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DART: Dropout Regularization in Boosting Ensembles

The dropout approach developed by Hinton has been widely employed in deep learnings to prevent the deep neural network from overfitting, as shown in https://statcompute.wordpress.com/2017/01/02/dropout-regularization-in-deep-neural-networks.

In the paper http://proceedings.mlr.press/v38/korlakaivinayak15.pdf, the dropout can also be used to address the overfitting in boosting tree ensembles, e.g. MART, caused by the so-called “over-specialization”. In particular, while first few trees added at the beginning of ensembles would dominate the model performance, the rest added later can only improve the prediction for a small subset, which increases the risk of overfitting. The idea of DART is to build an ensemble by randomly dropping boosting tree members. The percentage of dropouts can determine the degree of regularization for boosting tree ensembles.

Below is a demonstration showing the implementation of DART with the R xgboost package. First of all, after importing the data, we divided it into two pieces, one for training and the other for testing.

pkgs <- c('pROC', 'xgboost')
lapply(pkgs, require, character.only = T)
df1 <- read.csv("Downloads/credit_count.txt")
df2 <- df1[df1$CARDHLDR == 1, ]
set.seed(2017)
n <- nrow(df2)
sample <- sample(seq(n), size = n / 2, replace = FALSE)
train <- df2[sample, -1]
test <- df2[-sample, -1]

For the comparison purpose, we first developed a boosting tree ensemble without dropouts, as shown below. For the simplicity, all parameters were chosen heuristically. The max_depth is set to 3 due to the fact that the boosting tends to work well with so-called “weak” learners, e.g. simple trees. While ROC for the training set can be as high as 0.95, ROC for the testing set is only 0.60 in our case, implying the overfitting issue.

mart.parm <- list(booster = "gbtree", nthread = 4, eta = 0.1, max_depth = 3, subsample = 1, eval_metric = "auc")
mart <- xgboost(data = as.matrix(train[, -1]), label = train[, 1], params = mart.parm, nrounds = 500, verbose = 0, seed = 2017)
pred1 <- predict(mart, as.matrix(train[, -1]))
pred2 <- predict(mart, as.matrix(test[, -1]))
roc(as.factor(train$DEFAULT), pred1)
# Area under the curve: 0.9459
roc(as.factor(test$DEFAULT), pred2)
# Area under the curve: 0.6046

With the same set of parameters, we refitted the ensemble with dropouts, e.g. DART. As shown below, by dropping 10% tree members, ROC for the testing set can increase from 0.60 to 0.65. In addition, the performance disparity between training and testing sets with DART decreases significantly.

dart.parm <- list(booster = "dart", rate_drop = 0.1, nthread = 4, eta = 0.1, max_depth = 3, subsample = 1, eval_metric = "auc")
dart <- xgboost(data = as.matrix(train[, -1]), label = train[, 1], params = dart.parm, nrounds = 500, verbose = 0, seed = 2017)
pred1 <- predict(dart, as.matrix(train[, -1]))
pred2 <- predict(dart, as.matrix(test[, -1]))
roc(as.factor(train$DEFAULT), pred1)
# Area under the curve: 0.7734
roc(as.factor(test$DEFAULT), pred2)
# Area under the curve: 0.6517

Besides rate_drop = 0.1, a wide range of dropout rates have also been tested. In most cases, DART outperforms its counterpart without the dropout regularization.

Written by statcompute

August 20, 2017 at 5:50 pm

Model Operational Losses with Copula Regression

In the previous post (https://statcompute.wordpress.com/2017/06/29/model-operational-loss-directly-with-tweedie-glm), it has been explained why we should consider modeling operational losses for non-material UoMs directly with Tweedie models. However, for material UoMs with significant losses, it is still beneficial to model the frequency and the severity separately.

In the prevailing modeling practice for operational losses, it is often convenient to assume a functional independence between frequency and severity models, which might not be the case empirically. For instance, in the economic downturn, both the frequency and the severity of consumer frauds might tend to increase simultaneously. With the independence assumption, while we can argue that same variables could be included in both frequency and severity models and therefore induce a certain correlation, the frequency-severity dependence and the its contribution to the loss distribution might be overlooked.

In the context of Copula, the distribution of operational losses can be considered a joint distribution determined by both marginal distributions and a parameter measuring the dependence between marginals, of which marginal distributions can be Poisson for the frequency and Gamma for the severity. Depending on the dependence structure in the data, various copula functions might be considered. For instance, a product copula can be used to describe the independence. In the example shown below, a Gumbel copula is considered given that it is often used to describe the positive dependence on the right tail, e.g. high severity and high frequency. For details, the book “Copula Modeling” by Trivedi and Zimmer is a good reference to start with.

In the demonstration, we simulated both frequency and severity measures driven by the same set of co-variates. Both are positively correlated with the Kendall’s tau = 0.5 under the assumption of Gumbel copula.

library(CopulaRegression)
# number of observations to simulate
n <- 100
# seed value for the simulation
set.seed(2017)
# design matrices with a constant column
X <- cbind(rep(1, n), runif(n), runif(n))
# define coefficients for both Poisson and Gamma regressions
p_beta <- g_beta <- c(3, -2, 1)
# define the Gamma dispersion
delta <- 1
# define the Kendall's tau
tau <- 0.5
# copula parameter based on tau
theta <- 1 / (1 - tau)
# define the Gumbel Copula 
family <- 4
# simulate outcomes
out <- simulate_regression_data(n, g_beta, p_beta, X, X, delta, tau, family, zt = FALSE)
G <- out[, 1]
P <- out[, 2]

After the simulation, a Copula regression is estimated with Poisson and Gamma marginals for the frequency and the severity respectively. As shown in the model estimation, estimated parameters with related inferences are different between independent and dependent assumptions.

m <- copreg(G, P, X, family = 4, sd.error = TRUE, joint = TRUE, zt = FALSE)
coef <- c("_CONST", "X1", "X2")
cols <- c("ESTIMATE", "STD. ERR", "Z-VALUE")
g_est <- cbind(m$alpha, m$sd.alpha, m$alpha / m$sd.alpha)
p_est <- cbind(m$beta, m$sd.beta, m$beta / m$sd.beta)
g_est0 <- cbind(m$alpha0, m$sd.alpha0, m$alpha0 / m$sd.alpha0)
p_est0 <- cbind(m$beta0, m$sd.beta0, m$beta0 / m$sd.beta0)
rownames(g_est) <- rownames(g_est0) <- rownames(p_est) <- rownames(p_est0) <- coef
colnames(g_est) <- colnames(g_est0) <- colnames(p_est) <- colnames(p_est0) <- cols

# estimated coefficients for the Gamma regression assumed dependence 
print(g_est)
#          ESTIMATE  STD. ERR   Z-VALUE
# _CONST  2.9710512 0.2303651 12.897141
# X1     -1.8047627 0.2944627 -6.129003
# X2      0.9071093 0.2995218  3.028526

# estimated coefficients for the Gamma regression assumed dependence 
print(p_est)
#         ESTIMATE   STD. ERR   Z-VALUE
# _CONST  2.954519 0.06023353  49.05107
# X1     -1.967023 0.09233056 -21.30414
# X2      1.025863 0.08254870  12.42736

# estimated coefficients for the Gamma regression assumed independence 
# should be identical to GLM() outcome
print(g_est0)
#         ESTIMATE  STD. ERR   Z-VALUE
# _CONST  3.020771 0.2499246 12.086727
# X1     -1.777570 0.3480328 -5.107478
# X2      0.905527 0.3619011  2.502140

# estimated coefficients for the Gamma regression assumed independence 
# should be identical to GLM() outcome
print(p_est0)
#         ESTIMATE   STD. ERR   Z-VALUE
# _CONST  2.939787 0.06507502  45.17536
# X1     -2.010535 0.10297887 -19.52376
# X2      1.088269 0.09334663  11.65837

If we compare conditional loss distributions under different dependence assumptions, it shows that the predicted loss with Copula regression tends to have a fatter right tail and therefore should be considered more conservative.

df <- data.frame(g = G, p = P, x1 = X[, 2], x2 = X[, 3])
glm_p <- glm(p ~ x1 + x2, data = df, family = poisson(log))
glm_g <- glm(g ~ x1 + x2, data = df, family = Gamma(log))
loss_dep <- predict(m, X, X, independence = FALSE)[3][[1]][[1]]
loss_ind <- fitted(glm_p) * fitted(glm_g)
den <- data.frame(loss = c(loss_dep, loss_ind), lines = rep(c("DEPENDENCE", "INDEPENDENCE"), each = n))
ggplot(den, aes(x = loss, fill = lines)) + geom_density(alpha = 0.5)

loss2

Written by statcompute

August 20, 2017 at 5:22 pm

Model Operational Loss Directly with Tweedie GLM

In the development of operational loss forecasting models, the Frequency-Severity modeling approach, which the frequency and the severity of a Unit of Measure (UoM) are modeled separately, has been widely employed in the banking industry. However, sometimes it also makes sense to model the operational loss directly, especially for UoMs with non-material losses. First of all, given the low loss amount, the effort of developing two models, e.g. frequency and severity, might not be justified. Secondly, for UoMs with low losses due to low frequencies, modeling the frequency and the severity separately might overlook the internal connection between the low frequency and the subsequent low loss amount. For instance, when the frequency N = 0, then the loss L = $0 inevitably.

The Tweedie distribution is defined as a Poisson sum of Gamma random variables. In particular, if the frequency of loss events N is assumed a Poisson distribution and the loss amount L_i of an event i, where i = 0, 1 … N, is assumed a Gamma distribution, then the total loss amount L = SUM[L_i] would have a Tweedie distribution. When there is no loss event, e.g. N = 0, then Prob(L = $0) = Prob(N = 0) = Exp(-Lambda). However, when N > 0, then L = L_1 + … + L_N > $0 is governed by a Gamma distribution, e.g. sum of I.I.D. Gamma also being Gamma.

For the Tweedie loss, E(L) = Mu and VAR(L) = Phi * (Mu ** P), where P is called the index parameter and Phi is the dispersion parameter. When P approaches 1 and therefore VAR(L) approaches Phi * E(L), the Tweedie would be similar to a Poisson-like distribution. When P approaches 2 and therefore VAR(L) approaches Phi * (E(L) ** 2), the Tweedie would be similar to a Gamma distribution. When P is between 1 and 2, then the Tweedie would be a compound mixture of Poisson and Gamma, where P and Phi can be estimated.

To estimate a regression with the Tweedie distributional assumption, there are two implementation approaches in R with cplm and statmod packages respectively. With the cplm package, the Tweedie regression can be estimated directly as long as P is in the range of (1, 2), as shown below. In the example, the estimated index parameter P is 1.42.

> library(cplm)
> data(FineRoot)
> m1 <- cpglm(RLD ~ Zone + Stock, data = FineRoot)
> summary(m1)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.0611  -0.6475  -0.3928   0.1380   1.9627  

            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.95141    0.14643 -13.327  < 2e-16 ***
ZoneOuter   -0.85693    0.13292  -6.447 2.66e-10 ***
StockMM106   0.01177    0.17535   0.067    0.947    
StockMark   -0.83933    0.17476  -4.803 2.06e-06 ***
---
Estimated dispersion parameter: 0.35092
Estimated index parameter: 1.4216 

Residual deviance: 203.91  on 507  degrees of freedom
AIC:  -157.33 

The statmod package provides a more general and flexible solution with the two-stage estimation, which will estimate the P parameter first and then estimate regression parameters. In the real-world practice, we could do a coarse search to narrow down a reasonable range of P and then do a fine search to identify the optimal P value. As shown below, all estimated parameters are fairly consistent with ones in the previous example.

> library(tweedie)
> library(statmod)
> prof <- tweedie.profile(RLD ~ Zone + Stock, data = FineRoot, p.vec = seq(1.1, 1.9, 0.01), method = "series")
1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9 
.................................................................................Done.
> prof$p.max
[1] 1.426531
> m2 <- glm(RLD ~ Zone + Stock, data = FineRoot, family = tweedie(var.power = prof$p.max, link.power = 0))
> summary(m2)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.0712  -0.6559  -0.3954   0.1380   1.9728  

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.95056    0.14667 -13.299  < 2e-16 ***
ZoneOuter   -0.85823    0.13297  -6.454 2.55e-10 ***
StockMM106   0.01204    0.17561   0.069    0.945    
StockMark   -0.84044    0.17492  -4.805 2.04e-06 ***
---
(Dispersion parameter for Tweedie family taken to be 0.4496605)

    Null deviance: 241.48  on 510  degrees of freedom
Residual deviance: 207.68  on 507  degrees of freedom
AIC: NA

Written by statcompute

June 29, 2017 at 10:46 pm

GLM with H2O in R

Below is an example showing how to fit a Generalized Linear Model with H2O in R. The output is much more comprehensive than the one generated by the generic R glm().

> library(h2o)

> h2o.init(max_mem_size = "12g")

> df1 <- h2o.uploadFile("Documents/credit_count.txt", header = TRUE, sep = ",", parse_type = "CSV")

> df2 <- h2o.assign(df1[df1$CARDHLDR == 1, ], "glm_df")

> h2o.colnames(df2)
 [1] "CARDHLDR" "DEFAULT"  "AGE"      "ACADMOS"  "ADEPCNT"  "MAJORDRG"
 [7] "MINORDRG" "OWNRENT"  "INCOME"   "SELFEMPL" "INCPER"   "EXP_INC"
[13] "SPENDING" "LOGSPEND"

> Y <- "DEFAULT"

> X <- c("MAJORDRG", "MINORDRG", "INCOME", "OWNRENT")

> dist <- "binomial"

> link <- "logit"

> id <- "h2o_mdl01"

> mdl <- h2o.glm(X, Y, training_frame = h2o.getFrame("glm_df"), model_id = id, family = dist, link = link, lambda = 0, compute_p_values = TRUE, standardize = FALSE)

> show(h2o.getModel(id)@model$coefficients_table)
Coefficients: glm coefficients
      names coefficients std_error    z_value  p_value
1 Intercept    -1.204439  0.090811 -13.263121 0.000000
2  MAJORDRG     0.203135  0.069250   2.933370 0.003353
3  MINORDRG     0.202727  0.047971   4.226014 0.000024
4   OWNRENT    -0.201223  0.071619  -2.809636 0.004960
5    INCOME    -0.000442  0.000040 -10.942350 0.000000

> h2o.performance(h2o.getModel(id))
H2OBinomialMetrics: glm
** Reported on training data. **

MSE:  0.08414496
RMSE:  0.2900775
LogLoss:  0.3036585
Mean Per-Class Error:  0.410972
AUC:  0.6432189
Gini:  0.2864378
R^2:  0.02005004
Residual Deviance:  6376.221
AIC:  6386.221

Confusion Matrix (vertical: actual; across: predicted) for F1-optimal threshold:
          0    1    Error         Rate
0      7703 1800 0.189414   =1800/9503
1       630  366 0.632530     =630/996
Totals 8333 2166 0.231451  =2430/10499

Maximum Metrics: Maximum metrics at their respective thresholds
                        metric threshold    value idx
1                       max f1  0.126755 0.231499 142
2                       max f2  0.075073 0.376556 272
3                 max f0point5  0.138125 0.191828 115
4                 max accuracy  0.368431 0.905039   0
5                max precision  0.314224 0.250000   3
6                   max recall  0.006115 1.000000 399
7              max specificity  0.368431 0.999895   0
8             max absolute_mcc  0.126755 0.128940 142
9   max min_per_class_accuracy  0.106204 0.604546 196
10 max mean_per_class_accuracy  0.103730 0.605663 202

Written by statcompute

June 28, 2017 at 12:25 am

Using Tweedie Parameter to Identify Distributions

In the development of operational loss models, it is important to identify which distribution should be used to model operational risk measures, e.g. frequency and severity. For instance, why should we use the Gamma distribution instead of the Inverse Gaussian distribution to model the severity?

In my previous post https://statcompute.wordpress.com/2016/11/20/modified-park-test-in-sas, it is shown how to use the Modified Park test to identify the mean-variance relationship and then decide the corresponding distribution of operational risk measures. Following the similar logic, we can also leverage the flexibility of the Tweedie distribution to accomplish the same goal. Based upon the parameterization of a Tweedie distribution, the variance = Phi * (Mu ** P), where Mu is the mean and P is the power parameter. Depending on the specific value of P, the Tweedie distribution can accommodate several important distributions commonly used in the operational risk modeling, including Poisson, Gamma, Inverse Gaussian. For instance,

  • With P = 0, the variance would be independent of the mean, indicating a Normal distribution.
  • With P = 1, the variance would be in a linear form of the mean, indicating a Poisson-like distribution
  • With P = 2, the variance would be in a quadratic form of the mean, indicating a Gamma distribution.
  • With P = 3, the variance would be in a cubic form of the mean, indicating an Inverse Gaussian distribution.

In the example below, it is shown that the value of P is in the neighborhood of 1 for the frequency measure and is near 3 for the severity measure and that, given P closer to 3, the Inverse Gaussian regression would fit the severity better than the Gamma regression.

library(statmod)
library(tweedie)

profile1 <- tweedie.profile(Claim_Count ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE)
print(profile1$p.max)
# [1] 1.216327
# The P parameter close to 1 indicates that the claim_count might follow a Poisson-like distribution

profile2 <- tweedie.profile(Severity ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE)
print(profile2$p.max)
# [1] 2.844898
# The P parameter close to 3 indicates that the severity might follow an Inverse Gaussian distribution

BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = Gamma(link = log)))
# [1] 360.8064

BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = inverse.gaussian(link = log)))
# [1] 350.2504

Together with the Modified Park test, the estimation of P in a Tweedie distribution is able to help us identify the correct distribution employed in operational loss models in the context of GLM.

Written by statcompute

June 24, 2017 at 10:55 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

Double Poisson Regression in SAS

In the previous post (https://statcompute.wordpress.com/2016/11/27/more-about-flexible-frequency-models), I’ve shown how to estimate the double Poisson (DP) regression in R with the gamlss package. The hurdle of estimating DP regression is the calculation of a normalizing constant in the DP density function, which can be calculated either by the sum of an infinite series or by a closed form approximation. In the example below, I will show how to estimate DP regression in SAS with the GLIMMIX procedure.

First of all, I will show how to estimate DP regression by using the exact DP density function. In this case, we will approximate the normalizing constant by computing a partial sum of the infinite series, as highlighted below.

data poi;
  do n = 1 to 5000;
    x1 = ranuni(1);
    x2 = ranuni(2);
    x3 = ranuni(3);
    y = ranpoi(4, exp(1 * x1 - 2 * x2 + 3 * x3));
    output;
  end;
run;

proc glimmix data = poi;
  nloptions tech = quanew update = bfgs maxiter = 1000;
  model y = x1 x2 x3 / link = log solution;
  theta = exp(_phi_);
  _variance_ = _mu_ / theta;
  p_u = (exp(-_mu_) * (_mu_ ** y) / fact(y)) ** theta;
  p_y = (exp(-y) * (y ** y) / fact(y)) ** (1 - theta);
  f = (theta ** 0.5) * ((exp(-_mu_)) ** theta);  
  do i = 1 to 100;
    f = f + (theta ** 0.5) * ((exp(-i) * (i ** i) / fact(i)) ** (1 - theta)) * ((exp(-_mu_) * (_mu_ ** i) / fact(i)) ** theta);
  end;
  k = 1 / f;
  prob = k * (theta ** 0.5) * p_y * p_u;
  if log(prob) ~= . then _logl_ = log(prob);
run;

Next, I will show the same estimation routine by using the closed form approximation.

proc glimmix data = poi;
  nloptions tech = quanew update = bfgs maxiter = 1000;
  model y = x1 x2 x3 / link = log solution;
  theta = exp(_phi_);
  _variance_ = _mu_ / theta;
  p_u = (exp(-_mu_) * (_mu_ ** y) / fact(y)) ** theta;
  p_y = (exp(-y) * (y ** y) / fact(y)) ** (1 - theta);
  k = 1 / (1 + (1 - theta) / (12 * theta * _mu_) * (1 + 1 / (theta * _mu_)));
  prob = k * (theta ** 0.5) * p_y * p_u;
  if log(prob) ~= . then _logl_ = log(prob);
run;

While the first approach is more accurate by closely following the DP density function, the second approach is more efficient with a significantly lower computing cost. However, both are much faster than the corresponding R function gamlss().

Written by statcompute

April 20, 2017 at 12:48 am