# Linear VS Quadratic Discriminant Analysis in R

In this post we will look at linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). Discriminant analysis is used when the dependent variable is categorical. Another commonly used option is logistic regression but there are differences between logistic regression and discriminant analysis. Both LDA and QDA are used in situations in which there is a clear separation between the classes you want to predict. If the categories are fuzzier logistic regression is often the better choice.

For our example, we will use the “Mathlevel” dataset found in the “Ecdat” package. Our goal will be to predict the sex of a respondent based on SAT math score, major, foreign language proficiency, as well as the number of math, physic, and chemistry classes a respondent took. Below is some initial code to start our analysis.

library(MASS);library(Ecdat)
data("Mathlevel")

The first thing we need to do is clean up the data set. We have to remove any missing data in order to run our model. We will create a dataset called “math” that has the “Mathlevel” dataset but with the “NA”s removed use the “na.omit” function. After this, we need to set our seed for the purpose of reproducibility using the “set.seed” function. Lastly, we will split the data using the “sample” function using a 70/30 split. The training dataset will be called “math.train” and the testing dataset will be called “math.test”. Below is the code

math<-na.omit(Mathlevel)
set.seed(123)
math.ind<-sample(2,nrow(math),replace=T,prob = c(0.7,0.3))
math.train<-math[math.ind==1,]
math.test<-math[math.ind==2,]

Now we will make our model and it is called “lda.math” and it will include all available variables in the “math.train” dataset. Next we will check the results by calling the modle. Finally, we will examine the plot to see how our model is doing. Below is the code.

lda.math<-lda(sex~.,math.train)
lda.math
## Call:
## lda(sex ~ ., data = math.train)
##
## Prior probabilities of groups:
##      male    female
## 0.5986079 0.4013921
##
## Group means:
##        mathlevel.L mathlevel.Q mathlevel.C mathlevel^4 mathlevel^5
## male   -0.10767593  0.01141838 -0.05854724   0.2070778  0.05032544
## female -0.05571153  0.05360844 -0.08967303   0.2030860 -0.01072169
##        mathlevel^6      sat languageyes  majoreco  majoross   majorns
## male    -0.2214849 632.9457  0.07751938 0.3914729 0.1472868 0.1782946
## female  -0.2226767 613.6416  0.19653179 0.2601156 0.1907514 0.2485549
##          majorhum mathcourse physiccourse chemistcourse
## male   0.05426357   1.441860    0.7441860      1.046512
## female 0.07514451   1.421965    0.6531792      1.040462
##
## Coefficients of linear discriminants:
##                       LD1
## mathlevel.L    1.38456344
## mathlevel.Q    0.24285832
## mathlevel.C   -0.53326543
## mathlevel^4    0.11292817
## mathlevel^5   -1.24162715
## mathlevel^6   -0.06374548
## sat           -0.01043648
## languageyes    1.50558721
## majoreco      -0.54528930
## majoross       0.61129797
## majorns        0.41574298
## majorhum       0.33469586
## mathcourse    -0.07973960
## physiccourse  -0.53174168
## chemistcourse  0.16124610
plot(lda.math,type='both')

Calling “lda.math” gives us the details of our model. It starts be indicating the prior probabilities of someone being male or female. Next is the means for each variable by sex. The last part is the coefficients of the linear discriminants. Each of these values is used to determine the probability that a particular example is male or female. This is similar to a regression equation.

The plot provides us with densities of the discriminant scores for males and then for females. The output indicates a problem. There is a great deal of overlap between male and females in the model. What this indicates is that there is a lot of misclassification going on as the two groups are not clearly separated. Furthermore, this means that logistic regression is probably a better choice for distinguishing between male and females. However, since this is for demonstrating purposes we will not worry about this.

We will now use the “predict” function on the training set data to see how well our model classifies the respondents by gender. We will then compare the prediction of the model with thee actual classification. Below is the code.

math.lda.predict<-predict(lda.math)
math.train$lda<-math.lda.predict$class
table(math.train$lda,math.train$sex)
##
##          male female
##   male    219    100
##   female   39     73
mean(math.train$lda==math.train$sex)
## [1] 0.6774942

As you can see, we have a lot of misclassification happening. A large amount of false negatives which is a lot of males being classified as female. The overall accuracy us only 59% which is not much better than chance.

We will now conduct the same analysis on the test data set. Below is the code.

lda.math.test<-predict(lda.math,math.test)
math.test$lda<-lda.math.test$class
table(math.test$lda,math.test$sex)
##
##          male female
##   male     92     43
##   female   23     20
mean(math.test$lda==math.test$sex)
## [1] 0.6292135

As you can see the results are similar. To put it simply, our model is terrible. The main reason is that there is little distinction between males and females as shown in the plot. However, we can see if perhaps a quadratic discriminant analysis will do better

QDA allows for each class in the dependent variable to have it’s own covariance rather than a shared covariance as in LDA. This allows for quadratic terms in the development of the model. To complete a QDA we need to use the “qda” function from the “MASS” package. Below is the code for the training data set.

math.qda.fit<-qda(sex~.,math.train)
math.qda.predict<-predict(math.qda.fit)
math.train$qda<-math.qda.predict$class
table(math.train$qda,math.train$sex)
##
##          male female
##   male    215     84
##   female   43     89
mean(math.train$qda==math.train$sex)
## [1] 0.7053364

You can see there is almost no difference. Below is the code for the test data.

math.qda.test<-predict(math.qda.fit,math.test)
math.test$qda<-math.qda.test$class
table(math.test$qda,math.test$sex)
##
##          male female
##   male     91     43
##   female   24     20
mean(math.test$qda==math.test$sex)
## [1] 0.6235955

Still disappointing. However, in this post we reviewed linear discriminant analysis as well as learned about the use of quadratic linear discriminant analysis. Both of these statistical tools are used for predicting categorical dependent variables. LDA assumes shared covariance in the dependent variable categories will QDA allows for each category in the dependent variable to have it’s own variance.

# Validating a Logistic Model in R

In this post, we are going to continue are analysis of the logistic regression model from the the post on logistic regression  in R. We need to rerun all of the code from the last post to be ready to continue. As such the code form the last post is all below

library(MASS);library(bestglm);library(reshape2);library(corrplot);
library(ggplot2);library(ROCR)
data(survey)
survey$Clap<-NULL survey$W.Hnd<-NULL
survey$Fold<-NULL survey$Exer<-NULL
survey$Smoke<-NULL survey$M.I<-NULL
survey<-na.omit(survey)
pm<-melt(survey, id.var="Sex")
ggplot(pm,aes(Sex,value))+geom_boxplot()+facet_wrap(~variable,ncol = 3)

pc<-cor(survey[,2:5]) corrplot.mixed(pc)

set.seed(123) ind<-sample(2,nrow(survey),replace=T,prob = c(0.7,0.3)) train<-survey[ind==1,] test<-survey[ind==2,] fit<-glm(Sex~.,binomial,train) exp(coef(fit))

train$probs<-predict(fit, type = 'response') train$predict<-rep('Female',123)
train$predict[train$probs>0.5]<-"Male"
table(train$predict,train$Sex)
mean(train$predict==train$Sex)
test$prob<-predict(fit,newdata = test, type = 'response') test$predict<-rep('Female',46)
test$predict[test$prob>0.5]<-"Male"
table(test$predict,test$Sex)
mean(test$predict==test$Sex)

Model Validation

We will now do a K-fold cross validation in order to further see how our model is doing. We cannot use the factor variable “Sex” with the K-fold code so we need to create a dummy variable. First, we create a variable called “y” that has 123 spaces, which is the same size as the “train” dataset. Second, we fill “y” with 1 in every example that is coded “male” in the “Sex” variable.

In addition, we also need to create a new dataset and remove some variables from our prior analysis otherwise we will confuse the functions that we are going to use. We will remove “predict”, “Sex”, and “probs”

train$y<-rep(0,123) train$y[train$Sex=="Male"]=1 my.cv<-train[,-8] my.cv$Sex<-NULL
my.cv$probs<-NULL We now can do our K-fold analysis. The code is complicated so you can trust it and double check on your own. bestglm(Xy=my.cv,IC="CV",CVArgs = list(Method="HTF",K=10,REP=1),family = binomial) ## Morgan-Tatar search since family is non-gaussian. ## CV(K = 10, REP = 1) ## BICq equivalent for q in (6.66133814775094e-16, 0.0328567092272112) ## Best Model: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) -45.2329733 7.80146036 -5.798014 6.710501e-09 ## Height 0.2615027 0.04534919 5.766425 8.097067e-09 The results confirm what we alreaedy knew that only the “Height” variable is valuable in predicting Sex. We will now create our new model using only the recommendation of the kfold validation analysis. Then we check the new model against the train dataset and with the test dataset. The code below is a repeat of prior code but based on the cross-validation reduce.fit<-glm(Sex~Height, family=binomial,train) train$cv.probs<-predict(reduce.fit,type='response')
train$cv.predict<-rep('Female',123) train$cv.predict[train$cv.probs>0.5]='Male' table(train$cv.predict,train$Sex) ## ## Female Male ## Female 61 11 ## Male 7 44 mean(train$cv.predict==train$Sex) ## [1] 0.8536585 test$cv.probs<-predict(reduce.fit,test,type = 'response')
test$cv.predict<-rep('Female',46) test$cv.predict[test$cv.probs>0.5]='Male' table(test$cv.predict,test$Sex) ## ## Female Male ## Female 16 7 ## Male 1 22 mean(test$cv.predict==test$Sex) ## [1] 0.826087 The results are consistent for both the train and test dataset. We are now going to create the ROC curve. This will provide a visual and the AUC number to further help us to assess our model. However, a model is only good when it is compared to another model. Therefore, we will create a really bad model in order to compare it to the original model, and the cross validated model. We will first make a bad model and store the probabilities in the “test” dataset. The bad model will use “age” to predict “Sex” which doesn’t make any sense at all. Below is the code followed by the ROC curve of the bad model. bad.fit<-glm(Sex~Age,family = binomial,test) test$bad.probs<-predict(bad.fit,type='response')
pred.bad<-prediction(test$bad.probs,test$Sex)
plot(perf.bad,col=1)

The more of a diagonal the line is the worst it is. As, we can see the bad model is really bad.

What we just did with the bad model we will now repeat for the full model and the cross-validated model.  As before, we need to store the prediction in a way that the ROCR package can use them. We will create a variable called “pred.full” to begin the process of graphing the the original full model from the last blog post. Then we will use the “prediction” function. Next, we will create the “perf.full” variable to store the performance of the model. Notice, the arguments ‘tpr’ and ‘fpr’ for true positive rate and false positive rate. Lastly, we plot the results

pred.full<-prediction(test$prob,test$Sex)
perf.full<-performance(pred.full,'tpr','fpr')
plot(perf.full, col=2)

We repeat this process for the cross-validated model

pred.cv<-prediction(test$cv.probs,test$Sex)
perf.cv<-performance(pred.cv,'tpr','fpr')
plot(perf.cv,col=3)

Now let’s put all the different models on one plot

plot(perf.bad,col=1)
legend(.7,.4,c("BAD","FULL","CV"), 1:3)

Finally, we can calculate the AUC for each model

auc.bad<-performance(pred.bad,'auc')
auc.bad@y.values
## [[1]]
## [1] 0.4766734
auc.full<-performance(pred.full,"auc")
auc.full@y.values
## [[1]]
## [1] 0.959432
auc.cv<-performance(pred.cv,'auc')
auc.cv@y.values
## [[1]]
## [1] 0.9107505

The higher the AUC the better. As such, the full model with all variables is superior to the cross-validated or bad model. This is despite the fact that there are many high correlations in the full model as well. Another point to consider is that the cross-validated model is simpler so this may be a reason to pick it over the full model. As such, the statistics provide support for choosing a model but the do not trump the ability of the research to pick based on factors beyond just numbers.

# Logistic Regression in R

In this post, we will conduct a logistic regression analysis. Logistic regression is used when you want to predict a categorical dependent variable using continuous or categorical dependent variables. In our example, we want to predict Sex (male or female) when using several continuous variables from the “survey” dataset in the “MASS” package.

library(MASS);library(bestglm);library(reshape2);library(corrplot)
data(survey)
?MASS::survey #explains the variables in the study

The first thing we need to do is remove the independent factor variables from our dataset. The reason for this is that the function that we will use for the cross-validation does not accept factors. We will first use the “str” function to identify factor variables and then remove them from the dataset. We also need to remove in examples that are missing data so we use the “na.omit” function for this. Below is the code

survey$Clap<-NULL survey$W.Hnd<-NULL
survey$Fold<-NULL survey$Exer<-NULL
survey$Smoke<-NULL survey$M.I<-NULL
survey<-na.omit(survey)

We now need to check for collinearity using the “corrplot.mixed” function form the “corrplot” package.

pc<-cor(survey[,2:5])
corrplot.mixed(pc)
corrplot.mixed(pc)

We have extreme correlation between “We.Hnd” and “NW.Hnd” this makes sense because people’s hands are normally the same size. Since this blog post  is a demonstration of logistic regression we will not worry about this too much.

We now need to divide our dataset into a train and a test set. We set the seed for. First we need to make a variable that we call “ind” that is randomly assigns 70% of the number of rows of survey 1 and 30% 2. We then subset the “train” dataset by taking all rows that are 1’s based on the “ind” variable and we create the “test” dataset for all the rows that line up with 2 in the “ind” variable. This means our data split is 70% train and 30% test. Below is the code

set.seed(123)
ind<-sample(2,nrow(survey),replace=T,prob = c(0.7,0.3))
train<-survey[ind==1,]
test<-survey[ind==2,]

We now make our model. We use the “glm” function for logistic regression. We set the family argument to “binomial”. Next, we look at the results as well as the odds ratios.

fit<-glm(Sex~.,family=binomial,train)
summary(fit)
##
## Call:
## glm(formula = Sex ~ ., family = binomial, data = train)
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -1.9875  -0.5466  -0.1395   0.3834   3.4443
##
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)
## (Intercept) -46.42175    8.74961  -5.306 1.12e-07 ***
## Wr.Hnd       -0.43499    0.66357  -0.656    0.512
## NW.Hnd        1.05633    0.70034   1.508    0.131
## Pulse        -0.02406    0.02356  -1.021    0.307
## Height        0.21062    0.05208   4.044 5.26e-05 ***
## Age           0.00894    0.05368   0.167    0.868
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 169.14  on 122  degrees of freedom
## Residual deviance:  81.15  on 117  degrees of freedom
## AIC: 93.15
##
## Number of Fisher Scoring iterations: 6
exp(coef(fit))
##  (Intercept)       Wr.Hnd       NW.Hnd        Pulse       Height
## 6.907034e-21 6.472741e-01 2.875803e+00 9.762315e-01 1.234447e+00
##          Age
## 1.008980e+00

The results indicate that only height is useful in predicting if someone is a male or female. The second piece of code shares the odds ratios. The odds ratio tell how a one unit increase in the independent variable leads to an increase in the odds of being male in our model. For example, for every one unit increase in height there is a 1.23 increase in the odds of a particular example being male.

We now need to see how well our model does on the train and test dataset. We first capture the probabilities and save them to the train dataset as “probs”. Next we create a “predict” variable and place the string “Female” in the same number of rows as are in the “train” dataset. Then we rewrite the “predict” variable by changing any example that has a probability above 0.5 as “Male”. Then we make a table of our results to see the number correct, false positives/negatives. Lastly, we calculate the accuracy rate. Below is the code.

train$probs<-predict(fit, type = 'response') train$predict<-rep('Female',123)
train$predict[train$probs>0.5]<-"Male"
table(train$predict,train$Sex)
##
##          Female Male
##   Female     61    7
##   Male        7   48
mean(train$predict==train$Sex)
## [1] 0.8861789

Despite the weaknesses of the model with so many insignificant variables it is surprisingly accurate at 88.6%. Let’s see how well we do on the “test” dataset.

test$prob<-predict(fit,newdata = test, type = 'response') test$predict<-rep('Female',46)
test$predict[test$prob>0.5]<-"Male"
table(test$predict,test$Sex)
##
##          Female Male
##   Female     17    3
##   Male        0   26
mean(test$predict==test$Sex)
## [1] 0.9347826

As you can see, we do even better on the test set with an accuracy of 93.4%. Our model is looking pretty good and height is an excellent predictor of sex which makes complete sense. However, in the next post we will use cross-validation and the ROC plot to further assess the quality of it.

# Data Wrangling in R

Collecting and preparing data for analysis is the primary job of a data scientist. This experience is called data wrangling. In this post, we will look at an example of data wrangling using a simple artificial data set. You can create the table below in r or excel. If you created it in excel just save it as a csv and load it into r. Below is the initial code

library(readr)
apple <- read_csv("~/Desktop/apple.csv")
## # A tibble: 10 × 2
##        weight      location
##         <chr>         <chr>
## 1         3.2        Europe
## 2       4.2kg       europee
## 3      1.3 kg          U.S.
## 4  7200 grams           USA
## 5          42 United States
## 6         2.3       europee
## 7       2.1kg        Europe
## 8       3.1kg           USA
## 9  2700 grams          U.S.
## 10         24 United States

This a small dataset with the columns of “weight” and “location”. Here are some of the problems

• Weights are in different units
• Weights are written in different ways
• Location is not consistent

In order to have any success with data wrangling you need to state specifically what it is you want to do. Here are our goals for this project

• Convert the “Weight variable” to a numerical variable instead of character
• Remove the text and have only numbers in the “weight variable”
• Change weights in grams to kilograms
• Convert the “location” variable to a factor variable instead of character
• Have consistent spelling for Europe and United States in the “location” variable

We will begin with the “weight” variable. We want to convert it to a numerical variable and remove any non-numerical text. Below is the code for this

corrected.weight<-as.numeric(gsub(pattern = "[[:alpha:]]","",apple$weight)) corrected.weight ## [1] 3.2 4.2 1.3 7200.0 42.0 2.3 2.1 3.1 2700.0 24.0 Here is what we did. 1. We created a variable called “corrected.weight” 2. We use the function “as.numeric” this makes whatever results inside it to be a numerical variable 3. Inside “as.numeric” we used the “gsub” function which allows us to substitute one value for another. 4. Inside “gsub” we used the argument pattern and set it to “[[alpha:]]” and “” this told r to look for any lower or uppercase letters and replace with nothing or remove it. This all pertains to the “weight” variable in the apple dataframe. We now need to convert the weights in grams to kilograms so that everything is the same unit. Below is the code gram.error<-grep(pattern = "[[:digit:]]{4}",apple$weight)
corrected.weight[gram.error]<-corrected.weight[gram.error]/1000
corrected.weight
##  [1]  3.2  4.2  1.3  7.2 42.0  2.3  2.1  3.1  2.7 24.0

Here is what we did

1. We created a variable called “gram.error”
2. We used the grep function to search are the “weight” variable in the apple data frame for input that is a digit and is 4 digits in length this is what the “[[:digit:]]{4}” argument means. We do not change any values yet we just store them in “gram.error”
3. Once this information is stored in “gram.error” we use it as a subset for the “corrected.weight” variable.
4. We tell r to save into the “corrected.weight” variable any value that is changeable according to the criteria set in “gram.error” and to divided it by 1000. Dividing it by 1000 converts the value from grams to kilograms.

We have completed the transformation of the “weight” and will move to dealing with the problems with the “location” variable in the “apple” dataframe. To do this we will first deal with the issues related to the values that relate to Europe and then we will deal with values related to United States. Below is the code.

europe<-agrep(pattern = "europe",apple$location,ignore.case = T,max.distance = list(insertion=c(1),deletions=c(2))) america<-agrep(pattern = "us",apple$location,ignore.case = T,max.distance = list(insertion=c(0),deletions=c(2),substitutions=0))
corrected.location<-apple$location corrected.location[europe]<-"europe" corrected.location[america]<-"US" corrected.location<-gsub(pattern = "United States","US",corrected.location) corrected.location ## [1] "europe" "europe" "US" "US" "US" "europe" "europe" ## [8] "US" "US" "US" The code is a little complicated to explain but in short We used the “agrep” function to tell r to search the “location” to look for values similar to our term “europe”. The other arguments provide some exceptions that r should change because the exceptions are close to the term europe. This process is repeated for the term “us”. We then store are the location variable from the “apple” dataframe in a new variable called “corrected.location” We then apply the two objects we made called “europe” and “america” to the “corrected.location” variable. Next we have to make some code to deal with “United States” and apply this using the “gsub” function. We are almost done, now we combine are two variables “corrected.weight” and “corrected.location” into a new data.frame. The code is below cleaned.apple<-data.frame(corrected.weight,corrected.location) names(cleaned.apple)<-c('weight','location') cleaned.apple ## weight location ## 1 3.2 europe ## 2 4.2 europe ## 3 1.3 US ## 4 7.2 US ## 5 42.0 US ## 6 2.3 europe ## 7 2.1 europe ## 8 3.1 US ## 9 2.7 US ## 10 24.0 US If you use the “str” function on the “cleaned.apple” dataframe you will see that “location” was automatically converted to a factor. This looks much better especially if you compare it to the original dataframe that is printed at the top of this post. # Principal Component Analysis in R This post will demonstrate the use of principal component analysis (PCA). PCA is useful for several reasons. One it allows you place your examples into groups similar to linear discriminant analysis but you do not need to know beforehand what the groups are. Second, PCA is used for the purpose of dimension reduction. For example, if you have 50 variables PCA can allow you to reduce this while retaining a certain threshold of variance. If you are working with a large dataset this can greatly reduce the computational time and general complexity of your models. Keep in mind that there really is not a dependent variable as this is unsupervised learning. What you are trying to see is how different examples can be mapped in space based on whatever independent variables are used. For our example, we will use the “Carseats” dataset form the “ISLR”. Our goal is to understanding the relationship among the variables when examining the shelve location of the car seat. Below is the initial code to begin the analysis library(ggplot2) library(ISLR) data("Carseats") We first need to rearrange the data and remove the variables we are not going to use in the analysis. Below is the code. Carseats1<-Carseats Carseats1<-Carseats1[,c(1,2,3,4,5,6,8,9,7,10,11)] Carseats1$Urban<-NULL
Carseats1$US<-NULL Here is what we did 1. We made a copy of the “Carseats” data called “Careseats1” 2. We rearranged the order of the variables so that the factor variables are at the end. This will make sense later 3.We removed the “Urban” and “US” variables from the table as they will not be a part of our analysis We will now do the PCA. We need to scale and center our data otherwise the larger numbers will have a much stronger influence on the results than smaller numbers. Fortunately, the “prcomp” function has a “scale” and a “center” argument. We will also use only the first 7 columns for the analysis as “sheveLoc” is not useful for this analysis. If we hadn’t moved “shelveLoc” to the end of the dataframe it would cause some headache. Below is the code. Carseats.pca<-prcomp(Carseats1[,1:7],scale. = T,center = T) summary(Carseats.pca) ## Importance of components: ## PC1 PC2 PC3 PC4 PC5 PC6 PC7 ## Standard deviation 1.3315 1.1907 1.0743 0.9893 0.9260 0.80506 0.41320 ## Proportion of Variance 0.2533 0.2026 0.1649 0.1398 0.1225 0.09259 0.02439 ## Cumulative Proportion 0.2533 0.4558 0.6207 0.7605 0.8830 0.97561 1.00000 The summary of “Carseats.pca” Tells us how much of the variance each component explains. Keep in mind that number of components is equal to the number of variables. The “proportion of variance” tells us the contribution each component makes and the “cumulative proportion”. If your goal is dimension reduction than the number of components to keep depends on the threshold you set. For example, if you need around 90% of the variance you would keep the first 5 components. If you need 95% or more of the variance you would keep the first six. To actually use the components you would take the “Carseats.pca$x” data and move it to your data frame.

Keep in mind that the actual components have no conceptual meaning but is a numerical representation of a combination of several variables that were reduce using PCA to fewer variables such as going form 7 variables to 5 variables.

This means that PCA is great for reducing variables for prediction purpose but is much harder for explanatory studies unless you can explain what the new components represent.

For our purposes, we will keep 5 components. This means that we have reduce our dimensions from 7 to 5 while still keeping almost 90% of the variance. Graphing our results is tricky because we have 5 dimensions but the human mind can only conceptualize 3 at the best and normally 2. As such we will plot the first two components and label them by shelf location using ggplot2. Below is the code

scores<-as.data.frame(Carseats.pca$x) pcaplot<-ggplot(scores,(aes(PC1,PC2,color=Carseats1$ShelveLoc)))+geom_point()
pcaplot

From the plot you can see there is little separation when using the first two components of the PCA analysis. This makes sense as we can only graph to components so we are missing a lot of the variance. However for demonstration purposes the analysis is complete.

# Linear Discriminant Analysis in R

In this post we will look at an example of linear discriminant analysis (LDA). LDA is used to develop a statistical model that classifies examples in a dataset. In the example in this post, we will use the “Star” dataset from the “Ecdat” package. What we will do is try to predict the type of class the students learned in (regular, small, regular with aide) using their math scores, reading scores, and the teaching experience of the teacher. Below is the initial code

library(Ecdat)
library(MASS)
data(Star)

We first need to examine the data by using the “str” function

str(Star)
## 'data.frame':    5748 obs. of  8 variables:
##  $tmathssk: int 473 536 463 559 489 454 423 500 439 528 ... ##$ treadssk: int  447 450 439 448 447 431 395 451 478 455 ...
##  $classk : Factor w/ 3 levels "regular","small.class",..: 2 2 3 1 2 1 3 1 2 2 ... ##$ totexpk : int  7 21 0 16 5 8 17 3 11 10 ...
##  $sex : Factor w/ 2 levels "girl","boy": 1 1 2 2 2 2 1 1 1 1 ... ##$ freelunk: Factor w/ 2 levels "no","yes": 1 1 2 1 2 2 2 1 1 1 ...
##  $race : Factor w/ 3 levels "white","black",..: 1 2 2 1 1 1 2 1 2 1 ... ##$ schidkn : int  63 20 19 69 79 5 16 56 11 66 ...
##  - attr(*, "na.action")=Class 'omit'  Named int [1:5850] 1 4 6 7 8 9 10 15 16 17 ...
##   .. ..- attr(*, "names")= chr [1:5850] "1" "4" "6" "7" ...

We will use the following variables

• dependent variable = classk (class type)
• independent variable = tmathssk (Math score)
• independent variable = totexpk (Teaching experience)

We now need to examine the data visually by looking at histograms for our independent variables and a table for our dependent variable

hist(Star$tmathssk) hist(Star$treadssk)

hist(Star$totexpk) prop.table(table(Star$classk))
##
##           regular       small.class regular.with.aide
##         0.3479471         0.3014962         0.3505567

The data mostly looks good. The results of the “prop.table” function will help us when we develop are training and testing datasets. The only problem is with the “totexpk” variable. IT is not anywhere near to be normally distributed. TO deal with this we will use the square root for teaching experience. Below is the code

star.sqrt<-Star
star.sqrt$totexpk.sqrt<-sqrt(star.sqrt$totexpk)
hist(sqrt(star.sqrt$totexpk)) Much better. We now need to check the correlation among the variables as well and we will use the code below. cor.star<-data.frame(star.sqrt$tmathssk,star.sqrt$treadssk,star.sqrt$totexpk.sqrt)
cor(cor.star)
##                        star.sqrt.tmathssk star.sqrt.treadssk
## star.sqrt.tmathssk             1.00000000          0.7135489
## star.sqrt.totexpk.sqrt         0.08647957          0.1045353
##                        star.sqrt.totexpk.sqrt
## star.sqrt.tmathssk                 0.08647957
## star.sqrt.totexpk.sqrt             1.00000000

None of the correlations are too bad. We can now develop our model using linear discriminant analysis. First, we need to scale are scores because the test scores and the teaching experience are measured differently. Then, we need to divide our data into a train and test set as this will allow us to determine the accuracy of the model. Below is the code.

star.sqrt$tmathssk<-scale(star.sqrt$tmathssk)
star.sqrt$treadssk<-scale(star.sqrt$treadssk)
star.sqrt$totexpk.sqrt<-scale(star.sqrt$totexpk.sqrt)
train.star<-star.sqrt[1:4000,]
test.star<-star.sqrt[4001:5748,]

Now we develop our model. In the code before the “prior” argument indicates what we expect the probabilities to be. In our data the distribution of the the three class types is about the same which means that the apriori probability is 1/3 for each class type.

train.lda<-lda(classk~tmathssk+treadssk+totexpk.sqrt, data =
train.star,prior=c(1,1,1)/3)
train.lda
## Call:
## lda(classk ~ tmathssk + treadssk + totexpk.sqrt, data = train.star,
##     prior = c(1, 1, 1)/3)
##
## Prior probabilities of groups:
##           regular       small.class regular.with.aide
##         0.3333333         0.3333333         0.3333333
##
## Group means:
## regular           -0.04237438 -0.05258944  -0.05082862
## small.class        0.13465218  0.11021666  -0.02100859
## regular.with.aide -0.05129083 -0.01665593   0.09068835
##
## Coefficients of linear discriminants:
##                      LD1         LD2
## tmathssk      0.89656393 -0.04972956
## totexpk.sqrt -0.49061950  0.80051026
##
## Proportion of trace:
##    LD1    LD2
## 0.7261 0.2739

The printout is mostly readable. At the top is the actual code used to develop the model followed by the probabilities of each group. The next section shares the means of the groups. The coefficients of linear discriminants are the values used to classify each example. The coefficients are similar to regression coefficients. The computer places each example in both equations and probabilities are calculated. Whichever class has the highest probability is the winner. In addition, the higher the coefficient the more weight it has. For example, “tmathssk” is the most influential on LD1 with a coefficient of 0.89.

The proportion of trace is similar to principal component analysis

Now we will take the trained model and see how it does with the test set. We create a new model called “predict.lda” and use are “train.lda” model and the test data called “test.star”

predict.lda<-predict(train.lda,newdata = test.star)

We can use the “table” function to see how well are model has done. We can do this because we actually know what class our data is beforehand because we divided the dataset. What we need to do is compare this to what our model predicted. Therefore, we compare the “classk” variable of our “test.star” dataset with the “class” predicted by the “predict.lda” model.

table(test.star$classk,predict.lda$class)
##
##                     regular small.class regular.with.aide
##   regular               155         182               249
##   small.class           145         198               174
##   regular.with.aide     172         204               269

The results are pretty bad. For example, in the first row called “regular” we have 155 examples that were classified as “regular” and predicted as “regular” by the model. In rhe next column, 182 examples that were classified as “regular” but predicted as “small.class”, etc. To find out how well are model did you add together the examples across the diagonal from left to right and divide by the total number of examples. Below is the code

(155+198+269)/1748
## [1] 0.3558352

Only 36% accurate, terrible but ok for a demonstration of linear discriminant analysis. Since we only have two-functions or two-dimensions we can plot our model.  Below I provide a visual of the first 50 examples classified by the predict.lda model.

plot(predict.lda$x[1:50]) text(predict.lda$x[1:50],as.character(predict.lda$class[1:50]),col=as.numeric(predict.lda$class[1:100]))
abline(h=0,col="blue")
abline(v=0,col="blue")

The first function, which is the vertical line, doesn’t seem to discriminant anything as it off to the side and not separating any of the data. However, the second function, which is the horizontal one, does a good of dividing the “regular.with.aide” from the “small.class”. Yet, there are problems with distinguishing the class “regular” from either of the other two groups.  In order improve our model we need additional independent variables to help to distinguish the groups in the dependent variable.

# Bagging in R

In this post, we will explore the potential of bagging. Bagging is a process in which the original data is boostrapped to make several different datasets. Each of these datasets are used to generate a model and voting is used to classify an example or averaging is used for numeric prediction.

Bagging is especially useful for unstable learners. These are algorithms who generate models that can change a great deal when the data is modified a small amount. In order to complete this example, you will need to load the following packages, set the seed, as well as load the dataset “Wages1”. We will be using a decision tree that was developed in an earlier post. Below is the initial code

library(caret); library(Ecdat);library(ipred);library(vcd)
set.seed(1)
data(Wages1)

We will now use the “bagging” function from the “ipred” package to create our model as well as tell R how many bags to make.

theBag<-bagging(sex~.,data=Wages1,nbagg=25)

Next, we will make our predictions. Then we will check the accuracy of the model looking at a confusion matrix and the kappa statistic. The “kappa” function comes from the “vcd” package.

bagPred<-predict(theBag, Wages1)
keep<-table(bagPred, Wages1$sex) keep ## ## bagPred female male ## female 1518 52 ## male 51 1673 Kappa(keep) ## value ASE z Pr(>|z|) ## Unweighted 0.9373 0.006078 154.2 0 ## Weighted 0.9373 0.006078 154.2 0 The results appearing exciting with almost 97% accuracy. In addition, the Kappa was almost 0.94 indicating a well-fitted model. However, in order to further confirm this we can cross-validate the model instead of using bootstrap aggregating as bagging does. Therefore we will do a 10-fold cross-validation using the functions from the “caret” package. Below is the code. ctrl<-trainControl(method="cv", number=10) trainModel<-train(sex~.,data=Wages1, method="treebag",trControl=ctrl) trainModel ## Bagged CART ## ## 3294 samples ## 3 predictors ## 2 classes: 'female', 'male' ## ## No pre-processing ## Resampling: Cross-Validated (10 fold) ## Summary of sample sizes: 2965, 2965, 2965, 2964, 2964, 2964, ... ## Resampling results ## ## Accuracy Kappa Accuracy SD Kappa SD ## 0.5504128 0.09712194 0.02580514 0.05233441 ## ##  Now the results are not so impressive. In many ways the model is terrible. The accuracy has fallen significantly and the kappa is almost 0. Remeber that cross-validation is an indicator of future performance. This means that our current model would not generalize well to other datasets. Bagging is not limited to decision trees and can be used for all machine learning models. The example used in this post was one that required the least time to run. For real datasets, the processing time can be quite long for the average laptop. # Ensemble Learning for Machine Models One way to improve a machine learning model is to not make just one model. Instead, you can make several models that all have different strengths and weaknesses. This combination of diverse abilities can allow for much more accurate predictions. The use of multiple models is know as ensemble learning. This post will provide insights into ensemble learning as they are used in developing machine models. The Major Challenge The biggest challenges in creating an ensemble of models is deciding what models to develop and how the various models are combined to make predictions. To deal with these challenges involves the use of training data and several different functions. The Process Developing an ensemble model begins with training data. The next step is the use of some sort of allocation function. The allocation function determines how much data each model receives in order to make predictions. For example, each model may receive a subset of the data or limit how many features each model can use. However, if several different algorithms are used the allocation function may pass all the data to each model with making any changes. After the data is allocated, it is necessary for the models to be created. From there, the next step is to determine how to combine the models. The decision on how to combine the models is made with a combination function. The combination function can take one of several approaches for determining final predictions. For example, a simple majority vote can be used which means that if 5 models where develop and 3 vote “yes” than the example is classified as a yes. Another option is to weight the models so that some have more influence then others in the final predictions. Benefits of Ensemble Learning Ensemble learning provides several advantages. One, ensemble learning improves the generalizability of your model. With the combine strengths of many different models and or algorithms it is difficult to go wrong Two, ensemble learning approaches allow for tackling large datasets. The biggest enemy to machine learning is memory. With ensemble approaches, the data can be broken into smaller pieces for each model. Conclusion Ensemble learning is yet another critical tool in the data scientist’s toolkit. The complexity of the world today makes it too difficult to lean on a singular model to explain things. Therefore, understanding the application of ensemble methods is a necessary step. # Developing a Customize Tuning Process in R In this post, we will learn how to develop customize criteria for tuning a machine learning model using the “caret” package. There are two things that need to be done in order to complete assess a model using customized features. These two steps are… • Determine the model evaluation criteria • Create a grid of parameters to optimize The model we are going to tune is the decision tree model made in a previous post with the C5.0 algorithm. Below is code for loading some prior information. library(caret); library(Ecdat) data(Wages1) DETERMINE the MODEL EVALUATION CRITERIA We are going to begin by using the “trainControl” function to indicate to R what re-sampling method we want to use, the number of folds in the sample, and the method for determining the best model. Remember, that there are many more options but these are the onese we will use. All this information must be saved into a variable using the “trainControl” function. Later, the information we place into the variable will be used when we rerun our model. For our example, we are going to code the following information into a variable we will call “chck” for re sampling we will use k-fold cross-validation. The number of folds will be set to 10. The criteria for selecting the best model will be the through the use of the “oneSE” method. The “oneSE” method selects the simplest model within one standard error of the best performance. Below is the code for our variable “chck” chck<-trainControl(method = "cv",number = 10, selectionFunction = "oneSE") For now this information is stored to be used later CREATE GRID OF PARAMETERS TO OPTIMIZE We now need to create a grid of parameters. The grid is essential the characteristics of each model. For the C5.0 model we need to optimize the model, number of trials, and if winnowing was used. Therefore we will do the following. • For model, we want decision trees only • Trials will go from 1-35 by increments of 5 • For winnowing, we do not want any winnowing to take place. In all we are developing 8 models. We know this based on the trial parameter which is set to 1, 5, 10, 15, 20, 25, 30, 35. To make the grid we use the “expand.grid” function. Below is the code. modelGrid<-expand.grid(.model ="tree", .trials= c(1,5,10,15,20,25,30,35), .winnow="FALSE") CREATE THE MODEL We are now ready to generate our model. We will use the kappa statistic to evaluate each model’s performance set.seed(1) customModel<- train(sex ~., data=Wages1, method="C5.0", metric="Kappa", trControl=chck, tuneGrid=modelGrid) customModel ## C5.0 ## ## 3294 samples ## 3 predictors ## 2 classes: 'female', 'male' ## ## No pre-processing ## Resampling: Cross-Validated (10 fold) ## Summary of sample sizes: 2966, 2965, 2964, 2964, 2965, 2964, ... ## Resampling results across tuning parameters: ## ## trials Accuracy Kappa Accuracy SD Kappa SD ## 1 0.5922991 0.1792161 0.03328514 0.06411924 ## 5 0.6147547 0.2255819 0.03394219 0.06703475 ## 10 0.6077693 0.2129932 0.03113617 0.06103682 ## 15 0.6077693 0.2129932 0.03113617 0.06103682 ## 20 0.6077693 0.2129932 0.03113617 0.06103682 ## 25 0.6077693 0.2129932 0.03113617 0.06103682 ## 30 0.6077693 0.2129932 0.03113617 0.06103682 ## 35 0.6077693 0.2129932 0.03113617 0.06103682 ## ## Tuning parameter 'model' was held constant at a value of tree ## ## Tuning parameter 'winnow' was held constant at a value of FALSE ## Kappa was used to select the optimal model using the one SE rule. ## The final values used for the model were trials = 5, model = tree ## and winnow = FALSE. The actually output is similar to the model that “caret” can automatically create. The difference here is that the criteria was set by us rather than automatically. A close look reveals that all of the models perform poorly but that there is no change in performance after ten trials. CONCLUSION This post provided a brief explanation of developing a customize way of assessing a models performance. To complete this, you need configure your options as well as setup your grid in order to assess a model. Understanding the customization process for evaluating machine learning models is one of the strongest ways to develop supremely accurate models that retain generalizability. # Developing an Automatically Tuned Model in R In this post, we are going to learn how to use the “caret” package to automatically tune a machine learning model. This is perhaps the simplest way to evaluate the performance of several models. In a later post, we will explore how to perform custom tuning to a model. The model we are trying to tune is the decision tree we made using the C5.0 algorithm in a previous post. Specifically we were trying to predict sex based on the variables available in the “Wages1” dataset in the “Ecdat” package. In order to accomplish our goal we will need to load the “caret” and “Ecdat”package, load the “Wages1” dataset as well as set the seed. Setting the seed will allow us to reproduce our results. Below is the code for these steps. library(caret); library(Ecdat) data(Wages1) set.seed(1) We will now build and display our model using the code below. tuned_model<-train(sex ~., data=Wages1, method="C5.0")  tuned_model ## C5.0 ## ## 3294 samples ## 3 predictors ## 2 classes: 'female', 'male' ## ## No pre-processing ## Resampling: Bootstrapped (25 reps) ## Summary of sample sizes: 3294, 3294, 3294, 3294, 3294, 3294, ... ## Resampling results across tuning parameters: ## ## model winnow trials Accuracy Kappa Accuracy SD Kappa SD ## rules FALSE 1 0.5892713 0.1740587 0.01262945 0.02526656 ## rules FALSE 10 0.5938071 0.1861964 0.01510209 0.03000961 ## rules FALSE 20 0.5938071 0.1861964 0.01510209 0.03000961 ## rules TRUE 1 0.5892713 0.1740587 0.01262945 0.02526656 ## rules TRUE 10 0.5938071 0.1861964 0.01510209 0.03000961 ## rules TRUE 20 0.5938071 0.1861964 0.01510209 0.03000961 ## tree FALSE 1 0.5841768 0.1646881 0.01255853 0.02634012 ## tree FALSE 10 0.5930511 0.1855230 0.01637060 0.03177075 ## tree FALSE 20 0.5930511 0.1855230 0.01637060 0.03177075 ## tree TRUE 1 0.5841768 0.1646881 0.01255853 0.02634012 ## tree TRUE 10 0.5930511 0.1855230 0.01637060 0.03177075 ## tree TRUE 20 0.5930511 0.1855230 0.01637060 0.03177075 ## ## Accuracy was used to select the optimal model using the largest value. ## The final values used for the model were trials = 10, model = rules ## and winnow = TRUE. There is a lot of information that is printed out. The first column is the type of model developed. Two types of models were developed either a rules-based classification tree or a normal decision tree. Next, is the winnow column. This column indicates if a winnowing process was used to remove poor predictor variables. The next two columns are accuracy and kappa which have been explained previously. The last two columns are the standard deviations of accuarcy and kappa. None of the models are that good but the purpose here is for teaching. At the bottom of the printout, r tells you which model was the best. For us, the best model was the fifth model from the top which was a rule-based, 10 trial model with winnow set to “TRUE”. We will now use the best model (the caret package automatically picks it) to make predictions on the training data. We will also look at the confusion matrix of the correct classification followed by there proportions. Below is the code. predict_model<-predict(tuned_model, Wages1) table(predict_model, Wages1$sex)
##
## predict_model female male
##        female    936  590
##        male      633 1135
prop.table(table(predict_model, Wages1$sex)) ## ## predict_model female male ## female 0.2841530 0.1791135 ## male 0.1921676 0.3445659 In term of prediction, the model was correct 62% of the time (.28 + .34 = .62). If we want to know, can also see the probabilities for each example using the following code. probTable<-(predict(tuned_model, Wages1, type="prob")) head(probTable) ## female male ## 1 0.6191287 0.38087132 ## 2 0.2776770 0.72232303 ## 3 0.2975327 0.70246734 ## 4 0.7195866 0.28041344 ## 5 1.0000000 0.00000000 ## 6 0.9092993 0.09070072  Conclusion In this post, we looked at an automated way to determine the best model among many using the “caret” package. Understanding how to improve the performance of a model is critical skill in machine learning. # Improving the Performance of Machine Learning Model For many, especially beginners, making a machine learning model is difficult enough. Trying to understand what to do, how to specify the model, among other things is confusing in itself. However, after developing a model it is necessary to assess ways in which to improve performance. This post will serve as an introduction to understanding how to improving model performance. In particular, we will look at the following • When it is necessary to improve performance • Parameter tuning When to Improve It is not always necessary to try and improve the performance of a model. There are times when a model does well and you know this through the evaluating it. If the commonly used measures are adequate there is no cause for concern. However, there are times when improvement is necessary. Complex problems, noisy data, and trying to look for subtle/unclear relationships can make improvement necessary. Normally, real-world data has the problems so model improvement is usually necessary. Model improvement requires the application of scientific means in an artistic manner. It requires a sense of intuition at times and also brute trial-and-error effort as well. The point is that there is no singular agreed upon way to improve a model. It is better to focus on explaining how you did it if necessary. Parameter Tuning Parameter tuning is the actual adjustment of model fit options. Different machine learning models have different options that can be adjusted. Often, this process can be automated in r through the use of the “caret” package. When trying to decide what to do when tuning parameters it is important to remember the following. • What machine learning model and algorithm you are using for your data. • Which parameters you can adjust. • What criteria you are using to evaluate the model Naturally, you need to know what kind of model and algorithm you are using in order to improve the model. There are three types of models in machine learning, those that classify, those that employ regression, and those that can do both. Understanding this helps you to make decision about what you are trying to do. Next, you need to understand what exactly you or r are adjusting when analyzing the model. For example, for C5.0 decision trees “trials” is one parameter you can adjust. If you don’t know this, you will not know how the model was improved. Lastly, it is important to know what criteria you are using to compare the various models. For classifying models you can look at the kappa and the various information derived from the confusion matrix. For regression based models you may look at the r-square, the RMSE (Root mean squared error), or the ROC curve. Conclusion As you can perhaps tell there is an incredible amount of choice and options in trying to improve a model. As such, model improvement requires a clearly developed strategy that allows for clear decision-making. In a future post, we will look at an example of model improvement. # K-Fold Cross-Validation In this post, we are going to look at k-fold cross-validation and its use in evaluating models in machine learning. K-fold cross-validation is use for determining the performance of statistical models. How it works is the data is divided into a pre-determined number of folds (called ‘k’). One fold is used to determine the model estimates and the other folds are used for evaluating. This is done k times and the results are average based on a statistic such as kappa to see how the model performs. In our example we are going to review a model we made using the C5.0 algorithm. In that post, we were trying to predict gender based on several other features. First, we need to load several packages into R as well as the dataset we are going to use. All of this is shared in the code below library(caret);library(C50);library(irr);library(Ecdat) data("Wages1") We now need to set the seed. This is important for allowing us to reproduce the results. Every time a k-fold is perform the results can be slightly different but setting the seed prevents this. The code is as follows set.seed(1) We will now create are folds. How many folds to create is up to the researcher. For us, we are going to create ten folds. What this means is that R will divide are sample into ten equal parts. To do this we use the “createFolds” function from the “caret” package. After creating the folds, we will view the results using the “str” function which will tell us how many examples are in each fold. Below is the code to complete this. folds<-createFolds(Wages1$sex, k=10)
str(folds)
## List of 10
##  $Fold01: int [1:328] 8 13 18 37 39 57 61 67 78 90 ... ##$ Fold02: int [1:329] 5 27 47 48 62 72 76 79 85 93 ...
##  $Fold03: int [1:330] 2 10 11 31 32 34 36 64 65 77 ... ##$ Fold04: int [1:330] 19 24 40 45 55 58 81 96 99 102 ...
##  $Fold05: int [1:329] 6 14 28 30 33 84 88 91 95 97 ... ##$ Fold06: int [1:330] 4 15 16 38 43 52 53 54 56 63 ...
##  $Fold07: int [1:330] 1 3 12 22 29 50 66 73 75 82 ... ##$ Fold08: int [1:330] 7 21 23 25 26 46 51 59 60 83 ...
##  $Fold09: int [1:329] 9 20 35 44 49 68 74 94 100 105 ... ##$ Fold10: int [1:329] 17 41 42 71 101 107 117 165 183 184 ...

As you can see, we normally have about 330 examples per fold.

In order to get the results that we need. We have to take fold 1 to make the model and fold 2-10 to evaluate it. We repeat this process until every combination possible is used. First fold 1 is used and 2-10 are the test data, then fold 2 is used and then folds 1, 3-10 are the test data etc.  Manually coding this would take great deal of time. To get around this we will use the “lapply” function.

Using “lapply” we will create a function that takes “x” (one of our folds) and makes it the “training set” shown here as “Wages1_train”. Next we assigned the rest of the folds to be the “test” (Wages1_test) set as depicted with the “-x”. The next two lines of code should look familiar as it is the code for developing a decision tree. The “Wages_actual” are the actual labels for gender in the “Wages_1” testing set.

The “kappa2” function is new and it comes form the “irr” package. The kappa statistic is a measurement of accuracy of a  model while taking into account chance. The closer the value is to 1 the better. Below is the code for what has been discussed.

results<-lapply(folds, function(x) {
Wages1_train<-Wages1[x, ]
Wages1_test<-Wages1[-x, ]
Wages1_model<-C5.0(sex~.,data=Wages1)
Wages1_pred<-predict(Wages1_model, Wages1_test)
Wages1_actual<-Wages1_test$sex Wages1_kappa<-kappa2(data.frame(Wages1_actual, Wages1_pred))$value
return(Wages1_kappa)
})

To get our results, we will use the “str” function again to display them. This will tell us the kappa for each fold. To really see how are model does we need to calculate the mean kappa of the ten models. This is done with the “unlist” and “mean” function as shown below

str(results)
## List of 10
##  $Fold01: num 0.205 ##$ Fold02: num 0.186
##  $Fold03: num 0.19 ##$ Fold04: num 0.193
##  $Fold05: num 0.202 ##$ Fold06: num 0.208
##  $Fold07: num 0.196 ##$ Fold08: num 0.202
##  $Fold09: num 0.194 ##$ Fold10: num 0.204
mean(unlist(results))
## [1] 0.1978915

The final mean kappa was 0.19 which is really poor. It indicates that the model is no better at predicting then chance alone. However, for illustrative purposes we now understand how to perfrom a k-fold cross-validation

The receiver operating characteristic curve (ROC curve) is a tool used in statistical research to assess the trade-off of detecting true positives and true negatives. The origins of this tool goes all the way back to WWII when engineers were trying to distinguish between true and false alarms. Now this technique is used in machine learning

This post will explain the ROC curve and provide and example using R.

Below is a diagram of an ROC curve

On the X axis we have the false positive rate. As you move to the right the false positive rate increases which is bad. We want to be as close to zero as possible.

On the y axis we have the true positive rate. Unlike the x axis we want the true positive rate to be as close to 100 as possible. In general we want a low value on the x-axis and a high value on the y-axis.

In the diagram above, the diagonal line called “Test without diagnostic benefit” represents a model that cannot tell the difference between true and false positives. Therefore, it is not useful for our purpose.

The L-shaped curve call “Good diagnostic test” is an example of an excellent model. This is because  all the true positives are detected .

Lastly, the curved-line called “Medium diagonistic test” represents an actually model. This model is a balance between the perfect L-shaped model and the useless straight-line model. The curved-line model is able to moderately distinguish between false and true positives.

Area Under the ROC Curve

The area under an ROC curve is literally called the “Area Under the Curve” (AUC). This area is calculated with a standardized value ranging from 0 – 1. The closer to 1 the better the model

We will now look at an analysis of a model using the ROC curve and AUC. This is based on the results of a post using the KNN algorithm for nearest neighbor classification. Below is the code

predCollege <- ifelse(College_test_pred=="Yes", 1, 0)
realCollege <- ifelse(College_test_labels=="Yes", 1, 0)
pr <- prediction(predCollege, realCollege)
collegeResults <- performance(pr, "tpr", "fpr")
plot(collegeResults, main="ROC Curve for KNN Model", col="dark green", lwd=5)
abline(a=0,b=1, lwd=1, lty=2)
aucOfModel<-performance(pr, measure="auc")
unlist(aucOfModel@y.values)
1. The first to variables (predCollege & realCollege) is just for converting the values of the prediction of the model and the actual results to numeric variables
2. The “pr” variable is for storing the actual values to be used for the ROC curve. The “prediction” function comes from the “ROCR” package
3. With the information information of the “pr” variable we can now analyze the true and false positives, which are stored in the “collegeResults” variable. The “performance” function also comes from the “ROCR” package.
4. The next two lines of code are for plot the ROC curve. You can see the results below

6. The curve looks pretty good. To confirm this we use the last two lines of code to calculate the actually AUC. The actual AUC is 0.88 which is excellent. In other words, the model developed does an excellent job of discerning between true and false positives.

Conclusion

The ROC curve provides one of many ways in which to assess the appropriateness of a model. As such, it is yet another tool available for a person who is trying to test models.

# Using Confusion Matrices to Evaluate Performance

The data within a confusion matrix can be used to calculate several different statistics that can indicate the usefulness of a statistical model in machine learning. In this post, we will look at several commonly used measures, specifically…

• accuracy
• error
• sensitivity
• specificity
• precision
• recall
• f-measure

Accuracy

Accuracy is probably the easiest statistic to understand. Accuracy is the total number of items correctly classified divided by the total number of items below is the equation

accuracy =   TP + TN
TP + TN + FP  + FN

TP =  true positive, TN =  true negative, FP = false positive, FN = false negative

Accuracy can range in value from 0-1 with one representing 100% accuracy. Normally, you don’t want perfect accuracy as this is an indication of overfitting and your model will probably not do well with other data.

Error

Error is the opposite of accuracy and represent the percentage of examples that are incorrectly classified it’s equation is as follows.

error =   FP + FN
TP + TN + FP  + FN

The lower the error the better in general. However, if error is 0 it indicates overfitting. Keep in mind that error is the inverse of accuracy. As one increases the other decreases.

Sensitivity

Sensitivity is the proportion of true positives that were correctly classified.The formula is as follows

sensitivity =       TP
TP + FN

This may sound confusing but high sensitivity is useful for assessing a negative result. In other words, if I am testing people for a disease and my model has a high sensitivity. This means that the model is useful telling me a person does not have a disease.

Specificity

Specificity measures the proportion of negative examples that were correctly classified. The formula is below

specificity =       TN
TN + FP

Returning to the disease example, a high specificity is a good measure for determining if someone has a disease if they test positive for it. Remember that no test is foolproof and there are always false positives and negatives happening. The role of the researcher is to maximize the sensitivity or specificity based on the purpose of the model.

Precision

Precision is the proportion of examples that are really positive. The formula is as follows

precision =       TP
TP + FP

The more precise a model is the more trustworthy it is. In other words, high precision indicates that the results are relevant.

Recall

Recall is a measure of the completeness of the results of a model. It is calculated as follows

recall =       TP
TP + FN

This formula is the same as the formula for sensitivity. The difference is in the interpretation. High recall means that the results have a breadth to them such as in search engine results.

F-Measure

The f-measure uses recall and precision to develop another way to assess a model. The formula is below

sensitivity =      2 * TP
2 * TP + FP + FN

The f-measure can range from 0 – 1 and is useful for comparing several potential models using one convenient number.

Conclusion

This post provide a basic explanation of various statistics that can be used to determine the strength of a model. Through using a combination of statistics a researcher can develop insights into the strength of a model. The only mistake is relying exclusively on any single statistical measurement.

# Using Probability of the Prediction to Evaluate a Machine Learning Model

In evaluating a model when employing machine learning techniques, there are three common types of data used for evaluation.

• The actual classification values
• The predicted classification values
• The estimated probability of the prediction

The first two types of data (actual and predicted) are used for assessing the accuracy of a model in several different ways such as error rate, sensitivity, specificity, etc.

The benefit of the probabilities of prediction is that it is a measure of a model’s confidence in its prediction. If you need to compare to models and one is more confident in it’s prediction of its classification of examples, the more confident model is the better learner.

In this post, we will look at examples of the probability predictions of several models that have been used in this blog in the past.

Prediction Probabilities for Decision Trees

Our first example come from the decision tree we made using the C5.0 algorithm. Below is the code for calculating the probability of the correct classification of each example in the model followed by an output of the first

Wage_pred_prob<-predict(Wage_model, Wage_test, type="prob")

 head(Wage_pred_prob)
female      male
497  0.2853016 0.7146984
1323 0.2410568 0.7589432
1008 0.5770177 0.4229823
947  0.6834378 0.3165622
695  0.5871323 0.4128677
1368 0.4303364 0.5696636

The argument “type” is added to the “predict” function so that R calculates the probability that the example is classified correctly. A close look at the results using the “head” function provides a list of 6 examples from the model.

• For example 497, there is a 28.5% probability that this example is female and a 71.5% probability that this example is male. Therefore, the model  predicts that this example is male.
• For example 1322, there is a 24% probability that this example is female and a 76% probability that this example is male. Therefore, the model  predicts that this example is male.
• etc.

Prediction Probabilities for KNN Nearest Neighbor

Below is the code for finding the probilities for KNN algorithm.

College_test_pred_prob<-knn(train=College_train, test=College_test,
+                        cl=College_train_labels, k=27, prob=TRUE)

College_test_pred_prob

The print for this is rather long. However, you can match the predict level with the actual probability by looking carefully at the data.

• For example 1, there is a 77% probability that this example is a yes and a 23% probability that this example is a no. Therefore, the model  predicts that this example as yes.
• For example 2, there is a 71% probability that this example is no and a 29% probability that this example is yes. Therefore, the model  predicts that this example is a no.

Conclusion

One of the primary purposes of the probabilities option is in comparing various models that are derived from the same data. This information combined with other techniques for evaluating models can help a researcher in determining the most appropriate model of analysis.

# Kmeans Analysis in R

In this post, we will conduct a kmeans analysis on some data on student alcohol consumption. The data is available at the UC Irvine Machine Learning Repository and is available at https://archive.ics.uci.edu/ml/datasets/STUDENT+ALCOHOL+CONSUMPTION

We want to see what segments are within the sample of students who participated in this study on several factors in addition to alcohol consumption. Understanding the characteristics that groups of students have in common could be useful in reaching out to them for various purposes.

We will begin by loading the “stats” package for the kmeans analysis. Then we will combine the data at it is in two different files and we will explore the data using tables and histograms. I will not print the results of the exploration of the data here as there are too many variables. Lastly, we need to set the seed in order to get the same results each time. The code is still found below.

library(stats)
student.mat <- read.csv("~/Documents/R working directory/student-mat.csv", sep=";")
student.por <- read.csv("~/Documents/R working directory/student-por.csv", sep=";")
student_alcohol <- rbind(student.mat, student.por)
set.seed(123)
options(digits = 2)
• str(student_alcohol)
• hist(student_alcoholage)
• table(studentalcoholage)
• table(student_alcoholfamsize)
• table(studentalcoholfamsize)
• table(studentalcoholPstatus)
• hist(student_alcoholMedu)
• hist(studentalcoholMedu)
• hist(studentalcoholFedu)
• hist(student_alcoholtraveltime)
• hist(studentalcoholtraveltime)
• hist(studentalcoholstudytime)
• hist(student_alcoholfailures)
• table(studentalcoholfailures)
• table(studentalcoholschoolsup)
• table(student_alcoholfamsup)
• table(studentalcoholfamsup)
• table(studentalcoholpaid)
• table(student_alcoholactivities)
• table(studentalcoholactivities)
• table(studentalcoholnursery)
• table(student_alcoholhigher)
• table(studentalcoholhigher)
• table(studentalcoholinternet)
• hist(student_alcoholfamrel)
• hist(studentalcoholfamrel)
• hist(studentalcoholfreetime)
• hist(student_alcoholgoout)
• hist(studentalcoholgoout)
• hist(studentalcoholDalc)
• hist(student_alcoholWalc)
• hist(studentalcoholWalc)
• hist(studentalcoholhealth)
• hist(student_alcohol$absences) The details about the variables can be found at the website link in the first paragraph of this post. The study look at students alcohol use and other factors related to school and family life. Before we do the actual kmeans clustering we need to normalize the variables. This is because are variables are measured using different scales. Some are Likert with 5 steps, while others are numeric going from 0 to over 300. The different ranges have an influence on the results. To deal with this problem we will use the “scale” for the variables that will be included in the analysis. Below is the code student_alcohol_clean<-as.data.frame(student_alcohol) student_alcohol_clean$age<-scale(student_alcohol$age) student_alcohol_clean$address<-scale(as.numeric(student_alcohol$address)) student_alcohol_clean$famsize<-scale(as.numeric(student_alcohol$famsize)) student_alcohol_clean$Pstatus<-scale(as.numeric(student_alcohol$Pstatus)) student_alcohol_clean$Medu<-scale(student_alcohol$Medu) student_alcohol_clean$Fedu<-scale(student_alcohol$Fedu) student_alcohol_clean$traveltime<-scale(student_alcohol$traveltime) student_alcohol_clean$studytime<-scale(student_alcohol$studytime) student_alcohol_clean$failures<-scale(student_alcohol$failures) student_alcohol_clean$schoolsup<-scale(as.numeric(student_alcohol$schoolsup)) student_alcohol_clean$famsup<-scale(as.numeric(student_alcohol$famsup)) student_alcohol_clean$paid<-scale(as.numeric(student_alcohol$paid)) student_alcohol_clean$activities<-scale(as.numeric(student_alcohol$activities)) student_alcohol_clean$internet<-scale(as.numeric(student_alcohol$internet)) student_alcohol_clean$famrel<-scale(student_alcohol$famrel) student_alcohol_clean$freetime<-scale(student_alcohol$freetime) student_alcohol_clean$goout<-scale(student_alcohol$goout) student_alcohol_clean$Dalc<-scale(student_alcohol$Dalc) student_alcohol_clean$Walc<-scale(student_alcohol$Walc) student_alcohol_clean$health<-scale(student_alcohol$health) student_alcohol_clean$absences<-scale(student_alcohol$absences) student_alcohol_clean$G1<-scale(student_alcohol$G1) student_alcohol_clean$G2<-scale(student_alcohol$G2) student_alcohol_clean$G3<-scale(student_alcohol$G3) We also need to create a matrix in order to deal with the factor variables. All factor variables need to be converted so that they have dummy variables for the analysis. To do this we use the “matrix” function as shown in the code below. student_alcohol_clean_matrix<-(model.matrix(~.+0, data=student_alcohol_clean)) We are now ready to conduct our kmeans cluster analysis using the “kmeans” function. We have to determine how many clusters to develop before the analysis. There are statistical ways to do this but another method is domain knowledge. Since we are dealing with teenagers, it is probably that there will be about four distinct groups because of how high school is structured. Therefore, we will use four segments for our analysis. Our code is below. alcohol_cluster<-kmeans(student_alcohol_clean_matrix, 4) To view the results we need to view two variables in are “alcohol_cluster” list. The “size” variable will tell us how many people are in each cluster and the “centers” variable describes a clusters characteristics on a particular variable. Below is the code alcohol_cluster$size # size of clusters
## [1] 191 381 325 147
alcohol_cluster$centers #center of clusters ## schoolGP schoolMS sexM age address famsize Pstatus Medu Fedu ## 1 0.74 0.26 0.70 0.06 0.0017 0.265 -0.030 0.25 0.29 ## 2 0.69 0.31 0.27 -0.15 -0.1059 -0.056 -0.031 -0.53 -0.45 ## 3 0.88 0.12 0.45 -0.13 0.3363 0.005 0.016 0.73 0.58 ## 4 0.56 0.44 0.48 0.59 -0.4712 -0.210 0.086 -0.55 -0.51 ## Mjobhealth Mjobother Mjobservices Mjobteacher Fjobhealth Fjobother ## 1 0.079 0.30 0.27 0.199 0.0471 0.54 ## 2 0.031 0.50 0.17 0.024 0.0210 0.62 ## 3 0.154 0.26 0.28 0.237 0.0708 0.48 ## 4 0.034 0.46 0.22 0.041 0.0068 0.60 ## Fjobservices Fjobteacher reasonhome reasonother reasonreputation ## 1 0.33 0.042 0.27 0.120 0.20 ## 2 0.25 0.031 0.27 0.113 0.20 ## 3 0.28 0.123 0.24 0.077 0.34 ## 4 0.29 0.034 0.17 0.116 0.15 ## guardianmother guardianother traveltime studytime failures schoolsup ## 1 0.69 0.079 0.17 -0.32 -0.12 -0.079 ## 2 0.70 0.052 0.10 0.10 -0.26 0.269 ## 3 0.73 0.040 -0.37 0.29 -0.35 -0.213 ## 4 0.65 0.170 0.33 -0.49 1.60 -0.123 ## famsup paid activities nurseryyes higheryes internet romanticyes ## 1 -0.033 0.253 0.1319 0.79 0.92 0.228 0.34 ## 2 -0.095 -0.098 -0.2587 0.76 0.93 -0.387 0.35 ## 3 0.156 0.079 0.2237 0.88 1.00 0.360 0.31 ## 4 -0.057 -0.250 0.0047 0.73 0.70 -0.091 0.49 ## famrel freetime goout Dalc Walc health absences G1 G2 G3 ## 1 -0.184 0.43 0.76 1.34 1.29 0.273 0.429 -0.23 -0.17 -0.129 ## 2 -0.038 -0.31 -0.35 -0.37 -0.40 -0.123 -0.042 -0.17 -0.12 -0.053 ## 3 0.178 -0.01 -0.14 -0.41 -0.35 -0.055 -0.184 0.90 0.87 0.825 ## 4 -0.055 0.25 0.22 0.11 0.14 0.087 -0.043 -1.24 -1.40 -1.518 The size of the each cluster is about the same which indicates reasonable segmentation of the sample. The output for “centers” tells us how much above or below the mean a particular cluster is. For example, for the variable “age” we see the following age 1 0.06 2 -0.15 3 -0.13 4 0.59 What this means is that people in cluster one have an average age 0.06 standard deviations above the mean, cluster two is -0.14 standard deviations below the mean etc. To give our clusters meaning we have to look at the variables and see which one the clusters are extremely above or below the mean. Below is my interpretation of the clusters. The words in parenthesis is the variable from which I made my conclusion Cluster 1 doesn’t study much (studytime), lives in the biggest families (famsize), requires litle school support (schoolsup), has a lot of free time (freetime), and consumes the most alcohol (Dalc, Walc), lives in an urban area (address), loves to go out the most (goout), and has the most absences. This is the underachieving party alcoholics of the sample Cluster 2 have parents that are much less educated (Medu, Fedu), requires the most school support (schoolsup), while receiving the less family support (famsup), have the least involvement in extra-curricular activities (activities), has the least internet access at home (internet), socialize the least (goout), and lowest alcohol consumption. (Dalc, Walc) This cluster is the unsupported non-alcoholic loners of the sample Cluster 3 has the most educated parents (Medu, Fedu), live in an urban setting (address) choose their school based on reputation (reasonreputation), have the lowest travel time to school (traveltime), study the most (studytime), rarely fail a course (failures), have the lowest support from the school while having the highest family support and family relationship (schoolsup, famsup, famrel), most involve in extra-curricular activities (activities), best internet acess at home (internet), least amount of free time (freetime) low alcohol consumption (Dalc, Walc). This cluster represents the affluent non-alcoholic high achievers. CLuster 4 is the oldest (age), live in rural setting (address), has the smallest families (famsize), the least educated parents (Medu, Fedu), spends the most time traveling to school (traveltime), doesnt study much (studytime), has the highest failure rate (failures), never pays for extra classes (paid), most likely to be in a relationship (romanticyes), consumes alcohol moderately (Dalc, Walc), does poorly in school (G3). These students are the students in the greatest academic danger. To get better insights, we can add the cluster results to our original dataset that was not normalize we can then identify what cluster each student belongs to ad calculate unstandardized means if we wanted. student_alcohol$cluster<-alcohol_cluster$cluster # add clusters back to original data normalize does not mean much View(student_alcohol) aggregate(data=student_alcohol, G3~cluster, mean) ## cluster G3 ## 1 1 10.8 ## 2 2 11.1 ## 3 3 14.5 ## 4 4 5.5 The “aggregate function” tells us the average for each cluster on this question. We could do this for all of our variables to learn more about our clusters. Kmeans provides a researcher with an understanding of the homogeneous characteristics of individuals within a sample. This information can be used to develop intervention plans in education or marketing plans in business. As such, kmeans is another powerful tool in machine learning # K-Means Clustering There are times in research when you neither want to predict nor classify examples. Rather, you want to take a dataset and segment the examples within the dataset so that examples with similar characteristics are gather together. When your goal is to great several homogeneous groups within a sample it involves clustering. One type of clustering used in machine learning is k-means clustering. In this post we will learn the following about k-means clustering. • The purpose of k-means • Pros and cons of k-means The Purpose of K-means K-means has a number of applications. In the business setting, k-means has been used to segment customers. Businesses use this information to adjusts various aspects of their strategy for reaching their customers. Another purpose for k-means is in simplifying large datasets so that you have several smaller datasets with similar characteristics. This subsetting could be useful in finding distinct patterns K-means is a form of unsupervised classification. This means that the results label examples that the researcher must give meaning too. When R gives the results of an analysis it just labels the clusters as 1,2,3 etc. It is the researchers job to look at the clusters and give a qualitative meaning to them. Pros and Cons of K-Means The pros of k-means is that it is simple, highly flexible, and efficient. The simplicity of k-means makes it easy to explain the results in contrast to artificial neural networks or support vector machines. The flexibility of k-means allows for easy adjust if there are problems. Lastly, the efficiency of k-means implies that the algorithm is good at segmenting a dataset. Some drawbacks to k-means is that it does not allows develop the most optimal set of clusters and that the number of clusters to make must be decided before the analysis. When doing the analysis, the k-means algorithm will randomly selecting several different places from which to develop clusters. This can be good or bad depending on where the algorithm chooses to begin at. From there, the center of the clusters is recalculated until an adequate “center” is found for the number of clusters requested. How many clusters to include is left at the discretion of the researcher. This involves a combination of common sense, domain knowledge, and statistical tools. Too many clusters tells you nothing because the groups becoming very small and there are too many of them. There are statistical tools that measure within group homogeneity and with group heterogeneity. In addition, there is a technique called a dendrogram. The results of a dendrogram analysis provides a recommendation of how many clusters to use. However, calculating a dendrogram for a large dataset could potential crash a computer due to the computational load and the limits of RAM. Conclusion. K-means is an analytical tool that helps to separate apples from oranges to give you one example. If you are in need of labeling examples based on the features in the dataset this method can be useful. # Market Basket Analysis in R In this post, we will conduct a market basket analysis on the shopping habits of people at a grocery store. Remember that a market basket analysis provides insights through indicating relationships among items that are commonly purchased together. The first thing we need to do is load the package that makes association rules, which is the “arules” package. Next, we need to load our dataset groceries. This dataset is commonly used as a demonstration for market basket analysis. However, you don’t won’t to load this dataset as dataframe because it leads to several technical issues during the analysis. Rather you want to load it as a sparse matrix. The function for this is “read.transactions” and is available in the “arules” pacakge library(arules) ## Loading required package: Matrix ## ## Attaching package: 'arules' ## ## The following objects are masked from 'package:base': ## ## abbreviate, write #make sparse matrix groceries<-read.transactions("/home/darrin/Documents/R working directory/Machine-Learning-with-R-datasets-master/groceries.csv", sep = ",") Please keep in mind that the location of the file on your computer will be different from my hard drive. We will now explore the data set by using several different functions. First, we will use the “summary” function as indicated below. summary(groceries) ## transactions as itemMatrix in sparse format with ## 9835 rows (elements/itemsets/transactions) and ## 169 columns (items) and a density of 0.02609146 ## ## most frequent items: ## whole milk other vegetables rolls/buns soda ## 2513 1903 1809 1715 ## yogurt (Other) ## 1372 34055 ## ## element (itemset/transaction) length distribution: ## sizes ## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ## 2159 1643 1299 1005 855 645 545 438 350 246 182 117 78 77 55 ## 16 17 18 19 20 21 22 23 24 26 27 28 29 32 ## 46 29 14 14 9 11 4 6 1 1 1 1 3 1 ## ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 1.000 2.000 3.000 4.409 6.000 32.000 ## ## includes extended item information - examples: ## labels ## 1 abrasive cleaner ## 2 artif. sweetener ## 3 baby cosmetics The output tells us the number of rows in our dataset (9835) columns (169) as well as the density, which is the percentage of columns that are not empty (2.6%). This may seem small but remember that the number of purchases varies from person to person so this affects how many empty columns there are. Next, we have the most commonly purchased items. Milk and other vegetables were the two most common followed by other foods. After the most frequent items we have the size of each transaction. For example, 2159 people purchased one item during a transaction. While one person purchased 32 items in a transaction. Lastly, we summary statistics about transactions. On average, a person would purchased 4.4 items per transaction. We will now look at the support of different items. Remember, that the support is the frequency of an item in the dataset. We will use the “itemFrequencyPlot” function to do this and we will add the argument “topN” to sort the items from most common to less for the 15 most frequent transactions. Below is the code itemFrequencyPlot(groceries, topN=15) The plot that is produce gives you an idea of what people were purchasing. We will now attempt to develop association rules using the “apriori” function. For now we will use the default settings for support and confidence (confidence is the proportion of transactions that have they same item(s)). The default for support is 0.1 and for confidence it is 0.8. Below is the code. apriori(groceries) ## Apriori ## ## Parameter specification: ## confidence minval smax arem aval originalSupport support minlen maxlen ## 0.8 0.1 1 none FALSE TRUE 0.1 1 10 ## target ext ## rules FALSE ## ## Algorithmic control: ## filter tree heap memopt load sort verbose ## 0.1 TRUE TRUE FALSE TRUE 2 TRUE ## ## Absolute minimum support count: 983 ## ## set item appearances ...[0 item(s)] done [0.00s]. ## set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s]. ## sorting and recoding items ... [8 item(s)] done [0.00s]. ## creating transaction tree ... done [0.00s]. ## checking subsets of size 1 2 done [0.00s]. ## writing ... [0 rule(s)] done [0.00s]. ## creating S4 object ... done [0.00s]. ## set of 0 rules As you can see from the printout, nothing meets the criteria of a support of 0.1 and confidence of 0.8. How to play with these numbers is a matter of experience as there are few strong rules for this matter. Below, I set the support to 0.006, confidence to 0.25, and the minimum number of rules items to 2. The support of 0.006 means that this item must have been purchased at least 60 times out of 9835 items and the confidence of 0.25 means that rule needs to be accurate 25% of the time. Lastly, I want at least two items in each rule that is produce as indicated by minlen = 2. Below is the code with the “summary” as well. groceriesrules<-apriori(groceries, parameter = list(support=0.006, confidence = 0.25, minlen=2)) ## Apriori ## ## Parameter specification: ## confidence minval smax arem aval originalSupport support minlen maxlen ## 0.25 0.1 1 none FALSE TRUE 0.006 2 10 ## target ext ## rules FALSE ## ## Algorithmic control: ## filter tree heap memopt load sort verbose ## 0.1 TRUE TRUE FALSE TRUE 2 TRUE ## ## Absolute minimum support count: 59 ## ## set item appearances ...[0 item(s)] done [0.00s]. ## set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s]. ## sorting and recoding items ... [109 item(s)] done [0.00s]. ## creating transaction tree ... done [0.00s]. ## checking subsets of size 1 2 3 4 done [0.00s]. ## writing ... [463 rule(s)] done [0.00s]. ## creating S4 object ... done [0.00s]. summary(groceriesrules) ## set of 463 rules ## ## rule length distribution (lhs + rhs):sizes ## 2 3 4 ## 150 297 16 ## ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 2.000 2.000 3.000 2.711 3.000 4.000 ## ## summary of quality measures: ## support confidence lift ## Min. :0.006101 Min. :0.2500 Min. :0.9932 ## 1st Qu.:0.007117 1st Qu.:0.2971 1st Qu.:1.6229 ## Median :0.008744 Median :0.3554 Median :1.9332 ## Mean :0.011539 Mean :0.3786 Mean :2.0351 ## 3rd Qu.:0.012303 3rd Qu.:0.4495 3rd Qu.:2.3565 ## Max. :0.074835 Max. :0.6600 Max. :3.9565 ## ## mining info: ## data ntransactions support confidence ## groceries 9835 0.006 0.25 Are current analysis has 463 rules. This is a major improvement from 0. We can also see how many rules contain 2 (150), 3 (297), and 4 items (16). We also have a summary of of the average number of items per rule. The next is descriptive stats on the support and confidence of the rules generated. Something that is new for us is the “lift” column. Lift is a measure how much more likely an item is to be purchased above its number rate. Anything above 1 means that the likelihood of purchase is higher than chance. We are now going to look for useful rules from our dataset. We are looking for rules that we can use to make decisions. It takes industry experiences in the field of your data to really glean useful rules. For now, we can only determine this statistically by sorting the rules by their lift. Below is the code for this inspect(sort(groceriesrules, by="lift")[1:7]) ## lhs rhs support confidence lift ## 1 {herbs} => {root vegetables} 0.007015760 0.4312500 3.956 ## 2 {berries} => {whipped/sour cream} 0.009049314 0.2721713 3.796 ## 3 {other vegetables, ## tropical fruit, ## whole milk} => {root vegetables} 0.007015760 0.4107143 3.768 ## 4 {beef, ## other vegetables} => {root vegetables} 0.007930859 0.4020619 3.688 ## 5 {other vegetables, ## tropical fruit} => {pip fruit} 0.009456024 0.2634561 3.482 ## 6 {beef, ## whole milk} => {root vegetables} 0.008032537 0.3779904 3.467 ## 7 {other vegetables, ## pip fruit} => {tropical fruit} 0.009456024 0.3618677 3.448 The first three rules rules are translated into simply English below as ordered by lift. 1. If herbs are purchased then root vegetables are purchased 2. If berries are purchased then whipped sour cream is purchased 3. If other vegetables, tropical fruit and whole milk are purchased then root vegetables are purchased Conclusion Since we are making no predictions, there is no way to really objectively improve the model. This is normal when the learning is unsupervised. If we had to make a recommendation based on the results we could say that the store should place all vegetables near each other. The power of market basket analysis is allowing the researcher to identify relationships that may not have been noticed any other way. Naturally, insights gained from this approach must be use for practical actions in the setting in which they apply. # Understanding Market Basket Analysis Market basket analysis a machine learning approach that attempts to find relationships among a group of items in a data set. For example, a famous use of this method was when retailers discovered an association between beer and diapers. Upon closer examination, the retailers found that when men came to purchase diapers for their babies they would often buy beer in the same trip. With this knowledge, the retailers placed beer and diapers next to each other in the store and this further increased sales. In addition, many of the recommendation systems we experience when shopping online use market basket analysis results to suggest additional products to us. As such, market basket analysis is an intimate part of our lives with us even knowing. In this post, we will look at some of the details of market basket analysis such as association rules, apriori, and the role of support and confidence. Association Rules The heart of market basket analysis are association rules. Association rules explain patterns of relationship among items. Below is an example {rice, seaweed} -> {soy sauce} Everything in curly braces { } is an itemset, which is some form of data that occurs often in the dataset based on a criteria. Rice and seaweed is our itemset on the left and soy sauce is our itemset on the right. The arrow -> indicates what comes first as we read from left to right. If we put this association rule in simply English it would say “if someone buys rice and seaweed then they will buy soy sauce”. The practical application of this rule is to place rice, seaweed and soy sauce near each other in order to reinforce this rule when people come to shop. The Algorithm Market basket analysis uses a apriori algorithm. This algorithm is useful for unsupervised learning that does not require any training and thus no predictions. The apriori algorithm is especially useful with large datasets but it employs simple procedures to find useful relationships among the items. The shortcut that this algorithm uses is the “apriori property” which states that all sugsets of a frequent itemset must also be frequent. What this means in simply English is that the items in an itemset need to be common in the overall dataset. This simple rule saves a tremendous amount of computational time. Support and Confidence To key pieces of information that can further refine the work of the apriori algorithm is support and confidence. Support is a measure of the frequency of an itemset ranging from 0 (no support) to 1 (highest support). High support indicates the importance of the itemset in the data and contributes to the itemset being used to generate association rule(s). Returning to our rice, seaweed, and soy sauce example. We can say that the support for soy sauce is 0.4. This means that soy sauce appears in 40% of the purchases in the dataset which is pretty high. Confidence is a measure of the accuracy of an association rule which is measured from 0 to 1. The higher the confidence the more accurate the association rule. If we say that our rice, seaweed, and soy sauce rule has a confidence of 0.8 we are saying that when rice and seaweed are purchased together, 80% of the time soy sauce is purchased as well. Support and confidence can be used to influence the apriori algorithm by setting cutoff values to be searched for. For example, if we setting a minimum support of 0.5 and a confidence of 0.65 we are telling the computer to only report to us association rules that are above these cutoff points. This helps to remove useless rules that are obvious or useless. Conclusion Market basket analysis is a useful tool for mining information from large datasets. The rules are easy to understanding. In addition, market basket analysis can be used in many fields beyond shopping and can include relationships within DNA, and other forms of human behavior. As such, care must be made so that unsound conclusions are not drawn from random patterns in the data # Support Vector Machines in R In this post, we will use support vector machine analysis to look at some data available on kaggle. In particular, we will predict what number a person wrote by analyze the pixels that were used to make the number. The file for this example is available at https://www.kaggle.com/c/digit-recognizer/data To do this analysis you will need to use the ‘kernlab’ package. While playing with this dataset I noticed a major problem, doing the analysis with the full data set of 42000 examples took forever. To alleviate this problem. We are going to practice with a training set of 7000 examples and a test set of 3000. Below is the code for the first few sets. Remember that the dataset was download separately #load packages library(kernlab) digitTrain<-read.csv(file="digitTrain.csv",head=TRUE,sep=",") #split data digitRedux<-digitTrain[1:7000,] digitReduxTest<-digitTrain[7001:10000,] #explore data str(digitRedux) ## 'data.frame': 7000 obs. of 785 variables: ##$ label   : int  1 0 1 4 0 0 7 3 5 3 ...
##  $pixel0 : int 0 0 0 0 0 0 0 0 0 0 ... ##$ pixel1  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $pixel2 : int 0 0 0 0 0 0 0 0 0 0 ... ##$ pixel3  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $pixel4 : int 0 0 0 0 0 0 0 0 0 0 ... ##$ pixel5  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $pixel6 : int 0 0 0 0 0 0 0 0 0 0 ... ## [list output truncated] From the “str” function you can tell we have a lot of variables (785). This is what slowed the analysis down so much when I tried to run the full 42000 examples in the original dataset. SVM need a factor variable as the predictor if possible. We are trying to predict the “label” variable so we are going to change this to a factor variable because that is what it really is. Below is the code #convert label variable to factor digitRedux$label<-as.factor(digitRedux$label) digitReduxTest$label<-as.factor(digitReduxTest$label) Before we continue with the analysis we need to scale are variables. This makes all variables to be within the same given range which helps to equalizes the influence of them. However, we do not want to change our “label” variable as this is the predictor variable and scaling it would make the results hard to understand. Therefore, we are going to temporarily remove the “label” variable from both of our data sets and save them in a temporary data frame. The code is below. #temporary dataframe for the label results keep<-as.data.frame(digitRedux$label)
keeptest<-as.data.frame(digitReduxTest$label) #null label variable in both datasets digitRedux$label<-NA
digitReduxTest$label<-NA Next, we scale the remaining variable and reinsert the label variables for each data set as show in our code below. digitRedux<-as.data.frame(scale(digitRedux)) digitRedux[is.na(digitRedux)]<- 0 #replace NA with 0 digitReduxTest<-as.data.frame(scale(digitReduxTest)) digitReduxTest[is.na(digitReduxTest)]<- 0 #add back label digitRedux$label<-keep$digitRedux$label
digitReduxTest$label<-keeptest$digitReduxTest$label Now we make our model using the “ksvm” function in the “kernlab” package. We set the kernel to “vanilladot” which is a linear kernel. We will aslo print the results. However, the results do not make any sense on their own and the model can only be assess througn other means. Below is the code. If you get a warning message about scaling do not worry about this as we scaled the data ourselves. #make the model number_classify<-ksvm(label~., data=digitRedux, kernel="vanilladot") ## Setting default kernel parameters ## Warning in .local(x, ...): Variable(s) ' constant. Cannot scale data. #look at the results number_classify ## Support Vector Machine object of class "ksvm" ## ## SV type: C-svc (classification) ## parameter : cost C = 1 ## ## Linear (vanilla) kernel function. ## ## Number of Support Vectors : 2218 ## ## Objective Function Value : -0.0623 -0.207 -0.1771 -0.0893 -0.3207 -0.4304 -0.0764 -0.2719 -0.2125 -0.3575 -0.2776 -0.1618 -0.3408 -0.1108 -0.2766 -1.0657 -0.3201 -1.0509 -0.2679 -0.4565 -0.2846 -0.4274 -0.8681 -0.3253 -0.1571 -2.1586 -0.1488 -0.2464 -2.9248 -0.5689 -0.2753 -0.2939 -0.4997 -0.2429 -2.336 -0.8108 -0.1701 -2.4031 -0.5086 -0.0794 -0.2749 -0.1162 -0.3249 -5.0495 -0.8051 ## Training error : 0 We now need to use the “predict” function so that we can determine the accuracy of our model. Remember that for predicting, we use the answers in the test data and compare them to what our model would guess based on what it knows. number_predict<-predict(number_classify, digitReduxTest) table(number_predict, digitReduxTest$label)
##
## number_predict   0   1   2   3   4   5   6   7   8   9
##              0 297   0   3   3   1   4   6   1   1   1
##              1   0 307   4   1   0   4   2   5  11   1
##              2   0   2 268  10   5   1   3  10  12   3
##              3   0   1   7 291   1  11   0   1   8   3
##              4   0   1   3   0 278   4   3   2   0   9
##              5   2   0   1  10   1 238   4   1  11   1
##              6   2   1   1   0   2   1 287   1   0   0
##              7   0   1   1   0   1   0   0 268   3  10
##              8   1   3   4  10   0  11   1   0 236   2
##              9   0   0   2   2   9   2   0  14   2 264
accuracy<-number_predict == digitReduxTest$label prop.table(table(accuracy)) ## accuracy ## FALSE TRUE ## 0.08866667 0.91133333 The table allows you to see how many were classified correctly and how they were misclassified. The prop.table allows you to see an overall percentage. This particular model was highly accurate at 91%. It would be difficult to improve further. Below is code for a model that is using a different kernel with results that are barely better. However, if you ever enter a data science competition any improve ususally helps even if it is not practical for everyday use. number_classify_rbf<-ksvm(label~., data=digitRedux, kernel="rbfdot") ## Warning in .local(x, ...): Variable(s) ' constant. Cannot scale data. #evaluate improved model number_predict_rbf<-predict(number_classify_rbf, digitReduxTest) table(number_predict_rbf, digitReduxTest$label)
##
## number_predict_rbf   0   1   2   3   4   5   6   7   8   9
##                  0 294   0   2   3   1   1   2   1   1   0
##                  1   0 309   1   0   1   0   1   3   4   2
##                  2   4   2 277  12   4   4   8   9   9   8
##                  3   0   1   3 297   1   3   0   1   3   3
##                  4   0   1   3   0 278   4   1   5   0   6
##                  5   0   0   1   6   1 254   4   0   9   1
##                  6   2   1   0   0   2   6 289   0   2   0
##                  7   0   0   3   2   3   0   0 277   2  13
##                  8   2   2   4   3   0   3   1   0 253   3
##                  9   0   0   0   4   7   1   0   7   1 258
accuracy_rbf<-number_predict_rbf == digitReduxTest\$label
prop.table(table(accuracy_rbf))
## accuracy_rbf
##      FALSE       TRUE
## 0.07133333 0.92866667


Conclusion

From this demonstration we can see the power of support vector machines with numerical data. This type of analysis can be used for things beyond the conventional analysis and can be used to predict things such as hand written numbers. As such, SVM is yet another tool available for the data scientist.