Regression: Credit Limit Prediction
WQD7004 - Group 4
2026-01-16
Import Package
library(dslabs)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionlibrary(tidyr)
library(lubridate)##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
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## date, intersect, setdiff, unionlibrary(caret)## Loading required package: ggplot2## Loading required package: latticelibrary(stats)
library(randomForest)## randomForest 4.7-1.2## Type rfNews() to see new features/changes/bug fixes.##
## Attaching package: 'randomForest'## The following object is masked from 'package:ggplot2':
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## margin## The following object is masked from 'package:dplyr':
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## combinelibrary(xgboost)
library(Ckmeans.1d.dp)
library(gbm)## Loaded gbm 2.2.2## This version of gbm is no longer under development. Consider transitioning to gbm3, https://github.com/gbm-developers/gbm3library(e1071)##
## Attaching package: 'e1071'## The following object is masked from 'package:ggplot2':
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## element1. Data Transformation
1.1 Load Data
credit_limit_data <- readRDS("CleanedDataSet/credit_limit_data.rds")1.2 GEO CLUSTERING (K-MEANS)
Latitude and longitude are continuous numbers which the machine learning models do not naturally understand spatial proximity and treat them as independent numbers. Hence, by using K-means to convert them into meaningful features which helps the machine learning models to capture geographical patterns.
# ---------- GEO CLUSTERING (K-MEANS) ----------
set.seed(42)
coords <- credit_limit_data %>%
select(latitude, longitude)
kmeans_model <- kmeans(coords, centers = 8)
credit_limit_data$geo_cluster <- NA
credit_limit_data$geo_cluster[as.numeric(rownames(coords))] <- kmeans_model$cluster
credit_limit_data$geo_cluster <- as.factor(credit_limit_data$geo_cluster)
# One-hot encode geo_cluster
geo_dummies <- model.matrix(~ geo_cluster - 1, data = credit_limit_data)
credit_limit_data <- bind_cols(
credit_limit_data %>% select(-geo_cluster),
as.data.frame(geo_dummies)
)1.3 ONE-HOT ENCODING
To convert the non-ordinal categorical variables into binary numerical format which suitable for machine learning models and avoid false ordering issue.
card_type_dummies <- model.matrix(~ card_type - 1, data = credit_limit_data)
card_brand_dummies <- model.matrix(~ card_brand - 1, data = credit_limit_data)
colnames(card_type_dummies) <- gsub("^card_type", "", colnames(card_type_dummies))
colnames(card_brand_dummies) <- gsub("^card_brand", "", colnames(card_brand_dummies))
credit_limit_data <- bind_cols(
credit_limit_data,
as.data.frame(card_type_dummies),
as.data.frame(card_brand_dummies)
)
head(credit_limit_data[, c("card_type", "Credit", "Debit", "Debit (Prepaid)")], 20)## card_type Credit Debit Debit (Prepaid)
## <char> <num> <num> <num>
## 1: Credit 1 0 0
## 2: Credit 1 0 0
## 3: Debit 0 1 0
## 4: Credit 1 0 0
## 5: Debit 0 1 0
## 6: Debit 0 1 0
## 7: Credit 1 0 0
## 8: Debit 0 1 0
## 9: Credit 1 0 0
## 10: Credit 1 0 0
## 11: Debit 0 1 0
## 12: Debit 0 1 0
## 13: Credit 1 0 0
## 14: Credit 1 0 0
## 15: Debit 0 1 0
## 16: Debit 0 1 0
## 17: Debit 0 1 0
## 18: Debit 0 1 0
## 19: Credit 1 0 0
## 20: Debit 0 1 0
## card_type Credit Debit Debit (Prepaid)head(credit_limit_data[, c("card_brand", "Amex", "Discover", "Mastercard", "Visa")], 20)## card_brand Amex Discover Mastercard Visa
## <char> <num> <num> <num> <num>
## 1: Amex 1 0 0 0
## 2: Mastercard 0 0 1 0
## 3: Mastercard 0 0 1 0
## 4: Visa 0 0 0 1
## 5: Mastercard 0 0 1 0
## 6: Visa 0 0 0 1
## 7: Amex 1 0 0 0
## 8: Mastercard 0 0 1 0
## 9: Mastercard 0 0 1 0
## 10: Amex 1 0 0 0
## 11: Mastercard 0 0 1 0
## 12: Mastercard 0 0 1 0
## 13: Visa 0 0 0 1
## 14: Mastercard 0 0 1 0
## 15: Mastercard 0 0 1 0
## 16: Mastercard 0 0 1 0
## 17: Visa 0 0 0 1
## 18: Visa 0 0 0 1
## 19: Mastercard 0 0 1 0
## 20: Mastercard 0 0 1 0
## card_brand Amex Discover Mastercard Visa1.4 DATA CONVERSION
- New variable “retired” is created if current age is greater than or
equal to the retirement age then classified as retired (1)
- New variable “age_group” is created by grouping the exact ages into
ranges to represent different life stages
- New variable “credit_score_category” is created by categorizing
credit scores into standard risk levels
- New variable “gender_binary” is created by converting gender from
string variable into a numeric format
- New variable “acct_tenure_years” is created by calculating the time difference between the current date and the account opening date to represent the length of time of the account has been active
credit_limit_data <- credit_limit_data %>%
mutate(
retired = ifelse(current_age >= retirement_age, 1, 0)
)
head(credit_limit_data[, c("retired", "current_age", "retirement_age")], 20)## retired current_age retirement_age
## <num> <int> <int>
## 1: 0 58 67
## 2: 1 76 70
## 3: 0 46 66
## 4: 1 65 62
## 5: 0 48 65
## 6: 0 54 65
## 7: 0 44 66
## 8: 0 66 68
## 9: 1 71 69
## 10: 0 44 66
## 11: 0 52 70
## 12: 1 81 70
## 13: 0 56 69
## 14: 0 47 67
## 15: 0 62 68
## 16: 0 54 61
## 17: 0 60 66
## 18: 0 64 67
## 19: 0 39 67
## 20: 0 61 69
## retired current_age retirement_agecredit_limit_data <- credit_limit_data %>%
mutate(
age_group = cut(
current_age,
breaks = c(0, 24, 40, 60, Inf),
labels = c(1, 2, 3, 4),
right = TRUE
)
)
credit_limit_data$age_group <- as.integer(as.character(credit_limit_data$age_group))
head(credit_limit_data[, c("current_age", "age_group")], 20)## current_age age_group
## <int> <int>
## 1: 58 3
## 2: 76 4
## 3: 46 3
## 4: 65 4
## 5: 48 3
## 6: 54 3
## 7: 44 3
## 8: 66 4
## 9: 71 4
## 10: 44 3
## 11: 52 3
## 12: 81 4
## 13: 56 3
## 14: 47 3
## 15: 62 4
## 16: 54 3
## 17: 60 3
## 18: 64 4
## 19: 39 2
## 20: 61 4
## current_age age_groupcredit_limit_data <- credit_limit_data %>%
mutate(
credit_score_category = cut(
credit_score,
breaks = c(0, 579, 669, 739, 799, 850),
labels = c(1, 2, 3, 4, 5),
right = TRUE,
include.lowest = TRUE
)
)
credit_limit_data$credit_score_category <- as.integer(
as.character(credit_limit_data$credit_score_category)
)
head(credit_limit_data[, c("credit_score", "credit_score_category")], 20)## credit_score credit_score_category
## <int> <int>
## 1: 727 3
## 2: 763 4
## 3: 715 3
## 4: 667 2
## 5: 702 3
## 6: 748 4
## 7: 547 1
## 8: 661 2
## 9: 698 3
## 10: 547 1
## 11: 749 4
## 12: 728 3
## 13: 752 4
## 14: 700 3
## 15: 735 3
## 16: 726 3
## 17: 715 3
## 18: 812 5
## 19: 714 3
## 20: 748 4
## credit_score credit_score_categorycredit_limit_data <- credit_limit_data %>%
mutate(gender_binary = ifelse(gender == "Male", 1, 0))
head(credit_limit_data[, c("gender", "gender_binary")], 20)## gender gender_binary
## <char> <num>
## 1: Male 1
## 2: Male 1
## 3: Male 1
## 4: Female 0
## 5: Male 1
## 6: Male 1
## 7: Male 1
## 8: Male 1
## 9: Female 0
## 10: Male 1
## 11: Male 1
## 12: Female 0
## 13: Female 0
## 14: Male 1
## 15: Male 1
## 16: Male 1
## 17: Male 1
## 18: Male 1
## 19: Female 0
## 20: Male 1
## gender gender_binaryREFERENCE_DATE <- as.Date("2020-03-01")
credit_limit_data <- credit_limit_data %>%
mutate(
acct_open_date = as.Date(acct_open_date),
acct_tenure_years =
as.numeric(difftime(REFERENCE_DATE, acct_open_date, units = "days")) / 365.25
)
head(credit_limit_data[, c("acct_open_date", "acct_tenure_years")], 20)## acct_open_date acct_tenure_years
## <Date> <num>
## 1: 1991-01-01 29.16359
## 2: 1994-01-01 26.16290
## 3: 1995-01-01 25.16359
## 4: 1995-01-01 25.16359
## 5: 1997-01-01 23.16222
## 6: 1997-01-01 23.16222
## 7: 1998-01-01 22.16290
## 8: 1998-01-01 22.16290
## 9: 1998-01-01 22.16290
## 10: 1999-01-01 21.16359
## 11: 1999-01-01 21.16359
## 12: 1999-01-01 21.16359
## 13: 1999-01-01 21.16359
## 14: 2000-01-01 20.16427
## 15: 2000-01-01 20.16427
## 16: 2000-01-01 20.16427
## 17: 2000-01-01 20.16427
## 18: 2000-01-01 20.16427
## 19: 2001-01-01 19.16222
## 20: 2001-01-01 19.16222
## acct_open_date acct_tenure_yearssummary(credit_limit_data$acct_tenure_years)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0794 3.7906 10.0780 9.1243 13.4155 29.16361.5 DROP COLUMNS
By removing redundant or no longer needed features to ensure the dataset contains only relevant features for modeling
# ---------- DROP ORIGINAL STRING / SENSITIVE COLUMNS ----------
columns_to_drop <- c(
"gender", "acct_open_date", "card_number", "expires", "cvv",
"has_chip", "birth_year", "birth_month", "address", "latitude",
"longitude", "transaction_frequency", "card_type", "card_brand",
"retirement_age", "current_age","year_pin_last_changed", "avg_errors",
"total_errors","num_refunds", "total_refunded",
"min_transaction_amount", "credit_score"
)
# Drop only columns that exist
columns_to_drop <- intersect(columns_to_drop, names(credit_limit_data))
credit_limit_data <- credit_limit_data %>%
select(-all_of(columns_to_drop))
colnames(credit_limit_data)## [1] "id" "client_id" "num_cards_issued"
## [4] "per_capita_income" "yearly_income" "total_debt"
## [7] "num_credit_cards" "debt_to_income_ratio" "total_transactions"
## [10] "avg_transaction_amount" "max_transaction_amount" "total_spent"
## [13] "credit_limit" "geo_cluster1" "geo_cluster2"
## [16] "geo_cluster3" "geo_cluster4" "geo_cluster5"
## [19] "geo_cluster6" "geo_cluster7" "geo_cluster8"
## [22] "Credit" "Debit" "Debit (Prepaid)"
## [25] "Amex" "Discover" "Mastercard"
## [28] "Visa" "retired" "age_group"
## [31] "credit_score_category" "gender_binary" "acct_tenure_years"2. Data Splitting
To split the dataset into training (80%) and testing (20%) at the client level to ensure that all transactions for a given client stay in the same set. This is done to avoid data leakage and ensures the same client won’t appear in both training and testing.
set.seed(42)
clients <- unique(credit_limit_data$client_id)
n <- length(clients)
clients <- sample(clients)
# Split indices
train_index <- 1:floor(0.8 * n)
test_index <- (floor(0.8*n) + 1):n
# Assign clients to each set
train_clients <- clients[train_index]
test_clients <- clients[test_index]
# Subset dataset
train_data <- credit_limit_data %>% filter(client_id %in% train_clients)
test_data <- credit_limit_data %>% filter(client_id %in% test_clients)
# Separate features and target
x_train <- train_data %>% select(-credit_limit, -id, -client_id)
y_train <- train_data$credit_limit
x_test <- test_data %>% select(-credit_limit, -id, -client_id)
y_test <- test_data$credit_limit3. MULTICOLLINEARITY CHECK
To check whether any highly correlated numeric features exists and removes them from both training and testing data if the correlations is greater than 0.8
num_data <- credit_limit_data %>%
select(where(is.numeric))
cor_matrix <- cor(num_data, use = "complete.obs")
cor_df <- as.data.frame(as.table(cor_matrix)) %>%
rename(
var1 = Var1,
var2 = Var2,
correlation = Freq
) %>%
filter(
abs(correlation) > 0.8,
var1 != var2
)
cor_df <- cor_df %>%
rowwise() %>%
mutate(pair = paste(sort(c(var1, var2)), collapse = "_")) %>%
ungroup() %>%
distinct(pair, .keep_all = TRUE) %>%
select(-pair)
knitr::kable(cor_df, digits = 3)| var1 | var2 | correlation |
|---|---|---|
| yearly_income | per_capita_income | 0.950 |
| total_spent | total_transactions | 0.890 |
| Debit | Credit | -0.818 |
| Visa | Mastercard | -0.817 |
x_train <- x_train %>%
select(-per_capita_income, -total_transactions)
x_test <- x_test %>%
select(-per_capita_income, -total_transactions)4. MODEL TRAINING
5 regression models are trained to predict the credit limit
- Linear Regression (LM)
- Random Forest (RF)
- eXtreme Gradient Boosting (XGBoost)
- Gradient Boosting Machine (GBM)
- Support Vector Regression (SVR)
Evaluation Metrics
- R-squared (R2): Measures the standard deviation of prediction errors.
Lower is better
- RMSE (Root Mean Squared Error): Average absolute difference between
predictions and actual values. Lower is better
- MAE (Mean Absolute Error): Average absolute difference between
predictions and actual values. Lower is better
# ---------------- Train Linear Regression model ----------------
model_lm <- train(
x = x_train,
y = y_train,
method = "lm"
)
lm_pred <- predict(model_lm, x_test)
lm_rmse <- RMSE(lm_pred, y_test)
lm_mae <- MAE(lm_pred, y_test)
lm_r2 <- R2(lm_pred, y_test)
# ---------------- Train Random Forest model ----------------
set.seed(42)
rf_model <- randomForest(
x = x_train,
y = y_train,
ntree = 500, # n_estimators
importance = TRUE
)
rf_pred <- predict(rf_model, x_test)
rf_rmse <- RMSE(rf_pred, y_test)
rf_mae <- MAE(rf_pred, y_test)
rf_r2 <- R2(rf_pred, y_test)
# ------------------- Train XGBoost model -------------------
# Convert numeric data frames to matrices
x_train_matrix <- as.matrix(x_train)
x_test_matrix <- as.matrix(x_test)
y_train_vector <- as.numeric(y_train)
y_test_vector <- as.numeric(y_test)
dtrain <- xgb.DMatrix(data = x_train_matrix, label = y_train_vector)
dtest <- xgb.DMatrix(data = x_test_matrix, label = y_test_vector)
set.seed(42)
xgb_params <- list(
objective = "reg:squarederror",
eta = 0.05,
max_depth = 6,
subsample = 0.8,
colsample_bytree = 0.8
)
xgb_model <- xgb.train(
params = xgb_params,
data = dtrain,
nrounds = 500,
verbose = 0
)
xgb_pred <- predict(xgb_model, newdata = dtest)
xgb_rmse <- RMSE(xgb_pred, y_test_vector)
xgb_mae <- MAE(xgb_pred, y_test_vector)
xgb_r2 <- R2(xgb_pred, y_test_vector)
# ------------------- Train GBM model -------------------
set.seed(42)
model_gbm <- gbm(
formula = credit_limit ~ . -id -client_id,
data = train_data,
distribution = "gaussian",
n.trees = 500,
interaction.depth = 6,
shrinkage = 0.1,
n.minobsinnode = 10,
verbose = FALSE
)
gbm_pred <- predict(model_gbm, newdata = test_data, n.trees = 500)
gbm_rmse <- RMSE(gbm_pred, y_test)
gbm_mae <- MAE(gbm_pred, y_test)
gbm_r2 <- R2(gbm_pred, y_test)
# ------------------- Train SVR model -------------------
model_svr <- svm(
x = x_train,
y = y_train,
type = "eps-regression", # SVR
kernel = "radial", # RBF kernel
cost = 10, # C parameter
gamma = 1 / ncol(x_train), # common default
epsilon = 0.1
)
svr_pred <- predict(model_svr, x_test)
svr_rmse <- RMSE(svr_pred, y_test)
svr_mae <- MAE(svr_pred, y_test)
svr_r2 <- R2(svr_pred, y_test)
# ------------------- Summary Table -------------------
results <- data.frame(
Model = c("Linear Regression", "Random Forest", "XGBoost", "GBM", "SVR"),
RMSE = c(lm_rmse, rf_rmse, xgb_rmse, gbm_rmse, svr_rmse),
MAE = c(lm_mae, rf_mae, xgb_mae, gbm_mae, svr_mae),
R2 = c(lm_r2, rf_r2, xgb_r2, gbm_r2, svr_r2)
)
results## Model RMSE MAE R2
## 1 Linear Regression 9160.614 5833.327 0.5480276
## 2 Random Forest 8781.928 5339.454 0.5908154
## 3 XGBoost 8393.193 5469.750 0.6144570
## 4 GBM 8448.865 5380.416 0.6106920
## 5 SVR 9222.758 6073.933 0.53604465. HYPERPARAMETER TUNING
In order to improve the prediction performance, hyperparameter tuning is performed on the XGBoost (best-performing model in previous section) by testing different learning rates (eta) and depths (max_depth)
# Convert numeric data frames to matrices
x_train_matrix <- as.matrix(x_train)
x_test_matrix <- as.matrix(x_test)
y_train_vector <- as.numeric(y_train)
y_test_vector <- as.numeric(y_test)
dtrain <- xgb.DMatrix(data = x_train_matrix, label = y_train_vector)
dtest <- xgb.DMatrix(data = x_test_matrix, label = y_test_vector)
# Set of 3 hyperparameter combinations
param_list <- list(
list(eta = 0.05, max_depth = 6),
list(eta = 0.01, max_depth = 6),
list(eta = 0.1, max_depth = 6),
list(eta = 0.05, max_depth = 5),
list(eta = 0.01, max_depth = 5),
list(eta = 0.1, max_depth = 5),
list(eta = 0.05, max_depth = 4),
list(eta = 0.01, max_depth = 4),
list(eta = 0.1, max_depth = 4)
)
# Store results
results <- data.frame(
Set = character(),
RMSE = numeric(),
MAE = numeric(),
R2 = numeric()
)
models_list <- list()
# Train and evaluate each set
for (i in seq_along(param_list)) {
params <- param_list[[i]]
set.seed(42)
xgb_params <- list(
objective = "reg:squarederror",
eta = params$eta,
max_depth = params$max_depth,
subsample = 0.8,
colsample_bytree = 0.8
)
model <- xgb.train(
params = xgb_params,
data = dtrain,
nrounds = 500,
verbose = 0
)
pred <- predict(model, newdata = dtest)
results <- rbind(
results,
data.frame(
Set = paste0("Set_", i),
RMSE = RMSE(pred, y_test_vector),
MAE = MAE(pred, y_test_vector),
R2 = R2(pred, y_test_vector)
)
)
models_list[[i]] <- model
}
# Show comparison
results## Set RMSE MAE R2
## 1 Set_1 8393.193 5469.750 0.6144570
## 2 Set_2 8213.909 5188.977 0.6325698
## 3 Set_3 8511.461 5611.461 0.6036345
## 4 Set_4 8299.762 5345.261 0.6229770
## 5 Set_5 8227.450 5197.688 0.6312360
## 6 Set_6 8585.872 5610.113 0.5973321
## 7 Set_7 8384.557 5342.359 0.6153398
## 8 Set_8 8285.508 5219.537 0.6273367
## 9 Set_9 8481.278 5481.965 0.60756706. FEATURE IMPORTANCE
- Feature importance is extracted to show which variables contributed
most to predictions
- Additional evaluation metrics including residuals, Median Absolute
Error (MedAE), and Mean Absolute Percentage Error (MAPE) are
calculated
- Prediction vs actual and residual plots are visualized for performance assessment
# ------------------- Find best model -------------------
best_index <- which.min(results$RMSE)
best_model <- models_list[[best_index]]
best_set_name <- results$Set[best_index]
cat("Best hyperparameter set:", best_set_name, "\n")## Best hyperparameter set: Set_2cat("RMSE:", round(results$RMSE[best_index],3),
"MAE:", round(results$MAE[best_index],3),
"R2:", round(results$R2[best_index],3), "\n")## RMSE: 8213.909 MAE: 5188.977 R2: 0.633# ----------- Feature importance for best model -------------------
importance_matrix <- xgb.importance(
feature_names = colnames(x_train_matrix),
model = best_model
)
n_features <- nrow(importance_matrix)
cat("Total features:", n_features, "\n")## Total features: 28# View top 10 features
head(importance_matrix, 10)## Feature Gain Cover Frequency
## <char> <num> <num> <num>
## 1: yearly_income 0.40550375 0.33351594 0.17772424
## 2: Debit 0.18822962 0.08916624 0.03567457
## 3: Credit 0.07301733 0.03852494 0.02548184
## 4: Debit (Prepaid) 0.05313758 0.03359398 0.01385285
## 5: acct_tenure_years 0.04504496 0.07449848 0.12059859
## 6: total_debt 0.04401212 0.06368296 0.09191994
## 7: debt_to_income_ratio 0.03432842 0.04528809 0.06801334
## 8: avg_transaction_amount 0.03380690 0.04983254 0.07968866
## 9: total_spent 0.02054469 0.06000097 0.07385100
## 10: max_transaction_amount 0.01950817 0.04603279 0.06893996# Plot top 10 features
xgb.plot.importance(importance_matrix[1:10, ])
# ------------------- Additional Results for XGBoost -------------------
# Predict using best model
best_pred <- predict(best_model, newdata = dtest)
# 1. Residuals
residuals <- y_test_vector - best_pred
residuals## [1] 8.791729e+02 3.423892e+03 -5.166446e+03 1.348534e+04 1.068193e+03
## [6] -8.772004e+03 1.532232e+04 -1.277378e+04 -8.669209e+02 -1.371560e+04
## [11] 1.624356e+03 8.222813e+02 -5.818965e+02 -5.369721e+02 3.942818e+03
## [16] -1.435929e+04 -2.009473e+03 1.312433e+03 -1.301137e+03 3.386204e+03
## [21] -4.552896e+03 -2.865277e+03 9.912140e+03 1.468184e+02 2.073516e+03
## [26] 4.420279e+03 1.436514e+03 -2.204127e+02 3.873193e+03 1.642603e+03
## [31] -4.024652e+03 2.080187e+03 -1.453548e+04 -1.692115e+03 -1.145421e+04
## [36] 5.304802e+00 -8.318438e+03 -9.965049e+02 1.974162e+03 -3.303841e+03
## [41] -2.071337e+03 6.253965e+02 5.543336e+03 5.964187e+03 -3.449731e+03
## [46] 3.562905e+04 -7.404768e+03 -6.201436e+02 -3.645439e+02 6.972570e+03
## [51] -1.422373e+02 -9.948844e+03 -7.567189e+02 4.427242e+03 2.817361e+02
## [56] -9.174583e+03 -8.514059e+03 -7.390403e+03 1.008323e+04 3.703107e+03
## [61] -4.735269e+02 8.940152e+03 1.733154e+03 -9.105049e+02 8.149453e+02
## [66] -2.444986e+04 4.827600e+03 4.105839e+03 -4.345310e+03 -5.160187e+03
## [71] -4.029090e+03 -4.462930e+02 -3.595960e+03 -1.117179e+03 1.724766e+02
## [76] -3.956524e+02 -4.653994e+02 -4.583994e+03 1.594521e+03 -7.064473e+02
## [81] -4.785098e+03 -4.494633e+03 3.145098e+03 -5.590732e+03 -6.227967e+03
## [86] -2.458545e+03 5.713556e+03 4.681611e+02 4.874917e+02 -4.161920e+03
## [91] -1.231660e+02 -1.466856e+03 -1.601239e+03 -2.008281e+02 2.500335e+03
## [96] -4.585651e+03 3.814727e+02 2.197224e+03 1.812422e+02 3.143452e+02
## [101] 5.383701e+02 -1.339129e+04 -1.303880e+04 -3.156867e+03 -1.177052e+03
## [106] -4.741596e+02 -3.475001e+02 -2.017337e+03 1.295412e+03 5.364473e+02
## [111] -3.503579e+02 -1.245545e+03 5.556299e+02 -4.250854e+02 -1.494244e+03
## [116] -3.580386e+03 -1.257793e+03 -2.129392e+03 -1.267723e+04 -7.503404e+03
## [121] -6.778934e+03 -5.085760e+02 -5.462988e+01 -1.339994e+04 4.379111e+02
## [126] -9.032897e+02 1.808523e+03 -2.788368e+03 1.469992e+03 -4.639775e+02
## [131] -1.273153e+03 6.232717e+03 1.818308e+03 -1.163322e+04 4.216988e+03
## [136] -3.437247e+03 5.745637e+03 5.486205e+03 -2.823422e+03 -2.404195e+03
## [141] -8.704456e+01 -2.136918e+03 -1.608776e+03 4.200685e+03 -1.731017e+03
## [146] 8.373496e+03 2.916348e+02 -2.007330e+03 -8.616461e+03 9.013652e+03
## [151] -4.400935e+03 3.577324e+02 -6.127246e+02 9.133875e+03 -8.299006e+02
## [156] 5.830060e+03 -2.028872e+03 5.175355e+03 -2.100576e+04 4.433182e+03
## [161] 6.981973e+02 8.698369e+02 6.437055e+03 7.803285e+03 7.122981e+03
## [166] -4.563424e+03 -1.662766e+03 -1.029766e+03 6.745715e+03 2.858398e+02
## [171] -8.641098e+03 2.275588e+03 -9.461289e+02 6.083871e+03 5.760206e+03
## [176] -5.028850e+03 -6.377850e+03 4.633057e+03 2.272584e+03 -2.011805e+03
## [181] -6.031956e+02 2.836717e+03 -1.002533e+03 -1.016335e+03 2.249049e+04
## [186] 4.926846e+02 1.184779e+04 -1.198504e+04 -6.733789e+01 8.981322e+03
## [191] -4.764771e+03 2.950905e+03 -1.396623e+04 -4.657548e+03 -8.830768e+03
## [196] -1.048489e+04 8.872070e+01 9.593848e+02 -4.692027e+03 1.161019e+04
## [201] 7.148174e+02 -1.233517e+03 -1.396847e+03 2.408022e+03 -1.957188e+03
## [206] -6.512850e+03 7.217511e+03 -4.947380e+03 -4.447239e+02 8.795434e+03
## [211] -1.832796e+03 3.042804e+03 9.651838e+03 -1.976034e+03 1.240216e+04
## [216] -5.630795e+03 -8.108850e+03 2.599872e+03 9.838867e+00 7.792795e+03
## [221] -1.358030e+04 -3.663030e+03 -4.013062e+03 -1.736693e+03 1.126501e+04
## [226] -5.126982e+03 -1.651300e+03 2.740006e+03 -2.894402e+03 4.350743e+03
## [231] -5.944760e+02 5.168102e+03 -1.123957e+04 2.041072e+03 -3.221469e+03
## [236] -8.225664e+02 -5.470297e+03 -4.405253e+03 -5.081441e+03 8.267520e+02
## [241] 1.085227e+04 -1.743965e+02 -3.701065e+03 -6.265220e+02 1.261276e+03
## [246] 3.829898e+03 7.726455e+03 -6.152851e+03 1.068299e+04 1.877451e+02
## [251] 5.251145e+03 1.081006e+04 2.588552e+03 -8.043561e+03 -1.013372e+03
## [256] 1.908866e+03 -1.422429e+03 -1.481228e+03 -9.679717e+02 -5.557008e+03
## [261] 2.007859e+03 -1.794398e+03 -4.565137e+01 3.458715e+03 -1.118238e+04
## [266] -8.242313e+02 -2.924292e+03 -3.074092e+03 -8.957402e+02 -1.334484e+03
## [271] -3.401168e+02 1.175836e+04 -4.558127e+03 -5.771439e+03 -2.611910e+03
## [276] 3.195605e+02 -1.150312e+03 -1.131868e+04 -6.260713e+02 -2.807459e+03
## [281] 3.239348e+03 5.698125e+02 1.473314e+03 1.253430e+04 1.045111e+04
## [286] -6.430770e+03 -3.323611e+03 -2.389774e+03 -7.789826e+02 -4.659283e+03
## [291] -1.126281e+04 3.185487e+03 -5.981894e+03 7.037420e+03 -1.685083e+03
## [296] -1.885195e+02 -9.453457e+03 -6.541848e+03 -5.015950e+03 -1.024370e+04
## [301] 2.395721e+03 4.572284e+04 5.000320e+03 7.867127e+03 1.258136e+03
## [306] -1.754461e+03 2.633533e+03 -4.564317e+03 -6.937485e+02 -4.691641e+02
## [311] -2.272229e+03 -2.696241e+03 -1.311527e+04 5.270807e+03 -3.374830e+03
## [316] 1.692330e+03 -5.104959e+03 7.647445e+03 -1.441765e+04 1.716271e+03
## [321] -1.420307e+03 1.306797e+03 -2.528656e+03 4.361344e+03 5.756605e+03
## [326] 2.045564e+03 -1.982324e+04 -1.986364e+03 -2.673715e+03 2.742541e+03
## [331] -4.098386e+02 2.732374e+03 -1.588073e+02 2.448681e+03 -1.738871e+03
## [336] -1.163228e+04 -4.821104e+03 -1.054705e+04 -2.644858e+03 1.607562e+04
## [341] 1.580613e+03 -1.043717e+03 1.355896e+03 -1.297768e+04 9.478703e+03
## [346] 1.296223e+03 4.050348e+03 -1.437313e+03 -8.709414e+02 1.759444e+03
## [351] 1.052775e+04 2.766104e+03 5.331592e+02 5.926727e+03 -5.303448e+02
## [356] 3.571840e+03 -9.139749e+03 2.806491e+03 -5.474062e+02 -5.298680e+01
## [361] 1.245380e+04 -7.474694e+02 -1.904884e+04 -1.166233e+04 2.193563e+03
## [366] -4.237975e+03 -7.289393e+02 -1.643938e+04 3.451991e+03 -5.880454e+03
## [371] 3.240402e+03 4.140280e+03 -1.593624e+03 -1.451266e+03 -3.436936e+03
## [376] -4.875559e+03 2.152214e+03 2.306109e+03 -2.969280e+03 -2.664947e+04
## [381] -8.126035e+02 -1.049222e+04 5.748461e+03 -7.026426e+03 8.898011e+03
## [386] -4.486107e+03 -5.870896e+03 1.832875e+03 2.286155e+03 -2.016282e+03
## [391] 9.817324e+02 -1.027935e+04 1.749789e+03 -1.120761e+04 -1.334613e+03
## [396] 4.310053e+03 -2.603015e+02 1.156453e+04 -7.356133e+02 1.303907e+03
## [401] -1.848405e+04 1.805168e+03 -1.544377e+04 1.684326e+03 1.723401e+04
## [406] 1.502029e+02 -1.248069e+04 1.242323e+04 -3.886900e+03 1.649440e+03
## [411] 3.994854e+03 -4.305961e+02 -7.334277e+02 -3.440690e+03 -1.045066e+04
## [416] -1.224350e+03 -9.284023e+02 9.804001e+03 -5.335865e+03 1.616748e+04
## [421] -1.904506e+03 4.236426e+02 -4.484631e+03 3.003742e+03 -7.363017e+02
## [426] -5.420736e+02 -1.364334e+04 2.878066e+02 -3.676940e+02 -8.439654e+03
## [431] -4.191822e+03 -1.721453e+03 7.251971e+03 -8.891381e+03 -9.100234e+02
## [436] -8.558351e+03 -6.473705e+03 3.733754e+03 -3.653947e+03 5.687408e+03
## [441] 1.318943e+03 6.825638e+03 4.891577e+03 1.990003e+03 7.040858e+03
## [446] 2.502089e+03 -2.503434e+03 -2.920803e+03 1.172739e+03 -9.287207e+02
## [451] -2.388603e+03 -8.928789e+02 8.448945e+02 -1.675847e+02 7.700444e+03
## [456] 2.667391e+03 -9.000937e+02 1.017513e+03 7.554043e+02 4.319619e+03
## [461] 4.574217e+03 -8.254459e+02 -3.230381e+03 -7.299326e+02 1.548836e+03
## [466] -1.277658e+03 7.228519e+03 -9.150342e+03 -7.050523e+03 -9.425077e+02
## [471] -9.702204e+03 -1.853452e+03 -2.282207e+03 1.269389e+03 4.623219e+03
## [476] -8.610801e+02 4.815028e+03 -3.308205e+03 1.546032e+04 1.857516e+03
## [481] -3.367240e+03 4.637281e+03 -7.019635e+03 -8.084480e+03 4.958292e+03
## [486] -4.895475e+03 1.570292e+03 2.844483e+04 -5.185156e+02 -5.931097e+03
## [491] -2.962666e+02 1.779516e+04 4.042864e+03 2.056116e+04 -3.669209e+02
## [496] -5.917605e+02 -9.325244e+02 3.855599e+03 -4.602423e+03 -2.202865e+03
## [501] 4.546113e+03 3.692176e+03 9.726234e+03 -2.038781e+03 5.615010e+02
## [506] -4.233685e+03 -6.906166e+03 5.863395e+03 -6.041615e+01 -2.587319e+02
## [511] -2.519037e+03 -5.191992e+02 -2.785547e+01 5.333548e+03 1.412236e+03
## [516] 1.231920e+03 -1.940836e+03 -1.895826e+03 1.094192e+03 3.772408e+03
## [521] -8.621783e+03 -1.600783e+03 -1.533395e+03 -7.029355e+02 -2.118425e+03
## [526] -3.034740e+03 -3.586613e+03 2.759180e+03 -4.912533e+02 1.861673e+04
## [531] -7.322701e+03 -2.907075e+02 -3.824061e+03 -5.946611e+02 -1.749335e+03
## [536] 2.128066e+03 -4.744497e+02 -2.756568e+03 -3.315787e+03 2.804891e+03
## [541] 5.312422e+02 3.801565e+03 -4.506637e+03 -2.751836e+03 -4.545381e+02
## [546] 2.367597e+04 8.314338e+03 -2.544029e+03 -1.380984e+03 -1.364561e+04
## [551] -1.856531e+02 -6.135824e+03 -6.734863e+02 -1.609043e+03 -1.592479e+03
## [556] 1.027760e+04 8.561472e+03 -3.546602e+03 4.648919e+03 -2.521781e+03
## [561] 4.847696e+03 3.290609e+03 3.885344e+03 1.835580e+03 3.754934e+03
## [566] 7.154824e+02 -2.761678e+03 7.399953e+03 -2.806345e+02 -2.451507e+03
## [571] 2.538130e+04 -2.002349e+03 1.427615e+03 -1.686797e+03 -6.034923e+03
## [576] -1.341427e+04 6.633730e+02 -9.647539e+02 -5.697304e+03 -1.358258e+03
## [581] -1.184449e+03 4.803220e+03 1.893929e+04 -1.269930e+04 6.276504e+02
## [586] -3.339593e+02 -4.678356e+03 3.217241e+02 -1.391233e+03 1.584655e+04
## [591] 6.321510e+03 8.209121e+03 -1.289479e+04 -2.024058e+04 -1.625813e+04
## [596] 1.080419e+03 -1.626199e+03 6.374891e+03 3.685514e+03 1.235604e+04
## [601] -1.596827e+04 2.212368e+03 9.293031e+03 4.171945e+03 -3.847760e+02
## [606] -4.744695e+03 1.689771e+04 -3.767115e+03 5.182411e+04 -7.702656e+02
## [611] 3.033780e+03 -4.105786e+03 1.530505e+04 6.995262e+03 1.593311e+01
## [616] -9.980137e+02 5.263418e+03 -3.502869e+03 1.408269e+04 1.256421e+03
## [621] -4.752992e+03 -1.217437e+03 3.683597e+03 4.650859e+03 3.467704e+04
## [626] -1.732793e+04 -1.000498e+03 -1.675295e+03 -2.418247e+03 4.235062e+03
## [631] -2.838648e+03 1.110820e+04 4.602248e+04 -2.871030e+03 1.292352e+03
## [636] 1.609543e+04 4.197266e+02 7.518309e+03 -4.399246e+02 -5.219530e+02
## [641] -1.309072e+03 -8.378148e+02 5.477631e+03 3.812461e+02 -1.077702e+04
## [646] -2.325443e+03 1.771257e+03 1.012756e+03 3.765117e+03 2.201523e+03
## [651] 6.732287e+04 -9.499812e+03 1.049441e+03 -2.904872e+03 -1.435171e+03
## [656] -7.779721e+03 -8.772568e+03 1.808675e+03 2.101096e+03 1.063939e+04
## [661] -6.986982e+02 -8.056982e+02 -3.040042e+03 3.650051e+03 -1.511968e+04
## [666] -1.740840e+03 5.188086e+01 1.201265e+03 -3.749902e+02 -1.870647e+03
## [671] -3.487752e+03 1.149556e+03 -7.324219e-01 2.344321e+03 -3.222790e+03
## [676] -9.179824e+03 5.873313e+03 -3.054131e+02 -4.237090e+03 -4.889121e+02
## [681] 5.284854e+02 3.194892e+03 -3.124416e+03 9.104912e+03 -6.258309e+02
## [686] -3.252258e+03 2.514354e+03 -7.461611e+02 1.545185e+04 -1.396776e+03
## [691] 6.007958e+03 1.103214e+03 8.319889e+03 -3.934180e+03 3.066488e+03
## [696] -3.105295e+03 4.841290e+03 -1.282374e+03 2.820137e+02 -1.911318e+03
## [701] 1.614999e+04 -2.618605e+03 5.361383e+03 3.668240e+03 1.087012e+03
## [706] -1.019843e+04 -5.865721e+03 -1.667889e+03 -5.342959e+02 1.329565e+04
## [711] 2.708311e+02 -4.033271e+03 2.509053e+03 -5.208027e+02 3.408770e+03
## [716] -1.130096e+04 -1.240982e+03 -3.301385e+03 -3.275781e+01 -2.702754e+02
## [721] 3.915039e+00 1.418770e+04 1.565743e+03 -7.671914e+02 -6.464437e+03
## [726] 1.213400e+04 -6.995562e+03 -2.181496e+03 -3.152953e+03 3.999970e+03
## [731] -4.004048e+02 -1.890764e+03 -1.892210e+04 -1.983730e+02 7.173691e+03
## [736] 5.961570e+03 -3.498809e+03 -1.321613e+03 -7.150388e+02 -7.021396e+02
## [741] -4.162988e+02 1.041144e+02 -1.255035e+04 8.539386e+03 -9.448101e+03
## [746] -5.425047e+03 -1.699157e+04 6.543375e+03 7.579334e+03 1.325879e+04
## [751] -6.918166e+03 1.914428e+03 7.631549e+03 -7.357915e+02 -7.617851e+03
## [756] -5.823213e+03 3.685547e+00 -3.707202e+03 4.911291e+03 8.735239e+02
## [761] -5.398004e+03 -1.955048e+03 3.286005e+04 2.734306e+03 1.122543e+03
## [766] 7.343400e+03 1.128114e+04 3.455273e+02 2.234609e+03 2.930912e+04
## [771] 4.751504e+02 -9.627356e+03 -9.075414e+03 1.551043e+04 -4.201051e+03
## [776] 8.376020e+03 8.916436e+02 -8.917721e+03 1.796787e+02 -8.581549e+03
## [781] 3.721582e+03 8.633406e+03 1.104066e+04 2.434456e+03 -1.113346e+04
## [786] -1.179867e+03 -4.461768e+01 -4.901757e+02 -6.834412e+03 -3.073768e+03
## [791] -2.360187e+03 4.992287e+03 -6.526071e+02 -1.347767e+03 9.142687e+03
## [796] -4.725852e+03 -4.654924e+03 1.553721e+03 1.697475e+04 -4.591851e+03
## [801] 1.341324e+04 -1.896899e+03 -2.494087e+03 -2.743848e+02 -4.051318e+03
## [806] -6.690952e+02 -2.484988e+03 -8.312730e+03 1.017365e+03 3.193475e+03
## [811] 4.446435e+03 -9.293262e+02 -1.813212e+03 1.480906e+03 -6.755118e+02
## [816] 8.726049e+03 3.826437e+03 -2.225098e+02 5.812305e+01 -4.769043e+01
## [821] 1.322055e+04 1.223184e+04 5.968134e+03 4.882142e+03 -1.829796e+04
## [826] -3.888114e+02 -4.354577e+02 9.130922e+03 8.248213e+03 1.622765e+04
## [831] 1.110599e+04 -8.699609e+01 1.192507e+04 -3.807676e+02 3.278742e+03
## [836] -2.459415e+03 -1.478960e+03 -8.398965e+02 9.216451e+03 -5.795384e+03
## [841] 4.100096e+03 -4.826326e+03 -8.348592e+03 -9.807645e+03 4.548635e+03
## [846] 1.148526e+04 6.222273e+03 1.414983e+04 -7.949342e+03 -1.202334e+02
## [851] -4.155660e+03 7.682505e+03 -1.167145e+04 -6.253052e+03 3.174320e+03
## [856] -6.396896e+03 -1.070360e+04 -1.010796e+04 -4.990629e+03 -5.939258e+02
## [861] 5.637712e+03 9.625953e+03 6.874779e+03 -1.264019e+03 -3.720742e+02
## [866] 1.619875e+03 -5.359453e+02 1.049635e+03 6.022156e+03 1.089827e+03
## [871] 8.185469e+02 -1.419541e+03 -7.788984e+02 -8.332711e+03 -5.068558e+03
## [876] 3.022922e+03 -2.404571e+03 -1.798912e+03 3.465405e+03 -3.589883e+03
## [881] -7.715566e+03 3.922976e+03 -6.999266e+03 -1.230286e+04 -3.802969e+02
## [886] 1.499613e+03 -2.578548e+03 -6.962648e+02 -1.822187e+03 5.055625e+04
## [891] 1.845192e+03 -1.375600e+04 -1.973804e+03 8.347354e+02 1.470470e+03
## [896] 6.415942e+03 2.961687e+04 -1.182854e+04 5.071334e+03 -2.340254e+03
## [901] 4.312402e+01 1.997608e+03 1.287934e+03 -2.464550e+03 -4.556336e+03
## [906] 1.485786e+04 -4.836990e+03 -1.861822e+03 -5.514511e+03 -5.970307e+02
## [911] -3.819852e+02 2.302952e+03 -1.007468e+04 -3.160722e+03 -1.626918e+03
## [916] 1.217689e+04 3.579446e+03 3.290788e+03 4.387910e+02 -1.433655e+04
## [921] -2.092342e+04 -1.244095e+04 4.001654e+03 -6.415381e+02 6.865332e+02
## [926] 1.345701e+04 -2.241381e+04 -3.707816e+03 -2.165200e+04 7.129326e+02
## [931] -2.312079e+04 -2.650947e+02 1.118526e+04 -2.781041e+03 -3.188339e+03
## [936] -2.387592e+03 1.886547e+04 -6.841713e+03 2.710695e+03 2.071075e+03
## [941] -2.269005e+03 -1.748879e+03 1.417319e+04 2.090006e+03 1.679922e+04
## [946] -5.625981e+02 9.941289e+02 -8.373788e+03 -6.718828e+03 -7.144678e+02
## [951] -4.099450e+02 -1.462547e+03 -6.518743e+02 6.449223e+03 2.808502e+03
## [956] 5.024121e+02 1.351502e+04 -9.326617e+03 1.338477e+02 -1.524562e+03
## [961] 9.142598e+02 -9.960361e+02 -4.275221e+03 2.681409e+03 -2.297760e+03
## [966] -3.938828e+03 -3.709798e+02 -4.520000e+02 -5.320850e+03 2.380662e+04
## [971] -1.232695e+02 -2.017100e+02 -4.154321e+02 2.850488e+01 5.089687e+02
## [976] 9.985781e+02 3.335461e+03 -5.335943e+03 -8.663050e+03 7.194291e+03
## [981] -6.609392e+03 -8.726980e+03 9.231758e+02 2.964496e+03 1.945311e+03
## [986] -1.154632e+03 1.813059e+03 1.675783e+04 -1.207681e+04 1.270217e+04
## [991] -8.748280e+03 4.804961e+02 -1.736584e+03 5.625266e+03 2.132125e+03
## [996] -1.223510e+03 -1.048595e+04 3.546792e+03 -1.704531e+03 4.560927e+03
## [1001] -1.592970e+04 -8.173018e+03 -1.559841e+04 -1.211536e+04 -4.232952e+03
## [1006] 1.785858e+03 -2.859508e+03 -7.447285e+03 3.522419e+03 8.537523e+03
## [1011] -2.762262e+03 -1.709866e+02 2.805801e+02 -1.174771e+04 -6.753779e+03
## [1016] -7.554484e+03 -3.200339e+03 9.339883e+02 4.709900e+03 3.773785e+03
## [1021] -4.506996e+03 9.044998e+03 -3.165205e+03 -1.523564e+04 -7.172441e+02
## [1026] 1.026037e+03 -3.502461e+02 -1.314147e+04 -1.433423e+04 -1.116499e+03
## [1031] 5.001854e+03 -2.004823e+04 -3.435292e+02 -1.493588e+04 -3.746182e+03
## [1036] 7.790029e+03 1.421804e+03 -3.620937e+02 9.727520e+03 7.877869e+03
## [1041] -3.564113e+02 2.232672e+04 -5.104018e+02 -2.535666e+03 1.459937e+03
## [1046] 7.180977e+03 4.949176e+03 -5.478156e+03 3.210938e+01 -4.684482e+03
## [1051] -1.472332e+03 1.003155e+04 -4.842426e+03 -7.740967e+03 3.819703e+03
## [1056] -5.840957e+02 -2.783428e+03 -7.231721e+02 3.522238e+03 -3.416732e+02
## [1061] -1.976454e+04 -2.673540e+04 1.549805e+00 -5.064775e+02 2.395631e+03
## [1066] -2.266753e+03 -1.769981e+03 -7.471338e+02 3.251229e+04 2.442489e+04
## [1071] 4.217631e+03 2.909768e+03 1.604404e+03 8.651758e+02 -3.961469e+04
## [1076] -7.381652e+03 -9.389922e+03 -6.403505e+03 -6.400587e+03 4.270650e+03
## [1081] 8.048205e+03 5.605351e+03 1.431030e+03 4.620187e+03 -8.614887e+03
## [1086] -1.164497e+04 4.689375e+02 1.687875e+04 -3.632871e+02 -1.211830e+04
## [1091] 6.954863e+02 4.866330e+03 -4.009203e+02 1.425398e+03 -6.899100e+03
## [1096] 6.129630e+03 -4.710537e+03 -4.292274e+02 4.294691e+03 -2.660321e+03
## [1101] 1.590699e+03 -3.278534e+02 -9.137629e+03 5.273092e+03 6.649898e+03
## [1106] 3.074121e+02 6.609461e+03 1.520235e+04 6.668015e+03 1.114545e+03
## [1111] -4.221356e+02 -4.067321e+02 1.058912e+04 -6.780100e+03 -1.505101e+03
## [1116] -3.481348e+01 -5.144460e+03 -6.152703e+02 -7.696484e+01 1.831756e+03
## [1121] 1.945081e+03 5.862535e+03 -3.255634e+04 -8.716289e+02 8.949679e+01
## [1126] -6.075199e+03 -8.907135e+03 -1.298167e+03 9.241323e+01 -1.235289e+04
## [1131] 4.885518e+02 -6.561332e+03 -4.877380e+03 1.710609e+03 -5.420441e+03
## [1136] -2.729829e+02 -3.975828e+03 -1.747496e+03 -9.354727e+02 -1.028718e+04
## [1141] 3.050282e+03 7.304203e+03 -3.667046e+03 -6.986420e+03 -2.683348e+03
## [1146] -3.085654e+03 6.620830e+02 1.030133e+03 2.383833e+03 2.746576e+03
## [1151] -5.534393e+03 1.818406e+04 6.494677e+03 6.745586e+03 -1.196015e+04
## [1156] 1.314545e+04 1.670462e+02 3.686781e+04 -2.427713e+04 1.891330e+04
## [1161] -8.270710e+03 1.391405e+03 -1.735552e+02 7.264322e+03 3.688907e+03
## [1166] -8.102441e+02 1.032449e+03 8.379910e+03 -3.018303e+03 -6.768027e+02
## [1171] 8.914156e+03 2.542258e+03 3.401509e+03 2.815338e+03 -9.638486e+02
## [1176] -8.226915e+03 1.971635e+04 -8.260156e+02 -1.572379e+04 1.161321e+02
## [1181] 1.093915e+03 -3.137088e+03 1.722996e+03 -6.727157e+03 -7.672695e+02
## [1186] 5.169922e+00 6.137934e+03 1.171627e+04 2.896218e+03 -1.187364e+04
## [1191] 1.381452e+04 3.009180e+01 -3.359119e+03 -4.992219e+02 -3.771495e+02
## [1196] 3.633853e+03 -6.073535e+02 -8.136309e+03 6.461490e+03 -3.290476e+03
## [1201] 4.453877e+04 1.773203e+02 -1.093334e+03 3.714187e+03 5.059618e+01
## [1206] 4.917271e+03 -4.190740e+03 5.132530e+03 7.212227e+02 -4.794092e+03
## [1211] 6.420437e+03 8.545707e+03 -1.056264e+03 1.800148e+03 -4.510952e+02
## [1216] -2.595576e+03 -3.186507e+03 2.296862e+03 7.592045e+03 1.510465e+03# 2. Median Absolute Error (MedAE)
medae <- median(abs(residuals))
medae## [1] 3169.763# 3. Mean Absolute Percentage Error (MAPE)
mape <- mean(abs(residuals / y_test_vector)) * 100
mape## [1] 199.1095# 4. Prediction vs Actual Plot
plot(y_test_vector, best_pred,
xlab = "Actual Credit Limit",
ylab = "Predicted Credit Limit",
main = paste("XGBoost (Best) Predicted vs Actual -", best_set_name))
abline(a = 0, b = 1, col = "red", lwd = 2)
# 5. Residual Plot
plot(best_pred, residuals,
xlab = "Predicted Credit Limit",
ylab = "Residuals",
main = paste("XGBoost (Best) Residuals vs Predicted -", best_set_name))
abline(h = 0, col = "red", lwd = 2)