KBO clustering

I refer to this MPML message and to the associated conversation, in particular to the interesting comments made by Aldo Vitagliano, author of Solex orbit simulator, about the difficulty to see a cluster of KBOs.

I do not know the answer but this prompted me to look at the KBO's orbit parameter distribution. What follows next is an exercise ... so I do not claim that it is correct!

I used the Web service made available by MPC to look for KBOs characterized by:

 250 AU < a < 1000 AU

I found this list:

Asteroid	a	e	i	w	om
2006 UL321	260.7642	0.9099041	37.36673	-6.20276	-17.0798
2012 VP113	268.2509	0.7005635	24.0183	-66.968	90.88303
1996 PW	271.4848	0.9905821	29.72461	-178.15845	144.61041
2011 OR17	287.1875	0.9892086	110.33772	14.03442	-88.40106
336756	319.0439	0.9704821	140.81617	132.96423	136.19752
2013 RF98	325.1039	0.8883746	29.57957	-43.45499	67.57105
2004 VN112	339.0837	0.8604115	25.51725	-32.76479	66.06033
2010 GB174	364.8513	0.8670264	21.53648	-12.61199	130.60394
2015 DB216	418.7213	0.9800038	40.48768	-119.1578	3.27749
2010 BK118	484.3816	0.987395	143.90195	179.11098	175.97502
2007 DA61	518.3954	0.9948766	76.71542	-10.32894	145.98047
90377	539.668	0.8588315	11.92852	-48.91693	144.50447
2007 TG422	546.4473	0.934885	18.57957	-74.16184	112.98074
87269	586.5961	0.9645518	20.07087	-147.4484	142.37805
2002 RN109	746.6725	0.9963802	57.99165	-147.53176	170.49258
308933	780.1764	0.969011	19.46659	122.49628	-162.61603
2013 AZ60	990.1281	0.992013	16.52296	157.9403	-10.77058

Then I used the R package to produce a hierarchical cluster as follows:
1) I scaled the above table so that every column has mean 0 and variance 1.
2) I calculated the distance between any two rows (manhattan distance).
3) I submitted the scaled table to the function hclust choosing clustering method complete.
4) I used a further R function ( see rect.hclust ) to display colored rectangles at different height: the purpose is to help visualize the various clusters at different levels.

Result

I do not know, whether these clusters have a statistical significance or not.

The left cluster maybe interesting: in fact, it consistently maintains its shape even when you cut the dendogram at a level where the second big cluster gets split into 5 subgroups.

The left cluster contains asteroid 2012 VP113 plus other 4 even more similar asteroids.

From now on I will refer to this cluster as "cluster 2" as opposed to "cluster1" made by all other KBOs without further distinctions.

A nice R function is cutree: you can tag the original table with a further column i.e. the cluster where it is supposed to belong. By doing this, you can use, for example, the function ggpairs of the GGally library to make a set of plots like these:

• in the diagonal, you can see the density distribution, each cluster being given a different color. Cluster 2 is coloured in blue. 
• above the diagonal: you can see the correlation between pairs of parameters, with cluster detail and total as well.
• below the diagonal: you can see a scatter plot diagram of each pair of parameters, again the colour represents a cluster.

.... and, if you are interested in a specifc plot, you can make it alone with the ggplot function.

For example:

• let's see a scatter plot of orbital parameters a and w where we add the name of the asteroids
• finally, let's see the density distribution of orbital parameter w

a versus w

w density distribution

Kind Regards,
Alessandro Odasso