# On the Role of Clustering and Visualization Techniques in Gene Microarray Data
## 3 Visualization Techniques for Microarray Gene Expression Data
### Visualization Methods: A Brief Review  
* **Principal Component Analysis (PCA)**[59]. A well established linear projection technique is used to map data from higher to lower dimensional spaces. PCA linearly transforms data, preserving as much as possible its variance.
* **Probabilistic PCA**[60]. The lack of a generative model in PCA gives no means to interpret its error function in a principled way. Probabilistic Principal Component Analysis (PPCA) was introduced to enhance PCA, i.e., turning PCA into a generative model by using a latent variable approach. PPCA consists of a Gaussian mixture model where each component has a diagonal covariance matrix with a single variance parameter for describing the variance in each Gaussian of the mixture.
* **Mixture of PPCA**[61]. Because PCA describes only a linear data projection, it is a method that is rather restricted. Using a set of local linear models is one way around this. This is attractive since each model is easier to comprehend and generally easier to accommodate.
* **Multidimensional Scaling (MDS)**[62]. MDS maps data points from an original high dimensional space to spaces of a lower dimension, likewise several other methods, but approaching the problem differently, i.e., on the basis of the dissimilarities between data points rather than the points themselves. In particular, MDS tries to find a lower-dimensional representation of the data preserving the pairwise distances as much as possible. A variant of MDS is the so-called Sammon mapping [62; 63].  
PCA, PPCA and mixture of PPCA are appropriate when the data are linear or approximately piecewise linear. An alternative approach is to use global nonlinear methods such as SOM in fact, due to its ease and its several plotting choices [48; 63; 64], it was used for a broad number of applications.  
However, SOM also does not provide any probability density function, suffering from other disadvantages that can be overcome using nonlinear, latent variable models such as the Generative Topographic Mapping (GTM) [65]. These latter models are able to effectively show the composition of simple and complex data sets, even though they are less effective when dealing with too complex data sets.  
Even if it is nonlinear, a single two-or three-dimensional projection may be inadequate to catch all of the information of interest within a data set. For instance, a projection that is capable of better separating two clusters might not be the one that better reveals the structure within one of those clusters. Hierarchical models, comprising multiple 2D or 3D visualization spaces, come into play here. The idea is to display an entire data set at the top-level projection in the hierarchy, where a clustering structure is possibly revealed, whereas lower-level projections enable displaying internal structures within single clusters, i.e., an arrangement of sub-clusters not evident in higher-level projections. The interested reader may refer to [66; 67] for the hierarchical versions of linear latent variable model and of GTM, respectively.