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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