Visual Analytics (Breast Cancer Analysis)

PCA of Breast Cancer

This PCA visualization plot presents the breast cancer data, showing that each point denotes a benign or malignant tumor data sample. The x-axis is the first principal component, containing the data's most significant variance, and the y-axis is the second principal component. The separation into two classes is witnessed by coloring the points with purple on the benign and yellow on the malignant classes. The vast majority of benign (purple) cases can be found on the left, with malignant (yellow) cases spread further to the right but with an overlap among them. This implies that PCA has the significant characteristics necessary to differentiate between benign and malignant tumors.

In the case of PCA, I decided to keep the first two principal components so that the dimensionality of the dataset would be reduced to 2D, allowing a clear graphical representation in two dimensions. The choice was made because it was assumed that the first two components would explain an important percentage variation in the data since PCA maximizes variance along these initial components. Plotting only the two components, I found a reasonable compromise of not too many parameters and the ability to capture their essential features. In addition, I did not even scale the components up, as I was sure PCA automatically rescales data according to variability.

t-SNE of Breast Cancer

The t-SNE plot shows separate benign and malignant breast cancer data. It is based on the first t-SNE component in the form of the x-axis and the second t-SNE component in the vertical y-axis. The benign cases (colored purple) have points tightly distributed on the left, whereas the malignant cases (coloured yellow) lie in a clear single cluster on the right. The extent of spreading the yellow points is broader than the purple points, which means that t-SNE has managed to place the two classes in clear areas. This implies that the non-linear nature of the data has been recreated by t-SNE, giving a distinguishing line between the two classes.

For t-SNE, I used the default perplexity of 30 and a learning rate of 200, a good starting point most of the time with such a dataset size. These parameters assist in striking the balance between the data's global and local structure considerations. The confusion will impact how many nearest neighbours will be considered per point, and the learning rate will affect the rate at which the algorithm will converge. Having tried various values, I established that these settings gave a plot that was densely separated with little noise, permitting easy distinction between benign and malignant categories.

UMAP of Breast Cancer

The UMAP plot reveals a strict separation of benign and malignant samples of the breast cancer dataset. The x-axis is the first component of UMAP, and the y-axis is the second component of UMAP. The benign cases (colored purple) are pointed out on the left side of the plot, whereas the malignant ones (colored yellow) reside on the right side. The clusters are highly separated, meaning that UMAP has successfully captured the underlying structure of the data and it is easy to tell the difference between the two classes. Also, the fact that the benign points are more compact than the malignant points further shows that the given method could extract some meaningful patterns in the data.

In UMAP, I applied the default settings associated with the number of neighbors (15) and the minimal distance (0.1). The maximum number of neighbors regulates the degree of focus on local over global relationships, and the minimum distance influences the density between the clusters. With the default options, I discovered UMAP offers a clear distinction between benign and malignant cases without overrating the data. The values of the parameters were selected to maintain a balance between calculations and the resolution of the separation between the two classes.

GTM of Breast Cancer Data

The GTM plot shows a more intricate pattern than the other dimensionality reduction methods. The x-axis is the first GTM component (GTM1), and the y-axis is the second GTM component (GTM2). The benign cases (purple) disperse in the lower left and center, whereas malignant cases (yellow) mainly form towards the top right. Although some prior knowledge overlaps between the two types, one can still feel the separation very well. This means the underlying structure of the data GTM has captured differs from that of methods such as PCA or UMAP. This sparse distribution of the benign cases indicates that GTM may have been concentrating more on non-linearities of the data rather than linear separations, as PCA can produce.

In the case of GTM, I took the parameters as default, that is, a grid size of 20 and the regularization parameter that checked the smoothness of the mapping. Such settings enabled a flexible model that could handle the non-linear relations in the data. The parametric selection aimed at a tradeoff between the preservation of both local and global structure and the smooth, interpretable projectability of the model. Although there was some overlap, the general distinction between the benign and malignant cases was apparent, indicating that the parameters employed were suitable for such kind of data.