Fueled by newly developed single-cell technologies such as single-cell transcriptomic [22, 23], genomic  and proteomic  analysis, many new methods have emerged which attempt to shed light on cellular heterogeneity [26–29].
Previous methods for the detection of heterogeneous subpopulations in biological data have largely focused on grouping observations according to expression level, and thus requires that subpopulations be readily separable. For instance, in FACS cellular subpopulations are often identified with manually determined compartments [30–32]. If the data are easily separated, clustering methods such as Gaussian mixture modeling and k-means clustering have proven well suited to this task .
Alternatively, methods such as principal component analysis attempts to identify the principal directions, along which the data are maximally separated . Data which cluster together in the reduced dimensional subspace spanned by the first few principal components are thought to be representative of subpopulations. A similar method was employed by Trott et al. when analyzing the Fgf5+/- and Nanog +/- compartments . Non-linear alternatives to PCA including Gaussian Process Latent Variable Modeling have also recently been shown to be useful for the identification of cellular subpopulations [29, 34].
None of the previously mentioned methods utilize correlation information in the identification of cellular subpopulations, with the exception of Gaussian mixture modeling which attempts to learn the correlation matrices of Gaussian distributions thought to have generated the data. However, as shown here, the local correlation structure provides additional insight into the existence of differentially regulated subpopulations and hence should not be disregarded.
To date, relatively few methods have addressed the possibility of local, state-dependent correlations. Chen et al. developed a method for analyzing the effect of local non-linear correlations in gene expression data, and applied it to a microarray dataset; a similar method was recently developed by Tjøstheim et al. for estimating local Gaussian correlation in the context of econometric data. However, these methods required the definition of a interaction scale for the computation of local correlations or consider only the relative distance between data points and not their absolute levels when computing local correlations.
Recently Cordeiro et al., developed a sophisticated algorithm for identifying clusters of arbitrary orientation, also in a multiresolution context. MCA is not as general in that it does not consider clusters aligned along arbitrary projections of the data but provides instead a comprehensive, multiresolution view of the correlation structure according to the measured covariates, preserving expression-level dependencies while not requiring any predefined bandwidth or interaction distance, and thus may provide more biological insight into the role of individual factors in differential regulation motifs.
MCA has the advantage of being easy to compute and intuitively interpretable; it is in effect a moving window correlation analysis simultaneously over many window sizes. The MCA plot provides a graphical diagnostic for detection of subpopulations points that contribute inordinately to the overall correlation, or outliers, and may provide biological insights that serve as hypotheses for further experimentation. Finally, although we have focused on biological data and in particular cellular subpopulations in single-cell transcriptional data, the method is more general and applicable to any multivariate data.
While the simplicity of MCA plots makes them easy to interpret, there are nonetheless shortcomings that must be mentioned. MCA plots are a graphical representation of the interaction of only two factors, sorted by a third. If there are many covariates, many such plots are possible, and it becomes increasingly more difficult to generate and search through all possible plots as the dimension increases. In such cases it is helpful to consider only those plots which may be of biological interest such as sorting variables thought to have a regulatory role, or pairs of factors that are suspected to interact. However, one may also use alternative sorting variables, such as products of covariates representing potential interactions, principal directions as determined by PCA, or even arbitrary non-linear functions of the covariates.
In the case of many variables, one may wish to sort the resultant plots according to arbitrary functions of the estimated correlation structures; i.e. one could filter for only those plots showing large significant regions or for plots for which a significant region of both positive and negative correlation are present. Although preliminary tests with such methods are successful in identifying such interesting plots, the results are not shown here as they are unnecessary when the number of dimensions is still manageable via manual inspection.
The correlation becomes difficult to estimate when the number of samples is small, or when the number of variables is relatively large compared to the number of observations. If the resolution is fine, then the MCA plot will contain many points for which the corresponding subpopulation only contains one or a few observations. Such points are omitted from the plot since the correlation cannot be robustly computed. This can sometimes give rise to small regions near the bottom of the MCA plots for which there are too few observations to compute the subpopulation correlation. These regions do not have biological significance.
Similarly, the stochastic nature of the data may give rise to “noise” in small subpopulations, leading to interspersed points on the MCA plot which are not part of a large, significant region. These points typically do not indicate robust subpopulations since a small perturbation away from them leads to a different correlation structure, and can safely be ignored. This “noise” also gives rise to the slight inhomogeneities in the regions identified in Figures 2 and 3.
Lastly, in the case of relatively many variables compared to the number of observations, correlations can be computed using shrinkage-based estimators , although this results in a different estimation of statistical significance, and increases computational complexity.