| title | author | date | output | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
loan prediction |
Reinp |
2020-05-30 |
|
## Set Chunk requirements
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE)
#echo=FALSE indicates that the code will not be shown in the final document
#(though any results/output would still be displayed).
#include=FALSE to have the chunk evaluated, but neither the code nor its output displayed
# warning=FALSE and message=FALSE suppress any R warnings or messages from being included
#in the final document#Import relevant packages
library(stats)
library(psych) #loading dataset
library(tidyverse)
setwd('E:/Documents/Reinp/GitHub Respositories/Loan_Prediction-with-RStudio')
ds_asses<-read.csv('Data Science assessment.csv', header=TRUE,sep=",")
#ds_asses[,-14] #Reads only the required columns in csv
View(ds_asses)
attach(ds_asses)#summary statistics
names(ds_asses) #display variable names## [1] "Product" "CustomerGender" "Location" "Region"
## [5] "TotalPrice" "StartDate" "Deposit" "DailyRate"
## [9] "TotalDays" "AmountPaid30" "AmountPaid60" "AmountPaid360"
## [13] "LoanStatus360"
summary(ds_asses) #summarizes the dataset## Product CustomerGender Location Region
## Length:9195 Length:9195 Length:9195 Length:9195
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## TotalPrice StartDate Deposit DailyRate
## Min. :13775 Length:9195 Min. :1000 Min. : 35.00
## 1st Qu.:16600 Class :character 1st Qu.:2000 1st Qu.: 40.00
## Median :16600 Mode :character Median :2000 Median : 40.00
## Mean :19266 Mean :2312 Mean : 46.45
## 3rd Qu.:20250 3rd Qu.:2000 3rd Qu.: 50.00
## Max. :69225 Max. :9000 Max. :165.00
## TotalDays AmountPaid30 AmountPaid60 AmountPaid360
## Min. :365 Min. :-3200 Min. :-3200 Min. :-3200
## 1st Qu.:365 1st Qu.: 2500 1st Qu.: 3438 1st Qu.:11351
## Median :365 Median : 3351 Median : 4400 Median :15140
## Mean :365 Mean : 3580 Mean : 4736 Mean :14674
## 3rd Qu.:365 3rd Qu.: 4601 3rd Qu.: 5668 3rd Qu.:16600
## Max. :365 Max. :26001 Max. :33501 Max. :69225
## LoanStatus360
## Length:9195
## Class :character
## Mode :character
##
##
##
describe(ds_asses) ## vars n mean sd median trimmed mad min max
## Product* 1 9195 NaN NA NA NaN NA Inf -Inf
## CustomerGender* 2 9195 NaN NA NA NaN NA Inf -Inf
## Location* 3 9195 NaN NA NA NaN NA Inf -Inf
## Region* 4 9195 NaN NA NA NaN NA Inf -Inf
## TotalPrice 5 9195 19266.26 6053.29 16600 18397.93 1482.60 13775 69225
## StartDate* 6 9195 NaN NA NA NaN NA Inf -Inf
## Deposit 7 9195 2311.53 1412.61 2000 2116.56 0.00 1000 9000
## DailyRate 8 9195 46.45 13.38 40 44.61 0.00 35 165
## TotalDays 9 9195 365.00 0.00 365 365.00 0.00 365 365
## AmountPaid30 10 9195 3580.48 2487.32 3351 3465.23 1555.25 -3200 26001
## AmountPaid60 11 9195 4735.67 2756.33 4400 4548.92 1482.60 -3200 33501
## AmountPaid360 12 9195 14673.83 6676.24 15140 14459.55 4062.32 -3200 69225
## LoanStatus360* 13 9195 NaN NA NA NaN NA Inf -Inf
## range skew kurtosis se
## Product* -Inf NA NA NA
## CustomerGender* -Inf NA NA NA
## Location* -Inf NA NA NA
## Region* -Inf NA NA NA
## TotalPrice 55450 4.88 26.51 63.13
## StartDate* -Inf NA NA NA
## Deposit 8000 2.63 9.39 14.73
## DailyRate 130 5.02 27.94 0.14
## TotalDays 0 NaN NaN 0.00
## AmountPaid30 29201 1.39 6.37 25.94
## AmountPaid60 36701 1.83 8.08 28.74
## AmountPaid360 72425 2.18 11.88 69.62
## LoanStatus360* -Inf NA NA NA
#learn more about the dataset
help(ds_asses)## No documentation for 'ds_asses' in specified packages and libraries:
## you could try '??ds_asses'
??ds_asses
str(ds_asses)## 'data.frame': 9195 obs. of 13 variables:
## $ Product : chr "Product B" "Product B" "Product C" "Product B" ...
## $ CustomerGender: chr "Male" "Male" "Female" "Male" ...
## $ Location : chr "Location 2" "Location 3" "Location 5" "Location 6" ...
## $ Region : chr "Region 2" "Region 3" "Region 5" "Region 1" ...
## $ TotalPrice : int 16600 16600 15600 16600 20250 16600 20250 16600 22250 20250 ...
## $ StartDate : chr "07-04-14" "01-02-15" "12-08-12" "2/20/2015" ...
## $ Deposit : int 2000 2000 1000 2000 2000 2000 2000 2000 4000 2000 ...
## $ DailyRate : int 40 40 40 40 50 40 50 40 50 50 ...
## $ TotalDays : int 365 365 365 365 365 365 365 365 365 365 ...
## $ AmountPaid30 : int 3121 3061 2160 3041 3470 3201 1851 3971 5750 1951 ...
## $ AmountPaid60 : int 4241 4171 3280 4241 4820 4141 3251 4131 7450 3251 ...
## $ AmountPaid360 : int 13621 15041 15340 15321 13720 14141 17701 15351 20250 17351 ...
## $ LoanStatus360 : chr "Active" "Active" "Active" "Active" ...
class(ds_asses)## [1] "data.frame"
typeof(ds_asses) ## [1] "list"
length(ds_asses) ## [1] 13
#attributes(ds_asses) #columnnames(ds_asses), class(ds_asses), row.names(ds_asses)#distribution of input variables
table(ds_asses$LoanStatus360)##
## Active Blocked Finished Payment
## 6742 849 1604
table(ds_asses$CustomerGender)##
## Female Male NotSpecified
## 3290 5358 547
table(ds_asses$Product)##
## Product A Product B Product C Product D Product E Product F
## 3141 3862 999 975 213 5
table(ds_asses$Region)##
## Region 1 Region 10 Region 11 Region 12 Region 13 Region 14 Region 15 Region 16
## 1232 38 12 39 62 25 22 11
## Region 17 Region 18 Region 2 Region 3 Region 4 Region 5 Region 6 Region 7
## 14 16 1364 1941 588 198 2934 622
## Region 8 Region 9
## 44 33
table(ds_asses$DailyRate)##
## 35 40 50 125 165
## 62 4825 4090 213 5
table(ds_asses$TotalPrice)##
## 13775 15600 16600 19250 20250 20750 22250 53625 54625 69225
## 62 937 3888 1008 1289 159 1634 2 211 5
table(ds_asses$Deposit)##
## 1000 2000 2500 4000 8000 9000
## 2007 5177 159 1634 2 216
#unique values per column
unique(ds_asses$ DailyRate)## [1] 40 50 125 165 35
unique(ds_asses$ TotalPrice)## [1] 16600 15600 20250 22250 20750 54625 19250 69225 13775 53625
unique(ds_asses$ Deposit)## [1] 2000 1000 4000 2500 9000 8000
unique(ds_asses$ TotalDays)## [1] 365
# Distribution of LoanStatus360 accross input variables
tabLoanStatus360gender <- xtabs(~LoanStatus360+CustomerGender,data= ds_asses)
tabLoanStatus360gender## CustomerGender
## LoanStatus360 Female Male NotSpecified
## Active 2534 3891 317
## Blocked 266 529 54
## Finished Payment 490 938 176
tabLoanStatus360product <- xtabs(~LoanStatus360+Product,data= ds_asses)
tabLoanStatus360product## Product
## LoanStatus360 Product A Product B Product C Product D Product E Product F
## Active 2550 2784 591 625 189 3
## Blocked 253 435 81 73 6 1
## Finished Payment 338 643 327 277 18 1
tabLoanStatus360region <- xtabs(~LoanStatus360+Region,data= ds_asses)
tabLoanStatus360region## Region
## LoanStatus360 Region 1 Region 10 Region 11 Region 12 Region 13 Region 14
## Active 857 29 8 25 55 20
## Blocked 155 4 0 4 0 2
## Finished Payment 220 5 4 10 7 3
## Region
## LoanStatus360 Region 15 Region 16 Region 17 Region 18 Region 2 Region 3
## Active 10 8 8 10 1004 1496
## Blocked 3 1 2 2 142 142
## Finished Payment 9 2 4 4 218 303
## Region
## LoanStatus360 Region 4 Region 5 Region 6 Region 7 Region 8 Region 9
## Active 422 126 2133 494 21 16
## Blocked 54 10 262 54 9 3
## Finished Payment 112 62 539 74 14 14
tabLoanStatus360deposit <- xtabs(~LoanStatus360+Deposit,data= ds_asses)
tabLoanStatus360deposit## Deposit
## LoanStatus360 1000 2000 2500 4000 8000 9000
## Active 1473 3696 140 1241 2 190
## Blocked 175 550 11 106 0 7
## Finished Payment 359 931 8 287 0 19
tabLoanStatus360dailyrate <- xtabs(~LoanStatus360+DailyRate,data= ds_asses)
tabLoanStatus360dailyrate## DailyRate
## LoanStatus360 35 40 50 125 165
## Active 55 3339 3156 189 3
## Blocked 0 520 322 6 1
## Finished Payment 7 966 612 18 1
tabLoanStatus360totalprice <- xtabs(~LoanStatus360+TotalPrice,data= ds_asses)
tabLoanStatus360totalprice## TotalPrice
## LoanStatus360 13775 15600 16600 19250 20250 20750 22250 53625 54625 69225
## Active 55 536 2803 882 893 140 1241 2 187 3
## Blocked 0 81 439 94 111 11 106 0 6 1
## Finished Payment 7 320 646 32 285 8 287 0 18 1
#convert categorical variables to integer data types for regression purpose
ds_asses$Product=as.factor(ds_asses$Product)
is.factor(ds_asses$Product)## [1] TRUE
ds_asses$Product=as.numeric(ds_asses$Product)
is.integer(ds_asses$Product)## [1] FALSE
is.numeric(ds_asses$Product)## [1] TRUE
is.character(ds_asses$Product)## [1] FALSE
ds_asses$CustomerGender=as.factor(ds_asses$CustomerGender)
is.factor(ds_asses$CustomerGender)## [1] TRUE
ds_asses$CustomerGender=as.numeric(ds_asses$CustomerGender)
is.integer(ds_asses$CustomerGender)## [1] FALSE
is.numeric(ds_asses$CustomerGender)## [1] TRUE
is.character(ds_asses$CustomerGender)## [1] FALSE
ds_asses$Location=as.factor(ds_asses$Location)
is.factor(ds_asses$Location)## [1] TRUE
ds_asses$Location=as.numeric(ds_asses$Location)
is.integer(ds_asses$Location)## [1] FALSE
is.numeric(ds_asses$Location)## [1] TRUE
is.character(ds_asses$Location)## [1] FALSE
ds_asses$Region=as.factor(ds_asses$Region)
is.factor(ds_asses$Region)## [1] TRUE
ds_asses$Region=as.numeric(ds_asses$Region)
is.integer(ds_asses$Region)## [1] FALSE
is.numeric(ds_asses$Region)## [1] TRUE
is.character(ds_asses$Region)## [1] FALSE
ds_asses$LoanStatus360=as.factor(ds_asses$LoanStatus360)
is.factor(ds_asses$LoanStatus360)## [1] TRUE
ds_asses$LoanStatus360=as.numeric(ds_asses$LoanStatus360)
is.integer(ds_asses$LoanStatus360)## [1] FALSE
is.numeric(ds_asses$LoanStatus360)## [1] TRUE
is.character(ds_asses$LoanStatus360)## [1] FALSE
cor_ds_asses1 <- cor(ds_asses [, c("Product", "CustomerGender", "Location",
"Region" ,"TotalPrice", "Deposit", "DailyRate", "TotalDays", "AmountPaid30",
"AmountPaid60", "AmountPaid360", "LoanStatus360")])
cor_ds_asses1## Product CustomerGender Location Region TotalPrice
## Product 1.00000000 0.215410318 -0.078985609 -0.078691471 0.27738511
## CustomerGender 0.21541032 1.000000000 -0.025930697 -0.001637478 -0.06148111
## Location -0.07898561 -0.025930697 1.000000000 0.068602167 0.02274191
## Region -0.07869147 -0.001637478 0.068602167 1.000000000 0.02078652
## TotalPrice 0.27738511 -0.061481106 0.022741907 0.020786524 1.00000000
## Deposit 0.08285576 -0.112198881 0.043638353 0.042175248 0.86554131
## DailyRate 0.31995111 -0.043765449 0.015570866 0.013569760 0.98944434
## TotalDays NA NA NA NA NA
## AmountPaid30 0.30512783 -0.019341926 0.016622021 0.037297000 0.63233502
## AmountPaid60 0.31947942 -0.019891374 0.019311098 0.044764604 0.68124418
## AmountPaid360 0.30005376 -0.001685120 0.020024004 0.050803703 0.69030820
## LoanStatus360 0.13397102 0.091076265 -0.002814233 -0.012374521 -0.06902098
## Deposit DailyRate TotalDays AmountPaid30 AmountPaid60
## Product 0.08285576 0.31995111 NA 0.30512783 0.31947942
## CustomerGender -0.11219888 -0.04376545 NA -0.01934193 -0.01989137
## Location 0.04363835 0.01557087 NA 0.01662202 0.01931110
## Region 0.04217525 0.01356976 NA 0.03729700 0.04476460
## TotalPrice 0.86554131 0.98944434 NA 0.63233502 0.68124418
## Deposit 1.00000000 0.78382689 NA 0.80411920 0.81250190
## DailyRate 0.78382689 1.00000000 NA 0.55135520 0.60957131
## TotalDays NA NA 1 NA NA
## AmountPaid30 0.80411920 0.55135520 NA 1.00000000 0.96219284
## AmountPaid60 0.81250190 0.60957131 NA 0.96219284 1.00000000
## AmountPaid360 0.66266283 0.66416437 NA 0.67589134 0.75124389
## LoanStatus360 -0.04673281 -0.07205609 NA 0.14305659 0.14556791
## AmountPaid360 LoanStatus360
## Product 0.30005376 0.133971022
## CustomerGender -0.00168512 0.091076265
## Location 0.02002400 -0.002814233
## Region 0.05080370 -0.012374521
## TotalPrice 0.69030820 -0.069020979
## Deposit 0.66266283 -0.046732808
## DailyRate 0.66416437 -0.072056093
## TotalDays NA NA
## AmountPaid30 0.67589134 0.143056589
## AmountPaid60 0.75124389 0.145567908
## AmountPaid360 1.00000000 0.109930606
## LoanStatus360 0.10993061 1.000000000
#Drop TotalDays column as it is not correlated with the rest of the columns
cor_ds_asses <- round(cor(ds_asses [, c("Product", "CustomerGender", "Location",
"Region" ,"TotalPrice", "Deposit", "DailyRate", "AmountPaid30", "AmountPaid60",
"AmountPaid360", "LoanStatus360")]),2)
cor_ds_asses## Product CustomerGender Location Region TotalPrice Deposit
## Product 1.00 0.22 -0.08 -0.08 0.28 0.08
## CustomerGender 0.22 1.00 -0.03 0.00 -0.06 -0.11
## Location -0.08 -0.03 1.00 0.07 0.02 0.04
## Region -0.08 0.00 0.07 1.00 0.02 0.04
## TotalPrice 0.28 -0.06 0.02 0.02 1.00 0.87
## Deposit 0.08 -0.11 0.04 0.04 0.87 1.00
## DailyRate 0.32 -0.04 0.02 0.01 0.99 0.78
## AmountPaid30 0.31 -0.02 0.02 0.04 0.63 0.80
## AmountPaid60 0.32 -0.02 0.02 0.04 0.68 0.81
## AmountPaid360 0.30 0.00 0.02 0.05 0.69 0.66
## LoanStatus360 0.13 0.09 0.00 -0.01 -0.07 -0.05
## DailyRate AmountPaid30 AmountPaid60 AmountPaid360 LoanStatus360
## Product 0.32 0.31 0.32 0.30 0.13
## CustomerGender -0.04 -0.02 -0.02 0.00 0.09
## Location 0.02 0.02 0.02 0.02 0.00
## Region 0.01 0.04 0.04 0.05 -0.01
## TotalPrice 0.99 0.63 0.68 0.69 -0.07
## Deposit 0.78 0.80 0.81 0.66 -0.05
## DailyRate 1.00 0.55 0.61 0.66 -0.07
## AmountPaid30 0.55 1.00 0.96 0.68 0.14
## AmountPaid60 0.61 0.96 1.00 0.75 0.15
## AmountPaid360 0.66 0.68 0.75 1.00 0.11
## LoanStatus360 -0.07 0.14 0.15 0.11 1.00
# we create a heatmap to visualize correlations
cor.plot((ds_asses [, c("Product", "CustomerGender", "Location", "Region" ,
"TotalPrice", "Deposit", "DailyRate", "AmountPaid30", "AmountPaid60",
"AmountPaid360", "LoanStatus360")]),numbers=TRUE,main="corlpot")
#or
# Get lower triangle of the correlation matrix
get_lower_tri <- function(cor_ds_asses){
cor_ds_asses[upper.tri(cor_ds_asses)]<- NA
return(cor_ds_asses)
}
lower_tri <- get_lower_tri(cor_ds_asses)
lower_tri## Product CustomerGender Location Region TotalPrice Deposit
## Product 1.00 NA NA NA NA NA
## CustomerGender 0.22 1.00 NA NA NA NA
## Location -0.08 -0.03 1.00 NA NA NA
## Region -0.08 0.00 0.07 1.00 NA NA
## TotalPrice 0.28 -0.06 0.02 0.02 1.00 NA
## Deposit 0.08 -0.11 0.04 0.04 0.87 1.00
## DailyRate 0.32 -0.04 0.02 0.01 0.99 0.78
## AmountPaid30 0.31 -0.02 0.02 0.04 0.63 0.80
## AmountPaid60 0.32 -0.02 0.02 0.04 0.68 0.81
## AmountPaid360 0.30 0.00 0.02 0.05 0.69 0.66
## LoanStatus360 0.13 0.09 0.00 -0.01 -0.07 -0.05
## DailyRate AmountPaid30 AmountPaid60 AmountPaid360 LoanStatus360
## Product NA NA NA NA NA
## CustomerGender NA NA NA NA NA
## Location NA NA NA NA NA
## Region NA NA NA NA NA
## TotalPrice NA NA NA NA NA
## Deposit NA NA NA NA NA
## DailyRate 1.00 NA NA NA NA
## AmountPaid30 0.55 1.00 NA NA NA
## AmountPaid60 0.61 0.96 1.00 NA NA
## AmountPaid360 0.66 0.68 0.75 1.00 NA
## LoanStatus360 -0.07 0.14 0.15 0.11 1
# Get upper triangle of the correlation matrix
get_upper_tri <- function(cor_ds_asses){
cor_ds_asses[lower.tri(cor_ds_asses)]<- NA
return(cor_ds_asses)
}
upper_tri <- get_upper_tri(cor_ds_asses)
upper_tri## Product CustomerGender Location Region TotalPrice Deposit
## Product 1 0.22 -0.08 -0.08 0.28 0.08
## CustomerGender NA 1.00 -0.03 0.00 -0.06 -0.11
## Location NA NA 1.00 0.07 0.02 0.04
## Region NA NA NA 1.00 0.02 0.04
## TotalPrice NA NA NA NA 1.00 0.87
## Deposit NA NA NA NA NA 1.00
## DailyRate NA NA NA NA NA NA
## AmountPaid30 NA NA NA NA NA NA
## AmountPaid60 NA NA NA NA NA NA
## AmountPaid360 NA NA NA NA NA NA
## LoanStatus360 NA NA NA NA NA NA
## DailyRate AmountPaid30 AmountPaid60 AmountPaid360 LoanStatus360
## Product 0.32 0.31 0.32 0.30 0.13
## CustomerGender -0.04 -0.02 -0.02 0.00 0.09
## Location 0.02 0.02 0.02 0.02 0.00
## Region 0.01 0.04 0.04 0.05 -0.01
## TotalPrice 0.99 0.63 0.68 0.69 -0.07
## Deposit 0.78 0.80 0.81 0.66 -0.05
## DailyRate 1.00 0.55 0.61 0.66 -0.07
## AmountPaid30 NA 1.00 0.96 0.68 0.14
## AmountPaid60 NA NA 1.00 0.75 0.15
## AmountPaid360 NA NA NA 1.00 0.11
## LoanStatus360 NA NA NA NA 1.00
# Reorder the correlation matrix
reorder_cor_ds_asses <- function(cor_ds_asses){
# Use correlation between variables as distance
dd <- as.dist((1-cor_ds_asses)/2)
hc <- hclust(dd)
cor_ds_asses <-cor_ds_asses[hc$order, hc$order]
}
cor_ds_asses2 <- reorder_cor_ds_asses(cor_ds_asses)
# Melt the correlation matrix
library(reshape2)
melted_cor_ds_asses <- melt(upper_tri, na.rm = TRUE)
library(ggplot2)
ggplot(data = melted_cor_ds_asses, aes(Var2, Var1, fill = value))+
geom_tile(color = "white")+
scale_fill_gradient2(low = "blue", high = "red", mid = "green",
midpoint = 0, limit = c(-1,1), space = "Lab",
name="Pearson\nCorrelation") +
theme_minimal()+
theme(axis.text.x = element_text(angle = 45, vjust = 1,
size = 12, hjust = 1))+
coord_fixed()
#additive model investigates only the main effects of predictors.
#It assumes that the relationship between a given predictor variable
#and the outcome is independent of the other predictor variables
#From the correlation table, we see that the AmountPaid360 is most
#strongly correlated with AmountPaid60. But in addition, we also see that
#TotalPrice and Deposit are positively correlated.
#We want to compute the association of the outcome variable AmountPaid360
#with each predictor while adjusting for the effect of the remaining predictors.
#Both the multivariate regression and the correlation measure how strongly
#the features are associated with the outcome (AmountPaid360). When comparing
#the regression coefficients with the correlation coefficients, we will see
#that they are slightly different.Ccorrelation computes bivariate relationship
#between the outcome variable and a single predictor (ignoring all other
#predictors) while a multiple regression asks if that predictor is related
#to the outcome after controlling for all the other predictors in the model.
mlm1a <- lm(AmountPaid360 ~ Product + CustomerGender + Location + Region
+ TotalPrice + Deposit + DailyRate + AmountPaid30 + AmountPaid60,
data=ds_asses) #multivariate regression
summary(mlm1a)##
## Call:
## lm(formula = AmountPaid360 ~ Product + CustomerGender + Location +
## Region + TotalPrice + Deposit + DailyRate + AmountPaid30 +
## AmountPaid60, data = ds_asses)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25651 -2483 1013 2934 14683
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -869.22408 236.30752 -3.678 0.000236 ***
## Product -98.95731 53.05113 -1.865 0.062168 .
## CustomerGender 174.88070 74.04345 2.362 0.018204 *
## Location 0.13513 0.16641 0.812 0.416790
## Region 27.16438 8.84915 3.070 0.002149 **
## TotalPrice 0.51187 0.01701 30.093 < 2e-16 ***
## Deposit -1.18053 0.09674 -12.203 < 2e-16 ***
## DailyRate NA NA NA NA
## AmountPaid30 -1.10490 0.06665 -16.577 < 2e-16 ***
## AmountPaid60 2.51516 0.05841 43.064 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3934 on 9186 degrees of freedom
## Multiple R-squared: 0.6531, Adjusted R-squared: 0.6528
## F-statistic: 2162 on 8 and 9186 DF, p-value: < 2.2e-16
#DropDailyRate
mlm2a <- lm(AmountPaid360 ~ Product + CustomerGender + Location +
Region + TotalPrice + Deposit + AmountPaid30 + AmountPaid60,
data=ds_asses) #multivariate regression
summary(mlm2a)##
## Call:
## lm(formula = AmountPaid360 ~ Product + CustomerGender + Location +
## Region + TotalPrice + Deposit + AmountPaid30 + AmountPaid60,
## data = ds_asses)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25651 -2483 1013 2934 14683
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -869.22408 236.30752 -3.678 0.000236 ***
## Product -98.95731 53.05113 -1.865 0.062168 .
## CustomerGender 174.88070 74.04345 2.362 0.018204 *
## Location 0.13513 0.16641 0.812 0.416790
## Region 27.16438 8.84915 3.070 0.002149 **
## TotalPrice 0.51187 0.01701 30.093 < 2e-16 ***
## Deposit -1.18053 0.09674 -12.203 < 2e-16 ***
## AmountPaid30 -1.10490 0.06665 -16.577 < 2e-16 ***
## AmountPaid60 2.51516 0.05841 43.064 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3934 on 9186 degrees of freedom
## Multiple R-squared: 0.6531, Adjusted R-squared: 0.6528
## F-statistic: 2162 on 8 and 9186 DF, p-value: < 2.2e-16
#Observed value in data is 13621
newdata <- data.frame(Product =2, CustomerGender=1, Location=109,
Region=11, TotalPrice=16600, Deposit=2000, AmountPaid30=3121,
AmountPaid60=4241)
p <- predict(mlm2a, newdata, interval="confidence")
p## fit lwr upr
## 1 12775.6 12615.51 12935.7
#Drop Location does not contribute to the model
mlm3a <- lm(AmountPaid360 ~ Product + CustomerGender + Region +
TotalPrice + Deposit + AmountPaid30 + AmountPaid60, data=ds_asses)
#multivariate regression
summary(mlm3a)##
## Call:
## lm(formula = AmountPaid360 ~ Product + CustomerGender + Region +
## TotalPrice + Deposit + AmountPaid30 + AmountPaid60, data = ds_asses)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25631 -2490 1004 2931 14661
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -825.91773 230.20676 -3.588 0.000335 ***
## Product -101.57837 52.95186 -1.918 0.055102 .
## CustomerGender 174.65458 74.04156 2.359 0.018351 *
## Region 27.59426 8.83313 3.124 0.001790 **
## TotalPrice 0.51190 0.01701 30.096 < 2e-16 ***
## Deposit -1.17973 0.09673 -12.196 < 2e-16 ***
## AmountPaid30 -1.10534 0.06665 -16.584 < 2e-16 ***
## AmountPaid60 2.51566 0.05840 43.076 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3934 on 9187 degrees of freedom
## Multiple R-squared: 0.6531, Adjusted R-squared: 0.6528
## F-statistic: 2471 on 7 and 9187 DF, p-value: < 2.2e-16
#Observed value in data is 13621
newdata1 <- data.frame(Product=2, CustomerGender=1, Region=11,
TotalPrice=16600, Deposit=2000, AmountPaid30=3121, AmountPaid60=4241)
p1 <- predict(mlm3a, newdata1, interval="confidence")
p1## fit lwr upr
## 1 12806.45 12664.74 12948.17
#Drop Product does not contribute to the model
#Final Model
mlm4a <- lm(AmountPaid360 ~ CustomerGender + Region + TotalPrice +
Deposit + AmountPaid30 + AmountPaid60, data=ds_asses)
#multivariate regression
summary(mlm4a)##
## Call:
## lm(formula = AmountPaid360 ~ CustomerGender + Region + TotalPrice +
## Deposit + AmountPaid30 + AmountPaid60, data = ds_asses)
##
## Residuals:
## Min 1Q Median 3Q Max
## -25651 -2488 1009 2920 14711
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -794.04659 229.63994 -3.458 0.000547 ***
## CustomerGender 150.33778 72.95903 2.061 0.039371 *
## Region 29.20511 8.79441 3.321 0.000901 ***
## TotalPrice 0.49460 0.01442 34.292 < 2e-16 ***
## Deposit -1.07204 0.07879 -13.607 < 2e-16 ***
## AmountPaid30 -1.13537 0.06480 -17.522 < 2e-16 ***
## AmountPaid60 2.51027 0.05834 43.027 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3934 on 9188 degrees of freedom
## Multiple R-squared: 0.653, Adjusted R-squared: 0.6527
## F-statistic: 2881 on 6 and 9188 DF, p-value: < 2.2e-16
#Observed value in data is 13621
newdata2 <- data.frame(CustomerGender=1, Region=11, TotalPrice=16600,
Deposit=2000, AmountPaid30=3121, AmountPaid60=4241)
p2 <- predict(mlm4a, newdata2, interval="confidence")
p2## fit lwr upr
## 1 12846.47 12710.76 12982.17
sigma(mlm4a) # Residual Standard Error (RSE). The lower the RSE, the more ## [1] 3934.305
#accurate the model (on the data in hand).
sigma(mlm4a)/mean(ds_asses$AmountPaid360) #The error rate## [1] 0.2681171
coefficients(mlm4a) # model coefficients## (Intercept) CustomerGender Region TotalPrice Deposit
## -794.0465916 150.3377772 29.2051094 0.4946036 -1.0720393
## AmountPaid30 AmountPaid60
## -1.1353656 2.5102698
confint(mlm4a, level=0.95) # CIs for model parameters## 2.5 % 97.5 %
## (Intercept) -1244.1918987 -343.9012846
## CustomerGender 7.3218671 293.3536872
## Region 11.9661084 46.4441103
## TotalPrice 0.4663306 0.5228766
## Deposit -1.2264811 -0.9175974
## AmountPaid30 -1.2623783 -1.0083530
## AmountPaid60 2.3959069 2.6246327
anova(mlm4a) # anova table## Analysis of Variance Table
##
## Response: AmountPaid360
## Df Sum Sq Mean Sq F value Pr(>F)
## CustomerGender 1 1.1637e+06 1.1637e+06 0.0752 0.7839
## Region 1 1.0576e+09 1.0576e+09 68.3246 <2e-16 ***
## TotalPrice 1 1.9545e+11 1.9545e+11 12626.8814 <2e-16 ***
## Deposit 1 7.3892e+09 7.3892e+09 477.3748 <2e-16 ***
## AmountPaid30 1 3.5026e+10 3.5026e+10 2262.8199 <2e-16 ***
## AmountPaid60 1 2.8656e+10 2.8656e+10 1851.3177 <2e-16 ***
## Residuals 9188 1.4222e+11 1.5479e+07
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
vcov(mlm4a) # covariance matrix for model parameters## (Intercept) CustomerGender Region TotalPrice
## (Intercept) 5.273450e+04 -8.396549e+03 -9.044206e+02 -1.9630901174
## CustomerGender -8.396549e+03 5.323020e+03 -3.338803e+00 -0.0993581245
## Region -9.044206e+02 -3.338803e+00 7.734168e+01 0.0050764365
## TotalPrice -1.963090e+00 -9.935812e-02 5.076437e-03 0.0002080335
## Deposit 6.052000e+00 1.030079e+00 -2.701815e-02 -0.0008656772
## AmountPaid30 -7.141471e-01 -2.487421e-01 1.979133e-02 0.0003067156
## AmountPaid60 2.152767e-02 -4.203525e-02 -1.941552e-02 -0.0002178811
## Deposit AmountPaid30 AmountPaid60
## (Intercept) 6.0520004193 -0.7141471317 0.0215276680
## CustomerGender 1.0300788573 -0.2487420904 -0.0420352501
## Region -0.0270181465 0.0197913250 -0.0194155166
## TotalPrice -0.0008656772 0.0003067156 -0.0002178811
## Deposit 0.0062075375 -0.0017222672 0.0002120275
## AmountPaid30 -0.0017222672 0.0041983941 -0.0033896522
## AmountPaid60 0.0002120275 -0.0033896522 0.0034037673
#fitted(mlm4a) # predicted values#residuals(mlm4a) # residuals#influence(mlm4a) # regression diagnostics $hat $coefficients $sigma
#$wt. residuals#Understanding Accuracy of the final model with Actual vs Predicted
plot(fitted(mlm4a), ds_asses$AmountPaid360,
ylab="Actual", xlab="Predicted",
main="AmountPaid360")
#Accuracy of the final model with Residual vs Predicted
#positive values for the residual (on the y-axis) mean the prediction
#was too low, and negative values mean the prediction was too high;
#0 means the guess was exactly correct.
plot(fitted(mlm4a), residuals(mlm4a),
ylab="Residuals", xlab="Predicted",
main="AmountPaid360")
abline(0,0)
#The final multiple regression model equation can be written as AmountPaid360 =
#-643.7088144 + 150.3377772*CustomerGender + 29.2051094*Region +
#0.4946036*TotalPrice - 1.0720393*Deposit - 1.1353656*AmountPaid30 +
#2.5102698*AmountPaid60.
#The overall quality of the model can be assessed by examining the R-squared (R2)
#and Residual Standard Error (RSE).
#R2 represents the correlation coefficient between the observed values of the
#outcome variable (y) and the predicted values of y. For this reason, the value of
#R will always be positive and will range from zero to one. R2 represents the
#proportion of variance, in the outcome variable y, that may be predicted by
#knowing the value of the x variables. An R2 value close to 1 indicates that the
#model explains a large portion of the variance in the outcome variable. Problem
#with the R2, is that, it will always increase when more variables are added to
#the model, even if those variables are only weakly associated with the response. A
#solution is to adjust the R2 by taking into account the number of predictor variables.
#The adjusted R2 = 0.6527, meaning that “65.27% of the variance in the measure of
#AmountPaid360 can be predicted by CustomerGender, Region, TotalPrice, Deposit,
#AmountPaid30 and AmountPaid60.
#In our multiple regression model, the RSE is 3934.305 corresponding to 26.8% error rate.#Classification model to predict LoanStatus360
ds_asses1 <- ds_asses [, c("Product", "CustomerGender", "Location", "Region" ,
"TotalPrice", "Deposit", "DailyRate", "AmountPaid30", "AmountPaid60",
"LoanStatus360")]
# see the stucture
head(ds_asses1)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 1 2 2 109 11 16600 2000 40
## 2 2 2 220 12 16600 2000 40
## 3 3 1 442 14 15600 1000 40
## 4 2 2 553 1 16600 2000 40
## 5 4 1 660 11 20250 2000 50
## 6 2 1 2 11 16600 2000 40
## AmountPaid30 AmountPaid60 LoanStatus360
## 1 3121 4241 1
## 2 3061 4171 1
## 3 2160 3280 1
## 4 3041 4241 1
## 5 3470 4820 1
## 6 3201 4141 1
tail(ds_asses1)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 9190 3 2 447 15 15600 1000 40
## 9191 1 1 221 11 20250 2000 50
## 9192 4 1 520 15 20250 2000 50
## 9193 2 1 763 11 16600 2000 40
## 9194 2 2 331 13 16600 2000 40
## 9195 4 2 67 1 20250 2000 50
## AmountPaid30 AmountPaid60 LoanStatus360
## 9190 2040 3400 3
## 9191 3601 4301 3
## 9192 4001 5501 3
## 9193 3401 4601 3
## 9194 3400 4600 3
## 9195 6950 6950 3
#1.Shuffle your data
#From the head and tail output, you can see if the data is shuffled or not.
#If not shuffled it will be a big issue! When you will split your data between
#a train set and test set, you will select only loanstatus 0 and 1 (No 2 are in
#the top 80 percent of the observations), which means the algorithm will never see
#the features of 2. This mistake will lead to poor prediction.
shuffle_index <- sample(1:nrow(ds_asses1)) #Generate a random list of index from
#1 to 9195
head(shuffle_index)## [1] 7462 7970 6570 505 890 4030
tail(shuffle_index)## [1] 2973 5301 6104 9144 4060 5808
ds_asses2 <- ds_asses1[shuffle_index, ]
head(ds_asses2)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 7462 3 3 548 11 15600 1000 40
## 7970 2 2 243 15 16600 2000 40
## 6570 4 2 423 15 20250 2000 50
## 505 2 2 354 13 16600 2000 40
## 890 1 2 784 15 20750 2500 50
## 4030 1 1 165 12 19250 1000 50
## AmountPaid30 AmountPaid60 LoanStatus360
## 7462 2040 2640 2
## 7970 3141 4201 3
## 6570 3301 4891 1
## 505 3041 4001 1
## 890 3400 3600 1
## 4030 500 2150 1
tail(ds_asses2)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 2973 2 1 553 1 16600 2000 40
## 5301 2 1 41 1 16600 2000 40
## 6104 3 2 96 15 15600 1000 40
## 9144 3 2 86 15 15600 1000 40
## 4060 3 2 373 17 15600 1000 40
## 5808 2 2 376 1 16600 2000 40
## AmountPaid30 AmountPaid60 LoanStatus360
## 2973 3251 3891 1
## 5301 3261 4251 1
## 6104 1600 2000 1
## 9144 2300 3400 3
## 4060 2200 3280 1
## 5808 3391 4351 1
#2.Create a train and test set: You train the model on the train set and test the
#prediction on the test set (i.e. unseen data)
#Split the data 80/20, 80 percent of the data serves to train the model, and 20
#percent to make predictions
#Function to create two separate data frames. You don't want to touch the test set
#until you finish building your model
set.seed(1729)
create_train_test <- function(data, size = 0.8, train = TRUE) #Add the arguments in
#the function
{
n_row = nrow(data) #Count number of rows in the dataset
total_row = size * n_row #Return the nth row to construct the train set
train_sample <- 1: total_row #Select the first row to the nth rows
if (train == TRUE) {
return (data[train_sample, ]) #If condition sets to true, return the
#train set, else the test set
} else {
return (data[-train_sample, ])
}
}
data_train <- create_train_test(ds_asses2, 0.8, train = TRUE)
data_test <- create_train_test(ds_asses2, 0.8, train = FALSE)
dim(data_train)## [1] 7356 10
dim(data_test)## [1] 1839 10
#Verify Randomization Process
prop.table(table(data_train$CustomerGender))##
## 1 2 3
## 0.35440457 0.58401305 0.06158238
prop.table(table(data_test$CustomerGender))##
## 1 2 3
## 0.37139750 0.57748777 0.05111474
#3.Buildthemodel
library(rpart)
fit <- rpart(LoanStatus360~., data = data_train, method = 'class')
fit## n= 7356
##
## node), split, n, loss, yval, (yprob)
## * denotes terminal node
##
## 1) root 7356 1977 1 (0.73123980 0.09284937 0.17591082) *
#4.Make a Prediction
predict_unseen <-predict(fit, data_test, type = 'class')
table_mat <- table(data_test$LoanStatus360, predict_unseen)
table_mat## predict_unseen
## 1 2 3
## 1 1363 0 0
## 2 166 0 0
## 3 310 0 0
library(gmodels)
#understand the relationship of the classes of your test data related to your model
CrossTable(x = data_test$LoanStatus360, y = predict_unseen, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 1839
##
##
## | predict_unseen
## data_test$LoanStatus360 | 1 | Row Total |
## ------------------------|-----------|-----------|
## 1 | 1363 | 1363 |
## | 0.741 | |
## ------------------------|-----------|-----------|
## 2 | 166 | 166 |
## | 0.090 | |
## ------------------------|-----------|-----------|
## 3 | 310 | 310 |
## | 0.169 | |
## ------------------------|-----------|-----------|
## Column Total | 1839 | 1839 |
## ------------------------|-----------|-----------|
##
##
#5.MeasurePerfomance
#compute an accuracy measure for classification task with the confusion matrix. The
#general idea is to count the number of times True instances are classified as False.
accuracy_Test <- sum(diag(table_mat)) / sum(table_mat)
print(paste('Accuracy for test', accuracy_Test))## [1] "Accuracy for test 0.741163675910821"
#Knn Classification model to predict LoanStatus360
# see the stucture
head(ds_asses1)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 1 2 2 109 11 16600 2000 40
## 2 2 2 220 12 16600 2000 40
## 3 3 1 442 14 15600 1000 40
## 4 2 2 553 1 16600 2000 40
## 5 4 1 660 11 20250 2000 50
## 6 2 1 2 11 16600 2000 40
## AmountPaid30 AmountPaid60 LoanStatus360
## 1 3121 4241 1
## 2 3061 4171 1
## 3 2160 3280 1
## 4 3041 4241 1
## 5 3470 4820 1
## 6 3201 4141 1
tail(ds_asses1)## Product CustomerGender Location Region TotalPrice Deposit DailyRate
## 9190 3 2 447 15 15600 1000 40
## 9191 1 1 221 11 20250 2000 50
## 9192 4 1 520 15 20250 2000 50
## 9193 2 1 763 11 16600 2000 40
## 9194 2 2 331 13 16600 2000 40
## 9195 4 2 67 1 20250 2000 50
## AmountPaid30 AmountPaid60 LoanStatus360
## 9190 2040 3400 3
## 9191 3601 4301 3
## 9192 4001 5501 3
## 9193 3401 4601 3
## 9194 3400 4600 3
## 9195 6950 6950 3
# Division of `LoanStatus360`
table(ds_asses1$LoanStatus360) ##
## 1 2 3
## 6742 849 1604
# Percentual division of `LoanStatus360`
round(prop.table(table(ds_asses1$LoanStatus360)) * 100, digits = 2)##
## 1 2 3
## 73.32 9.23 17.44
##Generate a random number that is 70% of the total number of rows in dataset.
ran <- sample(1:nrow(ds_asses1), 0.7 * nrow(ds_asses1))
##the normalization function is created
nor <-function(x) { (x -min(x))/(max(x)-min(x)) }
##Run nomalization on predictor coulumns of dataset
ds_asses1_norm <- as.data.frame(lapply(ds_asses[,c("Product", "CustomerGender",
"Location", "Region" ,"TotalPrice", "Deposit", "DailyRate", "AmountPaid30",
"AmountPaid60")], nor))
summary(ds_asses1_norm)## Product CustomerGender Location Region
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.1633 1st Qu.:0.5882
## Median :0.2000 Median :0.5000 Median :0.3848 Median :0.6471
## Mean :0.2102 Mean :0.3508 Mean :0.4291 Mean :0.6225
## 3rd Qu.:0.2000 3rd Qu.:0.5000 3rd Qu.:0.6987 3rd Qu.:0.8235
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
## TotalPrice Deposit DailyRate AmountPaid30
## Min. :0.00000 Min. :0.0000 Min. :0.00000 Min. :0.0000
## 1st Qu.:0.05095 1st Qu.:0.1250 1st Qu.:0.03846 1st Qu.:0.1952
## Median :0.05095 Median :0.1250 Median :0.03846 Median :0.2243
## Mean :0.09903 Mean :0.1639 Mean :0.08809 Mean :0.2322
## 3rd Qu.:0.11677 3rd Qu.:0.1250 3rd Qu.:0.11538 3rd Qu.:0.2671
## Max. :1.00000 Max. :1.0000 Max. :1.00000 Max. :1.0000
## AmountPaid60
## Min. :0.0000
## 1st Qu.:0.1809
## Median :0.2071
## Mean :0.2162
## 3rd Qu.:0.2416
## Max. :1.0000
##extract training set
ds_asses1_train <- ds_asses1_norm[ran,]
##extract testing set
ds_asses1_test <- ds_asses1_norm[-ran,]
dim(ds_asses1_train)## [1] 6436 9
dim(ds_asses1_test)## [1] 2759 9
#Verify Randomization Process
prop.table(table(ds_asses1_train$Product))##
## 0 0.2 0.4 0.6 0.8 1
## 0.34431324 0.41858297 0.11062772 0.10348042 0.02221877 0.00077688
prop.table(table(ds_asses1_test$Product))##
## 0 0.2 0.4 0.6 0.8
## 0.33526640 0.42334179 0.10402320 0.11199710 0.02537151
prop.table(table(ds_asses1_train$CustomerGender))##
## 0 0.5 1
## 0.35783095 0.58141703 0.06075202
prop.table(table(ds_asses1_test$CustomerGender))##
## 0 0.5 1
## 0.35773831 0.58571946 0.05654223
##extract 10th column of train dataset because it will be used as 'cl' argument
#in knn function.
ds_asses1_target_category <- ds_asses1[ran,10]
##extract 10th column if test dataset to measure the accuracy
ds_asses1_test_category <- ds_asses1[-ran,10]
##load the package class
library(class)
#To choose the appropiate number of k, the most widely used method is to do the
#square root of the rows
sqrt(nrow(ds_asses1))## [1] 95.89056
##run knn function
prk <- knn(ds_asses1_train,ds_asses1_test,cl=ds_asses1_target_category,k=96,prob = TRUE)
prk## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [149] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [223] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [260] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [297] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [334] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [371] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [408] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [445] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [482] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [519] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [556] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [593] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [630] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [667] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [704] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [741] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [778] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [815] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [852] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [889] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [926] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [963] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1000] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1037] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1074] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1111] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1148] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1185] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1222] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1259] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1296] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1333] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1370] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1407] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1444] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1481] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1518] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1555] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1592] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1629] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1666] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1703] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1740] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1777] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1814] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1851] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1888] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1925] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1962] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [1999] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2036] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2073] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2110] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2147] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2184] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2221] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2258] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2295] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2332] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2369] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2406] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2443] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2480] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2517] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2554] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2591] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2628] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2665] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2702] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [2739] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## attr(,"prob")
## [1] 0.6979167 0.7500000 0.8437500 0.7500000 0.8333333 0.8645833 0.5773196
## [8] 0.7755102 0.6391753 0.7083333 0.7708333 0.8229167 0.5773196 0.7812500
## [15] 0.6562500 0.7708333 0.7083333 0.9375000 0.6250000 0.5625000 0.7395833
## [22] 0.7395833 0.7708333 0.6145833 0.7187500 0.7812500 0.8333333 0.8125000
## [29] 0.7395833 0.8333333 0.7395833 0.7187500 0.8333333 0.6082474 0.8854167
## [36] 0.8020833 0.5625000 0.6224490 0.8854167 0.8333333 0.8854167 0.8125000
## [43] 0.8229167 0.8333333 0.6875000 0.8333333 0.8958333 0.7916667 0.8229167
## [50] 0.6082474 0.8125000 0.8333333 0.7708333 0.7604167 0.7916667 0.8541667
## [57] 0.8541667 0.8125000 0.8958333 0.7087379 0.5625000 0.7395833 0.9375000
## [64] 0.7291667 0.8645833 0.7916667 0.8437500 0.8333333 0.7395833 0.8333333
## [71] 0.7500000 0.6666667 0.7708333 0.9166667 0.7708333 0.9166667 0.9062500
## [78] 0.7500000 0.8958333 0.6770833 0.6979167 0.8645833 0.8762887 0.5520833
## [85] 0.8333333 0.6458333 0.6875000 0.8750000 0.7708333 0.6979167 0.6701031
## [92] 0.8333333 0.8020833 0.8437500 0.8854167 0.8437500 0.6875000 0.6597938
## [99] 0.7938144 0.7083333 0.7708333 0.6041667 0.7500000 0.8333333 0.6666667
## [106] 0.7500000 0.9583333 0.8125000 0.8437500 0.7395833 0.8020833 0.8020833
## [113] 0.6979167 0.6185567 0.7083333 0.6250000 0.8645833 0.5937500 0.8229167
## [120] 0.7083333 0.8437500 0.8333333 0.6354167 0.8125000 0.8645833 0.7708333
## [127] 0.6562500 0.6875000 0.9062500 0.6597938 0.8437500 0.7395833 0.8854167
## [134] 0.7291667 0.7812500 0.8333333 0.7187500 0.6458333 0.8437500 0.6979167
## [141] 0.7395833 0.8041237 0.8125000 0.8125000 0.7708333 0.7604167 0.7500000
## [148] 0.7291667 0.5625000 0.8229167 0.6666667 0.8958333 0.8020833 0.8333333
## [155] 0.7604167 0.7812500 0.8125000 0.7083333 0.7916667 0.7916667 0.8958333
## [162] 0.7916667 0.8854167 0.7708333 0.7083333 0.6979167 0.7291667 0.7812500
## [169] 0.8750000 0.8020833 0.6354167 0.8333333 0.6145833 0.7187500 0.7812500
## [176] 0.7812500 0.7812500 0.8854167 0.7708333 0.8645833 0.6666667 0.6458333
## [183] 0.7916667 0.8020833 0.7500000 0.7395833 0.7083333 0.7395833 0.7395833
## [190] 0.8541667 0.7916667 0.7916667 0.7500000 0.8750000 0.7083333 0.8645833
## [197] 0.6770833 0.7916667 0.7291667 0.8333333 0.7525773 0.8020833 0.7604167
## [204] 0.8854167 0.6875000 0.7916667 0.7708333 0.7604167 0.7083333 0.7291667
## [211] 0.7604167 0.7400000 0.6562500 0.7708333 0.6979167 0.7708333 0.7812500
## [218] 0.6979167 0.8333333 0.5833333 0.7187500 0.8020833 0.5208333 0.6458333
## [225] 0.8645833 0.5625000 0.6494845 0.8229167 0.7708333 0.8333333 0.8333333
## [232] 0.7291667 0.6562500 0.6804124 0.7916667 0.7604167 0.8854167 0.6701031
## [239] 0.7187500 0.8229167 0.7083333 0.7500000 0.7708333 0.7916667 0.8229167
## [246] 0.7916667 0.8437500 0.5416667 0.8229167 0.7708333 0.7291667 0.7916667
## [253] 0.8437500 0.7604167 0.6562500 0.7812500 0.7812500 0.7604167 0.6145833
## [260] 0.7083333 0.6979167 0.8645833 0.8247423 0.6875000 0.9166667 0.6875000
## [267] 0.7812500 0.8229167 0.7083333 0.8229167 0.6770833 0.8125000 0.6354167
## [274] 0.7604167 0.6145833 0.7272727 0.8958333 0.6562500 0.8645833 0.8865979
## [281] 0.5416667 0.7938144 0.7708333 0.7916667 0.6145833 0.7500000 0.6562500
## [288] 0.5416667 0.8437500 0.7291667 0.8333333 0.7916667 0.7395833 0.8958333
## [295] 0.6875000 0.7812500 0.8125000 0.6666667 0.6354167 0.7812500 0.8020833
## [302] 0.7708333 0.6770833 0.7604167 0.8645833 0.8125000 0.6770833 0.8437500
## [309] 0.5729167 0.6145833 0.7395833 0.8041237 0.9479167 0.7187500 0.7812500
## [316] 0.8229167 0.6354167 0.6979167 0.6354167 0.7291667 0.7083333 0.8125000
## [323] 0.7812500 0.8958333 0.7083333 0.8020833 0.7187500 0.8333333 0.8865979
## [330] 0.7291667 0.8125000 0.8020833 0.7916667 0.8020833 0.5729167 0.7604167
## [337] 0.7708333 0.8333333 0.7395833 0.8041237 0.8125000 0.8541667 0.8333333
## [344] 0.7812500 0.6979167 0.6979167 0.9062500 0.6458333 0.8645833 0.7916667
## [351] 0.7395833 0.9062500 0.7604167 0.6666667 0.6875000 0.7812500 0.7187500
## [358] 0.8229167 0.5000000 0.8229167 0.5000000 0.7604167 0.8437500 0.8645833
## [365] 0.6354167 0.7083333 0.7187500 0.6979167 0.9062500 0.6562500 0.7010309
## [372] 0.9062500 0.6597938 0.6288660 0.7187500 0.7500000 0.8125000 0.9166667
## [379] 0.5729167 0.7604167 0.8865979 0.5729167 0.8144330 0.8229167 0.6770833
## [386] 0.7291667 0.7812500 0.7604167 0.7604167 0.7083333 0.7291667 0.7916667
## [393] 0.7395833 0.7500000 0.7083333 0.7708333 0.7187500 0.6875000 0.7812500
## [400] 0.7500000 0.8958333 0.7395833 0.6770833 0.7500000 0.8333333 0.7400000
## [407] 0.7812500 0.8020833 0.5416667 0.7395833 0.7187500 0.6224490 0.7395833
## [414] 0.7291667 0.6979167 0.9062500 0.7291667 0.7916667 0.6770833 0.7187500
## [421] 0.7395833 0.7708333 0.7604167 0.7812500 0.6562500 0.6979167 0.7040816
## [428] 0.6145833 0.8020833 0.6562500 0.7812500 0.8645833 0.7916667 0.7938144
## [435] 0.6875000 0.7422680 0.5937500 0.7395833 0.7187500 0.5520833 0.7500000
## [442] 0.7812500 0.7291667 0.8541667 0.6979167 0.9166667 0.7187500 0.6458333
## [449] 0.8020833 0.7708333 0.8645833 0.8333333 0.5312500 0.7916667 0.5520833
## [456] 0.7500000 0.5729167 0.9062500 0.7812500 0.5773196 0.6875000 0.8854167
## [463] 0.6770833 0.6354167 0.8020833 0.8958333 0.7604167 0.8020833 0.6458333
## [470] 0.8958333 0.7708333 0.7604167 0.7604167 0.6979167 0.8020833 0.7708333
## [477] 0.9166667 0.7708333 0.9062500 0.7083333 0.8229167 0.7708333 0.7916667
## [484] 0.7291667 0.7708333 0.9375000 0.7395833 0.7395833 0.8453608 0.8020833
## [491] 0.8854167 0.7708333 0.8333333 0.6875000 0.8247423 0.8333333 0.8645833
## [498] 0.6562500 0.6250000 0.8333333 0.7916667 0.6458333 0.7812500 0.7604167
## [505] 0.6770833 0.8750000 0.8020833 0.7708333 0.5520833 0.7604167 0.9166667
## [512] 0.7395833 0.6875000 0.7604167 0.6770833 0.7187500 0.6666667 0.9062500
## [519] 0.7812500 0.8125000 0.7187500 0.7346939 0.8969072 0.8750000 0.7291667
## [526] 0.8645833 0.7916667 0.5833333 0.6770833 0.7604167 0.8020833 0.6666667
## [533] 0.7500000 0.7708333 0.6979167 0.8229167 0.6875000 0.6145833 0.9072165
## [540] 0.8541667 0.7291667 0.7187500 0.9166667 0.7291667 0.7083333 0.8020833
## [547] 0.8333333 0.7916667 0.6979167 0.7916667 0.8645833 0.7604167 0.7395833
## [554] 0.7187500 0.7395833 0.7187500 0.8125000 0.6770833 0.7604167 0.8854167
## [561] 0.8854167 0.8541667 0.7187500 0.7916667 0.8333333 0.7395833 0.6562500
## [568] 0.7916667 0.8229167 0.6979167 0.9278351 0.8020833 0.6354167 0.7812500
## [575] 0.8333333 0.8020833 0.7395833 0.7395833 0.8125000 0.7708333 0.6979167
## [582] 0.6770833 0.7916667 0.8645833 0.7708333 0.7291667 0.7187500 0.8854167
## [589] 0.8333333 0.8333333 0.6666667 0.7812500 0.8333333 0.7291667 0.6562500
## [596] 0.6979167 0.7708333 0.7916667 0.7708333 0.8541667 0.7938144 0.7708333
## [603] 0.7812500 0.7708333 0.8750000 0.8645833 0.6562500 0.6250000 0.6770833
## [610] 0.7395833 0.8958333 0.7916667 0.7812500 0.6562500 0.7187500 0.7422680
## [617] 0.6250000 0.6979167 0.6875000 0.7525773 0.8229167 0.8229167 0.8020833
## [624] 0.7916667 0.7395833 0.7812500 0.6666667 0.6666667 0.7500000 0.6041667
## [631] 0.7083333 0.7812500 0.6562500 0.8645833 0.6979167 0.8645833 0.8125000
## [638] 0.8437500 0.7500000 0.8125000 0.8333333 0.8125000 0.8229167 0.8020833
## [645] 0.8020833 0.8020833 0.8958333 0.6666667 0.6562500 0.7395833 0.7500000
## [652] 0.7500000 0.9375000 0.8020833 0.8854167 0.6875000 0.8125000 0.7708333
## [659] 0.7653061 0.7395833 0.5937500 0.8020833 0.8854167 0.7083333 0.5520833
## [666] 0.8125000 0.8229167 0.7272727 0.7187500 0.6145833 0.8020833 0.8958333
## [673] 0.8958333 0.7083333 0.7604167 0.7708333 0.8229167 0.8333333 0.8645833
## [680] 0.7291667 0.6145833 0.8437500 0.8125000 0.7187500 0.7083333 0.8229167
## [687] 0.8854167 0.7916667 0.8750000 0.7291667 0.6666667 0.8125000 0.7708333
## [694] 0.6875000 0.8125000 0.8020833 0.7395833 0.7422680 0.7500000 0.6041667
## [701] 0.6354167 0.6770833 0.6562500 0.5729167 0.7395833 0.6562500 0.7395833
## [708] 0.6875000 0.7500000 0.6666667 0.8125000 0.6666667 0.8659794 0.9270833
## [715] 0.8958333 0.7604167 0.5625000 0.7395833 0.7812500 0.8750000 0.5520833
## [722] 0.7604167 0.7083333 0.8020833 0.8144330 0.6875000 0.6979167 0.7395833
## [729] 0.7812500 0.8229167 0.7187500 0.7422680 0.7291667 0.8854167 0.6979167
## [736] 0.7216495 0.6875000 0.5416667 0.6804124 0.8229167 0.7395833 0.6562500
## [743] 0.7604167 0.8750000 0.7187500 0.9062500 0.5833333 0.7187500 0.7083333
## [750] 0.8125000 0.8750000 0.7083333 0.7604167 0.6979167 0.6875000 0.8958333
## [757] 0.7916667 0.6979167 0.6458333 0.8854167 0.8333333 0.8020833 0.8854167
## [764] 0.7812500 0.6666667 0.8750000 0.7604167 0.7395833 0.6145833 0.7040816
## [771] 0.8125000 0.7291667 0.6666667 0.6979167 0.6354167 0.5625000 0.8958333
## [778] 0.8333333 0.8854167 0.7812500 0.7187500 0.7187500 0.7916667 0.9175258
## [785] 0.8854167 0.7187500 0.8333333 0.7291667 0.8041237 0.8125000 0.8333333
## [792] 0.8041237 0.8020833 0.8958333 0.6562500 0.7812500 0.6979167 0.8645833
## [799] 0.8125000 0.7083333 0.6666667 0.7083333 0.6979167 0.8020833 0.6804124
## [806] 0.8125000 0.6875000 0.8020833 0.7812500 0.7500000 0.6391753 0.7187500
## [813] 0.8229167 0.8333333 0.5670103 0.8750000 0.8020833 0.7916667 0.7812500
## [820] 0.6875000 0.6562500 0.8020833 0.8750000 0.8125000 0.7916667 0.7708333
## [827] 0.7604167 0.8125000 0.7291667 0.7708333 0.7500000 0.6875000 0.7291667
## [834] 0.5729167 0.8958333 0.7916667 0.6458333 0.8229167 0.8125000 0.6875000
## [841] 0.6666667 0.6666667 0.6597938 0.7083333 0.7187500 0.8437500 0.8125000
## [848] 0.8333333 0.8020833 0.8437500 0.6979167 0.7346939 0.7708333 0.8125000
## [855] 0.8229167 0.8125000 0.8125000 0.8333333 0.7500000 0.8125000 0.5520833
## [862] 0.7500000 0.7187500 0.7291667 0.7604167 0.7708333 0.6458333 0.6770833
## [869] 0.7500000 0.6875000 0.8437500 0.7604167 0.6354167 0.7500000 0.7812500
## [876] 0.7812500 0.6770833 0.9166667 0.8958333 0.8229167 0.8229167 0.8020833
## [883] 0.7291667 0.7708333 0.7916667 0.7812500 0.8437500 0.8762887 0.6770833
## [890] 0.8229167 0.7319588 0.9270833 0.9270833 0.8437500 0.7395833 0.8020833
## [897] 0.7395833 0.7083333 0.7812500 0.7500000 0.8437500 0.8854167 0.8762887
## [904] 0.6875000 0.7395833 0.7604167 0.6875000 0.7604167 0.7500000 0.6804124
## [911] 0.7395833 0.7187500 0.7083333 0.7500000 0.6666667 0.7083333 0.7708333
## [918] 0.7083333 0.7291667 0.8958333 0.7708333 0.8125000 0.6770833 0.5833333
## [925] 0.6666667 0.6145833 0.8020833 0.7916667 0.8229167 0.6875000 0.8437500
## [932] 0.7395833 0.7916667 0.7604167 0.7500000 0.7916667 0.7812500 0.7187500
## [939] 0.8750000 0.8333333 0.8020833 0.7708333 0.8854167 0.6354167 0.8125000
## [946] 0.6875000 0.7083333 0.6979167 0.6354167 0.7812500 0.8333333 0.8958333
## [953] 0.8333333 0.7604167 0.6041667 0.8229167 0.8437500 0.7604167 0.8541667
## [960] 0.6562500 0.8958333 0.9166667 0.5625000 0.8020833 0.7708333 0.7708333
## [967] 0.7812500 0.7187500 0.7708333 0.8125000 0.8229167 0.8333333 0.6875000
## [974] 0.7604167 0.8125000 0.7916667 0.5520833 0.6770833 0.7812500 0.7812500
## [981] 0.6562500 0.6562500 0.5937500 0.7812500 0.7604167 0.6562500 0.8750000
## [988] 0.8541667 0.8958333 0.6458333 0.6562500 0.8125000 0.7291667 0.7307692
## [995] 0.8125000 0.8541667 0.9166667 0.6250000 0.7083333 0.8333333 0.7708333
## [1002] 0.7500000 0.8437500 0.8333333 0.6562500 0.6875000 0.6875000 0.8854167
## [1009] 0.7708333 0.6979167 0.7291667 0.8645833 0.7187500 0.8229167 0.7187500
## [1016] 0.6770833 0.7916667 0.8020833 0.8645833 0.7916667 0.7291667 0.7604167
## [1023] 0.7916667 0.7500000 0.7812500 0.8229167 0.8645833 0.6875000 0.6875000
## [1030] 0.8854167 0.8541667 0.7395833 0.7083333 0.6250000 0.5567010 0.7812500
## [1037] 0.7916667 0.6875000 0.7291667 0.7835052 0.8229167 0.5937500 0.7291667
## [1044] 0.7938144 0.8645833 0.7708333 0.7916667 0.7422680 0.6666667 0.7395833
## [1051] 0.8645833 0.6875000 0.8437500 0.6250000 0.7319588 0.6875000 0.5306122
## [1058] 0.5625000 0.7187500 0.6979167 0.8125000 0.9062500 0.6875000 0.8125000
## [1065] 0.7500000 0.8333333 0.7604167 0.8125000 0.8437500 0.7187500 0.6354167
## [1072] 0.7604167 0.7395833 0.9175258 0.7216495 0.7187500 0.6979167 0.7395833
## [1079] 0.7187500 0.8020833 0.8645833 0.6354167 0.8333333 0.7187500 0.8125000
## [1086] 0.7812500 0.7708333 0.8229167 0.8229167 0.7291667 0.8437500 0.5520833
## [1093] 0.9583333 0.8020833 0.7916667 0.8854167 0.7187500 0.8854167 0.6979167
## [1100] 0.8958333 0.9270833 0.8437500 0.7708333 0.7083333 0.7291667 0.8020833
## [1107] 0.7500000 0.7812500 0.6770833 0.7812500 0.7500000 0.7916667 0.6666667
## [1114] 0.8437500 0.7604167 0.8541667 0.7187500 0.6770833 0.8750000 0.8958333
## [1121] 0.7916667 0.7291667 0.8541667 0.6354167 0.5816327 0.7083333 0.6041667
## [1128] 0.8125000 0.6354167 0.7812500 0.6250000 0.7708333 0.7422680 0.7916667
## [1135] 0.7395833 0.7812500 0.8125000 0.9166667 0.7812500 0.8645833 0.6875000
## [1142] 0.8854167 0.7500000 0.6979167 0.8958333 0.7604167 0.8333333 0.7187500
## [1149] 0.8229167 0.7395833 0.8437500 0.7812500 0.7708333 0.8958333 0.7916667
## [1156] 0.7708333 0.6458333 0.8020833 0.7916667 0.8645833 0.8229167 0.8333333
## [1163] 0.7187500 0.8437500 0.8229167 0.7525773 0.8125000 0.7395833 0.7916667
## [1170] 0.6041667 0.7187500 0.7708333 0.7938144 0.7708333 0.7113402 0.8854167
## [1177] 0.7083333 0.7604167 0.7083333 0.8020833 0.7604167 0.7812500 0.6458333
## [1184] 0.7083333 0.6041667 0.9062500 0.7916667 0.5937500 0.6354167 0.8125000
## [1191] 0.8333333 0.6666667 0.8854167 0.7216495 0.7083333 0.8229167 0.8247423
## [1198] 0.8958333 0.7812500 0.8854167 0.8750000 0.8125000 0.8958333 0.7395833
## [1205] 0.6562500 0.8333333 0.7083333 0.7500000 0.8854167 0.9166667 0.7187500
## [1212] 0.7291667 0.7604167 0.6666667 0.6562500 0.8333333 0.6979167 0.7525773
## [1219] 0.8125000 0.7500000 0.7604167 0.8229167 0.7708333 0.7395833 0.8541667
## [1226] 0.7187500 0.6666667 0.7500000 0.7083333 0.7422680 0.8020833 0.6458333
## [1233] 0.7187500 0.7708333 0.8437500 0.6666667 0.8750000 0.6041667 0.6979167
## [1240] 0.6666667 0.9062500 0.7812500 0.8541667 0.9062500 0.6041667 0.8437500
## [1247] 0.6597938 0.8958333 0.7500000 0.7500000 0.8541667 0.7916667 0.7500000
## [1254] 0.5937500 0.8229167 0.8437500 0.8125000 0.7916667 0.7604167 0.6875000
## [1261] 0.6562500 0.9166667 0.7708333 0.8750000 0.7812500 0.5416667 0.8229167
## [1268] 0.8437500 0.8645833 0.7083333 0.7500000 0.8958333 0.7500000 0.7187500
## [1275] 0.7812500 0.7812500 0.6979167 0.8333333 0.7708333 0.8541667 0.7395833
## [1282] 0.7916667 0.8854167 0.8541667 0.8854167 0.5520833 0.8125000 0.6458333
## [1289] 0.6354167 0.7604167 0.8229167 0.6494845 0.8854167 0.8333333 0.8229167
## [1296] 0.7604167 0.6666667 0.8854167 0.8020833 0.7291667 0.8125000 0.6979167
## [1303] 0.7916667 0.7812500 0.8333333 0.7812500 0.6804124 0.7187500 0.7187500
## [1310] 0.7708333 0.8750000 0.6562500 0.7812500 0.8229167 0.7525773 0.6875000
## [1317] 0.8645833 0.7187500 0.6666667 0.6979167 0.7812500 0.8333333 0.7187500
## [1324] 0.6666667 0.7812500 0.6562500 0.7708333 0.7395833 0.7291667 0.7395833
## [1331] 0.8020833 0.8229167 0.7916667 0.8645833 0.8333333 0.9062500 0.7708333
## [1338] 0.8437500 0.7187500 0.7916667 0.8333333 0.8125000 0.8229167 0.6145833
## [1345] 0.7708333 0.8958333 0.8020833 0.8750000 0.7604167 0.7604167 0.7812500
## [1352] 0.7187500 0.8020833 0.7916667 0.6250000 0.6875000 0.7187500 0.8854167
## [1359] 0.6458333 0.8125000 0.5625000 0.8645833 0.8020833 0.6458333 0.7083333
## [1366] 0.7291667 0.7708333 0.9166667 0.7916667 0.7604167 0.6770833 0.6770833
## [1373] 0.7916667 0.6770833 0.7291667 0.8125000 0.7500000 0.7291667 0.7916667
## [1380] 0.7628866 0.8020833 0.8958333 0.8969072 0.7187500 0.8854167 0.7916667
## [1387] 0.8020833 0.8125000 0.7291667 0.8437500 0.8125000 0.5312500 0.8854167
## [1394] 0.7083333 0.7812500 0.9270833 0.7083333 0.7291667 0.6666667 0.7604167
## [1401] 0.8645833 0.7812500 0.8750000 0.7708333 0.6562500 0.8958333 0.8750000
## [1408] 0.7916667 0.6770833 0.7916667 0.7187500 0.8541667 0.8333333 0.8125000
## [1415] 0.7083333 0.6458333 0.8958333 0.7812500 0.7812500 0.8125000 0.8125000
## [1422] 0.6562500 0.7500000 0.5520833 0.8750000 0.7916667 0.7812500 0.7395833
## [1429] 0.7916667 0.6979167 0.7083333 0.8229167 0.8020833 0.8333333 0.7812500
## [1436] 0.8125000 0.8437500 0.8645833 0.6875000 0.7291667 0.6458333 0.6979167
## [1443] 0.8958333 0.8125000 0.8645833 0.6354167 0.8958333 0.7395833 0.7422680
## [1450] 0.8645833 0.7500000 0.8750000 0.7083333 0.6979167 0.8229167 0.6796117
## [1457] 0.7628866 0.7916667 0.7083333 0.7500000 0.7083333 0.8333333 0.7731959
## [1464] 0.7708333 0.6875000 0.8958333 0.6979167 0.7187500 0.7500000 0.8125000
## [1471] 0.8750000 0.7395833 0.8437500 0.6979167 0.5520833 0.5520833 0.7395833
## [1478] 0.7187500 0.8229167 0.8958333 0.8750000 0.6979167 0.7291667 0.7291667
## [1485] 0.6770833 0.6875000 0.7187500 0.7916667 0.6979167 0.8229167 0.6770833
## [1492] 0.7812500 0.6458333 0.7422680 0.8229167 0.7812500 0.8020833 0.7708333
## [1499] 0.7604167 0.8854167 0.7395833 0.8333333 0.6875000 0.7708333 0.7812500
## [1506] 0.8750000 0.8645833 0.6562500 0.9375000 0.5625000 0.8645833 0.8437500
## [1513] 0.8541667 0.8645833 0.8333333 0.6770833 0.8333333 0.7708333 0.8125000
## [1520] 0.7395833 0.8125000 0.9270833 0.8541667 0.8229167 0.5416667 0.7916667
## [1527] 0.8229167 0.5104167 0.5833333 0.7708333 0.8541667 0.8144330 0.7113402
## [1534] 0.9062500 0.8958333 0.8333333 0.7604167 0.8125000 0.8020833 0.7291667
## [1541] 0.8125000 0.7916667 0.6875000 0.7604167 0.8645833 0.8645833 0.7916667
## [1548] 0.6562500 0.8750000 0.7083333 0.6391753 0.7187500 0.6875000 0.8125000
## [1555] 0.6875000 0.6666667 0.7500000 0.9166667 0.8229167 0.6666667 0.6250000
## [1562] 0.7395833 0.7812500 0.6458333 0.8333333 0.7812500 0.7708333 0.8144330
## [1569] 0.7812500 0.7395833 0.7500000 0.8437500 0.5312500 0.5625000 0.8333333
## [1576] 0.5625000 0.8333333 0.8645833 0.7812500 0.8229167 0.5104167 0.8541667
## [1583] 0.7083333 0.7708333 0.5312500 0.8541667 0.7187500 0.6875000 0.7812500
## [1590] 0.9062500 0.7500000 0.5625000 0.8020833 0.6562500 0.7708333 0.6770833
## [1597] 0.6979167 0.8229167 0.7395833 0.8437500 0.8437500 0.8125000 0.8854167
## [1604] 0.6666667 0.8437500 0.6145833 0.8437500 0.7604167 0.7812500 0.7604167
## [1611] 0.7708333 0.8958333 0.6979167 0.7916667 0.7812500 0.8750000 0.7395833
## [1618] 0.5625000 0.6562500 0.8229167 0.6354167 0.6979167 0.8333333 0.9062500
## [1625] 0.6458333 0.8125000 0.7500000 0.8750000 0.9375000 0.8020833 0.8958333
## [1632] 0.8437500 0.8854167 0.6597938 0.7500000 0.8020833 0.9166667 0.6041667
## [1639] 0.9270833 0.8125000 0.8958333 0.5000000 0.8020833 0.8020833 0.8854167
## [1646] 0.6979167 0.5979381 0.8958333 0.8958333 0.7812500 0.7010309 0.5312500
## [1653] 0.7916667 0.6979167 0.8125000 0.8333333 0.8958333 0.7187500 0.5416667
## [1660] 0.7916667 0.9062500 0.6979167 0.7395833 0.5416667 0.6458333 0.8020833
## [1667] 0.6145833 0.7812500 0.6770833 0.6875000 0.8437500 0.5520833 0.7083333
## [1674] 0.8854167 0.6770833 0.7604167 0.7938144 0.6875000 0.7916667 0.7916667
## [1681] 0.7708333 0.7551020 0.8229167 0.8541667 0.6562500 0.6770833 0.6770833
## [1688] 0.5625000 0.8645833 0.7291667 0.6979167 0.7812500 0.7812500 0.8229167
## [1695] 0.8645833 0.7291667 0.8020833 0.7083333 0.5833333 0.6770833 0.8854167
## [1702] 0.8125000 0.7187500 0.6354167 0.7812500 0.7083333 0.7916667 0.8958333
## [1709] 0.8020833 0.7422680 0.8333333 0.9062500 0.6458333 0.6354167 0.8645833
## [1716] 0.7916667 0.8020833 0.8229167 0.6979167 0.8020833 0.7708333 0.7812500
## [1723] 0.7708333 0.7500000 0.7395833 0.5625000 0.8750000 0.7291667 0.7916667
## [1730] 0.7916667 0.6354167 0.8229167 0.8645833 0.6875000 0.8854167 0.8645833
## [1737] 0.6458333 0.7916667 0.5416667 0.6875000 0.7916667 0.6250000 0.7500000
## [1744] 0.8437500 0.6666667 0.6494845 0.7604167 0.6041667 0.8750000 0.7187500
## [1751] 0.8229167 0.7500000 0.8854167 0.8333333 0.7291667 0.5312500 0.7938144
## [1758] 0.6666667 0.7187500 0.7628866 0.7708333 0.7812500 0.8020833 0.7216495
## [1765] 0.8437500 0.8020833 0.6875000 0.7291667 0.6354167 0.8125000 0.5312500
## [1772] 0.7500000 0.6770833 0.7812500 0.7187500 0.7083333 0.8333333 0.8020833
## [1779] 0.6979167 0.8541667 0.7916667 0.7938144 0.8020833 0.8437500 0.7291667
## [1786] 0.8125000 0.7187500 0.8958333 0.5312500 0.6666667 0.8541667 0.6875000
## [1793] 0.6145833 0.7628866 0.8645833 0.7604167 0.6666667 0.6458333 0.7187500
## [1800] 0.8229167 0.7500000 0.8125000 0.6770833 0.7187500 0.6979167 0.5520833
## [1807] 0.7187500 0.7187500 0.7812500 0.7604167 0.6562500 0.8333333 0.8229167
## [1814] 0.5312500 0.8645833 0.7083333 0.7708333 0.8333333 0.7244898 0.9166667
## [1821] 0.8750000 0.6041667 0.8333333 0.7676768 0.6907216 0.7291667 0.8541667
## [1828] 0.7604167 0.6145833 0.6770833 0.8541667 0.7291667 0.6875000 0.7812500
## [1835] 0.6979167 0.9062500 0.5729167 0.8645833 0.6250000 0.6875000 0.7916667
## [1842] 0.7916667 0.7500000 0.7187500 0.7812500 0.7812500 0.6979167 0.7395833
## [1849] 0.7500000 0.8229167 0.7916667 0.8645833 0.7500000 0.6875000 0.7395833
## [1856] 0.7938144 0.9270833 0.6979167 0.8020833 0.5312500 0.7500000 0.8020833
## [1863] 0.6979167 0.8541667 0.7916667 0.7291667 0.8750000 0.7812500 0.9062500
## [1870] 0.7708333 0.7395833 0.6770833 0.8645833 0.8541667 0.7708333 0.8333333
## [1877] 0.8958333 0.7812500 0.7187500 0.9166667 0.7916667 0.8437500 0.6875000
## [1884] 0.7187500 0.7187500 0.7113402 0.5937500 0.7291667 0.8750000 0.7916667
## [1891] 0.8645833 0.7010309 0.8969072 0.7916667 0.7083333 0.8750000 0.8125000
## [1898] 0.7916667 0.8645833 0.8020833 0.6145833 0.8229167 0.9062500 0.8645833
## [1905] 0.7916667 0.8541667 0.7395833 0.8125000 0.6562500 0.6770833 0.8229167
## [1912] 0.7291667 0.7083333 0.8229167 0.6666667 0.7916667 0.5625000 0.6770833
## [1919] 0.8125000 0.5312500 0.8437500 0.7812500 0.8229167 0.8350515 0.8854167
## [1926] 0.7604167 0.6041667 0.6041667 0.6041667 0.8854167 0.7916667 0.7812500
## [1933] 0.7500000 0.7291667 0.6562500 0.7812500 0.7395833 0.8125000 0.6979167
## [1940] 0.7835052 0.7708333 0.7916667 0.7187500 0.7291667 0.7083333 0.6354167
## [1947] 0.9175258 0.7187500 0.7187500 0.7500000 0.7187500 0.7916667 0.5937500
## [1954] 0.7291667 0.8125000 0.7812500 0.5900000 0.9062500 0.7319588 0.7708333
## [1961] 0.8854167 0.8125000 0.7916667 0.8750000 0.8125000 0.7291667 0.7083333
## [1968] 0.6666667 0.7500000 0.9375000 0.7812500 0.7857143 0.7916667 0.6666667
## [1975] 0.8958333 0.7812500 0.6666667 0.7291667 0.8350515 0.8020833 0.6494845
## [1982] 0.7291667 0.7500000 0.6354167 0.7731959 0.8437500 0.7291667 0.8645833
## [1989] 0.6770833 0.8750000 0.7731959 0.8958333 0.8958333 0.8437500 0.7731959
## [1996] 0.8750000 0.9375000 0.6770833 0.7812500 0.7395833 0.7083333 0.6562500
## [2003] 0.7083333 0.7395833 0.7010309 0.8541667 0.7083333 0.6562500 0.6770833
## [2010] 0.9175258 0.7500000 0.8020833 0.6770833 0.8020833 0.8437500 0.6875000
## [2017] 0.8333333 0.7187500 0.7083333 0.8437500 0.8229167 0.6770833 0.7395833
## [2024] 0.7916667 0.7708333 0.6875000 0.8229167 0.6354167 0.8645833 0.5625000
## [2031] 0.7291667 0.7938144 0.7291667 0.9062500 0.6666667 0.8125000 0.6145833
## [2038] 0.8437500 0.8229167 0.8350515 0.7395833 0.6354167 0.8437500 0.5416667
## [2045] 0.6979167 0.7604167 0.7708333 0.6770833 0.7500000 0.7812500 0.6250000
## [2052] 0.7916667 0.9062500 0.6041667 0.7812500 0.9062500 0.6979167 0.6041667
## [2059] 0.6770833 0.6875000 0.6875000 0.6979167 0.7395833 0.8958333 0.9270833
## [2066] 0.8437500 0.7812500 0.8020833 0.8229167 0.8229167 0.7010309 0.8229167
## [2073] 0.8020833 0.6979167 0.7083333 0.8229167 0.9375000 0.7500000 0.6666667
## [2080] 0.7500000 0.7291667 0.7187500 0.7604167 0.7916667 0.6979167 0.9062500
## [2087] 0.7187500 0.7083333 0.6666667 0.6354167 0.6979167 0.6875000 0.7395833
## [2094] 0.5729167 0.7500000 0.8125000 0.8437500 0.7708333 0.6391753 0.7812500
## [2101] 0.6250000 0.8854167 0.7083333 0.6145833 0.8645833 0.6666667 0.7083333
## [2108] 0.7187500 0.8437500 0.6979167 0.7916667 0.5625000 0.6666667 0.8020833
## [2115] 0.8437500 0.8541667 0.8125000 0.9278351 0.7604167 0.8541667 0.7604167
## [2122] 0.8020833 0.6979167 0.7916667 0.7083333 0.8541667 0.8541667 0.6979167
## [2129] 0.7708333 0.6701031 0.5833333 0.7604167 0.7216495 0.7708333 0.6458333
## [2136] 0.7916667 0.7083333 0.6979167 0.7916667 0.7083333 0.7812500 0.9270833
## [2143] 0.6145833 0.7187500 0.7395833 0.8541667 0.7083333 0.8333333 0.7422680
## [2150] 0.7812500 0.5937500 0.7395833 0.6041667 0.8453608 0.8333333 0.7395833
## [2157] 0.8333333 0.7113402 0.7187500 0.8541667 0.8020833 0.8125000 0.8541667
## [2164] 0.7291667 0.8437500 0.6734694 0.9270833 0.8229167 0.9062500 0.6666667
## [2171] 0.6354167 0.5937500 0.7916667 0.7395833 0.8645833 0.7916667 0.6875000
## [2178] 0.8229167 0.8229167 0.8333333 0.6354167 0.6666667 0.8958333 0.7422680
## [2185] 0.6250000 0.8125000 0.7604167 0.7604167 0.7083333 0.8854167 0.8020833
## [2192] 0.7395833 0.8854167 0.7291667 0.8333333 0.6770833 0.7604167 0.7395833
## [2199] 0.6979167 0.7604167 0.7916667 0.8020833 0.7812500 0.6770833 0.8541667
## [2206] 0.7708333 0.7708333 0.6041667 0.7395833 0.7291667 0.8020833 0.8020833
## [2213] 0.7291667 0.6458333 0.6458333 0.6562500 0.6770833 0.8229167 0.7812500
## [2220] 0.6562500 0.6407767 0.8020833 0.6250000 0.8958333 0.6770833 0.6562500
## [2227] 0.7916667 0.7500000 0.6666667 0.5208333 0.7395833 0.5729167 0.7083333
## [2234] 0.8333333 0.5520833 0.7083333 0.5729167 0.7083333 0.6250000 0.7395833
## [2241] 0.8125000 0.8333333 0.6562500 0.7187500 0.7812500 0.7604167 0.6250000
## [2248] 0.6458333 0.8541667 0.6979167 0.8645833 0.7708333 0.5729167 0.6666667
## [2255] 0.6770833 0.6979167 0.7604167 0.6354167 0.7395833 0.7395833 0.7812500
## [2262] 0.8958333 0.8333333 0.7916667 0.7291667 0.6041667 0.7708333 0.6979167
## [2269] 0.7916667 0.6979167 0.6250000 0.7604167 0.7604167 0.6250000 0.7604167
## [2276] 0.7500000 0.6041667 0.5312500 0.5312500 0.6979167 0.8333333 0.6875000
## [2283] 0.6979167 0.7500000 0.7187500 0.6041667 0.8541667 0.7916667 0.7187500
## [2290] 0.7500000 0.6354167 0.7395833 0.6041667 0.8020833 0.7628866 0.7812500
## [2297] 0.7500000 0.6666667 0.6458333 0.8229167 0.6562500 0.7500000 0.7812500
## [2304] 0.5833333 0.7604167 0.7395833 0.7812500 0.7142857 0.6666667 0.6145833
## [2311] 0.7083333 0.7812500 0.8333333 0.7812500 0.7083333 0.6666667 0.6770833
## [2318] 0.6562500 0.7525773 0.7500000 0.6875000 0.7708333 0.8437500 0.7708333
## [2325] 0.6875000 0.7708333 0.5208333 0.7812500 0.7319588 0.7395833 0.5729167
## [2332] 0.7812500 0.7395833 0.7500000 0.8958333 0.8125000 0.7916667 0.6979167
## [2339] 0.6041667 0.6875000 0.7916667 0.7604167 0.6354167 0.7500000 0.7187500
## [2346] 0.7500000 0.7708333 0.7708333 0.7708333 0.6458333 0.7500000 0.6458333
## [2353] 0.7500000 0.7291667 0.6250000 0.5416667 0.7291667 0.4895833 0.6250000
## [2360] 0.7604167 0.7040816 0.5625000 0.5520833 0.7395833 0.7187500 0.5670103
## [2367] 0.7187500 0.7291667 0.8437500 0.8020833 0.6875000 0.7604167 0.7604167
## [2374] 0.7187500 0.5520833 0.5520833 0.8125000 0.6458333 0.5625000 0.5729167
## [2381] 0.6907216 0.6875000 0.8854167 0.6907216 0.6041667 0.7708333 0.7604167
## [2388] 0.7083333 0.7395833 0.7708333 0.6770833 0.5208333 0.6458333 0.7083333
## [2395] 0.6562500 0.6979167 0.7604167 0.7812500 0.7812500 0.7812500 0.6145833
## [2402] 0.5520833 0.6041667 0.7291667 0.7083333 0.6597938 0.7916667 0.6145833
## [2409] 0.5208333 0.7187500 0.8541667 0.6250000 0.7708333 0.9278351 0.5000000
## [2416] 0.5625000 0.7812500 0.8645833 0.8125000 0.6875000 0.7291667 0.7291667
## [2423] 0.9270833 0.8333333 0.5520833 0.7291667 0.8541667 0.8020833 0.7604167
## [2430] 0.7187500 0.8125000 0.7604167 0.7083333 0.6666667 0.5625000 0.7291667
## [2437] 0.5729167 0.7812500 0.6354167 0.6979167 0.6979167 0.5520833 0.7812500
## [2444] 0.6979167 0.7187500 0.6770833 0.7916667 0.8854167 0.7812500 0.6938776
## [2451] 0.7812500 0.6979167 0.6458333 0.7083333 0.6354167 0.7187500 0.5937500
## [2458] 0.6562500 0.7812500 0.6145833 0.6770833 0.6354167 0.6770833 0.7604167
## [2465] 0.7708333 0.6562500 0.6354167 0.7187500 0.6562500 0.5876289 0.6875000
## [2472] 0.7187500 0.7938144 0.7604167 0.8958333 0.8020833 0.7916667 0.7812500
## [2479] 0.7291667 0.7604167 0.8020833 0.7604167 0.5520833 0.6875000 0.8020833
## [2486] 0.7187500 0.6250000 0.6666667 0.8854167 0.6979167 0.4791667 0.6979167
## [2493] 0.6354167 0.6458333 0.7812500 0.7500000 0.6875000 0.6666667 0.7083333
## [2500] 0.5625000 0.6562500 0.7812500 0.8125000 0.5520833 0.6875000 0.5625000
## [2507] 0.8750000 0.5416667 0.5104167 0.8125000 0.6458333 0.5520833 0.7187500
## [2514] 0.7604167 0.6979167 0.7604167 0.6458333 0.7291667 0.8020833 0.8333333
## [2521] 0.7916667 0.8229167 0.6354167 0.8125000 0.7395833 0.7708333 0.5625000
## [2528] 0.6979167 0.7083333 0.5729167 0.5520833 0.7083333 0.8437500 0.8854167
## [2535] 0.7083333 0.7604167 0.7812500 0.8333333 0.6666667 0.7916667 0.8229167
## [2542] 0.5520833 0.7604167 0.7812500 0.6875000 0.8020833 0.5729167 0.5520833
## [2549] 0.8958333 0.7083333 0.7083333 0.7395833 0.7291667 0.8229167 0.7291667
## [2556] 0.6354167 0.7395833 0.6354167 0.8125000 0.7604167 0.5729167 0.7708333
## [2563] 0.7835052 0.8958333 0.5937500 0.8333333 0.8229167 0.5833333 0.7113402
## [2570] 0.7395833 0.5520833 0.6764706 0.7083333 0.6875000 0.8854167 0.5520833
## [2577] 0.5833333 0.7604167 0.7187500 0.5937500 0.7291667 0.5520833 0.7916667
## [2584] 0.7040816 0.6354167 0.7083333 0.5833333 0.8437500 0.6734694 0.6666667
## [2591] 0.7708333 0.6458333 0.7916667 0.5520833 0.5833333 0.6770833 0.7812500
## [2598] 0.5833333 0.7500000 0.7346939 0.6145833 0.6354167 0.6875000 0.8125000
## [2605] 0.5520833 0.7604167 0.5625000 0.7083333 0.6666667 0.6979167 0.7500000
## [2612] 0.6354167 0.7604167 0.7604167 0.6875000 0.8229167 0.8020833 0.5463918
## [2619] 0.7395833 0.5625000 0.7812500 0.5416667 0.6666667 0.6458333 0.6770833
## [2626] 0.7187500 0.5833333 0.6562500 0.7395833 0.8020833 0.6562500 0.7916667
## [2633] 0.5625000 0.8645833 0.5520833 0.6666667 0.6979167 0.7604167 0.7395833
## [2640] 0.5625000 0.7291667 0.6145833 0.7604167 0.6875000 0.5520833 0.8125000
## [2647] 0.7500000 0.5416667 0.8229167 0.8041237 0.7708333 0.7187500 0.8958333
## [2654] 0.8437500 0.6041667 0.7395833 0.6770833 0.5520833 0.7916667 0.8125000
## [2661] 0.7604167 0.7812500 0.5520833 0.8437500 0.7708333 0.7812500 0.6666667
## [2668] 0.6562500 0.6875000 0.7319588 0.8541667 0.7604167 0.7187500 0.5416667
## [2675] 0.7500000 0.6354167 0.6354167 0.7500000 0.6875000 0.8645833 0.8020833
## [2682] 0.7916667 0.6875000 0.7500000 0.5833333 0.6354167 0.8229167 0.6458333
## [2689] 0.6250000 0.6562500 0.7812500 0.6979167 0.6354167 0.6562500 0.8333333
## [2696] 0.7812500 0.7604167 0.6979167 0.7604167 0.7916667 0.7812500 0.6770833
## [2703] 0.6979167 0.5833333 0.6041667 0.8333333 0.6875000 0.5104167 0.5979381
## [2710] 0.7500000 0.8229167 0.5833333 0.7604167 0.7812500 0.5520833 0.7187500
## [2717] 0.7500000 0.6875000 0.5833333 0.7916667 0.6145833 0.7083333 0.8041237
## [2724] 0.7083333 0.7373737 0.6666667 0.8020833 0.7500000 0.7857143 0.6562500
## [2731] 0.6597938 0.7916667 0.8333333 0.7708333 0.6979167 0.6979167 0.6041667
## [2738] 0.7500000 0.6145833 0.6979167 0.8229167 0.7187500 0.5625000 0.7500000
## [2745] 0.7395833 0.7500000 0.7395833 0.8000000 0.7708333 0.6458333 0.8333333
## [2752] 0.6666667 0.7500000 0.7916667 0.7291667 0.8541667 0.6288660 0.8437500
## [2759] 0.6875000
## Levels: 1 2 3
#check the extent inside the knn model.
prk.prob <- attr(prk, "prob")
head(prk)## [1] 1 1 1 1 1 1
## Levels: 1 2 3
head(prk.prob)## [1] 0.6979167 0.7500000 0.8437500 0.7500000 0.8333333 0.8645833
#understand the relationship of the classes of your test data relate to your model
CrossTable(x = ds_asses1_test_category, y = prk, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 2759
##
##
## | prk
## ds_asses1_test_category | 1 | Row Total |
## ------------------------|-----------|-----------|
## 1 | 2016 | 2016 |
## | 0.731 | |
## ------------------------|-----------|-----------|
## 2 | 237 | 237 |
## | 0.086 | |
## ------------------------|-----------|-----------|
## 3 | 506 | 506 |
## | 0.183 | |
## ------------------------|-----------|-----------|
## Column Total | 2759 | 2759 |
## ------------------------|-----------|-----------|
##
##
##create confusion matrix
tabc <- table(prk,ds_asses1_test_category)
tabc## ds_asses1_test_category
## prk 1 2 3
## 1 2016 237 506
## 2 0 0 0
## 3 0 0 0
library(caret)
confusionMatrix(tabc)## Confusion Matrix and Statistics
##
## ds_asses1_test_category
## prk 1 2 3
## 1 2016 237 506
## 2 0 0 0
## 3 0 0 0
##
## Overall Statistics
##
## Accuracy : 0.7307
## 95% CI : (0.7137, 0.7472)
## No Information Rate : 0.7307
## P-Value [Acc > NIR] : 0.5099
##
## Kappa : 0
##
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: 1 Class: 2 Class: 3
## Sensitivity 1.0000 0.0000 0.0000
## Specificity 0.0000 1.0000 1.0000
## Pos Pred Value 0.7307 NaN NaN
## Neg Pred Value NaN 0.9141 0.8166
## Prevalence 0.7307 0.0859 0.1834
## Detection Rate 0.7307 0.0000 0.0000
## Detection Prevalence 1.0000 0.0000 0.0000
## Balanced Accuracy 0.5000 0.5000 0.5000
##this function divides the correct predictions by total number of predictions
#that tell us how accurate the model is.
accuracy <- function(x){sum(diag(x)/(sum(rowSums(x)))) * 100}
accuracy(tabc)## [1] 73.06995
#check which ‘K’ value will result in the most accurate model
i=1
k.optm=1
for (i in 1:96){
knn.mod <- knn(train=ds_asses1_train, test=ds_asses1_test, cl=ds_asses1_target_category, k=i)
k.optm[i] <- 100 * sum(ds_asses1_test_category == knn.mod)/NROW(ds_asses1_test_category)
k=i
cat(k,'=',k.optm[i],'\n')
}## 1 = 65.4585
## 2 = 65.20478
## 3 = 70.60529
## 4 = 71.22146
## 5 = 72.23632
## 6 = 72.85248
## 7 = 73.03371
## 8 = 73.35991
## 9 = 72.77999
## 10 = 73.21493
## 11 = 73.25118
## 12 = 73.86734
## 13 = 73.68612
## 14 = 73.54114
## 15 = 73.64987
## 16 = 73.32367
## 17 = 73.32367
## 18 = 73.61363
## 19 = 73.50489
## 20 = 73.35991
## 21 = 73.46865
## 22 = 73.50489
## 23 = 73.50489
## 24 = 73.54114
## 25 = 73.50489
## 26 = 73.54114
## 27 = 73.46865
## 28 = 73.61363
## 29 = 73.46865
## 30 = 73.46865
## 31 = 73.32367
## 32 = 73.35991
## 33 = 73.4324
## 34 = 73.39616
## 35 = 73.4324
## 36 = 73.35991
## 37 = 73.28742
## 38 = 73.21493
## 39 = 73.21493
## 40 = 73.17869
## 41 = 73.17869
## 42 = 73.21493
## 43 = 73.17869
## 44 = 73.21493
## 45 = 73.21493
## 46 = 73.25118
## 47 = 73.25118
## 48 = 73.17869
## 49 = 73.25118
## 50 = 73.17869
## 51 = 73.17869
## 52 = 73.21493
## 53 = 73.17869
## 54 = 73.14244
## 55 = 73.21493
## 56 = 73.17869
## 57 = 73.17869
## 58 = 73.21493
## 59 = 73.17869
## 60 = 73.14244
## 61 = 73.14244
## 62 = 73.14244
## 63 = 73.14244
## 64 = 73.14244
## 65 = 73.14244
## 66 = 73.17869
## 67 = 73.17869
## 68 = 73.17869
## 69 = 73.17869
## 70 = 73.17869
## 71 = 73.14244
## 72 = 73.1062
## 73 = 73.1062
## 74 = 73.1062
## 75 = 73.1062
## 76 = 73.1062
## 77 = 73.1062
## 78 = 73.1062
## 79 = 73.1062
## 80 = 73.1062
## 81 = 73.1062
## 82 = 73.1062
## 83 = 73.1062
## 84 = 73.1062
## 85 = 73.1062
## 86 = 73.1062
## 87 = 73.1062
## 88 = 73.1062
## 89 = 73.1062
## 90 = 73.1062
## 91 = 73.1062
## 92 = 73.1062
## 93 = 73.1062
## 94 = 73.1062
## 95 = 73.06995
## 96 = 73.06995
# Maximum accuracy at k=12, after that, it seems increasing K increases the classification
#but reduces success rate. It is worse to class LoanStatus360 as good when it is bad,
#than it is to class LoanStatus360 as bad when it is good.
plot(k.optm, type = 'b', xlab="K- Value", ylab="Accuracy level")
#Challenges faced in building the models
#1. categorical data are too many. Focus more on collecting integer data type features like
#the Amount of Income, size of the family, age, loan amount.