LASSO Regression의 장점
- 규제항을 통해 회귀계수를 “0”으로 추정하기 때문에 변수 선택이 가능하다.
LASSO Regression의 단점
- 예측 변수의 개수가 표본의 크기보다 큰 경우, 볼록 최적화 문제의 특성 때문에 표본의 크기보다 많은 예측 변수를 선택할 수 없다.
- 예측 변수 사이에 어떤 그룹 구조(쌍별 상관 관계가 매우 높은)가 있을 때, 그룹에서 하나의 예측 변수만 선택한다.
- 예측 변수의 개수가 표본의 크기보다 큰 상황에서 예측 변수들이 높은 상관관계를 가지고 있을 때,
Ridge Regression보다 예측 성능이 낮다.
실습 자료 : 유니버셜 은행의 고객 2,500명에 대한 자료(출처 : Data Mining for Business Intelligence, Shmueli et al. 2010)이며, 총 13개의 변수를 포함하고 있다. 이 자료에서 Target은
Personal Loan이다.
Contents
1. 데이터 불러오기
pacman::p_load("data.table",
"tidyverse",
"dplyr",
"ggplot2", "GGally",
"caret",
"doParallel", "parallel") # For 병렬 처리
registerDoParallel(cores=detectCores()) # 사용할 Core 개수 지정
UB <- fread("../Universal Bank_Main.csv") # 데이터 불러오기
UB %>%
as_tibble# A tibble: 2,500 × 14
ID Age Experience Income `ZIP Code` Family CCAvg Education
<int> <int> <int> <int> <int> <int> <dbl> <int>
1 1 25 1 49 91107 4 1.6 1
2 2 45 19 34 90089 3 1.5 1
3 3 39 15 11 94720 1 1 1
4 4 35 9 100 94112 1 2.7 2
5 5 35 8 45 91330 4 1 2
6 6 37 13 29 92121 4 0.4 2
7 7 53 27 72 91711 2 1.5 2
8 8 50 24 22 93943 1 0.3 3
9 9 35 10 81 90089 3 0.6 2
10 10 34 9 180 93023 1 8.9 3
# ℹ 2,490 more rows
# ℹ 6 more variables: Mortgage <int>, `Personal Loan` <int>,
# `Securities Account` <int>, `CD Account` <int>, Online <int>,
# CreditCard <int>2. 데이터 전처리
UB %<>%
data.frame() %>% # Data Frame 형태로 변환
mutate(Personal.Loan = ifelse(Personal.Loan == 1, "yes", "no")) %>% # Target을 문자형 변수로 변환
select(-1) # ID 변수 제거
# Convert to Factor
fac.col <- c("Family", "Education", "Securities.Account",
"CD.Account", "Online", "CreditCard",
# Target
"Personal.Loan")
UB <- UB %>%
mutate_at(fac.col, as.factor) # 범주형으로 변환
glimpse(UB) # 데이터 구조 확인Rows: 2,500
Columns: 13
$ Age <int> 25, 45, 39, 35, 35, 37, 53, 50, 35, 34, 6…
$ Experience <int> 1, 19, 15, 9, 8, 13, 27, 24, 10, 9, 39, 5…
$ Income <int> 49, 34, 11, 100, 45, 29, 72, 22, 81, 180,…
$ ZIP.Code <int> 91107, 90089, 94720, 94112, 91330, 92121,…
$ Family <fct> 4, 3, 1, 1, 4, 4, 2, 1, 3, 1, 4, 3, 2, 4,…
$ CCAvg <dbl> 1.6, 1.5, 1.0, 2.7, 1.0, 0.4, 1.5, 0.3, 0…
$ Education <fct> 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2,…
$ Mortgage <int> 0, 0, 0, 0, 0, 155, 0, 0, 104, 0, 0, 0, 0…
$ Personal.Loan <fct> no, no, no, no, no, no, no, no, no, yes, …
$ Securities.Account <fct> 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,…
$ CD.Account <fct> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ Online <fct> 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1,…
$ CreditCard <fct> 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,…3. 데이터 탐색
ggpairs(UB,
columns = c("Age", "Experience", "Income", # 수치형 예측 변수
"ZIP.Code", "CCAvg", "Mortgage"),
aes(colour = Personal.Loan)) + # Target의 범주에 따라 색깔을 다르게 표현
theme_bw() 
ggpairs(UB,
columns = c("Age", "Experience", "Income", # 수치형 예측 변수
"ZIP.Code", "CCAvg", "Mortgage"),
aes(colour = Personal.Loan)) + # Target의 범주에 따라 색깔을 다르게 표현
scale_color_brewer(palette="Purples") + # 특정 색깔 지정
scale_fill_brewer(palette="Purples") + # 특정 색깔 지정
theme_bw()
ggpairs(UB,
columns = c("Age", "Income", # 수치형 예측 변수
"Family", "Education"), # 범주형 예측 변수
aes(colour = Personal.Loan, alpha = 0.8)) + # Target의 범주에 따라 색깔을 다르게 표현
scale_colour_manual(values = c("purple","cyan4")) + # 특정 색깔 지정
scale_fill_manual(values = c("purple","cyan4")) + # 특정 색깔 지정
theme_bw()
4. 데이터 분할
# Partition (Training Dataset : Test Dataset = 7:3)
y <- UB$Personal.Loan # Target
set.seed(200)
ind <- createDataPartition(y, p = 0.7, list = T) # Index를 이용하여 7:3으로 분할
UB.trd <- UB[ind$Resample1,] # Training Dataset
UB.ted <- UB[-ind$Resample1,] # Test Dataset5. 모형 훈련
Package "caret"은 통합 API를 통해 R로 기계 학습을 실행할 수 있는 매우 실용적인 방법을 제공한다. Package "caret"를 통해 LASSO Regression을 수행하기 위해 옵션 method에 다양한 방법(Ex: "lasso", "blasso" 등)을 입력할 수 있지만, 대부분 회귀 문제에 대해서만 분석이 가능하다. 분류와 회귀 문제 모두 가능한 "glmnet"을 이용하려면 옵션 tuneGrid = expand.grid()을 통해 탐색하고자 하는 초모수 lambda의 범위를 직접 지정해줘야 한다.
fitControl <- trainControl(method = "cv", number = 5, # 5-Fold Cross Validation (5-Fold CV)
allowParallel = TRUE) # 병렬 처리
set.seed(200) # For CV
lasso.fit <- train(Personal.Loan ~ ., data = UB.trd,
trControl = fitControl ,
method = "glmnet",
tuneGrid = expand.grid(alpha = 1, # For LASSO Regression
lambda = seq(0, 1, 0.001)), # lambda의 탐색 범위
preProc = c("center", "scale")) # Standardization for 예측 변수
lasso.fitglmnet
1751 samples
12 predictor
2 classes: 'no', 'yes'
Pre-processing: centered (15), scaled (15)
Resampling: Cross-Validated (5 fold)
Summary of sample sizes: 1401, 1401, 1400, 1401, 1401
Resampling results across tuning parameters:
lambda Accuracy Kappa
0.000 0.9583053 0.751538993
0.001 0.9588767 0.752418331
0.002 0.9577354 0.743256100
0.003 0.9577354 0.742059920
0.004 0.9565926 0.733576889
0.005 0.9554514 0.726784857
0.006 0.9537387 0.712729772
0.007 0.9525958 0.702796871
0.008 0.9520244 0.697985312
0.009 0.9508816 0.688763877
0.010 0.9508816 0.688763877
0.011 0.9508816 0.688763877
0.012 0.9497387 0.679863757
0.013 0.9485958 0.670694570
0.014 0.9480260 0.665771867
0.015 0.9468832 0.656213995
0.016 0.9445975 0.636317627
0.017 0.9423134 0.619546346
0.018 0.9405991 0.606507131
0.019 0.9388848 0.590199261
0.020 0.9388848 0.590199261
0.021 0.9348848 0.559149435
0.022 0.9326007 0.538520693
0.023 0.9303183 0.523144315
0.024 0.9297501 0.519451170
0.025 0.9286072 0.512038401
0.026 0.9268930 0.498501525
0.027 0.9234693 0.465556693
0.028 0.9223264 0.454010402
0.029 0.9206138 0.436027387
0.030 0.9177582 0.406751134
0.031 0.9166170 0.393810038
0.032 0.9154758 0.382031101
0.033 0.9137615 0.363984391
0.034 0.9137615 0.363984391
0.035 0.9131884 0.352590508
0.036 0.9120456 0.338919927
0.037 0.9109027 0.326146558
0.038 0.9114742 0.324129628
0.039 0.9120456 0.325911450
0.040 0.9114725 0.309687237
0.041 0.9114725 0.303349935
0.042 0.9120440 0.305407784
0.043 0.9120440 0.299376432
0.044 0.9103297 0.278130229
0.045 0.9091868 0.264416018
0.046 0.9080440 0.249521822
0.047 0.9091868 0.252591102
0.048 0.9103280 0.256503169
0.049 0.9108995 0.258127513
0.050 0.9114693 0.255106230
0.051 0.9103297 0.241720391
0.052 0.9097582 0.234020160
0.053 0.9097582 0.228119255
0.054 0.9097599 0.216427058
0.055 0.9074758 0.184019046
0.056 0.9080472 0.185486079
0.057 0.9074758 0.177033425
0.058 0.9080472 0.178917305
0.059 0.9074774 0.171290986
0.060 0.9080488 0.172909816
0.061 0.9080488 0.172909816
0.062 0.9074774 0.164660050
0.063 0.9063346 0.147114559
0.064 0.9051933 0.129844066
0.065 0.9046235 0.121592703
0.066 0.9040537 0.112999601
0.067 0.9040537 0.112999601
0.068 0.9029109 0.093905651
0.069 0.9023394 0.084152584
0.070 0.9017680 0.075198298
0.071 0.9011982 0.066241776
0.072 0.9011982 0.066241776
0.073 0.9011982 0.066241776
0.074 0.9006268 0.057287490
0.075 0.8994872 0.038187899
0.076 0.8994872 0.038187899
0.077 0.8983443 0.019093950
0.078 0.8977729 0.009753067
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Tuning parameter 'alpha' was held constant at a value of 1
Accuracy was used to select the optimal model using the
largest value.
The final values used for the model were alpha = 1 and lambda = 0.001.plot(lasso.fit) # Plot
lasso.fit$bestTune # lambda의 최적값 alpha lambda
2 1 0.001Result! lambda = 0.001일 때 정확도가 가장 높은 것을 알 수 있으며, lambda = 0.001를 가지는 모형을 최적의 훈련된 모형으로 선택한다.
16 x 1 sparse Matrix of class "dgCMatrix"
s1
(Intercept) -4.550
Age 0.092
Experience .
Income 2.449
ZIP.Code 0.054
Family2 -0.129
Family3 0.716
Family4 0.666
CCAvg 0.354
Education2 1.373
Education3 1.420
Mortgage 0.042
Securities.Account1 -0.347
CD.Account1 0.963
Online1 -0.132
CreditCard1 -0.504Result! 데이터 “UB.trd”의 Target “Personal.Loan”은 “no”와 “yes” 2개의 클래스를 가지며, “Factor” 변환하면 알파벳순으로 수준을 부여하기 때문에 “yes”가 두 번째 클래스가 된다. 즉, “yes”에 속할 확률(= 개인 대출 제의를 수락할 확률)을 \(p\)라고 할 때, 추정된 회귀계수를 이용하여 다음과 같은 모형식을 얻을 수 있다.
\[
\begin{align*}
\log{\frac{p}{1-p}} = &-4.550 +0.092 Z_{\text{Age}} + 2.449 Z_{\text{Income}} \\
&+0.054 Z_{\text{ZIP.Code}} -0.129 Z_{\text{Family2}} + 0.716 Z_{\text{Family3}} + 0.666 Z_{\text{Family4}} \\
&+ 0.354 Z_{\text{CCAvg}} + 1.373 Z_{\text{Education2}} + 1.420 Z_{\text{Education3}} + 0.042 Z_{\text{Mortgage}} \\
&-0.347 Z_{\text{Securities.Account1}} + 0.963 Z_{\text{CD.Account1}} -0.132 Z_{\text{Online1}} -0.504 Z_{\text{CreditCard1}}
\end{align*}
\]
여기서, \(Z_{\text{예측 변수}}\)는 표준화한 예측 변수를 의미한다.
범주형 예측 변수(“Family”, “Education”, “Securities.Account”, “CD.Account”, “Online”, “CreditCard”)는 더미 변환이 수행되었는데, 예를 들어, Family2는 가족 수가 2명인 경우 “1”값을 가지고 2명이 아니면 “0”값을 가진다.
6. 모형 평가
Caution! 모형 평가를 위해 Test Dataset에 대한 예측 class/확률 이 필요하며, 함수 predict()를 이용하여 생성한다.
# 예측 class 생성
test.lasso.class <- predict(lasso.fit,
newdata = UB.ted[,-9]) # Test Dataset including Only 예측 변수
test.lasso.class %>%
as_tibble# A tibble: 749 × 1
value
<fct>
1 no
2 no
3 no
4 no
5 no
6 no
7 no
8 no
9 no
10 no
# ℹ 739 more rows6-1. ConfusionMatrix
CM <- caret::confusionMatrix(test.lasso.class, UB.ted$Personal.Loan,
positive = "yes") # confusionMatrix(예측 class, 실제 class, positive = "관심 class")
CMConfusion Matrix and Statistics
Reference
Prediction no yes
no 667 24
yes 6 52
Accuracy : 0.9599
95% CI : (0.9433, 0.9728)
No Information Rate : 0.8985
P-Value [Acc > NIR] : 3.394e-10
Kappa : 0.7546
Mcnemar's Test P-Value : 0.001911
Sensitivity : 0.68421
Specificity : 0.99108
Pos Pred Value : 0.89655
Neg Pred Value : 0.96527
Prevalence : 0.10147
Detection Rate : 0.06943
Detection Prevalence : 0.07744
Balanced Accuracy : 0.83765
'Positive' Class : yes
6-2. ROC 곡선
# 예측 확률 생성
test.lasso.prob <- predict(lasso.fit,
newdata = UB.ted[,-9], # Test Dataset including Only 예측 변수
type = "prob") # 예측 확률 생성
test.lasso.prob %>%
as_tibble# A tibble: 749 × 2
no yes
<dbl> <dbl>
1 1.00 0.000393
2 0.994 0.00633
3 0.999 0.000519
4 1.00 0.000159
5 0.993 0.00684
6 0.998 0.00225
7 0.975 0.0250
8 0.892 0.108
9 0.907 0.0927
10 0.980 0.0203
# ℹ 739 more rowstest.lasso.prob <- test.lasso.prob[,2] # "Personal.Loan = yes"에 대한 예측 확률
ac <- UB.ted$Personal.Loan # Test Dataset의 실제 class
pp <- as.numeric(test.lasso.prob) # 예측 확률을 수치형으로 변환1) Package “pROC”
pacman::p_load("pROC")
lasso.roc <- roc(ac, pp, plot = T, col = "gray") # roc(실제 class, 예측 확률)
auc <- round(auc(lasso.roc), 3)
legend("bottomright", legend = auc, bty = "n")
Caution! Package "pROC"를 통해 출력한 ROC 곡선은 다양한 함수를 이용해서 그래프를 수정할 수 있다.
# 함수 plot.roc() 이용
plot.roc(lasso.roc,
col="gray", # Line Color
print.auc = TRUE, # AUC 출력 여부
print.auc.col = "red", # AUC 글씨 색깔
print.thres = TRUE, # Cutoff Value 출력 여부
print.thres.pch = 19, # Cutoff Value를 표시하는 도형 모양
print.thres.col = "red", # Cutoff Value를 표시하는 도형의 색깔
auc.polygon = TRUE, # 곡선 아래 면적에 대한 여부
auc.polygon.col = "gray90") # 곡선 아래 면적의 색깔
# 함수 ggroc() 이용
ggroc(lasso.roc) +
annotate(geom = "text", x = 0.9, y = 1.0,
label = paste("AUC = ", auc),
size = 5,
color="red") +
theme_bw()
2) Package “Epi”
pacman::p_load("Epi")
# install_version("etm", version = "1.1", repos = "http://cran.us.r-project.org")
ROC(pp, ac, plot = "ROC") # ROC(예측 확률, 실제 class) 
3) Package “ROCR”
pacman::p_load("ROCR")
lasso.pred <- prediction(pp, ac) # prediction(예측 확률, 실제 class)
lasso.perf <- performance(lasso.pred, "tpr", "fpr") # performance(, "민감도", "1-특이도")
plot(lasso.perf, col = "gray") # ROC Curve
perf.auc <- performance(lasso.pred, "auc") # AUC
auc <- attributes(perf.auc)$y.values
legend("bottomright", legend = auc, bty = "n")
6-3. 향상 차트
1) Package “ROCR”
lasso.perf <- performance(lasso.pred, "lift", "rpp") # Lift Chart
plot(lasso.perf, main = "lift curve",
colorize = T, # Coloring according to cutoff
lwd = 2) 