---
title: "Regression and Other Stories: Simulation"
output: pdf_document
---

\renewcommand{\vec}[1]{\mathbf{#1}}

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, fig.height = 3, fig.width = 5)
library(tidyverse) 
library(gridExtra)
set.seed(09142020)
```

### Simulation

Simulation is important element of applied statistics (as well as probability/ mathematical statistics). 

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Why simulation?

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Simulation can be used for discrete or continuous outcomes, 

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- With discrete (binary) outcomes, 
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- With discrete (count) outcomes, 

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- With continuous outcomes, normal or lognormal are common, but there are others too.

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\newpage


### Simulation Activity

According to the MSU promotional materials [https://www.montana.edu/marketing/about-msu/bozeman/](https://www.montana.edu/marketing/about-msu/bozeman/), Bozeman has 300 "days of sunshine".

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1. Using this information, simulate the probability that a given week has all sunny days.



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2. Using this information, simulate the probability that the next three days are "not sunny days."



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3. What assumptions are you making with these simulations? Hint: do three consecutive "not sunny days" seem equally likely in the summer / winter?

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Now let's consider a simulation for that captures the likelihood that an air quality measurement will fall in the good level (< 12 $\mu$g/m):  [https://svc.mt.gov/deq/todaysair/](https://svc.mt.gov/deq/todaysair/).

4. Assume the hourly particulate measurement can be approximated with a lognormal distribution with parameters mean (log 10) and sd of 1. Note these are the `meanlog` and `sdlog` parameters in `rlnorm`. Create a data visualization to accompany this figure.


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5. Finally let's scrape actual air quality data from [https://svc.mt.gov/deq/todaysair/](https://svc.mt.gov/deq/todaysair/). And create a data visualization.

```{r, warning = F, message = F}
library(rvest)
scrape_BZNPM <- function(days){
  # Scrapes hourly PM2.5 readings in Bozeman for September 2020
  # inputs: sequence of days
  # outputs: data frame that contains day, hour, and hourly average PM2.5 concentration.
  smoke.df <- data.frame(day = NULL, hour = NULL, conc = NULL)
  for (d in days){
    bzn_aq <- read_html(paste("http://svc.mt.gov/deq/todaysair/AirDataDisplay.aspx?siteAcronym=BH&targetDate=9/",d,"/2020", sep=''))
    daily.smoke <- cbind(rep(d,24),html_table(bzn_aq)[[1]][,1:2])
    colnames(daily.smoke) <- c('day','hour','conc')
    daily.smoke$conc[daily.smoke$conc == 'DU'] <- 'NA'
    daily.smoke$conc <- as.numeric(daily.smoke$conc)
    smoke.df <- bind_rows(smoke.df,daily.smoke)
  }
  return(smoke.df)
}


air.quality <- scrape_BZNPM(1:13)

conc <- air.quality %>% 
  filter(!is.na(conc)) %>% 
  select(conc)

```

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### Bootstrap


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Example. Using the air quality data, we can implement the bootstrap.

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A single bootstrap replicate

```{r}
conc %>% sample_frac(1, replace = T) %>% head()
```

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Bootstrap replicates to simulate the sampling distribution of mean air quality measurement.

```{r}
num_replicates <- 1000

boot_mean <- rep(0, num_replicates)

for (i in 1:num_replicates){
  boot_mean[i] <- conc %>% sample_frac(1, replace = T) %>% summarise(mean(conc)) %>% pull()
}

tibble(boot_mean = boot_mean) %>% 
  ggplot(aes(x = boot_mean)) + 
  geom_histogram( bins = 30) + 
  theme_bw() + xlab('Mean Air Quality') + 
  ggtitle('Bootstrap distribution of Mean Air Quality') 
```

