Analytical Overview

1. Analytical Fieldmarks

Statistical and Relationship Fieldmarks compare features of individual khipus such as knots, and sums between cords. Analytical Fieldmarks, on the other hand, compare khipus to one another as a whole, using metrics derived from the whole khipu, as opposed to individual parts of the khipu. Three methods of analysis have been tried:

  • Clustering: Clustering, groups khipus that are similar, together. The first example of this is in the Khipu Field Guide’s Khipu Analysis, where khipu were converted into their textual representation and then textual machine learning techniques were used to cluster the data. Clustering showed promise as an investigatory tool. For example, the initial textual representation clustering showed that calendar khipus group together. Other approaches were used to discover duplicate khipu. These approaches, however proved to be less successful.

  • Benford’s Law: Benford’s Law provides a metric for a khipu’s numerical values distribute - do a khipus values distribute randomly, or do they distribute in a human fashion (i.e. lots of small digits, few large digits). Using Benford’s Law provides a surprising result and indicates that most khipus are of a human, numerical/accounting nature.

  • Zipf’s Law: Herman Zipf discovered that human language obeys certain distributional measures. For example, commonly used words, have shorter lengths than rarely used words; that is, word_length is inversely proportional to the frequency of the word being used. The distribution of human language by this measure is an inverse hyperbolic. I call this the relationship between ubiquity and uniquity; how common, or rare, are individual words, from the set of all words we have documented? Astonishingly, using Zipf’s Law of word frequencies, we can investigate if khipus encode language, without knowing the actual underlying language!

2. Clustering

2.1 What Constitutes “Similarity” and “Difference”?

Machine-Learning offers the tantalizing vision of being allowed to sort khipus based on “similarity”.

Through the course of this study I have performed various clustering techniques on the khipu database. The results have been reasonable, but unspectacular, and in the end, not very useful or productive in revealing information. Clustering, tried previously by Urton and Brezine (see Khipu Typologies in THEIR WAY OF WRITING - Scripts, Signs, and Pictographies in Pre-Columbian America by Boone & Urton), showed promise as an investigatory tool. But…

The first approach was to try Hierarchical Clustering on a document-based representation of a khipu. In this approach, the khipu describes itself as text into a document - ie. “red cord with three knots”, etc. Then modern linguistic techniques are used to organize documents by similarity, creating a hierarchical tree description of the documents (khipus). This approach showed some success, notably calendar khipus grouped together, etc.

Encouraged by the initial success of this approach, I tried to find similar and duplicate khipus. Those results proved to be amusingly bad, and educational. To wit:

  • Clustering by Cord Color and/or Value - The following approaches were tried. All were inconclusive.
    • Joining Orphaned Khipus - A search for a khipu similar to UR231
    • Identifying Duplicate Khipus - A search to identify khipus that were duplicates. A failure. Why? I suspect because transcription of khipus is notoriously inconsistent. A shoe leather approach, looking at museum #’s, etc., by Manuel Medrano, was used instead.
  • Clustering by Image - A clustering algorithm using the tiny thumbnail images of each khipu (generated in the khipu field guide) as input to an image clustering algorithm that looks at various “edge features” of an image to sort (ie. these all look like dogs, these cats, etc). This proved to be much more succesful at identifying duplicate khipus. Why? Because, at the very tiny thumbnail size, the most useful image feature to the clustering algorithm was the aspect ratio of the image. Duplicate khipus, while they may have not had similar cords, cord values, or cord colors, did have similar overall measurements. So much for sophisticated machine-learning algorithms!

When looking at duplicate khipus, it was astonishing how different they looked in the khipu field guide. It revealed, stunningly, how inconsistent khipu recordings can be. For example, compare the following two recordings/images, AS208 and UR083, both of the SAME khipu. Take a moment to see why simply comparing cord values and colors may fail:

AS208


UR083

Do you see what happened? A slow glance will reveal that Urton chose to read the khipu in the reverse direction from Ascher. While a clustering search based on comparing cord values and colors might be effective at locating duplicate khipu, it is unlikely to be more informative about the nature of khipu in general!

Incidentally, the study of white as a leading pendant color on sum clusters, indicates that Ascher’s reading is likely the correct direction.

3. Benford’s Law

Like, Hierarchical Clustering, Benford’s Law was previously examined in the Khipu Field Guide. A more in-depth analysis of Benford’s Law is provided here. The Benford Law analysis indicates that the majority of cataloged khipus are accounting in nature.

4. Zipf’s Law

Another approach to understanding if khipus are linguistic in nature, is an examination of their various distributions of color, cord-value, etc., according to Zipf’s law. Zipf’s Law, as a linguistic measure has been extensively studied, and it’s general conclusions apply to all forms of human language.

Zipf’s Law, like Benford’s Law is a similar power curve, although it is different in distribution than Benford’s Law. The study of khipus, using Zipf’s Law, once again, diminishes the argument that khipus are linguistic in nature.

5. Conclusions

  • Inferences on cord colors and values across khipus is difficult due to differences in measurement techniques and approaches by their measurer. The only things that seem consistently accurate are cord lengths and long-knot locations. As a consequence, clustering is not a very reliable technique for khipu analysis.
  • The Ascher sum relationships study shows that the majority of khipus have some sort of summation relationship, indicating that khipus are an “accounting” of something.
  • An study of Benford’s Law indicates that the majority of khipus are an “accounting” of something.
  • Applications of Zipf’s Law to various possible information encodings, such as color or cord value, indicates that the majority of khipus are not likely to be a linguistic representation.