Transforming association matrix to a functional matrix

In summary, the conversation discusses the use of association and functional matrices in relation to physical and biological systems, specifically in the context of unidirectional and two-way diffusion coupling between electrophysiological neurons. The speaker mentions using MATLAB commands to manipulate the association matrix into a functional shape, but is having trouble with a more general case. A solution is proposed by a co-worker, which involves taking the sum of rows of the association matrix and subtracting it from the transpose to obtain the functional matrix for diffusive coupling.
  • #1
Pythagorean
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I have n elements. Say n = 3.

Suppose I have an association matrix that gives the relationship between each element

[tex]
\begin{array}{cc}
0 & 0 & D3\\
D1 & 0 & 0\\
0 & D2 & 0
\end{array} [/tex]

I have a function in mind now, I want to operate and the physical variables representing my three elements, [tex]\vec{y} = [y_1 y_2 y_3] [/tex]. My functional matrix would look like:

[tex]
\begin{array}{cc}
-D1 & D1 & 0\\
0 & -D2 & D2\\
D3 & 0 & -D3
\end{array} [/tex]

so that I get [tex] \vec{y} = [D1(y_2 - y_1) D2(y_3-y_2) D3(y_1-y_3) [/tex]

If you're interested in the physical/biological motivation, we basically have a unidirectional diffusion coupling between electrophysiological neurons here so you're seeing a numerical second derivative. Now, using MATLAB commands (circshift and transpose) I do circular shifts and transposes on the association matrix to nudge it into my functional shape.But I'm having trouble with the more general case. What if I have two-way diffusion, but the diffusion is stronger in one direciton than in the other? Now the association matrix is:

[tex]
\begin{array}{cc}
0 & D4 & D3\\
D1 & 0 & D5\\
D6 & D2 & 0
\end{array} [/tex]

and what we for functional is:[tex]
\begin{array}{cc}
-(D1+D4) & D1 & D4\\
D5 & -(D2+D5) & D2\\
D3 & D6 & -(D3 + D6)
\end{array} [/tex]

so that [tex]\vec{y} = [D1(y_3-y_1) + D4(y2_y1); D2(y_1- y_2) + D5(y_3-y_2) + ...][/tex]
Now, I can design another series of circshifts and tranposes, but it won't work for the case above. I can't find a general set of operations that works for both

This should work in general, for an nxn matrix.

Thank you for your help.
 
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  • #2
A friend and co-worker came up with this (F = functional matrix, A = association matix)

F = transpose(A) - diag(sum(rows(A)))

that is, if you take the sum of the rows of A and place it on a diagonal of an otherwise 0 matrix, then subtract that from the transpose of A, you have the functional matrix of diffusive coupling for that association matrix!
 

1. What is a transforming association matrix?

A transforming association matrix is a mathematical representation of data that shows the relationships between different variables or factors. It is often used in statistical analysis to understand the patterns and connections between variables.

2. How is an association matrix different from a functional matrix?

An association matrix displays the strength and direction of the relationships between variables, while a functional matrix shows the specific functions or operations that relate the variables to each other. In other words, the association matrix provides a broad overview, while the functional matrix delves into the specific mechanisms at work.

3. What information can be derived from a functional matrix?

A functional matrix can provide insights into the underlying mechanisms and processes that drive the relationships between variables. It can also help identify any dependencies or interactions between variables, and reveal any patterns or trends that are not apparent in the association matrix.

4. How is a transforming association matrix useful in scientific research?

A transforming association matrix can be used to analyze complex data sets and identify patterns and relationships between variables. This can be particularly useful in fields such as biology, ecology, and social sciences, where understanding the interactions between different factors is crucial for making meaningful conclusions.

5. Are there any limitations to using a transforming association matrix?

While a transforming association matrix can provide valuable insights, it is important to note that it is only a representation of the data and does not necessarily reflect causality. Additionally, the accuracy and reliability of the results can be influenced by the quality of the data and the method used to construct the matrix.

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