Archetypal Analysis (AA) is a method for analyzing data by looking at its convex hull and not by looking "inside" it, as is the practice in most of other procedures.
For example in Cluster Analysis (CA) we seek for a finite number of clusters, each one with its centroid (some times called "medoid") that best describes all observations. So in CA we perform "centroid analysis".
In AA we perform "boundary analysis" since we search for the best boundary points that can describe all other points as convex combinations of the boundary ones.
(We suppose that readers of this question are familiar with AA)
There exist some methods for computationally AA in R programming.
The most common is that of package "archetypes".
But there exist a serious problem with that package:
*It uses matrix inversion during the computations, so it can crash even for a small data set
Our contribution is next package:
https://CRAN.R-project.org/package=archetypal
which is based on Morup and Hansen PCHA method.
What do you think about?
Please look at the following below links which may help you in your analysis:
https://www.jstor.org/stable/1269949?seq=1#page_scan_tab_contents
https://www.researchgate.net/profile/Christian_Bauckhage/publication/282733207_Archetypal_Analysis_as_an_Autoencoder/links/561a81e208ae78721f9f8a30.pdf
https://link.springer.com/chapter/10.1007/978-3-642-03798-6_28
https://arxiv.org/pdf/1901.09078.pdf
Thanks
Dear Isam,
Thank you for your list.
I was aware of all except last one which I found closer to my question.
Have you any practical experience on using AA?
If yes, what is your preferred method?
Let me give an example in R language:
> library(archetypal)
> library(archetypes)
> data("wd2")
> archetypes(wd2,9,verbose = T)
Archetypes object
archetypes(data = wd2, k = 9, verbose = T)
Convergence after 1 iterations
with RSS = NA.
Warning message:
In archetypes(wd2, 9, verbose = T) :
k=9: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
As you see package "archetypes" was not bale to give a solution for k=9.
This is critical because in order to find the optimal number of archetypes, we must run AA for k=1 to 12 or even 15 some times.
Lets run for k=1:15:
##################################################
> sink('example.txt',split=TRUE)
> for(k in 1:15){
+ cat(k,'\n')
+ archetypes(wd2,k=k,verbose =TRUE)
+ }
1
1: rss = 0.10254644, improvement = 0.02308369
2: rss = 0.10254644, improvement = 0.00000000
2
1: rss = 0.09971716, improvement = 0.00861427
2: rss = 0.09934102, improvement = 0.00037613
3: rss = 0.09896461, improvement = 0.00037642
4: rss = 0.09859682, improvement = 0.00036779
5: rss = 0.09825304, improvement = 0.00034377
6: rss = 0.09793678, improvement = 0.00031627
7: rss = 0.09765651, improvement = 0.00028026
8: rss = 0.09741469, improvement = 0.00024182
9: rss = 0.09721304, improvement = 0.00020165
10: rss = 0.09704850, improvement = 0.00016454
11: rss = 0.09692509, improvement = 0.00012342
12: rss = 0.09683079, improvement = 0.00009429
13: rss = 0.09675683, improvement = 0.00007397
14: rss = 0.09669771, improvement = 0.00005912
15: rss = 0.09664901, improvement = 0.00004869
16: rss = 0.09660917, improvement = 0.00003985
17: rss = 0.09657711, improvement = 0.00003205
18: rss = 0.09655110, improvement = 0.00002601
19: rss = 0.09652982, improvement = 0.00002128
20: rss = 0.09651227, improvement = 0.00001755
21: rss = 0.09649771, improvement = 0.00001456
22: rss = 0.09648555, improvement = 0.00001216
23: rss = 0.09647533, improvement = 0.00001022
24: rss = 0.09646670, improvement = 0.00000863
25: rss = 0.09645939, improvement = 0.00000732
26: rss = 0.09645574, improvement = 0.00000364
27: rss = 0.09645560, improvement = 0.00000014
28: rss = 0.09645584, improvement = -0.00000024
3
1: rss = 0.04696980, improvement = 0.07156993
2: rss = 0.03025714, improvement = 0.01671266
3: rss = 0.02373515, improvement = 0.00652199
4: rss = 0.01960032, improvement = 0.00413484
5: rss = 0.01842036, improvement = 0.00117995
6: rss = 0.01745469, improvement = 0.00096567
7: rss = 0.01696749, improvement = 0.00048721
8: rss = 0.01684073, improvement = 0.00012676
9: rss = 0.01680839, improvement = 0.00003234
10: rss = 0.01683802, improvement = -0.00002964
4
1: rss = 0.04078924, improvement = 0.06587195
2: rss = 0.03115873, improvement = 0.00963051
3: rss = 0.02505424, improvement = 0.00610449
4: rss = 0.02304118, improvement = 0.00201306
5: rss = 0.02156311, improvement = 0.00147807
6: rss = 0.02001002, improvement = 0.00155309
7: rss = 0.01826702, improvement = 0.00174300
8: rss = 0.01660279, improvement = 0.00166423
9: rss = 0.01517824, improvement = 0.00142455
10: rss = 0.01427966, improvement = 0.00089857
11: rss = 0.01384261, improvement = 0.00043705
12: rss = 0.01368893, improvement = 0.00015368
13: rss = 0.01368129, improvement = 0.00000764
14: rss = 0.01371003, improvement = -0.00002874
5
6
7
1: rss = 0.03651477, improvement = 0.06664168
2: rss = 0.02098797, improvement = 0.01552680
3: rss = 0.01387426, improvement = 0.00711371
4: rss = 0.01020294, improvement = 0.00367131
5: rss = 0.00819713, improvement = 0.00200581
6: rss = 0.00714054, improvement = 0.00105659
7: rss = 0.00667618, improvement = 0.00046436
8: rss = 0.00632375, improvement = 0.00035243
9: rss = 0.00601828, improvement = 0.00030547
10: rss = 0.00573725, improvement = 0.00028103
11: rss = 0.00548256, improvement = 0.00025468
12: rss = 0.00525102, improvement = 0.00023154
13: rss = 0.00504151, improvement = 0.00020951
14: rss = 0.00485215, improvement = 0.00018936
15: rss = 0.00468425, improvement = 0.00016790
16: rss = 0.00453256, improvement = 0.00015169
17: rss = 0.00439018, improvement = 0.00014238
18: rss = 0.00425074, improvement = 0.00013944
19: rss = 0.00411485, improvement = 0.00013589
20: rss = 0.00397799, improvement = 0.00013686
21: rss = 0.00384251, improvement = 0.00013548
22: rss = 0.00370991, improvement = 0.00013260
23: rss = 0.00358141, improvement = 0.00012850
24: rss = 0.00345766, improvement = 0.00012375
25: rss = 0.00333695, improvement = 0.00012071
26: rss = 0.00322116, improvement = 0.00011579
27: rss = 0.00311207, improvement = 0.00010909
28: rss = 0.00302353, improvement = 0.00008854
29: rss = 0.00297040, improvement = 0.00005314
30: rss = 0.00295257, improvement = 0.00001782
31: rss = 0.00295019, improvement = 0.00000239
32: rss = 0.00295283, improvement = -0.00000265
8
9
10
11
12
13
14
15
Warning messages:
1: In archetypes(wd2, k = k, verbose = TRUE) :
k=5: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
2: In archetypes(wd2, k = k, verbose = TRUE) :
k=6: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
3: In archetypes(wd2, k = k, verbose = TRUE) :
k=8: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
4: In archetypes(wd2, k = k, verbose = TRUE) :
k=9: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
5: In archetypes(wd2, k = k, verbose = TRUE) :
k=10: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
6: In archetypes(wd2, k = k, verbose = TRUE) :
k=11: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
7: In archetypes(wd2, k = k, verbose = TRUE) :
k=12: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
8: In archetypes(wd2, k = k, verbose = TRUE) :
k=13: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
9: In archetypes(wd2, k = k, verbose = TRUE) :
k=14: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
10: In archetypes(wd2, k = k, verbose = TRUE) :
k=15: Error in qr.solve(alphas %*% t(alphas)): singular matrix 'a' in solve
> sink()
################################################
As you see package "archetypes" failed for k in {5,6,8,9,10,11,12,13,14,15}
But with package "archetypal" we can find the optimal k by running 10 times for each k, in parallel computing:
> yy=find_optimal_kappas(wd2,15)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time difference of 3.109115 mins
> yy$kappas
[1] 3
Please see the vignette after installing the package:
vignette("archetypal", package = "archetypal")
Thank you.
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