Examples of principal nested torii (general \(d\)) – Part I

1 Examples in \(d=2\)

We optimize iteratively the objective function \[ \lambda\underbrace{\sum_{i=1}^n{\sum_{k=1}^{d-1}}\left(\frac{1}{\sqrt{d}}\|{\mathbf{P}_k}\mathbf{X}_i\|-1\right)^2}_{\substack{\text{Proj's distance to}\\ \sqrt{d}\times\mathbb{S}^{2 d-1}\supset(\mathbb{S}^1)^d}}+(1-\lambda)\underbrace{\sum_{i=1}^n\|(\mathbf{I}-\mathbf{P})\mathbf{X}_i\|^2}_{\substack{\text{Complement to}\\\text{proj's variation}}} \] We set \(\lambda=0.5\) in the next plots. We use the raw scores (no rescaling, see Issue 1). The legend for the next plots:

• Red vertical lines: Fréchet circular means of the marginals.
• Overlaid gray numbers: Fréchet standard deviations (with circular distance) with respect to the Fréchet circular mean.
• Dashed horizontal/vertical black lines: theoretical support of the scores (changes with the level of the scores). Periodicity is assumed in this support. Present if distanceScaled = FALSE (the default). See Issue 1.
• Dashed black curves: projections of the PC1 curve in pairs of variables. The backwards mean is represented with “*“, whereas the backwards antimean is represented by an”x“.

1.1 Steve’s dataset

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1.2 Uniform-like

Uniform in \([-\pi,\pi)^2\)

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Uniform vertical band

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1.3 Diagonal-like

Diagonal $y=x

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Diagonal \(y=x-2\)

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1.4 Trigonometric-like

\(\sin(x)\)

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Shorter-support \(\sin(x)\)

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1.5 Gaussian-like

Non-isotropic

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Rotated

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1.6 Clusters

Two separated clusters

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Three clusters

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2 \(\lambda\) effect

Legend for the next plots:

• Left plots corresponds to fits with \(\lambda=0\).
• Right to fits with \(\lambda=1\).

Three sparse separated clusters

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Three concentrated separated clusters

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Shorter-support \(\sin(x)\)

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3 Issue 1: re-scaling vs. not re-scaling the raw scores

In the way we construct the (raw) scores, we have that, in \(d=2\): \[\begin{align*} \text{Score}_1=&\,\mathrm{Distance\_along\_curve\_from\_projection\_to\_mean}\in\left[-l/2,l/2\right),l\leq4\pi,\\ \text{Score}_2=&\,\mathrm{Distance\_from\_point\_to\_curve\_projection}\in\left[-\frac{\sqrt{2}\pi}{2},\frac{\sqrt{2}\pi}{2}\right). \end{align*}\]

Therefore, \((\text{Score}_1,\text{Score}_2)\not\in[-\pi,\pi)^2\). Using distanceScaled = TRUE, these are scores are multiplied by \(\frac{\pi}{l/2}\) and \(\frac{\pi}{\sqrt{2} \pi / 2}\) in order to force them to be in \([-\pi,\pi)\).

Recall that for a random variable \(X\) with \(\mathrm{supp}(X)=(a,b)\), \(\mathbb{V}\mathrm{ar}[X]\leq\frac{(b-a)^2}{4}\). This means that:

• \(\mathbb{V}\mathrm{ar}[\text{Score}_1]\leq\frac{1}{4}l^2\leq4\pi^2=39.47=:C_1\).
• \(\mathbb{V}\mathrm{ar}[\text{Score}_2]\leq\frac{1}{4}2\pi^2=\frac{1}{2}\pi^2=4.93=:C_2\).

For general \(d\), \[ \text{Score}_d=\mathrm{Distance\_from\_point\_to\_surface\_projection}\in[-\frac{\sqrt{d}\pi}{2},\frac{\sqrt{d}\pi}{2}) \] and therefore \(\mathbb{V}\mathrm{ar}[\text{Score}_d]\leq\frac{1}{4}d\pi^2=\frac{d}{4}\pi^2=:C_d\). This means that \[ C_2<C_3<\ldots<C_{d-1}<C_{d} \] and that \[ C_{1}=C_{16}>C_{15}>\ldots>C_2,\quad C_1<C_{17}<C_{18}<\ldots<C_d. \] Setting distanceScaled = FALSE (raw scores) or distanceScaled = TRUE (applies rescaling) has consequences, as illustrated below.

Shorter-support \(\sin(x)\)

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Non-isotropic Gaussian \(\sin(x)\)

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Rotated Gaussian

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4 Issue 2: excentricity of the curves

Some choices of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) provide more squarish curves than others. Squarish fits tend to yield degenerate projections to the corners. This is likely more problematic in higher dimensions, since the moment the projections collapse, they will remain degenerate for lower-dimensional fits, hence producing a sequence of degenerate scores.

I do not know what is the parametrization of \(\mathbf{u}\) and \(\mathbf{v}\) that provides squarish curves (squarish can be characterized by the length of the curves, which is close to \(4\pi\) for the squarish ones). It is somehow related with the vectors \(\mathbf{u}\) and \(\mathbf{v}\) having entries close to \(0\) (respectively, close to \(1\)), as the following empirical evidence suggests (regressions of lengths on entries of the vectors):

# Sample random curves and evaluate their lengths
M <- 1e4
l <- numeric(M)
x <- matrix(nrow = M, ncol = 3)
y <- matrix(nrow = M, ncol = 2)
u <- v <- matrix(nrow = M, ncol = 4)
for (i in 1:M) {
    
  x[i, ] <- c(runif(2, min = 0, max = pi), runif(1, min = -pi, max = pi))
  y[i, ] <- c(runif(1, min = 0, max = pi), runif(1, min = -pi, max = pi))
  u[i, ] <- anglesToSphere(theta = x[i, ])
  v[i, ] <- c(anglesToSphere(theta = y[i, ]) %*% Hu(u = u[i, ])[-1, ])
  l[i] <- distPC1Curve(alpha = c(0, 2 * pi - 1e-4), u = u[i, ], v = v[i, ], 
                       N = 1e3, shortest = FALSE, der = FALSE)

}
df <- data.frame("l" = l, "u" = u, "v" = v)
df <- reshape(df, direction = "long", 
              varying = c("u.1", "u.2", "u.3", "u.4", "v.1", "v.2", "v.3", "v.4"),
              times = c("u", "v"), timevar = "k")
df <- reshape2::melt(df, measure.vars = c("u", "v"))
df$id <- NULL
df$k <- as.factor(df$k)
names(df) <- c("len", "k", "uv", "vec")
grid.arrange(
  ggplot(data = df[df$uv == "u", ], mapping = aes(x = vec, y = len, colour = k)) + 
    geom_smooth(se = FALSE) + ggtitle("Regression of the length curve on the entries of u"),
  ggplot(data = df[df$uv == "v", ], mapping = aes(x = vec, y = len, colour = k)) + 
    geom_smooth(se = FALSE) + ggtitle("Regression of the length curve on the entries of v")
)

If we could characterize when the curves are squarish and also the surface, we could add a small penalty in the obejective function to avoid degenerate solutions.