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Dear All, Thank you for this interesting discussion! I have a practical (but rather basic) question on this issue: Selecting two seed masks A and B for a single probtrackx command, the output is a single fdt_paths file, whereas waytotal contains two values (presumably # streamlines A->B and B->A). So my question is, doesn't the division of fdt_paths by sum(waytotal) already represent the combined probability of A->B and B->A? Ie, as there is only one fdt_paths, which files are you talking about to get the combination of A->B and B->A? Thank you, Andreas Markus Gschwind schrieb: > Dear Mark, > > Thank you for the helpful comment. > > What you say is true if one uses the /conditional/ probability that > "the path starting at A, passes through x and B, knowing that A > connects to B" > which, following Saad is: p(A->x->B | A->B) = p(A->x->B)/waytotal(A->B) > > But still: When combining p(A->x->B | A->B) and p(A<-x<-B | B->A), > when looking for the probability that either the path from A to B or > from B to A passes through x, it means that x should be the same > voxels, no? However, when there is minimal overlap, x is actually not > the same regions, which finally gives a wrongly inflated tract volume > - or do I miss something completely? > > Thank you so much! > Markus > > > > 2014-09-08 6:11 GMT+02:00 Mark Jenkinson <[log in to unmask] > <mailto:[log in to unmask]>>: > > Dear Markus, > > I think what you are not taking into account is that in case [2] > where p(A->x->B) and p(A<-x<-B) are similar then the OR value is > twice the value of the individual probabilities, whereas the > product (AND) will be less than the individual probabilities > (given they are always less than 1) which means that the result of > subtracting the AND from the OR gives a value that is actually > larger than the original probability and so will not take away > from the tract. > > I hope this helps. > All the best, > Mark > > > On 7 Sep 2014, at 04:35, Markus Gschwind <[log in to unmask] > <mailto:[log in to unmask]>> wrote: > >> Dear all, and especially Saad, >> >> I come back to this old discussion. >> >> You explained the understanding of the combination of two >> independent paths from a probabilistic point of view (c.f. below). >> >> Probtrackx calculates the probability that the path of least >> hindrance to diffusion from A passes through x: P(A->x). >> Using a waypoint B, the values become now the probability >> that the path of least hindrance to diffusion from A passes >> through x and B, i.e. p(A->x->B). >> >> Now combining p(A->x->B) and p(A<-x<-B), what you want is the >> probability that either the path from A to B or from B to A >> passes through x, which means: >> >> p( [A->x->B] or [A<-x<-B] ) = p( [A->x->B] ) + p( [A<-x<-B] >> ) - p( [A->x->B] and [A<-x<-B] ) >> >> Assuming the two events are independent (i.e. passing through >> x coming from A to B, or from B to A), the last probability >> is the product of both maps. >> >> >> >> And Cherif detailed it later on: >> >> p = [p( [A->x->B] )]/waytotal(a->b) + [p( [A<-x<-B] >> )]/waytotal(b->a) - ([p( [A->x->B] )]/waytotal(a->b)) * >> ([p( [A<-x<-B] )]/waytotal(b->a)) >> >> What I understand: >> The sum p(A->x->B)+p(A<-x<-B) describes the combination of both >> distributions, where the overlap is counted double (OR condition) >> >> The product p(A->x->B)*p(A<-x<-B) describes the intersection, >> i.e. the voxels where both direction pass (AND condition) >> >> However, I do not get the reason of the substraction of the >> product from the sum. >> >> For exemple, in a case [1] where p(A->x->B) and p(A<-x<-B) travel >> in very different voxels with only very few overlap, the >> OR-region (p(A->x->B)+p(A<-x<-B) ) will be very large, and the >> AND-region (p(A->x->B)*p(A<-x<-B) ) will be very small. >> >> However, in a case [2] where p(A->x->B) and p(A<-x<-B) travel in >> exactly the same voxels with maximal overlap, the OR-region >> (p(A->x->B)+p(A<-x<-B) ) will be (nearly) the same as the >> AND-region (p(A->x->B)*p(A<-x<-B) ) and the substraction of the >> product will thus take away a lot of the whole tract, although it >> was much a stronger tract with a clear definition than in case [1]. >> >> I am asking this because in one of our projects, we observed a >> much bigger volume of the OR-regions in one group compared to >> another group (potentially interesting), however, the AND-region >> has comparable size between both groups. >> >> Could you please explain a little on this? >> >> Thank you very much an advance! >> Markus >> >> >> >> >> >> 2008-10-01 10:55 GMT+02:00 Saad Jbabdi <[log in to unmask] >> <mailto:[log in to unmask]>>: >> >> Hi - >> It is important to keep in mind what is the probability that >> you want to calculate, it will tell you whether you want to >> add, substract, multiply or divide. >> >> Probtrackx calculates the probability that the path of least >> hindrance to diffusion from A passes through x: P(A->x). >> Using a waypoint B, the values become now the probability >> that the path of least hindrance to diffusion from A passes >> through x and B, i.e. p(A->x->B). >> >> Now combining p(A->x->B) and p(A<-x<-B), what you want is the >> probability that either the path from A to B or from B to A >> passes through x, which means: >> >> p( [A->x->B] or [A<-x<-B] ) = p( [A->x->B] ) + p( [A<-x<-B] >> ) - p( [A->x->B] and [A<-x<-B] ) >> >> Assuming the two events are independent (i.e. passing through >> x coming from A to B, or from B to A), the last probability >> is the product of both maps. >> >> So in order to calculate the probability that you want, you >> would need to add both maps, and substract their product. >> >> Cheers, >> Saad. >> >> >> On 30 Sep 2008, at 18:25, Cherif Sahyoun wrote: >> >>> Hi Markus, >>> >>> Matt and Saad will correct me if I'm wrong, but I'd say the >>> way you're proposing will cause you to lose any meaningful >>> relationship between the waytotal and your image. i.e. if >>> you add, then essentially the statement that waytotal is the >>> total number of streamlines that "made it" between your >>> masks is preserved, but once you multiply your images, then >>> it would seem like your waytotal is not as helpful... >>> (i like the idea of multiplying to get rid of outliers >>> though... i've been thresholding as per previous posts). >>> >>> Cherif. >>> ------------------------------------------------------------------------------------------ >>> Cherif P. Sahyoun >>> HST-MEMP >>> >>> Developmental Neuroimaging of Cognitive Functions >>> >>> C: 617 688 8048 <tel:617%20688%208048> >>> H: 617 424 6956 <tel:617%20424%206956> >>> [log in to unmask] <mailto:[log in to unmask]> >>> >>> "Live as if this were your last day. Learn as if you'll live >>> forever" >>> Ghandi >>> ------------------------------------------------------------------------------------------- >>> >>> >>> On Tue, Sep 30, 2008 at 3:41 AM, Markus Gschwind >>> <[log in to unmask] <mailto:[log in to unmask]>> >>> wrote: >>> >>> Hi! >>> I am following your discussion with a lot of interest! >>> Here I have a question: >>> Why wouldn't you MULTIPLY (instead of add) the paths AB >>> and BA? >>> I thought it is a more conservative measure of what is >>> in common and trancking outliers will dorp out. (And >>> take the mean of waytotals). >>> >>> Thanks, >>> Markus >>> >> >> >> >> >> >> >> > > > > > -- > Dr Markus Gschwind > Functional Brain Mapping Laboratory | Campus Biotech - Neuroscience > Department | phone +41.22.37.90.886 > postal adress: 9 Chemin des Mines | Case postale 60 | 1211 Genève 20, > Switzerland > https://sites.google.com/site/fbmlab/ > > -- Andreas Hahn, MSc PhD Functional, Molecular & Translational Neuroimaging Department of Psychiatry and Psychotherapy Medical University of Vienna, Austria Phone: +43-1-40400-23200 Email: [log in to unmask] Web: http://www.meduniwien.ac.at/neuroimaging/