Visualising your fitted non-linear dimension reduction model in the high-dimensional data space

Jayani P.G. Lakshika

Joint work with Prof. Dianne Cook, Dr. Paul Harrison, Dr. Michael Lydeamore, Dr. Thiyanga S. Talagala

High-dimensional data

\(X = \begin{bmatrix} \textbf{x}_{11} & \textbf{x}_{12} & \cdots & \textbf{x}_{1p}\\ \textbf{x}_{21} & \textbf{x}_{22} & \cdots & \textbf{x}_{2p} \\\vdots & \vdots & \ddots & \vdots & \\ \textbf{x}_{n1} & \textbf{x}_{n2} & \cdots & \textbf{x}_{np}\\\end{bmatrix}\)

Peripheral Blood Mononuclear Cells (PBMC)

Tour

What is a tour?

Interactive and dynamic graphics to visualise high-dimensional data.

Why is the tour technique employed?

Tour shows a sequence of linear projections as a movie.
It involves mentally assembling multiple low-dimensional views to comprehend the structure in higher dimensions.

Software: langevitour

Non-linear dimensional reduction (NLDR) techniques

NLDR techniques designed to capture the complex and non-linear relationships present within high-dimensional data.

Match-a-roo (1/4)

The data shown in the two displays is the

SAME

DIFFERENT

Match-a-roo (2/4)

The data shown in the two displays is the

SAME

DIFFERENT

Match-a-roo (3/4)

The data shown in the two displays is the

SAME

DIFFERENT

Match-a-roo (4/4)

The data shown in the two displays is the

SAME

DIFFERENT

Motivation

Single-cell gene expression: same data, different NLDR + hyper-parameters

How do you decide which is the most reasonable representation?

This is the published figure.

Here is the 9D data viewed using a grand tour, linear projections into 2D.

Show “model-in-the-data-space”

data-in-the-model-space

model-in-the-data-space

S-curve in 7D

\(\theta \sim U(-3\pi/2, 3\pi/2)\)

\(X_1 = \sin(\theta)\)

\(X_2 \sim U(0, 2)\)

\(X_3 = \text{sign}(\theta) \times (\cos(\theta) - 1)\)

True model: \(T=(X_1, X_2, X_3)\)

\(X_4, X_5, X_6, X_7\) are additional noise dimensions

data-in-the-model-space

What is the model?

data-in-the-model-space

model-in-the-data-space

Overview of method

1. Construct the \(2\text{-}D\) model

2. Lift the model into high-dimensions

Steps of the algorithm

1. Construct the \(2\text{-}D\) model

NLDR layout, b. hex bin , c. bin centers, d. triangulation wire frame.

Steps of the algorithm

2. Lift the model into high-dimensions

Factors for fitting and measuring fit

NLDR layout, different methods and different hyper-parameters
Number of bins
Bin start position
Low density removal
Long edge removal

MSE in high-dimensions: mean sum of squared differences between observed and fitted values

\[\frac{1}{n}\sum_{h = 1}^{b}\sum_{i = 1}^{n_h}\sum_{j = 1}^{p} ({x}_{hij} - C^{(p)}_{hj})^2\] \(n =\) the number of observations,

\(b =\) the number of bins,

\(n_h =\) the number of observations in \(h^{th}\) bin,

\(p =\) the number of variables,

\({x}_{hij} =\) the \(j^{th}\) dimensional data of \(i^{th}\) observation in \(h^{th}\) hexagon.

Candidates for NLDR layout

tSNE, b. UMAP, c. PHATE, d. TriMAP, e. PaCMAP

MSE of candidates

PHATE not competitive
Not much difference between any other method based on Error
No elbow, just gradual decrease as number of (non-empty) bins increase

Chosen fit for S-curve

tSNE with perplexity: 27

Fills out the width of the S

Pretty good! Can you see the twist??

PBMC data set

MSE of candidates

Chosen fit for PBMC data set

tSNE with perplexity: 30

Clusters with small separations, non-linear clusters

Densed points, filled out clusters

Five Gaussian clusters in 4D

tSNE

PaCMAP

Linked plots

Assess the model fits the points everywhere, better in some places, simply mismatches the pattern

quollr

questioning how a high-dimensional object looks in low-dimensions using r

Summary

Note

Provided a method to create a model from a NLDR layout that can be displayed with the data to assess the fit.

Make it easier for researchers to make better decisions on which NLDR layout is best for their work.

Jayani P.G. Lakshika

Collaborators: Prof. Dianne Cook, Dr. Paul Harrison, Dr. Michael Lydeamore, Dr. Thiyanga S. Talagala