[Traminer-users] violation of triangle inequality with future based substitution costs

[Jeremy Reynolds]

Dear Traminer Experts,

I have been using future based substitution costs as described in Studer
and Ritschard 2014 and implemented in Traminer to make dissimilarity
matrices.  Unfortunately, the results often violate the triangle
inequality.  It seems to depend on what subset of my data I use, but I
don't really know why the triangle inequality is violated in some cases and
not in others.

Is there some appropriate way to fix this (e.g. alter one of the
substitution costs manually)?  Does the violation suggest that there is a
problem somewhere else in my analysis or suggest something about my data?
I have pasted the output from my latest analysis below in case it is
useful.

Thanks,

Jeremy


> ###########> # Calculate Substitution Costs using future similarity > ###########> > ######> #default lag of 1> ######> future <- seqcost(seq.hc, method="FUTURE", lag=1)   [>] creating substitution-cost matrix using common future... [>] computing transition rates for states 1/2/3/4/5 ...> dimnames(future) = list( c("M", "S", "F", "O", "U"), c("M", "S", "F", "O", "U"))> round(future, 4)       M      S      F      O      U
M 0.0000 0.1701 0.4147 0.6304 0.3960
S 0.1701 0.0000 0.2375 0.6125 0.3910
F 0.4147 0.2375 0.0000 0.7700 0.5552
O 0.6304 0.6125 0.7700 0.0000 0.4573
U 0.3960 0.3910 0.5552 0.4573 0.0000

> #largest substitution cost (F vs O)> maxsub <- max (max (future))> maxsub    [1] 0.7699533> #smallest substitution cost (M vs S)> minsub <- min(future[lower.tri(future)])> minsub     [1] 0.1701276



> ###### > #lag of 2 > ###### > future2 <- seqcost(seq.hc, method="FUTURE",
lag=2) [>] creating substitution-cost matrix using common future... [>]
computing transition rates for states 1/2/3/4/5 ... > dimnames(future2) =
list( c("M", "S", "F", "O", "U"), c("M", "S", "F", "O", "U")) > round(future2,
4) M S F O U M 0.0000 0.1015 0.2740 0.4063 0.2184 S 0.1015 0.0000 0.1692
0.4116 0.2428 F 0.2740 0.1692 0.0000 0.5508 0.3660 O 0.4063 0.4116 0.5508
0.0000 0.2577 U 0.2184 0.2428 0.3660 0.2577 0.0000

> #largest substitution cost (F vs O)> maxsub2 <- max (max (future2))> maxsub2     [1] 0.5508451> #smallest substitution cost (M vs S)> minsub2 <- min(future2[lower.tri(future2)])> minsub2    [1] 0.1014867



> ##########################################> # make distance matrices> #########################################
> disomf1i5 <- seqdistOO(seq.hc, method = "OM", indel = .5*maxsub, sm = future) [>] 10923 sequences with 5 distinct events/states [>] 10923 distinct sequences [>] min/max sequence length: 18/18 [>] computing distances using OM metric [>] total time: 59.34 secsWarning message: [!] at least, one substitution cost doesn't respect the triangle inequality.
 [!] replacing 1 with 2 (cost=0.1701276) and then 2 with 3 (cost=0.2375203)
 [!] costs less than replacing directly 1 with 3 (cost=0.4147297)
 [!] total difference ([1=>2] + [2=>3] - [1=>3]): -0.007081773
> disomf2i5 <- seqdistOO(seq.hc, method = "OM", indel = .5*maxsub2, sm = future2) [>] 10923 sequences with 5 distinct events/states [>] 10923 distinct sequences [>] min/max sequence length: 18/18 [>] computing distances using OM metric [>] total time: 59.04 secsWarning message: [!] at least, one substitution cost doesn't respect the triangle inequality.
 [!] replacing 1 with 2 (cost=0.1014867) and then 2 with 3 (cost=0.1691506)
 [!] costs less than replacing directly 1 with 3 (cost=0.27397)
 [!] total difference ([1=>2] + [2=>3] - [1=>3]): -0.003332678



-- 
********************
Dr. Jeremy Reynolds
Professor
Department of Sociology
116 Baldwin Hall
University of Georgia
Athens, GA 30602-1611
Phone: (706) 583-8072
Web: http://uga.edu/soc/people/faculty/reynolds_jeremy.php
Fax: (706) 542-4320
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