Dear all,
I'd like to compute the distance between two brain states. Each brain state is represented as a symmetric correlation matrix of X by X regions. As a result, I'd like a "distance matrix" having the same size as my inout matrices. What would you suggest? I was thinking about geodesic distance, however, I couldn't find any scripts (preferably matlab) that calculates it.
Thanks in advance!
Hello,
This link talk about: Calculating the distance between two matrices Matlab
https://www.mathworks.com/matlabcentral/answers/165027-calculating-the-distance-between-two-matrices-matlab
Hi Nina von Schwanenflug,
If you are looking for a pairwise distance between two brain states A and B (X*X sets of observations), I will suggest you use the MATLAB function pdist2.
dist = pdist2(A, B, 'type');
here, dist will return the distance between each pair of observations in A and B for specified distance type where,
type belong to euclidean or hamming or minkowski or cityblock etc.
I hope this will be helpful to you.
Cheers!
Dear Nina. I will suggest you a pipeline. Then, I will use a stepwise structure: a) you get two 'brain spaces' matrix with the same dimensions if you register both 'brain spaces' in a template ---> b) 'brain space' isn't a defined concept ---> c) define which sequence of MR has been used to obtain 'brain spaces' (for instance, T1) ---> d) 'distance of brain spaces' is a geometrical representation, not a well-defined concept. All native brain spaces are different, except you registered the same brain to the same template --> e) be conscious that your sentence " I couldn't find any scripts to calculate it" is the result of 'brain space distance' is a geometrical representation about brain A (registered in Z template) is different to brain B (registered in Z template). It is the same that 'moon.png A of Sunday' in a matrix code is different to 'moon.png B (Monday)' ---> then, you will not find any specific script about a gross concept like A is different to B. 'pdist2' is not different to (matrixA - matrix B) ---> you are calling 'brain distance' similar to 'brain difference', anything mode to obtain different values is useful for a non- well-defined concept --> then, at this stage, matrix with the same dimensions through different values using arithmetic, 'distance functions', but all of that make me hypothesize that you have more than two brain spaces (A, B, C, D). I conjecture that you want to represent in a geometric space A, B, C, D... ---> to do that you need conceptual dimensions (vectors, to form a k- space) because 'pdist2' is only useful to compare two matrices in a simple two-dimensional space ---> please, write again which is the 'scientific' problem you are in, as you expressed that it means --> 'I can't find any specific script to represent that two whole entire brain spaces are different' --> all arithmetic basic functions are the scripts you are looking for but think again which is the scientific problem you are in. Consider your question is a not well- description of what you are involved in (are the 'brain spaces' clinic ones? 'brain spaces include whole-brain voxel-wise data? do you have more than two brain-spaces and you want to represent all brain spaces in a geometrical k-space? I can help you, but, at first, is necessary to walk to state a scientific problem --> if the problem isn't in the field of science---> you can rest- sum- multiply- divide- you will get different values, then do a plot of that, call 'distance' to different values, average differences ---> consider it has no meaning
Looks like you will need to generalize your metric function
We want a function (our metric) to satisfy for all x,y,zx,y,z in our space
- d(x,y)=0⇔x=yd(x,y)=0⇔x=y
- d(x,y)=d(y,x)d(x,y)=d(y,x)
- d(x,y)≤d(x,z)+d(z,y)d(x,y)≤d(x,z)+d(z,y)
If we have a norm we also have a metric: d(x,y)=∥x−y∥d(x,y)=‖x−y‖
Some metrics come to mind for matrices with the same size
- 1 norm induced: Sum up the differences of the entries
- d(A,B)=∑i∑j|aij−bij|d(A,B)=∑i∑j|aij−bij|
- Matrix euclidean, or Frobenius norm induced (2 norm induced): Sum up the squares of the differences, then take the square root.
- d(A,B)=∑i∑j|aij−bij|2−−−−−−−−−−−−−−√d(A,B)=∑i∑j|aij−bij|2
- Some other metric induced by a matrix norm.
If our matrices are different sizes we will need to find some way of making them comparible. It seems to me that the first step would be to resize the matrices by adding extra rows and columns of zeroes.
dr(A,B)=d(Ar,Br)dr(A,B)=d(Ar,Br)
Where A,BA,B are the original matrices and Ar,BrAr,Br are the resized ones.
As an example
A=[a11a21a12a22a13a23]A=[a11a12a13a21a22a23]
B=⎡⎣⎢b11b21b31b12b22b32⎤⎦⎥B=[b11b12b21b22b31b32]
Ar=⎡⎣⎢a11a210a12a220a13a230⎤⎦⎥Ar=[a11a12a13a21a22a23000]
Br=⎡⎣⎢b11b21b31b12b22b32000⎤⎦⎥Br=[b11b120b21b220b31b320]
Now if we stopped here it wouldn’t be a proper metric because we would have more than one zero object, and we can have the distance between two matrices be zero even if they are different (They end up the same after the resizing).
We can work with the equivalence class of matrices where after resizing they are the same and stop it there, or we can add an extra term to our metric that takes into account the difference in the matrix dimensions before we resize.
If we go with the latter it would look something like this
dr(A,B)=d(Ar,Br)+|n1−n2|+|m1−m2|dr(A,B)=d(Ar,Br)+|n1−n2|+|m1−m2|
If A,BA,B are the same dimension it reduces to whatever metric you were using before.
-- Eric Platt, Ph.D in Mathematics
Hello,
This link talk about: Calculating the distance between two matrices Matlab
https://www.mathworks.com/matlabcentral/answers/165027-calculating-the-distance-between-two-matrices-matlab
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