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.

• $x_i\in R^p$, $i^{th}$ data vector
• $d$ projection dimension
• $F$ is a $p\times d$ orthonormal frame, $F^{\prime }F=I_d$
• The projection of $x$ onto $F$ is $y_i=F^{\prime }{x}_i$.
• Paths of projections are given by continuous one-parameter families $F\left(t\right)$, where $t\in [a,z]$. Starting and target frame denoted as $F_a=F\left(a\right),F_z=F\left(t\right)$.
• The animation of the projected data is given by a path $y_i\left(t\right)=F^{\prime }\left(t\right)x_i$.

• Given a starting frame $F_a$, create a new target frame $F_z$.
• Initialize interpolation. Generate planar rotations, $R\left(\tau \right)=R_m\left({\tau }_m\right)...R_1\left({\tau }_1\right),\tau =({\tau }_1,...,{\tau }_m)$ such that $F_z=R\left(\tau \right)F_a$.
• Execute interpolation
  • $t\leftarrow min(1,t)$
  • $F\left(t\right)=R\left(\tau t\right)F_a$ (gives frame)
  • $y_i\left(t\right)=F(t{)}^{\prime }{x}_i$
  • If $t=1$ break iteration, else $t\leftarrow t+\delta$
• Set $F_a=F_z$, start again

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 $F_a^{\prime }{F}_z=V_a\Lambda V_z^{\prime }$,

$$G_a=F_a{V}_a,G_z=F_z{V}_z$$.

• 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

• Holes: finds projections with hollow centres $I\left(F\right)=\frac{1-\frac{1}{n}{\sum }_{i=1}^n\exp (-\frac{1}{2}{y}_i{y}_i^{\prime })}{1-\exp (-\frac{p}{2})}$
• LDA: finds separations between classes, classically $I\left(F\right)=1-\frac{|F^{\prime }WF|}{|F^{\prime }(W+B)F|}$, where $B={\sum }_{i=1}^g{n}_i({\overline{y}}_{i.}-{\overline{y}}_{..})({\overline{y}}_{i.}-{\overline{y}}_{..}{)}^{\prime }$ , $W={\sum }_{i=1}^g{\sum }_{j=1}^{n_i}(y_{ij}-{\overline{y}}_{i.})(y_{ij}-{\overline{y}}_{i.}{)}^{\prime }$
• PDA: finds separations between classes, when there are many variables and few points $I(F,\lambda )=1-\frac{|F^{\prime }\left(\right(1-\lambda )W+n\lambda I_p)F|}{|F^{\prime }\left(\right(1-\lambda \left)\right(B+W)+n\lambda I_p)F|}$

• 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

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))

• Cube:
  • vertices: vectors of length $p$, with all combinations of $0,1$
  • edges: connect all the vertices of length 1 apart
• Sphere hollow: $x_i\sim N_p(0,I_p)$, $\frac{x_i}{||x_i||}$