# On the Role of Clustering and Visualization Techniques in Gene Microarray Data
## 3 Visualization Techniques for Microarray Gene Expression Data
### Visualization Based on Probabilistic Principal Surfaces  
We focus on the visualization capabilities of PPS mentioned earlier when describing NEC clustering in Section 2.4. PPS is a spherical version of the latent variable model defined in GTM (for details on latent variable models and GTM, the reader may refer to [65; 68], in particular, if the latent space is chosen to be three-dimensional, then it is possible to construct a spherical manifold uniformly arranging PPS nodes \(\{\mathbf{x}_{m}\}_{m=1}^{M}\), on a sphere surface in the \(\mathbb{R}^{3}\) latent space, where a set of latent basis functions are evenly located on the sphere with a lower density [69]. Indeed, the sphere gives us a continuous manifold where data can be projected, and it is especially adequate to visualize high-dimensional data due to its natural tendency for characterizing data sparsity and periphery particularly at increasing dimensionality (curse of dimensionality). This way, a decision maker is capable of (a) projecting, visualizing, rotating and manipulating his data on the surface of the sphere; (b) carrying out deeper analysis by interacting with data and discovering areas of interest on the sphere; and (c) choosing points of interest, looking at their neighbors, analogous points, and showing all related information, etc.  
After a spherical PPS model is fitted to the data, the data themselves are projected into the latent space as points onto a sphere. Furthermore, some important information is obtained by drawing the data density function (i.e., through the responsibilities for each latent variable) with varying intensity on the spherical manifold [2]. The sphere will comprise areas that are denser than others and this visual information is easy to see and understand (Figure 1); interestingly, denser areas might suggest the existence of more clusters, thus requiring further investigations. Given a set of data samples \(\mathbf{t}_{i}\), \(i=1,\ldots,N\), the latent manifold coordinates \(\mathbf{\hat{x}}_{i}\) of each data point \(\mathbf{t}_{i}\) are computed as  
\[\mathbf{\hat{x}}_{i}\equiv\langle\mathbf{x}|\mathbf{t}_{i}\rangle=\int\mathbf{ x}p(\mathbf{x}|\mathbf{t})d\mathbf{x}=\sum_{m=1}^{M}r_{mi}\mathbf{x}_{m},\]  
where \(r_{mi}\) are the latent variable responsibilities, defined as  
\[r_{mi}=p(\mathbf{x}_{m}|\mathbf{t}_{i}) =\frac{p(\mathbf{t}_{i}|\mathbf{x}_{m})P(\mathbf{x}_{m})}{\sum_{ mt=1}^{M}p(\mathbf{t}_{i}|\mathbf{x}_{mt})P(\mathbf{x}_{mt})}\] \[=\frac{p(\mathbf{t}_{i}|\mathbf{x}_{m})}{\sum_{mt=1}^{M}p(\mathbf{ t}_{i}|\mathbf{x}_{mt})}.\]  
The responsibility \(r_{mi}\) corresponds to the posterior probability that that the \(i\)th data point was generated by the \(m\)th component. Since \(\|\mathbf{x}_{m}\|=1\) and \(\sum_{m}r_{mi}=1\), for \(i=1,\ldots,N\), these coordinates lie within a unit sphere, i.e., \(\|\mathbf{\hat{x}}_{i}\|\leq 1\).  
Cluster Visualization  
Another helpful plot for an information miner is cluster plotting and the data points therein. All of the plots on the spherical manifold can be viewed readily by moving the globe interactively. Once a decision maker has a general view of the number of clusters on the globe, he can use this information to discover the clusters by using agglomerative hierarchical clustering on the Gaussian centers in the data space. The points for which the corresponding centers are in the same group belong to the same cluster. The point projections into the latent space are then used to display the groupings onto the latent sphere [2; 54] (see Figure 2). PPS has been used effectively in a number of microarraydata, such as in a human cancer cell line (HeLa) using cDNA microarrays [53; 70], for studying cell cycle in yeast [13; 71].