DiamondsDataModel
Kai Tsyr Lee
2019-07-05
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 11.4ggplot(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 5.25Question #1
In the plot of lcarat vs. lprice, there are some bright vertical strips. What do they represent?
Ans : Bright means higher count. Hence, those bright strips means a high density of population is aggregated at some specific log of carat in respect to a specific price range. That strips are deemed to be the popular choices when people buying diamonds.
Question #2
If log(price) = a_0 + a_1 * log(carat), what does that say about the relationship between price and carat?
Ans : The equation describes a linear relationship between log of price and log of carat. log transformation means price and carat doesn’t have a linear relationship directly, but through a transformation. It means that 1% change in carat leads to a_1% change 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
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$lresid)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.17388 -0.12437 -0.00094 0.00000 0.11920 2.78322diamonds3 <- diamonds2 %>% filter(lresid > quantile(lresid)[[3]] | lresid < quantile(lresid)[[1]] )
table(diamonds3$cut)##
## Fair Good Very Good Premium Ideal
## 780 2562 6020 7048 10497table(diamonds3$clarity)##
## I1 SI2 SI1 VS2 VS1 VVS2 VVS1 IF
## 391 5032 6898 5879 3810 2395 1686 816diamonds3 %>%
ggplot(aes(clarity,price))+
geom_boxplot()+
facet_grid(~cut)
diamonds2 <-
diamonds %>%
mutate(lprice = log2(price),
lcarat = log2(carat))
mod1 <- lm(lprice ~ lcarat + color + clarity + cut, data = diamonds2)
bottom <-
diamonds2 %>%
add_residuals(mod1) %>%
arrange(resid) %>%
slice(1:10)
top <-
diamonds2 %>%
add_residuals(mod1) %>%
arrange(-resid) %>%
slice(1:10)
bind_rows(bottom, top) %>%
select(price, carat, resid)## # A tibble: 20 x 3
## price carat resid
## <int> <dbl> <dbl>
## 1 6512 3 -1.46
## 2 10470 2.46 -1.17
## 3 10453 3.05 -1.14
## 4 14220 3.01 -1.12
## 5 9925 3.01 -1.12
## 6 18701 3.51 -1.09
## 7 1262 1.03 -1.04
## 8 8040 3.01 -1.02
## 9 12587 3.5 -0.990
## 10 8044 3 -0.985
## 11 2160 0.34 2.81
## 12 1776 0.290 2.10
## 13 1186 0.25 2.06
## 14 1186 0.25 2.06
## 15 1013 0.25 1.94
## 16 2366 0.3 1.61
## 17 1715 0.32 1.57
## 18 4368 0.51 1.36
## 19 10011 1.01 1.31
## 20 3807 0.61 1.31Ans : There seems to be no pricing errors. Generally speaking, it is observed that high residuals occur mostly in low carat dimonds, while low residuals occur in high carat dimonds.
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
mod_diamond2 <- lm(
lprice ~ lcarat + color + cut + clarity, diamonds2
)
diamonds4 <- diamonds2 %>%
add_predictions(mod_diamond2)
ggplot(diamonds4, aes(lprice, pred)) +
geom_point() +
geom_abline(slope=1, color="red")
summary(mod_diamond2)##
## Call:
## lm(formula = lprice ~ lcarat + color + cut + clarity, data = diamonds2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.45867 -0.12459 -0.00033 0.12033 2.81005
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.200915 0.001685 7242.225 < 2e-16 ***
## lcarat 1.883718 0.001129 1668.750 < 2e-16 ***
## color.L -0.634174 0.002925 -216.828 < 2e-16 ***
## color.Q -0.137955 0.002687 -51.335 < 2e-16 ***
## color.C -0.021328 0.002515 -8.481 < 2e-16 ***
## color^4 0.017098 0.002310 7.403 1.35e-13 ***
## color^5 -0.003176 0.002182 -1.455 0.146
## color^6 0.003450 0.001984 1.739 0.082 .
## cut.L 0.174154 0.003396 51.284 < 2e-16 ***
## cut.Q -0.050660 0.002989 -16.950 < 2e-16 ***
## cut.C 0.019446 0.002595 7.494 6.77e-14 ***
## cut^4 -0.002253 0.002079 -1.084 0.278
## clarity.L 1.322709 0.005161 256.274 < 2e-16 ***
## clarity.Q -0.350630 0.004804 -72.982 < 2e-16 ***
## clarity.C 0.191013 0.004118 46.387 < 2e-16 ***
## clarity^4 -0.095368 0.003294 -28.955 < 2e-16 ***
## clarity^5 0.039556 0.002689 14.711 < 2e-16 ***
## clarity^6 -0.002624 0.002342 -1.120 0.263
## clarity^7 0.048375 0.002066 23.412 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.193 on 53921 degrees of freedom
## Multiple R-squared: 0.9826, Adjusted R-squared: 0.9826
## F-statistic: 1.693e+05 on 18 and 53921 DF, p-value: < 2.2e-16Ans : Based on the figure, the predicition of price is relative closed to the true value. And based on the summary table, most of the coefficients’ p-values are statistically significant at an alpha level of 0.05. The adjusted R-squared is 0.98, which is better than 0.93 from the simple model. However, the r-squared may indicate an overfittng model, thus furthur analysis needs to be conducted. For now, I would not depende on the model entirely for diamond purchasing.