A 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.
xi∈Rp
, ith
data vectord
projection dimensionF
is a p×d
orthonormal frame, F′F=Id
x
onto F
is yi=F′xi
.F(t)
, where t∈[a,z]
. Starting and target frame denoted as Fa=F(a),Fz=F(t)
.yi(t)=F′(t)xi
.Fa
, create a new target frame Fz
.R(τ)=Rm(τm)...R1(τ1), τ=(τ1,...,τm)
such that Fz=R(τ)Fa
.t←min(1,t)
F(t)=R(τt)Fa
(gives frame)yi(t)=F(t)′xi
t=1
break iteration, else t←t+δ
Fa=Fz
, start againRotation out of the projection frame, is defined by the principal basis in Fa
and Fz
, defining the shortest distance between the planes, computed using singular value decomposition of F′aFz=VaΛV′z
,
Ga=FaVa , Gz=FzVz
.
d
of the p
variablesI(F)=1−1n∑ni=1exp(−12yiy′i)1−exp(−p2)
I(F)=1−|F′WF||F′(W+B)F|
, where
B=∑gi=1ni(¯yi.−¯y..)(¯yi.−¯y..)′
,
W=∑gi=1∑nij=1(yij−¯yi.)(yij−¯yi.)′
I(F,λ)=1−|F′((1−λ)W+nλIp)F||F′((1−λ)(B+W)+nλIp)F|
d=1,...,p
, using density plots, scatterplots, parallel coordinates, stereo 3D, scatterplot matrix, chernoff faces, stars, andrews curves, and imagesSelf-reflection: Example path of a tour for d=1,p=3
, and d=1,p=4
.
animate(s3_tp, grand_tour(), display_xy( axes = "bottomleft", col=col, pch=pch, edges=edges))
Different types of displays, and projection dimension.
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))
Guided tour, LDA index.
animate_xy(flea[, 1:6], guided_tour( lda_pp(flea$species)), sphere = TRUE, col=col, axes = "bottomleft")
The geozoo
package is a library of high-dimensional shapes, and code to generate them. This includes cubes, spheres, simplices, mobius strips, torii, boy surface, enneper surface, dini surface, klein bottles, cones, ...
Web site: http://schloerke.com/geozoo/all/
c3 <- cube.iterate(p = 3)animate(c3$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$edges))
s4h <- sphere.hollow(p = 4, n = 4 * 500)colnames(s4h$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$points, grand_tour(), display_xy(axes = "bottomleft"))
sp3 <- simplex(p = 3)colnames(sp3$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$edges))
olive_rf <- randomForest(area~., data=olive_sub)votes <- f_composition(olive_rf$votes)animate(votes[,-4], grand_tour(), display_xy(axes = "bottomleft", col=col, edges=sp3$edges))
Any requests? What would you like to look at? A torus, a klein, ... ?
p
, with all combinations of 0,1
xi∼Np(0,Ip)
,
xi||xi||
Schloerke et al (2016) "Escape from Boxland" The R Journal
Wickham et al (2015) Visualizing statistical models: Removing the blindfold, SAM
Dendrogram: data in the model space
Model in the data space
Contact: dicook@monash.edu, visnut, dicook
Slides made with Rmarkdown, xaringan package by Yihui Xie, and lorikeet theme using the ochRe package. Available at https://github.com/dicook/IMS-Singapore-talk
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Keyboard shortcuts
↑, ←, Pg Up, k | Go to previous slide |
↓, →, Pg Dn, Space, j | Go to next slide |
Home | Go to first slide |
End | Go to last slide |
Number + Return | Go to specific slide |
b / m / f | Toggle blackout / mirrored / fullscreen mode |
c | Clone slideshow |
p | Toggle presenter mode |
t | Restart the presentation timer |
?, h | Toggle this help |
Esc | Back to slideshow |