A detailed view at Similarity, Complexity and their calculation.
What is a Similarity Matrix
The Similarity Matrix feature in FluoroFinder calculates and displays spectral similarity data between different fluorophores on a panel. This information helps researchers understand how similar the emission spectra of different fluorophores are, which helps them avoid potential issues related to spectral overlap.
The similarity data is represented as an n x n grid, where n is the number of colors on the panel. Each cell in the grid represents the spectral similarity between two fluorophores. A score of 0 means they are not at all similar. A score of 1.0 is perfectly similar. In FluoroFinder’s Similarity Matrix, pairs of fluorophores with a high similarity score are highlighted in read as a warning that this combination of fluorophores may not be ideal.
In FluoroFinder's Panel Builder app, a similarity matrix will look something like this:
While in the Spectra Viewer it will appear as follows:
Calculating Color Similarity: Cosine Similarity Formula
The cosine similarity formula is a common way to measure the similarity between two vectors, in this case, the emission spectra of two fluorophores. The cosine similarity measures the cosine of the angle between the two vectors. If the vectors are similar, the angle between them is small, and the cosine is close to 1. Conversely, if the vectors are dissimilar, the angle between them is large, and the cosine is close to 0 or negative.
The Similarity value between two fluorophores on a machine is the cosine similarity between those two fluorophores' emissions arrays on that machine.
What is the "Complexity Score" or "Complexity Index" associated with my Similarity Matrix?
The Matrix Score calculation is based on the condition number of the similarity matrix. The condition number measures how well-conditioned a problem is, with a low condition number representing a well-conditioned problem and a high condition number representing an ill-conditioned problem.
To compute the Matrix Score for a given similarity matrix, FluoroFinder follows these steps:
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Perform eigenvalue decomposition on the similarity matrix.
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Find the diagonal values, which are the eigenvalues.
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Calculate the ratio of the maximum eigenvalue to the minimum eigenvalue. This ratio is the condition number of the matrix.
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Compute the square root of the condition number to obtain the Matrix Score.
What is Eigenvalue decomposition?
Eigenvalue decomposition, also known as eigendecomposition, is a technique in linear algebra that breaks down a square matrix into a set of simpler, more manageable components. It is used in many applications, such as data analysis, computer graphics, and engineering.
Eigenvalue decomposition helps simplify complex matrix operations and reveal the underlying properties of the matrix. In many applications, it helps analyze data, solve systems of linear equations, and perform dimensionality reduction, among other uses.