## 2 Methods
### Adjusting Spread Factor for Balance between Distance-Stress and Local Topology
Thus, all the data sets used here had class labels which helped in evaluating the performance of clustering after dimension-reduction. To evaluate the usability of dimension reduction methods we have used two different approaches of clustering namely k-means and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) [18]. ## 3 Results

To evaluate TPDS we used have data sets where both classification and visualization are needed for making a useful inference. The first data set used here for evaluation was generated by [PERSON] et al. [14], and it consists of feature extracted from speech recording of normal individuals and Parkinson's disease patients. For every patient or individual, there are 4 replicate of data points which have 44 features. In the original manuscript, authors have used supervised classification approach to classify the normal and Parkinson's cases. We applied our unsupervised method of reducing the dimension. As shown in Figure 1 for [PERSON] et al. 's data set [14] the data-points belonging to normal and [PERSON]'s disease are shown with two colours. Sammon mapping, non-metric MDS and t-SNE are not able to show separability of data-points belonging to two classes. However, TPDS revealed clear separability between Parkinson's and normal cases and had lower MDS-cost than t-SNE. Furthermore, we performed k-mean clustering (k = 2) on the output from TPDS and other tested methods. We calculated clustering purity using the adjusted Rand index (ARI) and normalized mutual information (NMI). K-mean clustering on output from TPDS had the highest purity as the ARI and NMI scores were almost 1.7-2 times greater than other method used (tSNE, non-metric MDS and Sammon mapping) (see Figure 1). We also checked clustering purity after applying DBSCAN to cluster data-points in low-dimension. Even with DBSCAN-based classification, TPDS provided best clustering purity among tested methods (see Table 1). \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Data-Set** & **TPDS** & **t-SNE** & **Non-Metric MDS** & **Sammon Mapping** \\ \hline Parkinson’s & ARI: 0.43 & ARI: 0 & ARI: 0.0 & ARI: 0.0 \\ [PERSON] et al. & NMI: 0.33 & NMI: 0 & NMI: 0.0 & NMI: 0.0 \\ \hline Mouse Protein & ARI: 0.133 & ARI: 0 & ARI: 0.004 & ARI: 0.002 \\ [PERSON] et al. & NMI: 0.23 & NMI: 0 & NMI: 0.078 & NMI: 0.068 \\ \hline SCADI & ARI: 0.288 & ARI: 0 & ARI: 0.058 & ARI: 0.002 \\ [PERSON] et al. & NMI: 0.355 & NMI: 0 & NMI: 0.060 & NMI: 0.068 \\ \hline single cell Expression & ARI: 0.027 & ARI: 0.0022 & ARI: 0 & ARI: 0 \\ [PERSON] et al. & NMI: 0.068 & NMI: 0.024 & NMI: 0 & NMI: 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Result of clustering purity after applying DBSCANThe second data sets we used for evaluation consist of the expression levels of 77 proteins/protein modifications from the nuclear fraction of cortex of mouse [16]. This data set of protein expression was generated using 38 control mice and 34 trisomic mice with Down syndrome. There are 15 replicates per sample/mouse such that control mice, there are \(38\times 15\), or 570 measurements, and for trisomic mice, there are \(34\times 15\), or 510 measurements. In the original manuscript authors recommended to consider each replicate as a separate sample. The mice themselves belonged to eight different groups depending on genotype, behaviour and treatment. Using genotype mice could group separately as control or trisomic [16]. According to behaviour, the mice could be grouped as stimulated or non-stimulated to learn the context-shock. Similarly, mice could be grouped as treated with drug or untreated. First, we visualised data points with reduced dimension, using different colours for eight different groups. Visually, it was non-trivial to judge about which method performed better. The t-SNE method avoided crowding but spread out samples too much to cause mixing between different groups. With TPDS there seem to be crowding among some samples but for the majority of samples, co-localization was according to group (Figure 2).