mixture-distributionmultivariate analysisnormal distribution
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.
A mixture distribution combines different component distributions with weights that typically sum to one (or can be renormalized). A gaussian-mixture is the special case where the components are Gaussians.
For instance, here is a mixture of 25% $N(-2,1)$ and 75% $N(2,1)$, which you could call "one part $N(-2,1)$ and three parts $N(2,1)$":
xx <- seq(-5,5,by=.01)
plot(xx,0.25*dnorm(xx,-2,1)+0.75*dnorm(xx,2,1),type="l",xlab="",ylab="")
Essentially, it's like a recipe. Play around a little with the weights, the means and the variances to see what happens, or look at the two tags on CV.
Best Answer
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.