Continious variables involved in the following:

• Parallel coordinate plots
• Heatmaps
• Star charts

Construction:

• Vertical axis: Values
• Horizontal Axis : Variables
• 1 trace line = 1 observation

Analysis: - Clusters - Outliers - Correlated variables

• Positive correlation between two adjacent variables: almost all segments are parallel to each other
• Clusters in some variable space: several trace lines that are near each other and have similar pattern
• Outliers: trace lines that have unusual pattern and/or fall out outside the common plot area

• Trace lines overlap each other -> difficult to find patterns, difficult to follow a specific trace line
• Analysis depends much on the order of variables (correlation, clusters) -> a proper reordering may improve the analysis

Example: Iris dataset - How many clusters do you see?

• Sometimes clusters overlap with categories given by some variable
  • Non-mixing groups is not the same as clustering!

• Problem of ordering (variables, observations) is one of the key problems in multidimensional visualization
  • Sometimes has a huge impact on perception (heatmaps)
• A lot of approaches exist

Problem formulation: Given data set \(\chi=\left( x_{ij}| i=1, \ldots, n, j=1, \ldots, p \right)\)

• Select order \(\Psi={i_1, \ldots, i_p}\) that optimizes visual perception (analysis) -> this defines reordering of data columns \(\Psi: \chi \rightarrow \chi'\)

Note: \(p!\) possible orderings exist…

• early approaches (for ex. Ankerst et al. 1998):

1. Choose a distance (proximity) matrix \(D=\left\{d_{ij}=d(x^i, x^j)\right\}\) between variables (columns)
   • Euclidian distance on scaled columns
   • 1- correlation
2. This defines graph with vertices \(1, \ldots, p\) and edge weights \(d_{ij}\) -> Hamiltonian path (Traveling Salesman Problem)

\(\min_{\Psi} {\sum_{j=1}^{p-1} d'_{j,j+1}}\)

• TSP is NP-complete -> Approximate solutions are used

Solution: modern approaches

• Based on:
  • Decreasing visual clutter
  • Clustering data points/dimensions
  • Outlier detection
  • Dimensionality reduction (for ex. MDS)
  • …
• Note: most of these can be applied both for ordering observations and ordering variables
  • Just transpose the data matrix…

Objective functions:

• Gradient measures (anti-Robinson)
• Hamiltonian path length
• Least squares
• …

They based on \(\min_{\Psi} {L(\Psi(D))}\)

Optimization algorithms:

• Partial enumeration
• Traveling salesman solvers
• Hierarchical clustering
• …

Aim: distances should increase from diagonal

\[ d_{ik}\leq d_{ij} \mbox{ for } 1\leq i<k<j\leq n \] \[ d_{kj}\leq d_{ij} \mbox{ for } 1\leq i<k<j\leq n \] Objective function:

\[ L(D)= \sum_{i<k<j}f(d_{ik}, d_{ij})+f(d_{kj}, d_{ij}) \] where \[ f(z,y)=sign(z-y) \mbox{ or } f(z,y)=z-y \]

Hamiltonian path length:

\[ L(D)=\sum_{i=1}^{n-1} d_{i,i+1} \]

Least squares criterion (PCA)

• Solution is similar to first PCA component

\[ L(D)= \sum_i \sum_j (d_{ij}-|i-j|)^2 \]

Partial enumeration methods

• Ex: Branch and bounds and dynamic programming
• Constructing solutions by parts

TSP solver

• Suitable for hamiltonian path objective
• Find shortest path by dynamic programming or heuristics

Hierarchical clustering

• Observations are joined into clusters
• Clusters are joined in larger clusters
• Until only one cluster left
• Leaves and branches are permuted to minimize given objective

A heat map visualizes a matrix [ n x m]

• Normally rows=observations, columns= parameters
• Heatmap has the corresponding size
• Each cell of the matrix corresponds a cell in the heatmap
• High values correspond intense colors in this map (or visa versa for other color schemes!)
• Names of variables and observations are shown

Analysis:

• Compare the values of a parameter for different observations (row)
• Compare the values for a single observation (column)
• Compare the patterns for different rows or columns
• Find similar observations (areas with the same color intensity)
• Find which variables define similarity for a group
• Find correlated variables (similar pattern within a column)

• Exercise (last picture):
  • How many clusters do you see?
  • Which variables define clusters?
  • Which variables are correlated?

• Gradient measure objective used
• See new analysis possibilities

• Use polar coordinate system
• Map column value as a coordinate in certain direction

If juxtaposed, analyse:

• Clusters
• Outliers
• Outlying directions

If superimposed,

• Comparing variable length
• Seeing similar and outlying observations

• Difficult to judge orientations
• Number of dimensions are observations is very limited
  • Number of observations is extremely limited if superimposed
• More close radar charts easier to compare
• Perception is much affected by observation ordering

Ordering:

• Same as before plus
• Dimensions can be sorted to promote more symmetric charts

• Now with reordering by Gradient Measures

Other positioning possible - PCA/MDS

Idea:

1. Make same kind of plot for subsets of data
2. Plot together
3. See patterns/differences

Analogy: cutting a sausage

• Example: Barley data
  • Anything strange?

• Faceting = one more aesthetics

• What can be analysed?
  • Patterns within/between plots
  • Conditional dependence \(Y \sim X | Z\)
  • Variable interaction, additivity

–> Useful tool for modeling!

• Compare : 3D- scatter plots

• Another car data: is there additivity?

• Design issues:
  • How to order rows/columns in trellis?
    • A: X=one var, Y0=another var (facet_grid)
    • B: independently of aes (facet_wrap)
  • How to handle categorical vars?
    • One value/panel
    • Group
    • Ordering? (R: decide factor levels)
  • How to handle real-valued vars?
    • Split equal size/length
    • Shingles

• Creates overlap
• To avoid boundary effects

Example: Aids data (Age, Time of Death, Time of Diag)

• Aids data: conclusions?

• Chapter 5

• Paper "Hahsler, M., Hornik, K., & Buchta, C. (2008). Getting things in order: an introduction to the R package seriation. Journal of Statistical Software, 25(3), 1-34".

• (Browse through) paper "Ankerst, M., Berchtold, S., & Keim, D. A. (1998, October). Similarity clustering of dimensions for an enhanced visualization of multidimensional data. In Information Visualization, 1998. Proceedings. IEEE Symposium on (pp. 52-60). IEEE."

• Becker, R. A., Cleveland, W. S., & Shyu, M. J. (1996). The visual design and control of trellis display. Journal of computational and Graphical Statistics, 5(2), 123-155.