1 Diamonds

Diamonds are the hardest element found in nature and their atoms are arranged crystal structure. Moreover, diamonds are one of the most popular and preferable jewelry. They, on the other hand, are one of the most expensive jewelry. Thus, the characteristics of the diamond are very important to define price. According to the Yadav Diamond information, there are four important features. They are called 4C. Especially, these 4C’s of diamond constitutes the vast majority of the price.

  • Carat: The carat is the weight of the diamond and this weight is important for the diamond
    • One carat is equal to 1/5 of a gram
    • One carat is divided into a hundred equal points
  • Color: This feature give the color feature into the diamond.
    • Grade range of D to F has colorless diamonds. Moreover, the value of these type of diamonds is higher than the others
    • Diamonds, which have lower than F, are also shine. But these diamonds have strong color tones
    • A factor which is related with color is named as Fluorescence which is caused by boron trace amount inside the diamond. Fluorescence is activated by ultraviolet light. It is believed that a diamond with intense fluorescence has positive effect.
    • Colors:
      • Colorless Diamonds,the rarest and highest quality with a pure icy look, D-F Color Diamonds
      • Near-colorless diamonds: No discernible color; great value for the quality, G-H Color Diamonds and I-J Color Diamonds
      • Faint color diamonds: Budget-friendly pick; pairs beautifully with yellow gold, K Color Diamonds

Figure 1. Color classification of the diamonds

  • Clarity: Clarity defines the extent or degree of imperfections that are present in a diamond.There is 11 point diamond clarity scale that is creating by Gemologist Institute of America or GIA. his scale starts from flawless point and ends at prominent inclusions. Flawless means a diamond with no inclusion. This diamond is extremely rare.

Figure 2. Classification of Diamonds by Clarity

  • Cut: Cut is the most important characteristic of the diamond.
    • It determines how the light which enters into the diamond from the above will be reflected back to the eye of observer
    • A perfect cut diamond reflects light to its optimum

Figure 3. Quality of Cut

2 Data and Required Packages

2.1 Data Information

In this assignment, there are few objective to improve data analysis skills. These objectives can be listed as follow:

  1. To provide useful exploratory data analysis (EDA) by using visuals and tables
  2. To present a model for estimation of the diamond price

To fulfill these objectives, (i) data is analyzed and prepared for creating prediction model and analysis, (ii) the meaningful EDA is presented by using some useful packages such as ggplot2, dplyr, tidyverse etc.

Note that this packages are used to prepare EDA, if more packages are required, they are installed and loaded when they are required.

library(tidyverse) # used for data analyses
#as the tidyverse includes ggplot2 and dplyr packages, we do not need to add these packages additionally.
  # library(ggplot2) # for visualization
  # library(dplyr)   # to make calculations 
library(tm) # provides a set of predefined sources
library(knitr) # to provide report creation
library(kableExtra)  # to provide arrangement of the tables
library(patchwork)   # for combine plots in one lines
library(data.table)  # for easy implementation of some loops and functions
library(rpart)
library(rpart.plot)
library(caret)
library(e1071)
library(rattle)
library(corrplot)    # to get correlation plot 

The dataset in this analysis is obtained from the ggplot2 packages. Before the load this data set, by using ?diamonds we can obtain information about this dataset. This data set contains more than 50.000 round cut diamonds, whose features like are located in this data set. There are nrow(diamonds) rows and ncol(diamonds) variables. By using this glimpse() function, we obtain these variables.

#variables of the dataset
diamonds %>%
  glimpse()
## Rows: 53,940
## Columns: 10
## $ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, 0.23,...
## $ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very Good, ...
## $ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, J, J,...
## $ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI1, VS...
## $ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, 59.4,...
## $ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54, 62,...
## $ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339, 340,...
## $ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, 4.00,...
## $ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, 4.05,...
## $ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, 2.39,...

Like glimpse() function, we can also get similar results by using structure function which is the function of Base R, i.e., str().

diamonds %>%
  str()
## tibble [53,940 x 10] (S3: tbl_df/tbl/data.frame)
##  $ carat  : num [1:53940] 0.23 0.21 0.23 0.29 0.31 0.24 0.24 0.26 0.22 0.23 ...
##  $ cut    : Ord.factor w/ 5 levels "Fair"<"Good"<..: 5 4 2 4 2 3 3 3 1 3 ...
##  $ color  : Ord.factor w/ 7 levels "D"<"E"<"F"<"G"<..: 2 2 2 6 7 7 6 5 2 5 ...
##  $ clarity: Ord.factor w/ 8 levels "I1"<"SI2"<"SI1"<..: 2 3 5 4 2 6 7 3 4 5 ...
##  $ depth  : num [1:53940] 61.5 59.8 56.9 62.4 63.3 62.8 62.3 61.9 65.1 59.4 ...
##  $ table  : num [1:53940] 55 61 65 58 58 57 57 55 61 61 ...
##  $ price  : int [1:53940] 326 326 327 334 335 336 336 337 337 338 ...
##  $ x      : num [1:53940] 3.95 3.89 4.05 4.2 4.34 3.94 3.95 4.07 3.87 4 ...
##  $ y      : num [1:53940] 3.98 3.84 4.07 4.23 4.35 3.96 3.98 4.11 3.78 4.05 ...
##  $ z      : num [1:53940] 2.43 2.31 2.31 2.63 2.75 2.48 2.47 2.53 2.49 2.39 ...

Since we check the order of the variables of color categorical variable, there is wrong order,i.e., the D is actually the best color in the explanation, on the other hand, it is the worst in the data set. Moreover, when we search the data in the internet, we can obtain that the explanation is true. For this reason, to interpret correctly, by using following code we can rearrange the order.

diamonds$color <- factor(diamonds$color, levels = c("J", "I", "H", "G", "F", "E", "D"))

According to the above code, we define new levels, in other words we rearranged correctly.Thus, from the glimpse() and str() functions and explanation of the dataset, we obtain following variables and their definition.

Price: price in US dollars ($326–$18,823)
Carat: weight of the diamond (0.2–5.01)
Cut: quality of the cut (Fair, Good, Very Good, Premium, Ideal)
Color: diamond color, from D (best) to J (worst)
Clarity: a measurement of how clear the diamond is I1 (worst), SI2, SI1, VS2, VS1, VVS2, VVS1, IF (best)
x: length in mm (0–10.74)
y: width in mm (0–58.9)
z: depth in mm (0–31.8)
Depth: total depth percentage = z / mean(x, y) = 2 * z / (x + y) (43–79)
Table: width of top of diamond relative to widest point (43–95)

As we know the data, we can make EDA and create a prediction model. However, we need to check the dataset whether it has pre-processing or not. For thi purpose, the following section is used as data pre-processing.

2.2 Data Preprocessing

In the previous section, detailed information about investigated dataset is presented. To make pre processing, basic examinations will be made.Some of them are

  • NA values,
  • Duplicated values,
  • Control of the variables (as the depth formulation is given, we control variables which take place in this formulation)
  • Controlling of the outliers.

Moreover, the level control is made in the previous section by using str() and it is determined that the level of the color is wrong. Then, by using factor() function and its property level, the correct order is defined for the color.

#to control NA values in the dataset
sum(any(is.na(diamonds)))
## [1] 0

By using above function we can obtain number of NA values in this dataset. When we check the dataset, there is 0 NA value. This means that there is no missing value in any row or column.

sum(as.numeric(duplicated(diamonds)))
## [1] 146

Another important control point is that checking of the duplicated values. There are 146 duplicated lines. Before the analysis, we need to extract these data from the data frame. Because these duplicated values can manipulate our analysis, for example one of the duplicated value go to train data set, the other one can be in the test data. As a result, the same value become in both data set and this can prevent the good prediction. For this reason, duplicated values should be removed from the data set. This process can be made by using various function, i.e., duplicated(), unique(), and distinct(). In this assignment, the unique() function is used.

#taking only unique rows form the data set
diamonds <- unique(diamonds)
#control of the duplicated lines after removing of the duplicated lines
sum(as.numeric(duplicated(diamonds)))
## [1] 0

According to the usage of unique()function, there is no duplicated line. As a results, we have 53794 rows and 10 columns. While the rows show that the number of trials, the columns give the number of variables/features in the dataset. After this process, we can use summary() function and head() function to get more information about the dataset.

summary(diamonds) # summary of each variable in the dataset
##      carat               cut        color        clarity          depth      
##  Min.   :0.2000   Fair     : 1598   J: 2802   SI1    :13032   Min.   :43.00  
##  1st Qu.:0.4000   Good     : 4891   I: 5407   VS2    :12229   1st Qu.:61.00  
##  Median :0.7000   Very Good:12069   H: 8272   SI2    : 9150   Median :61.80  
##  Mean   :0.7978   Premium  :13748   G:11262   VS1    : 8156   Mean   :61.75  
##  3rd Qu.:1.0400   Ideal    :21488   F: 9520   VVS2   : 5056   3rd Qu.:62.50  
##  Max.   :5.0100                     E: 9776   VVS1   : 3647   Max.   :79.00  
##                                     D: 6755   (Other): 2524                  
##      table           price             x                y         
##  Min.   :43.00   Min.   :  326   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.:56.00   1st Qu.:  951   1st Qu.: 4.710   1st Qu.: 4.720  
##  Median :57.00   Median : 2401   Median : 5.700   Median : 5.710  
##  Mean   :57.46   Mean   : 3933   Mean   : 5.731   Mean   : 5.735  
##  3rd Qu.:59.00   3rd Qu.: 5327   3rd Qu.: 6.540   3rd Qu.: 6.540  
##  Max.   :95.00   Max.   :18823   Max.   :10.740   Max.   :58.900  
##                                                                   
##        z         
##  Min.   : 0.000  
##  1st Qu.: 2.910  
##  Median : 3.530  
##  Mean   : 3.539  
##  3rd Qu.: 4.030  
##  Max.   :31.800  
## 
head(diamonds) #first 6 rows of the data
## # A tibble: 6 x 10
##   carat cut       color clarity depth table price     x     y     z
##   <dbl> <ord>     <ord> <ord>   <dbl> <dbl> <int> <dbl> <dbl> <dbl>
## 1 0.23  Ideal     E     SI2      61.5    55   326  3.95  3.98  2.43
## 2 0.21  Premium   E     SI1      59.8    61   326  3.89  3.84  2.31
## 3 0.23  Good      E     VS1      56.9    65   327  4.05  4.07  2.31
## 4 0.290 Premium   I     VS2      62.4    58   334  4.2   4.23  2.63
## 5 0.31  Good      J     SI2      63.3    58   335  4.34  4.35  2.75
## 6 0.24  Very Good J     VVS2     62.8    57   336  3.94  3.96  2.48

2.3 Accuracy of the Values

The negative price values in the data set can be control.

diamonds %>%
  filter(price <= 0) %>%
  summarise(NegativePrice = n())
## # A tibble: 1 x 1
##   NegativePrice
##           <int>
## 1             0

Logically, price should be greater than zero. For this reason, we check dataset to control negative price values. According to the results there is no negative price value. This is the expected situation. Moreover, we can control accuracy of the data by using other variables. According to the research results, the relationship between x,y,z and depth can be found. In other words, the depth variable is obtain from the calculation which is obtained by using x,y, and z variable. This explanation also can be obtain from the R package explanation by using ?diamonds.

The calculation is that

total depth percentage = z / mean(x, y)
= 2 * z / (x + y)

According to the this calculation, the depth variable and x,y,z variables are compared and outliers or manipulated data can be removed. For this purpose following calculations and visualizations are made. Thus, we can control negativity of the x,y, and z variables.

diamonds %>%
  filter(x <= 0 & y > 0 & z > 0) # provide to find x variables which is smaller than equal to zero whereas the x and y is greater than zero
## # A tibble: 0 x 10
## # ... with 10 variables: carat <dbl>, cut <ord>, color <ord>, clarity <ord>,
## #   depth <dbl>, table <dbl>, price <int>, x <dbl>, y <dbl>, z <dbl>
diamonds %>%
  filter(y <= 0 & x > 0 & z > 0) # provide to find y variables which is smaller than equal to zero whereas the x and y is greater than zero
## # A tibble: 0 x 10
## # ... with 10 variables: carat <dbl>, cut <ord>, color <ord>, clarity <ord>,
## #   depth <dbl>, table <dbl>, price <int>, x <dbl>, y <dbl>, z <dbl>
diamonds %>%
  filter(z <= 0 & x > 0 & y > 0) # provide to find z variables which is smaller than equal to zero whereas the x and y is greater than zero
## # A tibble: 12 x 10
##    carat cut     color clarity depth table price     x     y     z
##    <dbl> <ord>   <ord> <ord>   <dbl> <dbl> <int> <dbl> <dbl> <dbl>
##  1  1    Premium G     SI2      59.1    59  3142  6.55  6.48     0
##  2  1.01 Premium H     I1       58.1    59  3167  6.66  6.6      0
##  3  1.1  Premium G     SI2      63      59  3696  6.5   6.47     0
##  4  1.01 Premium F     SI2      59.2    58  3837  6.5   6.47     0
##  5  1.5  Good    G     I1       64      61  4731  7.15  7.04     0
##  6  1.15 Ideal   G     VS2      59.2    56  5564  6.88  6.83     0
##  7  2.18 Premium H     SI2      59.4    61 12631  8.49  8.45     0
##  8  2.25 Premium I     SI1      61.3    58 15397  8.52  8.42     0
##  9  2.2  Premium H     SI1      61.2    59 17265  8.42  8.37     0
## 10  2.02 Premium H     VS2      62.7    53 18207  8.02  7.95     0
## 11  2.8  Good    G     SI2      63.8    58 18788  8.9   8.85     0
## 12  1.12 Premium G     I1       60.4    59  2383  6.71  6.67     0

All x and y values are positive, for now we don’t need to do anything. But, there are some missing values in the x,y, and z columns, to detect these values, we can use following codes. When we find the missing values, by using depth formulation we can fill some lines.

diamonds = diamonds %>%
  mutate(z = ifelse(z == 0 & x != 0 & y != 0,round(depth * mean(c(x, y)) / 100, 2), z)) 

Up until now, we control the negativity of the values, and moreover we find the zero values in the z column. Then, we fulfill these lines by using formulation. However, we need to control one more thing in this data set that is whether the two of these variable are zero or not. In this situation, we do not calculate the values because there two unknown value and one formula. In this situation, we can remove these trials from the data set. For this purpose, the binary combinations are controlled with filter operation.

diamonds %>%
  filter(x <= 0 & y <= 0 ) 
## # A tibble: 6 x 10
##   carat cut       color clarity depth table price     x     y     z
##   <dbl> <ord>     <ord> <ord>   <dbl> <dbl> <int> <dbl> <dbl> <dbl>
## 1  1    Very Good H     VS2      63.3    53  5139     0     0     0
## 2  1.14 Fair      G     VS1      57.5    67  6381     0     0     0
## 3  1.56 Ideal     G     VS2      62.2    54 12800     0     0     0
## 4  1.2  Premium   D     VVS1     62.1    59 15686     0     0     0
## 5  2.25 Premium   H     SI2      62.8    59 18034     0     0     0
## 6  0.71 Good      F     SI2      64.1    60  2130     0     0     0

When we control the data set we obtain trials which have both x and y are zero.We control other combinations.

diamonds %>%
  filter(x <= 0 & z <= 0)
## # A tibble: 7 x 10
##   carat cut       color clarity depth table price     x     y     z
##   <dbl> <ord>     <ord> <ord>   <dbl> <dbl> <int> <dbl> <dbl> <dbl>
## 1  1.07 Ideal     F     SI2      61.6    56  4954     0  6.62     0
## 2  1    Very Good H     VS2      63.3    53  5139     0  0        0
## 3  1.14 Fair      G     VS1      57.5    67  6381     0  0        0
## 4  1.56 Ideal     G     VS2      62.2    54 12800     0  0        0
## 5  1.2  Premium   D     VVS1     62.1    59 15686     0  0        0
## 6  2.25 Premium   H     SI2      62.8    59 18034     0  0        0
## 7  0.71 Good      F     SI2      64.1    60  2130     0  0        0
diamonds %>%
  filter(y <= 0 & z <= 0)
## # A tibble: 6 x 10
##   carat cut       color clarity depth table price     x     y     z
##   <dbl> <ord>     <ord> <ord>   <dbl> <dbl> <int> <dbl> <dbl> <dbl>
## 1  1    Very Good H     VS2      63.3    53  5139     0     0     0
## 2  1.14 Fair      G     VS1      57.5    67  6381     0     0     0
## 3  1.56 Ideal     G     VS2      62.2    54 12800     0     0     0
## 4  1.2  Premium   D     VVS1     62.1    59 15686     0     0     0
## 5  2.25 Premium   H     SI2      62.8    59 18034     0     0     0
## 6  0.71 Good      F     SI2      64.1    60  2130     0     0     0

As a result there are 7 rows that both x and z values are 0. As we do not calculate values of these variable, we can remove from the data set. Because this number is too small when we compare the all data set.

#by using following code we find values which do not have negative values. 
diamonds <- diamonds %>%
  filter(!(x == 0 & z == 0))
#control of the data set according to the zero values of x, y, and z variables.
diamonds %>%
  filter(x == 0 | y == 0 | z == 0)
## # A tibble: 0 x 10
## # ... with 10 variables: carat <dbl>, cut <ord>, color <ord>, clarity <ord>,
## #   depth <dbl>, table <dbl>, price <int>, x <dbl>, y <dbl>, z <dbl>

The last filter shows that, there is no zero values in the x,y, and z variables’ column.Now, we can have more clear data set. But,there is one more examination about the data set which is finding outliers of the data set. For this purpose, we can control numeric variables x,y, and z with range() function.

#In this report this code do not run because of the space. But following codes can be run and the results can be shown.
#this codes provide the range of the data set and values in the defined column
range(diamonds$x)  # finding of the outlier for x variable
diamonds$x %>% 
  unique() %>% 
  sort(decreasing = TRUE)

range(diamonds$y) # finding of the outlier for x variable
diamonds$y %>% 
  unique() %>% 
  sort(decreasing = TRUE) 

When we control the output of the above lines, we obtain that there is huge difference between the highest value of the y and the second highest value of the y.The reason behind this huge different, the data can be entered wrong or the calculation may be wrong. For this reason, we can remove this line or we can calculate by using depth formulation.

#first we can try to add calculation for the outlier y variable with the following code.
diamonds %>%
  filter(y > 15) %>%
  mutate(new_y = (2 * z / depth) / 100 - x) %>%
  select(depth, x, z, y, new_y)
## # A tibble: 2 x 5
##   depth     x     z     y new_y
##   <dbl> <dbl> <dbl> <dbl> <dbl>
## 1  58.9  8.09  8.06  58.9 -8.09
## 2  61.8  5.15  5.12  31.8 -5.15
#when we calculate this value, we can not obtain proper value. For this reason we can remove this data from the data set.
diamonds = diamonds %>%
  filter(y < 15)

Lastly, we control the z variable range.

range(diamonds$z)
diamonds$z %>% 
  unique() %>% 
  sort(decreasing = TRUE)

Since, we also obtain outlier trial for the z variable, we can remove or calculate new value for this trial. There are one variable, for this according to the manual calculation, the wrong entering process is detected. For this reason, instead of the removing of the line, the calculation is preferred.

#changing of the outlier with the calculated value.
diamonds$z[diamonds$z == 31.8] <- diamonds$z[diamonds$z == 31.8] / 10
#Now, we also need to check the depth column. Then, we need to change the depth value, with the new calculated depth value.
diamonds$calculated_depth <- 2 * diamonds$z / (diamonds$x + diamonds$y) * 100

By using calculated depth, which is obtained from the x,y and z values with using formulation, and depth that is given in the dataset, we can compare the results. It is expected that, both values have linear relationship.

# diamonds[, calculate := 2 * z / (x + y)]
diamonds %>%
  ggplot(., aes(x = calculated_depth, y = depth)) +
  geom_point() + 
  geom_abline(intercept = 0, slope = 1, color="red", size=1.5) +
  theme_minimal() +
  labs(title = "The Depth vs Calculated Depth",
       x = "Calculated Depth",
       y = "Depth (given in the dataset)")

According to the results, we can say that almost all variables have linear relationship, but some of them is not. When we analyzed the information about the diamonds, the difference between 56.5 and 65, whereas the ideal range is between 59.5 and 62.9. Hence, we can remove data which does not have this property. According to the web-page,the interval selected as 9. But this criteria can be changed according to the other implementations and variables.

diamonds %>%
  filter(!(abs(calculated_depth - depth) <9)) %>%
  select(calculated_depth, depth, x, y, z)
## # A tibble: 24 x 5
##    calculated_depth depth     x     y     z
##               <dbl> <dbl> <dbl> <dbl> <dbl>
##  1             51.2  62.8  6.26  6.19  3.19
##  2             63.1  43    6.32  6.27  3.97
##  3             65.7  44    6.31  6.24  4.12
##  4             51.7  64    7.15  7.04  3.67
##  5             63.0  43    6.53  6.55  4.12
##  6             49.5  59.2  6.88  6.83  3.39
##  7             16.1  60.6  6.62  6.67  1.07
##  8             65.0  55.9  6.77  6.71  4.38
##  9             56.9  67.3  7.85  5.75  3.87
## 10             20.5  61.9  7.43  7.5   1.53
## # ... with 14 more rows

There are 24 observations. When we compare it with the number of all observations in the dataset, it is very small value. So, we can remove them from the dataset.

diamonds = diamonds %>%
  filter((abs(calculated_depth - depth) < 9))
diamonds = subset(diamonds, select = -c(calculated_depth))

According to the depth formulation and web information, for ideal diamonds, there should be relationship between variables x,y, and z. For this reason, as a final interpretation we can compare the x, y and z values with each other.

#by using ggplot function, we can realize the relationship between x and y.
#it is expected that there linear relationship between these two variable. 
p1 <- diamonds %>%
        ggplot(., aes(x = x, y = y)) + 
        geom_point() + 
        geom_smooth(method = "lm", color = "red") + # provide adding smooth line for data
        theme_minimal() +
        labs(title = "Comparison of \n the x and y variable",
             x = "X variable",
             y = "Y variable")

p2 <- diamonds %>%
        ggplot(., aes(x = z, y = y)) + #comparison of the z and y variables
        geom_point() + 
        geom_smooth(method = "lm", color = "red")+ # provide adding smooth line for data
        theme_minimal() +
        labs(title = "Comparison of \n the x and y variable",
             x = "Z variable",
             y = "Y variable")

p3 <- diamonds %>%
        ggplot(., aes(x = x, y = z)) +
        geom_point() + 
        geom_smooth(method = "lm", color = "red")+
        theme_minimal() +
        labs(title = "Comparison of \n the x and y variable",
             x = "Z variable",
             y = "Y variable")

(p1 | p2 | p3) #provide combination of the plots in one line

As we mentioned before, the relationship between variables show that there is high correlation. But, we can assume that these x, y and z values are valid values.

3 Exploratory Data Analysis

After the pre-processing of the data, we make EDA for the diamonds dataset by using different variables.

3.1 Cut

As we mentioned before, cut is one of the most important feature of diamonds. This variable shows the quality of cut. In this data set, quality of cut is given as categorical variables. Fair is the lowest quality, whereas, ideal is the highest quality. For this reason, to examine diamond dataset by using this feature, first I illustrate the number of trials grouped according to the cut feature. Note that the color of the pie chart can be change by using scale_fill_manual

diamonds %>%
  #mutate(cut = factor(cut)) %>%
  group_by(cut) %>%
  summarise(count = n()) %>%
  mutate(percentage = 100*count/sum(count)) %>%

ggplot(., aes(x = '', y = count, fill = cut)) + 
  geom_bar(width = 1, stat = "identity", alpha = 0.85) +
  coord_polar("y") +
  theme_void() +
  theme(plot.title = element_text(vjust = 0.5),) +
  scale_fill_manual(values=c("seashell2", "lightpink1", "seashell3", "paleturquoise3", "paleturquoise4")) +
  geom_text(aes(label = paste(format(percentage,digits=3), "%")), size=3, color = "black", position = position_stack(vjust = 0.3)) +
  labs(title = "Quality of Cut Percentage ",
       subtitle = st,
       fill = "Quality of the Cut")

The pie chart illustrates that, most of the diamonds are cut as ideal form. Moreover, percentage of premium and very good are almost equal. On the other hand, there is a little fair cutting type.

diamonds %>%
  group_by(cut) %>%
  summarise(cut_count = n(),
            MinPrice = min(price),
            AveragePrice = mean(price),
            MaxPrice = max(price)) %>%
  #mutate(percentage = 100*count/sum(count)) %>%
  arrange(desc(cut_count)) %>%
  kable(col.names = c("Cut", "Number of Cut Quality", "Minimum Price", "Average Price", "Maximum Price")) %>%
  kable_minimal(full_width = F)
Cut Number of Cut Quality Minimum Price Average Price Maximum Price
Ideal 21479 326 3461.600 18806
Premium 13738 326 4576.581 18823
Very Good 12065 336 3981.305 18818
Good 4887 327 3916.913 18707
Fair 1592 337 4330.162 18574

More information about cut type can be obtained by analyzing the above table. While, the pie chart presents the percentage of the dataset according to the cut type, the Above table shows the number of trial with cut type. Furthermore, the minimum, average and maximum prices are addressed in this table. Although the most popular cutting type is Ideal cut, its price is not the highest one. According to the average prices, the most expensive diamonds are belongs to Premium if we just look for the cut variable. So, it means that we can not explain the response (price) variable only with the cut type of a diamond.

3.2 Colors

Colors are the another important feature of the diamonds and also it affects the price of the diamonds. For this reason, in this data set, colors are also examined. - First, according to trials, we can group according to the color. - After this grouping process we can number of diamonds with this group.

diamonds %>%
  group_by(color)%>%
  summarise(count = n()) %>%
  
  ggplot(., aes(x=color, y = count, fill = count)) +
  geom_col() +
  scale_fill_gradient("count", low="paleturquoise1", high="paleturquoise4") +
  #geom_line(aes(x=color, y=count), color = "black") +
  geom_line(aes(y = count), size = 1.2, color="black", group = 1) +
  theme_minimal() +
  theme(legend.position = "none") +
  labs(title = "Number of Colors in the Data",
       subtitle = st,
       x = "Color",
       y = "Number of Color")

According to the results of the bar chart, while there is at most G color diamond, there is at least J color. To get exact value of the each color, we can use following table.

diamonds %>%
  group_by(color)%>%
  summarise(color_count = n(),
            MinPrice = min(price),
            AveragePrice = mean(price),
            MaxPrice = max(price)) %>%
  arrange(desc(color_count)) %>%
  kable(col.names = c("Color", "Count","Minimum Price", "Average Price", "Maximum Price")) %>%
  kable_minimal(full_width = F)
Color Count Minimum Price Average Price Maximum Price
G 11252 354 3997.035 18818
E 9772 326 3080.276 18731
F 9517 342 3726.559 18791
H 8262 337 4474.006 18803
D 6753 357 3172.009 18693
I 5405 334 5079.123 18823
J 2800 335 5327.337 18710

As we mentioned before, according to the diamond information given above D-F color interval is the highest purity. Then, G-F color scale follow. At least, K color takes place. Results,on the other hand, illustrates that

  • The number of G color is the highest.
  • The most beautiful color, i.e., actually colorless, is D-F. Although the D is close to bottom of the table, E and F take place in the second and third places of the table.

Like the classification of the color according to the number of color in the dataset, by using this table output, we can obtain minimum,maximum and average prices and compare the price of diamonds with each other by using color classification. According to the average prices, the most expensive diamonds are belongs to J if we just look for the color variable. However, we expect that, as the D color is the best color in the data set, it should be most expensive.So, by using only color variable, we can not explain the response (price) variable.

3.3 Clarity

Clarity gives information about diamonds whether it contains stain or not. In this report, to find clarity classification of the data set, a kind of scatter plot is used.

diamonds %>%
  mutate(clarity = factor(clarity)) %>%
  group_by(clarity) %>%
  summarise(clarity_count = n()) %>%
  
  ggplot(., aes(x=clarity, y = clarity_count, fill = clarity_count)) +
  geom_col() +
  scale_fill_gradient("clarity_count", low="peachpuff", high="peachpuff4") +
  geom_line(aes(y = clarity_count), size = 1.2, color="black", group = 1) +
  theme_minimal() +
  theme(legend.position = "none") +
  theme(axis.text.x = element_text(angle = 90), legend.position = "none")+
        labs(title = "The Number of Diamonds According to the Clarity",
         subtitle = st, 
         x = "Clarity",
         y = "Count")

According to the results,

  • The most of the data is occurred form the SI1 clarity type, VS2 and SI2 follow SI1, respectively. These clarity classes are worse when we compare the other clarity classes.
  • The best type of clarity, IF, takes place in the at the second place from the last.

To get more clear analysis, the following table, which shows the number of diamonds according to clarity, can be analyzed. Moreover, the minimum, maximum, and average prices can be also obtained.

diamonds %>%
  mutate(clarity = factor(clarity)) %>%
  group_by(clarity) %>%
  summarise(clarity_count = n(),
            MinPrice = min(price),
            AveragePrice = mean(price),
            MaxPrice = max(price)) %>%
  arrange(desc(clarity_count)) %>%
  kable(col.names = c("Clarity", "Count","Minimum Price", "Average Price", "Maximum Price")) %>%
  kable_minimal(full_width = F)
Clarity Count Minimum Price Average Price Maximum Price
SI1 13026 326 3992.932 18818
VS2 12221 334 3925.962 18823
SI2 9140 326 5051.592 18804
VS1 8151 327 3839.867 18795
VVS2 5055 336 3286.971 18768
VVS1 3645 336 2523.088 18777
IF 1784 369 2870.570 18806
I1 739 345 3924.185 18531

3.4 Carat

Carat has two meanings: (i) the purity of the diamonds and (ii) the weight of the diamonds. In this data set, the carat illustrates the weight of the carat. To see the most used carat, the number of data is group according to the carat variable. The clarity variable is numeric type. For this reason, we do not apply bar chart illustration. However, to see the most preferable carat type in the data set, first, the clarity variable is transformed into factor variable type. Then, by using bar chart, the the most 20 preferable carats are listed.

diamonds %>%
  mutate(carat = factor(carat)) %>%
  group_by(carat) %>%
  summarise(carat_count = n())%>%
  arrange(desc(carat_count)) %>%
  head(20) %>%
  
  ggplot(., aes(x=carat_count, y = reorder(carat, carat_count), fill= carat)) +
  geom_col() +
  geom_text(aes(label = carat_count), size=3, color = "black", position = position_stack(vjust = 0.95)) +
  theme_minimal() +
  theme(legend.position = "none") +
  labs(title = "The Most Preferable 20 Carat Size",
       subtitle = st,
       x = "Carat",
       y = "Number of Diamonds")

In the bar chart, there are the most 20 popular or preferable carat of diamonds are present.

  • According to the data set, the most preferable carat is 0.3 carat.

However, there are carats which are more than twenty. To see all carat according to the count, following table can be analyzed. In addition to the number of diamonds according to the carat classification, the price intervals and average price also can be investigated.

diamonds %>%
  group_by(carat) %>%
  summarise(carat_count = n(),
            inPrice = min(price),
            AveragePrice = mean(price),
            MaxPrice = max(price))%>%
  arrange(desc(carat_count)) %>%
  kable(col.names = c("Carat", "Number of Carats","Minimum Price", "Average Price", "Maximum Price")) %>%
  kable_styling("striped", full_width = T) %>%
  scroll_box(width = "100%", height = "300px")
Carat Number of Carats Minimum Price Average Price Maximum Price
0.30 2596 339 680.6367 2366
1.01 2240 2017 5507.1902 16234
0.31 2238 335 708.1997 1917
0.70 1981 945 2516.0717 5539
0.32 1827 345 719.1916 1715
1.00 1543 1681 5249.5846 16469
0.90 1484 1599 3939.5889 9182
0.41 1378 467 973.0530 1745
0.40 1291 484 933.0077 1908
0.71 1291 1305 2674.3106 5019
0.50 1254 584 1503.8453 3378
0.33 1179 366 765.1747 1389
0.51 1123 826 1623.5841 4368
0.34 908 413 755.7423 2346
1.02 883 2344 5598.4632 17100
0.52 813 870 1656.4785 3373
1.51 804 3497 10552.0211 18806
1.50 787 2964 10082.9835 18691
0.72 764 945 2705.7788 5086
0.53 707 844 1685.5248 3827
0.42 704 552 994.4545 1722
0.38 669 497 904.1973 1992
0.35 663 409 799.5098 1672
1.20 643 2360 6670.7387 16256
0.54 624 935 1704.3317 4042
0.36 570 410 781.4667 1718
0.91 570 1570 3981.1561 9636
1.03 522 1262 5583.8966 17590
0.55 495 959 1725.2283 3509
0.56 492 953 1831.9980 4632
0.73 492 1049 2791.2480 5154
0.43 486 452 995.2737 1827
1.04 474 2037 5774.6709 18542
1.21 470 2396 6905.4043 17353
2.01 433 5696 14772.7136 18741
0.57 429 929 1798.0443 4032
0.39 397 451 923.4761 1624
0.37 394 451 816.0711 1764
1.52 380 3105 10629.9868 17723
1.06 372 2080 5947.0081 15813
1.05 361 2066 5781.6814 13060
1.07 340 2260 6004.2118 18279
0.74 322 1632 2821.9006 5841
0.58 310 893 1798.6484 3741
1.11 308 3183 6005.5682 15464
1.22 299 2699 6971.5151 14414
0.23 293 326 486.1433 688
1.09 285 2708 6020.0947 18231
0.80 284 1232 3146.1901 6834
0.59 282 903 1831.4504 4916
1.23 277 3168 7290.8881 16253
1.10 276 3216 5913.0725 12748
2.00 261 5051 14107.1456 18818
0.24 254 336 505.1850 963
0.26 253 337 550.8972 814
0.76 251 1140 2889.5936 5056
0.77 251 1651 2884.3984 5251
1.12 251 2383 6044.4900 14407
0.75 249 1013 2763.4016 5288
1.08 246 2869 5774.3415 14032
1.13 246 2968 6193.4024 14525
1.24 235 2940 7147.1149 15026
0.27 233 361 574.7597 893
0.60 226 806 1956.2434 4209
0.92 226 2283 4035.2124 7544
1.53 219 5631 10703.4658 17223
1.70 215 5617 12110.3209 18718
0.25 212 357 550.9245 1186
0.44 212 648 1056.6226 2506
1.14 206 2327 6354.4078 18112
0.61 204 931 2069.2451 4279
0.81 200 1687 3094.4250 6338
0.28 198 360 580.1212 828
0.78 187 1720 2918.9091 4904
1.25 187 3276 7123.2513 16823
0.46 177 746 1240.2429 2139
2.02 176 6346 14767.4091 18731
1.54 174 4492 10865.0287 18416
1.16 172 2368 6274.2965 13595
0.79 151 1809 2993.2649 4858
1.15 148 3220 5962.5811 12895
1.26 146 3881 7490.7466 15247
0.93 142 2374 3994.2958 5992
0.82 140 2033 3083.6500 4931
0.62 135 933 1973.3704 3968
1.27 134 2845 7119.1194 15377
1.31 132 3697 7750.6364 17496
0.83 131 1863 3181.9924 7889
0.29 130 334 601.1923 1776
1.19 126 2892 6483.2381 15767
1.18 123 3219 6481.1057 14847
1.55 123 4965 10717.7642 18188
2.03 122 6002 14710.1148 18781
1.30 121 2512 7851.6612 15874
1.71 119 6320 12396.5546 18791
0.45 110 672 1093.3182 1833
1.17 110 2336 6224.8818 11886
1.56 108 5766 10205.9444 17455
1.28 106 3355 7711.8113 18700
1.57 105 5170 10444.5143 17548
0.96 103 1637 3623.4660 6807
0.63 102 1065 2244.7843 6607
1.29 101 2596 7033.0198 17932
0.47 99 719 1272.3131 2460
1.60 95 6272 11124.2526 18780
1.32 89 4140 7639.3708 15802
1.58 89 4459 11427.9888 18057
1.59 89 6010 10574.3596 17552
1.33 87 4634 7527.0805 18435
2.04 86 7403 14840.1744 18795
0.64 80 1080 2113.5875 4281
1.35 77 4082 7448.2857 14616
1.34 68 4352 7758.5000 17663
0.65 65 1214 2322.6154 5052
0.95 65 2195 4013.6615 6951
2.05 65 7026 15647.2462 18787
0.84 64 1749 3294.9844 6399
1.61 64 6915 10992.8438 18318
0.48 63 841 1322.0476 2677
0.85 62 1250 3204.1774 7230
1.62 61 5157 11705.8852 18281
0.94 59 2571 4195.7797 7120
0.97 59 1547 4026.2373 7415
2.06 59 5203 14348.6780 18779
1.72 57 5240 12040.3158 18730
1.73 52 6007 11875.9615 18377
2.10 51 6597 14464.3725 18648
1.36 50 4158 8591.5200 15386
1.40 50 4939 9108.5200 16808
1.63 50 7100 11556.3200 18179
1.75 50 7276 12673.7000 17343
2.07 50 6503 15384.2800 18804
0.66 48 1188 2235.7292 4311
0.67 48 1244 2067.8542 3271
2.14 48 5405 14905.0417 18528
1.37 46 5063 8044.0870 13439
0.49 45 900 1306.3333 2199
2.09 45 10042 15287.3111 18640
1.64 43 4849 10738.0000 17604
2.11 43 7019 14563.4186 18575
2.08 41 6150 15128.6829 18760
1.74 40 4677 12270.3250 18430
1.41 39 4145 9452.2051 17216
1.39 36 3914 8335.1111 13061
0.86 34 1828 3289.9706 5029
1.65 32 5914 10973.8750 17729
0.87 31 1827 3342.3226 5640
0.98 31 1712 3469.7742 4950
2.20 31 7934 15226.8710 18364
1.66 30 5310 11084.5000 16294
2.18 30 10340 14901.7000 18717
1.76 28 9053 12398.5714 18178
2.22 27 5607 14691.5185 18706
0.69 26 1636 2206.6154 3784
1.38 26 4598 8164.3077 17598
0.68 25 1341 2334.3200 4325
1.42 25 6069 9487.2000 18682
1.67 25 7911 12281.4800 17351
2.12 25 7508 15007.6800 18656
2.16 25 8709 15860.4800 18678
1.69 24 5182 11878.0833 17803
0.88 23 1964 3431.3478 5595
0.99 23 1789 4406.1739 6094
2.21 23 6535 15287.9565 18483
2.15 22 5430 14187.1818 18791
2.19 22 10179 15521.0909 18693
0.89 21 1334 3291.2381 4854
1.47 21 4898 7833.7143 12547
2.13 21 12356 15220.4286 18442
1.80 20 7862 12955.1500 17273
2.28 20 9068 15422.9500 18055
2.30 20 7226 14861.5500 18304
1.43 19 6086 9234.9474 14429
1.68 19 5765 11779.1053 16985
1.44 18 4064 8021.6111 12093
1.46 18 5421 8237.4444 12261
1.83 18 5083 11930.0000 18525
2.17 18 6817 14416.6111 17805
1.77 17 7771 11103.1765 15278
2.29 17 11502 15834.7647 18823
2.51 17 14209 16616.8824 18308
2.24 16 10913 15573.9375 18299
2.25 16 6653 13403.5625 18034
2.32 16 6860 16042.1875 18532
2.50 16 7854 14316.1875 18447
1.45 15 4320 8709.3333 17545
1.79 15 9122 13144.0667 18429
2.26 15 11226 15330.2000 18426
1.82 13 5993 12986.1538 15802
2.23 13 7006 13947.4615 18149
2.31 13 7257 15017.3846 17715
2.40 13 11988 16362.4615 18541
0.20 12 345 365.1667 367
1.78 12 8889 12504.5833 18128
1.91 12 6632 12365.8333 17509
2.27 12 5733 14198.4167 18343
3.01 12 8040 15521.6667 18710
1.49 11 5407 9174.2727 18614
0.21 9 326 380.2222 394
1.81 9 5859 11374.4444 14177
1.86 9 9791 12164.3333 17267
2.33 9 8220 14089.7778 18119
2.48 9 12883 16537.1111 18692
2.52 9 10076 16242.1111 18252
2.36 8 11238 15756.7500 18745
2.38 8 10716 15481.3750 18559
2.53 8 14659 16548.8750 18254
2.54 8 12095 15125.2500 17996
3.00 8 6512 12378.5000 16970
1.48 7 6216 8791.2857 15164
1.87 7 9955 12933.5714 17761
1.90 7 8576 11477.1429 13919
2.35 7 14185 16090.4286 17999
2.39 7 15917 17182.4286 17955
2.42 7 16198 16923.5714 18615
1.93 6 13278 16828.1667 18306
2.37 6 14837 16525.1667 18508
2.43 6 9716 15356.0000 18692
0.22 5 337 391.4000 470
1.98 5 12923 14667.8000 16171
2.34 5 8491 11922.0000 15818
2.41 5 13563 15808.2000 17923
1.84 4 7922 11108.5000 13905
1.88 4 8048 10946.2500 13607
1.89 4 10055 13969.0000 17553
1.96 4 5554 8642.2500 13099
1.97 4 14410 15966.0000 17535
2.44 4 13027 16365.5000 18430
2.45 4 11830 14593.7500 18113
1.85 3 5688 11436.0000 14375
1.94 3 9344 14189.3333 18735
1.95 3 5045 12340.6667 17374
1.99 3 11486 14179.0000 17713
2.46 3 10470 14227.0000 16466
2.47 3 14970 15644.0000 16532
2.49 3 6289 13843.0000 18325
2.55 3 14351 15964.0000 18766
2.56 3 15231 16723.3333 17753
2.57 3 17116 17841.6667 18485
2.58 3 12500 14481.3333 16195
2.60 3 17027 17535.0000 18369
2.61 3 13784 16583.0000 18756
2.63 3 10437 12659.6667 16914
2.72 3 6870 12088.3333 17801
2.74 3 8807 14385.0000 17184
1.92 2 15364 15418.0000 15472
2.66 2 16239 17367.0000 18495
2.68 2 8419 9042.0000 9665
2.75 2 13156 14285.5000 15415
3.04 2 15354 16956.5000 18559
4.01 2 15223 15223.0000 15223
2.59 1 16465 16465.0000 16465
2.64 1 17407 17407.0000 17407
2.65 1 16314 16314.0000 16314
2.67 1 18686 18686.0000 18686
2.70 1 14341 14341.0000 14341
2.71 1 17146 17146.0000 17146
2.77 1 10424 10424.0000 10424
2.80 1 15030 15030.0000 15030
3.02 1 10577 10577.0000 10577
3.05 1 10453 10453.0000 10453
3.11 1 9823 9823.0000 9823
3.22 1 12545 12545.0000 12545
3.24 1 12300 12300.0000 12300
3.40 1 15964 15964.0000 15964
3.50 1 12587 12587.0000 12587
3.51 1 18701 18701.0000 18701
3.65 1 11668 11668.0000 11668
3.67 1 16193 16193.0000 16193
4.00 1 15984 15984.0000 15984
4.13 1 17329 17329.0000 17329
4.50 1 18531 18531.0000 18531
5.01 1 18018 18018.0000 18018

Like the number of carat, we also count the number of diamonds according to the cut,clarity and color. There is one difference between carat and these variables. The carat is numeric and these are not. For this reason, when we count these vraibles we do not need to transform our data from factor to numeric.

Color <- diamonds %>%
  group_by(color) %>%
  summarise(count = n())%>%

  ggplot(., aes(x=count, y = reorder(color, count), fill= color)) +
  geom_col() +
  geom_text(aes(label = count), size=3, color = "black", position = position_stack(vjust = 0.95)) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90), legend.position = "none")+
  labs(title = "The Number of Diamonds \n According to  Color",
       subtitle = st,
       y = "Color",
       x = "Number of Diamonds")

Cut <- diamonds %>%
  group_by(cut) %>%
  summarise(count = n())%>%

  ggplot(., aes(x=count, y = reorder(cut, count), fill= cut)) +
  geom_col() +
  geom_text(aes(label = count), size=3, color = "black", position = position_stack(vjust = 0.95)) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90), legend.position = "none")+
  labs(title = "The Number of Diamonds \n According to Cut",
       subtitle = st,
       y = "Cut",
       x = "Number of Diamonds")

Clarity <- diamonds %>%
  group_by(clarity) %>%
  summarise(count = n())%>%

  ggplot(., aes(x=count, y = reorder(clarity, count), fill= clarity)) +
  geom_col() +
  geom_text(aes(label = count), size=3, color = "black", position = position_stack(vjust = 0.95)) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90), legend.position = "none")+
  labs(title = "The Number of Diamonds \n According to Clarity",
       subtitle = st,
       y = "Clarity",
       x = "Number of Diamonds")

(Color | Cut  | Clarity)

The counting process shows that the G type in color, ideal in cut, and SI1 in clarity are the most used in this data set. Up until now, there are most important features of the diamonds are presented as a count analysis. After that point, relationship between variables and prices are illustrated.

3.5 Price vs Variables

3.5.1 Price and X,Y,Z

Since the x,y, and z variables are continuous numeric values, we can plot price according to these variables. In the graph, three plots are given in the same line.

XP <- diamonds%>%
  ggplot(., aes(x,price)) +
  geom_point(alpha=0.85)+
  geom_smooth( color = "red")+
  theme_minimal() +
  labs(title = "X vs Price",
       subtitle = st,
       x= "X",
       y = "Price")

YP <- diamonds%>%
  ggplot(., aes(y,price)) +
  geom_point(alpha=0.85)+
  geom_smooth( color = "red")+
  theme_minimal() +
  labs(title = "Y vs Price",
       subtitle = st,
       x= "Y",
       y = "Price")

ZP <- diamonds%>%
  ggplot(., aes(z,price)) +
  geom_point(alpha=0.85)+
  geom_smooth(color = "red")+
  theme_minimal() +
  labs(title = "Z vs Price",
       subtitle = st,
       x= "Z",
       y = "Price")

(XP | YP | ZP)

Results show that there relationship bewteen price and x,y, and z variables respectively. After tat point, the price and 4C feature of the data is analyzed.

3.5.2 Price and Clarity by Color

diamonds %>%
  group_by(clarity, color) %>%
  summarise(MeanPrice = mean(price)) %>%
  ggplot(aes(x=clarity, y = MeanPrice, fill = color)) +
  geom_col(alpha = 0.8) +
  theme_minimal() +
  facet_wrap(~color) +
  theme(axis.text.x = element_text(angle = 90))+
  labs(title = "Price and Clarity by Color",
       subtitle = st,
       x = "Clarity",
       y = "Average Price",
       fill = "Color")

To make interpretation, in this graphs, price and two variable is used. For this purpose, in the first EDA, the clarity and color are selected. In the D color, the highest price belongs to highest level of clarity as we expect. However, when we analyze the other colors, we obtain that the average price is getting smaller when the quality of clarity is getting better. This shows that, these two variable are not only variables which are affect the price.

3.5.3 Price and Cut, Color and Clarity

Another examination is made by using cut and color. When we group our data according to the cut and color, we can obtain bar charts and moreover we can also shows average price values according to the classification. To make bar chart, the factor variable should be used. For this reason, three of 4C is analyzed. Thus, there is one more combination to analyze relationship between variables and price.

b <- diamonds %>%
  group_by(cut, color) %>%
  summarise(avg_price = mean(price)) %>%
  
  ggplot(aes(x=cut, y= avg_price, fill = cut)) +
  geom_col() +
  facet_wrap(~color) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90), legend.position = "none") +
  labs(title = "Price and Cut by Color",
       subtitle = st, 
       y = "Average Price",
       x = "Quality of Cut")

c<-diamonds %>%
  group_by(clarity, cut) %>%
  summarise(MeanPrice = mean(price)) %>%
  ggplot(aes(x=clarity, y = MeanPrice, fill = cut)) +
  geom_col(alpha = 0.8) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90)) +
  facet_wrap(~cut) +
  labs(title = "Price and Clarity by Cut",
       subtitle = st,
       x = "Clarity",
       y = "Average Price",
       fill = "Cut")

(b | c) #combination of the two plot in one line

When we analyze the given plots, the premium cut type has the highest average price. In addition, the best quality of cut, ideal, on the other hand nearly the lowest. This results also shows that, like the previous analysis, by using these two variable we do not explain the price. Like the previous two analysis, we get same result that is when the quality of the diamonds increase and at the same time clarity increases, the price does not increase in the same way. Moreover, with the best cut and clarity, the price is lowest. Thus, we can not explain the price variable with these two types.

Lastly, we can summarize Price response variable by using 4C variables.

diamonds%>%
  group_by(carat,color,clarity, cut) %>%
  summarise(MinPrice = min(price),
            AveragePrice = mean(price),
            MaxPrice = max(price)) %>%
  arrange(desc(AveragePrice))%>%
  select(carat, color, clarity, cut, MinPrice, AveragePrice, MaxPrice) %>%
  kable(col.names = c("Carat", "Color", "Clarity","Cut", "Minimum Price","Average Price", "Maximum Price")) %>%
  kable_styling("striped", full_width = T) %>%
  scroll_box(width = "100%", height = "300px")
Carat Color Clarity Cut Minimum Price Average Price Maximum Price
1.51 G IF Ideal 18806 18806.0000 18806
2.29 I SI1 Premium 18797 18797.0000 18797
2.15 G SI2 Ideal 18791 18791.0000 18791
2.05 G SI1 Ideal 18787 18787.0000 18787
2.05 F SI2 Premium 18784 18784.0000 18784
2.03 H SI1 Very Good 18781 18781.0000 18781
1.51 F VVS1 Very Good 18777 18777.0000 18777
2.55 I VS1 Premium 18766 18766.0000 18766
2.61 I SI2 Ideal 18756 18756.0000 18756
2.36 H SI2 Premium 18745 18745.0000 18745
2.29 I VS2 Premium 18653 18738.0000 18823
1.94 F SI1 Premium 18735 18735.0000 18735
2.02 E SI1 Very Good 18731 18731.0000 18731
1.72 G VVS2 Premium 18730 18730.0000 18730
2.18 F SI1 Premium 18717 18717.0000 18717
3.01 J SI2 Premium 18710 18710.0000 18710
2.00 E SI1 Very Good 18709 18709.0000 18709
2.22 J VS1 Premium 18706 18706.0000 18706
3.51 J VS2 Premium 18701 18701.0000 18701
1.28 E IF Ideal 18700 18700.0000 18700
2.19 D SI2 Ideal 18693 18693.0000 18693
2.43 J VS2 Premium 18692 18692.0000 18692
2.48 F SI2 Very Good 18692 18692.0000 18692
2.67 F SI2 Good 18686 18686.0000 18686
1.71 G VVS2 Ideal 18599 18683.5000 18768
1.42 F VVS1 Ideal 18682 18682.0000 18682
2.16 F SI2 Very Good 18678 18678.0000 18678
1.60 F VS1 Ideal 18571 18675.5000 18780
2.01 D SI2 Ideal 18674 18674.0000 18674
2.05 H VS2 Ideal 18659 18659.0000 18659
2.12 H SI1 Premium 18656 18656.0000 18656
1.49 F VVS2 Ideal 18614 18614.0000 18614
2.07 F SI2 Premium 18611 18611.0000 18611
3.01 H SI2 Good 18593 18593.0000 18593
2.01 G SI1 Fair 18574 18574.0000 18574
2.02 H VS2 Fair 18565 18565.0000 18565
3.04 I SI2 Premium 18559 18559.0000 18559
2.01 H VS2 Good 18474 18558.3333 18640
1.72 E VS2 Very Good 18557 18557.0000 18557
1.50 F IF Very Good 18552 18552.0000 18552
1.04 D IF Very Good 18542 18542.0000 18542
2.40 I SI1 Good 18541 18541.0000 18541
2.32 G SI2 Ideal 18532 18532.0000 18532
4.50 J I1 Fair 18531 18531.0000 18531
2.14 H SI1 Ideal 18528 18528.0000 18528
2.14 D SI2 Very Good 18526 18526.0000 18526
1.83 G VS2 Premium 18525 18525.0000 18525
2.38 H VS1 Premium 18522 18522.0000 18522
2.00 G VS2 Fair 18515 18515.0000 18515
2.29 J IF Premium 18426 18510.0000 18594
2.32 H SI2 Very Good 18508 18508.0000 18508
2.37 J SI1 Ideal 18508 18508.0000 18508
2.57 J SI1 Premium 18485 18485.0000 18485
2.21 F SI2 Very Good 18483 18483.0000 18483
2.16 I VS2 Premium 18481 18481.0000 18481
2.10 H SI1 Very Good 18480 18480.0000 18480
1.70 F VS1 Premium 18398 18475.3333 18598
2.19 H SI1 Ideal 18475 18475.0000 18475
2.09 F SI2 Good 18472 18472.0000 18472
2.15 H SI1 Ideal 18470 18470.0000 18470
2.04 D SI1 Good 18468 18468.0000 18468
2.03 G SI1 Very Good 18257 18464.6667 18630
2.40 H SI2 Ideal 18374 18457.5000 18541
2.08 I VS2 Premium 18447 18447.0000 18447
2.22 I VS2 Ideal 18363 18447.0000 18531
2.50 I SI2 Premium 18447 18447.0000 18447
2.09 H VS2 Premium 18443 18443.0000 18443
1.33 F IF Ideal 18435 18435.0000 18435
1.72 G VVS2 Ideal 18431 18431.0000 18431
1.74 F VS2 Very Good 18430 18430.0000 18430
1.79 F VS2 Very Good 18429 18429.0000 18429
2.26 H SI1 Premium 18426 18426.0000 18426
2.03 G VS2 Premium 18166 18423.0000 18680
1.79 G VS1 Premium 18419 18419.0000 18419
2.04 G SI1 Ideal 18153 18408.0000 18663
2.11 D SI2 Very Good 18407 18407.0000 18407
2.08 D SI2 Very Good 18405 18405.0000 18405
2.03 E SI2 Premium 18310 18393.5000 18477
2.09 H SI1 Very Good 18392 18392.0000 18392
2.01 G VS2 Ideal 18172 18389.5000 18607
2.02 G VS2 Ideal 18077 18388.5000 18700
1.73 G VVS2 Premium 18377 18377.0000 18377
2.06 D SI2 Ideal 18371 18371.0000 18371
2.32 I SI1 Very Good 18371 18371.0000 18371
2.60 J SI2 Ideal 18369 18369.0000 18369
1.70 G VVS1 Ideal 18279 18362.0000 18445
2.13 H SI1 Ideal 18275 18358.5000 18442
1.83 G VS2 Very Good 18358 18358.0000 18358
2.27 G SI1 Ideal 18343 18343.0000 18343
2.16 I VS1 Ideal 18342 18342.0000 18342
2.01 H VS1 Good 18340 18340.0000 18340
2.49 I SI2 Premium 18325 18325.0000 18325
2.50 J VS2 Good 18325 18325.0000 18325
1.61 D VS2 Ideal 18318 18318.0000 18318
2.00 G SI1 Very Good 17724 18313.0000 18818
2.10 G SI2 Ideal 18312 18312.0000 18312
2.51 H SI2 Fair 18308 18308.0000 18308
1.93 F SI1 Ideal 18306 18306.0000 18306
2.24 I SI1 Ideal 18299 18299.0000 18299
2.14 E SI2 Premium 18291 18291.0000 18291
2.01 G SI1 Good 17804 18290.4000 18625
1.62 F VS2 Ideal 18281 18281.0000 18281
2.08 I VS2 Very Good 18281 18281.0000 18281
1.07 D IF Premium 18279 18279.0000 18279
2.07 I VS2 Premium 18193 18276.0000 18359
2.51 I SI2 Premium 18255 18255.0000 18255
2.53 H SI1 Premium 18254 18254.0000 18254
2.52 I SI1 Premium 18252 18252.0000 18252
3.01 I SI2 Fair 18242 18242.0000 18242
3.01 I SI2 Good 18242 18242.0000 18242
3.01 I SI2 Premium 18242 18242.0000 18242
2.30 I VS2 Premium 18239 18239.0000 18239
2.02 E SI2 Good 18236 18236.0000 18236
2.19 E SI2 Premium 18232 18232.0000 18232
1.09 D IF Very Good 18231 18231.0000 18231
2.09 I VS1 Very Good 18215 18215.0000 18215
1.73 G VVS2 Very Good 18211 18211.0000 18211
2.05 E SI1 Ideal 18193 18193.0000 18193
2.20 G SI1 Premium 18193 18193.0000 18193
2.30 I VS2 Ideal 18190 18190.0000 18190
1.51 E VS1 Ideal 17649 18189.0000 18729
1.55 E VVS2 Ideal 18188 18188.0000 18188
1.54 F VS1 Ideal 17949 18182.5000 18416
1.63 F VS1 Ideal 18179 18179.0000 18179
2.52 F SI2 Premium 18179 18179.0000 18179
1.76 G VS1 Ideal 18178 18178.0000 18178
2.12 G SI2 Premium 18164 18164.0000 18164
1.50 F VVS2 Ideal 18159 18159.0000 18159
2.23 I VS2 Premium 18149 18149.0000 18149
2.04 F SI2 Premium 18066 18148.5000 18231
2.02 H VS2 Very Good 17887 18142.5000 18398
1.70 E VS2 Very Good 18107 18141.5000 18176
2.20 I VS2 Premium 17905 18134.5000 18364
2.02 I VS1 Ideal 17936 18130.0000 18324
1.78 F VS2 Ideal 18128 18128.0000 18128
2.08 E SI2 Very Good 18128 18128.0000 18128
2.08 E SI2 Ideal 18128 18128.0000 18128
2.04 I VS1 Very Good 18127 18127.0000 18127
2.14 E SI2 Very Good 18125 18125.0000 18125
2.08 G SI2 Ideal 18124 18124.0000 18124
2.10 F SI2 Ideal 18124 18124.0000 18124
2.33 I SI1 Very Good 18119 18119.0000 18119
2.12 H SI1 Very Good 18118 18118.0000 18118
2.45 J VS2 Ideal 18113 18113.0000 18113
1.14 D IF Very Good 18112 18112.0000 18112
2.00 F SI1 Good 17869 18104.6667 18376
2.04 H VS1 Premium 18104 18104.0000 18104
1.51 F IF Premium 18102 18102.0000 18102
2.02 H SI1 Very Good 17672 18095.0000 18236
2.51 I SI2 Very Good 18090 18090.0000 18090
2.03 H VS2 Premium 17975 18084.3333 18139
2.32 H SI1 Ideal 18080 18080.0000 18080
2.11 F SI2 Very Good 18071 18071.0000 18071
2.29 I SI1 Very Good 18068 18068.0000 18068
2.19 E SI2 Very Good 18067 18067.0000 18067
2.21 I SI1 Ideal 18062 18062.0000 18062
2.51 I SI2 Ideal 18037 18037.0000 18037
2.11 E SI2 Premium 18034 18034.0000 18034
2.25 I SI1 Premium 18034 18034.0000 18034
2.16 I SI1 Ideal 18029 18029.0000 18029
2.51 H SI2 Very Good 18029 18029.0000 18029
2.26 D SI2 Very Good 18028 18028.0000 18028
2.04 H SI1 Good 18026 18026.0000 18026
2.32 I SI1 Premium 18026 18026.0000 18026
2.32 H SI1 Fair 18026 18026.0000 18026
2.51 J VS2 Premium 18020 18020.0000 18020
5.01 J I1 Fair 18018 18018.0000 18018
1.76 G VS1 Very Good 18014 18014.0000 18014
1.07 D IF Very Good 17909 18011.5000 18114
2.25 D SI2 Premium 18007 18007.0000 18007
2.16 G SI2 Ideal 18001 18001.0000 18001
2.09 I VS2 Ideal 17999 17999.0000 17999
2.35 H SI1 Premium 17999 17999.0000 17999
1.93 F VS1 Fair 17995 17995.0000 17995
2.24 F SI2 Ideal 17989 17989.0000 17989
2.08 J IF Ideal 17986 17986.0000 17986
2.20 I SI1 Ideal 17895 17977.0000 18059
1.58 E VS1 Ideal 17894 17975.5000 18057
1.70 G VVS2 Premium 17228 17973.0000 18718
2.39 I VS2 Ideal 17955 17955.0000 17955
2.40 I SI2 Very Good 17955 17955.0000 17955
2.00 D SI2 Very Good 17953 17953.0000 17953
2.04 H SI1 Very Good 17952 17952.0000 17952
1.29 D VVS1 Very Good 17932 17932.0000 17932
2.57 D SI2 Premium 17924 17924.0000 17924
2.41 I SI2 Good 17923 17923.0000 17923
2.18 G SI2 Premium 17841 17922.0000 18003
2.39 J VS1 Very Good 17920 17920.0000 17920
2.01 G VS2 Fair 17917 17917.0000 17917
2.08 I VS2 Ideal 17916 17916.0000 17916
2.01 H VS2 Very Good 16677 17915.0000 18561
2.01 H VS1 Ideal 17838 17910.5000 17983
1.74 G VS1 Premium 17904 17904.0000 17904
2.08 H SI1 Ideal 17469 17899.3333 18760
2.00 G VS2 Good 17898 17898.0000 17898
2.48 H SI1 Very Good 17893 17893.0000 17893
2.42 J VS2 Premium 17168 17891.5000 18615
2.16 J VVS1 Premium 17891 17891.0000 17891
2.32 J VS1 Ideal 17891 17891.0000 17891
1.76 G VVS2 Very Good 17888 17888.0000 17888
2.11 E SI2 Very Good 17871 17871.0000 17871
2.43 I SI2 Ideal 17856 17856.0000 17856
2.07 G SI2 Ideal 17095 17837.6667 18804
2.10 I VS2 Very Good 17837 17837.0000 17837
2.00 I VS1 Premium 17278 17836.3333 18795
1.70 E VS2 Premium 17323 17832.5000 18342
2.00 F SI1 Premium 17172 17832.5000 18493
2.36 J VS1 Ideal 17829 17829.0000 17829
1.63 E VS2 Ideal 17825 17825.0000 17825
2.16 H SI1 Ideal 17820 17820.0000 17820
2.11 G SI1 Premium 17816 17816.0000 17816
2.05 H SI1 Very Good 17811 17811.0000 17811
2.17 H SI2 Ideal 17805 17805.0000 17805
1.69 E VS2 Very Good 17803 17803.0000 17803
2.03 F SI1 Premium 17803 17803.0000 17803
2.72 J SI2 Ideal 17801 17801.0000 17801
2.07 H SI1 Very Good 17019 17800.3333 18489
2.21 I SI1 Very Good 17784 17784.0000 17784
1.71 F VS2 Premium 16779 17778.6667 18791
2.05 D SI1 Premium 17776 17776.0000 17776
1.55 D VS1 Premium 17773 17773.0000 17773
2.16 I SI1 Very Good 17772 17772.0000 17772
1.87 H VS1 Ideal 17761 17761.0000 17761
2.00 G VS2 Ideal 17760 17760.0000 17760
2.00 D VS2 Good 17760 17760.0000 17760
2.56 J SI1 Ideal 17753 17753.0000 17753
2.03 G SI2 Very Good 17752 17752.0000 17752
2.00 G SI1 Good 17405 17744.2500 17930
2.14 F SI2 Premium 17742 17742.0000 17742
2.12 F SI2 Ideal 17730 17730.0000 17730
1.51 D VS1 Premium 17515 17725.5000 17936
1.52 D VS1 Premium 17723 17723.0000 17723
2.21 I SI1 Premium 17714 17714.0000 17714
1.99 H VS2 Fair 17713 17713.0000 17713
2.08 H SI1 Premium 17617 17697.5000 17778
2.12 F SI2 Very Good 17694 17694.0000 17694
1.51 E VVS2 Very Good 17689 17689.0000 17689
2.52 G SI2 Very Good 17689 17689.0000 17689
1.71 F VS1 Very Good 17685 17685.0000 17685
1.72 G VS1 Premium 17605 17685.0000 17765
2.28 I SI1 Premium 17673 17673.0000 17673
1.50 E VVS2 Premium 17667 17667.0000 17667
2.29 H SI2 Premium 17666 17666.0000 17666
1.34 E IF Premium 17663 17663.0000 17663
1.52 D VS1 Ideal 17659 17659.0000 17659
1.65 G IF Ideal 17569 17649.0000 17729
2.39 I VS1 Premium 17642 17642.0000 17642
2.16 I SI1 Premium 17313 17623.5000 17934
1.70 F VS2 Ideal 17114 17616.6000 18055
1.74 G VS2 Premium 17614 17614.0000 17614
2.52 G SI2 Good 17608 17608.0000 17608
2.48 I SI2 Premium 17607 17607.0000 17607
2.10 J VVS1 Premium 17606 17606.0000 17606
1.64 G VVS2 Ideal 17604 17604.0000 17604
1.38 F IF Very Good 17598 17598.0000 17598
1.70 F VS2 Good 17597 17597.0000 17597
1.70 F VS2 Very Good 17597 17597.0000 17597
2.32 I SI2 Ideal 17516 17596.0000 17676
1.03 D IF Ideal 17590 17590.0000 17590
2.14 F SI2 Very Good 17582 17582.0000 17582
2.19 H SI2 Premium 17554 17554.0000 17554
1.89 H VS1 Ideal 17553 17553.0000 17553
2.05 G SI2 Ideal 16446 17548.0000 18181
2.10 H SI2 Ideal 16629 17545.5000 18462
1.45 D VVS2 Very Good 17545 17545.0000 17545
2.29 H VS2 Premium 17539 17539.0000 17539
1.97 H VS2 Premium 17535 17535.0000 17535
2.36 J VS2 Ideal 17534 17534.0000 17534
2.01 G VS2 Premium 17057 17522.0000 17987
2.52 H SI2 Premium 17522 17522.0000 17522
2.05 H SI1 Premium 17521 17521.0000 17521
1.91 F SI1 Very Good 17509 17509.0000 17509
1.14 D IF Good 17499 17499.0000 17499
1.31 D VVS1 Premium 17496 17496.0000 17496
1.70 D VS2 Good 17485 17485.0000 17485
2.44 J VS2 Very Good 16533 17481.5000 18430
1.59 E VS1 Premium 17393 17472.5000 17552
2.44 J SI1 Very Good 17472 17472.0000 17472
2.20 H VS2 Ideal 17460 17460.0000 17460
1.56 F VS1 Ideal 17455 17455.0000 17455
2.51 J SI1 Premium 17452 17452.0000 17452
2.31 J VS1 Premium 17451 17451.0000 17451
2.21 G SI2 Ideal 15948 17450.6667 18276
1.50 E VVS2 Good 17449 17449.0000 17449
2.48 I SI2 Very Good 17448 17448.0000 17448
1.76 G VS2 Ideal 17442 17442.0000 17442
2.19 H SI2 Ideal 17433 17433.0000 17433
1.65 F VS1 Very Good 17425 17425.0000 17425
2.03 H SI1 Ideal 16900 17422.5000 18026
2.38 J VS2 Premium 16273 17416.0000 18559
2.05 I VS1 Fair 17408 17408.0000 17408
2.64 H SI2 Ideal 17407 17407.0000 17407
2.50 F SI2 Fair 17405 17405.0000 17405
1.69 E VS1 Good 17400 17400.0000 17400
2.01 G SI1 Very Good 15038 17399.2000 18741
2.03 G SI2 Premium 17393 17393.0000 17393
1.97 H VS2 Very Good 17377 17377.0000 17377
1.95 H VS2 Very Good 17374 17374.0000 17374
2.66 H SI2 Good 16239 17367.0000 18495
1.59 F VS1 Ideal 17366 17366.0000 17366
2.01 I VS1 Good 17365 17365.0000 17365
2.39 J VS1 Ideal 17365 17365.0000 17365
1.79 G VS2 Premium 17358 17358.0000 17358
1.21 E IF Ideal 17353 17353.0000 17353
1.93 H VS2 Premium 17353 17353.0000 17353
1.69 G VS1 Good 17338 17338.0000 17338
1.61 F VS1 Ideal 17256 17335.0000 17414
1.70 F VS1 Good 17330 17330.0000 17330
1.58 G IF Ideal 17329 17329.0000 17329
4.13 H I1 Fair 17329 17329.0000 17329
2.04 J IF Ideal 17327 17327.0000 17327
2.01 E SI1 Very Good 16956 17316.0000 17676
2.26 H SI2 Ideal 17312 17312.0000 17312
2.35 I SI2 Premium 17294 17294.0000 17294
1.60 G VVS1 Premium 17204 17282.0000 17360
1.50 F VVS1 Very Good 17279 17279.0000 17279
2.05 H VS2 Premium 16235 17276.5000 18318
1.80 E SI1 Very Good 17273 17273.0000 17273
2.01 I VS1 Ideal 17068 17272.6667 17523
1.67 F VS2 Ideal 17194 17272.5000 17351
1.86 H VVS2 Very Good 17267 17267.0000 17267
2.17 I SI1 Ideal 16716 17267.0000 17747
2.06 G SI2 Ideal 16235 17264.0000 18293
2.16 E SI2 Ideal 17263 17263.0000 17263
2.42 I VS1 Premium 17262 17262.0000 17262
2.08 H VS2 Premium 17258 17258.0000 17258
2.31 I SI1 Very Good 16801 17258.0000 17715
2.12 I VS2 Premium 17254 17254.0000 17254
2.06 I VS2 Ideal 15691 17235.0000 18779
2.25 H SI1 Premium 17233 17233.0000 17233
2.00 H SI1 Good 16650 17232.3333 17600
2.52 I SI1 Ideal 17231 17231.0000 17231
1.53 D VS2 Ideal 17223 17223.0000 17223
1.41 D VVS2 Premium 17216 17216.0000 17216
2.05 J VVS2 Ideal 17214 17214.0000 17214
2.14 I SI1 Ideal 17213 17213.0000 17213
2.60 J SI2 Premium 17209 17209.0000 17209
2.61 J SI2 Ideal 17209 17209.0000 17209
1.71 G VS1 Very Good 17206 17206.0000 17206
1.50 F VVS2 Premium 17203 17203.0000 17203
2.02 G SI1 Premium 16323 17198.0000 17692
2.30 I SI1 Ideal 17193 17193.0000 17193
1.21 D VVS1 Premium 17192 17192.0000 17192
2.09 F SI2 Very Good 15773 17191.2000 18640
1.75 G VS1 Premium 17191 17191.0000 17191
2.56 J SI1 Premium 17186 17186.0000 17186
2.74 H SI2 Very Good 17184 17184.0000 17184
1.57 E VS1 Premium 17175 17175.0000 17175
2.05 E SI2 Premium 17066 17169.5000 17294
2.74 J SI2 Very Good 17164 17164.0000 17164
2.51 H SI2 Ideal 17162 17162.0000 17162
2.22 I SI1 Very Good 17160 17160.0000 17160
2.22 I SI1 Good 17151 17151.0000 17151
2.05 H SI1 Ideal 17006 17150.0000 17294
2.71 E SI2 Premium 17146 17146.0000 17146
2.05 E SI2 Very Good 17138 17138.0000 17138
2.01 H SI1 Ideal 14725 17137.4545 18295
2.14 D SI2 Premium 17127 17127.0000 17127
2.02 G SI2 Ideal 15814 17125.0000 18296
2.57 E SI2 Premium 17116 17116.0000 17116
2.11 I VS2 Ideal 16512 17112.0000 17712
1.51 D VS1 Good 17111 17111.0000 17111
2.31 I SI2 Premium 17000 17109.0000 17218
1.56 F VS2 Ideal 17108 17108.0000 17108
2.53 H SI2 Fair 17103 17103.0000 17103
2.01 G SI1 Premium 15105 17099.8000 18741
2.02 H VS2 Premium 15987 17097.0000 18207
1.71 G VS1 Ideal 16618 17095.0000 17595
1.75 F VS2 Premium 17017 17094.5000 17172
2.16 J VS1 Ideal 15714 17088.0000 18462
2.28 H SI2 Premium 15413 17078.5000 18055
1.70 G VVS2 Very Good 17073 17073.0000 17073
2.27 J VS1 Premium 16994 17071.5000 17149
2.12 I VS2 Ideal 17068 17068.0000 17068
2.15 H SI2 Very Good 17065 17065.0000 17065
2.00 F SI1 Very Good 15851 17063.3333 18759
1.50 G VVS2 Ideal 16948 17062.0000 17176
2.03 F SI2 Ideal 12677 17062.0000 18578
2.31 I SI2 Very Good 17062 17062.0000 17062
1.57 E VS2 Ideal 16570 17059.0000 17548
2.02 F SI1 Very Good 15987 17058.3333 17658
1.53 F VVS2 Very Good 17057 17057.0000 17057
2.32 J VS2 Premium 17053 17053.0000 17053
2.09 G VS2 Ideal 17051 17051.0000 17051
2.27 J VS1 Good 17051 17051.0000 17051
2.07 I VS2 Ideal 17045 17045.0000 17045
1.07 D IF Ideal 17042 17042.0000 17042
1.75 G VS1 Very Good 17036 17036.0000 17036
1.67 G VVS2 Very Good 17028 17028.0000 17028
2.01 F SI2 Ideal 13201 17028.0000 18395
2.50 J SI1 Very Good 17028 17028.0000 17028
2.60 G SI2 Fair 17027 17027.0000 17027
1.93 H VS1 Premium 16941 17018.5000 17096
1.71 G VS1 Premium 16813 17017.0000 17197
2.26 J VS2 Ideal 17010 17010.0000 17010
1.70 G VS1 Premium 16521 17006.5000 17492
2.01 I VS1 Very Good 14792 17000.3000 18500
2.41 I SI2 Ideal 16987 16987.0000 16987
1.68 G VS1 Ideal 16985 16985.0000 16985
1.75 G VS2 Ideal 16783 16976.3333 17343
1.73 F VS2 Premium 16975 16975.0000 16975
3.00 I SI2 Fair 16970 16970.0000 16970
3.00 I SI2 Premium 16970 16970.0000 16970
2.28 E SI2 Ideal 16969 16969.0000 16969
1.73 G VS2 Premium 16960 16960.0000 16960
2.07 E SI1 Premium 16957 16957.0000 16957
2.50 I VS2 Ideal 16955 16955.0000 16955
2.01 G SI1 Ideal 12829 16949.5000 18741
2.37 H SI2 Ideal 16937 16937.0000 16937
2.04 I VS2 Very Good 16616 16935.0000 17315
2.53 J SI2 Premium 16934 16934.0000 16934
2.00 F SI2 Ideal 13351 16927.6000 18172
2.05 H VS2 Very Good 15701 16926.5000 18152
1.54 G VVS1 Ideal 16169 16926.3333 18294
2.01 I VS1 Premium 15475 16916.3333 18206
1.52 E VS1 Ideal 16916 16916.0000 16916
2.49 J SI1 Ideal 16915 16915.0000 16915
2.63 J SI2 Ideal 16914 16914.0000 16914
1.51 F VVS2 Very Good 16613 16913.0000 17317
2.07 I VS2 Good 15119 16913.0000 18707
2.09 I SI1 Ideal 16704 16909.0000 17114
2.26 I SI1 Ideal 16904 16904.0000 16904
2.04 I VS2 Premium 16768 16898.0000 17028
2.11 D SI2 Premium 15198 16886.5000 18575
1.76 G VS2 Premium 16879 16879.0000 16879
2.30 H SI2 Ideal 14215 16875.6667 18304
2.03 G SI1 Premium 15322 16872.5000 18423
2.04 D SI2 Premium 16872 16872.0000 16872
2.06 F SI2 Premium 16857 16857.0000 16857
2.13 G SI2 Premium 16778 16854.5000 16931
2.08 G VS2 Premium 16854 16854.0000 16854
2.51 J SI1 Very Good 16842 16842.0000 16842
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0.54 G VVS1 Premium 2236 2236.0000 2236
0.56 F VVS2 Very Good 2236 2236.0000 2236
0.77 J VS2 Ideal 2159 2236.0000 2292
0.71 I VS1 Premium 2104 2234.5000 2327
0.62 E VS2 Ideal 2208 2234.0000 2260
0.70 I VS1 Fair 2234 2234.0000 2234
0.59 E VS1 Ideal 2172 2233.0000 2294
0.62 H VVS1 Premium 2232 2232.0000 2232
0.75 J VS1 Ideal 2161 2230.5000 2300
0.61 D VS2 Ideal 2147 2230.0000 2364
0.75 D SI2 Premium 1858 2229.2500 2415
0.80 D SI2 Good 2229 2229.0000 2229
0.50 F IF Very Good 2219 2228.5000 2238
0.74 H VS2 Good 1989 2228.3333 2369
0.71 D SI2 Very Good 1758 2227.4000 2519
0.69 D SI1 Very Good 2227 2227.0000 2227
0.57 D VS1 Premium 2093 2226.5000 2360
0.71 J VVS2 Very Good 2226 2226.0000 2226
0.73 G SI1 Premium 2163 2225.6667 2351
0.72 G SI2 Ideal 1866 2225.0000 2501
0.64 D VS2 Ideal 2224 2224.0000 2224
0.68 E SI1 Premium 2224 2224.0000 2224
0.80 J SI1 Very Good 2224 2224.0000 2224
0.70 I SI1 Ideal 1790 2223.5000 2623
0.71 F SI1 Fair 2088 2223.2500 2458
0.51 F VVS1 Very Good 2223 2223.0000 2223
0.90 E I1 Very Good 2200 2222.5000 2245
0.71 I VS1 Good 2007 2222.1250 2327
0.58 H IF Ideal 2211 2222.0000 2233
0.64 F VS1 Good 2222 2222.0000 2222
0.64 F VS1 Ideal 2222 2222.0000 2222
0.71 D SI2 Good 2015 2222.0000 2436
0.83 J SI1 Good 2159 2221.6667 2311
0.85 J SI1 Good 2163 2220.5000 2278
1.03 J I1 Good 2220 2220.0000 2220
0.77 G SI2 Ideal 2162 2218.5000 2275
0.52 G IF Ideal 2055 2218.2500 2387
0.56 F VVS1 Premium 2218 2218.0000 2218
0.71 J VVS1 Very Good 2218 2218.0000 2218
0.73 I SI2 Ideal 2218 2218.0000 2218
0.71 F SI2 Fair 2103 2215.0000 2327
0.50 D VVS2 Very Good 2115 2212.5000 2310
0.70 G SI1 Very Good 2010 2211.5000 2674
0.51 E VVS2 Very Good 2056 2211.4286 2365
0.47 D IF Fair 2211 2211.0000 2211
0.66 G VVS2 Very Good 2211 2211.0000 2211
0.67 F VS2 Premium 2211 2211.0000 2211
0.71 I VS1 Fair 2104 2211.0000 2318
0.77 H SI2 Very Good 2129 2211.0000 2293
0.65 H VVS2 Ideal 2210 2210.0000 2210
0.75 I SI1 Premium 2122 2208.0000 2295
0.62 I VVS1 Ideal 2207 2207.0000 2207
0.70 J IF Premium 2150 2207.0000 2264
0.53 E VVS2 Very Good 2043 2206.6667 2297
0.66 F VS2 Good 2138 2206.5000 2275
0.74 H SI2 Very Good 2138 2205.5000 2264
0.76 J VS1 Premium 2148 2205.0000 2262
0.72 J VVS1 Premium 2203 2203.0000 2203
0.57 H IF Ideal 2195 2202.5000 2210
0.74 J VS1 Ideal 1985 2202.3333 2468
0.69 G VS1 Good 2202 2202.0000 2202
0.77 J SI1 Ideal 2147 2202.0000 2257
0.75 G SI2 Premium 2070 2201.0000 2268
0.83 I VS2 Ideal 2201 2201.0000 2201
0.49 E VVS2 Ideal 2199 2199.0000 2199
0.90 F I1 Very Good 2199 2199.0000 2199
0.58 E VS1 Premium 2105 2198.0000 2297
0.71 D SI2 Fair 2198 2198.0000 2198
0.62 H VVS1 Ideal 2197 2197.0000 2197
0.72 E SI2 Premium 1808 2197.0000 2376
0.58 E VS2 Premium 1862 2196.3333 2535
0.62 E VS1 Premium 2196 2196.0000 2196
0.64 H VVS1 Ideal 2084 2196.0000 2308
0.50 G IF Ideal 2104 2195.5000 2287
0.58 F VVS2 Very Good 2189 2195.0000 2201
0.95 J I1 Premium 2195 2195.0000 2195
0.71 I VS2 Premium 1923 2194.7500 2362
0.71 J VVS2 Ideal 2162 2194.0000 2226
0.72 J VVS1 Ideal 2139 2194.0000 2249
0.59 H VVS1 Ideal 2193 2193.0000 2193
0.74 I VS1 Premium 2193 2193.0000 2193
0.74 H VS1 Fair 2193 2193.0000 2193
0.58 E VS1 Fair 2192 2192.0000 2192
0.71 H SI1 Very Good 1986 2191.5625 2471
0.76 H SI1 Premium 1915 2189.0000 2425
0.80 F SI2 Good 2189 2189.0000 2189
0.56 D VS1 Very Good 2188 2188.0000 2188
0.63 F VS1 Good 2188 2188.0000 2188
0.77 E SI2 Good 2005 2188.0000 2371
0.60 E VS1 Very Good 2112 2186.3333 2270
0.71 G SI2 Ideal 1840 2186.0000 2387
0.52 F VVS2 Ideal 1978 2185.6667 2489
0.77 I VS1 Fair 2184 2184.0000 2184
0.77 I VS1 Good 2184 2184.0000 2184
0.53 F VVS2 Ideal 1943 2182.1667 2291
0.63 F VS2 Ideal 2182 2182.0000 2182
0.72 I VS2 Premium 1991 2180.0000 2333
0.50 E VVS2 Good 1917 2179.0000 2437
0.65 G VVS2 Very Good 2178 2178.0000 2178
0.66 H VS2 Ideal 2178 2178.0000 2178
0.82 J SI1 Premium 2083 2178.0000 2283
0.78 I SI2 Very Good 2130 2177.6667 2271
0.63 H VVS1 Premium 2177 2177.0000 2177
0.65 H VVS1 Ideal 2177 2177.0000 2177
0.72 F SI2 Very Good 2126 2176.6667 2246
0.66 H VS1 Ideal 2131 2176.5000 2222
0.76 I VS2 Premium 2175 2175.0000 2175
0.82 J SI2 Very Good 2175 2175.0000 2175
0.70 H VS1 Fair 1811 2174.6000 2415
0.62 G VS1 Ideal 2141 2174.5000 2206
0.51 D VVS2 Good 2174 2174.0000 2174
0.72 H SI2 Ideal 1814 2173.1250 2410
0.71 G SI1 Fair 2045 2172.5000 2300
0.57 H VVS1 Premium 2172 2172.0000 2172
0.61 G VS1 Ideal 2041 2172.0000 2309
0.77 G SI2 Premium 1996 2172.0000 2391
0.60 F VS1 Ideal 2142 2171.5000 2201
0.59 E VS2 Premium 1641 2171.4000 2578
0.70 G SI2 Ideal 1814 2171.0833 2480
0.67 H VVS2 Premium 2171 2171.0000 2171
0.74 J VS1 Very Good 2170 2170.0000 2170
0.87 F I1 Premium 2170 2170.0000 2170
0.56 D VS1 Ideal 1924 2169.8889 2741
0.70 H SI1 Premium 1889 2169.3750 2383
0.78 J SI1 Ideal 2169 2169.0000 2169
0.64 E VS2 Very Good 2114 2168.0000 2222
0.56 G VVS1 Premium 2167 2167.0000 2167
0.60 G VVS2 Fair 2167 2167.0000 2167
0.76 J VVS2 Ideal 2167 2167.0000 2167
0.84 D SI2 Fair 2167 2167.0000 2167
0.62 D VS2 Very Good 2022 2166.5000 2311
0.83 J SI1 Premium 2022 2166.5000 2311
0.63 G VS2 Ideal 1821 2166.0000 2357
0.81 F SI2 Fair 2031 2166.0000 2301
0.74 F SI2 Very Good 1909 2165.8000 2319
0.71 H SI2 Ideal 1800 2164.3000 2376
0.63 D VS2 Ideal 2013 2164.0000 2373
0.71 H SI1 Good 2056 2163.7778 2354
0.83 F SI2 Good 1968 2163.0000 2358
0.58 G VVS2 Ideal 2132 2161.2500 2245
0.62 G VS1 Very Good 2161 2161.0000 2161
0.34 F I1 Fair 2160 2160.0000 2160
0.77 I SI2 Very Good 2159 2159.0000 2159
0.54 F VVS2 Very Good 2115 2156.5000 2198
0.66 G VS1 Good 2055 2156.3333 2275
0.51 G IF Ideal 2096 2155.0000 2208
0.59 G VVS2 Very Good 2155 2155.0000 2155
0.64 G VS1 Ideal 2155 2155.0000 2155
0.71 H SI2 Very Good 1972 2153.2000 2339
0.52 G VVS1 Premium 2153 2153.0000 2153
0.69 G VS1 Fair 2070 2152.5000 2235
0.75 H SI2 Very Good 2093 2151.7500 2228
0.70 I VS2 Good 2010 2151.1667 2268
0.72 J VVS1 Very Good 2093 2150.5000 2208
0.56 H IF Ideal 2135 2149.6667 2157
0.63 F VS1 Very Good 2149 2149.0000 2149
0.72 I SI1 Very Good 2026 2148.3333 2345
0.62 E VS1 Very Good 2147 2147.0000 2147
0.73 D SI2 Fair 2147 2147.0000 2147
0.56 G VVS1 Ideal 1993 2146.4000 2318
0.79 G SI2 Very Good 1809 2145.5000 2482
0.70 H VS2 Fair 1981 2145.2500 2351
0.55 G VVS1 Very Good 1838 2144.5000 2451
0.52 F IF Fair 2144 2144.0000 2144
0.72 J VVS2 Very Good 2144 2144.0000 2144
0.72 H SI2 Premium 2144 2144.0000 2144
0.73 I VS2 Good 2144 2144.0000 2144
0.51 F VVS2 Very Good 2041 2143.5000 2633
0.70 H SI1 Very Good 1830 2143.1364 2548
0.90 F I1 Good 2143 2143.0000 2143
0.60 H VVS1 Very Good 2142 2142.0000 2142
0.60 F VS1 Premium 2142 2142.0000 2142
0.59 F VS1 Ideal 2140 2140.0000 2140
0.71 I VS2 Very Good 1840 2139.4286 2384
0.46 D IF Good 2139 2139.0000 2139
0.58 D VS1 Ideal 2067 2139.0000 2211
0.61 F VS2 Ideal 2002 2138.6667 2306
0.90 E I1 Fair 2138 2138.0000 2138
0.73 J VS2 Ideal 2054 2137.6667 2258
0.54 E VS1 Premium 1939 2137.0000 2608
0.51 E VVS2 Premium 1882 2136.5000 2419
0.51 E VVS2 Good 1906 2136.3333 2523
0.61 E VS1 Very Good 2025 2134.0000 2243
0.65 D SI1 Premium 2134 2134.0000 2134
1.04 F I1 Fair 2134 2134.0000 2134
0.83 H SI2 Fair 2133 2133.0000 2133
0.70 E SI2 Good 1758 2131.2778 2453
0.62 F VVS2 Good 2131 2131.0000 2131
0.63 G VS2 Premium 1871 2131.0000 2391
0.64 F VS2 Premium 2131 2131.0000 2131
0.72 I VVS2 Fair 2131 2131.0000 2131
0.61 F VS1 Good 2076 2130.6667 2196
0.60 G VVS2 Very Good 2130 2130.0000 2130
0.81 J VS2 Premium 2129 2129.0000 2129
0.91 I SI2 Fair 2129 2129.0000 2129
0.54 G IF Very Good 2128 2128.0000 2128
0.59 D VS2 Ideal 1978 2127.7500 2352
0.63 H VVS1 Very Good 2127 2127.0000 2127
0.72 E SI2 Fair 2125 2125.0000 2125
0.55 F VVS2 Very Good 2102 2124.5000 2147
0.73 D SI2 Premium 1730 2124.5000 2519
0.90 G I1 Fair 1939 2124.5000 2310
0.61 E VS2 Good 2124 2124.0000 2124
0.50 F VVS1 Ideal 2018 2123.2500 2165
0.61 H VS1 Very Good 2123 2123.0000 2123
0.80 H SI2 Ideal 2122 2122.0000 2122
0.72 I VS2 Good 2068 2121.0000 2174
0.73 J VS1 Ideal 2121 2121.0000 2121
0.50 E VVS2 Very Good 2003 2120.2222 2275
0.63 G VS1 Very Good 1960 2119.3333 2241
0.56 G VVS2 Very Good 2010 2118.6667 2336
0.57 G VVS2 Ideal 1976 2118.3333 2206
0.70 D SI2 Very Good 1806 2118.2000 2354
0.60 G VS1 Ideal 2061 2117.2500 2147
0.70 I VVS2 Fair 2117 2117.0000 2117
0.59 G VVS2 Ideal 1966 2116.4444 2283
0.58 E VS1 Ideal 1893 2116.1250 2359
0.78 I SI1 Good 2116 2116.0000 2116
0.71 H SI1 Premium 1883 2115.5000 2364
0.67 G VS2 Very Good 2080 2114.0000 2131
0.68 G VS1 Very Good 2114 2114.0000 2114
0.58 E VS1 Good 2041 2113.0000 2185
0.57 F VVS2 Very Good 1902 2111.2500 2351
0.70 G SI1 Good 1915 2111.2000 2218
0.75 J SI1 Good 1965 2107.5000 2250
0.70 F SI1 Fair 1974 2107.2500 2167
0.73 H SI1 Premium 2107 2107.0000 2107
0.73 F SI2 Good 2023 2107.0000 2191
0.48 F IF Ideal 2059 2106.5000 2154
0.65 H VS1 Ideal 2106 2106.0000 2106
0.74 J VVS2 Ideal 2106 2106.0000 2106
0.60 G VVS2 Premium 2105 2105.0000 2105
0.72 G SI2 Very Good 2001 2104.6250 2246
0.51 F VVS2 Ideal 1998 2102.5500 2203
0.48 E VVS2 Ideal 2102 2102.0000 2102
0.62 E VS2 Premium 2102 2102.0000 2102
0.75 F SI2 Fair 2102 2102.0000 2102
0.66 G VS2 Good 2101 2101.0000 2101
0.81 J SI2 Ideal 2100 2100.0000 2100
0.78 J SI1 Very Good 2035 2099.5000 2164
0.56 G VVS2 Ideal 1889 2098.6667 2510
0.55 I IF Ideal 2098 2098.0000 2098
0.73 J VS2 Good 2098 2098.0000 2098
0.70 H SI2 Ideal 1715 2097.8000 2326
0.70 I VS2 Very Good 1666 2097.0000 2360
0.54 H IF Ideal 1981 2095.6667 2210
0.79 I SI1 Good 2095 2095.0000 2095
0.72 J VVS2 Ideal 1996 2094.0000 2150
0.52 G IF Good 2093 2093.0000 2093
0.73 I SI2 Very Good 2093 2093.0000 2093
0.71 E SI2 Good 1783 2091.6667 2390
0.54 G VVS2 Premium 2090 2090.0000 2090
0.54 F VS1 Good 2090 2090.0000 2090
0.77 I SI1 Premium 2089 2089.0000 2089
0.60 E VS2 Ideal 1897 2088.0000 2189
0.62 F VVS2 Fair 2088 2088.0000 2088
0.71 I SI1 Very Good 1883 2087.2500 2502
0.82 J SI1 Good 2033 2086.5000 2140
0.76 I SI1 Good 1914 2085.3333 2326
0.60 E VS2 Very Good 2085 2085.0000 2085
0.73 H SI2 Good 2085 2085.0000 2085
0.70 H SI1 Good 1771 2084.9286 2394
0.78 H SI2 Ideal 1909 2084.3333 2332
0.58 F VS1 Good 2084 2084.0000 2084
0.51 G IF Very Good 2063 2083.0000 2102
0.62 F VS1 Fair 2083 2083.0000 2083
0.51 F VVS2 Premium 1882 2082.8333 2310
0.54 G VVS1 Ideal 1982 2082.0000 2236
0.58 E VS1 Very Good 2082 2082.0000 2082
0.61 G VS1 Very Good 2082 2082.0000 2082
0.50 F VVS1 Premium 2081 2081.0000 2081
1.06 G I1 Very Good 2080 2080.0000 2080
0.52 E VVS2 Fair 2079 2079.0000 2079
0.66 D SI1 Ideal 2079 2079.0000 2079
0.76 J VS2 Very Good 1949 2077.5000 2206
0.71 G SI2 Premium 1717 2076.9167 2913
0.61 F VS2 Premium 2075 2075.0000 2075
0.82 I SI2 Ideal 2075 2075.0000 2075
0.64 D SI1 Premium 2074 2074.0000 2074
0.72 I VVS2 Premium 2074 2074.0000 2074
0.50 F VVS1 Very Good 1978 2073.6667 2122
0.50 F VVS2 Ideal 1850 2071.4375 2283
0.59 E VS1 Very Good 1993 2071.3333 2200
0.69 I VS2 Very Good 2070 2070.0000 2070
0.75 I SI2 Premium 2070 2070.0000 2070
0.55 H IF Ideal 1905 2069.5000 2133
0.74 E SI1 Fair 2069 2069.0000 2069
0.51 F VVS2 Good 1942 2068.5000 2195
0.96 G I1 Fair 2068 2068.0000 2068
0.67 I IF Very Good 2066 2066.0000 2066
1.05 E I1 Good 2066 2066.0000 2066
0.60 E VS1 Good 2065 2065.0000 2065
0.63 I VVS1 Ideal 2064 2064.0000 2064
0.98 I I1 Premium 2064 2064.0000 2064
0.64 F VS2 Fair 2063 2063.0000 2063
0.64 F VS2 Good 2063 2063.0000 2063
0.50 F VVS2 Premium 1814 2062.3333 2160
0.79 J SI1 Premium 1924 2061.5000 2199
0.52 F VVS2 Very Good 2035 2060.5000 2079
0.67 G VS2 Good 1990 2060.5000 2131
0.68 G VS2 Premium 2040 2060.5000 2081
0.64 H VVS2 Very Good 2060 2060.0000 2060
0.64 G VS1 Very Good 2060 2060.0000 2060
0.74 G SI1 Fair 2060 2060.0000 2060
0.70 G SI1 Fair 1915 2059.8750 2352
0.57 G VVS2 Premium 2059 2059.0000 2059
0.57 E VS1 Ideal 1884 2058.1538 2297
0.74 J SI1 Good 2058 2058.0000 2058
0.81 J SI1 Premium 1917 2058.0000 2199
0.71 F SI2 Good 1818 2057.3333 2282
0.61 I IF Very Good 2057 2057.0000 2057
0.69 E VS2 Good 2057 2057.0000 2057
0.70 F SI2 Premium 1597 2056.7619 2373
0.57 D VS1 Ideal 1982 2056.2000 2122
0.52 D VS1 Ideal 1707 2056.1250 2550
0.64 E SI1 Ideal 1859 2054.3333 2215
0.61 G VS2 Ideal 1821 2053.5000 2286
0.54 G VVS1 Very Good 1983 2053.0000 2123
0.59 J IF Good 2053 2053.0000 2053
0.62 H VVS2 Ideal 2052 2052.0000 2052
0.76 J VS2 Premium 2052 2052.0000 2052
0.54 D VS1 Ideal 1941 2050.6667 2378
0.70 I VS2 Premium 1814 2050.5000 2268
0.70 G SI2 Good 1935 2050.3333 2268
0.67 G VS2 Premium 2050 2050.0000 2050
0.73 G SI2 Fair 2050 2050.0000 2050
0.71 G SI2 Very Good 1740 2049.8571 2171
0.52 I IF Ideal 1988 2049.5000 2111
0.79 J VS2 Good 2048 2048.0000 2048
0.80 J SI2 Premium 2022 2048.0000 2074
0.91 J I1 Very Good 2048 2048.0000 2048
0.57 F VS1 Ideal 1949 2047.9091 2193
0.70 G SI2 Premium 1636 2046.6429 2268
0.98 H I1 Premium 2046 2046.0000 2046
0.69 E VS2 Premium 2045 2045.0000 2045
0.80 J VS2 Premium 2045 2045.0000 2045
0.59 D VS2 Very Good 1948 2044.6667 2200
0.70 G SI2 Very Good 1715 2043.9167 2872
0.58 F VS1 Ideal 1772 2043.1538 2245
0.80 J SI1 Premium 1838 2042.2500 2227
0.70 J VVS1 Very Good 2042 2042.0000 2042
0.74 E SI1 Premium 2042 2042.0000 2042
0.71 I VS2 Good 1748 2040.8333 2270
0.52 H IF Ideal 1798 2040.5000 2441
0.72 F SI2 Premium 1676 2040.4000 2364
0.59 G VVS2 Premium 2040 2040.0000 2040
0.63 H VS1 Ideal 2040 2040.0000 2040
0.70 F SI2 Good 1792 2039.9167 2411
0.70 E SI2 Fair 1723 2039.8333 2600
0.53 D VS1 Very Good 1907 2038.5000 2170
0.51 D VS1 Premium 1873 2038.0000 2203
0.62 F VS2 Premium 2009 2037.0000 2065
0.73 J VS1 Premium 2037 2037.0000 2037
1.04 I I1 Fair 2037 2037.0000 2037
0.76 J VS1 Very Good 2036 2036.0000 2036
0.53 D VS1 Ideal 1813 2035.0000 2340
0.78 J SI1 Good 2035 2035.0000 2035
0.55 D VS1 Ideal 1890 2034.5714 2251
0.59 F VS1 Very Good 2034 2034.0000 2034
0.72 J VS1 Very Good 1914 2034.0000 2136
0.63 F VS2 Very Good 2032 2032.0000 2032
0.73 G SI1 Fair 2032 2032.0000 2032
0.72 J VS1 Ideal 1832 2031.7500 2177
0.52 G VVS1 Very Good 1913 2031.2000 2113
0.73 J SI1 Very Good 2031 2031.0000 2031
0.50 G IF Good 1976 2028.5000 2081
0.53 F VVS2 Very Good 1880 2027.5000 2175
0.73 J VS2 Very Good 1952 2026.5000 2101
0.76 J VS2 Ideal 2026 2026.0000 2026
0.63 F VS2 Premium 1905 2025.3333 2211
0.70 F SI2 Very Good 1792 2024.5385 2195
0.71 J VS1 Very Good 1981 2024.2500 2062
0.71 I VS2 Fair 1840 2024.0000 2208
0.58 F VS1 Very Good 2001 2023.5000 2046
0.50 G VVS1 Premium 1966 2023.2500 2070
0.70 I SI1 Good 1763 2023.2500 2276
0.55 G VVS2 Ideal 1799 2022.2000 2129
0.78 J SI2 Premium 2022 2022.0000 2022
0.72 G SI1 Good 1969 2021.5000 2074
0.73 G SI2 Very Good 1986 2021.5000 2057
0.51 E VVS1 Good 2020 2020.0000 2020
0.52 G VVS1 Ideal 1829 2019.8750 2387
0.60 D VS2 Premium 1808 2019.6667 2214
0.53 G VVS1 Ideal 1956 2019.4000 2153
0.63 D SI1 Ideal 1779 2017.8000 2313
0.66 F SI1 Good 2017 2017.0000 2017
0.71 J VS1 Ideal 1932 2017.0000 2085
1.01 I I1 Premium 2017 2017.0000 2017
0.70 H SI2 Very Good 1807 2016.4615 2306
0.56 F VS1 Premium 2016 2016.0000 2016
0.60 F VS2 Ideal 1866 2015.1667 2358
0.57 D VS2 Good 2015 2015.0000 2015
0.76 J VS1 Good 2015 2015.0000 2015
0.55 E VS1 Ideal 1764 2014.6250 2220
0.75 J VS2 Ideal 1923 2013.2500 2104
0.63 D SI1 Very Good 1885 2011.6667 2211
0.63 E VS2 Fair 2011 2011.0000 2011
0.72 J VS1 Good 1959 2010.5000 2062
0.71 G SI2 Fair 1691 2010.0000 2300
0.65 G VS2 Very Good 2009 2009.0000 2009
0.58 G VVS2 Premium 1975 2008.7500 2095
0.50 E VVS1 Good 2008 2008.0000 2008
0.67 D SI1 Very Good 2007 2007.0000 2007
0.76 D SI2 Very Good 2006 2006.0000 2006
0.57 D VS2 Ideal 1718 2005.5385 2546
0.77 J VS1 Fair 2005 2005.0000 2005
0.77 J VS1 Premium 2005 2005.0000 2005
0.76 G SI2 Premium 1676 2003.3333 2364
0.60 G VS1 Very Good 1912 2003.0000 2094
0.54 D VS1 Very Good 1988 2002.6667 2016
0.71 E SI2 Fair 2002 2002.0000 2002
0.72 J VS2 Ideal 1904 2001.5000 2099
0.72 H SI2 Fair 1799 2001.0000 2203
0.77 J SI2 Very Good 2001 2001.0000 2001
0.71 J VS2 Ideal 1899 1999.3333 2052
0.68 F SI1 Good 1924 1999.0000 2074
0.77 J VS2 Very Good 1843 1999.0000 2155
0.56 D VS2 Ideal 1753 1998.7742 2066
0.47 E VVS2 Ideal 1883 1998.0000 2113
0.75 J VS1 Premium 1998 1998.0000 1998
0.72 I SI1 Premium 1799 1997.8000 2262
0.51 G VVS1 Premium 1925 1997.6667 2111
0.56 E VS1 Ideal 1608 1996.5294 2352
0.71 H SI1 Fair 1828 1996.3333 2112
0.53 G VVS1 Very Good 1904 1996.0000 2136
0.60 F VS2 Very Good 1772 1994.0000 2270
0.70 D SI2 Fair 1696 1994.0000 2358
0.54 G VVS2 Ideal 1801 1991.0000 2424
0.59 G VS1 Premium 1991 1991.0000 1991
0.63 H VS1 Very Good 1991 1991.0000 1991
0.64 I IF Ideal 1991 1991.0000 1991
0.50 H IF Ideal 1844 1990.7500 2352
0.74 J VS2 Very Good 1978 1990.5000 2003
0.70 I SI1 Very Good 1763 1990.3571 2147
0.58 G VVS2 Very Good 1872 1990.3333 2078
0.56 D VS2 Premium 1910 1987.5000 2066
1.00 I I1 Fair 1805 1987.2500 2112
0.73 F SI2 Fair 1631 1986.3333 2164
0.76 H SI2 Premium 1985 1985.0000 1985
0.50 F VVS2 Very Good 1778 1984.0000 2153
0.74 J VS1 Fair 1982 1982.0000 1982
0.66 D VS2 Ideal 1981 1981.0000 1981
0.67 H VS1 Very Good 1981 1981.0000 1981
0.54 D VS1 Premium 1847 1980.3333 2100
0.50 D VVS2 Fair 1980 1980.0000 1980
0.55 H VVS1 Premium 1980 1980.0000 1980
0.73 J SI1 Ideal 1877 1979.7500 2184
0.58 G VS1 Premium 1884 1979.6667 2071
0.58 H VVS1 Very Good 1979 1979.0000 1979
0.64 D SI1 Ideal 1967 1979.0000 1991
0.66 H VS1 Very Good 1978 1978.0000 1978
0.58 H VVS1 Ideal 1961 1976.0000 2006
0.70 I VS2 Fair 1814 1975.6667 2117
0.70 H SI2 Premium 1749 1975.6000 2184
0.71 J SI1 Ideal 1609 1975.3333 2700
0.55 D VS1 Premium 1975 1975.0000 1975
0.71 I SI1 Premium 1774 1973.5000 2173
0.57 H VVS1 Ideal 1929 1972.7500 2052
0.71 H SI2 Premium 1567 1972.6667 2231
0.60 D SI1 Ideal 1698 1972.3333 2263
0.56 F VS1 Ideal 1787 1971.8182 2097
0.61 F VS2 Good 1971 1971.0000 1971
0.62 D SI1 Ideal 1927 1970.5000 2062
0.70 J VVS2 Very Good 1968 1968.0000 1968
0.77 J SI1 Very Good 1807 1967.6667 2086
0.51 G VVS1 Ideal 1810 1967.4667 2111
0.75 G SI2 Good 1819 1967.0000 2115
0.75 J SI1 Very Good 1859 1966.6667 2083
0.50 G VVS1 Good 1966 1966.0000 1966
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0.30 I SI2 Ideal 348 369.4000 405
0.23 H IF Very Good 369 369.0000 369
0.23 G VS2 Very Good 369 369.0000 369
0.23 G VVS2 Fair 369 369.0000 369
0.23 G SI1 Ideal 364 367.6667 375
0.23 E VS1 Good 327 367.3333 402
0.20 F VS2 Premium 367 367.0000 367
0.20 E VS2 Very Good 367 367.0000 367
0.20 E VS2 Premium 367 367.0000 367
0.20 E VS2 Ideal 367 367.0000 367
0.20 D VS2 Premium 367 367.0000 367
0.20 D VS2 Ideal 367 367.0000 367
0.30 J SI1 Ideal 367 367.0000 367
0.33 J SI2 Ideal 366 366.0000 366
0.24 G SI1 Ideal 364 364.0000 364
0.31 J SI1 Good 363 363.0000 363
0.26 H SI2 Ideal 362 362.0000 362
0.25 E VS1 Fair 361 361.0000 361
0.25 E VS1 Good 361 361.0000 361
0.27 G SI2 Ideal 361 361.0000 361
0.28 H SI1 Very Good 360 360.0000 360
0.23 E VS2 Good 357 357.0000 357
0.25 H SI1 Ideal 357 357.0000 357
0.24 I VS1 Premium 355 355.0000 355
0.30 I SI2 Good 351 351.0000 351
0.23 H VS1 Very Good 338 345.5000 353
0.20 E SI2 Premium 345 345.0000 345
0.31 J SI2 Ideal 344 344.0000 344
0.23 J VS1 Ideal 340 340.0000 340
0.22 E VS2 Fair 337 337.0000 337
0.26 H SI1 Very Good 337 337.0000 337
0.24 J VVS2 Very Good 336 336.0000 336
0.24 I VVS1 Very Good 336 336.0000 336
0.31 J SI2 Good 335 335.0000 335
0.21 E SI1 Premium 326 326.0000 326
0.23 E SI2 Ideal 326 326.0000 326

The shows that minimum, maximum and average price of the diamonds according to the cara,color, clarity and cut.

3.6 Price Group Analysis

Until know, we analyze variables and relationship with prices. Now, by slicing the dataset into five, we can make some interpretation by using these price groups.

quant <- quantile(diamonds$price, seq(0, 1, 0.2))

diamonds_price_group <- diamonds %>%
  mutate(price_group = case_when(
    price < quant[2] ~ "Very Low",
    price < quant[3] ~ "Low",
    price < quant[4] ~ "Medium",
    price < quant[5] ~ "High",
    TRUE ~ "Very High"
  )) %>%
  mutate(price_group = factor(price_group, levels = c("Very Low", "Low", "Medium", "High", "Very High")))

diamonds_price_group %>%
  group_by(color, price_group) %>%
  summarise(count=n()) %>%
  mutate(percentage = 100 * count / sum(count)) %>%
  ggplot(., aes(x = '', y = count, fill = price_group)) + 
  geom_bar(width = 1, stat = "identity", position = "fill") +
  coord_polar("y") +
  theme_void() +
  scale_fill_manual(values=c("seashell2", "lightpink1", "seashell3", "paleturquoise3", "paleturquoise4")) +
  theme(plot.title = element_text(vjust = 0.5)) +
  facet_wrap(~color) +
  labs(title = "Price Group Analyses of Color",
       subtitle = st,
       fill = "Price Group")

To make this analysis, the color variable is selected. Results shows that all price groups are distributed almost equally. However, when the color group is getting worst, the percentage of the high and very high price group increase. This situations shows the contrast with we expect. Moreover, this analysis supports our previous findings. Also, we can see the results in a table below.

#calculation of the number of 
PG_C = diamonds_price_group %>%
  group_by(color, price_group) %>%
  summarise(count=n()) %>%
  select(color, price_group, count)%>%
  pivot_wider(id_cols = color, names_from = price_group, values_from = count)

#calculation of the percentage of the color
ColorPercentage = diamonds_price_group %>%
  group_by(color) %>%
  summarise(count=n()) %>%
  mutate(percentage = round(100 * count / sum(count),2)) %>%
  select(color,percentage)

#by using left join we can merge these two table
PG_C %>%
  inner_join(ColorPercentage, by = "color") %>%
  kable(col.names = c("Color", "Very Low","Low", "Medium", "High", "Very High", "Percentage"))%>% 
  kable_minimal(full_width = F) 
Color Very Low Low Medium High Very High Percentage
J 394 242 518 768 878 5.21
I 1062 542 992 1268 1541 10.05
H 1763 1055 1321 2212 1911 15.37
G 2236 2421 2131 1820 2644 20.93
F 1691 2192 2081 1814 1739 17.70
E 2112 2686 2176 1576 1222 18.18
D 1488 1616 1533 1294 822 12.56

3.7 Price Histogram

Another price analysis can be made by using histogram. Histograms give information about the distribution of the variable in the first step. For this purpose, before preparing model, we can control the histogram of the price.

diamonds %>%
  ggplot(aes(x=price)) +
  geom_histogram(aes(y=..density..), position="identity", alpha=0.8, fill = "paleturquoise4", color="paleturquoise4") +
  geom_density(alpha=1, size = 1)+
  theme_minimal() +
  labs(title = "Price Distribution with Histogram",
       subtitle = st,
       x = "Price",
       y = "Count")

The histogram shows that the price in diamonds dataset is right skewed. Moreover, we can say that the price distribution has exponential distribution. For this reason,we can re-plot the histogram by using log transformation on the response variable price.

diamonds %>%
  ggplot(aes(x=log(price))) +
  geom_histogram(aes(y=..density..), position="identity", alpha=0.8, fill = "paleturquoise4", color="paleturquoise4") +
  geom_density(alpha=1, size = 1)+
  theme_minimal() +
  labs(title = "Price Distribution with Log Tranformation ",
       subtitle = st,
       x = "Price",
       y = "Count")

The transformed price histogram does not illustrate the specific shape of the normal distribution.To make prediction about the price, to create model becomes more suitable in this step. For this reason, in the next step, classification algorithm and prediction models are created. According to the models,we can predict price.

4 Clustering

To make model and predict the price variable. First, we need to divide data set into two pieces: train data and test data. By using train data, we can create a model for the data set, and then by using test data we can control the accuracy of the model obtained by using train data. Following codes provide the production of the train and test data set. Creating of the train and test data is random process, for this reason we use set.seed() function to get same two dataset.

#this data process is obtain from the course assignment page, for this reason set.seed value is not changed. 
set.seed(503)
#test data
diamonds_test <- diamonds %>% 
                    #to divide data set to, we add diamond id for each row
                    mutate(diamond_id = row_number()) %>% 
                    # we group data set according to the cut,color and clarity
                    group_by(cut, color, clarity) %>%   
                    # by using sample_frac() function, we get 20% of the data as test data
                    sample_frac(0.2) %>%                  
                    ungroup()    

#train data
#by using anti_join we get different rows from data set.
diamonds_train <- anti_join(diamonds %>% mutate(diamond_id = row_number()),  
    diamonds_test, by = "diamond_id")

#then to make clear analysis we need to extract diamond_id column by using following codes.
diamonds_train <- diamonds_train[, c(-ncol(diamonds_train))]
diamonds_test  <- diamonds_test[, c(-ncol(diamonds_test))]

After the division of the data set into test and train, we extract the diamonds_id column to make clear analysis. After that poin, we can transform factor variable to numeric variable by using as.numeric() function. That is, all the ordered columns, we will use them as numeric values.

#we transform our factor variable to numeric variable, then we assign new variable to their own variables by using following codes.
diamonds_train$cut <- as.numeric(diamonds_train$cut)
diamonds_train$clarity <- as.numeric(diamonds_train$clarity)
diamonds_train$color <- as.numeric(diamonds_train$color)
diamonds_test$cut <- as.numeric(diamonds_test$cut)
diamonds_test$clarity <- as.numeric(diamonds_test$clarity)
diamonds_test$color <- as.numeric(diamonds_test$color)

4.1 Principal Component Analysis

Principle component analysis (PCA) is used to provide less variable from the data set with huge number of variables. In other words, by using PCA, we can decrease number of columns. To make PCCA, we use princomp() function which belongs to base R.

set.seed(157)
#into numeric variable. But to provide accuracy, by using sapply function we select only numeric variable to implement PCA.
diamonds_pca <- princomp(as.matrix(diamonds_train[,sapply(diamonds_train, class) == "numeric"]),cor=T)
summary(diamonds_pca,loadings=TRUE)
## Importance of components:
##                           Comp.1    Comp.2    Comp.3    Comp.4     Comp.5
## Standard deviation     2.0743446 1.1893677 1.1061659 0.9926647 0.83123137
## Proportion of Variance 0.4781006 0.1571773 0.1359559 0.1094870 0.07677173
## Cumulative Proportion  0.4781006 0.6352779 0.7712338 0.8807208 0.95749254
##                            Comp.6      Comp.7       Comp.8       Comp.9
## Standard deviation     0.59250776 0.172964901 0.0341800209 2.040935e-02
## Proportion of Variance 0.03900727 0.003324095 0.0001298082 4.628241e-05
## Cumulative Proportion  0.99649981 0.999823909 0.9999537176 1.000000e+00
## 
## Loadings:
##         Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## carat    0.471                       0.126         0.868              
## cut     -0.109  0.595  0.452  0.240 -0.176  0.584                     
## color   -0.158 -0.203         0.803  0.537                            
## clarity -0.222  0.208  0.212 -0.506  0.778                            
## depth           0.209 -0.828         0.117  0.492                0.101
## table    0.129 -0.698  0.226 -0.182         0.642                     
## x        0.475                       0.115        -0.278 -0.699  0.425
## y        0.474                       0.120        -0.299  0.714  0.381
## z        0.474  0.103                0.132        -0.282        -0.815

The summary shows that four components can almost explain 88% of the data. By using these four component we can prepare variables and them into the data sets.

#by using following function we can add this pca variables into data set
#since we select the four component, we add four variables into data set

#Train data set
PCAResults = predict(diamonds_pca, newdata = diamonds_train) # prediction by using data set
diamonds_train$PCA1 <- PCAResults[,1] # adding variable into data
diamonds_train$PCA2 <- PCAResults[,2]
diamonds_train$PCA3 <- PCAResults[,3]
diamonds_train$PCA4 <- PCAResults[,4]

#Test data set
PCAResultsTest = predict(diamonds_pca, newdata = diamonds_test) # prediction by using data set
diamonds_test$PCA1 <- PCAResultsTest[,1] # adding variable into data
diamonds_test$PCA2 <- PCAResultsTest[,2]
diamonds_test$PCA3 <- PCAResultsTest[,3]
diamonds_test$PCA4 <- PCAResultsTest[,4]

4.2 Clustering

Like PCA, we can also* make clustering process by using this data sets. As the difference between values in each variable is different, we need to use scale process in the data set to implement K-means. Because if a column would have much higher values respect to other columns, it can dominate the rest.By using this clustering, we can create a feature with using the clustering.To be able to scale the data we need the minimum and maximum values of the train dataset. We will scale both train and test set with the same values.

MinValues = sapply(diamonds_train[,c("cut", "clarity", "color", "carat", "depth", "table", "x", "y", "z")], min)
MaxValues = sapply(diamonds_train[,c("cut", "clarity", "color", "carat", "depth", "table", "x", "y", "z")], max)

We get each variables minimum and maximum variables, now we need to implement scaling process into our data sets. In this process, we try to subtract vector form each row of the matrix data set. To make it easy, we can use sweep() function. By using the sweep function we can make calculation with vector and matrix.

diamonds_train_scale <- sweep(sweep(diamonds_train[,c("cut", "clarity", "color", "carat", "depth", "table", "x", "y", "z")], 2, MinValues), 2, MaxValues - MinValues, "/")

By using this process, we can obtain scaled data set. In other words, every value in each column is between 0 and 1. Thus, one value does not dominate each other. By using K-means algorithm, we can create different clusters, i.e., we can prepare or select number of cluster. Like we learn in the class, we can find number of cluster for data set by comparing errors. Then, we plot these errors value and choose a number of cluster.

#by using following functions we obtain new column namely cluster. 
#this column shows that tows cluster
ErrorsKMEANS = c()
for (i in (1:17)){
  #provide getting same results with random function
  set.seed(157)  
  # application of the k-means function with i number of group size
  cluster<-kmeans(diamonds_train_scale,centers=i)
  # calculation of the fulfillment of the clusters to data.
  ErrorsKMEANS = c(ErrorsKMEANS, 100*round(1 - (cluster$tot.withinss/cluster$totss), digits = 3)) 
}

# creating data frame with errors.
Errors_df <- data.frame(x=c(1:17), y=ErrorsKMEANS) 

#by using ggplot function we visualize the errors between 1:17.
ggplot(Errors_df, aes(x=x, y=y)) +
  geom_point(color = "paleturquoise4") +
  geom_line(color="paleturquoise4") +
  geom_text(aes(label = ErrorsKMEANS), size=3, color = "black", position = position_stack(vjust = 0.95))+
  theme_minimal() +
  labs(title = "Erros' Line Plot",
       subtitle = st,
       x = "X",
       y = "Y")

When we analyze the proportion of the of the increase of the errors, the decrease in errors are slowly changing after the cluster with 6 centers. For this reason, we can select six as a number of cluster.

set.seed(157)
BC <- kmeans(diamonds_train_scale,centers=6) # BC: Best cluster obtained form the k-means
#we assign our cluster values into the train data set
diamonds_train$cluster <- as.factor(BC$cluster)

The classification of the train data set according to the new defined data set, cluster. In this data set, there are at most second cluster, whereas third cluster has the lowest value in the data set.

diamonds_train %>%
  group_by(cluster)%>%
  summarise(cluster_count = n()) %>%
  
  ggplot(., aes(x=cluster_count, y = reorder(cluster, cluster_count))) +
  geom_col(fill = "paleturquoise4")+
  theme_minimal() +
  labs(title = "Number of Cluster in Data Set",
       subtitle = st,
       x = "Number of cluster",
       y = "Cluster")

diamonds_train %>%
  group_by(cluster)%>%
  summarise(cluster_count = n()) %>%
  select(cluster, cluster_count) %>%
  kable(col.names = c("Cluster", "Number of Cluster"))%>% 
  kable_minimal(full_width = F) 
Cluster Number of Cluster
1 5838
2 9097
3 5458
4 8566
5 5904
6 8137

Up until now, we scale our train data set and create cluster for this data set by using k-means algorithm. Then we define best number of cluster, 6. Then, we need to control our clustering by using test data set. To make this test, following code is used. This code is implemented from a Stackoverflow page. This page, can be examined in detail.

#we get variables of the data set and by using sweep function we can arrange data set as scaled one.
diamonds_test_scale <- sweep(sweep(diamonds_test[,c("cut", "clarity", "color", "carat", "depth", "table", "x", "y", "z")], 2, MinValues), 2, MaxValues - MinValues, "/")

closest.cluster <- function(x) {
  cluster.dist <- apply(BC$centers, 1, function(y) sqrt(sum((x-y)^2)))
  return(which.min(cluster.dist)[1])
}
diamonds_test$cluster <- as.factor(apply(diamonds_test_scale, 1, closest.cluster))

Now, we have enough data to create models for this data sets. In the next section, by using all outputs and process in the previous sections, the best predicting model is prepared.

5 Predicting Model Creation

Before we start to to prepare prediction model, we can control the correlation between variables.

a <- diamonds %>%group_by(price, carat) %>%
      ggplot(aes(x=carat, y = price)) +
      geom_point() + theme_minimal() +geom_smooth(color="red")
b <- diamonds %>%group_by(price, cut) %>%
      ggplot(aes(x=cut, y = price)) +
      geom_point()+ theme_minimal() 
c <- diamonds %>%
      group_by(price, color) %>%
      ggplot(aes(x=color, y = price)) +
      geom_point()+ theme_minimal() 
d <- diamonds %>%group_by(price, clarity) %>%
      ggplot(aes(x=clarity, y = price)) +
      geom_point()+ theme_minimal() 
e <- diamonds %>%group_by(price, depth) %>%
      ggplot(aes(x=depth, y = price)) +
      geom_point()+ theme_minimal() 
f <- diamonds %>%group_by(price, table) %>%
      ggplot(aes(x=table, y = price)) +
      geom_point()+ theme_minimal() 
g <- diamonds %>%group_by(price, x) %>%
      ggplot(aes(x=x, y = price)) +
      geom_point()+ theme_minimal() +geom_smooth(color="red")
h <- diamonds %>%group_by(price, y) %>%
      ggplot(aes(x=y, y = price)) +
      geom_point()+ theme_minimal() +geom_smooth(color="red")
ı <- diamonds %>%group_by(price, z) %>%
      ggplot(aes(x=z, y = price)) +
      geom_point()+ theme_minimal() + geom_smooth(color="red")

(a | b | c) /
(d | e | f) /
(g | h | ı)  

The graph shows relationship between price and other variables, moreover we can obtain the relationship between price and other variables by using cor() and corrplot() function.

diamonds_cor <-cor(diamonds_train[-c(11:15)])
corrplot(diamonds_cor, method="number", type = "upper")

Like the scatter plot results, the correlation matrix also shows that there is correlation between price and carat, x, y, and z variables. Moreover, there are high correlation between independent variables. This means that, when the change occurred in one of the independent variable from carat, x,y,z, the change becomes in other variable.
After the correlation, we can try to make prediction models.

5.1 Linear Regression Model

There are some prediction models: some of them unsupervised model, whereas, some them are named as supervised models. PCA and K-means classifications are located in the unsupervised models. In this model, we do not know the how we make prediction. In contrast, we know the response variable. Linear regression models belongs to supervised models.

For this purpose, first linear regression model (LM) is implemented. To obtain prepare model by using linear regression model, we need to check assumptions. If the model provide each assumption, this model can be used. Hence, first we create lm model and then we plot the result of the linear regression. Two important assumptions are

  1. Normality of data
  2. Randomness of variance
#lm() function is used to construct linear model.
#since we add the pca results, we need to extract this variable from the dataset. 
#to create a model we use train data set.
LinearModel <- lm(price ~ . - PCA1 - PCA2 - PCA3 - PCA4, data = diamonds_train)
# to control the variables and p-value of regression results we can use summary() function
summary(LinearModel)

#after the creating of the regression model, by using visualization we can control basic two assumptions
ResidualPlot <- plot(LinearModel,1) #Residuals vs Fitted graph
QQplot <- plot(LinearModel,2) #Normal Q-Q plot

In this analysis, illustration of the regression is not given, but when the plots analyzed the increased variance is observed. By using log, 1/y, or square root transformation on the y (response variable). But to make this transformation we can use glm() function to construct generalized linear model

5.2 Generalized Linear Model

As we mentioned before, to use generalized linear model is not good way. As we have continuous response variable, to use generalized linear model (GLM). There are two different family and four link function. In this analysis, Gaussian is used as family function and identity is used as link function.

#By using train data and glm()  we can create regression model.
#But in contrast to normal implementation. In this process we need to starting point. 
#for this purpose we assign value into state for each variable.
#when the internet shows the 0.5 for the example, I use this value but it can be changed.

#Model1
GLModel1 <- glm(price ~ . - PCA1 - PCA2 - PCA3 - PCA4, data = diamonds_train, family = Gamma(link = "identity"), start = c(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5))
summary(GLModel1)
## 
## Call:
## glm(formula = price ~ . - PCA1 - PCA2 - PCA3 - PCA4, family = Gamma(link = "identity"), 
##     data = diamonds_train, start = c(0.5, 0.5, 0.5, 0.5, 0.5, 
##         0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5))
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.0946  -0.1909  -0.0552   0.1059  13.2252  
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.058e+04  6.210e+02 -33.136  < 2e-16 ***
## carat        7.527e+03  4.527e+01 166.288  < 2e-16 ***
## cut          1.872e+00  3.571e+00   0.524    0.600    
## color        3.494e+01  1.458e+00  23.958  < 2e-16 ***
## clarity      8.478e+01  1.174e+00  72.195  < 2e-16 ***
## depth        3.427e+02  9.935e+00  34.492  < 2e-16 ***
## table       -8.404e+00  9.984e-01  -8.417  < 2e-16 ***
## x            4.198e+03  8.287e+01  50.652  < 2e-16 ***
## y            9.321e+02  8.065e+01  11.558  < 2e-16 ***
## z           -9.164e+03  2.172e+02 -42.185  < 2e-16 ***
## cluster2     4.218e+01  8.130e+00   5.188 2.14e-07 ***
## cluster3    -6.723e+01  1.059e+01  -6.350 2.18e-10 ***
## cluster4     2.659e+01  1.957e+01   1.359    0.174    
## cluster5     2.226e+02  1.987e+01  11.202  < 2e-16 ***
## cluster6    -3.660e+01  8.347e+00  -4.385 1.16e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for Gamma family taken to be 0.08464894)
## 
##     Null deviance: 42124.3  on 42999  degrees of freedom
## Residual deviance:  3130.4  on 42985  degrees of freedom
## AIC: 679461
## 
## Number of Fisher Scoring iterations: 25

When we analyze the results, the p-value of the cut variable is too high. It is far away from the 0.05. For this reason, we need to remove this variable from the dataset and then we need to control the variables with the following lines.

#Second glm
#since the cut is not significant we remove from the data set.
GLModel2 <- glm(price ~ . - PCA1 - PCA2 - PCA3 - PCA4 - cut, data = diamonds_train, family = Gamma(link = "identity"),  start = c(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5))
summary(GLModel2)
## 
## Call:
## glm(formula = price ~ . - PCA1 - PCA2 - PCA3 - PCA4 - cut, family = Gamma(link = "identity"), 
##     data = diamonds_train, start = c(0.5, 0.5, 0.5, 0.5, 0.5, 
##         0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5))
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -1.13741  -0.19613  -0.05402   0.11302   2.99701  
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -23131.095    779.931 -29.658  < 2e-16 ***
## carat         7005.518     71.862  97.486  < 2e-16 ***
## color           34.633      2.015  17.188  < 2e-16 ***
## clarity         82.877      1.275  64.998  < 2e-16 ***
## depth          376.393     12.521  30.061  < 2e-16 ***
## table           -8.472      1.039  -8.150 3.74e-16 ***
## x             4628.373    112.732  41.057  < 2e-16 ***
## y             1169.083    100.364  11.648  < 2e-16 ***
## z            -9999.342    278.572 -35.895  < 2e-16 ***
## cluster2        43.784      6.733   6.503 7.97e-11 ***
## cluster3       -60.343     15.541  -3.883 0.000103 ***
## cluster4        61.880     26.743   2.314 0.020679 *  
## cluster5       256.168     28.196   9.085  < 2e-16 ***
## cluster6       -42.300      6.887  -6.142 8.21e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for Gamma family taken to be 0.1696263)
## 
##     Null deviance: 42124.3  on 42999  degrees of freedom
## Residual deviance:  3149.5  on 42986  degrees of freedom
## AIC: 679723
## 
## Number of Fisher Scoring iterations: 25

According to results, the model is significant. We can use this model. However, this model can be analyzed with more detailed way. Because, in this model, correlated variables are used in the model. This can be create conflict. Another options can be tried. Note that the another family function, Gaussian, can be used.

In addition to this best model, we can also create by using PCA values. For this following columns are used. Since, the PCA values are created by using variables of the data set, we only used this values to create different values.

#Model created by using PCA values
GLModel3 = glm(price ~ PCA1 + PCA2 + PCA3 + PCA4 , data = diamonds_train, family = Gamma(link = "sqrt"))
summary(GLModel3)
## 
## Call:
## glm(formula = price ~ PCA1 + PCA2 + PCA3 + PCA4, family = Gamma(link = "sqrt"), 
##     data = diamonds_train)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -1.13519  -0.28561  -0.06969   0.17035   2.10984  
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 55.92043    0.05615 995.881   <2e-16 ***
## PCA1        11.48255    0.02578 445.484   <2e-16 ***
## PCA2         2.76955    0.03353  82.608   <2e-16 ***
## PCA3         2.15136    0.03709  58.007   <2e-16 ***
## PCA4         0.38487    0.03955   9.732   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for Gamma family taken to be 0.1508171)
## 
##     Null deviance: 42124.3  on 42999  degrees of freedom
## Residual deviance:  5840.4  on 42995  degrees of freedom
## AIC: 706708
## 
## Number of Fisher Scoring iterations: 11

By using PCA components, we get new regression model. The last model provides that by using less component according to the real data set. For this reason, we expect that higher AIC value, which provide the definition of the best model in the AIC. The lowest AIC value is good.Now,there is two different glm model. By using AIC value,we can select one of them.

  • AIC value of the best model, GLModel2, is 679723
  • AIC value of the PCA model, GLModel2, is 706708.

As the lowest AIC value belongs to GLModel2, it is selected to test data set.

5.3 Classification And Regression Tree

Another supervised model is Classification and Regression Tree models, i.e., CART. It is a predicted model and it is a model that explains how an outcome variable’s values can be predicted based on other values. To create CART model, we use rpart() function which belongs to base R. To create a CARt model, again we use diamonds_train dataset. After the creating of the model, there are several visualization methods like prp(), fancyRpartPlot, and so on. In this example, fancyRpartPlot plot is used.

#creating of the CART model
ModelCART <- rpart(price~. -PCA1 - PCA2 - PCA3 - PCA4 - cluster, data=diamonds_train)
#visualization of CARt model 
fancyRpartPlot(ModelCART, 
               palettes=c( "Oranges"),
               main = "Decision Tree \n",
               type = 2,
               sub = " \n Ebru Geçici") 

By using obtain tree, we can examine nodes and define the divisions. According to the results, some variables are used as division parameters. In this output, we shows that y and clarity are main parameters or affecting values in the dataset. In other words, these variables are better features to reduce the variance in the dataset with default argument. This tree is obtain by using default variables, by changing the default values like cp, we can obtain different models and we can select the best of them. In this analysis, the default values are used.

Now we create two different models, one of them obtain from the glm function and the other one is obtained from the CART, tree analysis. However, we also need to test our models by using test data set for each model. First, We predict glm model, then we control the CART model. Then by using error, we can compare the result and select the best model.

#GLM prediction model
#To predict price we can use predict function with the best glm model
GLModel2Prediction <-predict(GLModel2, newdata=diamonds_test)

#PRS: Prediction R Square
GLModel2PRS <- 1 - (sum((GLModel2Prediction-diamonds_test$price)^2)/sum((diamonds_test$price-mean(diamonds_train$price))^2))
GLModel2PRS
## [1] 0.8121283
PredRealGLM <- as.data.frame(cbind(GLModel2Prediction, diamonds_test$price))
colnames(PredRealGLM) = c("Prediction", "Real")

PredRealGLM %>%
  ggplot(.,aes(x=Prediction, y=Real)) +
  geom_point(alpha = 0.7)  +
  theme_minimal()+
  geom_abline(intercept = 0, slope = 1, color="red",size=1) +
  labs(title = "Prediction vs Actual ",
       subtitle = st,
       x = "Predictions Values of GLM ",
       y = "Real Values of GLM")

According to the results, we obtain R Square value as 0.81. This means that our model explains our data set with 81%. We need to make same calculation for CART model.

#GLM prediction model
#To predict price we can use predict function with the best glm model
ModelCARTPrediction <-predict(ModelCART, newdata=diamonds_test)

#PRS: Prediction R Square
ModelCARTPRS <- 1 - (sum((ModelCARTPrediction-diamonds_test$price)^2)/sum((diamonds_test$price-mean(diamonds_train$price))^2))
ModelCARTPRS
## [1] 0.8768922

Also we obtain R square value for the CART model and the result shows that the CART model R Square value is higher than the GLM. This means that, the CART model explains the data set instead of GLM.

Moreover we can also visualize this result.

The R Square value of the glm model is smaller than the CART model. That is the CART model is the best model in the obtained model. In this model only y and clarity variables are used. This is also good for us because in the glm model almost all variables are located in the model. However, there are correlation between independent variables.

6 Conclusion

In this analysis, first, the data set is prepared for the analysis. In other words,

  • NA values,
  • Duplicated values,
  • Logic errors are defined.

If necessary, some of the data are removed from the dataset. After this process some analysis and prediction models are created. In the analysis part Exploratory Data Analysis, i.e., EDA, are made. By using these analyses, the some table outputs and visual representations are given. In this data set both numeric and ordered/categorical variables take place. For this reason, to get visualization of categorical variables, bar chart and this kind of plots are used. For numeric variables, the scatter plots are chosen. According to the this result, we can list following output:

  1. When we analyze each variable one by one, the price decrease from the best type of diamond to worse.
  2. The most preferable carat type is 0.3 carat. Moreover, the G type in color, ideal in cut, and SI1 in clarity are the most used in this data set.
  3. Price has right skewed histogram. When we implement log transformation into we do not obtain normal distribution of the price variable.

Moreover, by using unsupervised and supervised techniques, we create classification for diamond dataset by using k-means algorithm and PCA algorithm. Then the output of this classification methods are added into the model as a new variable. These models present unsupervised model. Then to make prediction about price, the supervised models are used. First, linear regression (LM) is implemented. Unless, the assumptions of the linear regression are not provided. For this reason generalized regression model (GLM) is implemented by using family Gamma and identity link function instead of making transformation of y in linear regression. Then the best model is obtained and this model is tested by using test dataset. After that, another model is implemented for the same data set with Classification And Regression Tree (CART). The results of the both models are compared by using R Square and the best model is selected. As a result of model prediction,

  1. Assumptions of the linear regression are not fulfilled.
  2. Generalized Regression model with Gamma family and Identity link function is implemented and the best model is obtain without cut variable
  3. Second model is implemented by using Classification And Regression Tree.
  4. According to the comparison of the GLM and CART models CART model is selected as best model with 0.8768922.