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?

Answer: From the plot of lcarat vs lprice, we could know that they have linear relationship and the distribution has a higher density on y-axis. They represent many diamonds have similar carat but with different price.

Question #2

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

Answer: From the log-transformation model, the linear relationship between log(price) and log(carat) means that for every 1% increase in carat, the price will increase by 1%.

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?

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

mod_diamond <- lm(lprice ~ lcarat + color + clarity + cut, data = diamonds2)

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

summary(diamonds2$lresid)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -1.17388 -0.12437 -0.00094  0.00000  0.11920  2.78322
diamonds3 <- diamonds2 %>% filter(lresid > quantile(lresid)[[3]] | lresid < quantile(lresid)[[1]] )

table(diamonds3$cut)
## 
##      Fair      Good Very Good   Premium     Ideal 
##       780      2562      6020      7048     10497
table(diamonds3$clarity)
## 
##   I1  SI2  SI1  VS2  VS1 VVS2 VVS1   IF 
##  391 5032 6898 5879 3810 2395 1686  816
diamonds3 %>% 
  ggplot(aes(clarity,price))+
  geom_boxplot()+
  facet_grid(~cut)

Conclusion: Yes, it has few errors. From the graph of the clarity vs price, some diamonds have lower price but with better clarity.

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?

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()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Conclusion: Based on the plots of the residuals, the prediction is good for most of the diamonds. However, the P-value is greater than 5%, and it may result in some error with unusual data. Therefore, I may not use it to buy a diamond.