In neurodata/r-mgc: Multiscale Graph Correlation

MGC Simulations

In this experiment, we will take a look at the 20 canonical simulations analyzed in the MGC Paper repository. For details on the simulations, please check out the appendix to the paper, which can be found on arXiv MGC paper link, with available pseudo code.

Data

Simulations

In the below simulations, we let each dimension $i$ be uniformly distributed $X_i \sim Unif(-1, 1)$, and $x_i$ is $n$ samples for the $i^{th}$ dimension, then $x \in \mathbb{R}^{n \times D}$. $A$ is a decaying vector where $A_i = \frac{1}{i}$, $A \in \mathbb{R}^D$. $\nu_i$ is the error for each measurement and is sampled from the RV $E \sim \mathcal{N}(0, \epsilon I_n)$. Below, we let $\kappa = \mathbb{I}\left{D = 1\right}$.

gen.coefs <- function(d) {
  A = as.array(1/1:d, dim=c(d, 1))
  return(A)
}

gen.x <- function(n, d) {
  x <- as.array(replicate(d, runif(n, -1, 1)))
  return(t(t(x)))
}

plot.one_d <- function(x, y, title="", eqn="") {
  dat = data.frame(x=x, y=y)
  ggplot(dat, aes(x=x, y=y)) +
    geom_point() +
    xlab("x") +
    ylab(eqn) +
    ggtitle(sprintf("1-D Simulated %s", title))
}

plot.mgc_result <- function(res, title="") {
  mgc.plot.plot_matrix(res$localCorr, title=TeX(sprintf("%s Corr Map, statMGC=%.3f", title, res$statMGC)), xlabel = "l, X neighbors", ylabel = "k, Y neighbors",
                      legend.name="statMGC")
}

ld <- 1  # low dimensional simulations are in 1 dimension
hd <- 50  # high dimensional simulations are in 50 dimensions

ln <- 100  # low dimensional simulations have 100 samples
hn <- 250  # high dimensional simulations have 250 samples

Linear Relationship

For our linear simulations, we allow a linear dependence of $y$ on $x$ with error of variance $\epsilon$, and $\nu$ is the per-sample residual:

\begin{align} y = xA + \Kappa\nu \ s_{\nu} = \epsilon \end{align}

and we let $\epsilon = 0.2$.

1 Dimensional

gen.y.linear <- function(x, A, eps) {
  er <- rnorm(n, mean=0, sd=1)
  return(x%*%A + eps*er)  # y = xA + nu  
}

eps <- 0.2
xld <- gen.x(ln, ld)
Ald <- gen.coefs(ld)
yld <- gen.y.linear(xld, Ald, 0.2)

simn <- "Linear Relationship"
plot.one_d(xld, yld, title=simn, eqn=TeX("$y = xA + \\epsilon$"))

We obtain a Linear Relationship:

res <- mgc.sample(xld, yld)
plot.mgc_result(res, title=simn)

High Dimensional

We repeat the same experiment with $50$ dimensions:

xhd <- gen.x(hn, hd)
Ahd <- gen.coefs(hd)
yhd <- gen.y.linear(xhd, Ahd, 0.2)
res <- mgc.sample(xhd, yhd)
plot.mgc_result(res, title=simn)

Exponential

For our linear simulations, we allow a linear dependence of $y$ on $x$ with error of variance $\epsilon$, and $\nu$ is the per-sample residual:

\begin{align} y = e^{xA + \kappa \nu} \ s_{\nu} = \epsilon \ \kappa = \begin{cases} 1 & D = 1 0 & otherwise \end{cases} \end{align}

and we let $\epsilon = 0.05$ and $\kappa = 0$.

1 Dimensional

gen.y.linear <- function(x, A, eps) {
  er <- rnorm(n, mean=0, sd=1)
  return(exp(x%*%A))  # y = exp(xA)
}

eps <- 0.2
xld <- gen.x(ln, ld)
Ald <- gen.coefs(ld)
yld <- gen.y.linear(xld, Ald, 0.2)

simn <- "Linear Relationship"
plot.one_d(xld, yld, title=simn, eqn=TeX("$y = xA + \\epsilon$"))

We obtain a Linear Relationship:

res <- mgc.sample(xld, yld)
plot.mgc_result(res, title=simn)

High Dimensional

We repeat the same experiment with $50$ dimensions:

xhd <- gen.x(hn, hd)
Ahd <- gen.coefs(hd)
yhd <- gen.y.linear(xhd, Ahd, 0.2)
res <- mgc.sample(xhd, yhd)
plot.mgc_result(res, title=simn)

Cubic

Joint Normal

Step Function

Quadratic

W shape

Spiral

Bernoulli

Logarithmic

Fourth Root

Circle

Ellipse

Diamond

Multiplicative

Independence

Comparison

1 Dimensional

High Dimensional

Geometries

neurodata/r-mgc documentation built on March 12, 2021, 9:45 a.m.