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Subject:

Re: Combining probability - fdt_path directions

From:

Andreas Hahn <[log in to unmask]>

Reply-To:

FSL - FMRIB's Software Library <[log in to unmask]>

Date:

Mon, 8 Sep 2014 17:26:53 +0200

Content-Type:

text/plain

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text/plain (235 lines)

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/

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