1 RESUMEN EJECUTIVO

1.1 El problema

Queremos automatizar el proceso de toma de decisiones para la aprobación de líneas de crédito con garantía hipotecaria.

Se creará un modelo de puntuación de crédito empíricamente derivado y estadísticamente sólido.

El modelo se basará en los datos recopilados de solicitantes recientes que recibieron crédito a través del proceso actual de suscripción de préstamos.

El modelo debe ser lo suficientemente interpretable para proporcionar una razón para cualquier acción adversa (rechazos).

1.2 La solución

  • Modelo de clasificación Logit o Regresion Logistica
  • Modelo de puntuación
  • Cuidados de la solucion:
    • Variables con skewness, Missing values, muestra desbalanceada en variable Outcome
  • Analisis de Correlaciones
  • Estrategia de seleccion de variables:

2 ANÁLISIS EXPLORATORIO DE LAS VARIABLES

Aqui preparamos los datos OJO ## Carga y primera revisión de los datos

Revisamos los datos

inspect_types(data)
## # A tibble: 3 x 4
##   type        cnt  pcnt col_name 
##   <chr>     <int> <dbl> <list>   
## 1 integer       6  46.2 <chr [6]>
## 2 numeric       5  38.5 <chr [5]>
## 3 character     2  15.4 <chr [2]>
str(data)
## 'data.frame':    5960 obs. of  13 variables:
##  $ BAD    : int  1 1 1 1 0 1 1 1 1 1 ...
##  $ LOAN   : int  1100 1300 1500 1500 1700 1700 1800 1800 2000 2000 ...
##  $ MORTDUE: num  25860 70053 13500 NA 97800 ...
##  $ VALUE  : num  39025 68400 16700 NA 112000 ...
##  $ REASON : chr  "HomeImp" "HomeImp" "HomeImp" NA ...
##  $ JOB    : chr  "Other" "Other" "Other" NA ...
##  $ YOJ    : num  10.5 7 4 NA 3 9 5 11 3 16 ...
##  $ DEROG  : int  0 0 0 NA 0 0 3 0 0 0 ...
##  $ DELINQ : int  0 2 0 NA 0 0 2 0 2 0 ...
##  $ CLAGE  : num  94.4 121.8 149.5 NA 93.3 ...
##  $ NINQ   : int  1 0 1 NA 0 1 1 0 1 0 ...
##  $ CLNO   : int  9 14 10 NA 14 8 17 8 12 13 ...
##  $ DEBTINC: num  NA NA NA NA NA ...
glimpse(data)
## Observations: 5,960
## Variables: 13
## $ BAD     <int> 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0,...
## $ LOAN    <int> 1100, 1300, 1500, 1500, 1700, 1700, 1800, 1800, 2000, 2000,...
## $ MORTDUE <dbl> 25860, 70053, 13500, NA, 97800, 30548, 48649, 28502, 32700,...
## $ VALUE   <dbl> 39025, 68400, 16700, NA, 112000, 40320, 57037, 43034, 46740...
## $ REASON  <chr> "HomeImp", "HomeImp", "HomeImp", NA, "HomeImp", "HomeImp", ...
## $ JOB     <chr> "Other", "Other", "Other", NA, "Office", "Other", "Other", ...
## $ YOJ     <dbl> 10.5, 7.0, 4.0, NA, 3.0, 9.0, 5.0, 11.0, 3.0, 16.0, 18.0, 1...
## $ DEROG   <int> 0, 0, 0, NA, 0, 0, 3, 0, 0, 0, NA, 0, 0, 0, 0, 0, 2, NA, 0,...
## $ DELINQ  <int> 0, 2, 0, NA, 0, 0, 2, 0, 2, 0, NA, 1, 0, 0, 1, 1, 6, NA, 0,...
## $ CLAGE   <dbl> 94.36667, 121.83333, 149.46667, NA, 93.33333, 101.46600, 77...
## $ NINQ    <int> 1, 0, 1, NA, 0, 1, 1, 0, 1, 0, NA, 1, 2, 0, 0, 0, 1, NA, 1,...
## $ CLNO    <int> 9, 14, 10, NA, 14, 8, 17, 8, 12, 13, NA, 9, 25, 24, 16, 8, ...
## $ DEBTINC <dbl> NA, NA, NA, NA, NA, 37.113614, NA, 36.884894, NA, NA, NA, N...
head(data)
##   BAD LOAN MORTDUE  VALUE  REASON    JOB  YOJ DEROG DELINQ     CLAGE NINQ CLNO
## 1   1 1100   25860  39025 HomeImp  Other 10.5     0      0  94.36667    1    9
## 2   1 1300   70053  68400 HomeImp  Other  7.0     0      2 121.83333    0   14
## 3   1 1500   13500  16700 HomeImp  Other  4.0     0      0 149.46667    1   10
## 4   1 1500      NA     NA    <NA>   <NA>   NA    NA     NA        NA   NA   NA
## 5   0 1700   97800 112000 HomeImp Office  3.0     0      0  93.33333    0   14
## 6   1 1700   30548  40320 HomeImp  Other  9.0     0      0 101.46600    1    8
##    DEBTINC
## 1       NA
## 2       NA
## 3       NA
## 4       NA
## 5       NA
## 6 37.11361
tail(data)
##      BAD  LOAN MORTDUE VALUE  REASON   JOB YOJ DEROG DELINQ    CLAGE NINQ CLNO
## 5955   0 88900   48919 93371 DebtCon Other  15     0      1 205.6502    0   15
## 5956   0 88900   57264 90185 DebtCon Other  16     0      0 221.8087    0   16
## 5957   0 89000   54576 92937 DebtCon Other  16     0      0 208.6921    0   15
## 5958   0 89200   54045 92924 DebtCon Other  15     0      0 212.2797    0   15
## 5959   0 89800   50370 91861 DebtCon Other  14     0      0 213.8927    0   16
## 5960   0 89900   48811 88934 DebtCon Other  15     0      0 219.6010    0   16
##       DEBTINC
## 5955 34.81826
## 5956 36.11235
## 5957 35.85997
## 5958 35.55659
## 5959 34.34088
## 5960 34.57152
summary(data)
##       BAD              LOAN          MORTDUE           VALUE       
##  Min.   :0.0000   Min.   : 1100   Min.   :  2063   Min.   :  8000  
##  1st Qu.:0.0000   1st Qu.:11100   1st Qu.: 46276   1st Qu.: 66076  
##  Median :0.0000   Median :16300   Median : 65019   Median : 89236  
##  Mean   :0.1995   Mean   :18608   Mean   : 73761   Mean   :101776  
##  3rd Qu.:0.0000   3rd Qu.:23300   3rd Qu.: 91488   3rd Qu.:119824  
##  Max.   :1.0000   Max.   :89900   Max.   :399550   Max.   :855909  
##                                   NA's   :518      NA's   :112     
##     REASON              JOB                 YOJ             DEROG        
##  Length:5960        Length:5960        Min.   : 0.000   Min.   : 0.0000  
##  Class :character   Class :character   1st Qu.: 3.000   1st Qu.: 0.0000  
##  Mode  :character   Mode  :character   Median : 7.000   Median : 0.0000  
##                                        Mean   : 8.922   Mean   : 0.2546  
##                                        3rd Qu.:13.000   3rd Qu.: 0.0000  
##                                        Max.   :41.000   Max.   :10.0000  
##                                        NA's   :515      NA's   :708      
##      DELINQ            CLAGE             NINQ             CLNO     
##  Min.   : 0.0000   Min.   :   0.0   Min.   : 0.000   Min.   : 0.0  
##  1st Qu.: 0.0000   1st Qu.: 115.1   1st Qu.: 0.000   1st Qu.:15.0  
##  Median : 0.0000   Median : 173.5   Median : 1.000   Median :20.0  
##  Mean   : 0.4494   Mean   : 179.8   Mean   : 1.186   Mean   :21.3  
##  3rd Qu.: 0.0000   3rd Qu.: 231.6   3rd Qu.: 2.000   3rd Qu.:26.0  
##  Max.   :15.0000   Max.   :1168.2   Max.   :17.000   Max.   :71.0  
##  NA's   :580       NA's   :308      NA's   :510      NA's   :222   
##     DEBTINC        
##  Min.   :  0.5245  
##  1st Qu.: 29.1400  
##  Median : 34.8183  
##  Mean   : 33.7799  
##  3rd Qu.: 39.0031  
##  Max.   :203.3121  
##  NA's   :1267

Observamos que REASON y JOB son variables categóricas. Reemplazaremos los valores NA por un string “MISSING”, para que cuando transformemos los valores a factores, quede un nivel para los valores faltantes.

data_original=data
data$REASON[is.na(data$REASON)]="Missing"
data$JOB[is.na(data$JOB)]="Missing"
data$REASON=as.factor(data$REASON)
data$JOB=as.factor(data$JOB)

2.0.1 Revision de variables

2.0.1.1 Boxplot, Histogramas y Densidades

####REVISAR:¿Hay outliers?

2.1 Manejo de valores faltantes

inspect_na(data_original)
## # A tibble: 13 x 3
##    col_name   cnt  pcnt
##    <chr>    <int> <dbl>
##  1 DEBTINC   1267 21.3 
##  2 DEROG      708 11.9 
##  3 DELINQ     580  9.73
##  4 MORTDUE    518  8.69
##  5 YOJ        515  8.64
##  6 NINQ       510  8.56
##  7 CLAGE      308  5.17
##  8 JOB        279  4.68
##  9 REASON     252  4.23
## 10 CLNO       222  3.72
## 11 VALUE      112  1.88
## 12 BAD          0  0   
## 13 LOAN         0  0
inspect_na(data)
## # A tibble: 13 x 3
##    col_name   cnt  pcnt
##    <chr>    <int> <dbl>
##  1 DEBTINC   1267 21.3 
##  2 DEROG      708 11.9 
##  3 DELINQ     580  9.73
##  4 MORTDUE    518  8.69
##  5 YOJ        515  8.64
##  6 NINQ       510  8.56
##  7 CLAGE      308  5.17
##  8 CLNO       222  3.72
##  9 VALUE      112  1.88
## 10 BAD          0  0   
## 11 LOAN         0  0   
## 12 REASON       0  0   
## 13 JOB          0  0

Observamos que 9 de las 13 variables presentan valores faltantes. Asimismo, 2445 casos presentan datos faltantes. Para las variables categóricas, el 4,22% de la variable REASON y el 4.68% de la variable JOB tenian datos faltantes. El caso de DEBTINC es especialmente preocupante, pues es un valor faltante en el 21,3% de los casos

missing = filter(inspect_na(data),cnt>0)
missmap(data, col= c('white','grey')  , main ='missmap of data')

Tratamos de observar algun otro patron en los datos faltantes:

Se toma la decisión de generar una categoria para los valores faltantes en las variables categoricas, y en las numericas transformalas en categóricas usando WOE binning, con la libreria Scorecard.

bins=NULL
for(i in missing$col_name){
  bins = append(bins,woebin(data[,c(i,"BAD")], y = 'BAD'))
}
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ... 
## [INFO] creating woe binning ...
data_woe = woebin_ply(data[,c("MORTDUE","VALUE","YOJ","DEROG","DELINQ","CLAGE","NINQ","CLNO","DEBTINC","BAD", "LOAN","REASON","JOB")], bins ) %>% as_tibble()
## [INFO] converting into woe values ...
str(data_woe)
## Classes 'tbl_df', 'tbl' and 'data.frame':    5960 obs. of  13 variables:
##  $ BAD        : int  1 1 1 1 0 1 1 1 1 1 ...
##  $ LOAN       : int  1100 1300 1500 1500 1700 1700 1800 1800 2000 2000 ...
##  $ REASON     : Factor w/ 3 levels "DebtCon","HomeImp",..: 2 2 2 3 2 2 2 2 2 2 ...
##  $ JOB        : Factor w/ 7 levels "Mgr","Missing",..: 4 4 4 2 3 4 4 4 4 6 ...
##  $ DEBTINC_woe: num  1.88 1.88 1.88 1.88 1.88 ...
##  $ DEROG_woe  : num  -0.221 -0.221 -0.221 -0.576 -0.221 ...
##  $ DELINQ_woe : num  -0.43 1.673 -0.43 -0.564 -0.43 ...
##  $ MORTDUE_woe: num  0.3715 0.0124 0.3715 0.0319 -0.3771 ...
##  $ YOJ_woe    : num  0.08 -0.333 0.245 -0.545 0.245 ...
##  $ NINQ_woe   : num  -0.0626 -0.2954 -0.0626 -0.3684 -0.2954 ...
##  $ CLAGE_woe  : num  0.344 0.344 0.344 0.308 0.344 ...
##  $ CLNO_woe   : num  0.623 -0.141 -0.141 0.23 -0.141 ...
##  $ VALUE_woe  : num  0.7752 -0.0891 0.7752 4.0975 -0.4624 ...
##  - attr(*, ".internal.selfref")=<externalptr>
inspect_na(data_woe)
## # A tibble: 13 x 3
##    col_name      cnt  pcnt
##    <chr>       <int> <dbl>
##  1 BAD             0     0
##  2 LOAN            0     0
##  3 REASON          0     0
##  4 JOB             0     0
##  5 DEBTINC_woe     0     0
##  6 DEROG_woe       0     0
##  7 DELINQ_woe      0     0
##  8 MORTDUE_woe     0     0
##  9 YOJ_woe         0     0
## 10 NINQ_woe        0     0
## 11 CLAGE_woe       0     0
## 12 CLNO_woe        0     0
## 13 VALUE_woe       0     0
summary(data_woe)
##       BAD              LOAN           REASON          JOB      
##  Min.   :0.0000   Min.   : 1100   DebtCon:3928   Mgr    : 767  
##  1st Qu.:0.0000   1st Qu.:11100   HomeImp:1780   Missing: 279  
##  Median :0.0000   Median :16300   Missing: 252   Office : 948  
##  Mean   :0.1995   Mean   :18608                  Other  :2388  
##  3rd Qu.:0.0000   3rd Qu.:23300                  ProfExe:1276  
##  Max.   :1.0000   Max.   :89900                  Sales  : 109  
##                                                  Self   : 193  
##   DEBTINC_woe        DEROG_woe          DELINQ_woe       MORTDUE_woe      
##  Min.   :-1.5106   Min.   :-0.57598   Min.   :-0.5644   Min.   :-0.37710  
##  1st Qu.:-1.5106   1st Qu.:-0.22079   1st Qu.:-0.4299   1st Qu.:-0.19160  
##  Median :-1.1663   Median :-0.22079   Median :-0.4299   Median : 0.01237  
##  Mean   :-0.4943   Mean   :-0.07684   Mean   :-0.1234   Mean   :-0.01680  
##  3rd Qu.: 0.7299   3rd Qu.:-0.22079   3rd Qu.:-0.4299   3rd Qu.: 0.03186  
##  Max.   : 1.8805   Max.   : 1.30940   Max.   : 1.6729   Max.   : 0.37149  
##                                                                           
##     YOJ_woe            NINQ_woe          CLAGE_woe            CLNO_woe       
##  Min.   :-0.55033   Min.   :-0.36841   Min.   :-0.818391   Min.   :-0.50266  
##  1st Qu.:-0.33291   1st Qu.:-0.29536   1st Qu.:-0.303563   1st Qu.:-0.14080  
##  Median : 0.08002   Median :-0.29536   Median : 0.009838   Median :-0.14080  
##  Mean   :-0.02641   Mean   :-0.04212   Mean   :-0.074206   Mean   :-0.02595  
##  3rd Qu.: 0.24523   3rd Qu.: 0.26795   3rd Qu.: 0.343828   3rd Qu.: 0.08780  
##  Max.   : 0.24523   Max.   : 1.11319   Max.   : 0.945192   Max.   : 0.62319  
##                                                                              
##    VALUE_woe       
##  Min.   :-1.01542  
##  1st Qu.:-0.46238  
##  Median :-0.08914  
##  Mean   :-0.05448  
##  3rd Qu.: 0.05520  
##  Max.   : 4.09749  
## 
glimpse(data_woe)
## Observations: 5,960
## Variables: 13
## $ BAD         <int> 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1...
## $ LOAN        <int> 1100, 1300, 1500, 1500, 1700, 1700, 1800, 1800, 2000, 2...
## $ REASON      <fct> HomeImp, HomeImp, HomeImp, Missing, HomeImp, HomeImp, H...
## $ JOB         <fct> Other, Other, Other, Missing, Office, Other, Other, Oth...
## $ DEBTINC_woe <dbl> 1.880533, 1.880533, 1.880533, 1.880533, 1.880533, -1.16...
## $ DEROG_woe   <dbl> -0.2207900, -0.2207900, -0.2207900, -0.5759797, -0.2207...
## $ DELINQ_woe  <dbl> -0.4299469, 1.6728610, -0.4299469, -0.5643720, -0.42994...
## $ MORTDUE_woe <dbl> 0.37148892, 0.01237060, 0.37148892, 0.03185905, -0.3770...
## $ YOJ_woe     <dbl> 0.08001654, -0.33291270, 0.24523224, -0.54541700, 0.245...
## $ NINQ_woe    <dbl> -0.06255769, -0.29536441, -0.06255769, -0.36841461, -0....
## $ CLAGE_woe   <dbl> 0.3438278, 0.3438278, 0.3438278, 0.3080728, 0.3438278, ...
## $ CLNO_woe    <dbl> 0.6231919, -0.1408016, -0.1408016, 0.2298365, -0.140801...
## $ VALUE_woe   <dbl> 0.77522339, -0.08914024, 0.77522339, 4.09749351, -0.462...

2.2 Análisis de correlación de variables

MEJORAR: miramos la correlacion de las variables numericas, para los casos completos.

2.2.1 Multicolinealidad

Revisar si hay multicolinealidad Sospecho de multicolinealidad entre mortdue y value

2.3 Analisis de la variable resultado “BAD”

BAD es nuestra variable resultado: “1 = applicant defaulted on loan or seriously delinquent; 0 = applicant paid loan”.

Observamos que la ocurrencia de no pago del credito es aproximadamente del 20% de los casos en la muestra. Este desbalance puede causar algunos problemas de reducción de precision en los modelos, por lo que trataremos de balancear la muestra usando undersampling

table(data$BAD)
## 
##    0    1 
## 4771 1189
prop.table(table(data$BAD))
## 
##         0         1 
## 0.8005034 0.1994966

3 MODELAMIENTO

3.1 Modelo A: el modelo de casos completos

roc_auc_test <- function(model, data_test){
  mod_predictions = predict(model, newdata=data_test, type="response") # Generar predicciones con el modelo
  mod_rocpred = prediction(mod_predictions, data_test$BAD) # Comparar la prediccion con la data de TEST
  mod_ROCRperf = performance(mod_rocpred, measure = "tpr", x.measure = "fpr") # Calcular el performance ROC
  mod_AUCperf = performance(mod_rocpred, measure = "auc") # Calcular el performance AUC
  mod_AUCperf = mod_AUCperf@y.values[[1]] # Extraer area bajo la curva
  cat(c("AUC Performance: ", mod_AUCperf))
  
  return(c(mod_ROCRperf,mod_AUCperf))
}

Entrenaremos un modelo con todas las variables, sin tratamiento de valores faltantes, para tener una referencia de un modelo a mejorar.

set.seed(1234)
do_complete=data_raw[complete.cases(data_raw),]
trainIndex = createDataPartition(do_complete$BAD, p=0.8, list=FALSE)
data_trainA = do_complete[trainIndex,]
data_testA = do_complete[-trainIndex,]

mod_complete_cases = glm(BAD~.,family=binomial,data=data_trainA)
mod_complete_cases[["terms"]][[3]]
## LOAN + MORTDUE + VALUE + REASON + JOB + YOJ + DEROG + DELINQ + 
##     CLAGE + NINQ + CLNO + DEBTINC
summary(mod_complete_cases)
## 
## Call:
## glm(formula = BAD ~ ., family = binomial, data = data_trainA)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.8525  -0.3975  -0.2802  -0.1841   3.5823  
## 
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   -4.845e+00  5.303e-01  -9.135  < 2e-16 ***
## LOAN          -2.558e-05  9.809e-06  -2.608  0.00911 ** 
## MORTDUE       -3.938e-06  3.917e-06  -1.005  0.31470    
## VALUE          4.125e-06  3.374e-06   1.222  0.22158    
## REASONHomeImp -1.565e-01  1.824e-01  -0.858  0.39092    
## JOBOffice     -7.314e-01  3.034e-01  -2.411  0.01592 *  
## JOBOther      -1.907e-01  2.315e-01  -0.824  0.40987    
## JOBProfExe    -2.410e-01  2.678e-01  -0.900  0.36817    
## JOBSales       1.233e+00  4.877e-01   2.529  0.01145 *  
## JOBSelf        5.676e-01  4.837e-01   1.173  0.24063    
## YOJ           -9.516e-03  1.157e-02  -0.822  0.41086    
## DEROG          7.780e-01  1.169e-01   6.658 2.77e-11 ***
## DELINQ         7.789e-01  7.741e-02  10.063  < 2e-16 ***
## CLAGE         -6.173e-03  1.250e-03  -4.938 7.88e-07 ***
## NINQ           1.311e-01  4.138e-02   3.168  0.00154 ** 
## CLNO          -9.222e-03  9.091e-03  -1.014  0.31039    
## DEBTINC        1.020e-01  1.157e-02   8.814  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1641.4  on 2691  degrees of freedom
## Residual deviance: 1224.1  on 2675  degrees of freedom
## AIC: 1258.1
## 
## Number of Fisher Scoring iterations: 6
roc_auc_test(mod_complete_cases, data_testA)
## AUC Performance:  0.730160601149254
## [[1]]
## An object of class "performance"
## Slot "x.name":
## [1] "False positive rate"
## 
## Slot "y.name":
## [1] "True positive rate"
## 
## Slot "alpha.name":
## [1] "Cutoff"
## 
## Slot "x.values":
## [[1]]
##   [1] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
##   [7] 0.000000000 0.001620746 0.003241491 0.003241491 0.004862237 0.006482982
##  [13] 0.008103728 0.008103728 0.008103728 0.008103728 0.008103728 0.008103728
##  [19] 0.009724473 0.009724473 0.009724473 0.009724473 0.011345219 0.012965964
##  [25] 0.014586710 0.014586710 0.016207455 0.017828201 0.019448947 0.021069692
##  [31] 0.022690438 0.024311183 0.025931929 0.027552674 0.027552674 0.029173420
##  [37] 0.029173420 0.029173420 0.030794165 0.032414911 0.034035656 0.035656402
##  [43] 0.037277147 0.038897893 0.040518639 0.042139384 0.043760130 0.045380875
##  [49] 0.047001621 0.048622366 0.050243112 0.051863857 0.053484603 0.055105348
##  [55] 0.056726094 0.056726094 0.058346840 0.059967585 0.061588331 0.061588331
##  [61] 0.063209076 0.064829822 0.064829822 0.066450567 0.068071313 0.068071313
##  [67] 0.069692058 0.071312804 0.072933549 0.074554295 0.076175041 0.077795786
##  [73] 0.079416532 0.081037277 0.082658023 0.084278768 0.085899514 0.087520259
##  [79] 0.089141005 0.090761750 0.092382496 0.094003241 0.095623987 0.097244733
##  [85] 0.098865478 0.100486224 0.102106969 0.103727715 0.105348460 0.106969206
##  [91] 0.108589951 0.110210697 0.111831442 0.113452188 0.115072934 0.116693679
##  [97] 0.116693679 0.118314425 0.119935170 0.121555916 0.123176661 0.124797407
## [103] 0.126418152 0.128038898 0.129659643 0.129659643 0.129659643 0.131280389
## [109] 0.132901135 0.134521880 0.136142626 0.137763371 0.139384117 0.141004862
## [115] 0.142625608 0.142625608 0.144246353 0.145867099 0.147487844 0.149108590
## [121] 0.150729335 0.152350081 0.153970827 0.155591572 0.155591572 0.157212318
## [127] 0.158833063 0.160453809 0.162074554 0.163695300 0.165316045 0.166936791
## [133] 0.168557536 0.170178282 0.171799028 0.173419773 0.175040519 0.176661264
## [139] 0.178282010 0.179902755 0.181523501 0.183144246 0.184764992 0.186385737
## [145] 0.188006483 0.189627229 0.191247974 0.192868720 0.194489465 0.196110211
## [151] 0.197730956 0.199351702 0.200972447 0.202593193 0.204213938 0.205834684
## [157] 0.207455429 0.209076175 0.209076175 0.210696921 0.212317666 0.213938412
## [163] 0.215559157 0.217179903 0.217179903 0.218800648 0.220421394 0.222042139
## [169] 0.223662885 0.225283630 0.226904376 0.228525122 0.230145867 0.231766613
## [175] 0.231766613 0.233387358 0.235008104 0.236628849 0.238249595 0.239870340
## [181] 0.241491086 0.243111831 0.244732577 0.246353323 0.247974068 0.249594814
## [187] 0.251215559 0.252836305 0.254457050 0.256077796 0.257698541 0.259319287
## [193] 0.260940032 0.262560778 0.264181524 0.265802269 0.267423015 0.269043760
## [199] 0.270664506 0.272285251 0.273905997 0.275526742 0.277147488 0.278768233
## [205] 0.280388979 0.282009724 0.283630470 0.285251216 0.286871961 0.288492707
## [211] 0.290113452 0.290113452 0.291734198 0.293354943 0.294975689 0.296596434
## [217] 0.298217180 0.299837925 0.301458671 0.303079417 0.304700162 0.306320908
## [223] 0.307941653 0.309562399 0.311183144 0.312803890 0.314424635 0.316045381
## [229] 0.317666126 0.319286872 0.320907618 0.322528363 0.324149109 0.325769854
## [235] 0.327390600 0.329011345 0.330632091 0.332252836 0.333873582 0.335494327
## [241] 0.337115073 0.338735818 0.340356564 0.340356564 0.341977310 0.343598055
## [247] 0.345218801 0.346839546 0.348460292 0.350081037 0.351701783 0.353322528
## [253] 0.354943274 0.356564019 0.358184765 0.359805511 0.361426256 0.363047002
## [259] 0.364667747 0.366288493 0.366288493 0.367909238 0.367909238 0.369529984
## [265] 0.369529984 0.371150729 0.372771475 0.372771475 0.374392220 0.376012966
## [271] 0.377633712 0.379254457 0.380875203 0.382495948 0.384116694 0.385737439
## [277] 0.387358185 0.387358185 0.388978930 0.390599676 0.392220421 0.393841167
## [283] 0.395461912 0.397082658 0.397082658 0.397082658 0.398703404 0.400324149
## [289] 0.401944895 0.403565640 0.405186386 0.406807131 0.408427877 0.410048622
## [295] 0.411669368 0.413290113 0.414910859 0.416531605 0.418152350 0.419773096
## [301] 0.419773096 0.421393841 0.423014587 0.424635332 0.426256078 0.427876823
## [307] 0.429497569 0.431118314 0.432739060 0.434359806 0.435980551 0.437601297
## [313] 0.439222042 0.440842788 0.442463533 0.444084279 0.445705024 0.447325770
## [319] 0.448946515 0.450567261 0.452188006 0.453808752 0.455429498 0.457050243
## [325] 0.458670989 0.460291734 0.461912480 0.463533225 0.465153971 0.466774716
## [331] 0.466774716 0.468395462 0.470016207 0.471636953 0.473257699 0.474878444
## [337] 0.476499190 0.478119935 0.479740681 0.481361426 0.482982172 0.484602917
## [343] 0.486223663 0.487844408 0.487844408 0.489465154 0.491085900 0.492706645
## [349] 0.494327391 0.495948136 0.497568882 0.497568882 0.499189627 0.500810373
## [355] 0.502431118 0.504051864 0.504051864 0.504051864 0.505672609 0.507293355
## [361] 0.508914100 0.510534846 0.512155592 0.513776337 0.515397083 0.517017828
## [367] 0.518638574 0.520259319 0.521880065 0.523500810 0.523500810 0.525121556
## [373] 0.526742301 0.528363047 0.529983793 0.531604538 0.533225284 0.534846029
## [379] 0.536466775 0.538087520 0.539708266 0.541329011 0.542949757 0.544570502
## [385] 0.546191248 0.547811994 0.549432739 0.549432739 0.551053485 0.552674230
## [391] 0.554294976 0.555915721 0.557536467 0.559157212 0.560777958 0.562398703
## [397] 0.564019449 0.565640194 0.567260940 0.568881686 0.570502431 0.572123177
## [403] 0.573743922 0.575364668 0.576985413 0.578606159 0.580226904 0.581847650
## [409] 0.583468395 0.585089141 0.586709887 0.588330632 0.589951378 0.591572123
## [415] 0.593192869 0.594813614 0.596434360 0.598055105 0.599675851 0.601296596
## [421] 0.602917342 0.604538088 0.606158833 0.607779579 0.609400324 0.611021070
## [427] 0.612641815 0.614262561 0.615883306 0.617504052 0.619124797 0.620745543
## [433] 0.622366288 0.623987034 0.625607780 0.627228525 0.628849271 0.630470016
## [439] 0.632090762 0.633711507 0.635332253 0.636952998 0.638573744 0.640194489
## [445] 0.641815235 0.643435981 0.645056726 0.646677472 0.648298217 0.648298217
## [451] 0.649918963 0.651539708 0.653160454 0.654781199 0.656401945 0.658022690
## [457] 0.659643436 0.661264182 0.662884927 0.664505673 0.666126418 0.667747164
## [463] 0.669367909 0.670988655 0.672609400 0.674230146 0.675850891 0.677471637
## [469] 0.679092382 0.680713128 0.682333874 0.683954619 0.685575365 0.687196110
## [475] 0.688816856 0.690437601 0.692058347 0.693679092 0.695299838 0.696920583
## [481] 0.698541329 0.700162075 0.701782820 0.703403566 0.705024311 0.706645057
## [487] 0.708265802 0.709886548 0.711507293 0.713128039 0.714748784 0.716369530
## [493] 0.717990276 0.719611021 0.721231767 0.722852512 0.724473258 0.726094003
## [499] 0.727714749 0.729335494 0.730956240 0.732576985 0.734197731 0.735818476
## [505] 0.737439222 0.739059968 0.740680713 0.742301459 0.742301459 0.743922204
## [511] 0.745542950 0.747163695 0.748784441 0.750405186 0.752025932 0.753646677
## [517] 0.755267423 0.756888169 0.758508914 0.760129660 0.760129660 0.761750405
## [523] 0.763371151 0.764991896 0.766612642 0.768233387 0.769854133 0.771474878
## [529] 0.773095624 0.774716370 0.776337115 0.777957861 0.779578606 0.781199352
## [535] 0.782820097 0.784440843 0.786061588 0.787682334 0.789303079 0.790923825
## [541] 0.792544571 0.794165316 0.795786062 0.797406807 0.799027553 0.800648298
## [547] 0.802269044 0.803889789 0.805510535 0.807131280 0.808752026 0.810372771
## [553] 0.811993517 0.813614263 0.815235008 0.816855754 0.818476499 0.820097245
## [559] 0.821717990 0.823338736 0.824959481 0.826580227 0.828200972 0.829821718
## [565] 0.831442464 0.833063209 0.834683955 0.836304700 0.837925446 0.839546191
## [571] 0.841166937 0.842787682 0.844408428 0.846029173 0.847649919 0.849270665
## [577] 0.850891410 0.852512156 0.854132901 0.855753647 0.857374392 0.858995138
## [583] 0.860615883 0.862236629 0.863857374 0.863857374 0.865478120 0.867098865
## [589] 0.868719611 0.870340357 0.871961102 0.873581848 0.875202593 0.876823339
## [595] 0.878444084 0.880064830 0.881685575 0.883306321 0.884927066 0.886547812
## [601] 0.888168558 0.889789303 0.891410049 0.893030794 0.894651540 0.896272285
## [607] 0.897893031 0.899513776 0.901134522 0.901134522 0.902755267 0.904376013
## [613] 0.905996759 0.907617504 0.909238250 0.910858995 0.912479741 0.914100486
## [619] 0.915721232 0.917341977 0.918962723 0.920583468 0.922204214 0.923824959
## [625] 0.925445705 0.927066451 0.927066451 0.928687196 0.930307942 0.931928687
## [631] 0.933549433 0.935170178 0.936790924 0.938411669 0.940032415 0.941653160
## [637] 0.943273906 0.944894652 0.946515397 0.948136143 0.949756888 0.951377634
## [643] 0.952998379 0.954619125 0.956239870 0.957860616 0.959481361 0.961102107
## [649] 0.962722853 0.964343598 0.965964344 0.967585089 0.969205835 0.970826580
## [655] 0.972447326 0.974068071 0.975688817 0.977309562 0.978930308 0.980551053
## [661] 0.982171799 0.983792545 0.985413290 0.987034036 0.988654781 0.990275527
## [667] 0.991896272 0.993517018 0.995137763 0.996758509 0.998379254 1.000000000
## [673] 1.000000000
## 
## 
## Slot "y.values":
## [[1]]
##   [1] 0.00000000 0.01818182 0.03636364 0.05454545 0.07272727 0.09090909
##   [7] 0.10909091 0.10909091 0.10909091 0.12727273 0.12727273 0.12727273
##  [13] 0.12727273 0.14545455 0.16363636 0.18181818 0.20000000 0.21818182
##  [19] 0.21818182 0.23636364 0.25454545 0.27272727 0.27272727 0.27272727
##  [25] 0.27272727 0.29090909 0.29090909 0.29090909 0.29090909 0.29090909
##  [31] 0.29090909 0.29090909 0.29090909 0.29090909 0.30909091 0.30909091
##  [37] 0.32727273 0.34545455 0.34545455 0.34545455 0.34545455 0.34545455
##  [43] 0.34545455 0.34545455 0.34545455 0.34545455 0.34545455 0.34545455
##  [49] 0.34545455 0.34545455 0.34545455 0.34545455 0.34545455 0.34545455
##  [55] 0.34545455 0.36363636 0.36363636 0.36363636 0.36363636 0.38181818
##  [61] 0.38181818 0.38181818 0.40000000 0.40000000 0.40000000 0.41818182
##  [67] 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182
##  [73] 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182
##  [79] 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182
##  [85] 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182
##  [91] 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182 0.41818182
##  [97] 0.43636364 0.43636364 0.43636364 0.43636364 0.43636364 0.43636364
## [103] 0.43636364 0.43636364 0.43636364 0.45454545 0.47272727 0.47272727
## [109] 0.47272727 0.47272727 0.47272727 0.47272727 0.47272727 0.47272727
## [115] 0.47272727 0.49090909 0.49090909 0.49090909 0.49090909 0.49090909
## [121] 0.49090909 0.49090909 0.49090909 0.49090909 0.50909091 0.50909091
## [127] 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091
## [133] 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091
## [139] 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091
## [145] 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091
## [151] 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091 0.50909091
## [157] 0.50909091 0.50909091 0.52727273 0.52727273 0.52727273 0.52727273
## [163] 0.52727273 0.52727273 0.54545455 0.54545455 0.54545455 0.54545455
## [169] 0.54545455 0.54545455 0.54545455 0.54545455 0.54545455 0.54545455
## [175] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [181] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [187] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [193] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [199] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [205] 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636 0.56363636
## [211] 0.56363636 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818
## [217] 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818
## [223] 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818
## [229] 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818
## [235] 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818 0.58181818
## [241] 0.58181818 0.58181818 0.58181818 0.60000000 0.60000000 0.60000000
## [247] 0.60000000 0.60000000 0.60000000 0.60000000 0.60000000 0.60000000
## [253] 0.60000000 0.60000000 0.60000000 0.60000000 0.60000000 0.60000000
## [259] 0.60000000 0.60000000 0.61818182 0.61818182 0.63636364 0.63636364
## [265] 0.65454545 0.65454545 0.65454545 0.67272727 0.67272727 0.67272727
## [271] 0.67272727 0.67272727 0.67272727 0.67272727 0.67272727 0.67272727
## [277] 0.67272727 0.69090909 0.69090909 0.69090909 0.69090909 0.69090909
## [283] 0.69090909 0.69090909 0.70909091 0.72727273 0.72727273 0.72727273
## [289] 0.72727273 0.72727273 0.72727273 0.72727273 0.72727273 0.72727273
## [295] 0.72727273 0.72727273 0.72727273 0.72727273 0.72727273 0.72727273
## [301] 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455
## [307] 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455
## [313] 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455
## [319] 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455
## [325] 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455 0.74545455
## [331] 0.76363636 0.76363636 0.76363636 0.76363636 0.76363636 0.76363636
## [337] 0.76363636 0.76363636 0.76363636 0.76363636 0.76363636 0.76363636
## [343] 0.76363636 0.76363636 0.78181818 0.78181818 0.78181818 0.78181818
## [349] 0.78181818 0.78181818 0.78181818 0.80000000 0.80000000 0.80000000
## [355] 0.80000000 0.80000000 0.81818182 0.83636364 0.83636364 0.83636364
## [361] 0.83636364 0.83636364 0.83636364 0.83636364 0.83636364 0.83636364
## [367] 0.83636364 0.83636364 0.83636364 0.83636364 0.85454545 0.85454545
## [373] 0.85454545 0.85454545 0.85454545 0.85454545 0.85454545 0.85454545
## [379] 0.85454545 0.85454545 0.85454545 0.85454545 0.85454545 0.85454545
## [385] 0.85454545 0.85454545 0.85454545 0.87272727 0.87272727 0.87272727
## [391] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [397] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [403] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [409] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [415] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [421] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [427] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [433] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [439] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727
## [445] 0.87272727 0.87272727 0.87272727 0.87272727 0.87272727 0.89090909
## [451] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [457] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [463] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [469] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [475] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [481] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [487] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [493] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [499] 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909 0.89090909
## [505] 0.89090909 0.89090909 0.89090909 0.89090909 0.90909091 0.90909091
## [511] 0.90909091 0.90909091 0.90909091 0.90909091 0.90909091 0.90909091
## [517] 0.90909091 0.90909091 0.90909091 0.90909091 0.92727273 0.92727273
## [523] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [529] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [535] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [541] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [547] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [553] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [559] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [565] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [571] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [577] 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273 0.92727273
## [583] 0.92727273 0.92727273 0.92727273 0.94545455 0.94545455 0.94545455
## [589] 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455
## [595] 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455
## [601] 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455 0.94545455
## [607] 0.94545455 0.94545455 0.94545455 0.96363636 0.96363636 0.96363636
## [613] 0.96363636 0.96363636 0.96363636 0.96363636 0.96363636 0.96363636
## [619] 0.96363636 0.96363636 0.96363636 0.96363636 0.96363636 0.96363636
## [625] 0.96363636 0.96363636 0.98181818 0.98181818 0.98181818 0.98181818
## [631] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [637] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [643] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [649] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [655] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [661] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [667] 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818 0.98181818
## [673] 1.00000000
## 
## 
## Slot "alpha.values":
## [[1]]
##   [1]          Inf 0.9999293221 0.9721298168 0.9667121870 0.9277615222
##   [6] 0.8816243564 0.8484769858 0.7406981538 0.6910705076 0.6858129079
##  [11] 0.6709038699 0.6494059255 0.6247875874 0.6149127323 0.6053373724
##  [16] 0.5770214058 0.5724456419 0.5627020927 0.5482299852 0.4932567950
##  [21] 0.4767996881 0.4092757001 0.4073887639 0.4071465683 0.4053019068
##  [26] 0.3903691098 0.3837766042 0.3773918412 0.3588321807 0.3577447071
##  [31] 0.3538475688 0.3485958431 0.3432111402 0.3378683504 0.3247552201
##  [36] 0.3065368097 0.2983874978 0.2885987018 0.2866362389 0.2851727434
##  [41] 0.2669234226 0.2654967284 0.2641416232 0.2620796134 0.2592932053
##  [46] 0.2493175926 0.2369871020 0.2319574498 0.2287026689 0.2282189184
##  [51] 0.2254231678 0.2210638750 0.2167219857 0.2114396420 0.2060955543
##  [56] 0.2038605278 0.2001587578 0.1976238971 0.1951065498 0.1927402476
##  [61] 0.1855776114 0.1839163428 0.1819823088 0.1752000792 0.1737208380
##  [66] 0.1727952588 0.1653509078 0.1594456386 0.1590432298 0.1579834407
##  [71] 0.1574444552 0.1574346149 0.1534926936 0.1516375795 0.1500387015
##  [76] 0.1475994941 0.1456720030 0.1452050024 0.1426162942 0.1412688322
##  [81] 0.1381125987 0.1377487186 0.1376808903 0.1373504377 0.1373069332
##  [86] 0.1303904877 0.1302985799 0.1301876562 0.1293499739 0.1287362082
##  [91] 0.1285359623 0.1275379443 0.1272229849 0.1266548867 0.1259246133
##  [96] 0.1256079982 0.1251760445 0.1242735696 0.1229335891 0.1216058956
## [101] 0.1188992864 0.1177046060 0.1154196672 0.1147582185 0.1132343825
## [106] 0.1131286747 0.1130462985 0.1127414460 0.1122807205 0.1120149665
## [111] 0.1104071404 0.1103609778 0.1092932549 0.1078711431 0.1077767384
## [116] 0.1068144788 0.1065859444 0.1063347577 0.1062225739 0.1060384207
## [121] 0.1053206816 0.1050642771 0.1044504076 0.1041547274 0.1009533283
## [126] 0.1001999482 0.1000451546 0.0999741113 0.0992482348 0.0990456502
## [131] 0.0989251230 0.0974613484 0.0974463083 0.0970082690 0.0958342666
## [136] 0.0958316873 0.0956779147 0.0954988640 0.0947714527 0.0945194123
## [141] 0.0944646965 0.0943736012 0.0940756386 0.0924945261 0.0921954389
## [146] 0.0917664826 0.0917442721 0.0917288534 0.0915096996 0.0913507257
## [151] 0.0911225036 0.0903409416 0.0903299155 0.0896869381 0.0895802212
## [156] 0.0894991463 0.0889519143 0.0882542561 0.0881109899 0.0880678790
## [161] 0.0879377614 0.0877677053 0.0866360707 0.0860851573 0.0860464898
## [166] 0.0857824672 0.0857221148 0.0850125944 0.0849499905 0.0845303781
## [171] 0.0838096956 0.0837626903 0.0831617873 0.0825491273 0.0823924716
## [176] 0.0823795194 0.0821067751 0.0818455747 0.0815007279 0.0812797126
## [181] 0.0811781486 0.0810062309 0.0808660745 0.0808236881 0.0802553028
## [186] 0.0799083742 0.0792955282 0.0791397898 0.0789955920 0.0788005188
## [191] 0.0786596292 0.0786273008 0.0779040715 0.0777139756 0.0776173622
## [196] 0.0772654600 0.0772310227 0.0769890850 0.0767589963 0.0766663553
## [201] 0.0764379946 0.0760559553 0.0759980210 0.0758551760 0.0755613239
## [206] 0.0754975741 0.0752331494 0.0751748728 0.0749548749 0.0747520505
## [211] 0.0746540150 0.0742830012 0.0738446660 0.0737210561 0.0735842467
## [216] 0.0733837447 0.0727997240 0.0727770798 0.0726507726 0.0726304048
## [221] 0.0725346997 0.0723334687 0.0722566655 0.0720918113 0.0714070927
## [226] 0.0712741934 0.0708528758 0.0707643811 0.0707589739 0.0702878207
## [231] 0.0693071051 0.0692164503 0.0687690408 0.0683589380 0.0680479974
## [236] 0.0674891930 0.0674328256 0.0672048800 0.0670039818 0.0668222949
## [241] 0.0660984700 0.0658892132 0.0658677187 0.0655687938 0.0652172000
## [246] 0.0652053554 0.0643692979 0.0640040695 0.0637481052 0.0635733588
## [251] 0.0635666090 0.0632994464 0.0631925301 0.0630980677 0.0629992295
## [256] 0.0628187493 0.0627794853 0.0623603626 0.0620888639 0.0619839343
## [261] 0.0614727957 0.0612840703 0.0606918101 0.0600687028 0.0600601210
## [266] 0.0596712258 0.0595988295 0.0594544589 0.0593572586 0.0593530471
## [271] 0.0588965185 0.0586466327 0.0584006262 0.0583232554 0.0582818163
## [276] 0.0580992164 0.0577105602 0.0575716891 0.0574898363 0.0569777671
## [281] 0.0569151935 0.0566649375 0.0565758935 0.0565679207 0.0564147672
## [286] 0.0561870725 0.0560438218 0.0556032347 0.0551827364 0.0550130981
## [291] 0.0546311200 0.0546156254 0.0543010437 0.0539565812 0.0535246547
## [296] 0.0533173751 0.0531793522 0.0528770984 0.0522600514 0.0512164282
## [301] 0.0511360693 0.0509632326 0.0508600915 0.0504487893 0.0500594191
## [306] 0.0498254970 0.0497103281 0.0496899413 0.0495130554 0.0494988585
## [311] 0.0493389907 0.0492973902 0.0492169408 0.0492084457 0.0489044070
## [316] 0.0484438256 0.0481205462 0.0478094489 0.0468237970 0.0464916079
## [321] 0.0464291267 0.0463270092 0.0462715376 0.0462506940 0.0460571566
## [326] 0.0459167092 0.0458589683 0.0458542541 0.0458372801 0.0454967219
## [331] 0.0454100972 0.0454090261 0.0453187255 0.0452480166 0.0451890516
## [336] 0.0450943120 0.0450284718 0.0449286327 0.0449040013 0.0447853326
## [341] 0.0446316156 0.0442055437 0.0441667547 0.0440731740 0.0437643287
## [346] 0.0436399033 0.0435548424 0.0435537800 0.0435511746 0.0435037088
## [351] 0.0434603825 0.0433746047 0.0432393721 0.0431118902 0.0421011632
## [356] 0.0419775103 0.0419452283 0.0418805975 0.0415977316 0.0415263554
## [361] 0.0415183932 0.0414203131 0.0413024982 0.0409186420 0.0407739718
## [366] 0.0407638706 0.0407343371 0.0405551719 0.0405450176 0.0402309936
## [371] 0.0402011442 0.0398316746 0.0398253234 0.0397827513 0.0396872654
## [376] 0.0393733784 0.0392954418 0.0392947247 0.0385890514 0.0384385696
## [381] 0.0381279997 0.0379098508 0.0378084512 0.0377952517 0.0377260363
## [386] 0.0377144136 0.0376835898 0.0376761814 0.0375341613 0.0375099054
## [391] 0.0374652621 0.0373104077 0.0372982039 0.0370674588 0.0369367311
## [396] 0.0368450246 0.0367028320 0.0366934243 0.0364533692 0.0362736807
## [401] 0.0359452445 0.0359110425 0.0358046739 0.0353390371 0.0352559549
## [406] 0.0352477404 0.0350068353 0.0347874083 0.0346048022 0.0345844002
## [411] 0.0345140219 0.0344262037 0.0343893672 0.0343265347 0.0342401296
## [416] 0.0341673131 0.0335623744 0.0334190338 0.0332818988 0.0330423205
## [421] 0.0325460385 0.0322854554 0.0322788090 0.0322577784 0.0321385401
## [426] 0.0321337941 0.0320589098 0.0319913550 0.0319464923 0.0318039810
## [431] 0.0317272375 0.0316374828 0.0315844537 0.0314726178 0.0310653190
## [436] 0.0310506598 0.0308869070 0.0305959277 0.0305846454 0.0305608673
## [441] 0.0304913556 0.0304822369 0.0304484393 0.0304011450 0.0298976192
## [446] 0.0298580820 0.0297970553 0.0297514921 0.0296825856 0.0296495633
## [451] 0.0296035519 0.0295764528 0.0295268537 0.0294541308 0.0294007795
## [456] 0.0293024479 0.0292716044 0.0292640713 0.0287405396 0.0286157766
## [461] 0.0285358227 0.0285225505 0.0282513391 0.0280699532 0.0280315078
## [466] 0.0279431516 0.0279216592 0.0278216363 0.0276678975 0.0276223543
## [471] 0.0273556185 0.0271193193 0.0270315752 0.0269798075 0.0269247753
## [476] 0.0268000858 0.0266900974 0.0266726290 0.0265493788 0.0263076926
## [481] 0.0262572722 0.0260611986 0.0260504185 0.0259352040 0.0255793905
## [486] 0.0251249078 0.0251141765 0.0250903626 0.0246558046 0.0246352146
## [491] 0.0246340137 0.0245660969 0.0245205389 0.0245108000 0.0244466939
## [496] 0.0242677403 0.0239827208 0.0239342462 0.0236276084 0.0236063472
## [501] 0.0235432290 0.0234693470 0.0232223376 0.0232144358 0.0231931821
## [506] 0.0231680811 0.0231435832 0.0231399385 0.0228947325 0.0227942155
## [511] 0.0227423894 0.0227259876 0.0227205715 0.0226668927 0.0225678840
## [516] 0.0225437027 0.0224340420 0.0222531840 0.0221640005 0.0220433546
## [521] 0.0219744314 0.0219713590 0.0216349177 0.0215414479 0.0214508904
## [526] 0.0213547971 0.0213154939 0.0212710036 0.0208047143 0.0204382762
## [531] 0.0202218312 0.0200379206 0.0199919569 0.0199601468 0.0199335259
## [536] 0.0196708242 0.0194979297 0.0194308547 0.0189914219 0.0186772667
## [541] 0.0186337165 0.0186261526 0.0185342713 0.0185226285 0.0184673994
## [546] 0.0181085130 0.0180895588 0.0179754260 0.0178035750 0.0177019861
## [551] 0.0177006656 0.0174142940 0.0173155029 0.0171304772 0.0171005880
## [556] 0.0169886910 0.0167977968 0.0162285055 0.0162158300 0.0161359608
## [561] 0.0161130577 0.0159825148 0.0159667213 0.0159528574 0.0158912462
## [566] 0.0158193529 0.0157435673 0.0156684196 0.0155640010 0.0154973044
## [571] 0.0154613858 0.0153376885 0.0152607338 0.0152551316 0.0151096800
## [576] 0.0147649409 0.0146891730 0.0145807991 0.0145639674 0.0144835640
## [581] 0.0144681749 0.0142306420 0.0139695543 0.0135692941 0.0135526685
## [586] 0.0132317080 0.0130931737 0.0130545994 0.0129097333 0.0127149217
## [591] 0.0124396629 0.0121063059 0.0119829828 0.0119303155 0.0118910747
## [596] 0.0117945212 0.0116461663 0.0114799916 0.0114370025 0.0113547358
## [601] 0.0112163106 0.0108401220 0.0107627221 0.0106289061 0.0106217607
## [606] 0.0103912891 0.0103723508 0.0103105010 0.0102695511 0.0102302005
## [611] 0.0102071544 0.0101666456 0.0099785924 0.0098558982 0.0097446100
## [616] 0.0097173808 0.0096613522 0.0094842716 0.0093687606 0.0093619625
## [621] 0.0093322688 0.0092563303 0.0092054703 0.0092023875 0.0090914127
## [626] 0.0089001790 0.0088027528 0.0087452028 0.0087290834 0.0086960311
## [631] 0.0083462498 0.0082533779 0.0082099804 0.0079858328 0.0079216532
## [636] 0.0079109397 0.0077352727 0.0077137049 0.0076471380 0.0075562382
## [641] 0.0072867583 0.0071876208 0.0070643460 0.0069651641 0.0069053504
## [646] 0.0068890716 0.0067358315 0.0066541425 0.0066180822 0.0063323322
## [651] 0.0061992435 0.0060590994 0.0058285451 0.0057818172 0.0055806622
## [656] 0.0054926878 0.0052643749 0.0052112495 0.0049823739 0.0048910020
## [661] 0.0048490361 0.0044145800 0.0040883528 0.0039897091 0.0036595090
## [666] 0.0034415582 0.0031016719 0.0030629909 0.0010952773 0.0005104153
## [671] 0.0004230532 0.0004093324 0.0002251355
## 
## 
## 
## [[2]]
## [1] 0.7301606

Hacemos selección de variables por inspección, eliminando iterativamente aquellas que no son (menor o igual a 0.05)

mod_ajustado1 = glm(BAD~LOAN + JOB + DEROG + DELINQ + CLAGE + NINQ + DEBTINC,family=binomial,data=data_trainA)
summary(mod_ajustado1)
## 
## Call:
## glm(formula = BAD ~ LOAN + JOB + DEROG + DELINQ + CLAGE + NINQ + 
##     DEBTINC, family = binomial, data = data_trainA)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.7804  -0.3972  -0.2769  -0.1852   3.5719  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.994e+00  5.121e-01  -9.752  < 2e-16 ***
## LOAN        -2.256e-05  8.769e-06  -2.573  0.01008 *  
## JOBOffice   -6.648e-01  2.980e-01  -2.231  0.02570 *  
## JOBOther    -1.388e-01  2.280e-01  -0.609  0.54261    
## JOBProfExe  -2.019e-01  2.627e-01  -0.769  0.44215    
## JOBSales     1.264e+00  4.872e-01   2.595  0.00945 ** 
## JOBSelf      5.755e-01  4.719e-01   1.220  0.22258    
## DEROG        7.584e-01  1.146e-01   6.615 3.72e-11 ***
## DELINQ       7.512e-01  7.406e-02  10.143  < 2e-16 ***
## CLAGE       -6.550e-03  1.202e-03  -5.451 5.02e-08 ***
## NINQ         1.326e-01  4.059e-02   3.268  0.00108 ** 
## DEBTINC      9.997e-02  1.135e-02   8.808  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1641.4  on 2691  degrees of freedom
## Residual deviance: 1228.4  on 2680  degrees of freedom
## AIC: 1252.4
## 
## Number of Fisher Scoring iterations: 6
anova(mod_complete_cases,mod_ajustado1,test='Chi')
## Analysis of Deviance Table
## 
## Model 1: BAD ~ LOAN + MORTDUE + VALUE + REASON + JOB + YOJ + DEROG + DELINQ + 
##     CLAGE + NINQ + CLNO + DEBTINC
## Model 2: BAD ~ LOAN + JOB + DEROG + DELINQ + CLAGE + NINQ + DEBTINC
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1      2675     1224.1                     
## 2      2680     1228.4 -5  -4.3633   0.4984

3.2 Modelo B: modelo con tratamiento de valores faltantes para todas las variables

Ahora construiremos un modelo para la data donde hemos: 1) creado un valor “Missing” para los factores con valores faltantes; y 2) apolicado Woe Binning para las variables númericas.

data_trainB = data_woe[trainIndex,]
data_testB = data_woe[-trainIndex,]

modB_completo = glm(BAD~.,family=binomial,data=data_trainB)
modB_completo[["terms"]][[3]]
## LOAN + REASON + JOB + DEBTINC_woe + DEROG_woe + DELINQ_woe + 
##     MORTDUE_woe + YOJ_woe + NINQ_woe + CLAGE_woe + CLNO_woe + 
##     VALUE_woe
summary(modB_completo)
## 
## Call:
## glm(formula = BAD ~ ., family = binomial, data = data_trainB)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -3.14463  -0.40268  -0.21875  -0.08064   3.07447  
## 
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   -9.643e-01  2.996e-01  -3.219 0.001287 ** 
## LOAN          -1.087e-05  1.876e-05  -0.580 0.562163    
## REASONHomeImp -3.094e-02  1.504e-01  -0.206 0.837003    
## REASONMissing -2.538e-01  4.826e-01  -0.526 0.599004    
## JOBMissing    -2.435e+00  5.378e-01  -4.527 5.99e-06 ***
## JOBOffice     -8.364e-01  2.681e-01  -3.120 0.001809 ** 
## JOBOther      -7.984e-02  2.117e-01  -0.377 0.706040    
## JOBProfExe    -1.620e-01  2.414e-01  -0.671 0.502173    
## JOBSales       1.184e-01  4.396e-01   0.269 0.787708    
## JOBSelf        1.314e-01  4.270e-01   0.308 0.758330    
## DEBTINC_woe    9.782e-01  4.683e-02  20.890  < 2e-16 ***
## DEROG_woe      8.472e-01  1.190e-01   7.117 1.10e-12 ***
## DELINQ_woe     7.854e-01  8.687e-02   9.041  < 2e-16 ***
## MORTDUE_woe    1.100e+00  3.196e-01   3.441 0.000579 ***
## YOJ_woe        7.477e-01  2.538e-01   2.946 0.003222 ** 
## NINQ_woe       1.968e-01  1.863e-01   1.057 0.290677    
## CLAGE_woe      1.266e+00  1.442e-01   8.781  < 2e-16 ***
## CLNO_woe       9.583e-01  2.228e-01   4.301 1.70e-05 ***
## VALUE_woe      1.060e+00  1.378e-01   7.690 1.47e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2881.2  on 2691  degrees of freedom
## Residual deviance: 1505.6  on 2673  degrees of freedom
## AIC: 1543.6
## 
## Number of Fisher Scoring iterations: 6

3.2.1 Selección de variables y regularización

Hacemos seleccion de atributos con un método de selección step-wise, en ambas direcciones.

modB_nulo = glm(BAD~1,family=binomial,data=data_trainB)
step(modB_nulo,scope=list(upper=modB_completo), data=data_trainB, direction="both")
## Start:  AIC=2883.15
## BAD ~ 1
## 
##               Df Deviance    AIC
## + DEBTINC_woe  1   1993.8 1997.8
## + DELINQ_woe   1   2675.2 2679.2
## + VALUE_woe    1   2686.9 2690.9
## + DEROG_woe    1   2714.4 2718.4
## + CLAGE_woe    1   2755.5 2759.5
## + JOB          6   2792.3 2806.3
## + MORTDUE_woe  1   2809.0 2813.0
## + LOAN         1   2816.8 2820.8
## + NINQ_woe     1   2816.9 2820.9
## + CLNO_woe     1   2848.7 2852.7
## + YOJ_woe      1   2865.7 2869.7
## + REASON       2   2872.4 2878.4
## <none>             2881.2 2883.2
## 
## Step:  AIC=1997.84
## BAD ~ DEBTINC_woe
## 
##               Df Deviance    AIC
## + DELINQ_woe   1   1880.8 1886.8
## + VALUE_woe    1   1881.7 1887.7
## + CLAGE_woe    1   1895.2 1901.2
## + DEROG_woe    1   1915.5 1921.5
## + JOB          6   1937.1 1953.1
## + MORTDUE_woe  1   1958.5 1964.5
## + CLNO_woe     1   1959.5 1965.5
## + YOJ_woe      1   1975.8 1981.8
## + NINQ_woe     1   1981.7 1987.7
## + LOAN         1   1986.7 1992.7
## + REASON       2   1989.1 1997.1
## <none>             1993.8 1997.8
## - DEBTINC_woe  1   2881.2 2883.2
## 
## Step:  AIC=1886.76
## BAD ~ DEBTINC_woe + DELINQ_woe
## 
##               Df Deviance    AIC
## + VALUE_woe    1   1770.2 1778.2
## + CLAGE_woe    1   1772.4 1780.4
## + CLNO_woe     1   1833.3 1841.3
## + DEROG_woe    1   1833.3 1841.3
## + JOB          6   1832.0 1850.0
## + MORTDUE_woe  1   1843.1 1851.1
## + YOJ_woe      1   1864.1 1872.1
## + LOAN         1   1873.7 1881.7
## + NINQ_woe     1   1873.8 1881.8
## <none>             1880.8 1886.8
## + REASON       2   1877.4 1887.4
## - DELINQ_woe   1   1993.8 1997.8
## - DEBTINC_woe  1   2675.2 2679.2
## 
## Step:  AIC=1778.21
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe
## 
##               Df Deviance    AIC
## + CLAGE_woe    1   1669.6 1679.6
## + DEROG_woe    1   1708.8 1718.8
## + JOB          6   1709.7 1729.7
## + CLNO_woe     1   1737.4 1747.4
## + YOJ_woe      1   1744.5 1754.5
## + REASON       2   1755.9 1767.9
## + MORTDUE_woe  1   1758.5 1768.5
## + NINQ_woe     1   1760.4 1770.4
## + LOAN         1   1767.7 1777.7
## <none>             1770.2 1778.2
## - VALUE_woe    1   1880.8 1886.8
## - DELINQ_woe   1   1881.7 1887.7
## - DEBTINC_woe  1   2487.0 2493.0
## 
## Step:  AIC=1679.57
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe
## 
##               Df Deviance    AIC
## + DEROG_woe    1   1606.2 1618.2
## + JOB          6   1606.3 1628.3
## + CLNO_woe     1   1646.3 1658.3
## + YOJ_woe      1   1653.0 1665.0
## + MORTDUE_woe  1   1654.3 1666.3
## + REASON       2   1655.2 1669.2
## + NINQ_woe     1   1663.0 1675.0
## <none>             1669.6 1679.6
## + LOAN         1   1669.3 1681.3
## - CLAGE_woe    1   1770.2 1778.2
## - VALUE_woe    1   1772.4 1780.4
## - DELINQ_woe   1   1789.0 1797.0
## - DEBTINC_woe  1   2363.7 2371.7
## 
## Step:  AIC=1618.17
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe
## 
##               Df Deviance    AIC
## + JOB          6   1548.3 1572.3
## + CLNO_woe     1   1586.1 1600.1
## + MORTDUE_woe  1   1591.2 1605.2
## + YOJ_woe      1   1594.2 1608.2
## + REASON       2   1593.2 1609.2
## + NINQ_woe     1   1603.2 1617.2
## <none>             1606.2 1618.2
## + LOAN         1   1605.7 1619.7
## - DEROG_woe    1   1669.6 1679.6
## - DELINQ_woe   1   1693.2 1703.2
## - CLAGE_woe    1   1708.8 1718.8
## - VALUE_woe    1   1723.7 1733.7
## - DEBTINC_woe  1   2242.0 2252.0
## 
## Step:  AIC=1572.29
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB
## 
##               Df Deviance    AIC
## + CLNO_woe     1   1527.5 1553.5
## + MORTDUE_woe  1   1537.5 1563.5
## + YOJ_woe      1   1539.1 1565.1
## <none>             1548.3 1572.3
## + NINQ_woe     1   1547.3 1573.3
## + LOAN         1   1547.8 1573.8
## + REASON       2   1547.0 1575.0
## - JOB          6   1606.2 1618.2
## - DEROG_woe    1   1606.3 1628.3
## - DELINQ_woe   1   1625.4 1647.4
## - CLAGE_woe    1   1653.1 1675.1
## - VALUE_woe    1   1682.0 1704.0
## - DEBTINC_woe  1   2159.6 2181.6
## 
## Step:  AIC=1553.47
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + CLNO_woe
## 
##               Df Deviance    AIC
## + MORTDUE_woe  1   1517.3 1545.3
## + YOJ_woe      1   1520.0 1548.0
## <none>             1527.5 1553.5
## + NINQ_woe     1   1525.8 1553.8
## + LOAN         1   1526.8 1554.8
## + REASON       2   1526.4 1556.4
## - CLNO_woe     1   1548.3 1572.3
## - JOB          6   1586.1 1600.1
## - DEROG_woe    1   1583.1 1607.1
## - DELINQ_woe   1   1611.7 1635.7
## - CLAGE_woe    1   1623.9 1647.9
## - VALUE_woe    1   1646.4 1670.4
## - DEBTINC_woe  1   2144.3 2168.3
## 
## Step:  AIC=1545.34
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + CLNO_woe + MORTDUE_woe
## 
##               Df Deviance    AIC
## + YOJ_woe      1   1507.5 1537.5
## <none>             1517.3 1545.3
## + NINQ_woe     1   1515.7 1545.7
## + LOAN         1   1517.1 1547.1
## + REASON       2   1516.4 1548.4
## - MORTDUE_woe  1   1527.5 1553.5
## - CLNO_woe     1   1537.5 1563.5
## - JOB          6   1573.0 1589.0
## - DEROG_woe    1   1573.7 1599.7
## - DELINQ_woe   1   1602.2 1628.2
## - VALUE_woe    1   1610.5 1636.5
## - CLAGE_woe    1   1616.7 1642.7
## - DEBTINC_woe  1   2120.7 2146.7
## 
## Step:  AIC=1537.46
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + CLNO_woe + MORTDUE_woe + YOJ_woe
## 
##               Df Deviance    AIC
## <none>             1507.5 1537.5
## + NINQ_woe     1   1506.3 1538.3
## + LOAN         1   1507.3 1539.3
## + REASON       2   1507.0 1541.0
## - YOJ_woe      1   1517.3 1545.3
## - MORTDUE_woe  1   1520.0 1548.0
## - CLNO_woe     1   1525.6 1553.6
## - JOB          6   1559.5 1577.5
## - DEROG_woe    1   1560.4 1588.4
## - DELINQ_woe   1   1593.3 1621.3
## - CLAGE_woe    1   1596.9 1624.9
## - VALUE_woe    1   1601.5 1629.5
## - DEBTINC_woe  1   2110.9 2138.9
## 
## Call:  glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
##     DEROG_woe + JOB + CLNO_woe + MORTDUE_woe + YOJ_woe, family = binomial, 
##     data = data_trainB)
## 
## Coefficients:
## (Intercept)  DEBTINC_woe   DELINQ_woe    VALUE_woe    CLAGE_woe    DEROG_woe  
##    -1.11734      0.98918      0.79374      1.05244      1.28796      0.85495  
##  JOBMissing    JOBOffice     JOBOther   JOBProfExe     JOBSales      JOBSelf  
##    -2.52369     -0.84130     -0.06617     -0.16659      0.12591      0.12786  
##    CLNO_woe  MORTDUE_woe      YOJ_woe  
##     0.93919      1.12142      0.78138  
## 
## Degrees of Freedom: 2691 Total (i.e. Null);  2677 Residual
## Null Deviance:       2881 
## Residual Deviance: 1507  AIC: 1537
modB_stepwise = glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
    DEROG_woe + JOB + CLNO_woe + MORTDUE_woe + YOJ_woe, family = binomial, 
    data = data_trainB)
summary(modB_stepwise)
## 
## Call:
## glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
##     DEROG_woe + JOB + CLNO_woe + MORTDUE_woe + YOJ_woe, family = binomial, 
##     data = data_trainB)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -3.06296  -0.40091  -0.21940  -0.07617   3.06704  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.11734    0.18759  -5.956 2.58e-09 ***
## DEBTINC_woe  0.98918    0.04574  21.624  < 2e-16 ***
## DELINQ_woe   0.79374    0.08667   9.159  < 2e-16 ***
## VALUE_woe    1.05244    0.13429   7.837 4.61e-15 ***
## CLAGE_woe    1.28796    0.14316   8.997  < 2e-16 ***
## DEROG_woe    0.85495    0.11828   7.228 4.90e-13 ***
## JOBMissing  -2.52369    0.51870  -4.865 1.14e-06 ***
## JOBOffice   -0.84130    0.26783  -3.141 0.001683 ** 
## JOBOther    -0.06617    0.21069  -0.314 0.753470    
## JOBProfExe  -0.16659    0.24119  -0.691 0.489764    
## JOBSales     0.12591    0.43641   0.289 0.772955    
## JOBSelf      0.12786    0.42361   0.302 0.762782    
## CLNO_woe     0.93919    0.22101   4.250 2.14e-05 ***
## MORTDUE_woe  1.12142    0.31779   3.529 0.000417 ***
## YOJ_woe      0.78138    0.25191   3.102 0.001923 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2881.2  on 2691  degrees of freedom
## Residual deviance: 1507.5  on 2677  degrees of freedom
## AIC: 1537.5
## 
## Number of Fisher Scoring iterations: 6

3.3 Modelo C: modelo con balanceo de casos para BAD

ggplot(data = data, aes(x = factor(BAD), y = ..count.., fill = factor(BAD))) +
  geom_bar() +
  scale_fill_manual(values = c("gray50", "orangered2")) +
  labs(title = "BAD (1 = Con mora; 0 = Préstamo pagado)") +
  theme_bw() +
  theme(legend.position = "bottom")

# Down sampling
data_woe.no = data_woe[data_woe$BAD==0,] #MUESTRA TODOS LOS VALORES BAD=0, sin BAD=1
muestra = sample(nrow(data_woe.no),nrow(data_woe[data_woe$BAD==1,])) 
data_woe.no = data_woe.no[muestra, ]
data_woe_balanceado = rbind(data_woe[data_woe$BAD==1,], data_woe.no)

Queda balanceado

ggplot(data = data_woe_balanceado, aes(x = factor(BAD), y = ..count.., fill = factor(BAD))) +
  geom_bar() +
  scale_fill_manual(values = c("gray50", "orangered2")) +
  labs(title = "BAD (1 = Con mora; 0 = Préstamo pagado)") +
  theme_bw() +
  theme(legend.position = "bottom")

trainIndexC = createDataPartition(data_woe_balanceado$BAD, p=0.8, list=FALSE)
data_trainC = data_woe_balanceado[trainIndexC,]
data_testC = data_woe_balanceado[-trainIndexC,]

modC_completo = glm(BAD~.,family=binomial,data=data_trainC)
summary(modC_completo)
## 
## Call:
## glm(formula = BAD ~ ., family = binomial, data = data_trainC)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -3.05331  -0.51526  -0.06198   0.47455   2.69823  
## 
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept)    1.749e-01  2.331e-01   0.750 0.453156    
## LOAN          -1.176e-05  6.374e-06  -1.845 0.065061 .  
## REASONHomeImp  2.255e-03  1.551e-01   0.015 0.988397    
## REASONMissing  2.271e-01  4.351e-01   0.522 0.601809    
## JOBMissing    -9.058e-01  4.994e-01  -1.814 0.069719 .  
## JOBOffice     -6.567e-01  2.631e-01  -2.496 0.012551 *  
## JOBOther       2.196e-01  2.185e-01   1.005 0.314891    
## JOBProfExe     2.537e-01  2.419e-01   1.049 0.294257    
## JOBSales       1.773e+00  5.366e-01   3.304 0.000953 ***
## JOBSelf        3.667e-01  4.140e-01   0.886 0.375708    
## DEBTINC_woe    9.601e-01  4.935e-02  19.455  < 2e-16 ***
## DEROG_woe      6.057e-01  1.208e-01   5.014 5.33e-07 ***
## DELINQ_woe     9.870e-01  9.976e-02   9.894  < 2e-16 ***
## MORTDUE_woe    4.129e-01  3.178e-01   1.299 0.193866    
## YOJ_woe        1.248e+00  2.544e-01   4.905 9.33e-07 ***
## NINQ_woe       2.880e-01  1.705e-01   1.689 0.091236 .  
## CLAGE_woe      1.027e+00  1.456e-01   7.052 1.76e-12 ***
## CLNO_woe       1.004e+00  2.331e-01   4.308 1.65e-05 ***
## VALUE_woe      9.923e-01  1.549e-01   6.407 1.49e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2639.5  on 1903  degrees of freedom
## Residual deviance: 1385.7  on 1885  degrees of freedom
## AIC: 1423.7
## 
## Number of Fisher Scoring iterations: 6

3.3.1 Selección de variables y regularización

Hacemos seleccion de atributos con un método de selección step-wise, en ambas direcciones.

modC_nulo = glm(BAD~1,family=binomial,data=data_trainC)
step(modC_nulo,scope=list(upper=modC_completo), data=data_trainC, direction="both")
## Start:  AIC=2641.5
## BAD ~ 1
## 
##               Df Deviance    AIC
## + DEBTINC_woe  1   1798.8 1802.8
## + DELINQ_woe   1   2392.9 2396.9
## + VALUE_woe    1   2462.8 2466.8
## + DEROG_woe    1   2505.4 2509.4
## + CLAGE_woe    1   2509.5 2513.5
## + JOB          6   2576.9 2590.9
## + NINQ_woe     1   2589.4 2593.4
## + CLNO_woe     1   2595.0 2599.0
## + YOJ_woe      1   2599.3 2603.3
## + LOAN         1   2611.7 2615.7
## + MORTDUE_woe  1   2623.2 2627.2
## <none>             2639.5 2641.5
## + REASON       2   2636.0 2642.0
## 
## Step:  AIC=1802.78
## BAD ~ DEBTINC_woe
## 
##               Df Deviance    AIC
## + DELINQ_woe   1   1681.6 1687.6
## + VALUE_woe    1   1695.5 1701.5
## + DEROG_woe    1   1734.7 1740.7
## + CLAGE_woe    1   1734.8 1740.8
## + CLNO_woe     1   1766.9 1772.9
## + YOJ_woe      1   1769.3 1775.3
## + JOB          6   1768.7 1784.7
## + LOAN         1   1785.1 1791.1
## + MORTDUE_woe  1   1786.2 1792.2
## + NINQ_woe     1   1790.3 1796.3
## <none>             1798.8 1802.8
## + REASON       2   1798.2 1806.2
## - DEBTINC_woe  1   2639.5 2641.5
## 
## Step:  AIC=1687.57
## BAD ~ DEBTINC_woe + DELINQ_woe
## 
##               Df Deviance    AIC
## + VALUE_woe    1   1588.6 1596.6
## + CLAGE_woe    1   1596.3 1604.3
## + CLNO_woe     1   1639.9 1647.9
## + DEROG_woe    1   1645.0 1653.0
## + YOJ_woe      1   1647.3 1655.3
## + JOB          6   1643.7 1661.7
## + MORTDUE_woe  1   1668.8 1676.8
## + LOAN         1   1669.0 1677.0
## + NINQ_woe     1   1677.1 1685.1
## <none>             1681.6 1687.6
## + REASON       2   1681.1 1691.1
## - DELINQ_woe   1   1798.8 1802.8
## - DEBTINC_woe  1   2392.9 2396.9
## 
## Step:  AIC=1596.57
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe
## 
##               Df Deviance    AIC
## + CLAGE_woe    1   1512.3 1522.3
## + YOJ_woe      1   1550.8 1560.8
## + DEROG_woe    1   1551.5 1561.5
## + JOB          6   1547.3 1567.3
## + CLNO_woe     1   1557.9 1567.9
## + LOAN         1   1578.6 1588.6
## + NINQ_woe     1   1582.1 1592.1
## <none>             1588.6 1596.6
## + MORTDUE_woe  1   1587.4 1597.4
## + REASON       2   1588.5 1600.5
## - VALUE_woe    1   1681.6 1687.6
## - DELINQ_woe   1   1695.5 1701.5
## - DEBTINC_woe  1   2235.1 2241.1
## 
## Step:  AIC=1522.33
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe
## 
##               Df Deviance    AIC
## + DEROG_woe    1   1480.0 1492.0
## + JOB          6   1471.8 1493.8
## + YOJ_woe      1   1483.2 1495.2
## + CLNO_woe     1   1491.6 1503.6
## + LOAN         1   1507.0 1519.0
## + NINQ_woe     1   1507.3 1519.3
## <none>             1512.3 1522.3
## + MORTDUE_woe  1   1511.1 1523.1
## + REASON       2   1512.2 1526.2
## - CLAGE_woe    1   1588.6 1596.6
## - VALUE_woe    1   1596.3 1604.3
## - DELINQ_woe   1   1637.1 1645.1
## - DEBTINC_woe  1   2090.4 2098.4
## 
## Step:  AIC=1491.95
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe
## 
##               Df Deviance    AIC
## + JOB          6   1441.1 1465.1
## + YOJ_woe      1   1452.5 1466.5
## + CLNO_woe     1   1458.7 1472.7
## + LOAN         1   1474.5 1488.5
## + NINQ_woe     1   1477.8 1491.8
## <none>             1480.0 1492.0
## + MORTDUE_woe  1   1479.1 1493.1
## + REASON       2   1479.6 1495.6
## - DEROG_woe    1   1512.3 1522.3
## - CLAGE_woe    1   1551.5 1561.5
## - VALUE_woe    1   1563.4 1573.4
## - DELINQ_woe   1   1577.2 1587.2
## - DEBTINC_woe  1   2030.3 2040.3
## 
## Step:  AIC=1465.11
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB
## 
##               Df Deviance    AIC
## + YOJ_woe      1   1412.6 1438.6
## + CLNO_woe     1   1420.0 1446.0
## + LOAN         1   1435.8 1461.8
## + NINQ_woe     1   1439.0 1465.0
## <none>             1441.1 1465.1
## + MORTDUE_woe  1   1440.0 1466.0
## + REASON       2   1440.6 1468.6
## - JOB          6   1480.0 1492.0
## - DEROG_woe    1   1471.8 1493.8
## - CLAGE_woe    1   1512.5 1534.5
## - VALUE_woe    1   1530.2 1552.2
## - DELINQ_woe   1   1544.0 1566.0
## - DEBTINC_woe  1   1963.8 1985.8
## 
## Step:  AIC=1438.57
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + YOJ_woe
## 
##               Df Deviance    AIC
## + CLNO_woe     1   1393.5 1421.5
## + LOAN         1   1408.8 1436.8
## + MORTDUE_woe  1   1410.3 1438.3
## <none>             1412.6 1438.6
## + NINQ_woe     1   1411.3 1439.3
## + REASON       2   1412.0 1442.0
## - DEROG_woe    1   1440.9 1464.9
## - YOJ_woe      1   1441.1 1465.1
## - JOB          6   1452.5 1466.5
## - CLAGE_woe    1   1474.4 1498.4
## - VALUE_woe    1   1503.2 1527.2
## - DELINQ_woe   1   1521.7 1545.7
## - DEBTINC_woe  1   1929.2 1953.2
## 
## Step:  AIC=1421.54
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + YOJ_woe + CLNO_woe
## 
##               Df Deviance    AIC
## + LOAN         1   1390.5 1420.5
## + NINQ_woe     1   1391.4 1421.4
## <none>             1393.5 1421.5
## + MORTDUE_woe  1   1391.7 1421.7
## + REASON       2   1393.2 1425.2
## - CLNO_woe     1   1412.6 1438.6
## - YOJ_woe      1   1420.0 1446.0
## - DEROG_woe    1   1422.2 1448.2
## - JOB          6   1432.6 1448.6
## - CLAGE_woe    1   1448.8 1474.8
## - VALUE_woe    1   1474.3 1500.3
## - DELINQ_woe   1   1507.7 1533.7
## - DEBTINC_woe  1   1905.2 1931.2
## 
## Step:  AIC=1420.51
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + YOJ_woe + CLNO_woe + LOAN
## 
##               Df Deviance    AIC
## + NINQ_woe     1   1387.7 1419.7
## <none>             1390.5 1420.5
## + MORTDUE_woe  1   1388.8 1420.8
## - LOAN         1   1393.5 1421.5
## + REASON       2   1390.2 1424.2
## - CLNO_woe     1   1408.8 1436.8
## - YOJ_woe      1   1415.5 1443.5
## - DEROG_woe    1   1419.2 1447.2
## - JOB          6   1429.2 1447.2
## - CLAGE_woe    1   1443.3 1471.3
## - VALUE_woe    1   1470.0 1498.0
## - DELINQ_woe   1   1502.6 1530.6
## - DEBTINC_woe  1   1899.4 1927.4
## 
## Step:  AIC=1419.73
## BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + YOJ_woe + CLNO_woe + LOAN + NINQ_woe
## 
##               Df Deviance    AIC
## <none>             1387.7 1419.7
## + MORTDUE_woe  1   1386.0 1420.0
## - NINQ_woe     1   1390.5 1420.5
## - LOAN         1   1391.4 1421.4
## + REASON       2   1387.4 1423.4
## - CLNO_woe     1   1407.1 1437.1
## - YOJ_woe      1   1411.6 1441.6
## - DEROG_woe    1   1414.1 1444.1
## - JOB          6   1426.5 1446.5
## - CLAGE_woe    1   1440.0 1470.0
## - VALUE_woe    1   1468.0 1498.0
## - DELINQ_woe   1   1496.9 1526.9
## - DEBTINC_woe  1   1880.8 1910.8
## 
## Call:  glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
##     DEROG_woe + JOB + YOJ_woe + CLNO_woe + LOAN + NINQ_woe, family = binomial, 
##     data = data_trainC)
## 
## Coefficients:
## (Intercept)  DEBTINC_woe   DELINQ_woe    VALUE_woe    CLAGE_woe    DEROG_woe  
##   1.684e-01    9.592e-01    9.868e-01    1.055e+00    1.023e+00    6.101e-01  
##  JOBMissing    JOBOffice     JOBOther   JOBProfExe     JOBSales      JOBSelf  
##  -7.846e-01   -6.428e-01    2.547e-01    2.620e-01    1.755e+00    4.070e-01  
##     YOJ_woe     CLNO_woe         LOAN     NINQ_woe  
##   1.217e+00    1.015e+00   -1.198e-05    2.822e-01  
## 
## Degrees of Freedom: 1903 Total (i.e. Null);  1888 Residual
## Null Deviance:       2640 
## Residual Deviance: 1388  AIC: 1420
modC_stepwise = glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
    DEROG_woe + JOB + CLNO_woe + MORTDUE_woe + YOJ_woe, family = binomial, 
    data = data_trainB)
summary(modC_stepwise)
## 
## Call:
## glm(formula = BAD ~ DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + 
##     DEROG_woe + JOB + CLNO_woe + MORTDUE_woe + YOJ_woe, family = binomial, 
##     data = data_trainB)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -3.06296  -0.40091  -0.21940  -0.07617   3.06704  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.11734    0.18759  -5.956 2.58e-09 ***
## DEBTINC_woe  0.98918    0.04574  21.624  < 2e-16 ***
## DELINQ_woe   0.79374    0.08667   9.159  < 2e-16 ***
## VALUE_woe    1.05244    0.13429   7.837 4.61e-15 ***
## CLAGE_woe    1.28796    0.14316   8.997  < 2e-16 ***
## DEROG_woe    0.85495    0.11828   7.228 4.90e-13 ***
## JOBMissing  -2.52369    0.51870  -4.865 1.14e-06 ***
## JOBOffice   -0.84130    0.26783  -3.141 0.001683 ** 
## JOBOther    -0.06617    0.21069  -0.314 0.753470    
## JOBProfExe  -0.16659    0.24119  -0.691 0.489764    
## JOBSales     0.12591    0.43641   0.289 0.772955    
## JOBSelf      0.12786    0.42361   0.302 0.762782    
## CLNO_woe     0.93919    0.22101   4.250 2.14e-05 ***
## MORTDUE_woe  1.12142    0.31779   3.529 0.000417 ***
## YOJ_woe      0.78138    0.25191   3.102 0.001923 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2881.2  on 2691  degrees of freedom
## Residual deviance: 1507.5  on 2677  degrees of freedom
## AIC: 1537.5
## 
## Number of Fisher Scoring iterations: 6

3.4 Medición de calidad predictiva y justificación del modelo seleccionado

3.4.1 Validez: calibración y discriminacion

mod_complete_cases[["terms"]][[3]]
## LOAN + MORTDUE + VALUE + REASON + JOB + YOJ + DEROG + DELINQ + 
##     CLAGE + NINQ + CLNO + DEBTINC
modB_completo[["terms"]][[3]]
## LOAN + REASON + JOB + DEBTINC_woe + DEROG_woe + DELINQ_woe + 
##     MORTDUE_woe + YOJ_woe + NINQ_woe + CLAGE_woe + CLNO_woe + 
##     VALUE_woe
modB_stepwise[["terms"]][[3]]
## DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + CLNO_woe + MORTDUE_woe + YOJ_woe
modC_completo[["terms"]][[3]]
## LOAN + REASON + JOB + DEBTINC_woe + DEROG_woe + DELINQ_woe + 
##     MORTDUE_woe + YOJ_woe + NINQ_woe + CLAGE_woe + CLNO_woe + 
##     VALUE_woe
modC_stepwise[["terms"]][[3]]
## DEBTINC_woe + DELINQ_woe + VALUE_woe + CLAGE_woe + DEROG_woe + 
##     JOB + CLNO_woe + MORTDUE_woe + YOJ_woe
ROC_M1=roc_auc_test(mod_complete_cases, data_testA)
## AUC Performance:  0.730160601149254
ROC_M2=roc_auc_test(modB_completo, data_testB)
## AUC Performance:  0.902133106733698
ROC_M3=roc_auc_test(modB_stepwise, data_testB)
## AUC Performance:  0.901244499595681
ROC_M4=roc_auc_test(modC_stepwise, data_testC)
## AUC Performance:  0.916911463618723
plot(ROC_M1[[1]])
abline(0,1)
plot(ROC_M2[[1]], add = T, col = 2)
plot(ROC_M3[[1]], add = T, col = 3)
plot(ROC_M4[[1]], add = T, col = 4)
legend(0.2, 0.3, 
       legend=c(paste("Modelo 1 (AUC:",round(ROC_M1[[2]],3),")"), 
                paste("Modelo 2 (AUC:",round(ROC_M2[[2]],3),")"), 
                paste("Modelo 3 (AUC:",round(ROC_M3[[2]],3),")"),
                paste("Modelo 4 (AUC:",round(ROC_M4[[2]],3),")")),
       col=   c(1,2,3,4),
       lty=1:2, cex=0.8)

hoslem.test(mod_complete_cases$y,fitted(mod_complete_cases),g=10)
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  mod_complete_cases$y, fitted(mod_complete_cases)
## X-squared = 17.942, df = 8, p-value = 0.02166
hoslem.test(modB_completo$y,fitted(modB_completo),g=10)
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  modB_completo$y, fitted(modB_completo)
## X-squared = 17.987, df = 8, p-value = 0.02133
hoslem.test(modB_stepwise$y,fitted(modB_stepwise),g=10)
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  modB_stepwise$y, fitted(modB_stepwise)
## X-squared = 11.53, df = 8, p-value = 0.1734
hoslem.test(modC_stepwise$y,fitted(modC_stepwise),g=10)
## 
##  Hosmer and Lemeshow goodness of fit (GOF) test
## 
## data:  modC_stepwise$y, fitted(modC_stepwise)
## X-squared = 11.53, df = 8, p-value = 0.1734

3.4.2 Generalizabilidad (generalizability)

3.5 Interpretación del modelo de puntuación derivado del modelo predictivo

3.5.1 Construcción del modelo de puntuación para el modelo seleccionado