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What is the relation between a dimension and a component in a Gaussian Mixture Model? And what are the meanings of dimension and component? Thank you.

Please correct me if Im wrong: my understanding is the observed data have many dimensions. Each dimension represents a feature/aspect of the collected data and has its own Gaussian distribution. I don't know where "component" fits into this picture and what it means.

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    $\begingroup$ I personally like this very concise description, by J.K. Vermunt: Latent Profile Model. $\endgroup$ – chl Mar 28 '11 at 7:43
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    $\begingroup$ The link above on latent profile model has been removed :( $\endgroup$ – Zhubarb Oct 2 '13 at 16:09
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A mixture of Gaussians is defined as a linear combination of multiple Gaussian distributions. Thus it has multiple modes. The dimension refers to the data (e.g. the color, length, width, height and material of a shoe) while the number of components refers to the model. Each Gaussian in your mixture is one component. Thus each component will correspond to one mode, in most of the cases.

I suggest you read up on mixture models on wikipedia.

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  • $\begingroup$ could you please explain the term "mode" that you have used in your explanation. I basically want to know how "mode" and "model" are different $\endgroup$ – Upendra Pratap Singh Sep 22 '16 at 10:37
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A mixture of Gaussians algorithm is a probabilistic generalization of the $k$-means algorithm. Each mean vector in $k$-means is component. The number of elements in each of the $k$ vectors is the dimension of the model. Thus, if you have $n$ dimensions, you have a $k\times n$ matrix of mean vectors.

It is no different in a mixture of Gaussians except that now you have to deal with covariance matrices in your model.

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    $\begingroup$ ... and that you have soft responsibilities instead of hard assignments. $\endgroup$ – bayerj Mar 29 '11 at 6:54

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