I have been reviewing Expectation Maximization (EM) in research papers such as this one:
http://pdf.aminer.org/000/221/588/fuzzy_k_means_clustering_with_crisp_regions.pdf
I have some doubts that I have not figured it out. For example, what would happen if we have many dimensions for each datapoint?
For example I have the following dataset with 6 datapoints and 4 dimensions:
>D1 D2 D3 D4
5, 19, 72, 5
6, 18, 14, 1
7, 22, 29, 4
3, 22, 51, 1
2, 21, 89, 2
1, 12, 28, 1
It means that for computing the expectation step, do I need to compute 4 standard deviations (one for each dimension)?
Do I also have to compute the variance for each cluster assuming k=3 (Do not know if it is necessary based on the formula from the paper...) or just the variances for each dimensions (4 attributes)?
Usually, you use a Covariance matrix, which also includes variances.
But it really depends on your chosen model. The simplest model does not use variances at all. A more complex model has a single variance value, the average variance over all dimensions. Next, you can have a separate variance for each dimension independently; and last but not least a full covariance matrix. That is probably the most flexible GMM in popular use.
Depending on your implementation, there can be many more.
From R's mclust documentation:
"E" = equal variance (one-dimensional)
"V" = variable variance (one-dimensional)
"EII" = spherical, equal volume
"VII" = spherical, unequal volume
"EEI" = diagonal, equal volume and shape
"VEI" = diagonal, varying volume, equal shape
"EVI" = diagonal, equal volume, varying shape
"VVI" = diagonal, varying volume and shape
"EEE" = ellipsoidal, equal volume, shape, and orientation
"EEV" = ellipsoidal, equal volume and equal shape
"VEV" = ellipsoidal, equal shape
"VVV" = ellipsoidal, varying volume, shape, and orientation
"X" = univariate normal
"XII" = spherical multivariate normal
"XXI" = diagonal multivariate normal
"XXX" = elliposidal multivariate normal