Initial Visualization

ggplot(diamonds, aes(cut,price)) + geom_boxplot()

ggplot(diamonds, aes(color,price)) + geom_boxplot()

ggplot(diamonds, aes(clarity,price)) + geom_boxplot()

ggplot(diamonds, aes(carat, price)) +
  geom_hex(bins=50)

Subset Data and replot

diamonds2 <- diamonds %>%
  filter(carat <= 2.5)  %>%
  mutate(lprice = log2(price), lcarat = log2(carat))

ggplot(diamonds2, aes(lcarat, lprice)) +
  geom_hex(bins=50)

Simple model and visualization

mod_diamond <- lm(lprice ~ lcarat, data = diamonds2)

grid <- diamonds2 %>%
  data_grid(carat = seq_range(carat, 20)) %>%
  mutate(lcarat = log2(carat)) %>%
  add_predictions(mod_diamond, "lprice") %>%
  mutate(price = 2 ^ lprice)

ggplot(diamonds2, aes(carat, price)) +
  geom_hex(bins = 50) +
  geom_line(data = grid, color = "green", size = 1)

Add residuals and plot

diamonds2 <- diamonds2 %>%
  add_residuals(mod_diamond, "lresid")

ggplot(diamonds2, aes(lcarat, lresid)) +
  geom_hex(bins = 50)

ggplot(diamonds2, aes(cut,lresid)) + geom_boxplot()

ggplot(diamonds2, aes(color,lresid)) + geom_boxplot()

ggplot(diamonds2, aes(clarity,lresid)) + geom_boxplot()

Four parameter model and visualization

mod_diamond2 <- lm(
  lprice ~ lcarat + color + cut + clarity, diamonds2
)

grid <- diamonds2 %>%
  data_grid(cut, .model = mod_diamond2) %>%
  add_predictions(mod_diamond2)
grid
## # A tibble: 5 x 5
##   cut       lcarat color clarity  pred
##   <ord>      <dbl> <chr> <chr>   <dbl>
## 1 Fair      -0.515 G     VS2      11.2
## 2 Good      -0.515 G     VS2      11.3
## 3 Very Good -0.515 G     VS2      11.4
## 4 Premium   -0.515 G     VS2      11.4
## 5 Ideal     -0.515 G     VS2      11.4
ggplot(grid, aes(cut, pred)) +
  geom_point()

Plot residuals of four parameter model

diamonds2 <- diamonds2 %>%
  add_residuals(mod_diamond2, "lresid2")

ggplot(diamonds2, aes(lcarat, lresid2)) +
  geom_hex(bins = 50)

diamonds2 %>%
  filter(abs(lresid2) > 1) %>%
  add_predictions(mod_diamond2) %>%
  mutate(pred = round(2^pred)) %>%
  select(price, pred, carat:table, x:z) %>%
  arrange(price)
## # A tibble: 16 x 11
##    price  pred carat cut       color clarity depth table     x     y     z
##    <int> <dbl> <dbl> <ord>     <ord> <ord>   <dbl> <dbl> <dbl> <dbl> <dbl>
##  1  1013   264 0.25  Fair      F     SI2      54.4    64  4.3   4.23  2.32
##  2  1186   284 0.25  Premium   G     SI2      59      60  5.33  5.28  3.12
##  3  1186   284 0.25  Premium   G     SI2      58.8    60  5.33  5.28  3.12
##  4  1262  2644 1.03  Fair      E     I1       78.2    54  5.72  5.59  4.42
##  5  1415   639 0.35  Fair      G     VS2      65.9    54  5.57  5.53  3.66
##  6  1415   639 0.35  Fair      G     VS2      65.9    54  5.57  5.53  3.66
##  7  1715   576 0.32  Fair      F     VS2      59.6    60  4.42  4.34  2.61
##  8  1776   412 0.290 Fair      F     SI1      55.8    60  4.48  4.41  2.48
##  9  2160   314 0.34  Fair      F     I1       55.8    62  4.72  4.6   2.6 
## 10  2366   774 0.3   Very Good D     VVS2     60.6    58  4.33  4.35  2.63
## 11  3360  1373 0.51  Premium   F     SI1      62.7    62  5.09  4.96  3.15
## 12  3807  1540 0.61  Good      F     SI2      62.5    65  5.36  5.29  3.33
## 13  3920  1705 0.51  Fair      F     VVS2     65.4    60  4.98  4.9   3.23
## 14  4368  1705 0.51  Fair      F     VVS2     60.7    66  5.21  5.11  3.13
## 15 10011  4048 1.01  Fair      D     SI2      64.6    58  6.25  6.2   4.02
## 16 10470 23622 2.46  Premium   E     SI2      59.7    59  8.82  8.76  5.25

Question #1

In the plot of lcarat vs. lprice, there are some bright vertical strips. What do they represent?

A: It means many diamonds are in the same weight but in different prices. There are more factors affecting the price.

Question #2

If log(price) = a_0 + a_1 * log(carat), what does that say about the relationship between price and carat?

A:This means the price and carat is positively related. The relationship is linear and 1% increase in carat leads to 1% increase in price.

Question #3

Extract the diamonds that have very high and very low residuals. Is there anything unusual about these diamonds? Are they particularly bad or good, or do you think these are pricing errors?

# Use this chunk to place your code for extracting the high and low residuals
diamond2=diamonds %>% 
  filter(carat<=2.5) %>% 
  mutate(lprice=log2(price), lcarat=log2(carat))
mod_diamond2=lm(lprice~lcarat+color+clarity+cut,data=diamond2)
diamond2=diamond2 %>%
  add_residuals(mod_diamond2,'lresid')
summary(diamond2$lresid)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -1.17388 -0.12437 -0.00094  0.00000  0.11920  2.78322
diamond3=diamond2 %>% filter(lresid>quantile(lresid)[[3]] | lresid<quantile(lresid)[[1]] )
table(diamond3$cut)
## 
##      Fair      Good Very Good   Premium     Ideal 
##       780      2562      6020      7048     10497
diamond3 %>% 
  ggplot(aes(clarity,price))+
  geom_boxplot()+
  facet_grid(~cut)

A: Diamonds with better clearity has lower price, which is not right. The reason could be that the price is determined not only by crearity, but more factors. If we ignore other factors, the data shows this unusual result.

Question #4

Does the final model, mod_diamonds2, do a good job of predicting diamond prices? Would you trust it to tell you how much to spend if you were buying a diamond and why?

# Use this chunk to place your code for assessing how well the model predicts diamond prices
diamonds2=diamonds2 %>% 
    add_predictions(mod_diamond) %>% 
    mutate(pred=round(2^pred),
           err=pred-price)
diamonds2 %>% 
  add_residuals(mod_diamond) %>% 
  mutate(resid=2^abs(resid)) %>% 
  ggplot(aes(resid)) +
  geom_histogram(color="black",fill="blue")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

A: Seeing the plot, I believe this average model is good for most cases and the prediction would be reliable. However, I still need to pay attention to many other factors when using this model to buy diamonds.