Mobile Price Classification using Tree-Based Algorithms - Bagging and Boosting

Introduction

As a mobile phone producer startup, one of the task at hand is to determine our brand’s price range. To complete this task, we are going to conduct a market price analysis. Using the data of competitor’s mobile phone specs and price, we

Using machine learning classification algorithms, we are going to clasify mobile phone price range according to its specs.

He does not know how to estimate price of mobiles his company creates. In this competitive mobile phone market you cannot simply assume things. To solve this problem he collects sales data of mobile phones of various companies.

Bob wants to find out some relation between features of a mobile phone(eg:- RAM,Internal Memory etc) and its selling price. But he is not so good at Machine Learning. So he needs your help to solve this problem.

In this problem you do not have to predict actual price but a price range indicating how high the price is

library(tidyverse)
library(GGally) # to make instant corr matrix in ggplot
library(caret) # classification and regression training
library(e1071) #naivebayes
library(xgboost)
# read data Customer_Behaviour.csv
train <- read.csv("data_input/train.csv", stringsAsFactors = T)
head(train)
##   battery_power blue clock_speed dual_sim fc four_g int_memory m_dep mobile_wt
## 1           842    0         2.2        0  1      0          7   0.6       188
## 2          1021    1         0.5        1  0      1         53   0.7       136
## 3           563    1         0.5        1  2      1         41   0.9       145
## 4           615    1         2.5        0  0      0         10   0.8       131
## 5          1821    1         1.2        0 13      1         44   0.6       141
## 6          1859    0         0.5        1  3      0         22   0.7       164
##   n_cores pc px_height px_width  ram sc_h sc_w talk_time three_g touch_screen
## 1       2  2        20      756 2549    9    7        19       0            0
## 2       3  6       905     1988 2631   17    3         7       1            1
## 3       5  6      1263     1716 2603   11    2         9       1            1
## 4       6  9      1216     1786 2769   16    8        11       1            0
## 5       2 14      1208     1212 1411    8    2        15       1            1
## 6       1  7      1004     1654 1067   17    1        10       1            0
##   wifi price_range
## 1    1           1
## 2    0           2
## 3    0           2
## 4    0           2
## 5    0           1
## 6    0           1
str(train)
## 'data.frame':    2000 obs. of  21 variables:
##  $ battery_power: int  842 1021 563 615 1821 1859 1821 1954 1445 509 ...
##  $ blue         : int  0 1 1 1 1 0 0 0 1 1 ...
##  $ clock_speed  : num  2.2 0.5 0.5 2.5 1.2 0.5 1.7 0.5 0.5 0.6 ...
##  $ dual_sim     : int  0 1 1 0 0 1 0 1 0 1 ...
##  $ fc           : int  1 0 2 0 13 3 4 0 0 2 ...
##  $ four_g       : int  0 1 1 0 1 0 1 0 0 1 ...
##  $ int_memory   : int  7 53 41 10 44 22 10 24 53 9 ...
##  $ m_dep        : num  0.6 0.7 0.9 0.8 0.6 0.7 0.8 0.8 0.7 0.1 ...
##  $ mobile_wt    : int  188 136 145 131 141 164 139 187 174 93 ...
##  $ n_cores      : int  2 3 5 6 2 1 8 4 7 5 ...
##  $ pc           : int  2 6 6 9 14 7 10 0 14 15 ...
##  $ px_height    : int  20 905 1263 1216 1208 1004 381 512 386 1137 ...
##  $ px_width     : int  756 1988 1716 1786 1212 1654 1018 1149 836 1224 ...
##  $ ram          : int  2549 2631 2603 2769 1411 1067 3220 700 1099 513 ...
##  $ sc_h         : int  9 17 11 16 8 17 13 16 17 19 ...
##  $ sc_w         : int  7 3 2 8 2 1 8 3 1 10 ...
##  $ talk_time    : int  19 7 9 11 15 10 18 5 20 12 ...
##  $ three_g      : int  0 1 1 1 1 1 1 1 1 1 ...
##  $ touch_screen : int  0 1 1 0 1 0 0 1 0 0 ...
##  $ wifi         : int  1 0 0 0 0 0 1 1 0 0 ...
##  $ price_range  : int  1 2 2 2 1 1 3 0 0 0 ...

Data Description

  • battery_power: total energy a battery can store in one time measured in mAh
  • blue: has bluetooth or not
  • clock_speed: speed at which microprocessor executes instructions
  • dual_sim: support dual sim or not
  • fc: front camera megapixel
  • four_g: support 4G or not
  • int_memory: Internal Memory in Gigabytes
  • m_dep: Mobile Depth in cm
  • mobile_wt: weight of mobile phone
  • n_cores: number of core processor
  • pc: primary camera megapixels
  • px_weight: pixel resolution height
  • px_width: pixel resolution width
  • ram: random access memory in megabytes
  • sc_h: screen height of mobile in cm
  • sc_w: screen width of mobile in cm
  • talk_time: longest time that a single battery charge will last when you are on call
  • three_g: support 3G or not
  • touch_screen: has touch screen or not
  • wifi: has wifi or not
  • price_range: this is the target variable with value of 0(low cost), 1(medium cost), 2(high cost) and 3(very high cost).
train <- train %>%
  mutate(across(c(blue, dual_sim, four_g, three_g, touch_screen, wifi, price_range), as.factor))
colSums(is.na(train))
## battery_power          blue   clock_speed      dual_sim            fc 
##             0             0             0             0             0 
##        four_g    int_memory         m_dep     mobile_wt       n_cores 
##             0             0             0             0             0 
##            pc     px_height      px_width           ram          sc_h 
##             0             0             0             0             0 
##          sc_w     talk_time       three_g  touch_screen          wifi 
##             0             0             0             0             0 
##   price_range 
##             0

EDA

  1. Target Class Distribution
ggplot(data = train, aes(x= price_range)) +
  geom_bar() +
  coord_flip() +
  labs(title = 'Target Class Distribution') +
  theme(plot.title = element_text(hjust = 0.5, face = "bold"))

attach(train)
par(mfrow = c(3,3))
hist(battery_power)
hist(clock_speed)
hist(fc)
hist(int_memory)
hist(m_dep)
hist(mobile_wt)
hist(n_cores)
hist(pc)
hist(ram)

attach(train)
par(mfrow = c(2,3))
plot(blue, main = "Bluetooth")
plot(wifi, main = "Wifi")
plot(dual_sim, main = "Dual SIM")
plot(four_g, main = "4G")
plot(three_g, main = "3G")
plot(touch_screen, main = "Touch Screen")

Training and Testing Model

Now we will make a Naive Bayes classsifier for our data as the benchmark model. We will make a 80/20 partitioning for the training and validation set. We used the createDataPartition from caret package to make a balanced partitioning. This is very important in this context because we are going to calculate the prior probabilities from the count of the training data. So, it should follow the proportion of the parent dataset.

# method 1: this method create a perfect partition of data
set.seed(7267166)
trainIndex <- createDataPartition(train$price_range, p=0.80)$Resample1
train <- train[trainIndex, ]
validation <- train[-trainIndex, ]
# method 2
# RNGkind(sample.kind = "Rounding")
# set.seed(100)
# index <- sample(x = nrow(train), size= nrow(train)*0.80)
# train <- train[index,] # subsetting data berdasarkan index data yang ada di variabel index 
# validation <- train[-index,] 
# Check the balance
print(table(train$price_range))
## 
##   0   1   2   3 
## 400 400 400 400

Naive Bayes

# Train Model
model_nb <- naiveBayes(formula = price_range~., data = train, laplace = 1)
# predict
nb_predClass <- predict(object = model_nb, newdata = validation, type = "class") 
head(nb_predClass)
## [1] 1 0 0 2 3 1
## Levels: 0 1 2 3
confusionMatrix(data = as.factor(nb_predClass), reference = as.factor(validation$price_range))
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1  2  3
##          0 71  9  0  0
##          1  4 62 12  0
##          2  0 20 53 13
##          3  0  0  6 65
## 
## Overall Statistics
##                                           
##                Accuracy : 0.7968          
##                  95% CI : (0.7481, 0.8399)
##     No Information Rate : 0.2889          
##     P-Value [Acc > NIR] : < 2.2e-16       
##                                           
##                   Kappa : 0.7293          
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: 0 Class: 1 Class: 2 Class: 3
## Sensitivity            0.9467   0.6813   0.7465   0.8333
## Specificity            0.9625   0.9286   0.8648   0.9747
## Pos Pred Value         0.8875   0.7949   0.6163   0.9155
## Neg Pred Value         0.9830   0.8776   0.9214   0.9467
## Prevalence             0.2381   0.2889   0.2254   0.2476
## Detection Rate         0.2254   0.1968   0.1683   0.2063
## Detection Prevalence   0.2540   0.2476   0.2730   0.2254
## Balanced Accuracy      0.9546   0.8049   0.8056   0.9040

Decision Tree

library(partykit)
## Loading required package: grid
## Loading required package: libcoin
## Loading required package: mvtnorm
model_dctree <- ctree(formula = price_range~., data = train)
model_dctree
## 
## Model formula:
## price_range ~ battery_power + blue + clock_speed + dual_sim + 
##     fc + four_g + int_memory + m_dep + mobile_wt + n_cores + 
##     pc + px_height + px_width + ram + sc_h + sc_w + talk_time + 
##     three_g + touch_screen + wifi
## 
## Fitted party:
## [1] root
## |   [2] ram <= 2235
## |   |   [3] ram <= 1105
## |   |   |   [4] px_height <= 1446
## |   |   |   |   [5] battery_power <= 1802
## |   |   |   |   |   [6] px_height <= 1222: 0 (n = 285, err = 2.5%)
## |   |   |   |   |   [7] px_height > 1222: 0 (n = 16, err = 31.2%)
## |   |   |   |   [8] battery_power > 1802
## |   |   |   |   |   [9] px_width <= 1532: 0 (n = 29, err = 10.3%)
## |   |   |   |   |   [10] px_width > 1532: 1 (n = 14, err = 21.4%)
## |   |   |   [11] px_height > 1446: 1 (n = 15, err = 20.0%)
## |   |   [12] ram > 1105
## |   |   |   [13] battery_power <= 1083
## |   |   |   |   [14] ram <= 1609
## |   |   |   |   |   [15] px_height <= 730
## |   |   |   |   |   |   [16] px_width <= 1643: 0 (n = 50, err = 0.0%)
## |   |   |   |   |   |   [17] px_width > 1643: 1 (n = 7, err = 42.9%)
## |   |   |   |   |   [18] px_height > 730
## |   |   |   |   |   |   [19] px_width <= 1164: 0 (n = 7, err = 14.3%)
## |   |   |   |   |   |   [20] px_width > 1164: 1 (n = 18, err = 11.1%)
## |   |   |   |   [21] ram > 1609
## |   |   |   |   |   [22] px_width <= 621: 0 (n = 7, err = 28.6%)
## |   |   |   |   |   [23] px_width > 621
## |   |   |   |   |   |   [24] px_height <= 1185
## |   |   |   |   |   |   |   [25] ram <= 1663: 1 (n = 9, err = 33.3%)
## |   |   |   |   |   |   |   [26] ram > 1663: 1 (n = 90, err = 4.4%)
## |   |   |   |   |   |   [27] px_height > 1185: 1 (n = 9, err = 44.4%)
## |   |   |   [28] battery_power > 1083
## |   |   |   |   [29] ram <= 1896
## |   |   |   |   |   [30] px_height <= 849
## |   |   |   |   |   |   [31] ram <= 1175: 1 (n = 11, err = 45.5%)
## |   |   |   |   |   |   [32] ram > 1175: 1 (n = 128, err = 7.0%)
## |   |   |   |   |   [33] px_height > 849
## |   |   |   |   |   |   [34] ram <= 1480: 1 (n = 38, err = 15.8%)
## |   |   |   |   |   |   [35] ram > 1480
## |   |   |   |   |   |   |   [36] battery_power <= 1656: 1 (n = 21, err = 47.6%)
## |   |   |   |   |   |   |   [37] battery_power > 1656: 2 (n = 13, err = 0.0%)
## |   |   |   |   [38] ram > 1896
## |   |   |   |   |   [39] px_width <= 1051
## |   |   |   |   |   |   [40] battery_power <= 1449: 1 (n = 17, err = 0.0%)
## |   |   |   |   |   |   [41] battery_power > 1449: 2 (n = 13, err = 46.2%)
## |   |   |   |   |   [42] px_width > 1051: 2 (n = 41, err = 4.9%)
## |   [43] ram > 2235
## |   |   [44] ram <= 3012
## |   |   |   [45] battery_power <= 1350
## |   |   |   |   [46] px_width <= 1351
## |   |   |   |   |   [47] ram <= 2651
## |   |   |   |   |   |   [48] battery_power <= 930: 1 (n = 31, err = 3.2%)
## |   |   |   |   |   |   [49] battery_power > 930
## |   |   |   |   |   |   |   [50] px_width <= 800: 1 (n = 14, err = 21.4%)
## |   |   |   |   |   |   |   [51] px_width > 800: 2 (n = 18, err = 0.0%)
## |   |   |   |   |   [52] ram > 2651: 2 (n = 52, err = 5.8%)
## |   |   |   |   [53] px_width > 1351
## |   |   |   |   |   [54] px_height <= 356: 2 (n = 22, err = 13.6%)
## |   |   |   |   |   [55] px_height > 356
## |   |   |   |   |   |   [56] battery_power <= 1021: 2 (n = 51, err = 3.9%)
## |   |   |   |   |   |   [57] battery_power > 1021: 2 (n = 17, err = 41.2%)
## |   |   |   [58] battery_power > 1350
## |   |   |   |   [59] px_height <= 642
## |   |   |   |   |   [60] battery_power <= 1441: 2 (n = 7, err = 28.6%)
## |   |   |   |   |   [61] battery_power > 1441
## |   |   |   |   |   |   [62] battery_power <= 1851: 2 (n = 58, err = 3.4%)
## |   |   |   |   |   |   [63] battery_power > 1851: 2 (n = 22, err = 36.4%)
## |   |   |   |   [64] px_height > 642
## |   |   |   |   |   [65] ram <= 2677: 2 (n = 34, err = 38.2%)
## |   |   |   |   |   [66] ram > 2677: 3 (n = 19, err = 0.0%)
## |   |   [67] ram > 3012
## |   |   |   [68] battery_power <= 909
## |   |   |   |   [69] px_height <= 575
## |   |   |   |   |   [70] ram <= 3731: 2 (n = 48, err = 16.7%)
## |   |   |   |   |   [71] ram > 3731: 3 (n = 19, err = 21.1%)
## |   |   |   |   [72] px_height > 575
## |   |   |   |   |   [73] ram <= 3187: 2 (n = 7, err = 42.9%)
## |   |   |   |   |   [74] ram > 3187: 3 (n = 49, err = 6.1%)
## |   |   |   [75] battery_power > 909
## |   |   |   |   [76] ram <= 3291
## |   |   |   |   |   [77] px_width <= 839: 2 (n = 17, err = 35.3%)
## |   |   |   |   |   [78] px_width > 839: 3 (n = 57, err = 10.5%)
## |   |   |   |   [79] ram > 3291: 3 (n = 220, err = 0.0%)
## 
## Number of inner nodes:    39
## Number of terminal nodes: 40
# prediksi kelas di data train
dctree_predClass_train <- predict(object = model_dctree, newdata = train, type = "response")
head(dctree_predClass_train)
## 1 2 3 5 6 7 
## 1 2 2 1 1 3 
## Levels: 0 1 2 3
# confusion matrix data train
confusionMatrix(data = as.factor(dctree_predClass_train),
                reference = train$price_range) 
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction   0   1   2   3
##          0 376  18   0   0
##          1  24 366  32   0
##          2   0  16 355  49
##          3   0   0  13 351
## 
## Overall Statistics
##                                           
##                Accuracy : 0.905           
##                  95% CI : (0.8896, 0.9189)
##     No Information Rate : 0.25            
##     P-Value [Acc > NIR] : < 2.2e-16       
##                                           
##                   Kappa : 0.8733          
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: 0 Class: 1 Class: 2 Class: 3
## Sensitivity            0.9400   0.9150   0.8875   0.8775
## Specificity            0.9850   0.9533   0.9458   0.9892
## Pos Pred Value         0.9543   0.8673   0.8452   0.9643
## Neg Pred Value         0.9801   0.9711   0.9619   0.9604
## Prevalence             0.2500   0.2500   0.2500   0.2500
## Detection Rate         0.2350   0.2288   0.2219   0.2194
## Detection Prevalence   0.2462   0.2637   0.2625   0.2275
## Balanced Accuracy      0.9625   0.9342   0.9167   0.9333
# prediksi kelas di data test
dctree_predClass <- predict(object = model_dctree, newdata = validation, type = "response")
head(dctree_predClass)
##  5 10 16 25 34 40 
##  1  0  0  1  3  1 
## Levels: 0 1 2 3
# confusion matrix data test
confusionMatrix(data = as.factor(dctree_predClass),
                reference = validation$price_range) 
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1  2  3
##          0 72  2  0  0
##          1  3 86  7  0
##          2  0  3 63  9
##          3  0  0  1 69
## 
## Overall Statistics
##                                          
##                Accuracy : 0.9206         
##                  95% CI : (0.8851, 0.948)
##     No Information Rate : 0.2889         
##     P-Value [Acc > NIR] : < 2.2e-16      
##                                          
##                   Kappa : 0.8938         
##                                          
##  Mcnemar's Test P-Value : NA             
## 
## Statistics by Class:
## 
##                      Class: 0 Class: 1 Class: 2 Class: 3
## Sensitivity            0.9600   0.9451   0.8873   0.8846
## Specificity            0.9917   0.9554   0.9508   0.9958
## Pos Pred Value         0.9730   0.8958   0.8400   0.9857
## Neg Pred Value         0.9876   0.9772   0.9667   0.9633
## Prevalence             0.2381   0.2889   0.2254   0.2476
## Detection Rate         0.2286   0.2730   0.2000   0.2190
## Detection Prevalence   0.2349   0.3048   0.2381   0.2222
## Balanced Accuracy      0.9758   0.9502   0.9191   0.9402

Pruning

Pruning paramater:

  • mincriterion: Nilai 1-\(\alpha\). Saat mincriterion 0.95, P-value harus < 0.05 untuk suatu node dapat membuat cabang. (default: 0.95)
  • minsplit: Jumlah minimal observasi di tiap cabang setelah pemisahan. Bila tidak terpenuhi, tidak dilakukan percabangan. (default: 20)
  • minbucket: Jumlah minimal observasi di terminal node. Bila tidak terpenuhi, tidak dilakukan percabangan. (default: 7)
# # parameter pruning
# model_dctree_new <- ctree(formula = price_range~., data = train,
#                        control = ctree_control(mincriterion = 0.07,
#                                                minsplit = 25,
#                                                minbucket = 7))

Random Forest

Random Forest is an ensemble technique that is a tree-based algorithm. It is a bagging technique that contains a number of decision trees on various subsets of the given dataset and takes the average to improve the predictive accuracy of that dataset. Instead of relying on one decision tree, the random forest takes the prediction from each tree and based on the majority votes of predictions, and it predicts the final output. The greater number of trees in the forest leads to higher accuracy and prevents the problem of overfitting.

library(randomForest)
## randomForest 4.6-14
## Type rfNews() to see new features/changes/bug fixes.
## 
## Attaching package: 'randomForest'
## The following object is masked from 'package:dplyr':
## 
##     combine
## The following object is masked from 'package:ggplot2':
## 
##     margin

By default, the number of decision trees in the forest is 500 and the number of features used as potential candidates for each split is 3. The model will automatically attempt to classify each of the samples in the Out-Of-Bag dataset and display a confusion matrix with the results.

Using tuneRF from randomForest package, we can find the best mtry, the number of variables randomly sampled

rownames(train) <- NULL
set.seed(123)
mtry <- tuneRF(train[,-21], train[,21], stepFactor=1.5)
## mtry = 4  OOB error = 17.5% 
## Searching left ...
## mtry = 3     OOB error = 23.69% 
## -0.3535714 0.05 
## Searching right ...
## mtry = 6     OOB error = 13.44% 
## 0.2321429 0.05 
## mtry = 9     OOB error = 11.38% 
## 0.1534884 0.05 
## mtry = 13    OOB error = 10.69% 
## 0.06043956 0.05 
## mtry = 19    OOB error = 11.5% 
## -0.07602339 0.05

# stepFactor is a magnitude by which the chosen mtry gets deflated or inflated.
best.m <- mtry[mtry[, 2] == min(mtry[, 2]), 1]
print(mtry)
##        mtry OOBError
## 3.OOB     3 0.236875
## 4.OOB     4 0.175000
## 6.OOB     6 0.134375
## 9.OOB     9 0.113750
## 13.OOB   13 0.106875
## 19.OOB   19 0.115000
print(best.m)
## [1] 13

The optimum mtry is 9. Now, we can proceed to model fitting.

set.seed(123)
model_rf <- randomForest(
  price_range ~ .,
  data=train,
  mtry = 9
)
model_rf
## 
## Call:
##  randomForest(formula = price_range ~ ., data = train, mtry = 9) 
##                Type of random forest: classification
##                      Number of trees: 500
## No. of variables tried at each split: 9
## 
##         OOB estimate of  error rate: 9.75%
## Confusion matrix:
##     0   1   2   3 class.error
## 0 376  24   0   0      0.0600
## 1  27 346  27   0      0.1350
## 2   0  27 349  24      0.1275
## 3   0   0  27 373      0.0675

\[Accuracy = 100 - 9.62 = 90.38 \%\]

# prediksi kelas di data test
rf_predClass <- predict(object = model_rf, newdata = validation, type = "response")
head(rf_predClass)
##  5 10 16 25 34 40 
##  1  0  0  1  3  2 
## Levels: 0 1 2 3
# confusion matrix data test
confusionMatrix(data = as.factor(rf_predClass),
                reference = validation$price_range) 
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1  2  3
##          0 75  0  0  0
##          1  0 91  0  0
##          2  0  0 71  0
##          3  0  0  0 78
## 
## Overall Statistics
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9884, 1)
##     No Information Rate : 0.2889     
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##                                      
##  Mcnemar's Test P-Value : NA         
## 
## Statistics by Class:
## 
##                      Class: 0 Class: 1 Class: 2 Class: 3
## Sensitivity            1.0000   1.0000   1.0000   1.0000
## Specificity            1.0000   1.0000   1.0000   1.0000
## Pos Pred Value         1.0000   1.0000   1.0000   1.0000
## Neg Pred Value         1.0000   1.0000   1.0000   1.0000
## Prevalence             0.2381   0.2889   0.2254   0.2476
## Detection Rate         0.2381   0.2889   0.2254   0.2476
## Detection Prevalence   0.2381   0.2889   0.2254   0.2476
## Balanced Accuracy      1.0000   1.0000   1.0000   1.0000
plot(model_rf)

importance(model_rf)
##               MeanDecreaseGini
## battery_power       115.389001
## blue                  1.760175
## clock_speed          14.425589
## dual_sim              1.765545
## fc                   14.772111
## four_g                2.200327
## int_memory           22.546449
## m_dep                14.042289
## mobile_wt            26.963784
## n_cores              12.686672
## pc                   17.702943
## px_height            69.430194
## px_width             69.487765
## ram                 759.647748
## sc_h                 16.336939
## sc_w                 16.534080
## talk_time            17.686105
## three_g               1.844758
## touch_screen          2.495804
## wifi                  1.515420

LEast important features: blue, dual_sim, three_g, wifi, four_g, touch_screen.

partialPlot(model_rf, train, ram, "3")

partialPlot(model_rf, train, battery_power, "3")

The inference should be:

  • if the RAM of a mobile phone is more than 1.25GB, then it has a higher chance of classifying into class 3 (very high cost).
  • if the battery power is more than 1,250 mAH, then it has a higher chance of classifying into class 3 very high cost).
train_reduced <- train %>%
  select(-c(blue, dual_sim, three_g, wifi, four_g, touch_screen))

validation_reduced <- validation %>%
  select(-c(blue, dual_sim, three_g, wifi, four_g, touch_screen))
set.seed(123)
model_rf_2 <- randomForest(
  price_range ~ .,
  data=train_reduced,
  mtry = 9
)
set.seed(123)
mtry <- tuneRF(train_reduced[,-15], train_reduced[,15], stepFactor=1.5)
## mtry = 3  OOB error = 15.69% 
## Searching left ...
## mtry = 2     OOB error = 21.44% 
## -0.3665339 0.05 
## Searching right ...
## mtry = 4     OOB error = 12.81% 
## 0.1832669 0.05 
## mtry = 6     OOB error = 10.88% 
## 0.1512195 0.05 
## mtry = 9     OOB error = 10.56% 
## 0.02873563 0.05

best.m <- mtry[mtry[, 2] == min(mtry[, 2]), 1]
print(mtry)
##       mtry OOBError
## 2.OOB    2 0.214375
## 3.OOB    3 0.156875
## 4.OOB    4 0.128125
## 6.OOB    6 0.108750
## 9.OOB    9 0.105625
print(best.m)
## [1] 9
model_rf_2
## 
## Call:
##  randomForest(formula = price_range ~ ., data = train_reduced,      mtry = 9) 
##                Type of random forest: classification
##                      Number of trees: 500
## No. of variables tried at each split: 9
## 
##         OOB estimate of  error rate: 9.62%
## Confusion matrix:
##     0   1   2   3 class.error
## 0 382  18   0   0       0.045
## 1  28 348  24   0       0.130
## 2   0  28 348  24       0.130
## 3   0   0  32 368       0.080
importance(model_rf_2)
##               MeanDecreaseGini
## battery_power       132.366696
## clock_speed           8.941181
## fc                   10.786739
## int_memory           16.061521
## m_dep                10.638421
## mobile_wt            20.812637
## n_cores               8.403486
## pc                   12.671719
## px_height            81.013497
## px_width             80.097889
## ram                 781.310306
## sc_h                 11.447211
## sc_w                 11.681159
## talk_time            13.008850

XGBoost Classification

The XGBoost library implements the gradient boosting decision tree algorithm. This algorithm goes by lots of different names such as gradient boosting, multiple additive regression trees, stochastic gradient boosting or gradient boosting machines. Boosting means combining a learning algorithm in series to achieve a strong learner from many sequentially connected weak learners. In case of gradient boosted decision trees algorithm, the weak learners are decision trees. Each tree attempts to minimize the errors of previous tree.

Gradient boosting re-defines boosting as a numerical optimization problem where the objective is to minimize the loss function of the model by adding weak learners using gradient descent. Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. As gradient boosting is based on minimizing a loss function, different types of loss functions can be used resulting in a flexible technique that can be applied to regression, multi-class classification, etc.

The name XGBoost (Extreme Gradient Boosting) actually refers to the engineering goal to push the limit of computations resources for boosted tree algorithms. Which is the reason why many people use xgboost. It uses sequentially-built shallow decision trees to provide accurate results and a highly-scalable training method that avoids overfitting.

The advantage of XGBoost compared to Random Forest:

rownames(train_reduced) <- NULL
train_reduced
##      battery_power clock_speed fc int_memory m_dep mobile_wt n_cores pc
## 1              842         2.2  1          7   0.6       188       2  2
## 2             1021         0.5  0         53   0.7       136       3  6
## 3              563         0.5  2         41   0.9       145       5  6
## 4             1821         1.2 13         44   0.6       141       2 14
## 5             1859         0.5  3         22   0.7       164       1  7
## 6             1821         1.7  4         10   0.8       139       8 10
## 7             1445         0.5  0         53   0.7       174       7 14
## 8              509         0.6  2          9   0.1        93       5 15
## 9              769         2.9  0          9   0.1       182       5  1
## 10            1520         2.2  5         33   0.5       177       8 18
## 11             803         2.1  7         17   1.0       198       4 11
## 12            1866         0.5 13         52   0.7       185       1 17
## 13             775         1.0  3         46   0.7       159       2 16
## 14             838         0.5  1         13   0.1       196       8  4
## 15             595         0.9  7         23   0.1       121       3 17
## 16            1131         0.5 11         49   0.6       101       5 18
## 17             682         0.5  4         19   1.0       121       4 11
## 18            1709         2.1  1         13   1.0       156       2  2
## 19            1949         2.6  4         47   0.3       199       4  7
## 20            1602         2.8  4         38   0.7       114       3 20
## 21             503         1.2  5          8   0.4       111       3 13
## 22             519         1.6  7         51   0.3       132       4 19
## 23             956         0.5  1         41   1.0       143       7  6
## 24            1453         1.6 12         52   0.3        96       2 18
## 25            1319         0.9  3         41   0.9       107       1 18
## 26            1310         2.2  0         51   0.6       100       4  0
## 27             725         1.3 16         60   0.4       160       8 17
## 28            1725         1.6  6          6   0.5       119       2 18
## 29             790         2.0 16         11   0.3        87       6 17
## 30             560         0.5 15         50   0.3       159       2 20
## 31            1347         2.9  5         44   0.6       132       1 14
## 32            1253         0.5  5          5   0.2       152       2 19
## 33            1656         1.0  5         34   0.1       166       3  7
## 34            1195         2.8  1         20   0.8       110       2 14
## 35            1514         2.9  0         27   0.2       118       3  1
## 36            1723         1.1  1         42   1.0       164       8 14
## 37            1054         1.8  3         40   0.8       196       7 10
## 38             596         2.1  9         64   0.8       111       8 15
## 39            1547         3.0  2         14   0.7       198       3 19
## 40            1760         1.4  5         63   0.8       127       8 19
## 41            1654         1.5  0         43   0.3       109       2  0
## 42            1073         0.5  0         51   0.5       145       7  0
## 43             823         2.7 13         60   0.5       148       8 19
## 44             987         1.3  0         61   0.4       107       3  9
## 45            1757         0.5  8         49   0.5       180       6 14
## 46             799         2.3  1         63   0.8       144       8  6
## 47            1156         1.2  0         50   0.8       159       2  0
## 48             702         2.6  2          9   0.7       141       3  3
## 49             616         1.9 13         44   0.8        81       3 17
## 50            1866         1.4  0         30   0.5       182       3  0
## 51            1242         1.1  0         10   0.6       165       2  1
## 52            1166         1.5  0         43   0.8        80       4  1
## 53            1448         0.5  6         45   0.8       138       7 11
## 54            1407         2.4  1         22   0.7       104       4  4
## 55             605         1.0  8          9   0.1       142       3 16
## 56            1038         1.2  3         43   0.7       141       1  5
## 57             797         2.9  4         38   0.5        90       4  6
## 58             819         0.6  8         42   0.9       188       6 20
## 59            1114         2.8  4          9   0.4       197       3 15
## 60            1589         0.6  0         58   0.9        85       7  7
## 61            1510         0.9  2         45   0.9       180       5  3
## 62            1008         0.8 11         61   1.0       114       3 15
## 63            1127         2.9  5         57   0.8       163       1 13
## 64            1412         2.4  5         25   0.8        96       8 17
## 65            1496         2.0  4         42   0.5       182       5  5
## 66            1309         1.1  0         33   0.5       100       4 14
## 67            1724         2.0  2         57   0.5       177       3  5
## 68            1977         2.0  7         54   1.0       171       7 18
## 69             885         2.3  0         15   0.4       103       7  0
## 70            1137         1.0 18          7   1.0       196       3 19
## 71            1355         2.3 10         23   0.2       132       5 16
## 72            1665         0.5  3         60   0.2       194       6 18
## 73            1883         0.5  3         10   0.3       146       6 13
## 74            1732         1.1  4         12   0.8       119       8  9
## 75             543         0.5  0         57   0.7       192       5  4
## 76            1939         2.8  0         55   0.6       199       4 10
## 77             553         2.4  1          8   0.5       121       7  3
## 78             832         0.7  2         39   0.7       103       4  3
## 79            1661         1.9  5         23   1.0       172       5 10
## 80            1135         2.0  0         46   0.3        83       1  4
## 81             617         2.3  0         32   0.5       199       7 20
## 82             867         1.4  0          4   0.7       135       6  1
## 83            1310         1.1 13          7   1.0       194       5 14
## 84            1804         0.5  9         22   0.6       153       7 19
## 85            1900         2.3  1         18   0.2        89       7  9
## 86             909         1.4  0         53   0.3        82       8  0
## 87            1084         2.6  3         40   0.9       107       3 11
## 88            1308         1.3  8         46   0.9       199       1 12
## 89            1778         0.5  0         15   0.5       102       3  0
## 90            1701         1.6  3         10   0.3       194       8  7
## 91             772         2.4  1         10   0.5        80       4  2
## 92             825         0.5  7         52   0.9       130       3 10
## 93            1166         2.0  3         25   0.8       198       5  4
## 94            1659         2.8  5         16   0.6        89       1 20
## 95             826         2.4  0         58   0.3        88       4  1
## 96            1957         0.9 11         31   0.6       181       3 16
## 97            1571         1.4  0         19   0.1       100       6  1
## 98            1790         2.5  9          2   0.1       184       1 14
## 99             668         2.9  7         30   0.2       195       8 12
## 100           1652         1.1  1         36   0.7        89       4  5
## 101           1213         0.5  0         16   0.5       108       5  0
## 102           1272         2.5  0         22   0.9       133       6  8
## 103            866         0.5  5         46   0.5       179       8  7
## 104            523         2.6  1         14   0.1       155       8  2
## 105           1753         0.5  0         31   0.1       134       1 18
## 106           1218         1.8  1         14   0.5       147       3  4
## 107           1537         2.5  4         43   0.3       137       1  5
## 108           1678         2.1  3         64   0.9       134       3 11
## 109           1101         1.8  0         31   0.2       148       2  7
## 110           1461         2.4  3         40   0.5       141       7  4
## 111           1216         3.0  4         47   0.4       199       3  7
## 112            506         1.6  6         41   0.8       159       1  7
## 113            742         2.2  4         56   0.2       190       7 14
## 114           1692         2.1  3         61   0.9       176       6 12
## 115           1485         1.0  2         32   0.7        84       2  9
## 116           1692         2.3  4         46   0.9       139       1  6
## 117            637         2.3  3         60   0.8        97       4  4
## 118           1224         0.5 16          6   0.4       109       6 18
## 119           1356         2.8  0         29   0.5       198       6  0
## 120           1840         0.5 12         34   0.7       142       1 16
## 121           1481         0.7  4         37   0.6        96       5 13
## 122            961         1.5 10         51   0.3       152       6 16
## 123           1193         0.9 11         26   0.5       166       1 20
## 124           1441         2.1 11          3   0.1       118       8 17
## 125            748         1.7  0         27   0.9       150       8  4
## 126           1569         2.8 17         44   0.3       110       4 20
## 127           1270         0.6  2         32   0.5       101       7  6
## 128           1854         3.0  2          8   0.1       101       5  5
## 129            625         1.2  5         10   0.2       183       4  8
## 130            946         1.7  8         25   0.8       143       6 15
## 131            685         0.5  2         29   0.8        93       7 10
## 132           1949         1.3  0         23   0.1       101       5  1
## 133            801         2.2  2         63   0.4       165       7  4
## 134            703         2.7 12         30   0.3       113       1 20
## 135           1165         2.4 11         41   0.2       178       6 12
## 136           1082         0.8  0         38   0.1       198       8  5
## 137            959         2.6 15         20   1.0        84       4 19
## 138           1502         1.2  0         54   0.2       138       5 12
## 139           1380         2.1  1          4   0.8        92       3 13
## 140           1266         0.5  0          5   0.1        95       6  9
## 141           1934         2.6  0         46   0.1       103       1 17
## 142           1905         0.5  3          6   0.5       151       1  5
## 143           1831         1.4  2         43   0.9       150       5  5
## 144            596         1.4  8         46   0.6       165       1 18
## 145           1562         1.3  1          7   0.2       190       5 15
## 146           1490         0.5  4         64   0.3       150       8  8
## 147           1523         2.4  5         57   0.2       117       2  6
## 148            640         0.6  0         20   0.1       163       4 15
## 149           1526         2.1  1         23   0.2       117       7  8
## 150           1989         2.5  0         41   0.8        94       3 13
## 151           1308         1.9  0         61   0.7       106       3  7
## 152           1703         0.5  0         22   0.6       192       1  0
## 153           1445         2.4  7         20   0.4       173       7 15
## 154           1087         1.3  0         16   0.3       166       3  9
## 155            671         0.9 10         30   0.7       105       7 17
## 156           1472         2.3  6         61   0.1       168       6 19
## 157            765         0.5  0         47   0.3       115       8  1
## 158           1642         0.5 16          8   0.3       171       6 17
## 159           1265         1.5  7         49   0.7       182       5 16
## 160            664         1.1  1         51   0.1       110       7  4
## 161           1277         0.5  0         35   0.4       142       6  6
## 162           1539         2.6  0         41   0.1       159       8  0
## 163           1827         1.7  4         39   0.5       128       5 16
## 164           1987         0.5  3         53   1.0       181       3  7
## 165           1154         1.2  0         49   0.2       181       8  2
## 166           1886         0.5  0         24   0.4       112       3  1
## 167           1551         1.1  4         51   0.1        88       5  6
## 168            850         1.6  6         29   0.5       133       5 19
## 169           1225         2.1  8         13   0.5       116       4 12
## 170           1452         0.5  1         25   0.9       123       6 11
## 171           1708         2.4 18         49   0.1       109       1 19
## 172           1151         2.9  0         31   0.6       128       8  1
## 173           1578         1.9  2         37   0.1       138       4 19
## 174           1689         1.8 17         24   0.3       127       3 18
## 175           1715         1.0  4         31   0.5        83       8 17
## 176           1523         1.8  6         11   0.1       129       1  8
## 177           1133         1.4  3         42   0.1       199       5 12
## 178           1330         0.5  0         38   1.0       183       7  1
## 179           1799         0.5  0         49   0.4       107       4  2
## 180            633         2.2  0         49   0.1       139       8  1
## 181            724         2.7  0         44   0.5       187       7  0
## 182            822         0.5  0         51   0.3       157       6 20
## 183           1918         1.9  2         10   0.6       110       5 10
## 184           1891         2.8  7         37   0.4       129       7 14
## 185           1373         1.0  2         50   0.2       127       7 12
## 186            862         0.8  3         23   0.4       165       2 15
## 187           1273         1.0  3         48   0.4        99       4 15
## 188           1617         1.3  3         30   0.7       105       2  4
## 189            893         2.5  0         55   0.6       183       2  1
## 190           1210         0.5  0         63   0.1        93       5  0
## 191            708         2.7  0         63   1.0       195       8  7
## 192            835         1.1  2          7   0.2       101       4  7
## 193           1450         2.7  9         19   0.6       131       1 17
## 194            973         1.5  5         57   0.2       190       7 16
## 195            601         1.4  4          4   0.1       154       4 13
## 196            507         0.5  1         32   0.5       141       7 11
## 197            564         1.8  0         51   0.5       168       3  1
## 198           1559         1.6  6          6   0.5       162       6 17
## 199           1431         0.5  0         56   0.6       191       1  1
## 200           1722         2.6  8         18   0.8       177       2 16
## 201           1992         1.6  5         57   1.0       100       2 15
## 202           1876         0.5 10         40   0.3        86       6 14
## 203           1054         1.1 10         38   0.4       136       6 12
## 204           1283         1.6  4         47   0.9        93       5  5
## 205           1066         0.6  2          7   0.9       114       7 18
## 206           1053         0.8  7         45   0.3       103       1 12
## 207           1611         0.5 11          3   0.6        98       3 14
## 208            793         1.0  5         30   0.9       133       2 15
## 209           1220         2.9  3         62   0.5       125       4  8
## 210           1281         2.8  3          8   0.3       144       4 19
## 211            538         0.8 12          2   0.8       177       7 13
## 212            823         0.5  0         39   0.4       187       8  4
## 213           1733         2.8  3         36   1.0       119       6 20
## 214            659         2.7  6          6   0.5       110       3  7
## 215           1172         1.9  0         27   0.7       164       1  3
## 216            615         0.5  0         42   0.6       163       6  4
## 217           1893         2.1  8         63   0.2       134       2 11
## 218           1563         1.7 10         16   0.1       151       7 13
## 219           1995         1.9  6          9   0.8       139       6 18
## 220           1517         0.5  6         48   0.6       143       5 14
## 221            832         1.6  1         34   0.3       108       5  5
## 222           1162         2.0  0         56   0.7       114       7  1
## 223           1329         1.0  1         52   0.9       181       6  2
## 224            928         0.5 11         56   0.7        80       8 13
## 225           1656         0.5  8         38   0.1       105       7 17
## 226           1937         1.7 17         58   0.6       189       1 19
## 227            977         1.1  2         57   0.6       181       6  7
## 228            768         0.9 16         56   0.1       113       1 20
## 229           1464         0.5 11         12   0.9       117       2 14
## 230           1177         0.9  1         29   0.4       191       2  8
## 231           1348         2.0 18         52   0.3        98       3 20
## 232           1956         2.2  0         37   0.5        87       3  2
## 233           1751         0.5  6          8   0.6       178       6 20
## 234           1530         2.0 11         42   0.8       177       6 13
## 235           1997         1.8  1         40   0.5       185       4  8
## 236           1414         1.5  5         44   0.5       180       2  6
## 237            894         0.5  1         50   0.5       199       3 14
## 238            645         1.2  6         41   0.7       184       8 17
## 239           1163         2.8  0         49   0.6       155       2  0
## 240           1126         0.7  3         49   0.9       165       8 17
## 241           1648         0.7  6         64   0.8       170       4  9
## 242            508         0.8  7         42   0.3        94       1  8
## 243            728         1.7  0          5   0.6       101       2 20
## 244           1379         2.1 14         63   1.0        83       1 17
## 245           1698         2.1  5         18   0.9       160       6 20
## 246            920         2.1  5         25   0.4       115       7 14
## 247           1995         0.5  6         26   0.2       129       1 15
## 248            662         0.5  5         20   0.8       138       1 10
## 249           1867         2.3  0          9   0.1       191       6  3
## 250           1730         2.0  0         20   0.1       143       4 10
## 251           1723         1.0  1          7   0.6       126       3 17
## 252           1882         1.3  8         47   0.2       140       2 13
## 253            668         2.3  2         48   0.3       114       1 19
## 254           1391         1.4  2         36   0.8       182       1  6
## 255           1560         2.3  0         22   0.8       169       7  1
## 256            821         1.8  5         54   0.1       109       3  9
## 257            811         2.4  5          2   0.3       106       6 13
## 258           1989         2.9  8         17   0.1       193       8 15
## 259           1034         2.7  6         37   0.7       120       7 20
## 260            618         2.1  2          6   0.3        86       1  4
## 261            667         0.6  6          3   0.7       185       6 13
## 262            869         1.0  7         57   0.6       118       1 15
## 263            635         0.5  2         42   0.4       149       3  4
## 264            609         3.0 15         44   0.3       117       1 16
## 265           1604         1.7  2         33   1.0       191       8  5
## 266           1182         0.5  7          8   0.5       138       8 16
## 267           1610         2.9  7         64   0.9       181       4 12
## 268           1828         2.1  0         17   0.9       199       1  1
## 269            880         2.0  1         50   0.5       194       7  3
## 270           1394         0.9 10         62   0.3       175       5 13
## 271           1960         2.3  1         61   0.3        86       7  3
## 272            809         0.8 10         21   0.9        91       7 19
## 273            899         0.5 13         57   0.9       172       2 18
## 274           1976         0.7  4         32   0.8       171       2  7
## 275            879         1.5  0         21   0.8       187       4  2
## 276            916         2.6  0         36   0.3       195       4 12
## 277            763         1.7  8          2   0.5        92       8 14
## 278            508         1.6  0          9   0.4       162       6  2
## 279            825         2.9  0         34   0.8        99       8 17
## 280           1864         1.4  8         46   0.7       178       1 12
## 281           1108         2.9 10         52   0.1       183       2 12
## 282           1011         1.1  4         36   0.2       131       7 18
## 283           1703         1.5 17         55   0.7       138       5 19
## 284           1067         1.7  0         58   0.5       135       7  0
## 285           1334         2.4  5         42   0.6       115       5 14
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## 388           1661         0.8  1          6   1.0       109       7  2
## 389           1438         2.2  0         32   0.6       176       7  6
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## 584           1303         1.0  0         15   0.1       138       3 10
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## 586           1972         1.2  0         37   0.7       102       2  1
## 587           1159         2.8  5         18   0.5        83       1  7
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## 594           1276         2.9  2         53   0.3       130       2  3
## 595           1331         1.6  0         16   0.5       165       2  7
## 596           1620         2.6  6          5   0.9       186       6 18
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## 598           1092         2.7  8         45   0.8       185       1 10
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## 600            545         0.5  0         11   0.3       131       3  1
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## 603           1030         1.6  3         58   0.2       138       6 20
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## 605            558         2.8  1         49   0.8       142       5  4
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## 619           1160         1.1  1         63   0.7       166       3  2
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## 622            774         0.6  2          8   0.4       136       1  4
## 623            819         2.5  0         11   0.2        83       8  1
## 624           1042         2.2 15         11   0.6       139       5 16
## 625           1592         2.1  6         50   0.1       181       2 17
## 626           1067         1.8 11         14   0.2       146       1 15
## 627            915         1.1  6         56   0.6       187       8 17
## 628           1614         1.9  8         55   0.3       188       3 17
## 629            911         2.9  0         46   0.1       181       5  1
## 630           1520         0.5  1         24   0.7       199       5  8
## 631           1647         1.4  3         38   0.3       102       6 15
## 632           1347         0.9  8         53   0.8       108       6 15
## 633            967         1.6  4         25   0.6        92       8  9
## 634           1442         2.2  0         56   0.7       145       5  3
## 635           1204         2.6  0         10   0.3       199       7  3
## 636           1320         2.1  0         41   0.6       144       7  2
## 637           1800         0.7  2          8   0.4       100       7  6
## 638           1567         2.1  3         22   0.8       153       3 19
## 639           1439         0.9 12         20   0.8       147       1 17
## 640           1422         1.0 15          3   0.1       112       1 17
## 641           1591         2.2  4         52   1.0       161       3 16
## 642            880         1.7  2          6   0.6       198       4 19
## 643           1929         0.5  0         19   1.0       163       6  2
## 644           1642         1.6  0          5   0.4       102       5  2
## 645           1511         2.5  3         21   0.2       114       3 18
## 646           1852         1.4  4         10   0.7       196       1 18
## 647            972         1.3  8         44   0.7        82       7 10
## 648            691         2.7  4         33   0.9       108       4  8
## 649            807         0.5  9          8   0.3       158       5 20
## 650           1113         0.6  0         25   0.3       145       1 10
## 651            721         1.3 10         16   0.9       125       1 18
## 652           1188         1.5  0         54   0.3       198       6  2
## 653           1512         0.5  1         52   0.1       131       1 14
## 654            918         0.9  1          8   0.1       123       7  5
## 655           1320         0.5  7         48   0.2       168       5 16
## 656           1236         0.9  2         57   0.1       188       1 14
## 657           1387         0.5  0         61   0.5        98       3  0
## 658           1883         2.7  6         30   0.5        95       2 10
## 659            839         2.0  0         14   0.4       175       7  3
## 660           1593         1.0  0         52   0.7       130       8  0
## 661           1722         1.0  7         25   0.8        88       6 19
## 662           1954         0.6  8          7   0.9       126       3  9
## 663           1628         0.5  5         33   0.9       101       6  8
## 664           1965         0.5  6          4   1.0        84       7  9
## 665            833         2.2  4         27   0.9       124       1  6
## 666           1808         0.5  1         31   0.5        92       2 16
## 667           1860         0.5  2         55   0.7        83       6  3
## 668           1368         1.6  4          9   0.1       115       7  7
## 669           1161         1.3  0         21   0.6       126       8 12
## 670            902         0.6  0         63   0.7       122       5 14
## 671            854         0.5  2         30   0.5       125       8  5
## 672           1184         1.5  1         48   0.6       191       1  6
## 673           1973         1.6  5         39   0.8       152       4  6
## 674            510         1.7  3         35   0.8       120       3  6
## 675            966         2.0  8         34   0.5       118       2 10
## 676           1438         1.8  3         16   0.6       169       8  7
## 677           1907         2.0 13         55   0.9       105       7 15
## 678           1489         2.4  8         32   0.6       200       1 16
## 679           1286         2.3 10         16   0.1        98       2 18
## 680            840         0.6  0         63   0.8       137       7 19
## 681            757         2.3  5         58   0.2       163       4  7
## 682            814         1.1  2         41   0.8       177       2 10
## 683           1195         1.1  2          8   0.9        80       6  6
## 684            767         0.8  7         10   0.5       107       4 12
## 685           1068         1.3  6         52   0.1        97       8  7
## 686           1550         2.5  6         21   0.8       133       2 11
## 687           1878         0.5  9         29   0.6        85       8 19
## 688            623         2.0  2          6   0.1       127       3  4
## 689           1829         0.5  0         15   0.4       160       5  7
## 690            525         2.4 11         35   0.8       190       7 20
## 691           1065         0.5  0         14   0.7        89       2  2
## 692           1425         1.2  9         35   0.7       127       8 14
## 693            642         2.8  3         34   0.8       197       2 15
## 694           1664         2.9  9         15   0.5       177       7 19
## 695           1498         0.7  3          8   0.1       170       7  4
## 696           1236         2.8  6         32   0.3       143       4 18
## 697           1673         2.8  7          9   0.1       187       3 10
## 698            712         1.6  6          9   0.8       172       2 14
## 699            603         0.7  0         64   0.2       156       6  0
## 700           1332         0.8  2         11   0.5       187       2  4
## 701           1366         1.7  5         53   0.2       161       5  9
## 702            906         2.1  7         18   0.1       136       2 11
## 703           1554         2.7  3         47   0.7       185       5 12
## 704           1187         2.3  0         21   0.2       122       6  6
## 705           1918         2.8  1         22   0.3       139       1  5
## 706            841         0.5  8         39   0.1       199       6 13
## 707            878         1.3 13         31   0.5       153       4 18
## 708           1578         2.9  8          3   0.9       200       1 20
## 709           1829         1.7  0         36   0.8       112       5  1
## 710           1652         0.6  4         40   1.0       198       7  6
## 711            912         0.5  0         58   0.3       107       5  1
## 712           1181         2.3 12         61   0.6       116       2 16
## 713           1497         0.7  5         32   0.7        92       3  9
## 714           1790         2.3  3         49   0.5       100       3 11
## 715           1742         1.6  3         49   0.8       100       1  9
## 716           1549         0.5  4          8   0.4       158       5  7
## 717           1372         2.7  7         34   0.4       193       4 17
## 718           1112         0.5  0         12   0.9       190       4  6
## 719           1005         0.5  0         46   0.4       133       8 17
## 720            783         1.7  9         27   0.3       150       6 11
## 721           1698         2.5 10         26   0.4       105       2 18
## 722            899         2.7  3         53   0.3       192       4 11
## 723           1062         1.5  3         45   0.2       151       3  4
## 724            989         2.0  4         17   0.2       166       3 19
## 725           1983         1.2  6         27   0.6       125       1 11
## 726           1800         3.0  5         53   0.3        87       1 12
## 727           1254         1.7  0         45   0.4       146       3  3
## 728            868         1.7  0         30   0.8        83       4  0
## 729           1205         0.5  3         12   0.6       175       4  6
## 730           1284         2.8  0         10   0.9       155       2 14
## 731            626         0.7  2         23   0.5       187       2  9
## 732           1763         2.9  1         53   0.2       101       4 20
## 733           1265         0.6  4         49   0.5        90       1  5
## 734            904         1.8  3         35   0.4       168       3  4
## 735            979         0.5  0          5   0.5       164       7  2
## 736           1531         2.0  0         60   0.4       162       6  1
## 737            934         2.0  2         30   0.7       182       8 15
## 738           1624         2.6  0         58   1.0       200       3 11
## 739            874         1.3  5          8   0.5       185       1 20
## 740           1027         2.2  0         63   0.8       102       5  4
## 741           1966         0.8 15         48   0.2       144       1 16
## 742           1048         1.5  2         29   1.0       182       1  4
## 743           1496         1.9 13         38   0.1       126       4 14
## 744           1349         2.2  5         55   0.1       105       7  7
## 745           1689         2.4  0         13   0.9       168       2  9
## 746            582         0.5  3         27   0.2       168       3 10
## 747           1485         2.8  3         44   0.1       138       4 14
## 748           1991         1.8 11          9   0.7        94       3 12
## 749            553         0.5  2         23   0.5       119       3  3
## 750           1231         0.7  2          7   0.9       113       1  7
## 751            764         1.2  1         13   1.0       152       8  2
## 752           1589         2.6  9         46   0.4        84       3 16
## 753           1456         1.6  9         39   0.1        80       8 10
## 754           1061         2.1  3         52   0.1       109       7  4
## 755           1428         1.6  0          7   0.1       153       4  2
## 756            674         2.8  0          8   0.2        93       3  0
## 757            832         1.1  0         46   0.3        93       6  0
## 758           1625         0.5  2         50   0.1       107       8 17
## 759            907         1.4  3         42   0.2       151       1 16
## 760           1631         0.5  2         13   0.5       166       2 16
## 761           1584         1.3  6         41   0.6       135       4 15
## 762           1382         0.5  7         13   0.3       199       2 18
## 763            633         1.4  0         46   0.7       102       6 10
## 764            852         1.6  6         54   0.4       147       5  7
## 765            922         0.7  0         47   1.0       157       8 14
## 766           1745         1.5  4         57   0.1       159       3 16
## 767            862         2.3  9          6   0.1       107       8 14
## 768           1754         2.1  1         52   1.0       101       4  3
## 769           1569         0.5  3         33   0.2       147       1 13
## 770           1596         2.9  7          7   0.7       149       2  9
## 771           1807         0.5  2         61   0.1       134       6  6
## 772           1660         2.8  4          7   0.8       184       1  6
## 773            648         1.9  4          8   1.0        91       5 19
## 774           1379         0.5  1         19   0.3       134       8 17
## 775           1807         2.0  0         13   0.5       155       8  0
## 776           1923         0.5  7         46   0.5       191       1 10
## 777           1345         3.0  7         38   0.1       162       3 11
## 778            904         1.6  5         14   0.2       102       8  9
## 779           1239         1.2  5         52   0.9       122       2 10
## 780           1330         0.9  6         52   0.8        84       4 12
## 781            516         0.7  1         30   0.9       138       1 12
## 782            892         0.5  0         47   0.4        94       1 18
## 783           1130         2.5  4         58   0.1       132       5 14
## 784           1359         0.6  4         59   0.3        86       2  6
## 785           1866         2.5  3         47   0.8        89       5  9
## 786           1597         2.3 11         34   0.4        89       1 14
## 787           1046         2.8  0         58   0.2       100       8  0
## 788           1035         0.6  2         44   0.5        99       3  3
## 789           1713         3.0  5         20   0.9       104       7 17
## 790           1688         2.5  0         21   0.2       170       7  0
## 791            559         0.5 10         11   0.4        82       5 14
## 792           1483         2.6 13         19   0.6       146       7 19
## 793            860         1.6 13         48   0.5       187       7 18
## 794           1090         1.3  5         27   0.1        91       4 16
## 795            686         0.5 11          3   0.3        91       6 15
## 796           1456         1.6  5         49   0.2       193       3 20
## 797           1068         0.5  0         19   0.9       197       8  9
## 798           1373         1.9  1         29   0.9       141       6 12
## 799           1402         0.6  3         46   0.3        87       7 12
## 800            965         0.5 15         53   1.0       131       8 19
## 801           1270         2.3  0         58   0.5       104       5  1
## 802            919         0.5 11         13   0.3       159       1 15
## 803           1963         1.0 16         34   0.1       157       2 17
## 804           1977         1.3  2         31   0.4       173       8  6
## 805           1881         2.2  4         51   0.3       158       5  7
## 806            772         2.6  7          6   0.1       100       2 10
## 807            536         1.4  0         53   0.7       135       3  0
## 808           1633         1.7  3         33   0.9       111       2 19
## 809           1371         0.5  4         52   0.5       167       4 19
## 810           1221         0.5  1         28   0.5       139       1  7
## 811           1000         0.5 12         63   0.7       179       8 18
## 812           1986         2.7  3         64   0.2       162       4 20
## 813           1699         0.5  1         59   0.5        97       4  3
## 814           1576         2.6  7         58   0.2        90       3 16
## 815           1081         2.5 13         49   1.0        99       3 17
## 816            606         2.5  1         53   0.2       146       4  2
## 817            502         1.5  7         37   0.2       199       2 12
## 818            957         0.7  0          4   0.1       160       2  0
## 819           1658         1.4  0         38   0.3       159       5  0
## 820           1681         2.5  6         18   0.3       167       1  7
## 821           1983         0.5  3         43   1.0        87       6 10
## 822           1753         1.2  8         12   0.9       113       7 17
## 823           1632         1.4  0         41   0.1       132       4  1
## 824            688         2.5  0         57   0.8       131       7  3
## 825           1104         1.7  1         60   0.4       199       2 13
## 826           1122         2.8  0         28   0.6       184       3  4
## 827            504         0.5  2         46   0.9       172       5 14
## 828           1674         2.8  0         52   0.2        89       3  7
## 829           1948         2.8  5         18   0.2       135       2 14
## 830           1218         2.8  3         39   0.8       150       7 14
## 831            531         1.1 10         63   0.7       189       7 14
## 832           1057         2.1  4          5   0.5       154       5 13
## 833            576         1.2  1         30   0.8       162       8  5
## 834           1960         2.1  1          7   0.2       165       7  9
## 835            673         1.7  3         22   0.1        88       1  5
## 836            718         0.7  0         21   0.8        88       1  7
## 837            864         1.7  3         35   0.3       103       3  7
## 838            547         1.9  1         37   0.4       154       5  4
## 839            623         0.9  1          3   0.1       102       8  7
## 840           1720         0.7  0         19   0.1       137       2  3
## 841            980         0.5  0         33   0.9       160       6  0
## 842            643         1.0  9         18   0.2       187       8 12
## 843           1925         3.0  0         16   0.8       175       8 11
## 844           1408         2.5  3         32   0.1       151       4  6
## 845           1522         0.7  4         28   0.2       124       1  5
## 846            587         2.5 16         51   0.6       111       1 17
## 847           1959         0.5  5         18   0.7       191       6 20
## 848           1872         0.5  3         17   0.7        89       6  6
## 849           1043         0.5  7         42   0.1       111       3  8
## 850           1703         0.5  5         23   0.2        92       6  7
## 851           1414         2.6  1         27   0.6       137       4 20
## 852           1659         2.5  2         41   0.1        88       7 18
## 853           1540         0.7  1         60   0.8       197       7 12
## 854           1342         1.0  0         31   0.6        86       7  1
## 855           1189         2.3 13          8   0.5       198       3 14
## 856            518         2.4  0         62   0.6       128       6  0
## 857           1449         0.8  3         10   0.2       144       1 12
## 858            516         1.1  0         39   0.4        91       5  7
## 859           1312         0.8  0         24   0.9       156       6  0
## 860           1893         0.5  1         23   0.1       179       8  3
## 861           1076         2.5  3         14   0.2       105       5  4
## 862            530         2.6  1         40   0.9       118       6  2
## 863           1044         2.8  7         33   0.6       129       4 13
## 864           1330         1.1 11         37   0.7       126       7 20
## 865           1836         1.1  0         14   0.7       104       5  1
## 866           1741         0.5  1         55   0.1       140       5  3
## 867           1869         2.6  0         30   0.4       152       1 13
## 868            605         1.5  9         18   0.9       162       2 14
## 869           1245         0.6  2         30   0.7       105       5  3
## 870           1709         0.8  8         58   0.8       136       2 10
## 871            688         1.5  0          2   1.0        89       3  2
## 872            775         2.5  1         10   0.1       185       7 14
## 873           1154         2.0  6         35   0.8       159       5 16
## 874           1375         0.5  2         35   0.1       194       2  3
## 875           1827         0.8  4         61   0.9       130       7  9
## 876           1697         0.5  0         60   0.1        90       4  0
## 877            741         1.5  1         14   0.3       178       5 13
## 878           1672         0.5 13         32   0.7       174       1 20
## 879            514         0.7  4         57   0.1       186       3  7
## 880           1375         1.3  2         12   0.1       105       1 13
## 881            989         1.8  4         55   0.7       124       1 12
## 882           1510         2.5 11         47   0.7       138       2 13
## 883           1396         2.4  0         46   0.1       138       7  1
## 884            808         2.3  3         45   0.8       161       1  9
## 885           1542         2.3  1         43   0.8       182       2 11
## 886           1735         1.9  0         59   0.7       148       6  0
## 887            586         0.5  1         57   0.3       188       2  5
## 888           1341         2.1  0         36   0.4       107       8  1
## 889            781         1.3  1         64   0.2        88       2  7
## 890           1224         1.8  2         44   0.2       139       2  9
## 891           1970         1.6 11         58   0.3        80       8 19
## 892            618         2.2  0         63   0.9       138       2  1
## 893           1537         2.9 11         46   0.8       158       4 17
## 894           1018         2.2  8         16   0.6       153       8 10
## 895           1545         1.3 15         35   0.1       173       5 20
## 896           1871         2.1  4         41   0.4       144       8  5
## 897           1729         2.1  1         12   0.9       186       1  6
## 898            680         2.1  9          9   0.1       131       3 11
## 899           1106         1.9  4         54   0.7       107       3  9
## 900            618         0.6  8         16   0.2       143       1 14
## 901           1671         0.5 13          5   0.9       179       8 18
## 902           1975         1.9  2         31   0.9       151       1 17
## 903            831         0.7  2         62   0.7       134       8  9
## 904           1524         1.8  0         10   0.6       174       4  1
## 905           1528         1.8  3         55   0.8       129       5  6
## 906           1447         2.5  2          9   0.2       155       8 16
## 907            564         3.0  1          8   0.7       130       2  3
## 908            921         0.6  1         60   0.1        83       2  3
## 909            802         0.5  3         43   0.6       153       5  6
## 910           1923         1.3  3         24   1.0        86       4  4
## 911           1193         3.0 10         56   0.4       196       3 17
## 912            904         1.2 10         31   0.8       112       3 11
## 913           1285         2.4  1         51   1.0       140       7  2
## 914           1635         2.6  0          2   0.6       182       4  0
## 915           1923         1.0  5         14   0.9       154       4  6
## 916           1851         1.9  3         12   0.9       198       6 11
## 917            727         1.5  0         32   1.0       131       7  1
## 918           1396         1.5  0         44   0.1       118       6  1
## 919            877         1.3  9         46   0.8       135       1 14
## 920            696         2.4 11         35   0.2       103       3 13
## 921           1009         1.0  2          2   0.6       102       8 14
## 922           1526         1.1  2         38   0.3       116       5  6
## 923            987         2.0  1         20   0.8       192       7  3
## 924           1414         2.2  0         49   0.5       101       4  1
## 925           1732         0.8  2         61   0.3       172       5  3
## 926           1587         0.5  0         50   0.9       148       8 15
## 927           1097         0.8 10         21   0.1       160       7 15
## 928            504         2.8  0         40   0.5       178       3  0
## 929           1159         2.7 13         28   0.2        80       8 18
## 930           1910         1.4  0         29   0.6       146       7  2
## 931           1930         2.0 11         16   0.8       186       8 12
## 932            860         1.3  8         40   0.7       170       3 13
## 933           1268         2.9  9         42   0.9       182       6 17
## 934            999         1.8  0         27   0.1       167       4  1
## 935            622         0.7  0          9   0.3       127       2  0
## 936           1250         1.4  8         44   0.2       182       7 20
## 937            763         0.5  9         51   0.3       104       5 12
## 938           1100         1.6 12         15   0.1       112       8 14
## 939            553         2.2  0         18   0.7       136       2  1
## 940            673         2.2  2          3   0.4        84       2  9
## 941            563         2.0  3         47   0.3       115       7 12
## 942           1031         2.1  2         11   0.1       121       2  9
## 943           1880         0.5  0          7   0.6       104       4  1
## 944           1479         1.8  0         24   0.1       146       6  0
## 945           1271         1.5  0          2   0.9       161       1  1
## 946            709         1.9 11         13   0.1        81       8 15
## 947           1744         2.8  0         62   0.7       158       1  0
## 948           1314         2.2  0         40   0.8       116       7  1
## 949           1136         0.5  0          9   0.9       150       6  2
## 950           1039         2.7 11         56   0.3       125       8 13
## 951            732         2.3  7         39   0.6       188       7 11
## 952            511         0.7  1         52   0.7       180       2 10
## 953           1092         0.5 10         11   0.5       167       3 14
## 954           1030         0.5  4         37   0.7       147       1  8
## 955           1130         1.8  0         27   0.9       101       1  8
## 956           1299         2.8  2         60   0.8       187       8  6
## 957            708         1.9  0         26   0.3       116       7  1
## 958           1481         2.0  0         35   0.5       105       3  0
## 959           1068         2.7  0         45   0.6       194       5  1
## 960           1487         0.5  6         57   0.7        84       3 12
## 961           1816         1.6  0         50   0.1       186       5  2
## 962           1895         0.5  2          5   0.7       134       1 11
## 963            805         1.7  0         58   0.9       168       3 10
## 964            713         1.6  7          2   0.4       181       8  9
## 965            864         2.5 10         24   0.7       148       4 15
## 966            582         0.5  8         32   0.7       107       3 11
## 967           1898         1.8  3         56   0.2       185       7  4
## 968           1158         0.7  1         29   0.7       123       2 20
## 969           1451         0.5  5         57   0.6       181       8 12
## 970           1820         1.4  5         51   0.9       163       7 10
## 971            984         0.5  3          9   0.8       140       3 20
## 972            580         2.1  0         24   0.4       135       4  5
## 973            514         1.6  7         37   0.1       172       1  9
## 974           1515         2.6  0         56   0.9       184       3  1
## 975            721         0.5  6          6   0.4       194       4  9
## 976            603         2.5  3         35   0.7       126       3 16
## 977           1820         1.7  0         64   0.6       161       1  3
## 978           1215         1.0  4         16   0.1       130       4  8
## 979            717         0.5  4         10   0.6       198       1  5
## 980           1348         2.0  5         36   0.3       145       6  6
## 981           1164         2.6  0         61   0.3        88       1  6
## 982            578         0.5  9         62   0.3       170       5 13
## 983           1971         0.5  0         40   0.3       186       7 19
## 984           1794         0.5  5         55   0.5        86       7  7
## 985           1597         0.9  4          4   0.5        85       1 19
## 986           1240         1.6  0         39   0.1       146       2  9
## 987           1221         2.3  7         35   0.4       152       7 19
## 988           1180         1.2  5         50   1.0       144       4 20
## 989            718         2.8 10         59   0.8       113       6 13
## 990           1670         1.6  0         47   0.8       139       8 15
## 991            534         2.5  1         21   0.3       164       7  9
## 992           1566         0.5  0         20   0.5       126       2  0
## 993           1935         0.5  0         29   0.6       124       5  9
## 994            504         2.8  2         54   0.4       163       2 10
## 995            913         1.7  1         41   0.3       103       1 14
## 996           1317         1.8  2         43   0.5       113       5  3
## 997            917         0.5  1         31   0.5        88       6  2
## 998           1083         1.0  6         52   0.9       126       2 20
## 999           1039         0.6  5          6   0.1       136       6  8
## 1000          1449         0.5  0         52   0.6       105       6  0
## 1001          1796         1.6  5          7   0.1       101       4  6
## 1002          1097         1.4  0         60   0.8       181       1 17
## 1003          1562         0.5  1         20   0.8       172       8 10
## 1004          1433         2.3  8         27   0.9       153       7 12
## 1005           740         1.8  1         12   0.2       123       1 11
## 1006          1020         1.5 13          7   0.7        96       3 18
## 1007           896         1.0  3          4   0.3       111       1  4
## 1008          1824         0.8  4         19   0.9       143       2  5
## 1009          1512         0.6  1         54   0.4       134       3  2
## 1010          1053         2.9  7         42   0.9       126       6 17
## 1011          1944         1.8  0         31   0.8        87       4 10
## 1012          1174         0.7  1         25   0.8        91       2  8
## 1013           627         1.6  3         12   0.2       131       7 17
## 1014           586         1.4 10          8   0.5       142       8 14
## 1015          1528         2.4  2         37   0.2       156       8  3
## 1016           875         0.7  2         23   0.1       110       7  7
## 1017          1602         0.6 12         58   0.4       170       1 13
## 1018          1370         1.4  5          6   0.2       158       6  8
## 1019           609         0.5  4         10   0.1        81       6  6
## 1020           840         1.2  5         15   0.3       178       6  8
## 1021           991         2.0  2         12   0.3       158       5  6
## 1022          1724         1.8  0          9   0.8       100       2  1
## 1023           584         1.8  0         19   0.3       168       3  0
## 1024           860         1.3  1         49   0.7       141       6 13
## 1025          1541         2.8 10         28   0.2       165       5 15
## 1026           615         0.5  7         58   0.5       130       5  8
## 1027           912         2.8  6         46   0.4        84       4 11
## 1028          1365         0.6  0         31   0.3       125       4  0
## 1029          1347         1.5 10         56   0.3        95       1 19
## 1030          1644         2.8  0         44   1.0       137       8  0
## 1031           956         1.9  1         42   0.1       133       7 11
## 1032          1089         1.4 10          2   0.1       153       7 20
## 1033           664         1.5  5          6   0.9       164       2 19
## 1034          1874         0.6  1         19   0.8       192       2  9
## 1035          1928         2.6  5          5   0.8       162       5 16
## 1036           888         2.6  2         33   0.4       198       2 17
## 1037          1077         1.7  1         58   0.8        86       7 13
## 1038          1023         2.3  5         30   0.2        80       8  7
## 1039          1426         0.8  2         42   0.6        84       3 10
## 1040           831         1.7  7         26   0.7       177       5 11
## 1041          1496         1.3  0          6   0.5        85       3  9
## 1042           643         0.7  3         23   0.8       156       1  9
## 1043          1142         1.1  0          6   0.5       198       4  0
## 1044           730         0.5  3         27   0.1       112       5 10
## 1045          1901         0.5  5         41   0.6       131       2 17
## 1046          1510         0.6  1         32   0.7       135       3  8
## 1047          1924         2.6  0         59   0.9       136       6  0
## 1048          1275         0.5  3          5   0.2       152       1  9
## 1049           987         2.3  9         17   0.8       124       7 17
## 1050          1538         2.3  6         13   0.2       121       7  7
## 1051          1965         0.5  4         29   0.5       149       5 13
## 1052          1507         0.9  0         42   0.4       123       5  0
## 1053           912         0.7  3         47   0.9       182       3  6
## 1054           769         1.2  2         17   0.1        95       8  4
## 1055           948         2.3 14         22   0.1       173       5 20
## 1056           507         1.9  0         39   0.7       142       1  0
## 1057          1766         1.3  1         29   0.5       194       8 20
## 1058          1407         1.8  5         16   0.2       170       4  6
## 1059           614         2.9  3         24   0.2        94       4  7
## 1060          1972         2.9  9         14   0.4       196       7 18
## 1061          1811         2.5  4          5   0.6        98       4 16
## 1062          1159         1.5  3         36   0.4       104       4  8
## 1063          1848         1.3  7         12   0.1       178       2 19
## 1064          1988         0.5  0         16   0.1        94       8  1
## 1065          1423         0.6 12         44   0.8       131       5 13
## 1066          1974         0.5  9         10   0.8       145       6 12
## 1067           835         0.8  2         42   0.8       128       5 18
## 1068          1429         0.6  8         42   0.9       151       5  9
## 1069           947         0.5  1         51   0.1       144       2 15
## 1070          1446         2.8  0         18   0.3       178       8  1
## 1071          1332         1.3  5          7   0.8       168       7 19
## 1072          1498         1.2  0         14   0.1       191       4  1
## 1073          1576         0.9  0         14   0.4       173       2  4
## 1074          1398         1.6  8         26   0.8       150       1 12
## 1075           798         0.5  7          2   0.8       148       2 15
## 1076          1436         2.5  0         42   0.9       124       8  1
## 1077          1998         1.2  8         31   0.6       196       8 19
## 1078          1321         1.3  0         63   0.4       109       8  9
## 1079          1021         1.3  7          9   0.7       192       2 12
## 1080          1339         2.3  0         40   0.1       123       2  1
## 1081          1210         1.6  9         57   0.5       111       4 20
## 1082           882         0.7  9         28   0.2       151       6 16
## 1083          1549         1.5  0         27   0.4       113       5  7
## 1084          1991         2.2  1         42   0.7        82       1 13
## 1085           796         0.5  2         12   0.5       114       1  8
## 1086          1012         0.5  4         39   0.1       161       4  6
## 1087           598         0.8  8         62   0.1       117       8  9
## 1088           541         2.3  4         51   0.4       200       8 17
## 1089           818         0.5  2         23   0.2       192       5 17
## 1090          1414         0.5  0         15   0.8       168       2  0
## 1091           601         1.7  3          6   0.8       169       4 11
## 1092           561         2.8  2         40   0.1        99       7  6
## 1093          1616         2.3  3         52   0.1       100       5  8
## 1094          1263         1.4  1         32   0.4       103       2  2
## 1095          1604         1.9  0         63   0.6        93       5  2
## 1096           539         2.0  1         24   0.5        86       4  2
## 1097          1071         2.6  0         49   0.3       199       4  5
## 1098           826         0.5  0         57   0.5       135       2  0
## 1099          1811         2.9  0         33   0.2       132       8  2
## 1100          1842         2.2  7         56   0.4       179       4 10
## 1101          1420         2.8  1         50   0.1       100       8  8
## 1102          1763         0.5 12         10   0.2       184       3 19
## 1103          1163         1.9  6          2   0.9       133       5 20
## 1104          1805         0.5  0         29   0.2       182       4  2
## 1105           610         1.6  9         52   0.4        90       4 19
## 1106          1924         0.5  3         63   0.3       111       2 10
## 1107          1801         0.5  6         52   0.9       120       6 10
## 1108          1726         1.9  1         42   0.6       106       5  3
## 1109          1686         0.5  4         28   0.8       182       5  8
## 1110          1444         2.1  0          6   0.9       103       1  1
## 1111          1004         1.6  0         30   1.0       160       8  1
## 1112          1242         2.4  8         28   0.2       166       4  9
## 1113           672         0.5  7         35   0.1        99       8 17
## 1114          1325         0.5  0         13   1.0       133       4  8
## 1115           950         0.5  6         55   0.1       124       8  7
## 1116           948         1.5  6         18   0.3       119       4  7
## 1117          1328         0.5  5         27   0.3       144       3 18
## 1118          1447         0.5  3         40   0.7       184       2  5
## 1119          1973         0.5  3         21   0.8        95       8  7
## 1120          1731         2.3 18         60   0.5       171       4 20
## 1121          1617         1.5  1         63   0.7       111       2 10
## 1122          1791         0.5  1         33   0.5       110       5  4
## 1123           851         2.3  7         54   1.0       101       2  9
## 1124           856         1.9  0         14   0.6       115       1  7
## 1125           951         0.5  6         15   0.6        81       2 17
## 1126          1303         2.1  6         10   0.4       109       8 15
## 1127          1550         2.7  2         32   0.1       126       7  5
## 1128          1759         1.4 10         15   0.4       101       4 13
## 1129          1448         0.5 18          2   0.2       100       5 19
## 1130          1987         1.9  0         50   0.7       119       8 18
## 1131           908         2.1  1         44   0.6        95       5  2
## 1132           820         0.5  2         58   0.3       160       4  4
## 1133           904         2.7  4         19   0.6       147       6  7
## 1134           667         1.3  4          6   0.1       149       6 10
## 1135          1333         1.4  3         59   0.7       183       7  5
## 1136          1352         2.4  5          6   0.9        92       6  9
## 1137          1600         2.5  1         19   0.6        88       6  9
## 1138          1454         1.4  8         37   0.8        80       8 20
## 1139          1823         2.8 11         10   0.6        81       8 19
## 1140          1581         1.0  9         41   0.4       172       4 19
## 1141          1672         1.7  1         44   0.5       172       8  4
## 1142          1283         0.7  0         27   0.2        80       4  6
## 1143           630         2.0  9         14   0.5       180       8 13
## 1144           659         2.2  0         24   0.7       151       8  2
## 1145          1811         0.5  0         32   1.0       115       6  0
## 1146           688         2.5  9         40   0.2       109       5 20
## 1147           514         2.9  5         18   0.7       141       2  8
## 1148          1933         2.8  1         16   0.5       154       3  9
## 1149          1006         2.2  4         22   0.8       135       4  8
## 1150          1134         0.9  2         13   0.4        92       4 19
## 1151           599         2.9  0         64   0.3       186       8  1
## 1152           973         0.5  5         53   0.5       196       5  7
## 1153          1180         2.5  0         27   0.1        90       6  1
## 1154          1237         2.5  7         13   0.8        98       4 12
## 1155           659         0.5  7          5   0.3       169       8 10
## 1156          1002         2.7 16         11   0.4       185       6 18
## 1157           602         0.9  9         50   0.9       141       8 18
## 1158          1249         0.5  3         55   0.8       140       3  8
## 1159           531         0.5  0         12   0.5       121       4  3
## 1160          1450         2.1  1         31   0.6       114       5  4
## 1161          1342         2.8  1         50   0.1       173       1  8
## 1162           832         0.5  5         22   0.6       198       1  8
## 1163           867         1.5  0         57   0.1       159       4  0
## 1164          1404         2.1 16         44   0.9       151       1 20
## 1165           840         0.9  6         33   0.7       150       6  7
## 1166          1368         0.5 10         42   0.2        91       1 20
## 1167          1927         0.9  3         11   0.4       190       8 12
## 1168          1714         0.5  7         31   0.4       120       2 10
## 1169          1201         0.5  1         10   0.4       198       6  5
## 1170          1796         2.3  4         44   0.9       184       2 11
## 1171          1147         0.8 11         28   0.7       158       6 13
## 1172           776         1.7  0         20   1.0       146       4  9
## 1173           510         2.6  0         33   0.1       110       6 10
## 1174          1045         1.2  3         40   0.1       175       6  7
## 1175          1497         0.9  4         10   0.1       153       8 20
## 1176           726         2.2  0         40   0.8        94       4 20
## 1177          1576         2.0  0         20   0.6       148       3  0
## 1178           714         2.1  5         11   0.1       197       3  6
## 1179          1595         1.9  1         63   0.9        88       1 20
## 1180           541         1.0  0         10   0.4       127       4  0
## 1181          1154         1.1  2          5   0.2        98       1 13
## 1182          1638         1.0  6          2   0.2       189       1 11
## 1183          1150         2.7  0         39   0.4       151       8  1
## 1184          1254         2.3  0         44   0.2       149       1 10
## 1185           603         1.5  8         12   0.7       114       6 20
## 1186          1834         2.1  7         40   0.1        99       4 11
## 1187           520         2.3  0         31   0.8       145       6 14
## 1188           565         0.6  1         50   0.1       158       2  8
## 1189          1689         0.5 11          9   0.5       150       5 15
## 1190           742         2.3  0         21   0.7       104       8  1
## 1191           761         0.5  9          3   0.8       196       7 15
## 1192           959         1.3  0         36   0.4       196       5 14
## 1193           772         1.2  7          8   0.6       134       7  8
## 1194          1824         1.9  9          2   0.7       103       1 14
## 1195          1130         0.5  1         29   0.7       130       1 14
## 1196          1183         0.5  3         48   0.1       120       8 12
## 1197          1472         3.0  4         20   0.3       169       2  6
## 1198          1868         0.5  2         40   0.9       179       8 17
## 1199          1375         2.9  2         20   0.4       147       6  7
## 1200           881         1.0  1         64   0.5       160       3  6
## 1201          1742         0.5  5         43   0.9       176       3  7
## 1202          1225         0.7  6         60   0.1       107       2 15
## 1203          1970         0.5  0         15   1.0       132       2  0
## 1204          1186         0.5  2         21   0.4       160       8  4
## 1205          1762         0.7  7         60   0.1       157       4 10
## 1206           852         1.8  5          8   0.8       160       5  9
## 1207          1575         2.5  0         11   0.2       185       7 20
## 1208          1972         1.6  0         48   0.5        92       8  2
## 1209           827         0.6  9         35   0.2        87       6 17
## 1210          1063         1.5  7          3   0.9       155       4 10
## 1211          1695         2.3  1         34   0.4        84       5 10
## 1212          1343         2.4  5         14   0.9        81       8 10
## 1213           834         0.5  2         27   0.2       123       1 16
## 1214           595         2.6  0         20   0.4       122       2 18
## 1215           844         2.2  1         22   0.9       197       7  2
## 1216          1335         3.0  3         60   0.8       180       2  4
## 1217          1883         1.6  9         24   0.1        87       1 16
## 1218          1128         0.5  6         33   0.6       147       8  7
## 1219           826         2.4 14         57   0.6       117       2 17
## 1220          1650         2.3  0         38   0.7        89       1  1
## 1221          1162         1.0  0         50   0.5       104       4 10
## 1222          1517         2.5  2          8   0.8       169       4 18
## 1223           963         0.5  1         60   0.8       156       3  3
## 1224          1837         1.0  5         12   0.9       103       8  6
## 1225          1028         2.8  1         30   0.1       193       1 10
## 1226          1831         2.2  4         55   1.0       189       8  9
## 1227           571         2.0  5         58   0.6       101       6 11
## 1228          1770         2.3  0          9   0.6       183       6  5
## 1229           970         1.3  5         59   0.2        84       7 19
## 1230           642         0.5  0         38   0.8        86       5 10
## 1231           622         1.9 12         11   0.8       193       8 14
## 1232          1412         0.6  9         57   0.2        86       3 10
## 1233          1070         3.0  9         19   0.8       160       7 16
## 1234           875         0.7  1         30   0.7       146       8  8
## 1235          1994         0.7  5          5   0.4       189       2  7
## 1236           823         1.0  9         18   0.2       182       5 16
## 1237          1908         0.5  9         30   1.0       111       2 19
## 1238           790         0.5  7         44   0.1        89       8  8
## 1239          1330         1.8  1         40   0.5       177       1  5
## 1240          1660         0.5  2         15   0.7       100       8  8
## 1241          1410         0.5  0         15   0.3        92       3  1
## 1242          1772         1.6 17         45   0.5       159       2 18
## 1243          1280         1.3  0         39   1.0       108       7  1
## 1244          1712         0.9  2         23   1.0       155       7 19
## 1245          1562         0.7  0         55   0.1       140       8  1
## 1246           891         1.9  1         18   0.1       153       6  3
## 1247          1110         0.6  1         42   0.9       192       5  2
## 1248           875         1.4  1         43   0.6       104       8  3
## 1249          1211         1.2  0         28   0.3       130       2  3
## 1250           769         0.5 12         54   0.2       112       4 19
## 1251           671         2.9  7          2   0.1        95       6 19
## 1252          1872         0.5  5         26   0.5       172       4 10
## 1253           911         2.2  1         28   0.5       182       2  2
## 1254          1273         0.7  1         23   0.6       178       6 14
## 1255          1062         0.5  3         39   0.9       124       4  4
## 1256           940         2.9  0         16   0.7       115       4  1
## 1257           729         2.0  0         14   0.6       135       6  2
## 1258          1494         2.4  6          3   0.9        93       7 17
## 1259          1546         1.6  2         27   0.7       124       2  6
## 1260          1253         0.7  0         34   0.9       113       4 10
## 1261           895         0.5  6         16   0.3       164       8 14
## 1262           793         1.9  0         43   0.3       124       8  2
## 1263          1628         1.8 11         56   0.1       136       2 13
## 1264           625         2.7  2         42   0.8       138       8  3
## 1265          1110         1.5  2         12   0.3       153       8  5
## 1266           999         1.7  6         49   0.6       132       1 17
## 1267          1856         0.5  0         24   0.1       155       5  1
## 1268          1715         2.6  9         15   0.6       120       6 20
## 1269          1897         1.9  2         54   0.3       154       2  7
## 1270          1171         1.7  2         19   0.3       167       7 19
## 1271          1973         2.4  6         46   0.1       193       4 10
## 1272          1992         0.6 11          2   0.1       141       1 14
## 1273          1093         0.5  1         31   0.6       171       2 12
## 1274          1880         2.0 11         35   0.1       200       4 14
## 1275          1765         1.1  3         11   0.7        87       2  7
## 1276          1640         0.5  1         36   0.9        87       8 11
## 1277          1830         0.5  0          7   0.9       191       6 12
## 1278           826         0.6  3         62   0.7       187       4 19
## 1279          1864         2.2  0          7   0.1       142       1  2
## 1280           586         0.6 16         42   0.3       121       7 17
## 1281          1206         3.0 10         30   0.6       127       3 13
## 1282           832         0.5  1          5   0.1       165       5  2
## 1283           848         2.8  1          8   0.1       103       7  3
## 1284          1851         2.9  0         53   0.7       112       6  2
## 1285          1166         0.5  7         59   0.5        80       1 11
## 1286          1776         2.1 10         35   0.1       172       4 16
## 1287          1254         1.2  2         27   0.2       134       2 10
## 1288          1494         1.6  4         24   0.2       101       1 17
## 1289           984         2.2  0         39   0.5       119       8 18
## 1290           513         1.9  8         27   0.4       174       8 13
## 1291           557         0.5  3         17   0.2        89       6  4
## 1292           744         1.7  0         33   0.5       105       4  2
## 1293          1129         0.8  9         21   0.6       116       2 10
## 1294          1793         2.7 12         44   0.7       175       5 17
## 1295          1310         2.3  3         15   0.2       150       5 11
## 1296          1604         1.2  1          6   0.5       191       6 15
## 1297          1676         1.6  2         37   0.2       145       1  3
## 1298          1001         2.8  1         40   0.2       117       3  7
## 1299          1086         2.7  0         24   0.1       135       8 19
## 1300          1986         0.5  7         39   0.1        89       7 19
## 1301           634         0.8  5         41   0.2       192       1 12
## 1302           502         0.8  7         52   1.0        82       6  8
## 1303          1641         1.1  6         48   0.7        97       8  7
## 1304           623         1.4  0         15   0.2       189       3  1
## 1305           962         1.1  3         17   0.1       161       8 20
## 1306          1057         2.7  3         41   0.1        97       4 10
## 1307          1862         1.5  1         62   1.0       182       1  2
## 1308           865         2.9  5         16   0.1        89       4 12
## 1309           999         2.6 11         56   1.0       165       4 13
## 1310          1232         1.2 12         30   0.9       131       6 14
## 1311          1194         2.3  6         34   0.2       165       2 12
## 1312           612         0.5  1         20   0.8       179       8  6
## 1313          1362         2.9  0         47   0.5       186       6  4
## 1314          1469         1.7 11         44   0.5       169       2 14
## 1315           986         0.8 13         28   0.8       182       4 20
## 1316          1843         2.8  1         62   0.8       104       3  7
## 1317          1561         0.5 10          4   0.5       134       2 13
## 1318          1695         0.5  2         47   0.5       189       5 13
## 1319           667         2.6  3          8   0.8       185       1  7
## 1320          1768         2.0  0          2   0.8        88       7  2
## 1321          1269         2.8 16         41   1.0       180       8 17
## 1322          1109         2.8  6         57   0.9       151       1 12
## 1323           936         0.5 14         14   0.1       164       3 17
## 1324          1175         1.7  1         45   0.6       167       2  7
## 1325          1940         1.0  0          5   0.2       194       4  1
## 1326           771         0.5  0         33   0.1       157       3 17
## 1327           645         2.5 10         63   1.0       174       4 14
## 1328          1307         1.8  2          7   0.6        88       7  3
## 1329          1948         2.9  4          5   0.2       136       1 13
## 1330          1509         0.6  5         50   0.3        81       4  8
## 1331           626         2.8  3         11   0.8       109       2  9
## 1332          1190         2.0  0         40   0.2        93       5  1
## 1333          1727         2.1 10         57   0.3       158       6 18
## 1334          1670         3.0 14         62   0.5       114       8 20
## 1335          1150         0.5  4         27   1.0       134       1 17
## 1336          1208         2.9  5         16   1.0       175       1 19
## 1337          1812         1.3  4         42   1.0       162       7 15
## 1338          1559         1.3 10         10   0.3       114       8 17
## 1339          1751         2.5  1          4   0.6       146       8  9
## 1340          1003         2.9 16         16   0.6       119       8 19
## 1341          1715         0.5  1         18   0.1        96       6  3
## 1342          1967         0.5  0         46   0.6       199       1  7
## 1343           962         1.0 14          2   0.9       125       1 20
## 1344          1569         1.6  4         54   0.9       190       4  6
## 1345          1146         3.0  5         57   0.4       111       5 10
## 1346          1379         1.5  6          9   0.8       118       8 13
## 1347          1504         1.9  7         25   0.3       152       4  9
## 1348          1630         2.8  0         32   0.9        80       6  1
## 1349          1125         2.7 10         23   0.4       117       1 14
## 1350           614         0.7  4         51   0.1       178       7  7
## 1351           854         1.0  5         17   0.9       101       4  9
## 1352          1826         0.5  5         49   0.6       172       8  6
## 1353           807         0.5  0         37   0.2       127       4  0
## 1354          1996         2.8  0          7   0.1       138       5  4
## 1355          1083         2.0  3         41   0.7       158       5  5
## 1356          1035         0.5 14         49   0.3       178       2 15
## 1357          1521         1.6  5         62   0.3       143       1  9
## 1358          1314         2.8 13          5   1.0        87       4 18
## 1359           535         1.9  5         24   0.2       190       8 19
## 1360           673         1.9 13         35   0.4       142       6 14
## 1361          1219         1.6  5          4   0.3       169       6  7
## 1362          1603         0.5  1         17   0.5       114       2  8
## 1363           695         0.5 18         12   0.6       196       2 19
## 1364          1441         0.6  0         53   0.5       106       1 12
## 1365          1470         2.2  4         33   0.4       138       7 10
## 1366          1152         2.1  0         26   0.3        82       2  0
## 1367          1619         0.9  3         20   0.9       117       6  9
## 1368          1002         1.8  1         45   0.7       180       7  7
## 1369           742         0.5  0         36   0.7       139       3  1
## 1370          1306         2.1  2         33   0.4       174       3  9
## 1371          1424         2.9  0         20   0.6       180       5  0
## 1372           942         1.4  8         27   0.5        91       4  9
## 1373          1934         2.7  8          2   0.1       118       4 16
## 1374          1849         0.5  1         29   0.5       181       7 12
## 1375          1922         0.5 15         42   0.6       153       8 16
## 1376           548         1.0  0         56   0.4       146       4 14
## 1377           916         0.5  0         47   0.2        82       4  1
## 1378          1783         0.5  8         14   0.6       183       8 11
## 1379          1083         1.4  2         49   0.8       156       5  9
## 1380          1698         2.8 10          7   0.8       131       8 12
## 1381          1969         0.8  5          7   0.2        85       8 20
## 1382           869         1.9  0          2   0.6       197       7  3
## 1383          1337         2.7  0          5   0.7       200       4  1
## 1384           637         0.8  2         28   0.6       194       4  6
## 1385          1248         2.4  6          9   0.4        89       3  8
## 1386          1174         0.9  7         30   1.0       147       4 20
## 1387          1831         2.3  3         44   0.2       148       7 14
## 1388          1010         2.0  2          8   0.9       139       5  3
## 1389          1117         0.5  2         21   0.1       177       2 19
## 1390           990         2.7  3         15   0.9       153       2  7
## 1391          1065         2.8  0         21   0.1       158       5  3
## 1392          1982         1.1 12         24   0.1       129       7 16
## 1393          1066         3.0  6          5   0.5       167       5  7
## 1394           681         0.5 12         17   0.9       194       5 19
## 1395          1163         2.1  0         19   0.9       132       2  1
## 1396          1285         0.8  4         11   0.8       199       5  7
## 1397           753         0.5 11         48   0.1       121       8 20
## 1398          1779         2.6  2          5   0.8        90       1  3
## 1399           987         1.9  4         52   0.5        83       3  8
## 1400           511         0.9 15         24   0.6       136       3 18
## 1401          1044         1.8  4         12   0.7       104       6  5
## 1402          1855         0.5  5         32   0.6       187       8 18
## 1403           946         1.4  9         26   0.1       186       6 10
## 1404           657         1.9  2         21   0.1       110       8  8
## 1405          1673         1.6  2         42   0.5       182       5  6
## 1406           731         0.5  0         28   0.5       130       8  0
## 1407          1261         0.5  3         52   0.3        86       2  9
## 1408           618         1.1 11         45   0.2       188       3 20
## 1409           936         2.3  5         57   0.2       129       2  6
## 1410          1485         0.9  0         28   0.8       158       5  1
## 1411          1339         0.5 10         21   0.5        80       4 11
## 1412           508         1.3  1         50   0.7        82       5  9
## 1413          1938         0.5  5         44   0.5       121       2  9
## 1414          1322         2.3  1         35   0.1       157       1  3
## 1415          1864         1.9  2         57   0.1       181       6  6
## 1416          1944         0.5  1         26   0.5        92       4  9
## 1417           581         0.6  5         56   0.2       180       2 10
## 1418          1936         1.3  2         44   0.1       139       3  3
## 1419          1329         1.7  5         23   0.6       198       8  6
## 1420           808         0.5  3         46   0.5       105       8 15
## 1421           994         1.2  4         40   0.9       164       2 17
## 1422          1811         2.0  3         22   0.2        82       2  8
## 1423           993         2.3  0         16   0.4       106       7 13
## 1424          1136         1.6 15         32   0.5       150       6 19
## 1425           719         0.5  0         23   0.4       113       6  9
## 1426          1946         2.7  8          8   0.3       123       8 19
## 1427          1230         1.6  0         48   0.7       111       7  2
## 1428           579         2.5  3          7   1.0        89       8 10
## 1429           574         3.0  1         30   0.9       126       2  3
## 1430          1072         2.2  6         19   0.4       114       8 16
## 1431           843         1.3  1         16   0.9       119       7  4
## 1432          1492         1.5  5         11   0.9        85       3  7
## 1433          1807         2.9  0         16   0.2       125       3  0
## 1434           909         0.7  7         47   0.1       146       3  9
## 1435          1876         1.3  9         64   1.0        98       3 19
## 1436          1251         0.5 10         29   0.7       133       6 16
## 1437           636         0.5  8         28   0.9       152       2 15
## 1438          1354         0.7  3         26   0.5       136       4  6
## 1439           930         0.5  3         34   0.5       190       4 16
## 1440          1830         1.0  7         47   0.3       117       1 11
## 1441          1424         0.5 16         23   0.6        80       4 18
## 1442           583         0.9  1         51   0.9        88       8  2
## 1443           648         0.5  9         54   0.4       109       5 20
## 1444          1299         2.8  0         64   0.5        83       6  6
## 1445          1494         1.9  2         47   0.4       129       2  9
## 1446           527         0.7  4         53   0.9       128       1 11
## 1447           873         0.5  6          8   0.6       152       7  9
## 1448           722         1.1 12         12   1.0        93       6 20
## 1449          1358         1.7  0         60   0.6       147       3  1
## 1450          1035         1.9  8         11   1.0       173       6 10
## 1451          1289         0.8  3         16   0.9       161       6 17
## 1452          1445         2.2  2         37   0.4        82       2  6
## 1453          1702         2.3 12         47   0.6       107       2 19
## 1454           713         1.4  4         53   0.8        98       1 17
## 1455          1207         0.5  0         58   0.3       112       8  0
## 1456           683         2.1  7         30   0.1       197       7  8
## 1457           594         2.6  1         27   0.1       196       5  2
## 1458           537         2.0  1         55   0.3       103       7  2
## 1459          1565         0.5  0         38   0.1       121       5  1
## 1460          1052         0.9  1          6   0.8       195       4 16
## 1461          1766         0.5  1         16   0.6       141       5  7
## 1462           584         1.7  1         19   0.4       199       4  4
## 1463          1541         1.5  6          5   0.5       109       4 10
## 1464           935         0.5  3         58   0.1       155       5 12
## 1465           680         2.4 13         34   0.4       157       2 17
## 1466           877         1.5  0         17   0.1       195       7  0
## 1467          1493         1.5 13         53   0.5       102       5 16
## 1468          1762         0.6  3         26   0.1        83       6 14
## 1469           842         1.8  4         57   1.0       124       3 10
## 1470           925         2.1  9         56   0.2       196       1 14
## 1471          1576         2.2  8          5   0.7       139       3 17
## 1472          1065         1.7  4         48   0.3       162       6  8
## 1473           958         1.6  9         60   0.4       200       1 12
## 1474          1680         1.2  6         37   0.7       168       4  7
## 1475          1715         1.3  0         12   0.3        94       8  6
## 1476           983         0.6  0         45   0.1       130       4 11
## 1477           720         0.9 12         14   0.8       165       4 14
## 1478          1564         1.5  1         42   0.7       108       2  2
## 1479           969         1.8  2         44   0.6       158       3  7
## 1480          1356         0.5  0         35   0.8       131       1 19
## 1481           504         1.0  8         14   0.5       189       7  9
## 1482          1546         0.5  3          6   0.2       101       1  4
## 1483           689         2.9  3          7   0.8       147       2  6
## 1484          1685         0.6 14         16   0.6       137       7 15
## 1485          1786         1.2 11         61   0.1       123       3 14
## 1486          1944         0.6  9         59   0.2       132       7 10
## 1487          1077         2.5 13         49   0.6       131       4 14
## 1488           648         1.8  1         60   0.1       195       4  4
## 1489          1702         0.5  8          2   0.6       185       6 19
## 1490           856         1.1 14         19   0.7       125       6 19
## 1491          1786         2.6  6         14   1.0       115       4 15
## 1492           914         0.7  1         60   0.9       198       5  4
## 1493           649         1.4  4         19   0.8       190       7 12
## 1494          1646         2.5  3         25   0.6       200       2  5
## 1495          1043         2.7  3         29   0.2        83       8 17
## 1496          1920         0.6  3         19   0.5       131       3  8
## 1497          1220         2.8  1         54   0.7       109       7 20
## 1498          1748         0.5 11         53   0.5       114       4 20
## 1499          1766         0.5  1         22   0.8       122       8  4
## 1500          1278         1.0  7         24   0.8       170       3 16
## 1501          1745         2.9  0          3   0.9       105       3  0
## 1502           868         0.9  5          5   0.4       137       4  7
## 1503           864         1.6  0         53   0.3        88       6  1
## 1504          1142         1.2  0          2   0.6       114       6  0
## 1505           608         0.5  0         64   0.9        94       4  0
## 1506           983         1.8  0         53   0.9       187       1  4
## 1507          1720         1.6 18          2   0.8       188       5 20
## 1508           591         2.1 18         16   0.5       196       7 20
## 1509          1205         2.7  5         10   0.1       163       6 10
## 1510          1369         1.7  3         28   0.6       108       1 18
## 1511          1082         2.3 11          2   1.0       101       4 20
## 1512          1719         2.1  0         52   0.1       187       3  1
## 1513          1090         2.3  1          8   0.6       128       8  8
## 1514          1991         2.8  3         15   0.1       181       1 13
## 1515          1179         0.5  7         32   0.3       182       2 12
## 1516           721         2.7  1         15   0.2       102       5 18
## 1517          1549         1.8  0         62   0.7        91       4  4
## 1518          1349         2.1  3         46   0.9       134       6 12
## 1519          1799         1.0  1         30   0.6       123       3  9
## 1520          1075         0.5  2         11   0.2       122       1 16
## 1521           684         0.9  3         63   1.0       157       5  9
## 1522          1837         0.5  9         36   0.9       162       4 20
## 1523          1619         1.9  2         21   0.4       138       1  9
## 1524           965         1.4  1         13   0.9       197       8  2
## 1525           730         1.5  9          9   0.7       122       4 15
## 1526          1494         1.6  3         45   0.8       158       5  4
## 1527          1807         0.8  0         38   0.2        97       8  0
## 1528           709         2.4  1         11   0.1       130       5 11
## 1529           689         1.4  2         45   0.5       103       5 11
## 1530           955         0.5  2          2   0.6       155       4 10
## 1531          1872         1.9  5         35   0.6       111       3 14
## 1532          1259         1.2  0         13   0.5        97       2  0
## 1533           695         1.2  1         41   0.4       169       8  8
## 1534          1969         1.2  9         63   0.2       132       4 11
## 1535           759         2.5  3         39   0.3       162       2  8
## 1536           936         1.0  1         18   0.2       153       3 18
## 1537          1802         2.7  0         10   0.6       101       7  9
## 1538           772         1.0  0         23   0.6        97       7 11
## 1539           591         0.5  1         16   0.9        90       6 17
## 1540           569         2.5  0         41   0.3       124       1  1
## 1541          1056         2.1  8         37   0.7       101       3 11
## 1542          1982         1.6  2         12   0.2        80       5 20
## 1543          1122         0.5  0         40   0.3       156       7 14
## 1544          1023         0.7  2         63   0.7       153       7  8
## 1545          1283         2.4  8         33   1.0        93       3 15
## 1546          1602         1.6  0         35   0.1        99       7  7
## 1547           732         1.3  5         32   0.4       163       8  7
## 1548          1854         0.9  0         33   0.4       195       1  0
## 1549           897         2.0  3          2   0.6       154       8 10
## 1550          1405         1.7  0          8   0.3       107       4  7
## 1551          1035         2.7  1         40   0.2       157       5 15
## 1552          1396         0.6  0         37   0.7       134       4 15
## 1553          1749         0.5  7         49   0.8       178       1  8
## 1554          1522         2.2  4         21   0.7       162       6 11
## 1555          1279         2.5  5          7   0.5       105       6  6
## 1556           719         2.9  7          6   0.7       102       6  8
## 1557           544         0.5  2          7   0.2       129       5 15
## 1558          1007         2.9  5         21   0.8       185       8 12
## 1559           590         1.2  0         32   0.4       141       6  1
## 1560          1788         0.5  0          6   0.2       141       6 16
## 1561           748         1.4  0         25   1.0       110       7 14
## 1562           757         0.9  4         64   0.5       157       7  5
## 1563          1561         1.6  1         27   0.8       174       3 10
## 1564           797         1.5  4         18   0.6        81       8 20
## 1565           727         2.5  0         64   0.2       193       1  9
## 1566           686         2.7  0         36   0.1       194       4  1
## 1567          1515         2.1  4         24   0.9       176       5  6
## 1568           555         0.5  2         17   0.2       163       5 12
## 1569          1976         2.6  6         45   1.0       136       4 19
## 1570          1063         2.0  0         27   0.2        86       6  7
## 1571          1753         0.7  4         40   0.6       191       4 11
## 1572           659         1.4  2          3   0.1        85       8  6
## 1573          1583         1.2  0         14   0.2       148       2  4
## 1574          1667         1.0  0         21   0.9       138       2  0
## 1575           640         1.3  5          6   0.6       143       8 20
## 1576          1913         1.8  0         29   0.6       111       5  4
## 1577          1191         0.8  6         46   0.8        89       6  9
## 1578           816         3.0  2          9   0.1       117       1  3
## 1579           915         0.5  9         33   0.3       199       2 16
## 1580          1157         0.8  7         27   0.1        88       8  8
## 1581          1201         0.5  2         10   1.0        99       7  7
## 1582          1483         2.2  3         53   0.7       169       5 11
## 1583          1614         1.2  1          9   0.1       161       3  2
## 1584           930         1.0  4          4   0.9       144       8 10
## 1585          1784         1.6  4         41   0.4       164       6 20
## 1586          1262         1.8 12         34   0.1       149       5 16
## 1587           797         2.2  0         37   0.9       144       7  2
## 1588          1829         2.1  8         59   0.1        91       5 15
## 1589           618         1.0  9         13   0.1        80       4 15
## 1590          1547         2.9  2         57   0.4       114       1  3
## 1591           586         2.8  2         15   0.2        83       3 11
## 1592          1617         2.4  8         36   0.8        85       1  9
## 1593          1882         2.0 11         44   0.8       113       8 19
## 1594           674         2.9  1         21   0.2       198       3  4
## 1595          1467         0.5  0         18   0.6       122       5  0
## 1596           794         0.5  0          2   0.8       106       6 14
## 1597          1965         2.6  0         39   0.2       187       4  3
## 1598          1911         0.9  1         36   0.7       108       8  3
## 1599          1512         0.9  4         46   0.1       145       5  5
## 1600           510         2.0  5         45   0.9       168       6 16
##      px_height px_width  ram sc_h sc_w talk_time price_range
## 1           20      756 2549    9    7        19           1
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Preprocessing

train_x <-  data.matrix(train_reduced[,-15])
train_y <-  train_reduced[,15]
 
validation_x <- data.matrix(validation_reduced[,-15])
validation_y <- validation_reduced[,15]
validation_y
##   [1] 1 0 0 1 3 2 2 1 1 1 2 3 2 3 3 3 1 3 1 3 3 0 0 1 3 2 1 2 0 3 2 2 1 0 1 0 1
##  [38] 3 3 2 2 1 0 3 3 0 0 0 3 0 3 1 3 1 1 3 1 1 0 2 0 1 2 2 1 0 1 0 1 3 3 0 1 1
##  [75] 1 0 1 0 2 0 1 1 3 0 0 3 0 1 2 0 0 2 3 0 0 3 3 1 2 2 0 1 2 1 1 3 3 0 1 3 2
## [112] 0 0 3 2 1 2 0 2 2 2 3 2 3 0 2 3 1 3 0 3 1 3 2 0 3 0 2 3 3 1 3 3 3 0 0 2 0
## [149] 3 3 3 3 2 0 1 0 1 1 1 3 3 0 0 1 1 2 1 1 1 1 0 1 2 2 3 0 0 1 2 3 1 2 2 1 0
## [186] 2 2 3 2 0 1 0 2 0 1 0 2 3 1 1 3 1 1 0 3 3 2 3 2 0 1 0 3 3 1 3 2 2 2 1 0 2
## [223] 0 3 0 0 2 2 1 1 2 1 1 0 1 1 1 1 3 0 0 0 1 2 1 0 3 1 0 2 2 3 2 3 2 1 0 2 0
## [260] 0 1 3 3 2 2 3 2 3 1 0 1 3 1 1 1 1 3 2 3 3 0 2 2 3 3 1 2 1 0 2 2 0 0 1 2 3
## [297] 3 3 0 1 2 0 0 1 2 2 3 1 1 2 1 3 1 1 1
## Levels: 0 1 2 3

Next, we need to convert the train and test data into xgb matrix type.

xgb_train <-  xgb.DMatrix(data=train_x, label=train_y)
xgb_test <-  xgb.DMatrix(data=validation_x, label=validation_y)

Hyperparamaters

  • eta: learning rate/shrinkage. Default = 0.3

  • gamma: minimum loss reduction needed to make another partition in a given tree. The larger value, the more conservative the tree will be. Default = 0.

  • max.depth: The maximum depth that you allow the tree to grow to. The deeper you allow, the more complex your model will become. For training error, it is easy to see what will happen. If you increase max_depth , training error will always go down (or at least not go up). Default = 6.

  • subsample: Fraction of training samples to use in each “boosting iteration”. Default = 1 (no subsampling).

  • colsample_bytree: Fraction of columns to be used when constructing each tree. This is an idea used in RandomForests. Default = 1 (no sampling).

  • min_child_weight: This is the minimum number of instances that have to been in a node. It’s a regularization parameter So, if it’s set to 10, each leaf has to have at least 10 instances assigned to it. The higher the value, the more conservative the tree will be. Default = 1

  • nrounds : the number of decision trees in the final model. So, how many weak learners get added to the ensemble.

# booster = 'gbtree': Possible to also have linear boosters as your weak learners.
params_booster <- list(booster = 'gbtree', eta = 1, gamma = 0, max.depth = 2, subsample = 1, colsample_bytree = 1, min_child_weight = 1, objective='multi:softmax')
typeof(params_booster)
## [1] "list"

Let’s use xgb.cv() to determine how many rounds (nrounds) we should use for training.

train_reduced$price_range <- as.factor(train_reduced$price_range)
model_xgb <- xgboost(data= xgb_train, nrounds=50)
## [1]  train-rmse:1.625547 
## [2]  train-rmse:1.159670 
## [3]  train-rmse:0.833087 
## [4]  train-rmse:0.603965 
## [5]  train-rmse:0.445092 
## [6]  train-rmse:0.336870 
## [7]  train-rmse:0.263754 
## [8]  train-rmse:0.214578 
## [9]  train-rmse:0.181862 
## [10] train-rmse:0.160168 
## [11] train-rmse:0.145543 
## [12] train-rmse:0.135125 
## [13] train-rmse:0.126012 
## [14] train-rmse:0.119892 
## [15] train-rmse:0.115471 
## [16] train-rmse:0.111966 
## [17] train-rmse:0.107397 
## [18] train-rmse:0.103576 
## [19] train-rmse:0.100815 
## [20] train-rmse:0.096675 
## [21] train-rmse:0.094538 
## [22] train-rmse:0.092162 
## [23] train-rmse:0.090066 
## [24] train-rmse:0.085751 
## [25] train-rmse:0.083730 
## [26] train-rmse:0.080554 
## [27] train-rmse:0.078098 
## [28] train-rmse:0.076071 
## [29] train-rmse:0.074886 
## [30] train-rmse:0.072286 
## [31] train-rmse:0.070580 
## [32] train-rmse:0.068643 
## [33] train-rmse:0.067727 
## [34] train-rmse:0.067082 
## [35] train-rmse:0.064621 
## [36] train-rmse:0.062630 
## [37] train-rmse:0.062104 
## [38] train-rmse:0.061067 
## [39] train-rmse:0.060876 
## [40] train-rmse:0.059987 
## [41] train-rmse:0.059366 
## [42] train-rmse:0.058086 
## [43] train-rmse:0.056035 
## [44] train-rmse:0.055050 
## [45] train-rmse:0.053148 
## [46] train-rmse:0.051739 
## [47] train-rmse:0.049753 
## [48] train-rmse:0.048730 
## [49] train-rmse:0.047848 
## [50] train-rmse:0.047122
nc <- length(unique(train_y))
pred_xgb <-  predict(model_xgb, xgb_test)

pred_y <-  as.factor((levels(validation_y))[round(pred_xgb)])
confusionMatrix(data = pred_y, reference = as.factor(validation$price_range))
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1  2  3
##          0 75  0  0  0
##          1  0 91  0  0
##          2  0  0 71  0
##          3  0  0  0 78
## 
## Overall Statistics
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9884, 1)
##     No Information Rate : 0.2889     
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##                                      
##  Mcnemar's Test P-Value : NA         
## 
## Statistics by Class:
## 
##                      Class: 0 Class: 1 Class: 2 Class: 3
## Sensitivity            1.0000   1.0000   1.0000   1.0000
## Specificity            1.0000   1.0000   1.0000   1.0000
## Pos Pred Value         1.0000   1.0000   1.0000   1.0000
## Neg Pred Value         1.0000   1.0000   1.0000   1.0000
## Prevalence             0.2381   0.2889   0.2254   0.2476
## Detection Rate         0.2381   0.2889   0.2254   0.2476
## Detection Prevalence   0.2381   0.2889   0.2254   0.2476
## Balanced Accuracy      1.0000   1.0000   1.0000   1.0000

next time do a k-folds cv to avoid overfitting?? is this even an overfitting wtf me confused

K-folds Cross-validation to Estimate Error

The number of fold will typically be 5 or 10, but it really depends on data size, signal:noise ratio, and training time. For this dataset, we are going to use 5 fold cross validation.

# NB: keep in mind xgb.cv() is used to select the correct hyperparams.
# Here I'm only looking for a decent value for nrounds; We won't do full hyperparameter tuning.
# Once you have them, train using xgb.train() or xgboost() to get the final model.

# train.cv <- xgb.cv(data = xgb_train, 
#               label = train_reduced$price_range, 
#               params = params_booster,
#               num.class = length(levels(train_reduced$price_range)),
#               nrounds = 300, 
#               nfold = 5, 
#               print_every_n = 20,
#               verbose = 2)

# Note, we can also implement early-stopping: early_stopping_rounds = 3, so that if there has been no improvement in test accuracy for a specified number of rounds, the algorithm stops.

XGBoost Model Tuning

There are many ways of controlling overfitting, but they can mostly be summed up in four categories:

  • Regularization
  • Pruning
  • Sampling
  • Early stopping