## Please edit system and help pages ONLY in the moinmaster wiki! For more
## information, please see MoinMaster:MoinPagesEditorGroup.
##master-page:Unknown-Page
##master-date:Unknown-Date
#acl DanieleMerico:admin,read,write,delete,revert All:read
#format wiki
#language en

== Distances: my personal experience ==
Just a few suggestions according to my own experience. For a more formal and detailed treatment go to Wikipedia or a maths manual.

=== Quantitative Data Arrays ===
For quantitative data (e.g. transcription signals from microarray experiments) you can use
 * '''Euclidean distance''', which is D = √ ((x,,1,, - x,,2,,)^2^ + (y1,,1,, - y,,2,,)^2^ + (z,,1,, - z,,2,,) + (...)^2^ + ...)[[BR]]
 it is the geometric distance you are used to calculate in 2 dimensions, just extended over N-dimensions[[BR]]
 ''be careful, I'd suggest to use it only when:''
   a. ''you are sure that the data are in the same magnitude scale''
   a. ''you do not want to consider anti-correlated patterns similar''
 * '''Correlation-based''' [[BR]]
 this distance is ideal when you want to group objects with inter-dependent behavioral trend [[BR]]
 for inst, for D = 1 - Correlation Index => similar = whenever A goes up, always B goes up as well; dissimilar = whenever A goes up, B always goes down[[BR]]
 whereas,  for D = 1 - Abs (Correlation Index) => similar = whenever A goes up, always B goes up as well, AND whenever A goes up, B always goes down[[BR]]
 ''for these reason, I usually prefer Pearson-based distance when clustering genes by expression signals (Affymetrix arrays)''

=== Binary Arrays ===
 * '''Hamming''' = number of discordant array elements (e.g. 1010 vs 1100 => D = 2)[[BR]]
 this distance was developed to measure transmission errors in binary coded strings[[BR]]
 * '''Jaccard-based''' = 1 - (number of concordant 1s in the two arrays)/(number of positions with 1 in either of the two arrays)[[BR]]
 (e.g. 1010 vs 1100 => D = 1 - 1/3 = 2/3)[[BR]] 
 this distance is useful when array positions correspond to elements which can belong (value = 1) or not belong (value = 0) to the set associated to the array[[BR]]
 (e.g. given a list of Transcription Factors, you can associate to every gene a binary array to express whether they are regulated by each TF or not)[[BR]]
 (e.g. given a list of interaction partners, you can associate to every protein a binary array to express whether they are interacting with each partner or not)
~-''Hamming distance is not good, in my opinion, when the strings compared have a very unequal 1/0 content, and the meaning of 1s and 0s is related to set-membership (as in the examples above).''-~[[BR]]
~-''E.g., consider these strings, and the choice made by Hamming and Jaccard:''-~[[BR]]
 * ~-{{{11 000 000}}}[[BR]]-~
   * ~-{{{is more similar to 10 000 001 according to Hamming (D,,H,, = 2, D,,J,, = 2/3)}}}-~
   * ~-{{{is more similar to 10 000 001 according to Jaccard (D,,H,, = 3, D,,J,, = 3/5)}}}-~
 * ~-{{{110 100 000}}}-~
   * ~-{{{is more similar to 011 000 000 according to Hamming (D,,H,, = 3, D,,J,, = 3/4)}}}-~
   * ~-{{{is more similar to 110 111 101 according to Jaccard (D,,H,, = 4, D,,J,, = 4/7)}}}-~
~-Mapping the first problem to interactions, let's say [[BR]]
{{{A1 interacts with B, C}}}[[BR]]
{{{A2 interacts with B, D}}}[[BR]]
{{{A3 interacts with B, C, E, F, G}}}[[BR]]
is A1 neighborhood more similar to A2 (Hamming's choice) or A3 (Jaccard's choice)[[BR]]-~
[[BR]]
In addition, neither of these two distances is suited when some array positions are noisy, 
whereas others are information-rich, as in the following fictional examples:[[BR]]
{{{110 010 001}}}[[BR]]
{{{110 100 000}}}[[BR]]
{{{110 000 110}}}[[BR]]
{{{000 101 100}}}[[BR]]
{{{001 101 011}}}[[BR]]
{{{000 111 000}}}[[BR]]
clearly, 1st and 2nd position are always co-conserved, as well as 4th and 6th; 
the co-conservation of other residues is irregular; 
therefore, we may think that the co-conserved positions hold more information than the other ones.