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How do you cluster a feature with an asymmetrical distance measure?

For example, let's say you are clustering a dataset with days of the week as a feature - the distance from Monday to Friday is not the same as the distance from Friday to Monday.

How do you incorporate this into the clustering algorithm's distance measure?

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If the M-F distance is asymmetric because the future is different from the past, then a genuine asymmetric clustering is called for. First, an asymmetric distance function must be defined.

One way to to asymmetric clustering, given a distance function, is to embed the original data into a new coordinate space. See "Geometrical Structures of Some Non-Distance Models for Asymmetric MDS" by Naohito Chino and Kenichi Shiraiwa, Behaviormetrika, 1992 (pdf). This is called HCM (the Hermitian Canonical Model).

Find a Hermitian matrix $H$, where $$ H_{ij} = \frac 1 2 [d(x_i, x_j) + d(x_j, x_i)] + i \frac 1 2 [d(x_i, x_j) - d(x_j, x_i)] $$ Find the eigenvalues and eigenvectors, then scale each eigenvector by the square root of its corresponding eigenvalue.

This transforms the data into a space of complex numbers. Once the data is embedded, the distance between objects x and y is just x * y, where * is the conjugate transpose. At this point you can run k-means on the complex vectors.

Spectral asymmetric clustering has also been done, see the thesis by Stefan Emilov Atev, "Using Asymmetry in the Spectral Clustering of Trajectories," University of Minnesota, 2011, which gives MATLAB code for a special algorithm.

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You can take some sort of a mean (like an arithmetic mean or, for probability distributions, the square root of the Jensen–Shannon divergence.)

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You should have a look to circular statistics (if you want to work "within"a tunning week)

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If your distance function is not a valid Mercer kernel, then $X \neq X^T$, where $X$ is the Gram matrix. In this case want co-clustering, also called bi-clustering. Algorithms of this class produce cluster indicators simultaneously for the rows and columns.

The example you gave is the result of a poorly chosen distance metric. A better distance metric would be $|\text{days apart}|$

Generally your distance function should be a valid Mercer kernel. A valid Mercer kernel is any function taking two observations that is continuous, symmetric and has a positive definite covariance matrix $\forall x \in D$.

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