Abstract
Students of human cognitive and cultural processes, social networks, pattern recognition and machine intelligence often find that the coordinate systems resulting from commonly used measurement and analysis tools yield non-Euclidean configurations. Typically, researchers consider this unfortunate, and seek methods to return the spaces to Euclidean configurations. This article details all the known methods of such transformations, but presents evidence from multiple fields of inquiry that shows the non-Euclidean nature of the space is meaningful, and that all transformations to Euclidean form produce serious distortions to measured values. The article further presents methods for describing processes in the non-Euclidean spaces along with empirical examples of such uses.
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Notes
Not all investigators choose to make use of the spatial coordinate systems in their research. Some social network analysts, for example, prefer to use specifically network based models such as graph theory and the like, and eschew spatial representations (Lee and Tkach-Kawasaki 2018; Shapiro et al. 2018; Danowski and Park 2014). This paper does not advocate for spatial modelling over alternative approaches, but provides some cross-disciplinary findings that may be useful whenever spatial modelling is appropriate.
The extent of Thurstone’s focus on lines rather than spaces is evident from the title of his 1935 book and his presidential address to the APA: Vectors of the Mind.
The first employment of the concept of the field was provided by Kurt Lewin in 1935, but his main work on the topic didn’t emerge until 1951, and even then, his use of the term was vague and far from operational (Lewin 1951).
Psychologists, sociologists and communication scientists used the term “metric” to distinguish this method from newer “non-metric” methods, but the procedure seldom met the mathematical criteria for a metric space. See method 5 below.
In the social science community, “high dimensional” usually means “more than three” and often “more than two.”
This issue cropped up among factor analysts also in the presence of negative eigenvalues, which were virtually always treated as indicators of measurement error and a sign that further extraction of roots should cease.
This is not meant as an insult to the intelligence of social scientists. Virtually every member of Western culture believed this uncritically; cf. the physicist Erwin Schrödinger at about the same time (1950): “For the observing mind is not a physical system, it cannot interact with any physical system.” (Schrödinger 1996, Emphasis in original).
Another set of collective representations recognized by Durkheim are the artifacts created by the culture, Among the most important of these are texts, and the Galileo community usually examines these using Catpactm, an unsupervised neural network that calculates the synaptic connection weights among words in the text using a propinquity-based algorithm (Woelfel 2014). The matrix of connection weights is then projected onto orthogonal coordinates using the Galileo algorithm. Discussion of this second method of research is beyond the scope of the present paper.
The magnitude estimation complete paired comparison measurement method is chosen for two reasons: first, it is the most precise psychometric technique available, and second, it is the only psychometric measurement procedure consistent with the definition of measurement in physical science and engineering: comparison to some standard.
Most social scientists, in fact, do reject the standard definition of measurement—comparison to some standard—and replace it with a much broader definition: assignment of numbers to observations according to some rule. Most social science measurement devices, e.g., five-point scales, rank-orders, etc., would not be recognized as measurements by scientists and engineers (Torgerson 1958).
The following Figures show only the first three eigenvectors of these spaces solely for illustrative purposes. These are all multi-dimensional, non-Euclidean spaces, and all calculations are based on the entire eigenstructure.
Recent research with Word2Vec have shown that vector operations on the Wod2Vec space can produce substantively meaningful combinations of words (Mikolov et al. 2013).
Rotation of non-Euclidean spaces is a generalization of earlier “Procrustes” procedures, modified to deal with imaginary coordinate values (Hsieh 2004; Woelfel and Fink 1980; Woelfel et al. 1980, 1989; Woelfel and Barnett 1982). Unlike the original Procrustes myth, the legs of the guests are not stretched or amputated to fit the bed, but non-Euclidean distances remain invariant under these rotations.
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Woelfel, J. Convergences in cognitive science, social network analysis, pattern recognition and machine intelligence as dynamic processes in non-Euclidean space. Qual Quant 54, 263–278 (2020). https://doi.org/10.1007/s11135-019-00852-2
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DOI: https://doi.org/10.1007/s11135-019-00852-2