How to determine number of profile in a dataset derivated from repeated measures? I'm currently working on datasets which have been derivated from repeated measures over time (blood concentrations). Actually the descriptors of these datasets are descripting the shape of the curves (concentrations over time) for each individual. The purpose is to distinguish if there are only one type of curve shape or two signing equivalence or not between two products.
The hypothesis is :

*

*if there is only one type of shape the products are equivalent

*if there is more than one type of shape, the products are inequivalent

Indeed, I had to generate synthetic data (which follow the tendency of the real datas). I have generated datasets in which there are equivalence between products and some where there are no equivalence.
In first intention I tried to work on PCA and Kmeans clustering, searching for the optimal number of cluster to know if there products are equivalent. The fact is that this technic doesn't work at all.
Now I plan to work with some classifiers like Random Forests, SVMs or ANN, trained on synthetic datas, in order to estimate the number of shape type of a real dataset. The problem is the labelling of the datas. Indeed, in this case I can't use labeled from individuals beacause the question is not to classify individuals but datasets in their integrity.
I've heard about SOM and Growing Neural Gas but I don't really know how to use them in that way. I also tried to extract features from the dataset like means, sd but it doesn't seem relevant.
Does someone has already worked on this type of problems ?
 A: Several considerations here pertain to the background upon which the question is based rather than the text of the question itself. As the say in Maine, "You can't get there from here." In simplest terms, one must first have shape information in order to test for shape, and there isn't any shape information to test, as follows.
1) Concentrations are not random variables and the best method of fitting concentrations is to take a density function and scale that to fit the data. Although we typically call density functions pdf, there is no 'p' for probability in one for matching concentration. pdf can be $f(x)$ or $f(t)$ and can be used to model random variables, but are not themselves random variables. This can cause confusion, e.g., see What is a good name for a density function that does not relate to probability?. 
2) Remembering this, one can use pdf notation anyway and then $C(t)= \kappa\, \text{pdf}(t)$ becomes our fit equation, where $\kappa=\text{AUC},$ the area under the concentration curve from $t=0,\infty$. 
3) The pdf used either have shape parameters in them or there will not be derivative fitting or shape fitting, and all the shapes fit will be indistinguishable. 
4) The usual functions used for fitting concentrations are sums of exponential terms (SET). As pdf these are mixture distribution models (not of random variables, but of concentrations). Mixture models are always some $f(t)$  or $f(x)$ and this can cause confusion, as mixture models do not have to be models of random variables, e.g., see this answer, and for modeling concentration the density functions  do  not model probability. SET models have no shape parameters, as the exponential distribution from which these models arise have no shape parameters. 
5) Much better derivative fitting is achieved using gamma distributions, or gamma distribution convolutions, where the latter has no less than two shape parameters and allows for very high precision concentration and shape fitting for data up to about four hours for at least some drugs.  
6)  Once one has a model that actually follows the shape of the concentration curve, one should be able to just inspect the shape parameter(s) to see if they are statistically different for the two cases you are considering.
This is a topic that I am actively researching, and there are multiple conclusions that I cannot reveal at this time. However, I would also suggest reading this to develop a broader background on how these problems should be treated.
Edit More information on fit procedures, shape determination, etc. Consider, for example, that fitting with minimizing proportional error (or indeed any error minimization) is not robust for biexponential, E2, (and higher SET) models. Approximately 2% of those models converge to any one of $n$-tuple solutions in the complex plane. Also, shape information from Tikhonov regularization that minimizes AUC error over the whole ill-posed curve for gamma distributions to GFR markers allows one to extract more exact clearance (CL) and volume of distribution (V$_{\text{d}}$) information, such that unlike for SET functions, there is no correlation (i.e., no contamination) between fluid disturbance and CL. However, that is insufficient to separate the shape information from the total duration of sample collection, such that any comparisons of shapes would have to be undertaken under the same sample-time collection regimen. 
Finally, a full model of concentration that may obviate the necessity of using regularization, that is, a full bore well posed model, must contend with the actual shape of the right hand tail function, as well as eliminate all instant mixing assumptions. Indeed, there is probably more evidence to support power function right hand tails than exponential ones. I am working on such a model, and it is not trivial. For one thing, it necessitates variable volume modelling as well as half-life as a function of time. Most people have difficulty understanding that a half-life for a density function other than a memoryless exponential is not a constant, but 1) is a function of time and 2) is negative when concentration is increasing. Newton must be turning over in his grave that people do not generally understand the slope at a point in time of a logarithm of a nonexponential density function.
Maybe some other approach would be of interest to you, depending on what exactly it is you are doing. For example, take a look at this article.
