simulating vases

[Armand1]

Arianna and Stephen claim that their fancy math gives a better description of change in shape space than some simple-minded approach. But does it? How do they know that -- or even propose to show it? Surely, to do so they must have a set of shapes that they know a priori are a certain relative distance from each other --- and then show that method captures that distance better than some other one.

But they don't have such a set of shapes. So, let's generate one. Starting with an average Mantamados vase, and taking account of the covariance structure among the variables (the y values) I generated a 500 vase lineages of 200 time steps each by a random walk process = 10,000 vases in all. Some of the lineages diverge really far from the original, others do not. Here is one of the most extreme:

The red vase is the original. You can see that, over time, the vases in this lineage become ever more extreme: in fact, they're evolving from an amphora to something like an arabyllos!

start:

end:

Well, kinda. Anyway, here's the idea: you have 500 lineages, each of which is progressively diverging from the original vase. Since they're evolving by a Random Walk, I believe that they should diverge from the original as a linear function of time. Certainly the variance among lineages should increase linearly with time. I haven't looked at the latter yet, but here's a suggestion of what you might do. For one lineage --- vase series 1, shown above --- I estimate the distance between the vase at each time step and the original and plot it. I estimate the distance in the most simple minded way possible: by euclidean distance:

I find that the distance does not increase linearly with timestep (the best-fit linear regression, forced through the intercept is shown in red): instead, over time, the gain in distance per timestep clearly decreases. Hmmm. Could it be that Arianna and Stephen are right when they say that Euclidean distance really doesn't describe evolution in shape space accurately? Should we be looking at shape evolution in non-Euclidean space?!

Now, there was a certain amount of fiddling that went on to make these vases -- including moving averages to make the contours pretty. And I am not totally convinced that the way in which each vase, at each timestep, was generated, really does result in a random walk of shapes (even if each individual variable does so evolve). So I am not totally sure that the asymptoting distances of the lineages aren't an artifact of the way in which the vases were generated. However, for the moment I think not. And here's the thing: I have 10,000 vases of a known evolutionary history for Arianna to test her methods on -- any distance she pleases. Perhaps, on reflection, Arianna's Brunel supervisors where not so silly to insist that she study simulated shapes...

Now, my real interest in generating fake vases is to test evolutionary hypotheses about the production of vase diversity. The data upon which these simulations were based (the COV matrix among the vases) are clearly not good enough for that. But it's a step in the right direction.

Arianna can find the vase data here in a dropbox folder called
simulated vases for Arianna 131119

There are two files: RW_vases_long.csv which has the data in long format, and RW_vases_wide.csv in wide. The original vase is "vaseseries==0". All the rest are simulated vases: vaseseries designates the lineage; timestep designates the timestep; x and y, the coordinates that describe each vase.

[Armand1]

I can check whether the variance increases linearly for each individual variable among vase series as predicted by a random walk: reminder: there are 200 variables, the points along the y axis that describe each vase. They do:

So, at least the individual variables are diverging among vase series as you would expect in a random walk. Of course, some variables diverge rapidly, others slowly: how fast they do so depends on the structure of the original COV matrix. So, it really does seem to me that, if the individual variables are diverging linearly, then a good shape descriptor, that accurately captured the increase of their "variance" in multivariate space, would diverge linearly too.

I can also check whether (for vase series 1 --- above) the variance increases linearly for each individual variableover increasingly larger intervals. Here this is estimated as VAR(value_t=0, value_t+1), VAR(value_t=0, value_t+2), VAR(value_t=0, value_t+3) etc. This, too, should increase linearly in a RW and, as we can see, it does:

Equivalently, I can ask how the euclidean distance behaves for the same vase, same variables, over the same intervals, and we find that, like the multivariate counterpart (above) it does not increase linearly but asymptotes:

And we can drive this home by looking, for the same data, at the relationship between variance and euclidean distance (colours as intervals)

Which really shows how, although in a RW the variance increases linearly with interval, the euclidean distance does not.

So now I am reasonably convinced that my vases are evolving as Random Walks and that a good distance metric -- one that accurately captures how they are evolving into uncharted shape territory -- should show a linear increase time for these data and not asymptote as euclidean distance does. (My clever mathematician colleagues will say this is well known, that this is indeed the point, but I still think the exercise -- and the data -- are useful.)

[Armand1]

We can also build a phylogeny of our lineage (vase series 1) using Neighbor Joining based on euclidean distance. We should expect it to have a real step-ladder like topology since they're evolving one step at a time. The tree is rooted at the original vase. As expected, for the first half of the tree (green) the NJ tree accurately represents the relationships: adjacent time steps are sister taxa. However, as time goes by (red) it becomes increasingly inaccurate. That illustrates the consequences of a poor distance measure for phylogeny construction.

Actually, this suggests a way of testing distance measures --- but perhaps it is unduly complicated, and you'd surely want to simulate your phylogeny explicitly. (Here the "sisters" aren't really sisters.)

[Armand1]

I remain concerned, however, that the way I generated the shapes from the individual variables is not quite kosher.

Here is the problem. If you start with 200 x 1 (y, x) matrix of values that describes a vase, and allow the 200 variables to each evolve as a random walk, even if you constrain their evolution by a COV matrix, you'll find that the vase profile dissolves into a shapeless jagged line as y_1 adopts a high value, y_2 becomes negative etc.

But I wanted to assemble coherent vases from my 200 variable x 500 replicates x 200 timestep Random walks. To do this, in each generation, I sorted the values for each variable from largest to smallest. I then ``glued'' the largest y_1 value to the largest y_2 and largest y_3 value...largest y_200 values together to make a vase. And did the same with the second largest, and third largest values of each... all the way to the smallest. The result are the smoothly evolving "lineages" that you see. But they're not really lineages since the values that go into the "largest" vase (the one with the greatest x values) in each generation need bear no relation to those in the previous generation (they need not come from the same random walk). So any one pseudo-lineage really isn't doing a random walk in shape space. I mean, upon reflection, it's obvious: they don't LOOK like random walks. In fact, the values are sorted to necessarily result in directional evolution of each pseudo-lineage to varying degrees.

The simulated shapes may still be useful for Arianna, but they cannot be used to test evolutionary hypotheses.Still I remain convinced that it should be possible and desirable to genuinely evolve coherent shapes via a random walk or some other process --- perhaps from some shape parameterization. Here is a paper that alludes to sampling random shapes from a pre-defined space. I don't understand it.
Cho2019_Article_ElasticStatisticalShapeAnalysi.pdf

[Armand1]

New RW simulations! These, too, are based on the Mantamados vases (a combination of the small, medium and large ones that Arianna got the contours for). However, this time, as discussed, I projected them into PC space. It turns out that the first 3 PCs account for 94% of the variance. I then simulate an RW for PCs 1:3, starting with the average vase which of course, takes the values 0,0,0. I run 500 simulations for 500 time steps. Importantly, this time I do not sort the results before glueing each vase together: Here the vases from a given series (e.g., vase 1_1, vase 1_2, vase 1_3, where the first number designates the run, the second the timestep, are lineally related since the values for each come from the same run. So this should be a genuine RW in shape space. I then project the vases back into 200-variable space, where each variable is a y value, to retrieve intelligible shapes. This is what I get for Vase series 1:

Pretty nice! (the original average Mantamados vase is in red.) I have not smoothed anything: even so we get a coherent sequence of vases.

I have fiddled the data in one way. Vases are functionally constrained in that the x values at the lip (y_1) and base (y_1=200) must be at least as large as any other y value in-between. (A vase must, after all, have a lip and a base that closes. To ensure this I made the x values for y_1 and y_200 == max(x) for a given vase. I think this is perfectly defensible --- though it may change any prediction of how shape space evolves under RW a bit.

It is clear that most of the change is happening between y_75 and y_150 or so. That is exactly as predicted since that is indeed where the original Mantamados vases showed the highest variance.

[Armand1]

And this shows you the diversity of shapes, for all series, present at timestep 500. Many vases have much more extreme shapes than those that evolved in series 1.

[Armand1]

So let's look at how euclidean distances behaves in our PC RW series. First we'll look at vase series 1. We see that the distance, relative to the start, dances all over the place. But this is as expected in a RW!

These results make sense in terms of the vases. For example, we see a peak of distances around time step 25. Here are the vase profiles

Then we see a return to the original shape (more or less) around time step 400:

Finally, at the end of the series, the vases are moving out into novel space again:

So euclidean distance seems to be doing a qualitatively reasonable job of describing the difference between the original vase and any simulated vase. But is the problem that we diagnosed with euclidean distance still present? That is, does it fail to capture what should be a linear increase with distance over time? If we pool all 25,000 distances --- distance(original, evolved vase) --- and get the median (q50:red line), and CIs (10% and 90% quantiles: grey ribbon), as a function of timestep, we find:

So, again, euclidean distance does not increase linearly with time. Thus it still seems that we'd like a distance measure to do better than this -- but now we have a better dataset to test it on.

As discussed it would be really nice if Arianna could replicate these results independently on simulated vases of her own based, for example, on cups.

If she wants to study these vases they're in the data files PC_RW_long.csv and PC_RW_wide.csv in the dropbox folder

https://www.dropbox.com/sh/sqoikn0q1bmcbyl/AAADXx5iLmaGtcrCCyOXHUD-a?dl=0

I have made 25,000 pots. Does that make me the greatest potter of all time?

[Armand1]

Han has done something nice. He wondered why exactly we find that distance in our RW vases asymptotes with increasing interval, as observed above. So he did a bit of math and

[Armand1]

Thinking about how I generated the RW pots, I want to draw your attention to one funny thing. I don't believe that this is the cause of our problems with diffeomorphic methods, but it does have implications for comparing closed and open curves.

The issue concerns the way I construct whole pots from single curves. Now, RW curves (here in red) either swing to the right or the left of the original curve (here in blue). In both cases, to make a whole pot, I find the min(x) of the RW curve.

In A, below, the RW has swung to the right: min(x) is near the mouth. So I align it and get a pot that looks a lot like the original but has a wider belly.

In B the RW has swung to the left: min(x) is at the belly. So then I make that the centre, close the mouth and the base, and you get a very wide pot that is kinda the inverse of the original.

There is something weird about this. In my example, you are to suppose that the open RW curves in A and B are identical --- just mirror images of each other. If you were to do an SVRF on them, the difference would be obliterated (since they are rotationally identical). But do an SVRF on the closed pots, and they'd be very different.

I cannot believe that this is messing up our SVRF shape distances. I actually think that open curves A and B are different: to make B you have to travel further in space than to make A. But I raise this issue for the sake of clarity. And it may fuck up the expectation that, on average, the pots should be becoming more distant in a linear way.

Thoughts?