Sampling Contamination.Rmd

---
title: 'Sampling: Contamination'
author: "Xianbin Cheng"
date: "August 29, 2018"
output: html_document
---

# Method #

1. Load in libraries, visualization functions and check the session info.

```{r, warning = FALSE, message= FALSE}
source(file = "Sampling_libraries.R")
source(file = "Sampling_contamination.R")
source(file = "Sampling_visualization.R")
#library(plotly)
```

```{r}
sessionInfo()
```

2. Define the input paramaters and create a simulation function.

* `n_contam` = the number of contamination points  
* `x_lim` = the limits of the horizontal axis  
* `y_lim` = the limits of the vertical axis  
* `x` = the horizontal coordinate of the contamination center, which follows a uniform distribution (`U(0,10)`)
* `y` = the vertical coordinate of the contamination center, which follows a uniform distribution(`U(0,10)`)  
* `cont_level` = a vector that indicates the mean contamination level (logCFU/g or logCFU/mL) and the standard deviation in a log scale, assuming contamination level follows a log normal distribution $ln(cont\_level)$~$N(\mu, \sigma^2)$. 

**Mode 1: Discrete Spread**

* `n_affected` = the number of affected plants near the contamination spot, which follows a Poisson distribution ($Pois(\lambda = 5)$)   
* `covar_mat` = covariance matrix of `x` and `y`, which defines the spread of contamination. Assume the spread follows a 2D normal distribution with var(X) = 0.25, var(Y) = 0.25 and cov(X, Y) = 0  

**Mode 2: Continuous Spread**

* `spread_radius` = the radius of the contamination spread
* `LOC` = the limit of contribution of contamination. By default, it is set at 0.001.  

```{r}
## The input parameters
n_contam = rpois(n = 1, lambda = 3)
x_lim = c(0, 10)
y_lim = c(0, 10)
cont_level = c(7, 1)

### Mode 1
n_affected = rpois(n = 1, lambda = 5)
covar_mat = matrix(data = c(0.25, 0, 0, 0.25), nrow = 2, ncol = 2)

### Mode 2
spread_radius = 2.5
LOC = 10^(-3)
```

3. Simulate the contamination center and its spread.

    a) Generate `n_contam` contamination spots whose coordinates follow a uniform distribution.
    
    b) If `n_affected` > 0, then generate contamination spread points that distribute around the contamination spots. The contamination spread points follow a bivariate normal distribution. If `n_affected` = 0, then skip this step.
    
    c) Generate contamination levels for contamination spots. The contamination levels follow a log normal distribution ($ln(X)$~$N(`r cont_level[1]`, `r cont_level[2]`)$). The spread of contamination level is defined by a decay function. This function is affected by distance, `spread_radius` and `LOC`.
        
        i) `f_exp()`: $$ f(d) = exp(- \frac {d} {\theta}) $$, where d = distance, $\theta = - \frac {spread\_radius} {ln(LOC)}$
        
        ii) `f_norm()`: $$ f(d) = exp(- \frac {d^2} {\sigma^2})$$, where d = distance, $\sigma = \sqrt {- {\frac {spread\_radius^2} {ln(LOC)}}}$.
        
    d) Check for any points that fall outside the perimeter of the field.
    
    e) Remove those outliers.

```{r}
sim_contam
```

```{r}
# Run this if we want reproducibility
#set.seed(123)
```

```{r}
contam_xy = sim_contam(n_contam = n_contam, xlim = x_lim, ylim = y_lim, covariance = covar_mat, n_affected = n_affected, radius = spread_radius, cont_level = cont_level) 
```

```{r}
summary(contam_xy$label)
```


# Results #

1. Visualization of contamination spots.

**Mode 1: Discrete Spread**

```{r}
plot_contam_dis = contam_draw(data = contam_xy, spread = "discrete", xlim = x_lim, ylim = y_lim)
```

```{r}
plot_contam_dis
```

**Mode 2: Continuous Spread**

```{r}
plot_contam_cont = contam_draw(data = contam_xy, spread = "continuous", xlim = x_lim, ylim = y_lim)
```

```{r}
plot_contam_cont
```

```{r}
## Choose an exponential function to describe the contamination spread
contam_level_draw(dimension = "2d", method = "exp", spread_radius = spread_radius, LOC = LOC)

contam_level_draw(dimension = "3d", method = "exp", spread_radius = spread_radius, LOC = LOC, df_contam = contam_xy, xlim = x_lim, ylim = y_lim, interactive = FALSE)

## Choose a normal distribution-like function to describe the contamination spread
contam_level_draw(dimension = "2d", method = "norm", spread_radius = spread_radius, LOC = LOC)

contam_level_draw(dimension = "3d", method = "norm", spread_radius = spread_radius, LOC = LOC, df_contam = contam_xy, xlim = x_lim, ylim = y_lim, interactive = FALSE)
```

```{r, eval = FALSE, echo = FALSE}
### Visualize the covariance matrix
test = mvrnorm(n = 10000, mu = c(5,5), Sigma = covar_mat)
test = as.data.frame(test)
colnames(test) = c("X", "Y")

kernel_density = kde2d(x = test$X, y = test$Y, n = 50)
image(kernel_density)
contour(kernel_density, add = TRUE)
```