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Additional Thoughts on Estimating LGD with Proportional Odds Model

In my previous post (https://statcompute.wordpress.com/2018/01/28/modeling-lgd-with-proportional-odds-model), I’ve discussed how to use Proportional Odds Models in the LGD model development. In particular, I specifically mentioned that we would estimate a sub-model, which can be Gamma or Simplex regression, to project the conditional mean for LGD values in the (0, 1) range. However, it is worth pointing out that, if we would define a finer LGD segmentation, the necessity of this sub-model is completely optional. A standalone Proportional Odds Model without any sub-model is more than sufficient to serve the purpose of stress testing, e.g. CCAR.

In the example below, I will define 5 categories based upon LGD values in the [0, 1] range, estimate a Proportional Odds Model as usual, and then demonstrate how to apply the model outcome in the setting of stress testing with the stressed model input, e.g. LTV.

First of all, I defined 5 instead of 3 categories for LGD values, as shown below. Nonetheless, we could use a even finer category definition in practice to achieve a more accurate outcome.


df <- read.csv("lgd.csv")
df$lgd <- round(1 - df$Recovery_rate, 4)
l1 <- c(-Inf, 0, 0.0999, 0.4999, 0.9999, Inf)
l2 <- c("A", "B", "C", "D", "E")
df$lgd_cat <- cut(df$lgd, breaks = l1, labels = l2, ordered_result = T)
summary(df$lgd_cat)
m1 <- ordinal::clm(lgd_cat ~ LTV, data = df)
#Coefficients:
#    Estimate Std. Error z value Pr(>|z|)    
#LTV   2.3841     0.1083   22.02   <2e-16 ***
#
#Threshold coefficients:
#    Estimate Std. Error z value
#A|B  0.54082    0.07897   6.848
#B|C  2.12270    0.08894  23.866
#C|D  3.18098    0.10161  31.307
#D|E  4.80338    0.13174  36.460

After the model estimation, it is straightforward to calculate the probability of each LGD category. The only question remained is how to calculate the LGD projection for each individual account as well as for the whole portfolio. In order to calculate the LGD projection, we need two factors, namely the probability and the expected mean of each LGD category, such that

Estimated_LGD = SUM_i [Prob(category i) * LGD_Mean(category i)], where i = A, B, C, D, and E in this particular case.

The calculation is shown below with the estimated LGD = 0.23 that is consistent with the actual LGD = 0.23 for the whole portfolio.


prob_A <- exp(df$LTV * (-m1$beta) + m1$Theta[1]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[1])) 
prob_B <- exp(df$LTV * (-m1$beta) + m1$Theta[2]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[2])) - prob_A
prob_C <- exp(df$LTV * (-m1$beta) + m1$Theta[3]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[3])) - prob_A - prob_B
prob_D <- exp(df$LTV * (-m1$beta) + m1$Theta[4]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[4])) - prob_A - prob_B - prob_C
prob_E <- 1 - exp(df$LTV * (-m1$beta) + m1$Theta[4]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[4]))
pred <- data.frame(prob_A, prob_B, prob_C, prob_D, prob_E)
sum(apply(pred, 2, mean) * aggregate(df['lgd'], df['lgd_cat'], mean)[2])
#[1] 0.2262811

One might be wondering how to apply the model outcome with simple averages in stress testing that the model input is stressed, e.g. more severe, and might be also concerned about the lack of model sensitivity. In the demonstration below, let’s stress the model input LTV by 50% and then evaluate the stressed LGD.


df$LTV_ST <- df$LTV * 1.5
prob_A <- exp(df$LTV_ST * (-m1$beta) + m1$Theta[1]) / (1 + exp(df$LTV_ST * (-m1$beta) + m1$Theta[1])) 
prob_B <- exp(df$LTV_ST * (-m1$beta) + m1$Theta[2]) / (1 + exp(df$LTV_ST * (-m1$beta) + m1$Theta[2])) - prob_A
prob_C <- exp(df$LTV_ST * (-m1$beta) + m1$Theta[3]) / (1 + exp(df$LTV_ST * (-m1$beta) + m1$Theta[3])) - prob_A - prob_B
prob_D <- exp(df$LTV_ST * (-m1$beta) + m1$Theta[4]) / (1 + exp(df$LTV_ST * (-m1$beta) + m1$Theta[4])) - prob_A - prob_B - prob_C
prob_E <- 1 - exp(df$LTV_ST * (-m1$beta) + m1$Theta[4]) / (1 + exp(df$LTV_ST * (-m1$beta) + m1$Theta[4]))
pred_ST <- data.frame(prob_A, prob_B, prob_C, prob_D, prob_E)
sum(apply(pred_ST, 2, mean) * aggregate(df['lgd'], df['lgd_cat'], mean)[2])
#[1] 0.3600153

As shown above, although we only use a simple averages as the expected mean for each LGD category, the overall LGD still increases by ~60%. The reason is that, with the more stressed model input, the Proportional Odds Model is able to push more accounts into categories with higher LGD. For instance, the output below shows that, if LTV is stressed by 50% overall, ~146% more accounts would roll into the most severe LGD category without any recovery.


apply(pred_ST, 2, mean) / apply(pred, 2, mean)
#   prob_A    prob_B    prob_C    prob_D    prob_E 
#0.6715374 0.7980619 1.0405573 1.4825803 2.4639293

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Written by statcompute

February 7, 2018 at 12:40 am

Estimating Parameters of A Hyper-Poisson Distribution in SAS

Similar to COM-Poisson, Double-Poisson, and Generalized Poisson distributions discussed in my previous post (https://statcompute.wordpress.com/2016/11/27/more-about-flexible-frequency-models/), the Hyper-Poisson distribution is another extension of the standard Poisson and is able to accommodate both under-dispersion and over-dispersion that are common in real-world problems. Given the complexity of parameterization and computation, the Hyper-Poisson is somewhat under-investigated. To the best of my knowledge, there is no off-shelf computing routine in SAS for the Hyper-Poisson distribution and only a R function available in http://www4.ujaen.es/~ajsaez/hp.fit.r written by A.J. Sáez-Castillo and A. Conde-Sánchez (2013).

The SAS code presented below is the starting point of my attempt on the Hyper-Poisson and its potential applications. The purpose is to replicate the calculation result shown in the Table 6 of “On the Hyper-Poisson Distribution and its Generalization with Applications” by Bayo H. Lawal (2017) (http://www.journalrepository.org/media/journals/BJMCS_6/2017/Mar/Lawal2132017BJMCS32184.pdf). As a result, the parameterization employed in my SAS code will closely follow Bayo H. Lawal (2017) instead of A.J. Sáez-Castillo and A. Conde-Sánchez (2013).


data d1;
  input y n @@;
datalines;
0 121 1 85 2 19 3 1 4 0 5 0 6 1
;
run;

data df;
  set d1;
  where n > 0;
  do i = 1 to n;
    output;
  end;
run;

proc nlmixed data = df;
  parms lambda = 1 beta = 1;
  theta = 1;
  do k = 1 to 100;
    theta = theta + gamma(beta) * (lambda ** k) / gamma(beta + k);
  end;
  prob = (gamma(beta) / gamma(beta + y)) * ((lambda ** y) / theta);
  ll = log(prob);
  model y ~ general(ll);
run;

/*
                     Standard
Parameter  Estimate     Error    DF  t Value  Pr > |t|   Alpha
lambda       0.3752    0.1178   227     3.19    0.0016    0.05
beta         0.5552    0.2266   227     2.45    0.0150    0.05 
*/

As shown, the estimated Lambda = 0.3752 and the estimated Beta = 0.5552 are identical to what is presented in the paper. The next step is be to explore applications in the frequency modeling as well as its value in business cases.

HP

Written by statcompute

February 4, 2018 at 3:22 pm

Modeling LGD with Proportional Odds Model

The LGD model is an important component in the expected loss calculation. In https://statcompute.wordpress.com/2015/11/01/quasi-binomial-model-in-sas, I discussed how to model LGD with the quasi-binomial regression that is simple and makes no distributional assumption.

In the real-world LGD data, we usually would observe 3 ordered categories of values, including 0, 1, and in-betweens. In cases with a nontrivial number of 0 and 1 values, the ordered logit model, which is also known as Proportional Odds model, can be applicable. In the demonstration below, I will show how we can potentially use the proportional odds model in the LGD model development.

First of all, we need to categorize all numeric LGD values into three ordinal categories. As shown below, there are more than 30% of 0 and 1 values.

df <- read.csv("lgd.csv")
df$lgd <- round(1 - df$Recovery_rate, 4)
df$lgd_cat <- cut(df$lgd, breaks = c(-Inf, 0, 0.9999, Inf), labels = c("L", "M", "H"), ordered_result = T)
summary(df$lgd_cat)

#   L    M    H 
# 730 1672  143 

The estimation of a proportional odds model is straightforward with clm() in the ordinal package or polr() in the MASS package. As demonstrated below, in addition to the coefficient for LTV, there are 2 intercepts to differentiate 3 categories.

m1 <- ordinal::clm(lgd_cat ~ LTV, data = df)
summary(m1)

#Coefficients:
#    Estimate Std. Error z value Pr(>|z|)    
#LTV   2.0777     0.1267    16.4   <2e-16 ***
#---
#Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#
#Threshold coefficients:
#    Estimate Std. Error z value
#L|M  0.38134    0.08676   4.396
#M|H  4.50145    0.14427  31.201

It is important to point out that, in a proportional odds model, it is the cumulative probability that is derived from the linear combination of model variables. For instance, the cumulative probability of LGD belonging to L or M is formulated as

Prob(LGD <= M) = Exp(4.50 – 2.08 * LTV) / (1 + Exp(4.50 – 2.08 * LTV))

Likewise, we would have

Prob(LGD <= L) = Exp(0.38 – 2.08 * LTV) / (1 + Exp(0.38 – 2.08 * LTV))

With above cumulative probabilities, then we can calculate the probability of each category as below.

Prob(LGD = L) = Prob(LGD <= L)
Prob(LGD = M) = Prob(LGD <= M) – Prob(LGD <= L)
Prob(LGD = H) = 1 – Prob(LGD <= M)

The R code is showing the detailed calculation how to convert cumulative probabilities to probabilities of interest.

cumprob_L <- exp(df$LTV * (-m1$beta) + m1$Theta[1]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[1])) 
cumprob_M <- exp(df$LTV * (-m1$beta) + m1$Theta[2]) / (1 + exp(df$LTV * (-m1$beta) + m1$Theta[2])) 
prob_L <- cumprob_L
prob_M <- cumprob_M - cumprob_L
prob_H <- 1 - cumprob_M
pred <- data.frame(prob_L, prob_M, prob_H)
apply(pred, 2, mean)

#    prob_L     prob_M     prob_H 
#0.28751210 0.65679888 0.05568903 

After predicting the probability of each category, we would need another sub-model to estimate the conditional LGD for lgd_cat = “M” with either Beta or Simplex regression. (See https://statcompute.wordpress.com/2014/10/27/flexible-beta-modeling and https://statcompute.wordpress.com/2014/02/02/simplex-model-in-r) The final LGD prediction can be formulated as

E(LGD|X)
= Prob(Y = 0|X) * E(Y|X, Y = 0) + Prob(Y = 1|X) * E(Y|X, Y = 1) + Prob(0 < Y < 1|X) * E(Y|X, 0 < Y < 1)
= Prob(Y = 1|X) + Prob(0 < Y < 1|X) * E(Y|X, 0 < Y < 1)

where E(Y|X, 0 < Y < 1) can be calculated from the sub-model.

Written by statcompute

January 28, 2018 at 2:39 am

Monotonic WoE Binning for LGD Models

While the monotonic binning algorithm has been widely used in scorecard and PD model (Probability of Default) developments, the similar idea can be generalized to LGD (Loss Given Default) models. In the post below, two SAS macros performing the monotonic binning for LGD are demonstrated.

The first one tends to generate relatively coarse bins based on iterative grouping, which requires a longer computing time.


%macro lgd_bin1(data = , y = , x = );

%let maxbin = 20;

data _tmp1 (keep = x y);
  set &data;
  y = min(1, max(0, &y));
  x = &x;
run;

proc sql noprint;
  select
    count(distinct x) into :xflg
  from
    _last_;
quit;

%let nbin = %sysfunc(min(&maxbin, &xflg));

%if &nbin > 2 %then %do;
  %do j = &nbin %to 2 %by -1;
    proc rank data = _tmp1 groups = &j out = _data_ (keep = x rank y);
      var x;
      ranks rank;
    run;

    proc summary data = _last_ nway;
      class rank;
      output out = _tmp2 (drop = _type_ rename = (_freq_ = freq))
      sum(y) = bads  mean(y) = bad_rate 
      min(x) = minx  max(x)  = maxx;
    run;

    proc sql noprint;
      select
        case when min(bad_rate) > 0 then 1 else 0 end into :minflg
      from
        _tmp2;
 
      select
        case when max(bad_rate) < 1 then 1 else 0 end into :maxflg
      from
        _tmp2;              
    quit;

    %if &minflg = 1 & &maxflg = 1 %then %do;
      proc corr data = _tmp2 spearman noprint outs = _corr;
        var minx;
        with bad_rate;
      run;
      
      proc sql noprint;
        select
          case when abs(minx) = 1 then 1 else 0 end into :cor
        from
          _corr
        where
          _type_ = 'CORR';
      quit;
 
      %if &cor = 1 %then %goto loopout;
    %end;
  %end;
%end;

%loopout:

proc sql noprint;
create table
  _tmp3 as
select
  a.rank + 1                                           as bin,
  a.minx                                               as minx,
  a.maxx                                               as maxx,
  a.freq                                               as freq,
  a.freq / b.freq                                      as dist,
  a.bad_rate                                           as avg_lgd,
  a.bads / b.bads                                      as bpct,
  (a.freq - a.bads) / (b.freq - b.bads)                as gpct,
  log(calculated bpct / calculated gpct)               as woe,
  (calculated bpct - calculated gpct) / calculated woe as iv 
from
  _tmp2 as a, (select sum(freq) as freq, sum(bads) as bads from _tmp2) as b;
quit;

proc print data = _last_ noobs label;
  var minx maxx freq dist avg_lgd woe;
  format freq comma8. dist percent10.2;
  label
    minx    = "Lower Limit"
    maxx    = "Upper Limit"
    freq    = "Freq"
    dist    = "Dist"
    avg_lgd = "Average LGD"
    woe     = "WoE";
  sum freq dist;
run; 

%mend lgd_bin1;

The second one can generate much finer bins based on the idea of isotonic regressions and is more computationally efficient.


%macro lgd_bin2(data = , y = , x = );

data _data_ (keep = x y);
  set &data;
  y = min(1, max(0, &y));
  x = &x;
run;

proc transreg data = _last_ noprint;
  model identity(y) = monotone(x);
  output out = _tmp1 tip = _t;
run;
 
proc summary data = _last_ nway;
  class _tx;
  output out = _data_ (drop = _freq_ _type_) mean(y) = lgd;
run;

proc sort data = _last_;
  by lgd;
run;
 
data _tmp2;
  set _last_;
  by lgd;
  _idx = _n_;
  if lgd = 0 then _idx = _idx + 1;
  if lgd = 1 then _idx = _idx - 1;
run;

proc sql noprint;
create table
  _tmp3 as
select
  a.*,
  b._idx
from
  _tmp1 as a inner join _tmp2 as b
on
  a._tx = b._tx;

create table
  _tmp4 as
select
  min(a.x)                                             as minx,
  max(a.x)                                             as maxx,
  sum(a.y)                                             as bads,
  count(a.y)                                           as freq,
  count(a.y) / b.freq                                  as dist,
  mean(a.y)                                            as avg_lgd,
  sum(a.y) / b.bads                                    as bpct,
  sum(1 - a.y) / (b.freq - b.bads)                     as gpct,
  log(calculated bpct / calculated gpct)               as woe,
  (calculated bpct - calculated gpct) * calculated woe as iv
from
  _tmp3 as a, (select count(*) as freq, sum(y) as bads from _tmp3) as b
group by
  a._idx;
quit;

proc print data = _last_ noobs label;
  var minx maxx freq dist avg_lgd woe; 
  format freq comma8. dist percent10.2;
  label
    minx    = "Lower Limit"
    maxx    = "Upper Limit"
    freq    = "Freq"
    dist    = "Dist"
    avg_lgd = "Average LGD"
    woe     = "WoE";
  sum freq dist;
run; 

%mend lgd_bin2;

Below is the output comparison between two macros with the testing data downloaded from http://www.creditriskanalytics.net/datasets-private.html. Should you have any feedback, please feel free to leave me a message.

lgd1

Written by statcompute

September 30, 2017 at 5:50 pm

Posted in CCAR, Credit Risk, SAS, Statistical Models, Statistics

Tagged with

Granular Monotonic Binning in SAS

In the post (https://statcompute.wordpress.com/2017/06/15/finer-monotonic-binning-based-on-isotonic-regression), it is shown how to do a finer monotonic binning with isotonic regression in R.

Below is a SAS macro implementing the monotonic binning with the same idea of isotonic regression. This macro is more efficient than the one shown in (https://statcompute.wordpress.com/2012/06/10/a-sas-macro-implementing-monotonic-woe-transformation-in-scorecard-development) without iterative binning and is also able to significantly increase the binning granularity.

%macro monobin(data = , y = , x = );
options mprint mlogic;

data _data_ (keep = _x _y);
  set &data;
  where &y in (0, 1) and &x ~= .;
  _y = &y;
  _x = &x;
run;

proc transreg data = _last_ noprint;
  model identity(_y) = monotone(_x);
  output out = _tmp1 tip = _t;
run;

proc summary data = _last_ nway;
  class _t_x;
  output out = _data_ (drop = _freq_ _type_) mean(_y) = _rate;
run;

proc sort data = _last_;
  by _rate;
run;

data _tmp2;
  set _last_;
  by _rate;
  _idx = _n_;
  if _rate = 0 then _idx = _idx + 1;
  if _rate = 1 then _idx = _idx - 1;
run;
  
proc sql noprint;
create table
  _tmp3 as
select
  a.*,
  b._idx
from
  _tmp1 as a inner join _tmp2 as b
on
  a._t_x = b._t_x;
  
create table
  _tmp4 as
select
  a._idx,
  min(a._x)                                               as _min_x,
  max(a._x)                                               as _max_x,
  sum(a._y)                                               as _bads,
  count(a._y)                                             as _freq,
  mean(a._y)                                              as _rate,
  sum(a._y) / b.bads                                      as _bpct,
  sum(1 - a._y) / (b.freq - b.bads)                       as _gpct,
  log(calculated _bpct / calculated _gpct)                as _woe,
  (calculated _bpct - calculated _gpct) * calculated _woe as _iv
from 
  _tmp3 as a, (select count(*) as freq, sum(_y) as bads from _tmp3) as b
group by
  a._idx;
quit;

title "Monotonic WoE Binning for %upcase(%trim(&x))";
proc print data = _last_ label noobs;
  var _min_x _max_x _bads _freq _rate _woe _iv;
  label
    _min_x = "Lower"
    _max_x = "Upper"
    _bads  = "#Bads"
    _freq  = "#Freq"
    _rate  = "BadRate"
    _woe   = "WoE"
    _iv    = "IV";
  sum _bads _freq _iv;
run;
title;

%mend monobin;

Below is the sample output for LTV, showing an identical binning scheme to the one generated by the R isobin() function.

Screenshot from 2017-09-24 21-30-40

Written by statcompute

September 24, 2017 at 11:00 pm

Finer Monotonic Binning Based on Isotonic Regression

In my early post (https://statcompute.wordpress.com/2017/01/22/monotonic-binning-with-smbinning-package/), I wrote a monobin() function based on the smbinning package by Herman Jopia to improve the monotonic binning algorithm. The function works well and provides robust binning outcomes. However, there are a couple potential drawbacks due to the coarse binning. First of all, the derived Information Value for each binned variable might tend to be low. Secondly, the binned variable might not be granular enough to reflect the data nature.

In light of the aforementioned, I drafted an improved function isobin() based on the isotonic regression (https://en.wikipedia.org/wiki/Isotonic_regression), as shown below.

isobin <- function(data, y, x) {
  d1 <- data[c(y, x)]
  d2 <- d1[!is.na(d1[x]), ]
  c <- cor(d2[, 2], d2[, 1], method = "spearman", use = "complete.obs")
  reg <- isoreg(d2[, 2], c / abs(c) * d2[, 1])
  k <- knots(as.stepfun(reg))
  sm1 <-smbinning.custom(d1, y, x, k)
  c1 <- subset(sm1$ivtable, subset = CntGood * CntBad > 0, select = Cutpoint)
  c2 <- suppressWarnings(as.numeric(unlist(strsplit(c1$Cutpoint, " "))))
  c3 <- c2[!is.na(c2)]
  return(smbinning.custom(d1, y, x, c3[-length(c3)]))
}

Compared with the legacy monobin(), the isobin() function is able to significantly increase the binning granularity as well as moderately improve the Information Value.

LTV Binning with isobin() Function

   Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds LnOdds     WoE     IV
1     <= 46     81      78      3        81         78         3 0.0139   0.9630  0.0370 26.0000 3.2581  1.9021 0.0272
2     <= 71    312     284     28       393        362        31 0.0535   0.9103  0.0897 10.1429 2.3168  0.9608 0.0363
3     <= 72     22      20      2       415        382        33 0.0038   0.9091  0.0909 10.0000 2.3026  0.9466 0.0025
4     <= 73     27      24      3       442        406        36 0.0046   0.8889  0.1111  8.0000 2.0794  0.7235 0.0019
5     <= 81    303     268     35       745        674        71 0.0519   0.8845  0.1155  7.6571 2.0356  0.6797 0.0194
6     <= 83    139     122     17       884        796        88 0.0238   0.8777  0.1223  7.1765 1.9708  0.6149 0.0074
7     <= 90    631     546     85      1515       1342       173 0.1081   0.8653  0.1347  6.4235 1.8600  0.5040 0.0235
8     <= 94    529     440     89      2044       1782       262 0.0906   0.8318  0.1682  4.9438 1.5981  0.2422 0.0049
9     <= 95    145     119     26      2189       1901       288 0.0248   0.8207  0.1793  4.5769 1.5210  0.1651 0.0006
10   <= 100    907     709    198      3096       2610       486 0.1554   0.7817  0.2183  3.5808 1.2756 -0.0804 0.0010
11   <= 101    195     151     44      3291       2761       530 0.0334   0.7744  0.2256  3.4318 1.2331 -0.1229 0.0005
12   <= 110   1217     934    283      4508       3695       813 0.2085   0.7675  0.2325  3.3004 1.1940 -0.1619 0.0057
13   <= 112    208     158     50      4716       3853       863 0.0356   0.7596  0.2404  3.1600 1.1506 -0.2054 0.0016
14   <= 115    253     183     70      4969       4036       933 0.0433   0.7233  0.2767  2.6143 0.9610 -0.3950 0.0075
15   <= 136    774     548    226      5743       4584      1159 0.1326   0.7080  0.2920  2.4248 0.8857 -0.4702 0.0333
16   <= 138     27      18      9      5770       4602      1168 0.0046   0.6667  0.3333  2.0000 0.6931 -0.6628 0.0024
17    > 138     66      39     27      5836       4641      1195 0.0113   0.5909  0.4091  1.4444 0.3677 -0.9882 0.0140
18  Missing      1       0      1      5837       4641      1196 0.0002   0.0000  1.0000  0.0000   -Inf    -Inf    Inf
19    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804 1.3559  0.0000 0.1897

LTV Binning with monobin() Function

  Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate   Odds LnOdds     WoE     IV
1    <= 85   1025     916    109      1025        916       109 0.1756   0.8937  0.1063 8.4037 2.1287  0.7727 0.0821
2    <= 94   1019     866    153      2044       1782       262 0.1746   0.8499  0.1501 5.6601 1.7334  0.3775 0.0221
3   <= 100   1052     828    224      3096       2610       486 0.1802   0.7871  0.2129 3.6964 1.3074 -0.0486 0.0004
4   <= 105    808     618    190      3904       3228       676 0.1384   0.7649  0.2351 3.2526 1.1795 -0.1765 0.0045
5   <= 114    985     748    237      4889       3976       913 0.1688   0.7594  0.2406 3.1561 1.1493 -0.2066 0.0076
6    > 114    947     665    282      5836       4641      1195 0.1622   0.7022  0.2978 2.3582 0.8579 -0.4981 0.0461
7  Missing      1       0      1      5837       4641      1196 0.0002   0.0000  1.0000 0.0000   -Inf    -Inf    Inf
8    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049 3.8804 1.3559  0.0000 0.1628

Bureau_Score Binning with isobin() Function

   Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds  LnOdds     WoE     IV
1    <= 491      4       1      3         4          1         3 0.0007   0.2500  0.7500  0.3333 -1.0986 -2.4546 0.0056
2    <= 532     24       9     15        28         10        18 0.0041   0.3750  0.6250  0.6000 -0.5108 -1.8668 0.0198
3    <= 559     51      24     27        79         34        45 0.0087   0.4706  0.5294  0.8889 -0.1178 -1.4737 0.0256
4    <= 560      2       1      1        81         35        46 0.0003   0.5000  0.5000  1.0000  0.0000 -1.3559 0.0008
5    <= 572     34      17     17       115         52        63 0.0058   0.5000  0.5000  1.0000  0.0000 -1.3559 0.0143
6    <= 602    153      84     69       268        136       132 0.0262   0.5490  0.4510  1.2174  0.1967 -1.1592 0.0459
7    <= 605     56      31     25       324        167       157 0.0096   0.5536  0.4464  1.2400  0.2151 -1.1408 0.0162
8    <= 606     14       8      6       338        175       163 0.0024   0.5714  0.4286  1.3333  0.2877 -1.0683 0.0035
9    <= 607     17      10      7       355        185       170 0.0029   0.5882  0.4118  1.4286  0.3567 -0.9993 0.0037
10   <= 632    437     261    176       792        446       346 0.0749   0.5973  0.4027  1.4830  0.3940 -0.9619 0.0875
11   <= 639    150      95     55       942        541       401 0.0257   0.6333  0.3667  1.7273  0.5465 -0.8094 0.0207
12   <= 653    451     300    151      1393        841       552 0.0773   0.6652  0.3348  1.9868  0.6865 -0.6694 0.0412
13   <= 662    295     213     82      1688       1054       634 0.0505   0.7220  0.2780  2.5976  0.9546 -0.4014 0.0091
14   <= 665    100      77     23      1788       1131       657 0.0171   0.7700  0.2300  3.3478  1.2083 -0.1476 0.0004
15   <= 667     57      44     13      1845       1175       670 0.0098   0.7719  0.2281  3.3846  1.2192 -0.1367 0.0002
16   <= 677    381     300     81      2226       1475       751 0.0653   0.7874  0.2126  3.7037  1.3093 -0.0466 0.0001
17   <= 679     66      53     13      2292       1528       764 0.0113   0.8030  0.1970  4.0769  1.4053  0.0494 0.0000
18   <= 683    160     129     31      2452       1657       795 0.0274   0.8062  0.1938  4.1613  1.4258  0.0699 0.0001
19   <= 689    203     164     39      2655       1821       834 0.0348   0.8079  0.1921  4.2051  1.4363  0.0804 0.0002
20   <= 699    304     249     55      2959       2070       889 0.0521   0.8191  0.1809  4.5273  1.5101  0.1542 0.0012
21   <= 707    312     268     44      3271       2338       933 0.0535   0.8590  0.1410  6.0909  1.8068  0.4509 0.0094
22   <= 717    368     318     50      3639       2656       983 0.0630   0.8641  0.1359  6.3600  1.8500  0.4941 0.0132
23   <= 721    134     119     15      3773       2775       998 0.0230   0.8881  0.1119  7.9333  2.0711  0.7151 0.0094
24   <= 723     49      44      5      3822       2819      1003 0.0084   0.8980  0.1020  8.8000  2.1748  0.8188 0.0043
25   <= 739    425     394     31      4247       3213      1034 0.0728   0.9271  0.0729 12.7097  2.5424  1.1864 0.0700
26   <= 746    166     154     12      4413       3367      1046 0.0284   0.9277  0.0723 12.8333  2.5520  1.1961 0.0277
27   <= 756    234     218     16      4647       3585      1062 0.0401   0.9316  0.0684 13.6250  2.6119  1.2560 0.0422
28   <= 761    110     104      6      4757       3689      1068 0.0188   0.9455  0.0545 17.3333  2.8526  1.4967 0.0260
29   <= 763     46      44      2      4803       3733      1070 0.0079   0.9565  0.0435 22.0000  3.0910  1.7351 0.0135
30   <= 767     96      92      4      4899       3825      1074 0.0164   0.9583  0.0417 23.0000  3.1355  1.7795 0.0293
31   <= 772     77      74      3      4976       3899      1077 0.0132   0.9610  0.0390 24.6667  3.2055  1.8495 0.0249
32   <= 787    269     260      9      5245       4159      1086 0.0461   0.9665  0.0335 28.8889  3.3635  2.0075 0.0974
33   <= 794     95      93      2      5340       4252      1088 0.0163   0.9789  0.0211 46.5000  3.8395  2.4835 0.0456
34    > 794    182     179      3      5522       4431      1091 0.0312   0.9835  0.0165 59.6667  4.0888  2.7328 0.0985
35  Missing    315     210    105      5837       4641      1196 0.0540   0.6667  0.3333  2.0000  0.6931 -0.6628 0.0282
36    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804  1.3559  0.0000 0.8357

Bureau_Score Binning with monobin() Function

   Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds LnOdds     WoE     IV
1    <= 617    513     284    229       513        284       229 0.0879   0.5536  0.4464  1.2402 0.2153 -1.1407 0.1486
2    <= 642    515     317    198      1028        601       427 0.0882   0.6155  0.3845  1.6010 0.4706 -0.8853 0.0861
3    <= 657    512     349    163      1540        950       590 0.0877   0.6816  0.3184  2.1411 0.7613 -0.5946 0.0363
4    <= 672    487     371    116      2027       1321       706 0.0834   0.7618  0.2382  3.1983 1.1626 -0.1933 0.0033
5    <= 685    494     396     98      2521       1717       804 0.0846   0.8016  0.1984  4.0408 1.3964  0.0405 0.0001
6    <= 701    521     428     93      3042       2145       897 0.0893   0.8215  0.1785  4.6022 1.5265  0.1706 0.0025
7    <= 714    487     418     69      3529       2563       966 0.0834   0.8583  0.1417  6.0580 1.8014  0.4454 0.0144
8    <= 730    489     441     48      4018       3004      1014 0.0838   0.9018  0.0982  9.1875 2.2178  0.8619 0.0473
9    <= 751    513     476     37      4531       3480      1051 0.0879   0.9279  0.0721 12.8649 2.5545  1.1986 0.0859
10   <= 775    492     465     27      5023       3945      1078 0.0843   0.9451  0.0549 17.2222 2.8462  1.4903 0.1157
11    > 775    499     486     13      5522       4431      1091 0.0855   0.9739  0.0261 37.3846 3.6213  2.2653 0.2126
12  Missing    315     210    105      5837       4641      1196 0.0540   0.6667  0.3333  2.0000 0.6931 -0.6628 0.0282
13    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804 1.3559  0.0000 0.7810

Written by statcompute

June 15, 2017 at 5:24 pm

Monotonic Binning with Smbinning Package

The R package smbinning (http://www.scoringmodeling.com/rpackage/smbinning) provides a very user-friendly interface for the WoE (Weight of Evidence) binning algorithm employed in the scorecard development. However, there are several improvement opportunities in my view:

1. First of all, the underlying algorithm in the smbinning() function utilizes the recursive partitioning, which does not necessarily guarantee the monotonicity.
2. Secondly, the density in each generated bin is not even. The frequency in some bins could be much higher than the one in others.
3. At last, the function might not provide the binning outcome for some variables due to the lack of statistical significance.

In light of the above, I wrote an enhanced version by utilizing the smbinning.custom() function, shown as below. The idea is very simple. Within the repeat loop, we would bin the variable iteratively until a certain criterion is met and then feed the list of cut points into the smbinning.custom() function. As a result, we are able to achieve a set of monotonic bins with similar frequencies regardless of the so-called “statistical significance”, which is a premature step for the variable transformation in my mind.

monobin <- function(data, y, x) {
  d1 <- data[c(y, x)]
  n <- min(20, nrow(unique(d1[x])))
  repeat {
    d1$bin <- Hmisc::cut2(d1[, x], g = n)
    d2 <- aggregate(d1[-3], d1[3], mean)
    c <- cor(d2[-1], method = "spearman")
    if(abs(c[1, 2]) == 1 | n == 2) break
    n <- n - 1
  }
  d3 <- aggregate(d1[-3], d1[3], max)
  cuts <- d3[-length(d3[, 3]), 3]
  return(smbinning::smbinning.custom(d1, y, x, cuts))
}

Below are a couple comparisons between the generic smbinning() and the home-brew monobin() functions with the use of a toy data.

In the first example, we applied the smbinning() function to a variable named “rev_util”. As shown in the highlighted rows in the column “BadRate”, the binning outcome is not monotonic.

  Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds LnOdds     WoE     IV
1     <= 0    965     716    249       965        716       249 0.1653   0.7420  0.2580  2.8755 1.0562 -0.2997 0.0162
2     <= 5    522     496     26      1487       1212       275 0.0894   0.9502  0.0498 19.0769 2.9485  1.5925 0.1356
3    <= 24   1166    1027    139      2653       2239       414 0.1998   0.8808  0.1192  7.3885 1.9999  0.6440 0.0677
4    <= 40    779     651    128      3432       2890       542 0.1335   0.8357  0.1643  5.0859 1.6265  0.2705 0.0090
5    <= 73   1188     932    256      4620       3822       798 0.2035   0.7845  0.2155  3.6406 1.2922 -0.0638 0.0008
6    <= 96    684     482    202      5304       4304      1000 0.1172   0.7047  0.2953  2.3861 0.8697 -0.4863 0.0316
7     > 96    533     337    196      5837       4641      1196 0.0913   0.6323  0.3677  1.7194 0.5420 -0.8140 0.0743
8  Missing      0       0      0      5837       4641      1196 0.0000      NaN     NaN     NaN    NaN     NaN    NaN
9    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804 1.3559  0.0000 0.3352

Next, we did the same with the monobin() function. As shown below, the algorithm provided a monotonic binning at the cost of granularity. Albeit coarse, the result is directionally correct with no inversion.

  Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate   Odds LnOdds     WoE     IV
1    <= 30   2962    2495    467      2962       2495       467 0.5075   0.8423  0.1577 5.3426 1.6757  0.3198 0.0471
2     > 30   2875    2146    729      5837       4641      1196 0.4925   0.7464  0.2536 2.9438 1.0797 -0.2763 0.0407
3  Missing      0       0      0      5837       4641      1196 0.0000      NaN     NaN    NaN    NaN     NaN    NaN
4    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049 3.8804 1.3559  0.0000 0.0878

In the second example, we applied the smbinning() function to a variable named “bureau_score”. As shown in the highlighted rows, the frequencies in these two bins are much higher than the rest.

  Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds LnOdds     WoE     IV
1   <= 605    324     167    157       324        167       157 0.0555   0.5154  0.4846  1.0637 0.0617 -1.2942 0.1233
2   <= 632    468     279    189       792        446       346 0.0802   0.5962  0.4038  1.4762 0.3895 -0.9665 0.0946
3   <= 662    896     608    288      1688       1054       634 0.1535   0.6786  0.3214  2.1111 0.7472 -0.6087 0.0668
4   <= 699   1271    1016    255      2959       2070       889 0.2177   0.7994  0.2006  3.9843 1.3824  0.0264 0.0002
5   <= 717    680     586     94      3639       2656       983 0.1165   0.8618  0.1382  6.2340 1.8300  0.4741 0.0226
6   <= 761   1118    1033     85      4757       3689      1068 0.1915   0.9240  0.0760 12.1529 2.4976  1.1416 0.1730
7    > 761    765     742     23      5522       4431      1091 0.1311   0.9699  0.0301 32.2609 3.4739  2.1179 0.2979
8  Missing    315     210    105      5837       4641      1196 0.0540   0.6667  0.3333  2.0000 0.6931 -0.6628 0.0282
9    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804 1.3559  0.0000 0.8066

With the monobin() function applied to the same variable, we were able to get a set of more granular bins with similar frequencies.

   Cutpoint CntRec CntGood CntBad CntCumRec CntCumGood CntCumBad PctRec GoodRate BadRate    Odds LnOdds     WoE     IV
1    <= 617    513     284    229       513        284       229 0.0879   0.5536  0.4464  1.2402 0.2153 -1.1407 0.1486
2    <= 642    515     317    198      1028        601       427 0.0882   0.6155  0.3845  1.6010 0.4706 -0.8853 0.0861
3    <= 657    512     349    163      1540        950       590 0.0877   0.6816  0.3184  2.1411 0.7613 -0.5946 0.0363
4    <= 672    487     371    116      2027       1321       706 0.0834   0.7618  0.2382  3.1983 1.1626 -0.1933 0.0033
5    <= 685    494     396     98      2521       1717       804 0.0846   0.8016  0.1984  4.0408 1.3964  0.0405 0.0001
6    <= 701    521     428     93      3042       2145       897 0.0893   0.8215  0.1785  4.6022 1.5265  0.1706 0.0025
7    <= 714    487     418     69      3529       2563       966 0.0834   0.8583  0.1417  6.0580 1.8014  0.4454 0.0144
8    <= 730    489     441     48      4018       3004      1014 0.0838   0.9018  0.0982  9.1875 2.2178  0.8619 0.0473
9    <= 751    513     476     37      4531       3480      1051 0.0879   0.9279  0.0721 12.8649 2.5545  1.1986 0.0859
10   <= 775    492     465     27      5023       3945      1078 0.0843   0.9451  0.0549 17.2222 2.8462  1.4903 0.1157
11    > 775    499     486     13      5522       4431      1091 0.0855   0.9739  0.0261 37.3846 3.6213  2.2653 0.2126
12  Missing    315     210    105      5837       4641      1196 0.0540   0.6667  0.3333  2.0000 0.6931 -0.6628 0.0282
13    Total   5837    4641   1196        NA         NA        NA 1.0000   0.7951  0.2049  3.8804 1.3559  0.0000 0.7810

Written by statcompute

January 22, 2017 at 11:05 pm