Can somebody explains me the difference between "multimodal" and "multivariate"?
For example, I have a dataset which contains different information. All information objects are connected together by a timestamp. Is this dataset multimodal or multivariate? If I create an algorithm for clustering these data, should I call this algorithm multimodal or multivariate?
Put very simply, "multi-modal" refers to a dataset (variable) in which there is more than one mode, whereas "multi-variate" refers to a dataset in which there is more than one variable.
Here is a simple demonstration, coded with R:
set.seed(5104)
x1mm = c(rnorm(50, mean=-2), rnorm(50, mean=2))
x1um = rnorm(100, mean=0.5, sd=sqrt(3))
plot(density(x1mm), main="multimodal data")
plot(density(x1um), main="unimodal data")
y = .5*x1um + rnorm(100)
plot(x1um, y, xlab="X", ylab="Y", main="bivariate data")
That's the gist of it. When you have response and regressor variables, and you want to fit a model that maps them, the use of "multivariate" depends on the nature of the mapping. When there is only one response and one covariate, we say this is simple regression; if there is more than one covariate, we say it is multiple regression; and if there is more than one response variable, we call it multivariate regression. In your case, I gather you are interested in clustering / unsupervised learning, so these distinctions don't really apply.
However, the clustering aspect makes this a little more interesting. In order to cluster successfully, you generally want your data to be multimodal in the full data space. The clusters / latent groupings are found by finding a partition that separates the data into unimodal subsets that are more coherent than the original (unpartitioned) superset.
Multivariate refers to cases where you have more than one outcome variable (not levels). In cases where you have just one outcome variable, one speaks of an univariate problem. But as @gung already said, in practice and sometimes even in textbooks these terms get a little blurry and often refer to cases where you have more than two input variables, e.g. multiple regression etc.
Where multimodal refers to the experimental design. One says a model is multimodal if you measure one construct with different methods (e.g. questionnaire and observation). But it can also refer to the distribution of your data.
To conclude, the meaning of those terms depends heavily on the context.
