## 2 Methods
### Adjusting Spread Factor for Balance between Distance-Stress and Local Topology
Overall the algorithm used by TPDS can be written as ```
Use SOM to get winner node (clusters) of data-points ;  Calculate mean for SOM clusters ;  Calculate distances between cluster means (centers) as \(dist\);  Calculate distances between data-points as \(d\) ;  Calculate standard deviation(\(sd\)) of distances between data-points ;  Initialize a multiplying factor \(multif\) ; while\(multif<limit\)do  increase \(multif\) by a factor ;  calculate \(\rho=sd\)\(x\)\(multif\) ; for every pair of data-points \((i,j)\)do ifcluster\(c_{i}=c_{j}\)then  use distance between data-points \(d_{ij}\);  Calculate likelihood of proximity \(m_{ij}\) using eq(1) else  use distance between cluster means \(dist(c_{i},c_{j})\) ;  Calculate likelihood of proximity \(m_{ij}\) using eq(2)  end if  calculate spread factor \(S_{ij}\) using eq(3) ;  Calculate modified distance \(\hat{d}_{ij}\) by multiplcation with spread factor using eq(4) ;  end if  Use non-metric MDS with modified distance;  Estimate cost function \(CF\) using output from non-metric MDS with eq(5);  Store CF and low-dimension coordinates ;  end while Result: Report low-dimension coordinates with lowest \(CF\)
```

**Algorithm 1**The algorithm of TPDS

### Optimization of Parameters

Most of the internal parameters used by TPDS were tuned to provide satisfactory performance. Such parameters which do not need optimization include the number of iterations and step size to search the value of spread factor to minimize Equation (5) (see Algorithm 1). The grid search method used for different data-set here was done using 5-10 values for spread factor \(\rho\), however TPDS also allows more step size in case if it is needed. However, TPDS is dependent on two important external parameters which can influence the quality of results. The first external parameter is Lagrange multiplier \(\lambda\) shown in Equation (5). As explained above, the value of \(\lambda\) helps in calculation of spread factor. The second external parameter is grid size of SOM. The grid size of SOM should be kept according to the expected number of classes for better visualization; however, it is not necessary for downstream analysis methods such as supervised classification or machine learning. if the value of grid-size is large (above 50), TPDS can still produce a satisfactory result. Here we have kept the value of grid-size between 20-60 for the data-sets used for evaluation. ### Experimental Setup

We compared the performance of TPDS with two types of dimension reduction methods which were proven to be efficient on different kinds of data-sets. Moreover, target of TPDS is not exactly the same as t-SNE, i.e., to improve only visualization of data. Therefore for evaluation, we used four data-sets which could represent the application of dimension reduction in different branches of Biomedical data analysis. We used two data-sets with physiological features, one protein-expression profile and a large single-cell gene-expression matrix for evaluation. One of the physiological data-sets used here, consisted of features derived from the speech of normal individuals and Parkinson'sdisease [14]. Every replicate for an individual was considered as a data-point. The other physiological data-set called SCADI (Self-Care Activities Dataset based on ICF-CY) is based on ICF-CY ( International Classification of Functioning, Disability and Health for Children and Youth) [15]. The SCADI data-set contains self-care attributes of 70 children with physical and motor disability based on ICF-CY. A protein expression data-set was also used to evaluate TPDS. This protein expression data-set consists of the expression levels of 77 proteins/protein modifications using samples from the nuclear fraction of cortex of mouse [16]. We also used a data set of single-cell expression profile which contained RNA-seq read-counts from cells belonging to seven cell-lines [17]. Single-cell expression data set had outliers which added a new dimension of evaluation of the robustness of different dimension reduction methods.