Unsupervised and Supervised Machine Learning:
Karol Orozco & Charles Hanks
2023-04-11
INTRODUCTION - MODEL PROJECT 3
Prediction task is to determine whether a person makes over 50k a year.
Applying principal component analysis, k-means, and a gradient boosting model using data from the 1994 Census.
# https://www.openml.org/d/1590
raw_income = read_csv("https://raw.githubusercontent.com/chanks06/ml-model3/main/datasets/openml_1590.csv", na=c("?"))## Rows: 48842 Columns: 15
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (9): workclass, education, marital-status, occupation, relationship, rac...
## dbl (6): age, fnlwgt, education-num, capital-gain, capital-loss, hours-per-week
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.income = read_csv("https://raw.githubusercontent.com/chanks06/ml-model3/main/datasets/openml_1590.csv", na=c("?")) %>%
drop_na() %>%
mutate(income_above_50k = class==">50K") %>%
select(-class) %>%
dummy_cols(remove_selected_columns = T)## Rows: 48842 Columns: 15
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (9): workclass, education, marital-status, occupation, relationship, rac...
## dbl (6): age, fnlwgt, education-num, capital-gain, capital-loss, hours-per-week
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.#formatting col names:
raw_income = raw_income %>% rename_all(funs(str_replace_all(.,"-","_"))) %>%
rename_all(funs(tolower(.)))## Warning: `funs()` was deprecated in dplyr 0.8.0.
## ℹ Please use a list of either functions or lambdas:
##
## # Simple named list: list(mean = mean, median = median)
##
## # Auto named with `tibble::lst()`: tibble::lst(mean, median)
##
## # Using lambdas list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## Warning: `funs()` was deprecated in dplyr 0.8.0.
## ℹ Please use a list of either functions or lambdas:
##
## # Simple named list: list(mean = mean, median = median)
##
## # Auto named with `tibble::lst()`: tibble::lst(mean, median)
##
## # Using lambdas list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.income = income %>% rename_all(funs(str_replace_all(.,"-","_"))) %>%
rename_all(funs(tolower(.)))## Warning: `funs()` was deprecated in dplyr 0.8.0.
## ℹ Please use a list of either functions or lambdas:
##
## # Simple named list: list(mean = mean, median = median)
##
## # Auto named with `tibble::lst()`: tibble::lst(mean, median)
##
## # Using lambdas list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## Warning: `funs()` was deprecated in dplyr 0.8.0.
## ℹ Please use a list of either functions or lambdas:
##
## # Simple named list: list(mean = mean, median = median)
##
## # Auto named with `tibble::lst()`: tibble::lst(mean, median)
##
## # Using lambdas list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.options(scipen=999)Exploratory Data Analysis
Examining distibution of sex:
n_sex = raw_income %>%
group_by(sex) %>%
count()raw_income %>% group_by(sex, class) %>% count() %>% mutate(prop = n/nrow(raw_income))## # A tibble: 4 × 4
## # Groups: sex, class [4]
## sex class n prop
## <chr> <chr> <int> <dbl>
## 1 Female <=50K 14423 0.295
## 2 Female >50K 1769 0.0362
## 3 Male <=50K 22732 0.465
## 4 Male >50K 9918 0.203#Only 3.6 % of this dataset includes women who make more than 50k.
#may want to equalize these classes by upsampling/downsampling Occupation by sex:
raw_income %>%
group_by(occupation,sex) %>%
count() %>%
ggplot(aes(x = occupation, y = n, fill = sex)) + geom_col() + coord_flip()
#craft-repair, farming/fishing, executive mostly male, admin mostly female
#there are some ethical considerations to unpack here in building this model...raw_income %>% group_by(education, workclass) %>% count() %>% ggplot(aes(x = n, y = education, fill = workclass)) + geom_col()
# Age
ggplot(income, aes(x = income_above_50k, y = age, fill = income_above_50k)) + geom_boxplot()
income %>% group_by(income_above_50k) %>% summarize(med_age = median(age), avg_age = mean(age))## # A tibble: 2 × 3
## income_above_50k med_age avg_age
## <lgl> <dbl> <dbl>
## 1 FALSE 34 36.7
## 2 TRUE 43 44.0ggplot(income, aes(x = age)) + geom_histogram(binwidth = 10)
#income$age_bin = factor(income$age_bin, levels = c("teen", "20-29","30-39","40-50","50-65","65+"))
#distribution of people by age bin
#ggplot(income, aes(x = age_bin, fill = age_bin)) + geom_histogram(stat= 'count')
#the largest age group in this dataset is 30-39
#ggplot(income, aes(x = age_bin)) + geom_histogram(stat = 'count',aes(fill = income_above_50k)) + facet_wrap(~income_above_50k)
#People in their forties are most likely to be making above 50k. raw_income %>% group_by(occupation, class, sex) %>% count() %>% filter(class == '>50K' ) %>% arrange(desc(n))## # A tibble: 28 × 4
## # Groups: occupation, class, sex [28]
## occupation class sex n
## <chr> <chr> <chr> <int>
## 1 Exec-managerial >50K Male 2487
## 2 Prof-specialty >50K Male 2202
## 3 Craft-repair >50K Male 1350
## 4 Sales >50K Male 1341
## 5 Prof-specialty >50K Female 582
## 6 Transport-moving >50K Male 468
## 7 Adm-clerical >50K Male 459
## 8 Exec-managerial >50K Female 421
## 9 Tech-support >50K Male 353
## 10 Machine-op-inspct >50K Male 344
## # ℹ 18 more rowsGiven that this dataset is imbalanced in terms of sex, it is likely that a predictive model will select male, between 30-50 years old, in executive-managerial, prof-speciality, craft-repair, or sales.
Changing response variable to a factor:
income = income %>% mutate(income_above_50k = factor(income_above_50k)) %>% relocate(income_above_50k)Log Transforming capital gain and capital loss:
“You have a capital gain if you sell the asset for more than your adjusted basis. You have a capital loss if you sell the asset for less than your adjusted basis.” https://www.irs.gov/taxtopics/tc409#:~:text=You%20have%20a%20capital%20gain,%2C%20aren't%20tax%20deductible.
#ggplot(income, aes(x = capital_loss)) + geom_histogram()
#definition: capital gain refers to the increase in the value of a capital asset when it is sold. A capital gain occurs when you sell an asset for more than what you originally paid for it.
income = income %>% mutate(l_capital_gain = log(capital_gain),
l_capital_loss = log(capital_loss)) %>%
select(-capital_gain,-capital_loss) %>% relocate(income_above_50k, l_capital_gain, l_capital_loss)
ggplot(income, aes(x = l_capital_loss)) + geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 43082 rows containing non-finite values (`stat_bin()`).
#changing -Inf back to 0 for log transformed vars:
income = income %>% mutate(l_capital_gain = ifelse(l_capital_gain == -Inf, 0, l_capital_gain),
l_capital_loss = ifelse(l_capital_loss == -Inf, 0, l_capital_loss))
#Now checking out shape of log transformed data:
ggplot(income, aes(x = l_capital_gain)) + geom_histogram() + xlim(5,12)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 41440 rows containing non-finite values (`stat_bin()`).## Warning: Removed 2 rows containing missing values (`geom_bar()`).
ggplot(income, aes(x = l_capital_loss)) + geom_histogram() + xlim(5,12)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 43082 rows containing non-finite values (`stat_bin()`).
## Removed 2 rows containing missing values (`geom_bar()`).
income %>% group_by(income_above_50k) %>% summarize(avg_cap_gain = mean(l_capital_gain), avg_cap_loss = mean(l_capital_loss))## # A tibble: 2 × 3
## income_above_50k avg_cap_gain avg_cap_loss
## <fct> <dbl> <dbl>
## 1 FALSE 0.333 0.228
## 2 TRUE 1.98 0.743#people making above over 50k will have greater log capital gain/loss on average PART I Principal Component Analysis on Income Dataset
We will apply Principal Component Analysis to our dataset in order to reduce the number of features needed to explain the variation in the data. We reduce the number of dimensions in our feature space through a linear combination of features that share co-variance. This process is based on a calculation of distances within the feature space. Is is therefore essential that we scale and center our numerical data.
Instead of having to manually assess correlation among variables, we call the prcomp() function on our dataset to group our variables together.
The result is that this new combination feature - the ‘principal component’ - captures the variation in the data according to the variables that it represents. Each principal component does not correlate with another - they are as distinct from one another as possible. If we were to graph these linear combination of features, they would form a right angle extend out from the center of the data’s spread.
In practice, this allows us to use only a handful of principal components to explain how the data behaves.
pr_income = prcomp(x = select(income,-income_above_50k), scale = T, center = T)
summary(pr_income) ## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.13805 1.76182 1.6158 1.53246 1.39287 1.33471 1.30557
## Proportion of Variance 0.04395 0.02985 0.0251 0.02258 0.01865 0.01713 0.01639
## Cumulative Proportion 0.04395 0.07380 0.0989 0.12149 0.14014 0.15727 0.17366
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 1.25616 1.19824 1.1762 1.1494 1.13345 1.117 1.10106
## Proportion of Variance 0.01517 0.01381 0.0133 0.0127 0.01235 0.012 0.01166
## Cumulative Proportion 0.18883 0.20264 0.2159 0.2286 0.24100 0.253 0.26465
## PC15 PC16 PC17 PC18 PC19 PC20 PC21
## Standard deviation 1.09652 1.09033 1.0842 1.08208 1.06551 1.06356 1.05277
## Proportion of Variance 0.01156 0.01143 0.0113 0.01126 0.01092 0.01088 0.01066
## Cumulative Proportion 0.27621 0.28764 0.2989 0.31020 0.32112 0.33200 0.34265
## PC22 PC23 PC24 PC25 PC26 PC27 PC28
## Standard deviation 1.05137 1.04406 1.04257 1.04098 1.03841 1.03694 1.03226
## Proportion of Variance 0.01063 0.01048 0.01045 0.01042 0.01037 0.01034 0.01025
## Cumulative Proportion 0.35328 0.36376 0.37422 0.38463 0.39500 0.40534 0.41559
## PC29 PC30 PC31 PC32 PC33 PC34 PC35
## Standard deviation 1.03101 1.02961 1.0250 1.02352 1.02314 1.02122 1.01912
## Proportion of Variance 0.01022 0.01019 0.0101 0.01007 0.01007 0.01003 0.00999
## Cumulative Proportion 0.42581 0.43600 0.4461 0.45618 0.46624 0.47627 0.48626
## PC36 PC37 PC38 PC39 PC40 PC41 PC42
## Standard deviation 1.01495 1.01376 1.01236 1.01039 1.0097 1.00492 1.00395
## Proportion of Variance 0.00991 0.00988 0.00985 0.00982 0.0098 0.00971 0.00969
## Cumulative Proportion 0.49616 0.50604 0.51590 0.52572 0.5355 0.54523 0.55492
## PC43 PC44 PC45 PC46 PC47 PC48 PC49
## Standard deviation 1.00257 1.00213 1.00184 1.00143 1.00142 1.00099 1.00090
## Proportion of Variance 0.00966 0.00966 0.00965 0.00964 0.00964 0.00963 0.00963
## Cumulative Proportion 0.56458 0.57424 0.58389 0.59353 0.60318 0.61281 0.62244
## PC50 PC51 PC52 PC53 PC54 PC55 PC56
## Standard deviation 1.00049 1.00035 1.00015 1.00000 0.99954 0.9994 0.99877
## Proportion of Variance 0.00962 0.00962 0.00962 0.00962 0.00961 0.0096 0.00959
## Cumulative Proportion 0.63207 0.64169 0.65131 0.66093 0.67053 0.6801 0.68973
## PC57 PC58 PC59 PC60 PC61 PC62 PC63
## Standard deviation 0.99834 0.99739 0.99678 0.99479 0.99474 0.99436 0.99339
## Proportion of Variance 0.00958 0.00957 0.00955 0.00952 0.00951 0.00951 0.00949
## Cumulative Proportion 0.69931 0.70888 0.71843 0.72794 0.73746 0.74697 0.75646
## PC64 PC65 PC66 PC67 PC68 PC69 PC70
## Standard deviation 0.99163 0.98918 0.9887 0.98599 0.98565 0.98490 0.98190
## Proportion of Variance 0.00946 0.00941 0.0094 0.00935 0.00934 0.00933 0.00927
## Cumulative Proportion 0.76591 0.77532 0.7847 0.79407 0.80341 0.81273 0.82201
## PC71 PC72 PC73 PC74 PC75 PC76 PC77
## Standard deviation 0.98122 0.97881 0.97693 0.96958 0.96596 0.95790 0.95563
## Proportion of Variance 0.00926 0.00921 0.00918 0.00904 0.00897 0.00882 0.00878
## Cumulative Proportion 0.83126 0.84047 0.84965 0.85869 0.86766 0.87649 0.88527
## PC78 PC79 PC80 PC81 PC82 PC83 PC84
## Standard deviation 0.95348 0.94928 0.93713 0.92981 0.92669 0.91355 0.9006
## Proportion of Variance 0.00874 0.00866 0.00844 0.00831 0.00826 0.00802 0.0078
## Cumulative Proportion 0.89401 0.90267 0.91112 0.91943 0.92769 0.93571 0.9435
## PC85 PC86 PC87 PC88 PC89 PC90 PC91
## Standard deviation 0.89148 0.86012 0.85386 0.81333 0.78869 0.77364 0.73843
## Proportion of Variance 0.00764 0.00711 0.00701 0.00636 0.00598 0.00575 0.00524
## Cumulative Proportion 0.95115 0.95827 0.96528 0.97164 0.97762 0.98337 0.98862
## PC92 PC93 PC94 PC95 PC96
## Standard deviation 0.69662 0.65566 0.49844 0.14221 0.0000000000001363
## Proportion of Variance 0.00467 0.00413 0.00239 0.00019 0.0000000000000000
## Cumulative Proportion 0.99328 0.99742 0.99981 1.00000 1.0000000000000000
## PC97 PC98
## Standard deviation 0.0000000000000326 0.00000000000002328
## Proportion of Variance 0.0000000000000000 0.00000000000000000
## Cumulative Proportion 1.0000000000000000 1.00000000000000000
## PC99 PC100
## Standard deviation 0.000000000000007325 0.000000000000006332
## Proportion of Variance 0.000000000000000000 0.000000000000000000
## Cumulative Proportion 1.000000000000000000 1.000000000000000000
## PC101 PC102
## Standard deviation 0.00000000000000627 0.000000000000005439
## Proportion of Variance 0.00000000000000000 0.000000000000000000
## Cumulative Proportion 1.00000000000000000 1.000000000000000000
## PC103 PC104
## Standard deviation 0.000000000000003673 0.000000000000003176
## Proportion of Variance 0.000000000000000000 0.000000000000000000
## Cumulative Proportion 1.000000000000000000 1.000000000000000000We need 37 principal components to explain half of the variation of the data.
Visualizing Principle Components’ Ability to Explain Variation
Looking at at the plot below, we can see a visual ranking of how much of the variation each component captures. Each component is ranked according to its Eigenvalue. The eigenvalue is a measure of the proportion of variance explain by that component.
The scree plot helps us select a cut off point for determining how we can explain the most variation with the fewest components as possible. We evaluate the slope of the line connecting the components, and find the point when the absolute value of the slope becomes small, and each component following continues that trend. The ‘elbow’ in the plot below is at PC5. The components after PC5 do much less for us in accounting for the variance the income dataset.
screeplot(pr_income, type = "lines")
#elbow is around pc5Interpretation of the Top 5 Principal Components
We will look at the factor loadings for each component to determine what each component is telling us about the data.
We can interpret factor loadings as coefficients of our linear combination for each feature within the principal component. They tell us the relative (transformed) value of each original feature. For example, a factor loading of -.75 for age would mean that this component contains the observations of younger people.
Component Interpretation
We want to find the most influential original features within our top 5 PCs. To this we will filter to show only those where at least one of the PCs has a factor loading greater than or equal to the absolute value of 0.25. This gives us the highlights of each component. Here are descriptions of the type of workers these principal components:
PC1: middle-aged husbands PC2: young men with little formal education PC3: Asian and Pacific Islander PC4: Career-focus males without family PC5: Mexican people
Note: these components are all quite male-dominated.
rownames_to_column(as.data.frame(pr_income$rotation)) %>%
select(1:6) %>%
filter(abs(PC1) >= 0.25 | abs(PC2) >= 0.25 | abs(PC3) >= 0.25 | abs(PC4) >= 0.25 | abs(PC5) >= 0.25) %>% rename("husbands" = PC1, "low_ed_male_laborer" = PC2, "asian_pac_isl" = PC3, "male_yopros" = PC4, "mexican" = PC5)## rowname husbands low_ed_male_laborer
## 1 age 0.18536748 -0.091194280
## 2 education_num 0.10559165 -0.456305049
## 3 marital_status_divorced -0.12297307 -0.121245468
## 4 marital_status_married_civ_spouse 0.39812029 0.064962147
## 5 marital_status_never_married -0.27611175 0.026881327
## 6 occupation_prof_specialty 0.06030228 -0.283025307
## 7 relationship_husband 0.41562618 0.100400172
## 8 relationship_own_child -0.18619502 0.088025922
## 9 relationship_unmarried -0.15438415 -0.061035972
## 10 race_asian_pac_islander -0.01133707 0.006958696
## 11 race_black -0.11602195 0.029250746
## 12 race_white 0.11374015 -0.046471243
## 13 sex_female -0.33225474 -0.195130792
## 14 sex_male 0.33225474 0.195130792
## 15 native_country_mexico -0.01221202 0.197502680
## 16 native_country_united_states 0.02207168 -0.180839129
## asian_pac_isl male_yopros mexican
## 1 0.14546866 -0.351735676 0.043968457
## 2 0.02123203 0.229444433 -0.001380269
## 3 0.01730921 -0.254545406 0.075931750
## 4 0.06343668 -0.060996695 -0.082585727
## 5 -0.14483745 0.374119315 0.008549345
## 6 0.10002404 0.130027759 0.106651957
## 7 0.02193476 -0.015109831 -0.091256658
## 8 -0.16118057 0.290041931 -0.093754589
## 9 0.10824314 -0.277233657 -0.043232822
## 10 0.39994102 0.219542435 -0.113266352
## 11 0.16183171 -0.066337055 -0.451022089
## 12 -0.36865300 -0.048990952 0.459449764
## 13 0.08166583 -0.201765893 0.065149987
## 14 -0.08166583 0.201765893 -0.065149987
## 15 0.14470615 0.006009948 0.297613434
## 16 -0.44698474 -0.128490322 -0.288966406Let us now determine if these principle components are good predictors of income_above_50k.
prc = bind_cols(select(income, income_above_50k), as.data.frame(pr_income$x)) %>%
select(1:6) %>%
filter(abs(PC1) >= 0.25 | abs(PC2) >= 0.25 | abs(PC3) >= 0.25 | abs(PC4) >= 0.25 | abs(PC5) >= 0.25) %>% rename("husbands" = PC1, "low_ed_male_laborer" = PC2, "asian_pac_isl" = PC3, "male_yopros" = PC4, "mexican" = PC5)
prc %>%
pivot_longer(cols = -income_above_50k, names_to = "component", values_to = "loading") %>% mutate(income_above_50k = as.factor(income_above_50k)) %>%
ggplot(aes(loading, fill=income_above_50k)) +
geom_density(alpha = 0.5) +
facet_grid(.~component)
Based on the density plots above, PC1 ‘husbands’ followed by PC2 ‘low_ed_male_laborer’ seem most predictive of income_above_50k.
Making a Logistic Regression Model using PC1 and PC2
What if we used only PC1 and PC2 to predict class using a logistic regression model?
prc$income_above_50k = factor(ifelse(prc$income_above_50k == 'TRUE', 'yes','no'),levels = c('yes','no'))
prc.pc1and2 = prc %>% select(income_above_50k, husbands, low_ed_male_laborer)
ctrl <- trainControl(method = "cv", number = 3, classProbs = TRUE, summaryFunction = twoClassSummary)
#splitting our data
prc.pc1and2_index <- createDataPartition(prc.pc1and2$income_above_50k, p = 0.80, list = FALSE)
train <- prc.pc1and2[prc.pc1and2_index, ]
test <- prc.pc1and2[-prc.pc1and2_index, ]
fit.prc.pc1and2 = train(income_above_50k ~ .,
data = train,
method = "glm",
family = "binomial",
metric = "ROC",
trControl = ctrl)
confusionMatrix(predict(fit.prc.pc1and2, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1109 575
## no 1132 6227
##
## Accuracy : 0.8112
## 95% CI : (0.803, 0.8193)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.4476
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.4949
## Specificity : 0.9155
## Pos Pred Value : 0.6586
## Neg Pred Value : 0.8462
## Prevalence : 0.2478
## Detection Rate : 0.1226
## Detection Prevalence : 0.1862
## Balanced Accuracy : 0.7052
##
## 'Positive' Class : yes
## The Specificity (true negative rate) is much higher lower than the Sensitivity (true positive) rate. This makes sense given that there are roughly 3x more ‘<50k’ than ‘50k’ observations in the data.
income %>% group_by(income_above_50k) %>% count()## # A tibble: 2 × 2
## # Groups: income_above_50k [2]
## income_above_50k n
## <fct> <int>
## 1 FALSE 34014
## 2 TRUE 11208What if we add the rest of the principal components to the LR model?
prc_index <- createDataPartition(prc$income_above_50k, p = 0.80, list = FALSE)
train <- prc[prc_index, ]
test <- prc[-prc_index, ]
fit.allpc = train(income_above_50k ~ .,
data = train,
method = "glm",
family = "binomial",
metric = "ROC",
trControl = ctrl)
confusionMatrix(predict(fit.allpc, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1108 533
## no 1133 6269
##
## Accuracy : 0.8158
## 95% CI : (0.8076, 0.8237)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.4571
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.4944
## Specificity : 0.9216
## Pos Pred Value : 0.6752
## Neg Pred Value : 0.8469
## Prevalence : 0.2478
## Detection Rate : 0.1225
## Detection Prevalence : 0.1815
## Balanced Accuracy : 0.7080
##
## 'Positive' Class : yes
## Not a significance increase in LR model performance. We will bring along PC1 and PC2 as guides for the rest of our journey.
PART II: K-means
Code adapted from This article on clustering
Major Assumptions: flnwgt does not include age information
#scaling all cols:
income_scaled = as.data.frame(lapply(income %>% select(-income_above_50k), function(x) scale(x, center = TRUE, scale = TRUE)))
#adding PC1 and PC2
income_features = bind_cols(prc %>% select(2:3), income_scaled)Choosing number of clusters
set.seed(503)
# Create 10 models with 1 to 10 clusters
kclusts <- tibble(k = 1:10) %>%
mutate(
model = map(k, ~ kmeans(x = income_features, centers = .x)),
glanced = map(model, glance)) %>%
unnest(cols = c(glanced))
# View results
kclusts## # A tibble: 10 × 6
## k model totss tot.withinss betweenss iter
## <int> <list> <dbl> <dbl> <dbl> <int>
## 1 1 <kmeans> 5050066. 5050066. -2.22e-5 1
## 2 2 <kmeans> 5050066. 5002611. 4.75e+4 1
## 3 3 <kmeans> 5050066. 4599933. 4.50e+5 4
## 4 4 <kmeans> 5050066. 4525683. 5.24e+5 4
## 5 5 <kmeans> 5050066. 4497399. 5.53e+5 4
## 6 6 <kmeans> 5050066. 4397761. 6.52e+5 5
## 7 7 <kmeans> 5050066. 4321297. 7.29e+5 4
## 8 8 <kmeans> 5050066. 4312093. 7.38e+5 4
## 9 9 <kmeans> 5050066. 4302771. 7.47e+5 6
## 10 10 <kmeans> 5050066. 4215190. 8.35e+5 5# Plot Total within-cluster sum of squares (tot.withinss)
kclusts %>%
ggplot(mapping = aes(x = k, y = tot.withinss)) +
geom_line(size = 1.2, alpha = 0.5, color = "darkseagreen") +
geom_point(size = 2, color = "darkseagreen")+
theme_minimal()+
labs(title = "Total within-cluster sum of squares (tot.withinss)")## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
If we take a look at graph above, we can notice how the total
within-cluster sum of squares decreases as the number of clusters
increase. Considering the elbow method, which is the point where WCSS
decreases much slower after adding another cluster, the number of
clusters suggested is 7 (it appears that there is a bit of an elbow or
“bend” at that point). A smaller WithinSS (or SSW) means there is less
variance in that cluster’s data.
Analysis of cluster centers
set.seed(503)
# Fit and predict clusters with k = 3
final_kmeans <- kmeans(income_features, centers = 7, nstart = 100, iter.max = 1000)## Warning: Quick-TRANSfer stage steps exceeded maximum (= 2261100)final_kmeans## K-means clustering with 7 clusters of sizes 9215, 1477, 8481, 2048, 5784, 6832, 11385
##
## Cluster means:
## husbands low_ed_male_laborer l_capital_gain l_capital_loss age
## 1 -2.7297445 -0.2759099 -0.198069706 -0.119988857 -0.23157356
## 2 -0.6380062 4.2241956 -0.165536052 -0.107056924 -0.10619941
## 3 -1.0888871 0.9964358 -0.163410594 -0.087346410 -0.58354726
## 4 -0.5326338 -1.4560798 0.146150992 0.072624708 0.10967102
## 5 -0.8974086 -2.3127304 0.080449663 0.050697529 0.05030215
## 6 2.7423707 -1.3573604 0.402081043 0.240559416 0.45216429
## 7 2.0094291 1.1844507 -0.004924149 -0.007102903 0.31929125
## fnlwgt education_num hours_per_week workclass_federal_gov
## 1 -0.01332680 -0.3260648 -0.4788903 -0.037829069
## 2 0.63117287 -1.9232894 -0.1255393 -0.159627228
## 3 0.06999104 -0.3913752 -0.1691089 -0.084702680
## 4 -0.08759274 0.2108513 -0.2880302 -0.007523982
## 5 -0.06136248 1.0436116 0.1416459 0.151577484
## 6 -0.05390728 1.1317791 0.4471669 0.134580614
## 7 -0.04395475 -0.4423143 0.2413852 -0.041988765
## workclass_local_gov workclass_private workclass_self_emp_inc
## 1 -0.08404144 0.28598820 -0.17001422
## 2 -0.19626061 0.41059735 -0.14735394
## 3 -0.14296209 0.26861172 -0.12005897
## 4 0.15769330 -0.10463070 -0.03531089
## 5 0.32328507 -0.37992404 0.02074598
## 6 0.16952255 -0.53159418 0.54109066
## 7 -0.09435525 0.04599792 -0.08272872
## workclass_self_emp_not_inc workclass_state_gov workclass_without_pay
## 1 -0.2111338665 -0.03718753 -0.01148029
## 2 -0.1317983346 -0.19203433 -0.02155415
## 3 -0.0850016092 -0.10746588 0.01675586
## 4 0.0195238546 0.03818573 0.04643704
## 5 0.0003009409 0.27527159 -0.02155415
## 6 0.1269504799 0.15652025 -0.01476034
## 7 0.1714635215 -0.10557625 0.01106092
## education_1st_4th education_5th_6th education_7th_8th education_9th
## 1 -0.06402641 -0.09357358 -0.01599685 -0.006930832
## 2 2.04149225 2.78836734 0.40580600 0.451490424
## 3 -0.07023692 -0.10014062 -0.01706540 0.036167590
## 4 -0.07023692 -0.10014062 -0.05578506 -0.006496357
## 5 -0.07023692 -0.10014062 -0.13614714 -0.123186805
## 6 -0.07023692 -0.10014062 -0.13614714 -0.123186805
## 7 -0.07023692 -0.08242278 0.13391720 0.057769714
## education_10th education_11th education_12th education_assoc_acdm
## 1 0.06207240 0.105187277 0.036184036 -0.006702235
## 2 0.07535966 0.004087475 0.109515504 -0.166806888
## 3 0.09641115 0.179198932 0.117441342 -0.054279697
## 4 -0.06437647 -0.105964954 -0.048425577 0.083658239
## 5 -0.16565392 -0.191760019 -0.109062011 0.149547676
## 6 -0.16491509 -0.191114961 -0.109770898 0.075292969
## 7 0.06286507 0.012009616 -0.000989485 -0.068707634
## education_assoc_voc education_bachelors education_doctorate education_hs_grad
## 1 -0.001700628 -0.35071618 -0.108352936 0.19516671
## 2 -0.159579495 -0.39397735 -0.110343762 -0.35048421
## 3 -0.032662780 -0.35552841 -0.109262199 0.26558380
## 4 0.034253947 0.09700945 -0.002850822 -0.01716268
## 5 0.025850908 0.95342051 0.160841747 -0.62980551
## 6 -0.055333067 0.86581003 0.354200279 -0.64134340
## 7 0.060320160 -0.42156198 -0.110343762 0.39757489
## education_masters education_preschool education_prof_school
## 1 -0.2321989 -0.03721127 -0.131248225
## 2 -0.2307989 1.16573482 -0.132910001
## 3 -0.2385014 -0.03993306 -0.132007203
## 4 0.1111233 -0.03993306 0.001679232
## 5 0.5413438 -0.03993306 0.145079747
## 6 0.5680097 -0.03993306 0.467785946
## 7 -0.2403181 -0.03993306 -0.132910001
## education_some_college marital_status_divorced
## 1 0.26029978 0.4237359
## 2 -0.34762813 -0.2457561
## 3 0.14643227 0.1304589
## 4 -0.03696363 -0.4022052
## 5 -0.31153720 0.5950667
## 6 -0.31521844 -0.3945952
## 7 0.07941126 -0.4014441
## marital_status_married_af_spouse marital_status_married_civ_spouse
## 1 -0.018448501 -0.91163128
## 2 -0.026610254 -0.01585235
## 3 -0.022176192 -0.91376690
## 4 0.322266837 1.05274544
## 5 -0.026610254 -0.92714757
## 6 -0.015601683 1.05990015
## 7 -0.000185816 1.06623754
## marital_status_married_spouse_absent marital_status_never_married
## 1 0.07759594 0.40972990
## 2 0.45608826 0.04230271
## 3 0.04775919 0.86481240
## 4 -0.11116210 -0.69041612
## 5 0.07147842 0.48510266
## 6 -0.10982914 -0.68853779
## 7 -0.10796251 -0.69041612
## marital_status_separated marital_status_widowed occupation_adm_clerical
## 1 0.26681319 0.374567691 0.5964540
## 2 0.13206986 0.005279633 -0.2559369
## 3 0.09316722 -0.087898422 -0.1416838
## 4 -0.17945978 -0.170465271 0.3888486
## 5 0.02737565 0.107153057 -0.1120948
## 6 -0.17440859 -0.165163778 -0.2504275
## 7 -0.17945978 -0.163042094 -0.2067430
## occupation_armed_forces occupation_craft_repair occupation_exec_managerial
## 1 -0.01759752 -0.31711725 -0.19451677
## 2 -0.01759752 0.08845074 -0.32857404
## 3 0.03602117 0.24018832 -0.26176392
## 4 -0.01759752 -0.32862416 0.09510417
## 5 -0.00776998 -0.33842916 0.34625975
## 6 0.01568264 -0.30741735 0.58702042
## 7 -0.01260477 0.48180311 -0.15022113
## occupation_farming_fishing occupation_handlers_cleaners
## 1 -0.15039490 -0.11117038
## 2 0.39445923 0.32632590
## 3 0.09969586 0.37175561
## 4 -0.13728679 -0.15660200
## 5 -0.14604356 -0.20936570
## 6 -0.09180305 -0.19585259
## 7 0.15027052 0.02277991
## occupation_machine_op_inspct occupation_other_service
## 1 -0.00315603 0.44050381
## 2 0.43182742 0.34256639
## 3 0.11423801 0.12978797
## 4 -0.06407017 0.04475994
## 5 -0.25046782 -0.27480477
## 6 -0.24208035 -0.31404983
## 7 0.14547525 -0.17765077
## occupation_priv_house_serv occupation_prof_specialty
## 1 0.14388358 -0.3050936
## 2 0.51575325 -0.3634910
## 3 -0.05860598 -0.3379195
## 4 0.01020580 0.2688844
## 5 -0.05970940 0.8713049
## 6 -0.06976060 0.6237046
## 7 -0.06935049 -0.3194768
## occupation_protective_serv occupation_sales occupation_tech_support
## 1 -0.094005054 0.11537995 0.02404528
## 2 -0.115905631 -0.18284663 -0.14122811
## 3 0.011326791 -0.04405052 -0.04753537
## 4 -0.114918301 -0.08414215 0.02153444
## 5 0.008527286 -0.05791534 0.11636294
## 6 0.151639688 0.05817950 -0.01555116
## 7 0.008029442 -0.02720652 -0.01938833
## occupation_transport_moving relationship_husband relationship_not_in_family
## 1 -0.1880240 -0.83837728 0.1928860
## 2 0.0103104 -0.03527593 -0.1116144
## 3 0.1335390 -0.83813779 0.4381345
## 4 -0.1835986 -0.83837728 -0.5908446
## 5 -0.1970361 -0.83837728 1.1334730
## 6 -0.1984660 1.17342976 -0.5838262
## 7 0.3035973 1.18008735 -0.5872346
## relationship_other_relative relationship_own_child relationship_unmarried
## 1 0.13211372 0.3813776 0.70586148
## 2 0.66040658 -0.1577830 0.08717944
## 3 0.19060589 0.7179856 -0.06282825
## 4 -0.17534868 -0.4143332 -0.34411129
## 5 -0.09912792 -0.1801502 0.14025457
## 6 -0.15728061 -0.4097803 -0.33840271
## 7 -0.15831058 -0.4111043 -0.34211299
## relationship_wife race_amer_indian_eskimo race_asian_pac_islander race_black
## 1 -0.22017979 0.038200500 -0.015903516 0.30508023
## 2 -0.08154854 -0.001439862 0.005838621 -0.14905090
## 3 -0.22017979 0.033123502 -0.011536019 0.08913338
## 4 4.54164250 0.021510381 0.049592212 0.01261872
## 5 -0.22017979 -0.033012942 0.032393818 -0.11091717
## 6 -0.22017979 -0.070059167 0.092876433 -0.22210030
## 7 -0.22017979 -0.000463200 -0.060403887 -0.10663288
## race_other race_white sex_female sex_male native_country_cambodia
## 1 -0.006081141 -0.25770598 1.4292453 -1.4292453 -0.019457541
## 2 0.796014593 -0.07931044 -0.1734024 0.1734024 0.032503641
## 3 -0.001610612 -0.07819326 -0.6890225 0.6890225 0.020284844
## 4 -0.022118326 -0.03496430 1.4402511 -1.4402511 -0.023984566
## 5 -0.039584123 0.09684642 0.4520014 -0.4520014 -0.002347356
## 6 -0.058760033 0.17631898 -0.6934933 0.6934933 -0.023984566
## 7 -0.037796644 0.12840536 -0.6936183 0.6936183 0.016321270
## native_country_canada native_country_china native_country_columbia
## 1 -0.014875386 -0.030487492 0.018596382
## 2 -0.048847383 0.044877567 0.036948750
## 3 -0.010957480 -0.021709301 0.012808626
## 4 0.053921774 0.037971227 -0.008189861
## 5 0.023517538 0.005357299 -0.026365701
## 6 0.015568837 0.073083461 -0.021978353
## 7 -0.004450494 -0.018382367 -0.001329888
## native_country_cuba native_country_dominican_republic native_country_ecuador
## 1 0.0038040273 -0.02759825 0.011399735
## 2 0.1082241867 1.12437562 0.078982348
## 3 -0.0260046335 -0.04636310 -0.004071481
## 4 -0.0002097586 -0.01470085 -0.015008333
## 5 -0.0032278401 -0.03515212 -0.030850423
## 6 -0.0110637415 -0.04003559 -0.011854755
## 7 0.0105692540 -0.04446457 0.009046319
## native_country_el_salvador native_country_england native_country_france
## 1 -0.05329368 -0.004763749 -0.008987397
## 2 1.54862041 -0.024933558 -0.028225686
## 3 -0.05710656 -0.007635263 -0.019864368
## 4 -0.04852851 0.034414226 -0.010913105
## 5 -0.05103193 0.026254207 0.051464866
## 6 -0.04167813 0.037204333 0.034050951
## 7 -0.05556349 -0.029076433 -0.018882823
## native_country_germany native_country_greece native_country_guatemala
## 1 0.01110644 -0.016442371 -0.04115906
## 2 -0.06546779 -0.012355577 1.24617814
## 3 -0.02386707 -0.007847144 -0.04364986
## 4 0.01692380 -0.003251724 -0.04364986
## 5 0.01939979 -0.011914392 -0.03571324
## 6 0.01985320 0.020452950 -0.04364986
## 7 -0.00753085 0.015121219 -0.04364986
## native_country_haiti native_country_holand_netherlands
## 1 0.010953292 0.01837455
## 2 0.117022052 -0.00470246
## 3 0.018305298 -0.00470246
## 4 -0.001561846 -0.00470246
## 5 -0.034661529 -0.00470246
## 6 -0.031591004 -0.00470246
## 7 -0.000835547 -0.00470246
## native_country_honduras native_country_hong native_country_hungary
## 1 0.01656535 -0.007440721 -0.009073947
## 2 0.11164742 0.029544091 -0.019954600
## 3 0.01401980 -0.010670507 -0.019954600
## 4 0.00332456 0.073253908 0.004524212
## 5 -0.02050163 -0.010990113 0.049385104
## 6 -0.01335935 0.028066102 -0.012616689
## 7 -0.02050163 -0.014297638 0.006465746
## native_country_india native_country_iran native_country_ireland
## 1 -0.03804216 -0.022868719 -0.001292082
## 2 -0.04521229 -0.035211408 -0.028225686
## 3 -0.01360642 -0.008389600 0.017761559
## 4 -0.03137242 0.006440647 -0.028225686
## 5 0.02490100 0.018865211 0.002424527
## 6 0.13317745 0.060513534 -0.012656526
## 7 -0.04013280 -0.017728643 0.002917191
## native_country_italy native_country_jamaica native_country_japan
## 1 -0.021663616 0.068318450 0.007013230
## 2 0.183540289 -0.019373586 -0.029129701
## 3 -0.024484599 -0.005730369 -0.009820242
## 4 0.004898403 -0.017050416 0.021697307
## 5 -0.024992362 -0.033271648 -0.017099234
## 6 0.002780470 -0.029356111 0.061276815
## 7 0.022110041 -0.010928156 -0.026569537
## native_country_laos native_country_mexico native_country_nicaragua
## 1 -0.006443355 -0.1342064 0.020725246
## 2 0.135572848 4.0485100 0.112948566
## 3 -0.005135570 -0.1427395 0.003613831
## 4 0.023773309 -0.1322683 0.012388835
## 5 -0.021554146 -0.1316165 -0.021977588
## 6 -0.014760336 -0.1280912 -0.023606491
## 7 0.006984036 -0.1427395 -0.011017259
## native_country_outlying_us.guam_usvi_etc. native_country_peru
## 1 -0.0023769043 0.0200664672
## 2 -0.0220616160 -0.0100870832
## 3 0.0153680010 0.0058362525
## 4 0.0000812817 0.0148987369
## 5 0.0249805567 -0.0205935736
## 6 -0.0220616160 -0.0176335526
## 7 -0.0061288432 -0.0009168276
## native_country_philippines native_country_poland native_country_portugal
## 1 -0.0050469470 -0.014129410 -0.02532116
## 2 0.0580106465 -0.026347966 0.14592275
## 3 0.0043745421 0.010621771 0.01074652
## 4 0.0630537192 0.003830261 0.01573178
## 5 -0.0004305018 0.002616419 -0.03237977
## 6 0.0153036563 -0.025051940 -0.03309650
## 7 -0.0270068665 0.019957165 0.02703965
## native_country_puerto_rico native_country_scotland native_country_south
## 1 0.021567819 -0.0003894171 0.005560725
## 2 0.133956324 0.0111666352 -0.047311479
## 3 0.025030492 -0.0042106123 -0.007347582
## 4 0.047773229 0.0254118463 0.035436034
## 5 -0.048404666 0.0036341176 -0.003362592
## 6 -0.048182940 -0.0001499383 0.020901928
## 7 -0.008569669 -0.0043243968 -0.010098719
## native_country_taiwan native_country_thailand native_country_trinadad.tobago
## 1 -0.01932740 0.004674978 0.021285690
## 2 -0.03489522 0.001412851 0.004259537
## 3 -0.02474613 -0.006700918 -0.019065743
## 4 0.02114283 0.013244033 0.037123648
## 5 0.02959117 0.001986205 -0.016772163
## 6 0.08269283 0.003577598 -0.011772464
## 7 -0.02985499 -0.004513865 0.005328769
## native_country_united_states native_country_vietnam native_country_yugoslavia
## 1 0.09238820 0.0002202758 -0.0033054533
## 2 -2.82319400 0.0520271920 0.0074709787
## 3 0.11344850 0.0259885147 -0.0016392412
## 4 0.01729990 -0.0314725956 -0.0009012734
## 5 0.10719426 -0.0226840523 -0.0225576941
## 6 0.04038956 -0.0326213854 -0.0030821307
## 7 0.12516120 0.0104740572 0.0163991348
##
## Clustering vector:
## [1] 3 7 6 7 3 6 1 7 7 6 1 7 6 3 4 1 6 1 5 6 7 3 3 7 5 6 1 5 7 1 6 2 3 3 6 6
## [37] 7 7 3 4 2 7 1 5 3 3 1 3 2 1 7 6 6 7 5 3 5 1 1 6 3 3 7 6 7 2 1 2 7 7 7 1
## [73] 7 5 3 7 1 1 7 3 5 3 4 5 5 3 5 7 7 1 1 3 1 1 1 7 3 3 1 1 7 7 7 3 7 1 1 5
## [109] 5 1 3 1 1 1 4 3 7 6 7 1 6 7 5 7 7 4 7 6 1 7 6 5 1 1 7 6 3 3 4 7 3 6 1 3
## [145] 5 1 5 7 7 7 3 6 7 6 7 3 5 7 5 5 7 5 7 7 7 7 7 5 6 5 1 3 3 1 6 5 1 1 1 3
## [181] 4 1 7 6 6 5 5 6 6 7 7 4 7 3 1 5 6 1 7 3 2 3 5 6 1 3 7 7 7 6 7 3 6 5 7 7
## [217] 1 7 1 1 7 1 7 1 7 1 7 1 7 7 3 3 6 3 3 5 1 3 5 4 5 5 1 5 7 7 4 7 7 4 5 1
## [253] 3 3 6 3 7 7 1 1 6 1 3 7 7 3 7 7 7 1 7 3 2 3 5 7 7 3 1 1 3 1 3 6 7 6 3 7
## [289] 5 4 3 1 6 7 3 6 7 2 3 6 5 3 7 3 7 1 6 1 2 1 7 5 7 5 2 6 7 5 1 7 1 3 3 1
## [325] 4 7 6 5 6 5 1 7 3 3 5 7 6 7 7 4 1 7 2 7 6 3 7 4 1 3 7 5 4 1 7 7 1 1 4 7
## [361] 5 3 5 1 3 6 6 6 5 1 1 7 3 1 3 1 4 1 4 3 2 7 7 3 6 5 1 3 5 1 3 1 1 1 7 3
## [397] 4 1 6 6 7 6 7 6 1 7 1 6 7 5 3 3 7 7 6 3 5 3 3 6 1 7 4 7 6 7 7 3 6 7 7 3
## [433] 3 5 3 5 2 3 4 1 4 3 3 5 1 3 3 3 5 7 3 3 7 7 7 7 3 3 2 5 7 2 7 1 7 1 6 1
## [469] 7 3 1 3 1 3 1 6 2 5 7 6 3 6 1 6 7 5 6 5 5 1 7 7 4 3 7 7 3 7 6 5 1 6 6 6
## [505] 7 7 7 3 3 6 3 7 6 6 6 6 1 1 1 6 3 3 7 5 6 1 6 5 7 1 6 4 7 7 6 7 7 4 5 1
## [541] 3 3 7 6 1 3 2 7 1 1 6 1 1 6 6 3 4 3 5 6 1 1 7 3 5 7 6 3 7 2 3 7 2 6 7 1
## [577] 3 3 7 6 6 7 7 3 1 5 1 3 6 2 5 2 4 6 3 1 1 1 7 1 4 7 6 1 6 3 3 4 7 3 1 7
## [613] 1 3 6 1 1 1 6 1 1 3 2 1 6 5 3 7 7 7 3 6 1 7 7 2 6 7 1 1 3 6 6 3 6 5 1 1
## [649] 4 3 6 6 7 5 7 3 7 3 7 6 6 5 6 5 7 1 5 6 1 5 7 7 6 3 3 7 6 7 3 1 4 7 7 7
## [685] 5 1 7 5 4 7 4 6 3 1 5 7 5 5 7 5 5 4 5 3 1 1 1 6 1 1 3 1 7 6 6 3 5 2 1 7
## [721] 7 7 6 7 7 5 3 1 1 3 7 7 6 5 7 1 6 3 5 7 7 7 5 7 6 1 7 5 3 3 7 7 6 1 2 5
## [757] 5 1 3 5 7 1 1 1 7 6 4 3 7 5 7 3 7 7 7 7 7 3 3 1 1 6 6 6 3 5 6 3 7 3 6 7
## [793] 5 6 7 1 7 1 7 5 5 6 7 6 3 4 3 7 3 7 5 3 6 6 5 6 1 7 6 7 7 5 6 2 5 7 1 7
## [829] 3 5 7 3 6 7 7 1 3 3 6 6 7 7 7 3 1 7 6 3 7 7 3 6 6 6 7 3 7 3 7 7 1 6 7 7
## [865] 6 1 6 1 3 7 7 1 6 3 6 7 3 3 5 3 1 4 3 7 1 7 7 4 7 2 2 3 6 3 1 7 4 1 3 6
## [901] 7 7 7 5 5 7 3 6 6 5 7 6 3 7 5 1 3 4 6 6 6 1 1 1 3 6 5 4 3 7 7 5 3 3 6 3
## [937] 2 6 6 7 7 4 1 7 5 3 6 1 4 5 1 7 7 7 1 3 7 1 5 6 5 1 1 6 7 5 5 4 7 6 3 7
## [973] 6 3 6 3 2 5 2 1 3 1 7 1 6 6 2 3 2 1 6 6 1 7 6 1 3 3 7 4 3 1 6 7 1 6 6 7
## [1009] 3 7 2 3 7 6 5 3 1 6 7 2 1 7 1 3 5 7 6 7 6 7 1 7 3 5 7 7 6 6 7 7 7 3 1 5
## [1045] 6 7 2 1 3 7 5 3 1 6 7 5 3 1 1 3 7 1 1 7 7 4 6 5 6 6 3 5 7 3 1 1 5 7 6 3
## [1081] 5 7 1 1 1 1 3 5 5 5 4 5 1 2 7 7 3 6 1 3 3 5 3 7 5 5 1 1 6 3 7 7 5 1 3 3
## [1117] 6 5 3 6 5 7 7 7 1 3 7 6 3 1 3 6 6 7 7 1 7 3 6 3 2 5 7 7 6 5 1 1 3 6 1 5
## [1153] 3 6 4 1 7 4 3 5 7 6 7 5 7 1 3 4 1 7 3 7 3 1 6 3 3 7 7 5 2 5 7 2 1 1 7 1
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## [41725] 7 7 3 7 7 7 3 5 1 6 2 7 1 6 1 1 1 3 3 1 1 3 1 1 6 1 5 3 3 5 4 1 5 4 3 6
## [41761] 6 3 5 7 6 3 3 5 2 3 6 1 3 7 1 7 7 3 7 7 7 5 1 6 1 1 6 7 6 7 2 2 7 7 3 1
## [41797] 2 1 7 3 7 1 3 6 6 3 5 7 1 3 7 3 3 1 6 1 6 6 3 6 5 7 2 5 1 5 1 7 6 3 5 7
## [41833] 7 3 6 3 5 7 1 2 4 5 7 1 7 6 5 7 6 4 2 6 7 1 3 3 3 7 5 1 7 7 3 2 3 7 1 3
## [41869] 5 6 3 7 3 6 7 3 1 7 4 3 1 7 6 3 3 6 1 6 5 7 1 1 4 6 7 3 1 1 2 5 3 7 1 1
## [41905] 5 5 1 7 4 2 3 1 3 1 3 5 4 6 6 7 7 7 3 7 7 6 1 6 6 1 1 3 3 7 4 6 7 7 1 3
## [41941] 6 7 6 6 4 5 5 6 1 6 3 1 3 3 2 6 3 2 3 1 7 3 1 7 1 7 6 3 6 1 3 3 7 7 5 7
## [41977] 3 2 7 1 6 7 5 7 3 3 5 7 7 5 6 6 1 7 7 7 5 7 1 3 1 5 1 7 7 3 2 1 6 7 5 5
## [42013] 1 3 7 6 3 5 4 7 7 1 3 7 7 5 1 7 7 3 7 3 1 3 3 5 7 7 6 6 5 1 3 7 3 6 4 7
## [42049] 7 7 3 1 7 7 3 6 6 2 7 3 6 5 6 7 1 7 1 6 4 4 1 7 1 7 1 5 3 5 5 6 5 7 7 7
## [42085] 7 4 7 1 3 4 1 1 7 7 1 6 5 7 3 3 7 7 1 1 6 7 7 3 1 1 1 5 5 7 1 7 7 4 1 1
## [42121] 7 7 3 3 3 7 3 3 1 5 6 7 1 5 4 3 1 3 3 2 1 3 3 3 1 5 1 6 7 5 7 1 7 1 6 6
## [42157] 6 3 3 7 3 6 6 6 3 7 5 7 7 7 3 5 1 7 7 7 5 3 7 7 7 6 6 7 6 7 6 5 2 7 3 4
## [42193] 6 6 5 7 3 1 7 1 4 3 2 6 7 5 1 3 3 3 7 3 3 5 6 7 7 5 5 5 7 3 3 7 3 5 5 3
## [42229] 1 6 1 1 7 7 5 7 1 2 3 6 6 1 1 3 7 3 1 7 5 3 2 2 5 1 3 1 3 7 7 7 3 1 7 6
## [42265] 4 5 1 7 1 1 7 6 3 3 1 6 7 1 5 2 1 5 3 3 2 7 3 2 7 3 6 7 3 1 3 1 3 1 7 7
## [42301] 7 1 1 3 1 1 7 1 1 3 7 1 6 1 5 7 2 1 7 7 3 2 1 7 3 3 7 1 6 7 6 6 3 6 7 1
## [42337] 5 6 7 3 1 1 2 3 3 1 7 3 1 6 1 4 3 7 1 2 7 7 6 7 7 5 6 7 6 5 3 7 6 7 1 3
## [42373] 3 7 7 4 7 3 1 1 6 3 3 5 3 7 4 5 5 3 5 5 1 3 1 5 1 7 5 1 6 6 5 3 1 3 7 3
## [42409] 1 7 3 7 3 1 7 3 7 1 7 6 7 2 3 1 4 1 3 3 5 6 7 5 1 7 7 4 3 3 5 1 1 6 7 5
## [42445] 7 1 7 6 6 6 7 3 5 4 6 7 7 5 7 1 3 7 3 3 3 7 3 1 6 5 6 3 6 3 3 1 7 1 6 6
## [42481] 1 1 3 5 6 2 5 1 5 3 1 3 3 3 3 6 3 7 4 6 1 7 1 5 5 5 7 1 6 7 3 6 5 6 1 6
## [42517] 3 1 5 3 6 1 3 1 6 6 5 6 3 7 5 5 7 7 4 5 7 1 6 3 3 1 1 7 6 4 7 5 7 5 4 6
## [42553] 5 7 1 2 7 3 1 5 7 1 6 1 5 3 5 1 6 7 3 2 5 5 5 1 5 5 1 5 5 6 5 6 7 6 4 3
## [42589] 3 1 3 7 2 7 4 4 1 1 3 7 6 6 6 2 3 7 5 1 3 7 6 7 1 1 2 6 1 3 3 5 7 6 6 1
## [42625] 6 1 2 5 7 1 5 7 6 1 6 1 1 7 7 6 1 3 6 1 7 1 2 5 6 6 7 6 7 6 7 6 7 7 4 1
## [42661] 5 4 7 1 7 7 5 5 1 7 1 1 2 5 7 5 1 7 5 7 6 6 6 4 3 7 5 7 7 3 7 1 6 5 1 4
## [42697] 3 7 4 5 7 5 7 7 7 1 7 5 3 7 6 6 7 1 1 7 1 7 7 7 7 7 6 2 1 2 1 7 3 6 7 5
## [42733] 5 5 7 6 1 4 3 1 1 1 7 6 6 1 2 7 6 5 7 7 4 7 3 7 3 4 7 4 5 3 1 3 1 3 7 7
## [42769] 6 7 5 2 1 3 6 7 6 5 4 1 1 1 3 5 3 6 7 3 1 4 4 7 7 7 2 6 6 7 1 7 6 6 1 4
## [42805] 3 5 7 1 7 6 5 7 7 1 2 5 6 3 1 6 1 6 7 6 6 7 5 1 7 4 1 7 3 3 1 1 6 2 1 5
## [42841] 6 4 6 5 7 3 1 7 5 3 6 3 5 6 1 1 1 7 1 5 7 7 3 3 5 3 1 1 3 7 6 4 1 5 6 6
## [42877] 6 7 1 5 1 1 1 7 6 7 1 7 3 1 7 1 7 1 6 3 6 6 1 6 7 3 3 5 3 5 6 7 6 1 3 5
## [42913] 3 1 1 1 1 6 1 7 6 3 1 7 1 2 5 5 5 1 1 6 3 1 7 5 6 1 6 3 7 1 3 5 1 5 3 5
## [42949] 3 6 7 3 7 7 5 7 7 1 6 7 3 6 6 6 3 3 5 6 7 3 2 4 1 2 6 7 1 1 1 7 3 5 1 5
## [42985] 6 4 7 1 1 3 3 1 3 1 7 3 1 5 7 6 1 7 3 1 1 6 1 6 6 4 7 7 4 5 3 2 3 5 7 1
## [43021] 7 1 7 1 7 4 5 1 5 1 7 3 7 1 6 1 1 7 7 1 6 1 3 7 6 1 7 3 1 5 1 3 4 7 3 7
## [43057] 7 6 7 7 7 5 5 1 6 6 6 6 1 1 5 3 3 3 1 1 1 1 6 7 6 1 6 5 3 1 6 1 1 7 3 6
## [43093] 1 4 7 6 6 1 1 7 6 5 1 5 3 7 7 5 3 3 7 7 3 6 3 7 7 5 2 7 3 1 5 7 5 5 7 6
## [43129] 2 7 7 1 1 7 3 7 5 7 3 4 3 3 3 3 5 3 3 3 1 3 3 1 7 1 6 4 1 5 7 3 7 5 4 1
## [43165] 3 2 7 1 3 7 1 3 3 3 3 7 6 6 7 1 6 3 7 5 4 6 7 7 2 7 7 7 7 1 3 5 7 3 3 2
## [43201] 5 7 5 7 7 3 3 7 3 7 7 5 6 1 3 5 1 3 3 7 6 1 7 3 5 2 7 6 3 7 7 7 7 7 7 7
## [43237] 4 4 5 4 1 7 1 1 7 7 7 6 1 3 3 7 1 6 7 5 1 7 3 6 1 6 6 6 5 7 7 5 1 5 2 7
## [43273] 1 5 7 3 3 3 3 1 7 1 7 3 3 3 7 5 7 2 1 3 7 6 3 5 7 7 7 6 5 7 7 1 7 7 6 5
## [43309] 3 1 7 7 2 6 6 5 7 5 3 5 3 1 3 1 6 7 1 2 3 3 2 7 3 7 2 7 3 1 7 2 1 3 1 6
## [43345] 3 1 6 4 1 6 7 3 1 4 7 6 7 4 5 1 5 7 3 6 6 7 3 5 3 7 7 3 3 6 7 1 3 5 5 7
## [43381] 7 5 1 7 6 5 7 6 7 5 3 6 3 6 5 3 1 6 1 7 5 3 1 1 6 3 7 1 3 7 4 2 5 7 5 1
## [43417] 5 7 7 1 4 5 3 1 7 1 3 7 5 6 7 7 1 6 6 1 3 1 1 7 6 1 5 3 4 1 5 6 7 1 7 7
## [43453] 6 3 3 3 5 7 6 5 6 7 1 4 6 5 7 7 5 6 5 6 7 7 5 3 1 6 6 7 7 6 3 6 5 6 3 6
## [43489] 7 5 3 3 1 5 4 7 6 7 1 1 1 6 6 3 1 7 7 3 3 5 7 1 7 3 3 1 3 3 1 5 1 6 3 5
## [43525] 1 3 3 2 6 7 3 7 5 7 4 7 1 3 7 4 6 7 6 5 4 7 3 1 1 7 1 4 6 3 7 7 7 5 1 3
## [43561] 6 7 3 5 7 1 4 1 7 5 6 2 1 1 1 3 7 1 7 1 3 4 7 3 1 6 6 3 3 3 6 1 3 7 5 1
## [43597] 1 5 1 7 7 7 2 3 3 7 1 1 3 1 1 5 5 5 3 5 1 3 6 5 5 3 3 5 3 4 5 4 3 7 5 5
## [43633] 5 6 7 2 1 4 1 1 6 7 3 5 3 7 6 7 7 7 2 2 6 1 7 3 6 1 6 6 3 1 5 6 6 5 1 7
## [43669] 3 3 3 3 5 3 7 3 7 1 5 7 7 4 7 3 6 1 1 3 3 3 1 3 5 5 5 5 3 1 7 1 1 1 7 1
## [43705] 4 6 3 3 1 7 6 7 2 7 7 2 5 7 3 7 1 7 6 5 3 7 4 5 7 1 7 7 1 1 3 6 1 7 1 1
## [43741] 6 7 3 6 5 1 6 1 1 7 7 3 1 7 1 7 3 7 7 7 6 1 6 1 6 1 7 6 6 5 7 7 5 4 1 3
## [43777] 5 1 3 2 6 1 4 6 5 7 4 5 7 1 3 7 7 7 7 4 6 3 3 6 7 5 7 6 6 3 3 7 3 1 3 7
## [43813] 7 7 3 6 3 7 5 7 5 7 1 1 1 3 7 6 2 3 6 6 5 3 5 1 5 5 6 1 5 1 3 7 4 3 3 6
## [43849] 5 6 3 3 5 5 7 1 6 6 6 3 3 6 2 6 4 6 7 7 1 2 7 3 7 6 6 6 7 3 7 7 2 1 7 7
## [43885] 5 7 1 4 3 3 4 7 3 1 7 3 6 3 4 1 7 7 1 1 6 3 1 3 1 5 4 5 1 6 3 2 6 3 6 7
## [43921] 5 5 6 6 5 7 5 3 1 7 3 5 1 7 7 6 6 7 7 4 7 7 4 5 7 7 5 3 5 3 3 1 3 5 3 7
## [43957] 3 3 6 5 3 6 6 3 5 7 7 3 5 7 6 5 7 7 1 6 7 6 6 6 3 3 7 6 3 1 1 6 7 7 7 7
## [43993] 1 5 5 3 7 7 7 5 7 7 6 7 3 3 1 1 1 7 7 2 6 7 6 3 1 7 4 5 7 6 5 7 7 2 1 5
## [44029] 3 3 5 3 3 6 1 5 1 7 7 6 7 7 3 7 6 7 7 1 5 3 6 7 6 5 4 4 3 7 1 4 3 1 3 1
## [44065] 3 7 6 3 7 1 5 1 5 3 6 7 1 1 5 1 1 1 5 7 7 3 5 1 4 6 7 5 3 6 4 6 1 4 7 5
## [44101] 3 7 1 6 3 3 3 7 5 7 3 5 3 6 7 7 6 5 6 7 4 3 5 7 6 3 7 7 2 6 1 7 3 5 2 1
## [44137] 3 7 6 3 1 7 1 3 4 6 7 7 3 1 7 1 7 6 3 6 4 5 3 7 6 6 5 1 7 3 7 1 3 1 7 1
## [44173] 6 6 1 6 1 3 4 6 7 1 2 3 7 7 3 6 6 1 7 1 3 1 3 4 7 7 7 1 7 6 7 6 4 4 7 4
## [44209] 6 6 2 1 3 1 7 7 7 7 1 2 4 7 1 2 3 6 7 7 2 4 3 1 7 6 7 3 1 5 4 3 7 4 1 6
## [44245] 5 3 1 5 6 7 7 3 1 1 7 7 3 7 1 6 5 2 6 6 1 5 5 7 6 5 2 6 7 1 4 7 6 1 3 7
## [44281] 1 7 4 3 7 7 7 1 6 1 1 7 6 1 5 7 3 7 3 2 6 1 1 6 1 6 5 7 1 6 3 7 1 6 3 1
## [44317] 5 1 1 1 7 1 1 5 6 1 5 7 5 3 1 1 1 4 5 7 1 6 7 3 6 5 1 4 5 3 3 3 7 3 5 4
## [44353] 6 7 6 7 7 6 6 2 3 7 7 7 7 5 5 1 5 3 6 7 7 6 7 3 4 1 7 3 7 6 2 6 7 1 7 3
## [44389] 5 5 7 5 6 6 5 7 1 3 7 1 1 3 1 5 7 6 4 2 6 6 1 3 7 1 3 7 7 7 1 6 7 1 3 6
## [44425] 1 6 3 1 7 7 3 7 7 1 7 7 7 7 7 3 1 7 4 6 1 3 3 3 3 1 1 7 1 1 1 4 6 3 6 7
## [44461] 6 6 7 5 5 5 3 7 6 6 6 5 5 7 5 7 1 6 4 5 3 1 1 3 3 6 2 7 6 1 3 5 1 1 3 4
## [44497] 2 1 1 5 1 7 5 3 7 5 3 7 1 6 4 6 6 6 7 2 6 2 6 6 7 3 3 1 6 1 6 3 4 6 3 6
## [44533] 6 3 5 6 7 1 4 3 7 6 7 6 3 6 7 7 4 1 5 3 2 3 7 1 7 6 5 7 5 3 7 3 1 5 1 3
## [44569] 1 7 5 7 7 3 1 7 6 1 2 3 3 5 6 7 6 2 1 1 7 6 7 3 5 4 5 1 6 4 7 3 7 5 7 1
## [44605] 5 7 7 3 1 6 7 3 5 7 6 3 3 4 6 7 5 7 5 3 3 3 3 2 1 4 1 5 4 6 3 7 7 7 1 5
## [44641] 3 1 7 1 6 3 6 7 7 1 5 2 7 6 1 1 6 6 7 7 4 1 5 1 4 3 1 6 3 3 7 1 1 6 3 7
## [44677] 1 6 5 6 1 6 3 3 7 7 7 1 3 6 7 2 5 6 3 7 3 6 5 7 3 7 7 7 1 3 6 7 6 3 7 7
## [44713] 1 6 7 7 6 7 5 5 5 1 5 1 7 1 3 7 7 3 5 6 6 1 2 4 7 7 2 3 6 6 6 3 1 5 2 1
## [44749] 6 3 3 6 7 1 6 7 5 3 1 6 1 7 7 6 5 4 7 5 7 3 7 5 1 4 7 7 5 6 5 7 3 7 7 3
## [44785] 7 5 6 7 5 3 3 7 1 3 5 5 7 5 3 1 3 6 3 7 6 7 6 4 7 7 1 7 3 3 3 5 3 7 5 1
## [44821] 1 7 6 6 3 7 1 5 5 5 3 6 1 1 1 3 4 7 3 5 6 1 6 1 3 2 7 3 6 6 3 5 1 7 6 6
## [44857] 5 1 5 5 1 3 3 4 6 5 5 7 7 5 7 1 6 7 6 3 6 5 1 5 5 7 5 3 1 7 7 4 6 7 6 3
## [44893] 6 5 1 3 3 3 6 1 1 2 3 7 1 7 3 1 7 6 1 2 7 1 1 6 1 6 7 1 7 3 7 3 6 1 7 5
## [44929] 3 3 1 3 7 5 5 6 6 7 4 3 1 5 7 6 1 7 3 4 2 7 6 6 6 7 1 7 6 7 3 7 3 1 1 3
## [44965] 1 6 3 7 5 5 1 6 3 3 1 7 1 6 7 1 4 1 6 6 7 7 3 7 2 7 7 5 2 3 7 3 7 1 2 6
## [45001] 1 7 3 1 6 5 3 6 7 1 2 6 2 1 5 1 7 5 5 7 5 3 7 4 6 1 7 3 7 3 1 6 3 1 7 2
## [45037] 7 5 7 7 7 6 1 7 3 3 5 6 7 5 1 7 1 7 2 3 5 7 1 5 7 7 4 1 1 7 7 3 3 5 3 3
## [45073] 1 1 3 6 6 1 7 2 1 7 1 3 5 7 1 3 3 1 7 7 1 5 6 5 4 1 1 6 2 1 4 5 7 1 2 6
## [45109] 1 7 6 3 6 5 3 7 3 1 7 1 1 3 7 7 1 7 7 1 6 3 6 6 6 1 5 7 7 6 1 6 7 3 5 3
## [45145] 6 5 7 3 7 3 4 6 1 7 1 5 4 1 7 3 7 1 1 3 3 1 7 3 3 7 6 7 5 3 7 6 2 1 1 3
## [45181] 4 7 7 1 7 7 7 1 7 7 3 1 7 3 4 1 6 6 1 3 5 3 5 1 5 1 4 5 2 5 1 7 7 7 5 6
## [45217] 3 4 7 1 3 4
##
## Within cluster sum of squares by cluster:
## [1] 891850.9 458095.2 762949.4 219191.0 512456.9 609274.8 851664.8
## (between_SS / total_SS = 14.7 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"# Adding cluster to the data set
results_cluster <- augment(final_kmeans, income_features)
results_cluster %>% group_by(.cluster) %>% count()## # A tibble: 7 × 2
## # Groups: .cluster [7]
## .cluster n
## <fct> <int>
## 1 1 9215
## 2 2 1477
## 3 3 8481
## 4 4 2048
## 5 5 5784
## 6 6 6832
## 7 7 11385From the output, we see that three clusters have been found. For each cluster, the squared distances between the observations to the centroids are calculated. So, each observation will be assigned to one of the five clusters.
Bivariate chart(s) against meaningful variables and/or analysis of density plots
Now, I will visualize the scatter plot between husbands and fnlwgt and color the points based on the cluster id:
clust_spc_plot <- results_cluster %>%
ggplot(mapping = aes(x = low_ed_male_laborer, y = husbands)) +
geom_point(aes(shape = .cluster, color= .cluster),size = 2,alpha=0.3)+
scale_color_manual(values = c("darkslateblue","goldenrod","deeppink", "green", "red", "yellow", "skyblue"))+ theme_minimal()
clust_spc_plot## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 7. Consider
## specifying shapes manually if you must have them.## Warning: Removed 11385 rows containing missing values (`geom_point()`).
clust_spc_plot2 <- results_cluster %>%
ggplot(mapping = aes(x = age, y = education_num)) +
geom_point(aes(shape = .cluster, color= .cluster),size = 2,alpha=0.3)+
scale_color_manual(values = c("darkslateblue","goldenrod","deeppink", "green", "red", "yellow", "skyblue"))+ theme_minimal()
clust_spc_plot2## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 7. Consider
## specifying shapes manually if you must have them.## Warning: Removed 11385 rows containing missing values (`geom_point()`).
clust_spc_plot3 <- results_cluster %>%
ggplot(mapping = aes(x = education_num, y = fnlwgt)) +
geom_point(aes(shape = .cluster, color= .cluster),size = 2,alpha=0.3)+
scale_color_manual(values = c("darkslateblue","goldenrod","deeppink", "green", "red", "yellow", "skyblue"))+ theme_minimal()
clust_spc_plot3## Warning: The shape palette can deal with a maximum of 6 discrete values because
## more than 6 becomes difficult to discriminate; you have 7. Consider
## specifying shapes manually if you must have them.## Warning: Removed 11385 rows containing missing values (`geom_point()`).
Looking at clusters in terms of PC1 and PC2, we see good distinct among the clusters.
results_cluster %>%
pivot_longer(c(husbands, low_ed_male_laborer),names_to = "feature") %>%
ggplot(aes(value, fill=.cluster))+
geom_density(alpha=0.3)+
facet_wrap(~feature)
results_cluster %>%
pivot_longer(c(age, hours_per_week),names_to = "feature") %>%
ggplot(aes(value, fill=.cluster))+
geom_density(alpha=0.3)+
facet_wrap(~feature)
Meaningful interpretation / discussion of conclusions
We think that k-means is not suitable for this dataset. Some of these plots do not show clear definition of clusters and the plot clust_spc_plot shows non-spherical shape, the algorithm suggests that each point is close to each other (A to B and B to C, and so on). Analyzing the ratio between_SS / total_SS, we see that the differences between clusters explain 14.7% of the total variation in the dataset.
A negative value of between_SS is mathematically possible, but it can occur only in certain circumstances. Specifically, negative values of between_SS can occur when the clusters are too small and the variance within the clusters is greater than the variance between the clusters. In such cases, the total sum of squares can be smaller than the sum of squares due to the between-cluster differences, resulting in a negative value for between_SS.
However, negative values of between_SS are generally uncommon and may indicate issues with the cluster analysis, such as inappropriate choice of clustering algorithm or incorrect data preprocessing. It’s important to investigate the reasons behind the negative value and to ensure that the results of the analysis are valid and reliable.
We suggest using an algorithm that handle non-spherical shaped data as well as other forms, such as Gaussian Mixtures Models [https://jakevdp.github.io/PythonDataScienceHandbook/05.12-gaussian-mixtures.html]
Nevertheless, we will add our cluster column to our dataset to see if it helps our predictive modeling.
# adding PC1 and PC2 to ds
income = bind_cols(prc %>% select(2:3), income) %>% relocate(income_above_50k)
# adding clusters
income = bind_cols(income, results_cluster[107])
income = income %>% rename("cluster" = .cluster)PART III SUPERVISED LEARNING
Logistic Regression Model to examine log-odds of each feature
income = income %>% mutate(income_above_50k = factor(ifelse(income_above_50k == 'TRUE','yes','no'), levels = c('yes','no')))
income_index <- createDataPartition(income$income_above_50k, p = 0.80, list = FALSE)
train <- income[income_index, ]
test <- income[-income_index, ]
control <- trainControl(method = "cv", number = 5)
fit.lr <- train(income_above_50k ~ .,
data = train,
trControl = control,
method = "glm",
family = "binomial")## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading
## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading
## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading
## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading
## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleadingodds_ratio = exp(coef(fit.lr$finalModel))
data.frame(name = names(odds_ratio), odds_ratio = odds_ratio) %>%
arrange(desc(odds_ratio))## name
## relationship_husband relationship_husband
## education_7th_8th education_7th_8th
## education_9th education_9th
## marital_status_married_civ_spouse marital_status_married_civ_spouse
## education_1st_4th education_1st_4th
## education_10th education_10th
## education_5th_6th education_5th_6th
## native_country_laos native_country_laos
## education_preschool education_preschool
## relationship_other_relative relationship_other_relative
## native_country_greece native_country_greece
## native_country_columbia native_country_columbia
## native_country_taiwan native_country_taiwan
## native_country_south native_country_south
## occupation_farming_fishing occupation_farming_fishing
## native_country_thailand native_country_thailand
## workclass_self_emp_not_inc workclass_self_emp_not_inc
## education_11th education_11th
## relationship_own_child relationship_own_child
## native_country_india native_country_india
## education_hs_grad education_hs_grad
## native_country_china native_country_china
## native_country_iran native_country_iran
## native_country_scotland native_country_scotland
## cluster2 cluster2
## workclass_self_emp_inc workclass_self_emp_inc
## relationship_not_in_family relationship_not_in_family
## cluster4 cluster4
## low_ed_male_laborer low_ed_male_laborer
## education_12th education_12th
## education_assoc_voc education_assoc_voc
## education_num education_num
## marital_status_divorced marital_status_divorced
## relationship_unmarried relationship_unmarried
## hours_per_week hours_per_week
## age age
## education_assoc_acdm education_assoc_acdm
## native_country_japan native_country_japan
## fnlwgt fnlwgt
## education_bachelors education_bachelors
## l_capital_loss l_capital_loss
## education_masters education_masters
## marital_status_separated marital_status_separated
## l_capital_gain l_capital_gain
## native_country_hong native_country_hong
## marital_status_married_spouse_absent marital_status_married_spouse_absent
## occupation_craft_repair occupation_craft_repair
## marital_status_never_married marital_status_never_married
## `\\`native_country_trinadad&tobago\\`` `\\`native_country_trinadad&tobago\\``
## education_doctorate education_doctorate
## native_country_ecuador native_country_ecuador
## cluster3 cluster3
## native_country_peru native_country_peru
## native_country_cuba native_country_cuba
## occupation_prof_specialty occupation_prof_specialty
## workclass_state_gov workclass_state_gov
## native_country_hungary native_country_hungary
## cluster5 cluster5
## cluster6 cluster6
## native_country_germany native_country_germany
## workclass_local_gov workclass_local_gov
## cluster7 cluster7
## native_country_canada native_country_canada
## native_country_poland native_country_poland
## occupation_protective_serv occupation_protective_serv
## native_country_portugal native_country_portugal
## occupation_exec_managerial occupation_exec_managerial
## native_country_italy native_country_italy
## occupation_armed_forces occupation_armed_forces
## native_country_england native_country_england
## occupation_handlers_cleaners occupation_handlers_cleaners
## occupation_machine_op_inspct occupation_machine_op_inspct
## native_country_philippines native_country_philippines
## native_country_france native_country_france
## native_country_mexico native_country_mexico
## workclass_federal_gov workclass_federal_gov
## native_country_puerto_rico native_country_puerto_rico
## occupation_sales occupation_sales
## native_country_nicaragua native_country_nicaragua
## occupation_other_service occupation_other_service
## native_country_cambodia native_country_cambodia
## `\\`native_country_outlying_us(guam_usvi_etc)\\`` `\\`native_country_outlying_us(guam_usvi_etc)\\``
## race_amer_indian_eskimo race_amer_indian_eskimo
## occupation_tech_support occupation_tech_support
## native_country_ireland native_country_ireland
## native_country_dominican_republic native_country_dominican_republic
## race_asian_pac_islander race_asian_pac_islander
## native_country_honduras native_country_honduras
## (Intercept) (Intercept)
## native_country_el_salvador native_country_el_salvador
## occupation_priv_house_serv occupation_priv_house_serv
## occupation_adm_clerical occupation_adm_clerical
## marital_status_married_af_spouse marital_status_married_af_spouse
## native_country_haiti native_country_haiti
## race_black race_black
## workclass_private workclass_private
## race_other race_other
## native_country_jamaica native_country_jamaica
## native_country_guatemala native_country_guatemala
## husbands husbands
## sex_female sex_female
## workclass_without_pay workclass_without_pay
## education_prof_school education_prof_school
## education_some_college education_some_college
## marital_status_widowed marital_status_widowed
## occupation_transport_moving occupation_transport_moving
## relationship_wife relationship_wife
## race_white race_white
## sex_male sex_male
## native_country_holand_netherlands native_country_holand_netherlands
## native_country_united_states native_country_united_states
## native_country_vietnam native_country_vietnam
## native_country_yugoslavia native_country_yugoslavia
## odds_ratio
## relationship_husband 111.92485930
## education_7th_8th 15.89998293
## education_9th 6.08715806
## marital_status_married_civ_spouse 5.09319545
## education_1st_4th 4.76632739
## education_10th 3.95043675
## education_5th_6th 3.90842626
## native_country_laos 3.27887038
## education_preschool 2.93754411
## relationship_other_relative 2.81862172
## native_country_greece 2.81634597
## native_country_columbia 2.67660694
## native_country_taiwan 2.65177537
## native_country_south 2.61125693
## occupation_farming_fishing 2.54828776
## native_country_thailand 2.42030177
## workclass_self_emp_not_inc 2.26060578
## education_11th 2.19640010
## relationship_own_child 2.11881143
## native_country_india 2.05716960
## education_hs_grad 1.90169065
## native_country_china 1.89121141
## native_country_iran 1.80699331
## native_country_scotland 1.80575164
## cluster2 1.80115289
## workclass_self_emp_inc 1.75306434
## relationship_not_in_family 1.69047411
## cluster4 1.58461278
## low_ed_male_laborer 1.48220513
## education_12th 1.44999183
## education_assoc_voc 1.42040093
## education_num 1.38569228
## marital_status_divorced 1.34665460
## relationship_unmarried 1.33240597
## hours_per_week 1.02299576
## age 1.02169994
## education_assoc_acdm 1.00852201
## native_country_japan 1.00282314
## fnlwgt 0.99999822
## education_bachelors 0.99972625
## l_capital_loss 0.96896717
## education_masters 0.95751042
## marital_status_separated 0.93446943
## l_capital_gain 0.93071163
## native_country_hong 0.90162382
## marital_status_married_spouse_absent 0.89549989
## occupation_craft_repair 0.86951345
## marital_status_never_married 0.86334595
## `\\`native_country_trinadad&tobago\\`` 0.85760022
## education_doctorate 0.77384224
## native_country_ecuador 0.76741027
## cluster3 0.74081882
## native_country_peru 0.72954740
## native_country_cuba 0.72919511
## occupation_prof_specialty 0.72674162
## workclass_state_gov 0.69334869
## native_country_hungary 0.67490997
## cluster5 0.67271580
## cluster6 0.66906353
## native_country_germany 0.65404960
## workclass_local_gov 0.65307003
## cluster7 0.64216210
## native_country_canada 0.62870427
## native_country_poland 0.59660467
## occupation_protective_serv 0.58688656
## native_country_portugal 0.57641571
## occupation_exec_managerial 0.57066924
## native_country_italy 0.56061691
## occupation_armed_forces 0.53606846
## native_country_england 0.51176848
## occupation_handlers_cleaners 0.49219796
## occupation_machine_op_inspct 0.44070701
## native_country_philippines 0.39350929
## native_country_france 0.36362989
## native_country_mexico 0.35489051
## workclass_federal_gov 0.34820591
## native_country_puerto_rico 0.34819719
## occupation_sales 0.34391147
## native_country_nicaragua 0.32900078
## occupation_other_service 0.27262604
## native_country_cambodia 0.26764746
## `\\`native_country_outlying_us(guam_usvi_etc)\\`` 0.24681197
## race_amer_indian_eskimo 0.24554677
## occupation_tech_support 0.23725758
## native_country_ireland 0.20895086
## native_country_dominican_republic 0.20257562
## race_asian_pac_islander 0.18147064
## native_country_honduras 0.16717527
## (Intercept) 0.16385896
## native_country_el_salvador 0.15934610
## occupation_priv_house_serv 0.15233020
## occupation_adm_clerical 0.14381178
## marital_status_married_af_spouse 0.14294829
## native_country_haiti 0.13039464
## race_black 0.11704690
## workclass_private 0.11120250
## race_other 0.10601700
## native_country_jamaica 0.09828295
## native_country_guatemala 0.09742095
## husbands 0.04212538
## sex_female 0.02822186
## workclass_without_pay NA
## education_prof_school NA
## education_some_college NA
## marital_status_widowed NA
## occupation_transport_moving NA
## relationship_wife NA
## race_white NA
## sex_male NA
## native_country_holand_netherlands NA
## native_country_united_states NA
## native_country_vietnam NA
## native_country_yugoslavia NAconfusionMatrix(predict(fit.lr, test),factor(test$income_above_50k))## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1348 510
## no 893 6292
##
## Accuracy : 0.8449
## 95% CI : (0.8372, 0.8523)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.5585
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.6015
## Specificity : 0.9250
## Pos Pred Value : 0.7255
## Neg Pred Value : 0.8757
## Prevalence : 0.2478
## Detection Rate : 0.1491
## Detection Prevalence : 0.2055
## Balanced Accuracy : 0.7633
##
## 'Positive' Class : yes
## myRoc <- roc(test$income_above_50k, predict(fit.lr, test, type="prob")[,2])## Warning in predict.lm(object, newdata, se.fit, scale = 1, type = if (type == :
## prediction from a rank-deficient fit may be misleading## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc, main = 'AUC = .89')
auc(myRoc)## Area under the curve: 0.9002PC1, education, and marital status are meaningful variables in predicting income level.
Feature Engineering
age_dummies = income %>% mutate(age_bin = case_when(
age < 20 ~ "teen",
age >=20 & age <30 ~ "20-29",
age >=30 & age <40 ~ "30-39",
age >=40 & age <50 ~ "40-50",
age >=50 & age <66 ~ "50-65",
age >=65 ~ "65+")) %>% select(age_bin) %>% dummy_cols(remove_selected_columns = T)
#people who work over 40 hours a week will get overtime pay if an hourly worker.
income = bind_cols(income, age_dummies) %>% mutate(overtime = as.numeric(ifelse(hours_per_week > 40, 1, 0)))Gradient Boosted Model
#resplitting after addition of new features:
income_index <- createDataPartition(income$income_above_50k, p = 0.80, list = FALSE)
train <- income[income_index, ]
test <- income[-income_index, ]
ctrl <- trainControl(method = "cv", number = 3, classProbs=TRUE, summaryFunction = twoClassSummary)
fit.gbm <- train(income_above_50k ~ .,
data = train,
method = "gbm",
tuneLength = 4,
preProcess = c("center","scale"),
metric = "ROC",
trControl = ctrl)## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0802 nan 0.1000 0.0204
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0509 nan 0.1000 0.0335
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## 200 0.5624 nan 0.1000 -0.0000## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
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## 200 0.6240 nan 0.1000 0.0000## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0653 nan 0.1000 0.0268
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## 200 0.5921 nan 0.1000 0.0000## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
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## 200 0.5718 nan 0.1000 0.0000## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0527 nan 0.1000 0.0334
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0541 nan 0.1000 0.0329
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## 200 0.5684 nan 0.1000 -0.0000confusionMatrix(predict(fit.gbm, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1444 457
## no 797 6345
##
## Accuracy : 0.8613
## 95% CI : (0.854, 0.8684)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.6081
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.6444
## Specificity : 0.9328
## Pos Pred Value : 0.7596
## Neg Pred Value : 0.8884
## Prevalence : 0.2478
## Detection Rate : 0.1597
## Detection Prevalence : 0.2102
## Balanced Accuracy : 0.7886
##
## 'Positive' Class : yes
## myRoc <- roc(test$income_above_50k, predict(fit.gbm, test, type="prob")[,2])## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc)
auc(myRoc)## Area under the curve: 0.9159The results from the training above tells that we could further tune parameters for performance:
grid.gbm = expand.grid(interaction.depth = seq(4,8,1),
n.trees = seq(200,400,50),
shrinkage = 0.1,
n.minobsinnode = 10)
fit.gbm.2 <- train(income_above_50k ~ .,
data = train,
method = "gbm",
tuneGrid = grid.gbm,
preProcess = c("center","scale"),
metric = "ROC",
trControl = ctrl)## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0496 nan 0.1000 0.0352
## 2 0.9993 nan 0.1000 0.0245
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0439 nan 0.1000 0.0377
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0437 nan 0.1000 0.0384
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
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## 400 0.4904 nan 0.1000 -0.0001## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
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## 400 0.5322 nan 0.1000 -0.0000## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
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## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0465 nan 0.1000 0.0358
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## 400 0.5033 nan 0.1000 -0.0001## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0463 nan 0.1000 0.0364
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## 400 0.4930 nan 0.1000 -0.0001## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0432 nan 0.1000 0.0393
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##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.0447 nan 0.1000 0.0379
## 2 0.9866 nan 0.1000 0.0287
## 3 0.9420 nan 0.1000 0.0226
## 4 0.9029 nan 0.1000 0.0195
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## 160 0.5475 nan 0.1000 -0.0001
## 180 0.5426 nan 0.1000 -0.0000
## 200 0.5391 nan 0.1000 -0.0000confusionMatrix(predict(fit.gbm.2, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1463 469
## no 778 6333
##
## Accuracy : 0.8621
## 95% CI : (0.8548, 0.8691)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.6122
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.6528
## Specificity : 0.9310
## Pos Pred Value : 0.7572
## Neg Pred Value : 0.8906
## Prevalence : 0.2478
## Detection Rate : 0.1618
## Detection Prevalence : 0.2136
## Balanced Accuracy : 0.7919
##
## 'Positive' Class : yes
## #kappa - .60
myRoc <- roc(test$income_above_50k, predict(fit.gbm.2, test, type="prob")[,2])## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc, main = "AUC = .92")
auc(myRoc)## Area under the curve: 0.9183#AUC = .92
print(fit.gbm.2$bestTune)## n.trees interaction.depth shrinkage n.minobsinnode
## 21 200 8 0.1 10Downsampling training data to remove imbalance of income_above_50k:
income_index <- createDataPartition(income$income_above_50k, p = 0.80, list = FALSE)
train <- income[income_index, ]
test <- income[-income_index, ]
traindown = downSample(x = train[,-1], y= train$income_above_50k) %>% mutate(income_above_50k = Class) %>% select(-Class)
traindown %>% group_by(income_above_50k) %>% count()## # A tibble: 2 × 2
## # Groups: income_above_50k [2]
## income_above_50k n
## <fct> <int>
## 1 yes 8967
## 2 no 8967fit.gbm.3 <- train(income_above_50k ~ .,
data = traindown,
method = "gbm",
tuneGrid = fit.gbm.2$bestTune,
preProcess = c("center","scale"),
metric = "ROC",
trControl = ctrl)## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.2998 nan 0.1000 0.0423
## 2 1.2293 nan 0.1000 0.0348
## 3 1.1706 nan 0.1000 0.0284
## 4 1.1209 nan 0.1000 0.0244
## 5 1.0792 nan 0.1000 0.0206
## 6 1.0429 nan 0.1000 0.0177
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## 8 0.9838 nan 0.1000 0.0125
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## 20 0.8111 nan 0.1000 0.0035
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## 60 0.6980 nan 0.1000 0.0011
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## 160 0.6314 nan 0.1000 -0.0002
## 180 0.6219 nan 0.1000 -0.0002
## 200 0.6121 nan 0.1000 -0.0001## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlands## Warning in (function (x, y, offset = NULL, misc = NULL, distribution =
## "bernoulli", : variable 80: native_country_holand_netherlands has no variation.## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.3010 nan 0.1000 0.0427
## 2 1.2317 nan 0.1000 0.0346
## 3 1.1744 nan 0.1000 0.0284
## 4 1.1233 nan 0.1000 0.0257
## 5 1.0798 nan 0.1000 0.0214
## 6 1.0445 nan 0.1000 0.0171
## 7 1.0137 nan 0.1000 0.0148
## 8 0.9854 nan 0.1000 0.0137
## 9 0.9610 nan 0.1000 0.0116
## 10 0.9392 nan 0.1000 0.0105
## 20 0.8146 nan 0.1000 0.0042
## 40 0.7293 nan 0.1000 0.0011
## 60 0.6977 nan 0.1000 -0.0001
## 80 0.6766 nan 0.1000 0.0006
## 100 0.6614 nan 0.1000 -0.0002
## 120 0.6490 nan 0.1000 -0.0000
## 140 0.6373 nan 0.1000 -0.0003
## 160 0.6278 nan 0.1000 -0.0001
## 180 0.6174 nan 0.1000 -0.0002
## 200 0.6094 nan 0.1000 -0.0003
##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.3026 nan 0.1000 0.0422
## 2 1.2333 nan 0.1000 0.0338
## 3 1.1760 nan 0.1000 0.0286
## 4 1.1234 nan 0.1000 0.0260
## 5 1.0804 nan 0.1000 0.0211
## 6 1.0444 nan 0.1000 0.0174
## 7 1.0142 nan 0.1000 0.0146
## 8 0.9853 nan 0.1000 0.0137
## 9 0.9606 nan 0.1000 0.0117
## 10 0.9375 nan 0.1000 0.0113
## 20 0.8100 nan 0.1000 0.0033
## 40 0.7220 nan 0.1000 0.0008
## 60 0.6918 nan 0.1000 -0.0000
## 80 0.6722 nan 0.1000 0.0001
## 100 0.6581 nan 0.1000 -0.0000
## 120 0.6454 nan 0.1000 -0.0002
## 140 0.6336 nan 0.1000 -0.0001
## 160 0.6234 nan 0.1000 -0.0002
## 180 0.6139 nan 0.1000 -0.0002
## 200 0.6057 nan 0.1000 0.0000
##
## Iter TrainDeviance ValidDeviance StepSize Improve
## 1 1.2999 nan 0.1000 0.0428
## 2 1.2290 nan 0.1000 0.0348
## 3 1.1727 nan 0.1000 0.0281
## 4 1.1230 nan 0.1000 0.0245
## 5 1.0799 nan 0.1000 0.0209
## 6 1.0461 nan 0.1000 0.0168
## 7 1.0163 nan 0.1000 0.0148
## 8 0.9896 nan 0.1000 0.0132
## 9 0.9671 nan 0.1000 0.0109
## 10 0.9447 nan 0.1000 0.0109
## 20 0.8173 nan 0.1000 0.0044
## 40 0.7331 nan 0.1000 0.0007
## 60 0.7013 nan 0.1000 0.0000
## 80 0.6836 nan 0.1000 0.0001
## 100 0.6706 nan 0.1000 -0.0001
## 120 0.6606 nan 0.1000 -0.0001
## 140 0.6503 nan 0.1000 0.0000
## 160 0.6428 nan 0.1000 -0.0001
## 180 0.6349 nan 0.1000 0.0000
## 200 0.6287 nan 0.1000 -0.0001#checking performance of downsampled training on test data
confusionMatrix(predict(fit.gbm.3, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1962 1294
## no 279 5508
##
## Accuracy : 0.8261
## 95% CI : (0.8181, 0.8338)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.5949
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.8755
## Specificity : 0.8098
## Pos Pred Value : 0.6026
## Neg Pred Value : 0.9518
## Prevalence : 0.2478
## Detection Rate : 0.2170
## Detection Prevalence : 0.3601
## Balanced Accuracy : 0.8426
##
## 'Positive' Class : yes
## myRoc <- roc(test$income_above_50k, predict(fit.gbm.3, test, type="prob")[,2])## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc)
auc(myRoc)## Area under the curve: 0.923#AUV = 0.917Conclusion: the difference between the two training sets is negligible.
Given the relative success of a gradient boosted, let’s try XG Boost:
grid.xgb=expand.grid(nrounds=50, #number of trees in final model
eta=c(.1,.3,.7), # our shrinkage parameter
max_depth=seq(4,8,1),
gamma = 0,
min_child_weight = 1,
subsample = 1,
colsample_bytree = 1)
fit.xgb <- train(income_above_50k ~ .,
data = train,
method = "xgbTree",
tuneGrid = grid.xgb,
verbose=FALSE,
trControl = ctrl)## Warning in train.default(x, y, weights = w, ...): The metric "Accuracy" was not
## in the result set. ROC will be used instead.confusionMatrix(predict(fit.xgb, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1433 399
## no 808 6403
##
## Accuracy : 0.8665
## 95% CI : (0.8593, 0.8735)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.6186
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.6394
## Specificity : 0.9413
## Pos Pred Value : 0.7822
## Neg Pred Value : 0.8879
## Prevalence : 0.2478
## Detection Rate : 0.1585
## Detection Prevalence : 0.2026
## Balanced Accuracy : 0.7904
##
## 'Positive' Class : yes
## myRoc <- roc(test$income_above_50k, predict(fit.xgb, test, type="prob")[,2])## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc)
auc(myRoc) #0.926## Area under the curve: 0.9276fit.xgb$bestTune## nrounds max_depth eta gamma colsample_bytree min_child_weight subsample
## 7 50 5 0.3 0 1 1 1Upsampling training data to equalizes classes:
Note that we are not touching the test data!
set.seed(1001)
income_index <- createDataPartition(income$income_above_50k, p = 0.80, list = FALSE)
train <- income[income_index, ]
test <- income[-income_index, ]
train_up = upSample(x = train[,-1], y= train$income_above_50k) %>% mutate(income_above_50k = Class) %>% select(-Class)
fit.xgb.tu.2 <- train(income_above_50k ~ .,
data = train_up,
method = "xgbTree",
tuneGrid = fit.xgb$bestTune,
preProcess = c("center","scale"),
metric = "ROC",
verbose=FALSE,
trControl = ctrl)## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: native_country_holand_netherlandsconfusionMatrix(predict(fit.xgb.tu.2, test),factor(test$income_above_50k))## Confusion Matrix and Statistics
##
## Reference
## Prediction yes no
## yes 1936 1215
## no 305 5587
##
## Accuracy : 0.8319
## 95% CI : (0.824, 0.8396)
## No Information Rate : 0.7522
## P-Value [Acc > NIR] : < 0.00000000000000022
##
## Kappa : 0.6032
##
## Mcnemar's Test P-Value : < 0.00000000000000022
##
## Sensitivity : 0.8639
## Specificity : 0.8214
## Pos Pred Value : 0.6144
## Neg Pred Value : 0.9482
## Prevalence : 0.2478
## Detection Rate : 0.2141
## Detection Prevalence : 0.3484
## Balanced Accuracy : 0.8426
##
## 'Positive' Class : yes
## myRoc <- roc(test$income_above_50k, predict(fit.xgb.tu.2, test, type="prob")[,2])## Setting levels: control = yes, case = no## Setting direction: controls < casesplot(myRoc)
auc(myRoc)## Area under the curve: 0.9256Choice of Metric:
We chose ROC/AUC as our metric to determine model performance because this is a binary classification problem. Given that there are 3x more negative (<50K) observations, our priority was improving the true positive (sensitivity) rate of our model. We can most easily interpret our TP rate in terms of the ROC/AUC metric. The ROC curve is helpful in showing us our TP vs FP at many different cut offs (decision threshold) for classification.
Conlusion:
Applying PCA helped us find a signal in an otherwise noisy dataset. K-means unsupervised learning did not meaningfully assist us to make predictions. We were able to improve the amount of true positives in validating our model as a result of training on an a balanced training dataset.