24-10-2026, 04:20 PM
(24-10-2026, 02:08 PM)npcompl33t wrote: You are not allowed to view links. Register or Login to view.You known that the graph is just showing the 1st two dimensions of a path that moved through more than 2 dimensions right?I am not trolling.
The situation in the original n dimensions can only be worse than what we see on the plot. In R^n, each of those clusters may split into two or more clusters. The clusters that, in the plot, appear elongated in the direction tangent to the circle may be elongated away from it.
If there was indeed a clear alignment of the points along a parabola or a circle in R^n, the alignment would only be only more evident in the MDS1 x MDS2 plot. That plot must necessarily be the best view of that hypothetical curve.
As I wrote, even if we consider it significant that the four main clusters fall on a circle, and even if that is still true in R^d, the order of traversal of the circle and the first cluster in that ordering are not determined by the data.
Quote:It seems to me like he does indeed recover tomb chronology even if some ordering is ambiguous, or not exact, which was my claim.
I agreed that the the chrono order of the clusters does happen to agree with one of the eight possible orderings suggested by the geometry.
But, again, that need not have been the case. We do expect the data of burials from M well-distinguished epochs to form M well-distinguished clusters in R^n. By why would the style of burials change along a circle (or any other simple curve) with time? Even if we expect some continuity between the epochs, so that the clusters trace a sort of continuous path in R^n, that path can snake randomly in R^d, not confined to a plane (as a parabola or circle must be).
That reminded me that someone once analyzed a large set of time series and concluded that there was a strong correlation between the length of miniskirts and the incidence of earthquakes, or something of the sort.
There are 24 possible orderings of the 4 clusters, so there is a 33% chance that one of the 8 geometric orders would coincide with the chrono order just by luck.
In order for the n properties of the style of burials to vary along a parabola in R^n space, with time as the parameter, there should be two significant factors whose influence dominates all those attributes; and one of them should increase monotonically with time, while the other should increase for a while then decrease. Why should we expect such factors to exist?
And there is no sign that the order of burials within each cluster is correlated with position along the circle (except roughly in cluster D). Again, that can only be worse in R^n than in the MDS1 x MDS2 projection.
Quote:He still identifies a dominant overall time dimension, that holds up under radiocarbon dating.No, he doesn't. He identifies 8 possible geometric orderings, and one of them happens to coincide with the time ordering of the clusters.
And that is one case where the "horseshoe" theory sort of worked, kinda. How many cases are there where it does not work?
All the best, --stolfi
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