I am not entirely sure the following will address your problem, but could you for example cluster first on X and Y separately and then find the distribution in each cluster in X over the clusters of Y?

For example if you have x1, x2, ... x6 as your X clusters and y1...y4 as your Y clusters:

For each x in [x1, ... x6]: compute p(y1|x), p(y2|x), etc... This will allow your model to be predictive up to a certain confidence based on the distribution values.

To identify the optimal values of k (for each variable) you can test with different values for each and maintain the ones that provide the highest information gain (confidence levels).

However, predicting a Y cluster given an X cluster still means there might be correlation but not necessarily causation.

Hope you can get some clue in the paper:

http://proceedings.mlr.press/v33/rasiwasia14.pdf

.

kinds of reminds me of canonical analysis but in a clustering setting ...

maybe the following could be a useful read :

https://www.ml.cmu.edu/research/dap-papers/F17/dap-gitman-igor.pdf

Canonical Least Squares Clustering on Sparse Medical Data

I. Gitman et al

We explore different applications of Canonical Least Squares (CLS) clustering on a corpus of sparse medical claims from a particular health insurance provider. We find that there are several reasons why most conclusions based on CLS clusters might be misleading, especially when the data is significantly sparse. We illustrate these findings by performing a number of synthetic experiments with the most focus on the sparsity issue since it has not been explored in the literature before. Based on the insights from the synthetic experiments we show how CLS clustering can be potentially applied to identify hospital peer-groups: hospitals that share similar operational characteristics. In addition we demonstrate that CLS clustering can be used to improve prediction results for the patients length of stay in the hospitals.

.

First of all thanks everibody for the hints.

About the answer of Fouad Amer: as you say there is more correlation than cousation in your suggested (and interesting) approach. But in my mind I have a different scenario, such that the clustering on X induce a "good" clustering on Y such that once observed a new object on X I can be almost sure that it belongs to a cluster induced on Y (supposing, the number of cluster equal for X and Y).

About the answers of B. K. Hooda and of F. Clerot, yes, what I have in mind has the same raw idea, except the fact that I would like to adopt a causation approach (maybe, with some modifications the methods can be adapted) and not necessarily a linear causation model.

About the answer of Alireza Givehchi can you explain better. I know that SOM is purely unsupervised method for dimension reduction and, in certain way, clustering. How, can you suggest me a SOM used into a causation framework?

.

Sorry Antonio, i do not get the "causal" part of your question

.

going back to your example, we have a set of clients (one discrete variable with as many modalities as there are clients), each client is described by a variable X (it could be a set of variables but to keep things simple, i assume only one variable, be it discrete or continuous) and a variable Y (discrete or continuous)

.

so, my database is made of lines

clientId, valX, valY

i rip this database from the first column and keep only the last two

i call one line in the remaining database an "observation"

.

assume i perform a bi-clustering of this dataset, that is, finding a partition of the observations such that the contrast between the distributions in the parts and the distribution i would have under the independence hypothesis (constrained by the marginals) is maximized

this bi-clustering induces two partitions : a partition of the values of X and a partition of the values of Y ; calling such "partitions" "clusterings", i end up with two clusterings of X and Y which are consistent in the sense that they are maximally informative with respect to each other

this is usually done by alternating partition optimization on each dimension ; some bi-clustering techniques use k-means as a basic partitionning routine, we do it by MDL-regularized optimizaton techniques ... there are many ways to co-clustering !

.

i suspect you do not appreciate this solution since it is symetric in (X, Y) ; maybe this is what you call "causal" : you do not want the partition of Y to have an influence on the partition of X

.

if the above is true, then , why not use just a single step of the bi-clustering optimization : given a partiton on X (from k-means for instance), find the partition on Y which maximizes the mutual information between both partitions and stop ?

.

i admit there is nothing "causal" here, since the quantity which is maximized is the mutual information so there must be something i misunderstood in your question

.

sorry for being so long to reach the conclusion i must have misunderstood something ... very little progress since my first line, indeed :)

.

Dear Alizera, the used approach is rather "classical", in the sense that a category is assigned after a clustering step. In my idea, I have two sets of variables, the independent (on which a train a clustering) and a dependent (that are in a certain sense clustered according to the partition obtained from the independent ones). The idea is to find a strategy, a method, that is able to have compact clusters for the dependents but working only on the independent variables, such that the final results are good clusters for wich I can assign a new object on the base of independent variables and obtain a good "estimate" of its dependent variables according to the trained clustering.

.

this paper might be of interest

http://www.vincentlemaire-labs.fr/publis/AIS34_camera_ready.pdf

but this is a two-step process (modelling, then cluster in a representation space deduced from the model)

.