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Hashed Fingerprints and RNG’s

with 10 comments

In my previous post I looked at how many collisions in bit positions were observed when generating hashed fingerprints (using the CDK 1024-bit hashed fingerprint and the Java hashCode method). I summarized the results in the form of “bit collision plots” where I plotted the number of times a bit was set to 1 versus the bit position (for a given molecule). As expected, for a series of molecules we observe a number of collisions in multiple bits. What was a little surprising was that even for a symmetric molecule like triphenylphosphine (i.e., a relatively small number of topologically unique paths), we observed collisions in two bits. So I decided to look into this case in a little more detail.

As I noted, collisions could occur if a) different paths get hashed to the same int or b) two different hashes lead to the same random number. Modifying the Fingerprinter code, I was able to generate the list of paths calculated for triphenylphosphine, the hash code for each path and the bit position that was generated for that hash code. The data (hash value, path, bit position) is given below.

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-662168118  P-C:C-H 3
-1466409134 H-C:C-P-C:C:C   58
1279033458  C:C:C:C:C-P-C:C:C   78
1434821739  H-C:C:C:C:C:C   80
-429128489  C:C:C:C-P-C 85
-1779215129 C:C:C:C:C   95
-245205916  H-C:C:C-P-C:C:C:C   97
-475263438  C:C:C:C:C:C-P-C:C   111
-1466409532 H-C:C-P-C:C-H   114
1434821341  H-C:C:C:C:C-H   142
-428753296  C:C:C:C:C:C 161
-245206314  H-C:C:C-P-C:C:C-H   161
1730724873  H-C:C:C-P-C 167
43327460    H-C:C:C:C:C-P-C:C   180
1731099668  H-C:C:C:C-H 180
886716569   H-C:C:C:C   182
-1406488727 C:C:C:C:C-P-C:C 191
78342   P-C 193
1731100066  H-C:C:C:C:C 211
63670037    C:C:C   213
178815835   H-C:C:C:C:C-P-C 224
179190630   H-C:C:C:C:C:C-H 230
-469902821  H-C:C-P-C:C:C:C 244
1057365278  C:C:C:C 253
-181369445  H-C:C:C:C-P-C:C 266
1056990085  C:C-P-C 300
886341376   H-C:C-P-C   333
827737424   H-C:C:C 390
-912402715  P-C:C:C:C:C-H   406
1572927053  H-C:C:C-P-C:C-H 421
1434446546  H-C:C:C:C-P-C   440
403512740   H-C:C:C:C:C:C-P-C   442
827737026   H-C:C-H 448
1572927451  H-C:C:C-P-C:C:C 458
-645296786  P-C:C:C:C:C:C-H 493
-688017645  P-C:C:C-H   503
-763796814  C:C:C:C-P-C:C:C:C   512
66252   C:C 574
75288527    P-C:C   600
-605043036  H-C:C-P-C:C:C:C:C   604
1074261266  H-C:C:C-P-C:C   629
-789313769  C:C:C-P-C:C 639
-2139775602 C:C-P-C:C   675
1370539593  H-C:C-P-C:C 698
284569632   C:C:C:C:C-P-C   702
1797624356  H-C:C:C:C-P-C:C:C   710
63294844    C-P-C   719
-912402317  P-C:C:C:C:C:C   725
72  H   741
67  C   742
80  P   744
-1779590322 C:C:C-P-C   762
1797623958  H-C:C:C:C-P-C:C-H   774
347808169   C:C:C:C-P-C:C:C 788
-688017247  P-C:C:C:C   815
240390684   P-C:C:C:C-H 834
-75615648   C:C:C:C-P-C:C   859
240391082   P-C:C:C:C:C 866
886716171   H-C:C:C-H   888
67900359    H-C:C   951
-1046303447 C:C:C:C:C:C-P-C 957
-662167720  P-C:C:C 969
70654   H-C 971
1678681248  C:C:C-P-C:C:C   979

First, all the hash codes are unique. So clearly the issue lies in the RNG and indeed, we see the following two paths being mapped to the same random integer.

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-428753296    C:C:C:C:C:C       161
-245206314    H-C:C:C-P-C:C:C-H 161

Does this mean that the two hash values, when used as seeds to the RNG give the same sequence of random ints? Using the code below

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Random rng1 = new Random(-428753296);
Random rng2 = new Random(-245206314);
for (int i = 0; i < 5; i++) {
    System.out.println(rng1.nextInt(1024) + " " + rng2.nextInt(1024));
}

we generate the first five random integers and we see that they match at the first value but then differ.

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161 161
846 40
317 885
461 535
448 982

This suggests that instead of using the first random integer from the RNG seeded by a hash value, we use the second random integer. Modifying the code to do this still gives collisions in two bits. Once again, looking at the paths, hashes and bit positions, we see that now, two different paths get mapped to the same bit position.

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886341376      H-C:C-P-C      686
1434821341     H-C:C:C:C:C-H  686

As before, we look at the sequence of random ints obtained from RNG’s seeded using these hash values. The resultant sequence looks like:

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333 142
686 686
905 1022
70 571
177 384

So now, the two sequences match at the second value. OK, so what happens if we take the third value from the sequence and use that as a bit position? We get exactly the same behavior (collisions at two bit positions), except that now, when we look at the sequence of random int’s they match at the third value.

This behavior seems a little strange to me – as if there is a pair of seeds such that the “trajectory” of the sequences generated using those seeds will always (?) intersect at a certain point (where point actually corresponds to the n’th element of the sequences).

May be this is a property of random sequences? Or a feature of the Java RNG. I’d love to hear if anybody has insight into this behavior.

Written by Rajarshi Guha

October 4th, 2010 at 7:30 am

10 Responses to 'Hashed Fingerprints and RNG’s'

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  1. Picking 64 random positive integers below 1024, isn’t there only a 13% chance of avoiding a collision?

    In [1]: p = 1
    In [2]: for i in range(64):
    …: p *= (1024 – i) / 1024.0
    In [3]: p
    Out[3]: 0.13388743455337332

    Richard West

    4 Oct 10 at 5:37 pm

  2. True – but here we’re working two different random number sequences. So each time a bit position is required we’re instantiating a new RNG with a new seed (ie the hash value) and pulling a single random int from it.

    Ths I’d have expected the probability of two separate RNG sequences intersecting to be low – but this is not a rigorous conclusion. Hopefully the math will prove me wrong :)

    Rajarshi Guha

    4 Oct 10 at 7:16 pm

  3. How is this thought experiment?

    If we use a perfect (behaves ideally) random number generator ‘R1′, then the chance of avoiding collisions in the set of 64 unique hashes is 13%.

    If we define a new random number generator ‘R2′ with the algorithm “use R1 but burn the first number” then it too must be perfect, and we’d still have 13% chance of avoiding collisions.

    Generalize to Rn and I expect 13% of these ideal RNG’s will luckily avoid collisions :)

    Richard West

    4 Oct 10 at 10:18 pm

  4. But you’re still considering a single RNG for obtaining all the 64 bit positions, right? Thus the R2 you describe is still based on the same sequence as R1 – and so you’re still getting all 64 values from the same sequence.

    The fingerprinter code uses 64 *different* random number sequences. and pulls one value from each individual sequence.

    Rajarshi Guha

    5 Oct 10 at 12:45 am

  5. I think the logic remains for 64 different seeds to the RNG because my hypothetical random number generators are ideal, which means (by definition) you can’t tell the difference!
    Anyway, I’ve enjoyed this peek under the hood of fingerprinting in CDK. I assume other tools do it a similar way?

    Richard West

    5 Oct 10 at 3:12 am

  6. Yes, finally got my head around it – you’re right, different seeds shouldn’t make a difference

    Rajarshi Guha

    5 Oct 10 at 10:59 pm

  7. Hi
    Very interesting topic. I have faced these challenges while working with fingerprints and here are few observations from my end. By the way I agree that mathematically the best is ~ 13%.

    1) hashed FP (CDK) is good enough to separate patterns which are not common but on a very large dataset (in my case 10000+ mols), the performance dropped. Top 1% hits were good but then I started to loose specificity (esp when Tanimoto score was around 0.77).

    2) First I thought it was an Tanimoto score but I wasn’t convinced incases where we had rings (close vs open). I ended up writing new FP based on the pubchem patterns as coded in the CDK and added few more patterns to resize it to 1024 from 881. Well! It’s works like magic and I could find much more serialised hits than before. I think the extensions of the fingerprint which I made based on the patterns in my db also helped.

    At the end of the day I believe all these searches are heuristic and hashed FP is faster to generate but prone to bit clash where as SMARTS based fps are slower to generate as u sped time in matching patterns but are more sensitive and specific as u get what u see (patterns and bitset relationship is know).

    Just a thought…..

    Asad

    8 Oct 10 at 12:26 pm

  8. Please ignore my previous reply…First answer was a rough draft! Found a lot of mistakes/typos.. courtesy my iphone auto text filler. This one is from my comp .. Rajarshi might like to delete it ;-)

    Hi
    Very interesting topic. I have faced these challenges while working with fingerprints and here are few observations from my end. By the way I agree that mathematically the best bet is ~ 13%.

    1) The hashed FP (CDK) is good enough to separate patterns which are not common but on a large dataset (in my case 10000+ mols), the performance drops drastically. Top 1% hits were good but then rest of the started to loose specificity (esp when Tanimoto score was around 0.77).

    2) First I thought it was an artifact of the Tanimoto score… but I wasn’t convinced spl. in cases where we had rings (close vs open). I ended up writing a new FP based on the pubchem patterns as coded in the CDK and added few more patterns to resize it to 1024 from 881. Well! It’s works like magic and I could find much more serialised hits than before. I think the extensions of the fingerprint which I made based on the patterns in my db also helped.

    At the end of the day, I believe that all these searches are heuristic and hashed FP is faster to generate but prone to bit clashes where as SMARTS based FPs are slower to generate (as u spend time in MCS) in matching patterns but they are more sensitive and specific as you can trace the patterns (u get what u see) as the patterns and bitset relationship is know and static.

    Just a thought…..

    Asad

    8 Oct 10 at 1:54 pm

  9. Asad, if you don’t mind, I’d like tp post your comment as a blog post

    Rajarshi Guha

    8 Oct 10 at 11:14 pm

  10. [...] a comment to my previous post on bit collisions in hashed fingerprints, Asad reported on some interesting points which would be [...]

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