<i class="fa fa-cubes "></i> Visualising high-dimensional spaces with application to particle physics models

# <i class="fa  fa-cubes "></i> Visualising high-dimensional spaces with application to particle physics models
### <br><br> Di Cook <br><br> Monash University
### <a href="bit.ly/IMS-Singapore-Cook" class="uri">bit.ly/IMS-Singapore-Cook</a> <br><br> <br><br>Feb 13, 2018

---

# Outline

- Visualisation of high-dimensions using tours: the [tourr](https://cran.r-project.org/web/packages/tourr/index.html) package
- A library of high-dimensional shapes: the [geozoo](https://cran.r-project.org/web/packages/geozoo/index.html) package
- Model in the data-space vs data in the model-space
- Physics data visualisation

---

.left-column[
# Tours
### Definition
]
.right-column[
A .red[grand tour] is by definition a movie of low-dimensional projections constructed in such a way that it comes arbitrarily close to any low-dimensional projection; in other words, a grand tour is a space-filling curve in the manifold of low-dimensional projections of high-dimensional data spaces.
]

---

---

.left-column[
# Tours
### Definition
### Notation
]
.right-column[
- `${\mathbf x}_i \in \Re^p$`, `$i^{th}$` data vector
- `$d$` projection dimension
- `$F$` is a `$p\times d$` orthonormal frame, `$F'F=I_d$`
- The projection of `${\mathbf x}$` onto `$F$` is `${\mathbf y}_i=F'{\mathbf x}_i$`.
- Paths of projections are given by *continuous one-parameter* families `$F(t)$`, where `$t\in [a, z]$`. Starting and target frame denoted as `$F_a = F(a), F_z=F(t)$`.
- The animation of the projected data is given by a path `${\mathbf y}_i(t)=F'(t){\mathbf x}_i$`.
]

---

.left-column[
# Tours
### Definition
### Notation
### Algorithm
]
.right-column[
- Given a starting frame `$F_a$`, create a new target frame `$F_z$`.
- Initialize interpolation. Generate planar rotations, `$R({\mathbf \tau}) = R_m(\tau_m)...R_1(\tau_1), ~~~ {\mathbf \tau}=(\tau_1, ..., \tau_m)$` such that `$F_z=R({\mathbf \tau})F_a$`.
- Execute interpolation
    - `$t \leftarrow min(1, t)$`
    - `$F(t)=R({\mathbf \tau}t)F_a$` (gives frame)
    - `${\mathbf y}_i(t)=F(t)'{\mathbf x}_i$`
    - If `$t=1$` break iteration, else `$t\leftarrow t+\delta$`
- Set `$F_a=F_z$`, start again
]

---

.left-column[
# Tours
### Definition
### Notation
### Algorithm
### Avoiding whip-spin
]
.right-column[
![](img/tour_shortest_path.png)

Rotation out of the projection frame, is defined by the principal basis in `$F_a$` and `$F_z$`, defining the shortest distance between the planes, computed using singular value decomposition of `$F_a'F_z=V_a\Lambda V_z'$`,

`$$G_a=F_aV_a~~~,~~~ G_z=F_zV_z$$`.

]

---

.left-column[
# Tours
### Definition
### Notation
### Algorithm
### Avoiding whip-spin
### Choosing targets
]
.right-column[
- *Grand:* Randomly choose target
- *Little:* Basis of `$d$` of the `$p$` variables
- *Local:* Randomly within a small radius 
- *Guided:* Define structure of interest in projection, and optimise function
- *Manual:* Control the contribution of a single variable, and move along this axis  
]

---

.left-column[
# Tours
### Definition
### Notation
### Algorithm
### Avoiding whip-spin
### Choosing targets
### PP Guidance
]
.right-column[
- *Holes:* finds projections with hollow centres `$I(F)= \frac{1-\frac{1}{n}\sum_{i=1}^{n}\exp(-\frac{1}{2}{\mathbf y}_i{\mathbf y}_i')}{1-\exp(-\frac{p}{2})}$`
- *LDA:* finds separations between classes, classically `$I(F) = 1- \frac{|F'WF|}{|F'(W+B)F|}$`, where 
`$B=\sum_{i=1}^gn_i(\bar{{\mathbf y}}_{i.}-\bar{{\mathbf y}}_{..})(\bar{{\mathbf y}}_{i.}-\bar{{\mathbf y}}_{..})'$` ,
`$W=\sum_{i=1}^g\sum_{j=1}^{n_i}({\mathbf y}_{ij}-\bar{{\mathbf y}}_{i.})({\mathbf y}_{ij}-\bar{{\mathbf y}}_{i.})'$`
- *PDA:* finds separations between classes, when there are many variables and few points `$I(F, \lambda) = 1-\frac{|F'((1-\lambda)W+n\lambda I_p)F|}{|F'((1-\lambda)(B+W)+n\lambda I_p)F|}$`
]

---

.left-column[
# Tours
### Definition
### Notation
### Algorithm
### Avoiding whip-spin
### Choosing targets
### PP Guidance
### R package: tourr
]
.right-column[
- implements all of the tours except for manual
- display projection dimension `$d=1, ..., p$`, using density plots, scatterplots, parallel coordinates, stereo 3D, scatterplot matrix, chernoff faces, stars, andrews curves, and images
- guided tour using projection pursuit indices: holes, cmass, lda, pda
- possible to generate a path and play it back
]

---

.pull-left[
<iframe src="https://player.vimeo.com/video/255464691" width="300" height="285" frameborder="10" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe>
<iframe src="https://player.vimeo.com/video/255464393" width="300" height="285" frameborder="10" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe>
]
.pull-right[
# Tourr package

Self-reflection: Example path of a tour for `$d=1, p=3$`, and `$d=1, p=4$`.

```r
animate(s3_tp, 
  grand_tour(),
  display_xy(
    axes = "bottomleft",
    col=col, pch=pch, 
    edges=edges))
```
]

---
# Tourr package

Different types of displays, and projection dimension.

```r
animate_dist(flea[, 1:6])
animate_scatmat(flea[, 1:6], grand_tour(6))
animate_pcp(flea[, 1:6], grand_tour(3))
animate_faces(flea[sort(sample(1:74, 4)), 1:6], grand_tour(4))
animate_stars(flea[sort(sample(1:74, 16)), 1:6], grand_tour(5))
```

---

.pull-left[
<iframe src="https://player.vimeo.com/video/255467472" width="300" height="290" frameborder="0" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe>
]
.pull-right[
# Tourr package

Guided tour, LDA index.

```r
animate_xy(flea[, 1:6], 
  guided_tour(
    lda_pp(flea$species)), 
  sphere = TRUE, 
  col=col, 
  axes = "bottomleft")
```
]

---

.left-column[
# Library
### Overview
    
]
.right-column[
The `geozoo` package is a library of high-dimensional shapes, and code to generate them. This includes cubes, spheres, simplices, mobius strips, torii, boy surface, enneper surface, dini surface, klein bottles, cones, ...

Web site: [http://schloerke.com/geozoo/all/](http://schloerke.com/geozoo/all/)
]

---

```r
c3 <- cube.iterate(p = 3)
animate(c3$points, grand_tour(), 
        display_xy(axes = "bottomleft", 
                   edges=c3$edges))
c5 <- cube.iterate(p = 5)
animate(c5$points, grand_tour(), 
        display_xy(axes = "bottomleft", 
                   edges=c5$edges))
c5_face <- cube.face(p = 5)
animate(c5_face$points, grand_tour(), 
        display_xy(axes = "bottomleft", 
                   edges=c5_face$edges))
```

]

---

```r
s4h <- sphere.hollow(p = 4, n = 4 * 500)
colnames(s4h$points) <- paste0("V", 1:4)
animate(s4h$points, grand_tour(), 
        display_xy(axes = "bottomleft"))
s4s <- sphere.solid.random(p = 4, n = 4 * 500)
colnames(s4s$points) <- paste0("V", 1:4)
animate(s4s$points, grand_tour(), 
        display_xy(axes = "bottomleft"))
```

<img src="img/sphere_h.png" width="200px">
<img src="img/sphere_s.png" width="200px">
]

---

```r
sp3 <- simplex(p = 3)
colnames(sp3$points) <- paste0("V", 1:3)
sp3$edges <- as.matrix(sp3$edges)
animate(sp3$points, grand_tour(), 
        display_xy(axes = "bottomleft", edges=sp3$edges))
sp5 <- simplex(p = 5)
colnames(sp5$points) <- paste0("V", 1:5)
sp5$edges <- as.matrix(sp5$edges)
animate(sp5$points, grand_tour(), 
        display_xy(axes = "bottomleft", edges=sp5$edges))
```
<img src="img/simplex3.png" width="200px">
<img src="img/simplex5.png" width="200px">
]

---

.left-column[
# Library
### Overview
### Cubes
### Spheres
### Simplices
### Why simplices
    
]
.right-column[

```r
olive_rf <- randomForest(area~., 
                         data=olive_sub)
votes <- f_composition(olive_rf$votes)
animate(votes[,-4], grand_tour(), 
        display_xy(axes = "bottomleft", 
                   col=col, edges=sp3$edges))
```
<img src="img/votes.png" width="400px">
]

---

<i class="fa  fa-child "></i> Any requests? What would you like to look at? A torus, a klein, ... ?

---
.left-column[
# Library
### Overview
### Cubes
### Spheres
### Simplices
### Why simplices
### Generation    
]
.right-column[
- Cube:
    - vertices: vectors of length `$p$`, with all combinations of `$0,1$`
    - edges: connect all the vertices of length 1 apart
- Sphere hollow: 
`${\mathbf x}_i\sim N_p(0,I_p)$`, 
`$\frac{{\mathbf x}_i}{||{\mathbf x}_i||}$`
]

- It is common to show the data in the model space, for example, predicted vs observed plots for regression, linear discriminant plots, and principal components.
- By displaying the model in the high-d data space, rather than low-d
summaries of the data produced by the model, we expect to better understand the fit.

---
# Hierarchical clustering
Dendrogram: data in the model space

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# Hierarchical clustering

Model in the data space

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# Multidimensional physics

- Need to interpret and compare models with multiple parameters
- Predictions vs measurements, theorist A model vs theorist B model
- The average theorist resorts to dropping all but 1 or 2 parameters (variables)
- Potentially misses multivariate associations and differences

---

# Higgs boson

- Data from kaggle challenge.
- Two parameter view of physicists, and multiparameter view in the tour.

---

# How dark matter interacts

.pull-left[
<iframe src="https://player.vimeo.com/video/255469149" width="450" height="430" frameborder="0" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe>
]
.pull-right[
<img src="figure/unnamed-chunk-21-1.svg" style="display: block; margin: auto;" />
]

---
# Scagnostics shape differences

- [Graph-Theoretic Scagnostics, Leland Wilkinson, Anushka Anand, Robert Grossman](http://www.ncdm.uic.edu/publications/files/proc-094.pdf)
- Describe shapes of scatterplots: Outlying, Skewed, Clumpy, Sparse, Striated, Convex, Skinny, Stringy, Monotonic. 
- Designed to find most interesting pairs of variables in high-dimensional data

---

---

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# Discreteness of scagnostics

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# Summary

- The tourr package is available for you to look beyond 2D
- High-dimensional shapes, how they are defined, what they look like, how they differ is interesting
- Think about ways to look at the model in the data space
- .red[Challenge: new ideas for defining shape differences]

---

# Joint work!

- *Tours:* Andreas Buja, Debby Swayne, Heike Hofmann, Hadley Wickham
- *Library of high-d shapes:* Barret Schloerke
- *Physics application:* Ursula Laa, Michael Kipp, German Valencia

Contact: [<i class="fa  fa-envelope "></i>](http://www.dicook.org) dicook@monash.edu, [<i class="fa  fa-twitter "></i>](https://twitter.com/visnut) visnut, [<i class="fa  fa-github "></i>](https://github.com/dicook) dicook

.footnote[Slides made with Rmarkdown, xaringan package by Yihui Xie, and lorikeet theme using the [ochRe package](https://github.com/ropenscilabs/ochRe). Available at [https://github.com/dicook/IMS-Singapore-talk](https://github.com/dicook/IMS-Singapore-talk].)

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# Further reading

- Buja et al (2004) [Computational Methods for High-Dimensional Rotations in Data Visualization](http://stat.wharton.upenn.edu/~buja/PAPERS/paper-dyn-proj-algs.pdf) 
- Cook, D., and Swayne, D. [Interactive and Dynamic Graphics for Data Analysis with examples using R and GGobi](http://www.ggobi.org)
- Wickham et al (2011) [tourr: An R Package for Exploring Multivariate Data with Projections](http://www.jstatsoft.org/v40)
- Wickham et al (2015) Visualising Statistical Models: Removing the Blindfold (with Discussion), Statistical Analysis and Data Mining.
- Schloerke, et al (2016) [Escape from Boxland](https://journal.r-project.org/archive/2016/RJ-2016-044/ index.html)

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