---
title: "MGC Statistic"
author: "Eric Bridgeford"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{mgc}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, message=FALSE}
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)
}
```

# MGC Statistic

In this notebook, we show how to use the `MGC` statistic in a real and simulated context.

## Simulated Data

First, we use a simulated example, where `Y = XB + N`; that is, `Y` is linearly dependent on `X` with added gaussian noise.

```{r}
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:

```{r, fig.width=5, fig.height=3}
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:

```{r, fig.width=5, fig.height=4}
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`.


## Real Data


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:

```{r, fig.width=5, fig.height=4}
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.
