# import the CSV file
wbcd <- read.csv("wisc_bc_data.csv", stringsAsFactors = FALSE)
# examine the structure of the wbcd data frame
str(wbcd)## 'data.frame': 569 obs. of 32 variables:
## $ id : int 87139402 8910251 905520 868871 9012568 906539 925291 87880 862989 89827 ...
## $ diagnosis : chr "B" "B" "B" "B" ...
## $ radius_mean : num 12.3 10.6 11 11.3 15.2 ...
## $ texture_mean : num 12.4 18.9 16.8 13.4 13.2 ...
## $ perimeter_mean : num 78.8 69.3 70.9 73 97.7 ...
## $ area_mean : num 464 346 373 385 712 ...
## $ smoothness_mean : num 0.1028 0.0969 0.1077 0.1164 0.0796 ...
## $ compactness_mean : num 0.0698 0.1147 0.078 0.1136 0.0693 ...
## $ concavity_mean : num 0.0399 0.0639 0.0305 0.0464 0.0339 ...
## $ points_mean : num 0.037 0.0264 0.0248 0.048 0.0266 ...
## $ symmetry_mean : num 0.196 0.192 0.171 0.177 0.172 ...
## $ dimension_mean : num 0.0595 0.0649 0.0634 0.0607 0.0554 ...
## $ radius_se : num 0.236 0.451 0.197 0.338 0.178 ...
## $ texture_se : num 0.666 1.197 1.387 1.343 0.412 ...
## $ perimeter_se : num 1.67 3.43 1.34 1.85 1.34 ...
## $ area_se : num 17.4 27.1 13.5 26.3 17.7 ...
## $ smoothness_se : num 0.00805 0.00747 0.00516 0.01127 0.00501 ...
## $ compactness_se : num 0.0118 0.03581 0.00936 0.03498 0.01485 ...
## $ concavity_se : num 0.0168 0.0335 0.0106 0.0219 0.0155 ...
## $ points_se : num 0.01241 0.01365 0.00748 0.01965 0.00915 ...
## $ symmetry_se : num 0.0192 0.035 0.0172 0.0158 0.0165 ...
## $ dimension_se : num 0.00225 0.00332 0.0022 0.00344 0.00177 ...
## $ radius_worst : num 13.5 11.9 12.4 11.9 16.2 ...
## $ texture_worst : num 15.6 22.9 26.4 15.8 15.7 ...
## $ perimeter_worst : num 87 78.3 79.9 76.5 104.5 ...
## $ area_worst : num 549 425 471 434 819 ...
## $ smoothness_worst : num 0.139 0.121 0.137 0.137 0.113 ...
## $ compactness_worst: num 0.127 0.252 0.148 0.182 0.174 ...
## $ concavity_worst : num 0.1242 0.1916 0.1067 0.0867 0.1362 ...
## $ points_worst : num 0.0939 0.0793 0.0743 0.0861 0.0818 ...
## $ symmetry_worst : num 0.283 0.294 0.3 0.21 0.249 ...
## $ dimension_worst : num 0.0677 0.0759 0.0788 0.0678 0.0677 ...# drop the id feature
wbcd <- wbcd[-1]
#see the changes
wbcd# table of diagnosis
table(wbcd$diagnosis)##
## B M
## 357 212#this refers to the nature of the autopsy: B=Benign, M=Malignant
# recode diagnosis as a factor
wbcd$diagnosis <- factor(wbcd$diagnosis, levels = c("B", "M"),
labels = c("Benign", "Malignant"))
#confirm the changes
str(wbcd$diagnosis)## Factor w/ 2 levels "Benign","Malignant": 1 1 1 1 1 1 1 2 1 1 ...# table or proportions with more informative labels
round(prop.table(table(wbcd$diagnosis)) * 100, digits = 1)##
## Benign Malignant
## 62.7 37.3#62.7% of the outcomes of the biopsies were found to be benign tumours. The other 37.3 were cancerous tumours
# summarize three numeric features
summary(wbcd[c("radius_mean", "area_mean", "smoothness_mean")])## radius_mean area_mean smoothness_mean
## Min. : 6.981 Min. : 143.5 Min. :0.05263
## 1st Qu.:11.700 1st Qu.: 420.3 1st Qu.:0.08637
## Median :13.370 Median : 551.1 Median :0.09587
## Mean :14.127 Mean : 654.9 Mean :0.09636
## 3rd Qu.:15.780 3rd Qu.: 782.7 3rd Qu.:0.10530
## Max. :28.110 Max. :2501.0 Max. :0.16340#The results from above represent the different features from each biopsy.
#The radius of the largest biopsy was 28.11, but in average, the samples had an area of 654.9. we will assume that these metrics are in cm.
#Based on my limited domain knowledge, the less smooth the appearance of a sample, the higher are the chances of a the biopsy to reveal cancerous tumor.# create normalization function
normalize <- function(x) {
return ((x - min(x)) / (max(x) - min(x)))
}
#the idea here is to put all of the observations in a range from 0-1 to be able to manage any outliers. # test normalization function - result should be identical
normalize(c(1, 2, 3, 4, 5))## [1] 0.00 0.25 0.50 0.75 1.00normalize(c(10, 20, 30, 40, 50)) # this indicates that these two are in the same scale## [1] 0.00 0.25 0.50 0.75 1.00# normalize the wbcd data
#here we are normalizing all of the independent variables for our analysis, and we are leaving the target column or "diagnosis" as the dependent variable of these 31 features.
wbcd_n <- as.data.frame(lapply(wbcd[2:31], normalize))
wbcd_n #these are all now in a range from 0-1 on each category# confirm that normalization worked
summary(wbcd_n$area_mean)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.1174 0.1729 0.2169 0.2711 1.0000#here we can see that the average size of a biopsy is 21.69 mm# create training and test data
wbcd_train <- wbcd_n[1:469, ] # this is the range for us to train the data. This is about 82% of the data.
wbcd_test <- wbcd_n[470:569, ] # the data will be tested in 18% of the data# create labels for training and test data
wbcd_train_labels <- wbcd[1:469, 1]
wbcd_test_labels <- wbcd[470:569, 1]#see the labels for the training data
wbcd_train_labels ## [1] Benign Benign Benign Benign Benign Benign Benign
## [8] Malignant Benign Benign Malignant Benign Benign Benign
## [15] Malignant Benign Benign Benign Malignant Benign Benign
## [22] Benign Benign Benign Benign Malignant Benign Malignant
## [29] Benign Benign Benign Malignant Malignant Benign Benign
## [36] Benign Malignant Benign Malignant Malignant Malignant Malignant
## [43] Malignant Benign Benign Malignant Benign Malignant Benign
## [50] Benign Malignant Benign Benign Benign Malignant Benign
## [57] Benign Benign Malignant Malignant Malignant Malignant Malignant
## [64] Malignant Malignant Benign Benign Benign Benign Benign
## [71] Malignant Benign Benign Benign Benign Benign Malignant
## [78] Benign Benign Malignant Benign Benign Benign Benign
## [85] Benign Benign Benign Benign Malignant Benign Benign
## [92] Benign Malignant Benign Malignant Malignant Benign Benign
## [99] Benign Malignant Benign Benign Benign Malignant Benign
## [106] Benign Benign Benign Benign Benign Benign Benign
## [113] Malignant Malignant Benign Malignant Malignant Malignant Malignant
## [120] Benign Benign Malignant Benign Malignant Benign Benign
## [127] Malignant Malignant Malignant Malignant Malignant Benign Benign
## [134] Malignant Benign Malignant Benign Benign Malignant Benign
## [141] Malignant Malignant Malignant Malignant Malignant Benign Malignant
## [148] Benign Benign Benign Benign Benign Benign Benign
## [155] Benign Benign Benign Malignant Benign Benign Benign
## [162] Benign Benign Benign Malignant Malignant Benign Malignant
## [169] Benign Benign Malignant Malignant Malignant Benign Malignant
## [176] Benign Malignant Benign Malignant Benign Benign Malignant
## [183] Benign Malignant Benign Benign Malignant Malignant Malignant
## [190] Benign Malignant Benign Benign Malignant Malignant Malignant
## [197] Malignant Benign Benign Benign Malignant Benign Benign
## [204] Benign Benign Malignant Benign Malignant Benign Malignant
## [211] Benign Malignant Malignant Benign Benign Benign Benign
## [218] Benign Malignant Malignant Benign Benign Malignant Benign
## [225] Benign Benign Benign Malignant Malignant Benign Benign
## [232] Benign Benign Malignant Benign Benign Malignant Benign
## [239] Benign Benign Malignant Benign Benign Malignant Malignant
## [246] Benign Benign Benign Benign Benign Benign Malignant
## [253] Benign Malignant Benign Benign Benign Benign Benign
## [260] Benign Benign Benign Malignant Benign Malignant Malignant
## [267] Benign Benign Benign Malignant Benign Malignant Benign
## [274] Benign Malignant Malignant Benign Malignant Benign Benign
## [281] Malignant Benign Benign Benign Malignant Benign Malignant
## [288] Malignant Malignant Benign Benign Benign Malignant Benign
## [295] Malignant Malignant Malignant Malignant Benign Malignant Malignant
## [302] Benign Benign Malignant Malignant Malignant Malignant Benign
## [309] Benign Benign Malignant Malignant Benign Benign Malignant
## [316] Benign Malignant Malignant Benign Benign Benign Malignant
## [323] Malignant Benign Benign Malignant Benign Malignant Benign
## [330] Malignant Benign Benign Benign Benign Malignant Benign
## [337] Benign Benign Malignant Benign Benign Malignant Malignant
## [344] Benign Malignant Benign Benign Benign Malignant Benign
## [351] Benign Malignant Benign Benign Malignant Benign Benign
## [358] Benign Malignant Benign Malignant Malignant Benign Benign
## [365] Benign Benign Malignant Benign Benign Malignant Benign
## [372] Benign Benign Benign Benign Benign Benign Benign
## [379] Benign Benign Benign Benign Malignant Malignant Benign
## [386] Benign Malignant Benign Malignant Benign Malignant Benign
## [393] Benign Malignant Benign Benign Benign Malignant Malignant
## [400] Benign Benign Benign Malignant Malignant Benign Malignant
## [407] Benign Benign Malignant Benign Benign Benign Benign
## [414] Benign Benign Malignant Benign Benign Malignant Malignant
## [421] Benign Malignant Benign Benign Benign Malignant Benign
## [428] Benign Benign Malignant Malignant Malignant Benign Malignant
## [435] Benign Benign Malignant Benign Benign Benign Benign
## [442] Benign Malignant Benign Malignant Benign Benign Benign
## [449] Benign Malignant Benign Malignant Malignant Benign Benign
## [456] Benign Benign Malignant Benign Benign Malignant Malignant
## [463] Benign Benign Malignant Benign Benign Malignant Benign
## Levels: Benign Malignant#this will allow the model to be trained and to run tests on the trained data# load the "class" library
library(class)# here we are assigning the K-Nearest Neighbor algorithm which will use the euclidean distance to group the data and create predictions using the labels from the trained data.
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test,
cl = wbcd_train_labels, k = 21) #we have to specify the training and testing portions for the algorithm to be instantiated and assigned to the test_pred variable. Finally, the model will use at least 21 neighbors to assign each new data point to either a benign or malignant label.
#display the predictions created by knn
wbcd_test_pred## [1] Benign Benign Benign Benign Malignant Benign Malignant
## [8] Benign Malignant Benign Malignant Benign Malignant Malignant
## [15] Benign Benign Malignant Benign Malignant Benign Malignant
## [22] Malignant Malignant Malignant Benign Benign Benign Benign
## [29] Malignant Malignant Malignant Benign Malignant Malignant Benign
## [36] Benign Benign Benign Benign Malignant Malignant Benign
## [43] Malignant Malignant Benign Malignant Malignant Malignant Malignant
## [50] Malignant Benign Benign Benign Benign Benign Benign
## [57] Benign Benign Malignant Benign Benign Benign Benign
## [64] Benign Malignant Malignant Benign Benign Benign Benign
## [71] Benign Malignant Benign Benign Malignant Malignant Benign
## [78] Benign Benign Benign Benign Benign Benign Malignant
## [85] Benign Benign Malignant Benign Benign Benign Benign
## [92] Malignant Benign Benign Benign Benign Benign Malignant
## [99] Benign Malignant
## Levels: Benign Malignant# load the "gmodels" library
library(gmodels)
# Create the cross tabulation of predicted vs. actual
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred,
prop.chisq = FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 100
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 61 | 0 | 61 |
## | 1.000 | 0.000 | 0.610 |
## | 0.968 | 0.000 | |
## | 0.610 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 2 | 37 | 39 |
## | 0.051 | 0.949 | 0.390 |
## | 0.032 | 1.000 | |
## | 0.020 | 0.370 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 63 | 37 | 100 |
## | 0.630 | 0.370 | |
## -----------------|-----------|-----------|-----------|
##
## #our model predicted from the trained data that 61 of the 100 samples were benign correctly.The model also predicted that 2 of the biopsies were benign when in reality these were malignant, and it predicted accurately that 37 of the remaining samples were malignant tumors.
## Step 5: Improving model performance
# use the scale() function to z-score standardize a data frame
wbcd_z <- as.data.frame(scale(wbcd[-1])) #here we need to exclude the target variable from the scaling
# confirm that the transformation was applied correctly
summary(wbcd_z$area_mean)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.4532 -0.6666 -0.2949 0.0000 0.3632 5.2459# RECREATE labels for training and test data
wbcd_train_labels <- wbcd[1:512, 1]
wbcd_test_labels <- wbcd[513:569, 1]# create training and test data sets
wbcd_train <- wbcd_z[1:512, ] #80% training size
wbcd_test <- wbcd_z[513:569, ] #20% test size
# re-classify test cases
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test,
cl = wbcd_train_labels, k = 15) #here we will be using 15 neighbors to classify the results of the biopsies# Create the cross tabulation of predicted vs. actual
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred,
prop.chisq = FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 57
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 38 | 1 | 39 |
## | 0.974 | 0.026 | 0.684 |
## | 0.974 | 0.056 | |
## | 0.667 | 0.018 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 1 | 17 | 18 |
## | 0.056 | 0.944 | 0.316 |
## | 0.026 | 0.944 | |
## | 0.018 | 0.298 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 39 | 18 | 57 |
## | 0.684 | 0.316 | |
## -----------------|-----------|-----------|-----------|
##
## #using a sample of 57 biopsies, the model only had one false positive and one false negative
# try several different values of k
# RECREATE labels for training and test data
wbcd_train_labels <- wbcd[1:500, 1]
wbcd_test_labels <- wbcd[501:569, 1]
# try several different values of k
wbcd_train <- wbcd_n[1:500, ]
wbcd_test <- wbcd_n[501:569, ]#K=1
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=1)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 43 | 2 | 45 |
## | 0.956 | 0.044 | 0.652 |
## | 0.977 | 0.080 | |
## | 0.623 | 0.029 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 1 | 23 | 24 |
## | 0.042 | 0.958 | 0.348 |
## | 0.023 | 0.920 | |
## | 0.014 | 0.333 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 44 | 25 | 69 |
## | 0.638 | 0.362 | |
## -----------------|-----------|-----------|-----------|
##
## #when k=1 the model predicts 2 false negatives and one false positive. The quality of our model decreased but not significantly
#k=5
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=5)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 45 | 0 | 45 |
## | 1.000 | 0.000 | 0.652 |
## | 0.978 | 0.000 | |
## | 0.652 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 1 | 23 | 24 |
## | 0.042 | 0.958 | 0.348 |
## | 0.022 | 1.000 | |
## | 0.014 | 0.333 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 46 | 23 | 69 |
## | 0.667 | 0.333 | |
## -----------------|-----------|-----------|-----------|
##
## #Here the model produced only 1 false positive which is an improvement compared to K=1
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=11)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE) ##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 45 | 0 | 45 |
## | 1.000 | 0.000 | 0.652 |
## | 0.938 | 0.000 | |
## | 0.652 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 3 | 21 | 24 |
## | 0.125 | 0.875 | 0.348 |
## | 0.062 | 1.000 | |
## | 0.043 | 0.304 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 48 | 21 | 69 |
## | 0.696 | 0.304 | |
## -----------------|-----------|-----------|-----------|
##
## wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=15)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 45 | 0 | 45 |
## | 1.000 | 0.000 | 0.652 |
## | 0.938 | 0.000 | |
## | 0.652 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 3 | 21 | 24 |
## | 0.125 | 0.875 | 0.348 |
## | 0.062 | 1.000 | |
## | 0.043 | 0.304 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 48 | 21 | 69 |
## | 0.696 | 0.304 | |
## -----------------|-----------|-----------|-----------|
##
## #similar results compared to the ones at k=11
wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=21)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 45 | 0 | 45 |
## | 1.000 | 0.000 | 0.652 |
## | 0.938 | 0.000 | |
## | 0.652 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 3 | 21 | 24 |
## | 0.125 | 0.875 | 0.348 |
## | 0.062 | 1.000 | |
## | 0.043 | 0.304 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 48 | 21 | 69 |
## | 0.696 | 0.304 | |
## -----------------|-----------|-----------|-----------|
##
## wbcd_test_pred <- knn(train = wbcd_train, test = wbcd_test, cl = wbcd_train_labels, k=27)
CrossTable(x = wbcd_test_labels, y = wbcd_test_pred, prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 69
##
##
## | wbcd_test_pred
## wbcd_test_labels | Benign | Malignant | Row Total |
## -----------------|-----------|-----------|-----------|
## Benign | 45 | 0 | 45 |
## | 1.000 | 0.000 | 0.652 |
## | 0.938 | 0.000 | |
## | 0.652 | 0.000 | |
## -----------------|-----------|-----------|-----------|
## Malignant | 3 | 21 | 24 |
## | 0.125 | 0.875 | 0.348 |
## | 0.062 | 1.000 | |
## | 0.043 | 0.304 | |
## -----------------|-----------|-----------|-----------|
## Column Total | 48 | 21 | 69 |
## | 0.696 | 0.304 | |
## -----------------|-----------|-----------|-----------|
##
## #based on all of the tests ran, the ideal number of neighbors for our model is k=5