---
title: "Regression"
author: "Øystein Olav Skaar"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Regression}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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


## Regression

Enjoy this brief demonstration of the regression module



### First we simulate some data
```{r simulatedata}
# Create normal distributed data with mean = 0 and standard deviation = 1
## r = 0.5
data <- MASS::mvrnorm(n=100,
                      mu=c(0, 0),
                      Sigma=matrix(c(1, 0.5, 0.5, 1), 2),
                      empirical=TRUE)
# Add names
colnames(data) <- c("x","y")
```

### Check the correlation and regression results using frequentist methods

```{r freq1}
# Correlation
stats::cor(data)[2]

# Regression
summary(stats::lm(y ~ x, data=data.frame(data)))
```

### Then the regression results using the Bayesian model
```{r baeys1, eval = FALSE}
mcmc <- bfw::bfw(project.data = data,
            y = "y",
            x = "x",
            saved.steps = 50000,
            jags.model = "regression",
            jags.seed = 100,
            silent = TRUE)
# Print the results            
round(mcmc$summary.MCMC[,3:7],3)
#>                        Mode   ESS  HDIlo HDIhi   n
#> beta0[1]: Intercept  -0.008 50000 -0.172 0.173 100
#> beta[1]: Y vs. X      0.492 51970  0.330 0.674 100
#> sigma[1]: Y vs. X     0.863 28840  0.760 1.005 100
#> zbeta0[1]: Intercept -0.008 50000 -0.172 0.173 100
#> zbeta[1]: Y vs. X     0.492 51970  0.330 0.674 100
#> zsigma[1]: Y vs. X    0.863 28840  0.760 1.005 100
#> R^2 (block: 1)        0.246 51970  0.165 0.337 100
```

### Now we add some noise
```{r noise}
# Create noise with mean = 10 / -10 and sd = 1
## r = -1.0
noise <- MASS::mvrnorm(n=2,
                       mu=c(10, -10),
                       Sigma=matrix(c(1, -1, -1, 1), 2),
                       empirical=TRUE)
# Combine data
biased.data <- rbind(data,noise)
```

### Repeat
```{r freq2}
# Correlation
stats::cor(biased.data)[2]

# Regression
summary(stats::lm(y ~ x, data=data.frame(biased.data)))
```

### Finally, using Bayesian model with robust estimates
```{r baeys2, eval = FALSE}
mcmc.robust <- bfw::bfw(project.data = biased.data,
            y = "y",
            x = "x",
            saved.steps = 50000,
            jags.model = "regression",
            jags.seed = 100,
            run.robust = TRUE,
            silent = TRUE)
# Print the results            
round(mcmc.robust$summary.MCMC[,3:7],3)
#>                        Mode   ESS  HDIlo HDIhi   n
#> beta0[1]: Intercept  -0.026 29844 -0.204 0.141 102
#> beta[1]: Y vs. X      0.430 29549  0.265 0.604 102
#> sigma[1]: Y vs. X     0.671 16716  0.530 0.842 102
#> zbeta0[1]: Intercept  0.138 28442  0.042 0.244 102
#> zbeta[1]: Y vs. X     0.430 29549  0.265 0.604 102
#> zsigma[1]: Y vs. X    0.392 16716  0.310 0.492 102
#> R^2 (block: 1)        0.236 29549  0.145 0.331 102
```