Presented below are two patterns created from the data set: 1024 cities that follow precisely Zipf’s law, which implies that the first largest city is size 1, the second largest city is size 1/2, the third largest city is size 1/3, and the smallest city is size 1/1024. The left pattern was created from head/tail breaks (https://en.wikipedia.org/wiki/Head/tail_Breaks) that is a new classification scheme for data with a heavy-tailed distribution (Jiang 2013). The classification scheme partitions the data around its average into two unbalanced parts: those above the average for big values in the head (a minority), and those below the average for small values in the tail (a majority); this partition process continues recursively for the head or the large values until the notion of far more small values than large ones is violated. For this particular data, it ends up with 5 classes. The right pattern is created by natural breaks (Jenks 1967), with which variance within classes is minimized, while the variance between classes are maximized. The natural breaks is in fact the same as k-means clustering (https://en.wikipedia.org/wiki/K-means_clustering).
Jenks G. F. (1967), The data model concept in statistical mapping, International Yearbook of Cartography, 7, 186–190.
Jiang B. (2013), Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution, The Professional Geographer, 65 (3), 482 – 494.
I have argued that the head/tail breaks helps revealing the underlying scaling of far more small values than large ones for data with a heavy-tailed distribution (Jiang 2013). My question to you is, do you feel more alive with the left pattern that is created by head/tail breaks? Or alternatively, do you feel the left pattern is more beautiful? do you feel you are more comfortable with the left rather than right pattern?
Yes, i do feel that i prefer the left one, but i can't argue if somebody disagrees.
The left one have more random big circles, while the right one have one at the edge, and three others along a line.
To me the left side is nicer to look at. The data seem spread better with lots of small dots and a handful of mid sized dots and few large dots. The right images seems like a lot of zeros with a few values and does not seem as informative...
What I am asking or conducting is in effect the mirror of the self test (Alexander 2002-2005), with which we can get to know which pattern is more beautiful. In the theory of centers, Christopher Alexander (2002-2005) argued that this kind of feeling belongs to 90 percentage of people. In other words, the feeling is NOT idiosyncratic, but as a connection to human beings. I have recently developed a mathematical model of wholeness, with which we can not only know why a structure or design is beautiful, but also how much beautiful it is.
Alexander C. (2002-2005), The Nature of Order: An essay on the art of building and the nature of the universe, Center for Environmental Structure: Berkeley, CA.
Jiang B. (2015), Wholeness as a hierarchical graph to capture the nature of space, Preprint: http://arxiv.org/ftp/arxiv/papers/1502/1502.03554.pdf
@Saleh Dinparvar
Yes, i do feel that i prefer the left one, but i can't argue if somebody disagrees.
If someone disagrees, you could show him/her the histograms of the two classifications or patterns (see blow). The left histogram corresponds the left pattern by head/tail breaks, while the right histogram for the right pattern by natural breaks. I believe you feel more comfortable with the left bar chart rather than the right bar chart.
Alternatively, you could conduct the head/tail breaks yourself with the tool: https://github.com/digmaa/HeadTailBreaks
You will see that the scaling pattern of far more small values than large ones recurs four times, so ht-index is 5, or 5 classes so to speak.
In summary, this kind of beauty lies in the deep hierarchical structure, being objective rather than subjective.
I do also prefer the left pattern, it looks that it is actually more alive than the right one. When I looked to the right pattern, I thought there is not much to look at/explore, except the 6/7 biggest circles. However, the left pattern reveals that some of the "minor" features do also exist (visually speaking) and this pushed my eyes not to ignore them but to study the distance to the biggest features and see if they present some clustering/cluttering/overlap.
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