merged. The heights of the clusters indicate is a measure of how different they are with respect to each other. The clusters merge at the heights of the cumulative sums of standard deviations.

We see that, indeed, the decades 1810 and 1820 are merged first, and that the height at which they are merged corresponds to their standard deviation (0). The next two periods to be merged are [1810-1820] and 1830, which display the smallest standard deviation (3.46) in the second iteration of the algorithm. The first two standard deviations add up to 3.46, which is the height of the second cluster.

We are left with a question: how many development stages are considered relevant to the diachronic study? The scree plot below allows the linguist to decide by comparing the distances between successive mergers. This is measured, once again, with standard deviations.

As we move from left to right, the difference between mergers decreases gradually. We start from the peak of the slope (on the left), go down a steep decrease, and count the number of clusters until we reach a point from where the slope levels off. The graph shows that the greatest differences are found between the 5 leftmost clusters, which are in fact the last 5 clusters. The data suggest that partitioning the 1810-2000 period into 5 development stages is relevant. Of course, we can perfectly decide otherwise and add more clusters, because the farther right we go, the more information we get. But the farther right we go, the less each additional cluster captures substantial information.

We end up with the following time partitions:

- 1810-1860
- 1870-1910
- 1920-1960
- 1970-1980
- 1990
- 2000

Given that the decades 1990 and 2000 are characterized by dramatic increase, we can arguably merge them. It looks like the split infinitive did undergo some change between the 1960s and the 1970s. However, this shift is but the tip of the iceberg. The split infinitive has undergone other shifts at varying rates over the last two centuries.

There is more to VNC than what I have shown here. Extended applications such as VNC on the basis of multiple measurements, detection of outliers with VNC, and VNC and constructional change can be found in Hilpert (2013: Sect. 2.3).

#### References

Desagulier, G. (2017). *Corpus Linguistics and Statistics with R. Introduction to Quantitative Methods in Linguistics.* New York, Springer.

Gries, Stefan Th., and Martin Hilpert (2008). “The identification of stages in diachronic data: Variability-based Neighbor Clustering“. *Corpora 3*: 59-81. (help page)

Hilpert, Martin (2013). *Constructional Change in English. Developments in Allomorphy, Word Formation, and Syntax*. Cambridge: Cambridge University Press.

*Around the word*, 23/05/2019, https://corpling.hypotheses.org/2551.

- This was, in fact, my very first post on this blog. [↩]
- For details on this method, see Sect. 10.6 of my book. [↩]
- “The standard deviation (σ) is the most widely used measure of dispersion. It is the square root of the variance.” (Desagulier, 2017: 148) [↩]