Analyzing stacks of images using copula concept

CREATED: 200906120348 Speaker: Andrej Borissenko, PICB ** MELK data

• using protein specific antibodies to provide information on spatial distribution of proteins
• for each protein, there is a flourescent image ** Problems
• cell detection
• classification
• topological motifs (prescence/abscence of protein) ** Copulas
• function which combines two marginal distribution into a joint distribution
• H(x,y) = C(F(x), G(y))
• lower bound: C(u,v) = max(u+v-1, 0)
• upper bound: C(u,v) = min(u,v)
• independence: C(u,v) = uv
• can be estimated from given data easily by counting
• comparing given data against independence allows one to visualize how two proteins are dependent ** Conventional dependence measures
• Pearson’s correlation (depends on copula and marginals)
• Kendall’s tau (only depends on copula)
• Spearman’s rho (only depends on copula)
• Pearson’s correlation sensitive to outliers
• single value does not give enough information about dependence between two RV ** Topological motifs (how to select threshold for prescence/abscence of protein)
• idea is to maximize multi-information, which is essentially the KL divergence of independence from estimated distribution
• can be shown that in the limit, as the number of thesholds increases, this converged to mutual information of two random variables, I(X,Y)
• MI only depends on copula
• $MI_{coarse} \le MI_{fine} \le I(X,Y)$
• so this approach is to find the best estimate of I(X,Y) ** Clustering in one dimention
• equivalent to fitting mixture models to histograms
• using mixture of gamma distribution
• actual model is linear combination of gamma
• extension to higher dimensions using copula