str(nm.census)
head(nm.census, 3)
nrow(nm.census) #33 counties

# columns to use for analysis
ind.col <- c(4:6, 8:14)
```

##### __(1.)__ Below is the scatterplot matrix of variables of interest, describe qualitatively what you see.
```{R, fig.height = 8, fig.width = 8}
# Scatterplot matrix
library(ggplot2)
library(GGally)
p <- ggpairs(subset(nm.census, select = ind.col))
print(p)
```

Distribution of PovAll and Rent35 are almost normal and symmetical.
Variable owner and heatG also tend to follow the same, while other
variables are almost right skewed. HeatG, HeatE and no phone looks
multimodel distribution, there are very extreme values possibly
outliers in the NoPlumb and NoPhone data. Highest positive
correlation is between NoPlumb and NoPhone. HeatG and HeatW have strongest
negative correlation.

##### __(2.)__ One county has the least plumbing and phone service, in the following analysis, remove that county and redo the scatterplot matrix, describe qualitatively what you see.

```{r, fig.height = 8, fig.width = 8}
df <- nm.census[!(nm.census$NoPhone == min(nm.census$NoPhone) & nm.census$NoPlumb == min(nm.census$NoPlumb)),]
p <- ggpairs(subset(df, select = ind.col))
print(p)
```

The scatter plot shows almost same pattern and distribution as previously
with extreme country, however, the distribution of the variable PovAll
has become little narrow in spread as low poverty country has been removed from the data. Due to this effect the correlation between PovAll and other variables have increased little bit.

### In the following analysis, let's continue with data without the extreme county. Now perform PCA analysis using correlation matrix, answer the following questions.

##### __(3.)__ How many principal components would you retain to explain about 3/4ths of the total variability? How much variability is actually retained?

```{r}
#df <- subset(nm.census, select = ind.col)
df.scaled <- scale(subset(df, select = ind.col), center = TRUE, scale = TRUE)
df.pca <- princomp(subset(df, select = ind.col), cor = TRUE)
pca.var <- df.pca$sdev^2 / sum(df.pca$sdev^2)
cumsum(pca.var)
```

Four principal components are retained to explain 3/4 (75%) of the variability.
First four PC expains about 77% of the total variability.

##### __(4.)__ Interpret the principal components you retained in the previous step.

First PC expains maximum portion of the data of about 32% varibility, PC2 explains
about 19%, PC3 explains about 14% and PC4 is responsible for 11% of total
variability.

```{r}
plot(df.pca)
p1 <- ggplot(as.data.frame(df.pca$scores), aes(x = Comp.1, y = Comp.2, colour = df$PovAll))
p1 <- p1 + scale_colour_gradientn(colours=c("red", "blue"))
p1 <- p1 + geom_text(aes(label = df$County), vjust = -0.5, alpha = 0.25)
p1 <- p1 + geom_point(size = 3)
p1 <- p1 + theme(legend.position="bottom

Problem 1: NM County-level Poverty Data In this example we’ll use...