---
title: "Introduction to PlotNormTest"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{intro-PlotNormTest}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

options(rmarkdown.html_vignette.check_title = FALSE)
oldpar <- par(no.readonly = TRUE)
```

# Introduction 

This vignette shows how to use the `PlotNormTest` package to access the normality assumption of a multivariate dataset. 

# Basic example: The Cork data set

- Cork Boring dataset by Rao (1948). The weights of 𝑛 = 28 cork boring were measured in four
directions: North, East, South, and West, respectively. 

```{r}
library(PlotNormTest)
```

```{r The Cork dataset}
cork <- matrix(c(
  72, 66, 76, 77,
  60, 53, 66, 63,
  56, 57, 64, 58,
  41, 29, 36, 38,
  32, 32, 35, 36,
  30, 35, 34, 26,
  39, 39, 31, 27,
  42, 43, 31, 25,
  37, 40, 31, 25,
  33, 29, 27, 36,
  32, 30, 34, 28,
  63, 45, 74, 63,
  54, 46, 60, 52,
  47, 51, 52, 43,
  91, 79, 100, 75,
  56, 68, 47, 50,
  79, 65, 70, 61,
  81, 80, 68, 58,
  78, 55, 67, 60,
  46, 38, 37, 38,
  39, 35, 34, 37,
  32, 30, 30, 32,
  60, 50, 67, 54,
  35, 37, 48, 39,
  39, 36, 39, 31,
  50, 34, 37, 40,
  43, 37, 39, 50,
  48, 54, 57, 43
), nrow = 28, ncol = 4, byrow = T)
colnames(cork) <- c("North", "East", "South", "West")

head(cork)
```


# Marginal Univariate Normality Assessment

This section illustration how to use `PlotNormTest` to assess univariate normality assumption. We will perform the assessment for each variables (North, East, South, West) of the Cork dataset. 

## Using Score function

In score plot, evidence of non-normality is curves different from the $45^\circ$ line $y = x$. 

```{r fig.width=6, fig.height=6, fig.align='center'}
library(ggplot2)
# Score function
lapply(1:4,  FUN = function(mycol) {
  re <- PlotNormTest::cox(matrix(sort(cork[, mycol])), x.dist = 0.0001)
  a <- re$a[, 1]
  p <- ggplot(data.frame(x = re$x, a = a), aes(x = x, y = a)) + 
    geom_point(color = "steelblue3", shape = 19, size = 1.5) + 
    ggtitle(paste("Score plot: ", colnames(cork)[mycol])) +
    coord_fixed() + xlab("y")+ 
    ylab("Score function") + 
    theme_bw() + 
    theme(aspect.ratio = 1/1, panel.grid = element_blank(),
          axis.line = element_line(colour = "black"), 
          axis.text=element_text(size=12),
          axis.title=element_text(size=14,face="bold"), 
          legend.background = element_rect( 
            size=0.5, linetype="solid"), 
          legend.text = element_text(size=12))
  p
  
}
)
```




## Using T3 plot 

In $T_3$ and $T_4$, evidence of non-normality is either curves crossing the $1 - \alpha = 95\%$ confidence region bands or curve with high slopes.


```{r fig.width=6, fig.height=6, fig.align='center'}
# T3 
lapply(1:4,  FUN = function(mycol) {
  x <- cork[, mycol]
  par(cex.axis = 1.2, cex.lab = 1.2,
               mar = c(4, 4.2, 2,1), cex.main = 1.2)
  PlotNormTest::dhCGF_plot1D(x, method = "T3") 
  namex <- colnames(cork)[mycol]
  title(main = bquote(T[3]~"plot: "~.(namex)), adj = 0)
}
)

```


## Using T4 plot 
```{r fig.width=6, fig.height=6, fig.align='center'}
# T4
 par(cex.axis = 1.2, cex.lab = 1.2,
             mar = c(4, 4.2, 2,1), cex.main = 1.2)
lapply(1:4,  FUN = function(mycol) {
  x <- cork[, mycol]
  PlotNormTest::dhCGF_plot1D(x, method = "T4") 
  namex <- colnames(cork)[mycol]
  title(main = bquote(T[4]~"plot: "~.(namex)), adj = 0)
}
)
```

# Multivariate Normality Assessment
## From multivariate normality to univariate normality 

Under the assumption that $n = 28$ samples Cork dataset follows a multivariate normal distribution in $p = 4$, standardization around sample mean and sample variance results in an $\tilde{n} = 28 \times 4 = 112$ sample approximately from $N(0,1)$. Hence normality evidence can be found via assessment of normality of this univariate sample. From this, any univariate normality testing method can be applied. 

Results below show weak evidence of non-normality, as score plot does not form a straight line and $T_3$ and $T_4$ plots show curves in the right tail. However as the weak nornality assumption here is ensured by large sample size, with $n = 28$, results may not be very convincing. Hence for those small sample, $MT_3$ and $MT_4$ plots below should be used. 



```{r fig.width=6, fig.height=6, fig.align='center'}
df <- Multi.to.Uni(cork)
# Cox
score_plot1D(df$x.new, ori.index = df$ind, x.dist = .001)$plot +
  theme(legend.position = "none")+ xlab("y") +
  ggtitle("Score plot")+
  ylab("Score function")

#T3 and T4
par(cex.axis = 1.2, cex.lab = 1.2, mar = c(4, 4.2, 2,1), cex.main = 1.2)
PlotNormTest::dhCGF_plot1D(df$x.new, method = "T3")
par(cex.axis = 1.2, cex.lab = 1.2, mar = c(4, 4.2, 2,1), cex.main = 1.2)
dhCGF_plot1D(df$x.new, method = "T4")
```

## MT3 plot

Accessing multivariate normality assumption of the Cork data set directly via plots of derivatives of cumlant generating functions, shown in $MT_3$ and $MT_4$ plot. 

The two figures from $MT_3$ and $MT_4$ plots support multivariate normality assumption. 

```{r fig.width=6, fig.height=6, fig.align='center'}
par(cex.axis = 1.2, cex.lab = 1.2, mar = c(4, 4.2, 2,1), cex.main = 1.2)
PlotNormTest::d3hCGF_plot(cork)
```

## MT4 plot
```{r fig.width=6, fig.height=6, fig.align='center'}
par(cex.axis = 1.2, cex.lab = 1.2, mar = c(4, 4.2, 2,1), cex.main = 1.2)
PlotNormTest::d4hCGF_plot(cork)
```

```{r Change par back to default, include = FALSE}
par(oldpar)
```