library(mgc) library(reshape2) library(ggplot2) plot_sim_func <- function(X, Y, Xf, Yf, name, geom='line') { if (!is.null(dim(Y))) { Y <- Y[, 1] Yf <- Yf[, 1] } if (geom == 'points') { funcgeom <- geom_point } else { funcgeom <- geom_line } data <- data.frame(x1=X[,1], y=Y) data_func <- data.frame(x1=Xf[,1], y=Yf) ggplot(data, aes(x=x1, y=y)) + funcgeom(data=data_func, aes(x=x1, y=y), color='red', size=3) + geom_point() + xlab("x") + ylab("y") + ggtitle(name) + theme_bw() } plot_mtx <- function(Dx, main.title="Local Correlation Map", xlab.title="# X Neighbors", ylab.title="# Y Neighbors") { data <- melt(Dx) ggplot(data, aes(x=Var1, y=Var2, fill=value)) + geom_tile() + scale_fill_gradientn(name="l-corr", colours=c("#f2f0f7", "#cbc9e2", "#9e9ac8", "#6a51a3"), limits=c(min(Dx), max(Dx))) + xlab(xlab.title) + ylab(ylab.title) + theme_bw() + ggtitle(main.title) }
In this notebook, we show how to use the MGC
statistic in a real and simulated context.
First, we use a simulated example, where Y = XB + N
; that is, Y
is linearly dependent on X
with added gaussian noise.
n=200 # 100 samples d=1 # simple 1-d case set.seed(12345) data <- mgc.sims.linear(n, d) # data with noise func <- mgc.sims.linear(n, d, eps=0) # source function
We first visualize the data:
plot_sim_func(data$X, data$Y, func$X, func$Y, name="Linear Simulation")
which clearly shows a slightly linear relationship.
visualizing the MGC
image with 100 replicates:
set.seed(12345) res <- mgc.test(data$X, data$Y, nperm=20) # 20 permutations test; typically should be run with >100 permutations plot_mtx(res$localCorr, main.title="Local Correlation Map") print(res$optimalScale) print(res$statMGC)
As we can see, the local correlation plot suggests a strongly linear relationship. This is because intuitively, having more and more neighbors will help in our identification of the linear relationship between x
and y
, and as we can see in the local correlation map, k=n=200
and l=n=200
shows the best smoothed p
.
In the below demo, we show the result of MGC
to determine the relationship between the first (sepal length) and third (petal length) dimensions of the iris
dataset:
set.seed(12345) res <- mgc.test(iris[,1,drop=FALSE], iris[,3,drop=FALSE], nperm=20) plot_mtx(res$localCorr, main.title="Local Correlation Map", xlab.title="Sepal Length Neighbors", ylab.title="Petal Length Neighbors") print(res$optimalScale) print(res$statMGC)
viewing the corr map above we see that the relationship betweel Sepal and Petal Length is strongly linear.
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