MCMC-revision.Rmd

---
title: "R Notebook"
output:
  pdf_document: default
  html_notebook: default
---
# Reading the data
```{r}
mcmc <- setNames(data.frame(matrix(ncol = 6, nrow = 1)), c("n", "k", "D", "kD", "Stop_Time", "nu2"))
mcmc


read.mcmc <- function(D, K, n, df) {
    tv.fileName = paste("data/TV-V", toString(n), "K", toString(K), "D", toString(D), ".csv", sep="")
    nu2.fileName = paste("data/NU2-V", toString(n), "K", toString(K), "D", toString(D), ".csv", sep="")
    KD <- paste(toString(K), toString(D), sep="/")
    stop <- nrow(read.csv(tv.fileName, skip = 2))
    nu2 <- tail(read.csv(nu2.fileName, skip=2), n=1)$NU2.est
    row <- c(n, K, D, KD, stop, nu2)
    df <- rbind(df, row) 
    return(df)
}

# degree 3
D = 3
for (K in 5:5){
  for (n in seq(4, 10, 2)){
    mcmc <- read.mcmc(D, K, n, mcmc)
  }
}

for (K in 6:7){
  for (n in seq(4, 8, 2)){
    mcmc <- read.mcmc(D, K, n, mcmc)
  }
}

# degree 4
D = 4
for (K in 6:7){
  for (n in seq(5, 9, 1)){
    mcmc <- read.mcmc(D, K, n, mcmc)
  }
}

# degree 4
D = 4
for (K in 6:6){
  for (n in seq(9, 11, 1)){
    mcmc <- read.mcmc(D, K, n, mcmc)
  }
}

# degree 4
D = 4
for (K in 7:7){
  for (n in seq(9, 9, 1)){ # change to 10
    mcmc <- read.mcmc(D, K, n, mcmc)
  }
}

```
# Read V12 K5 D3
```{r}
D = 3
n = 12
K = 5

    tv.fileName = paste("data/TV-V", toString(n), "K", toString(K), "D", toString(D), ".csv", sep="")
    nu2.fileName = paste("data/NU2-V", toString(n), "K", toString(K), "D", toString(D), ".csv", sep="")
    KD <- paste(toString(K), toString(D), sep="/")
    stop <- nrow(read.csv(tv.fileName, skip = 2))
    row <- c(n, K, D, KD, stop, NA)
    mcmc <- rbind(mcmc, row) 

```

# Ensure proper types
```{r}
mcmc <- mcmc[-1,]
mcmc$k <- as.integer(mcmc$k)
mcmc$D <- as.integer(mcmc$D)
mcmc$n <- as.integer(mcmc$n)
mcmc$Stop_Time <- as.integer(mcmc$Stop_Time)
mcmc$nu2 <- as.numeric(mcmc$nu2)
mcmc
```


```{r}
library(ggplot2)
ggplot(mcmc, aes(n, Stop_Time)) + geom_point(aes(colour = kD), size = 3) + geom_line(aes(colour = kD), size = 1.5) + ggtitle("Stop Time over n for k/D curves")
```


```{r}
ggplot(mcmc, aes(n, log(nu2))) + geom_point(aes(colour = kD), size = 3) + geom_line(aes(colour = kD), size = 1.5) + ggtitle("log(nu) for n and k/D")
# small nu => mixes more slowly 
# as size of graph gets bigger, then spectral gap gets smaller 
```
```{r}
# stopping time <= O(log(n / eps)/nu2) 
# this bound is very loose - much bigger than stopping time
# using bounds w nu2 not great for proving conjecture
# ordering is similar to stopping vs n graph
# nu2 picks one random starting vector, will benefit from more trials
ggplot(mcmc, aes(log(n/0.01)/nu2, Stop_Time)) + geom_point(aes(colour = kD), size = 3) + geom_line(aes(colour = kD), size = 1.5) + ggtitle("log(n/eps) / nu2 and stopping time")
```

```{r}
ggplot(mcmc, aes(log(nu2), Stop_Time)) + geom_point(aes(colour = kD), size = 3) + geom_line(aes(colour = kD), size = 1.5) + ggtitle("Stopping Time and log(nu2)")
```

```{r}
# nlogn coefficient

mcmc$nlogn <- mcmc$n * log(mcmc$n)

lm2 <- lm(Stop_Time ~ nlogn + D + k, data=mcmc)
summary(lm2)
plot(lm2)
```

```{r}
lm1 <- lm(Stop_Time ~ nlogn, data=mcmc)
summary(lm1)

lm.n <- lm(Stop_Time ~ n, data=mcmc)
lm.nlogn <- lm(Stop_Time ~ nlogn, data=mcmc)

plot(Stop_Time ~ nlogn, data=mcmc)

anova(lm.n, lm.nlogn)
```
```{r}
lm4 <- lm(Stop_Time ~ (nlogn + D + k)^2, data=mcmc)
summary(lm4)
```