---
title: "Homework 1"
author: "Nick Meyer"
output:
  word_document: default
  pdf_document: default
---

```{r setup, include=TRUE, echo=FALSE}
knitr::opts_chunk$set(echo = TRUE)
suppressMessages({library(smooth)
    library(tidyverse)
    library(car)
    library(leaps)
    library(bestglm)
    attach("./../Regression.Rdata", name="Regression")})
```

## King County Housing Prices


```{r DataAcqFmt, include=TRUE}
king <- read.csv("./../data/KingCountyHomes_train.csv")
king$waterfront <- king$waterfront %>% as.factor
king$renovated <- king$renovated %>% as.factor

king <- subset(king, select=-ID)
king.test <- read.csv("./../data/KingCountyHomes_test.csv")
king.test$waterfront <- king.test$waterfront %>% as.factor
king.test$renovated <- king.test$renovated %>% as.factor

king.test <- subset(king.test, select=-ID)
summary(king)
ggplot(king, aes(x=price)) +
    geom_density()

```

## Task 1
### Part A: Naive OLS

```{r NaiveOLS, include=TRUE}
king.baseOLS <-  lm(price~., data=king)

print(summary(king.baseOLS))

par(mfrow=c(2,2))
plot(king.baseOLS)
par(mfrow=c(1,1))

#print(VIF(king.baseOLS))
```
Our model does not account for colinearity, in fact, when I try to run `VIF`, it simply fails due to singularities (specifically the fact that `king$sqft_basement` can be zero messes with things.) It also does not take into account teh fact that `sqft_living = sqft_above + sqft_basement`, which is probably screwing up the model even more.
$R^2_{adj} =  0.6981$, which isn't terrible, but is certainly not great.

Let's try to find which variables are not independent: (ie find a linear combination)
```{r linCombs}
lcs <- findLinearCombos(data.matrix(king))
#which col are we getting rid of?
sprintf("There is a Linear Combination between the following columns: %s", lcs$linearCombos)
sprintf("Column[s] that needs to be removed is/are %s. (index %i)", names(king)[lcs$remove], lcs$remove)

king.fixed <- king %>% select(-.[[lcs$remove]])
```
So then the `sqft_basement` column is redundant, and in task 2, we will eliminate it.


### Part B: Stepwise Models

```{r stepOLS, include=TRUE}
king.full <- regsubsets(price ~., data=king, nvmax=length(names(king)))

summary(king.full)
king.full.summary <- summary(king.full)
names(king.full.summary)

which.max(king.full.summary$adjr2)
which.min(king.full.summary$cp)
which.min(king.full.summary$bic)

#par(mfrow=c(2,2))
#plot(king.full, scale='r2')
#plot(king.full, scale='adjr2')
#plot(king.full, scale='Cp')
#plot(king.full, scale='bic')
#par(mfrow=c(1,1))
```
```{r bestOLS, include=TRUE}
#king.xs <-subset(king, select=-price)
#king.bestOLS <- bestglm(cbind(king.xs, king$price))
#attributes(king.bestOLS)
```