Adv Quant: K-means classification in R

The explanatory variables in the logistic regression are both the type of loan and the borrowing amount.

4dbf1.PNG

Figure 1: The summary output of the logistic regression based on the type of loan and the borrowing amount.

The logistic equation shows statistical significance at the 0.01 level when the variables amount, and when the type of loan is used for a used car and a radio/television (Figure 1).  Thus, the regression equation comes out to be:

default = -0.9321 + 0.0001330(amount) – 1.56(Purpose is for used car) – 0.6499(purpose is for radio/television)

4dbf2.PNG

Figure 2: The comparative output of the logistic regression prediction versus actual results.

When comparing the predictions to the actual values (Figure 2), the mean and minimum scores between both of them are similar.  However, all other values are not. When the prediction values are rounded to the nearest whole number the actual prediction rate is 73%.

K-means classification, on the 3 continuous variables: duration, amount, and installment.

In K-means classification the data is clustered by the mean Euclidean distance between their differences (Ahlemeyer-Stubbe & Coleman, 2014).  In this exercise, there are two clusters. Thus, the cluster size is 825 no defaults, 175 defaults, where the within-cluster sum of squares for between/total is 69.78%.  The matrix of cluster centers is shown below (Figure 3).

4dbf3

Figure 3: K means center values, per variable

Cross-validation with k = 5 for the nearest neighbor.

K-nearest neighbor (K =5) is when a data point is clustered into a group, by having 5 of the nearest neighbors vote on that data point, and it is particularly useful if the data is a binary or categorical (Berson, Smith, & Thearling, 1999).  In this exercise, the percentage of correct classifications from the trained and predicted classification is 69%.  However, logistic regression in this scenario was able to produce a much higher prediction rate of 73%, this for this exercise and this data set, logistic regression was quite useful in predicting the default rate than the k-nearest neighbor algorithm at k=5.

Code

#

## The German credit data contains attributes and outcomes on 1,000 loan applications.

## Data source: https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.data

## Metadata file: https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.doc

#

## Reading the data from source and displaying the top five entries.

credits=read.csv(“https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.data”, header = F, sep = ” “)

head(credits)

#

##

### ———————————————————————————————————-

## The two outcomes are success (defaulting on the loan) and failure (not defaulting).

## The explanatory variables in the logistic regression are both the type of loan and the borrowing amount.

### ———————————————————————————————————-

##

#

## Defining and re-leveling the variables (Taddy, n.d.)

default = credits$V21 – 1 # set default true when = 2

amount = credits$V5

purpose = factor(credits$V4, levels = c(“A40″,”A41″,”A42″,”A43″,”A44″,”A45″,”A46″,”A48″,”A49″,”A410”))

levels(purpose) = c(“newcar”, “usedcar”, “furniture/equip”, “radio/TV”, “apps”, “repairs”, “edu”, “retraining”, “biz”, “other”)

## Create a new matrix called “cred” with the 8 defined variables (Taddy, n.d.)

credits$default = default

credits$amount  = amount

credits$purpose = purpose

cred = credits[,c(“default”,”amount”,”purpose”)]

head(cred[,])

summary(cred[,])

## Create a design matrix, such that factor variables are turned into indicator variables

Xcred = model.matrix(default~., data=cred)[,-1]

Xcred[1:5,]

## Creating training and prediction datasets: Select 900 rows for esitmation and 100 for testing

set.seed(1)

train = sample(1:1000,900)

## Defining which x and y values in the design matrix will be for training and for testing

xtrain = Xcred[train,]

xtest = Xcred[-train,]

ytrain = cred$default[train]

ytest = cred$default[-train]

## logistic regresion

datas=data.frame(default=ytrain,xtrain)

creditglm=glm(default~., family=binomial, data=datas)

summary(creditglm)

percentOfCorrect=100*(sum(ytest==round(testingdata$defaultPrediction))/100)

percentOfCorrect

## Predicting default from the test data (Alice, 2015; UCLA: Statistical Consulting Group., 2007)

testdata=data.frame(default=ytest,xtest)

testdata[1:5,]

testingdata=testdata[,2:11] #removing the variable default from the data matrix

testingdata$defaultPrediction = predict(creditglm, newdata=testdata, type = “response”)

results = data.frame(ytest,testingdata$defaultPrediction)

summary(results)

head(results,10)

#

##

### ———————————————————————————————————-

##  K-means classification, on the 3 continuous variables: duration, amount, and installment.

### ———————————————————————————————————-

##

#

install.packages(“class”)

library(class)

## Defining and re-leveling the variables (Taddy, n.d.)

default = credits$V21 – 1 # set default true when = 2

duration = credits$V2

amount = credits$V5

installment = credits$V8

## Create a new matrix called “cred” with the 8 defined variables (Taddy, n.d.)

credits$default = default

credits$amount  = amount

credits$installment = installment

credits$duration = duration

creds = credits[,c(“duration”,”amount”,”installment”,”default”)]

head(creds[,])

summary(creds[,])

## K means classification (R, n.b.a)

kmeansclass= cbind(creds$default,creds$duration,creds$amount,creds$installment)

kmeansresult= kmeans(kmeansclass,2)

kmeansresult$cluster

kmeansresult$size

kmeansresult$centers

kmeansresult$betweenss/kmeansresult$totss

#

##

### ———————————————————————————————————-

##  Cross-validation with k = 5 for the nearest neighbor. 

### ———————————————————————————————————-

##

#

## Create a design matrix, such that factor variables are turned into indicator variables

Xcreds = model.matrix(default~., data=creds)[,-1]

Xcreds[1:5,]

## Creating training and prediction datasets: Select 900 rows for esitmation and 100 for testing

set.seed(1)

train = sample(1:1000,900)

## Defining which x and y values in the design matrix will be for training and for testing

xtrain = Xcreds[train,]

xtest = Xcreds[-train,]

ytrain = creds$default[train]

ytest = creds$default[-train]

## K-nearest neighbor clustering (R, n.d.b.)

nearestFive=knn(train = xtrain[,2,drop=F],test=xtest[,2,drop=F],cl=ytrain,k=5)

knnresults=cbind(ytest+1,nearestFive) # The addition of 1 is done on ytest because when cbind is applied to nearestFive it adds 1 to each value.

percentOfCorrect=100*(sum(ytest==nearestFive)/100)

References

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Adv Quant: Use of Bayesian Analysis in research

Just using knowledge before data collection and the knowledge gained from data collection doesn’t tell the full story until they are combined, hence establishing the need for Bayesian analysis (Hubbard, 2010).  Bayes’ theory is a conditional probability that takes into account prior knowledge, but updates itself when new data becomes available (Hubbard, 2010; Smith, 2015).  Bayesian analysis aids in avoiding overconfidence and underconfidence because it doesn’t ignore prior or new data (Hubbard, 2010).  There are many examples of how Bayesian analysis can be used in the context of social media data.  Below are just three ways of many,

  • With high precision, Bayesian Analysis was able to detect spam twitter accounts from legitimate users, based on their followers/following ration information and their most 100 recent tweets (McCord & Chuah, 2011). McCord and Chuah (2011) was able to use Bayesian analysis to achieve a 75% accuracy in detecting spam just by using user-based features, and ~90% accuracy in detecting spam when using both user and content based features.
  • Boulle (2014) used Bayesian Analysis off of 60,000 URLs in 100 websites. The goal was to predict the number of visits and messages on Twitter and Facebook after 48 hours, and Boulle (2014) was able to come close to the actual numbers through using Bayesian Analysis, showcasing the robustness of the approach.
  • Zaman, Fox, and Bradlow (2014), was able to use Bayesian analysis for predicting the popularity of tweets by measuring the final count of retweets a source tweet gets.

An in-depth exploration of Zaman, et al. (2014)

Goal:

The researchers aimed to predict how popular a tweet can become a Bayesian model to analyze the time path of retweets a tweet receives, and the eventual number of retweets of a tweet one week later.

  • They were analyzing 52 tweets varying among different topics like music, politics, etc.
    • They narrowed down the scope to analyzing tweets with a max of 1800 retweets per root tweets.

Defining the parameters:

  • Twitter = microblogging site
  • Tweets = microblogging content that is contained in up to 140 characters
  • Root tweets = original tweets
  • Root user = generator of the root tweet
  • End user = those who read the root tweet and retweeted it
  • Twitter followers = people who are following the content of a root user
  • Follower graph = resulting connections into a social graph from known twitter followers
  • Retweet = a twitter follower’s sharing of content from the user for their followers to read
  • Depth of 1 = how many end users retweeted a root tweet
  • Depth of 2 = how many end users retweeted a retweet of the root tweet

Exploration of the data:

From the 52 sampled root tweets, the researchers found that the tweets had anywhere between 21-1260 retweets associated with them and that the last retweet that could have occurred between a few hours to a few days from the root tweet’s generation.  The researchers calculated the median times from the last retweet, yielding scores that ranged from 4 minutes to 3 hours.  The difference between the median times was not statistically significant to reject a null hypothesis, which involved a difference in the median times.  This gave potentially more weight to the potential value of the Bayesian model over just descriptive/exploratory methods, as stated by the researchers.

The researchers explored the depth of the retweets and found that 11,882 were a depth of 1, whereas 314 were a depth of 2 or more in those 52 root tweets, which suggested that root tweets get more retweets than retweeted tweets.  It was suggested by the researchers that the depth seemed to have occurred because of a large number of followers from the retweeter’s side.

It was noted by the researchers that retweets per time path decays similarly to a log-normally distribution, which is what was used in the Bayesian analysis model.

Bayesian analysis results:

The researchers partitioned their results randomly into a training set with 26 observations, and a testing set of 26 observations, and varied the amount of retweets observations from 10%-100% of the last retweet.  Their main results are plotted in boxplots, where the whiskers cover 90% of the posterior solution (Figure 10).

IP3F12.png

The figure above is directly from Zaman, et al. (2014). The authors mentioned that as the observation fraction increased the absolute percent errors decreased.    For future work, the researchers suggested that their analysis could be parallelized to incorporate more data points, take into consideration the time of day the root tweet was posted, as well as understanding the content within the tweets and their retweet-ability because of it.

References

  • Boullé, M. (2014). Selective Naive Bayes Regressor with Variable Construction for Predictive Web Analytics.
  • Hubbard, D. W. (2010). How to measure anything: Finding the values of “intangibles” in business. (2nd e.d.) New Jersey, John Wiley & Sons, Inc.
  • Mccord, M., & Chuah, M. (2011). Spam detection on twitter using traditional classifiers. In International Conference on Autonomic and Trusted Computing (pp. 175-186). Springer Berlin Heidelberg.
  • Smith, M. (2015). Statistical analysis handbook. Retrieved from http://www.statsref.com/HTML/index.html?introduction.html
  • Zaman, T., Fox, E. B., & Bradlow, E. T. (2014). A Bayesian approach for predicting the popularity of tweets. The Annals of Applied Statistics8(3), 1583-1611.

Adv Quant: Bayesian Analysis

Uncertainty in making decisions

Generalizing something that is specific from a statistical standpoint, is the problem of induction, and that can cause uncertainty in making decisions (Spiegelhalter & Rice, 2009). Uncertainty in making a decision could also arise from not knowing how to incorporate new data with old assumptions (Hubbard, 2010).

According to Hubbard (2010) conventional statistics assumes:

(1)    The researcher has no prior information about the range of possible values (which is never true) or,

(2)    The researcher does have prior knowledge that the distribution of the population and it is never any of the messy ones (which is not true more often than not)

Thus, knowledge before data collection and the knowledge gained from data collection doesn’t tell the full story until they are combined, hence the need for Bayes’ analysis (Hubbard, 2010).  Bayes’ theory can be reduced to a conditional probability that aims to take into account prior knowledge, but updates itself when new data becomes available (Hubbard, 2010; Smith, 2015; Spiegelhalter & Rice, 2009; Yudkowsky, 2003).  Bayesian analysis avoids overconfidence and underconfidence from ignoring prior data or ignoring new data (Hubbard, 2010), through the implementation of the equation below:

 eq4                           (1)

Where P(hypothesis|data) is the posterior data, P(hypothesis) is the true probability of the hypothesis/distribution before the data is introduced, P(data) marginal probability, and P(data|hypothesis) is the likelihood that the hypothesis/distribution is still true after the data is introduced (Hubbard, 2010; Smith, 2015; Spiegelhalter & Rice, 2009; Yudkowsky, 2003).  This forces the researcher to think about the likelihood that different and new observations could impact a current hypothesis (Hubbard, 2010). Equation (1) shows that evidence is usually a result of two conditional probabilities, where the strongest evidence comes from a low probability that the new data could have led to X (Yudkowsky, 2003).  From these two conditional probabilities, the resultant value is approximately the average from that of the prior assumptions and the new data gained (Hubbard, 2010; Smith, 2015).  Smith (2015) describe this approximation in the following simplified relationship (equation 2):

 eq5.PNG                                            (2)

Therefore, from equation (2) the type of prior assumptions influence the posterior resultant. Prior distributions come from Uniform, Constant, or Normal distribution that results in a Normal posterior distribution and a Beta or Binomial distribution results in a Beta posterior distribution (Smith, 2015).  To use Bayesian Analysis one must take into account the analysis’ assumptions.

Basic Assumptions of Bayesian Analysis

Though these three assumptions are great to have for Bayesian Analysis, it has been argued that they are quite unrealistic when real life data, particularly unstructured text-based data (Lewis, 1998; Turhan & Bener, 2009):

  • Each of the new data samples is independent of each other and identically distributed (Lewis, 1998; Nigam & Ghani, 2000; Turhan & Bener, 2009)
  • Each attribute has equation importance (Turhan & Bener, 2009)
  • The new data is compatible with the target posterior (Nigam & Ghani, 2000; Smith 2015).

Applications of Bayesian Analysis

There are typically three main situations where Bayesian Analysis is used (Spiegelhalter, & Rice, 2009):

  • Small data situations: The researcher has no choice but to include prior quantitative information, because of a lack of data, or lack of a distribution model.
  • Moderate size data situations: The researcher has multiple sources of data. They can create a hierarchical model on the assumption of similar prior distributions
  • Big data situations: where there are huge join probability models, with 1000s of data points or parameters, which can then be used to help make inferences of unknown aspects of the data

Pros and Cons

Applying Bayesian Analytics to data has its advantages and disadvantages.  Those Advantages and Disadvantages with Bayesian Analysis as identified by SAS (n.d.) are:

Advantages

+    Allows for a combination of prior information with data, for a strong decision-making

+    No reliance on asymptotic approximation, thus the inferences are conditional on the data

+    Provides easily interpretive results.

Disadvantages

– Posteriors are heavily influenced by their priors.

– This method doesn’t help the researcher to select the proper prior, given how much influence it has on the posterior.

– Computationally expensive with large data sets.

The key takeaway from this discussion is that the prior knowledge can heavily influence the posterior, which can easily be seen in equation (2).  That is because knowledge before data collection and the knowledge gained from data collection doesn’t tell the full story unless they are combined.

Reference

Adv Quant: Logistic Regression in R

Introduction

The German credit data contains attributes and outcomes on 1,000 loan applications. The data are available at this Web site, where datasets are provided for the machine learning community.

Results

IP3F1.PNG

Figure 1: Image shows the first six entries in the German credit data.

IP3F2.png

Figure 2: Matrix scatter plot, showing the 2×2 relationships between all the variables within the German credit data.

IP3F3.png

Figure 3: A summary of the credit data with the variables of interest.

IP3F3.png

Figure 4: Shows the entries in the designer matrix which will be used for logistical analysis.

IP3F4

Figure 5: Summarized logistic regression information based on the training data.

IP3F6.1.pngIP3F6.2.png

Figure 6: The coeficients’ confidence interval at the 95% level using log-likelihood vlaues, with values to the right including the standard errors values.

IP3F7.png

Figure 7: Wald Test statistic to test the significance level of the entire ranked variable.

IP3F8.png

Figure 8: The Odds Ratio for each independent variable along with the 95% confidence interval for those odds ratio.

IP3F9.png

Figure 9: Part of the summarized test data set for the logistics regression model.

IP3F10.png

Figure 10: The ROC curve, which illustrates the false positive rate versus the true positive rate of the prediction model.

Discussion

The results from Figure 1 means that the data needs to be formatted before any analysis could be conducted on the data.  Hence, the following lines of code were needed to redefine the variables in the German data set.   Given the data output (Figure 1), the matrix scatter plot (Figure 2) show that duration, amount, and age are continuous variables, while the other five variables are factor variables, which have categorized scatterplots.  Even though installment and default show box plot data in the summary (Figure 3), the data wasn’t factored like history, purpose, or rent, thus it won’t show a count.  From the count data (Figure 3), the ~30% of the purpose of loans are for cars, where as 28% is for TVs.  In this German credit data, about 82% of those asking for credit do not rent and about 53% of borrowers have an ok credit history with 29.3% having a horrible credit history.  The mean average default rate is 30%.

Variables (Figure 5) that have statistical significance at the 0.10 include duration, amount, installment, age, history (per category), rent, and some of the purposes categories.  Though it is preferred to see a large difference in the null deviance and residual deviance, it is still a difference.  The 95% confidence interval for all the logistic regression equation don’t show much spread from their central tendencies (Figure 6).  Thus, the logistic regression model is (from figure 5):

IP3F11.PNG

The odds ratio measures the constant strength of association between the independent and dependent variables (Huck, 2011; Smith, 2015).  This is similar to the correlation coefficient (r) and coefficient of determination (r2) values for linear regression.  According to UCLA: Statistical Consulting Group, (2007), if the P value is less than 0.05, then the overall effect of the ranked term is statistically significant (Figure 7), which in this case the three main terms are.  The odds ratio data (Figure 8) is promising, as values closer to one is desirable for this prediction model (Field, 2013). If the value of the odds ratio is greater than 1, it will show that as the independent variable value increases, so do the odds of the dependent variable (Y = n) occurs increases and vice versa (Fields, 2013).

Moving into the testing phase of the logistics regression model, the 100 value data set needs to be extracted, and the results on whether or not there will be a default or not on the loan are predicted. Comparing the training and the test data sets, the maximum values between the both are not the same for durations and amount of the loan.  All other variables and statistical distributions are similar to each other between the training and the test data.  Thus, the random sampling algorithm in R was effective.

The area underneath the ROC curve (Figure 10), is 0.6994048, which is closer to 0.50 than it is to one, thus this regression does better than pure chance, but it is far from perfect (Alice, 2015).

In conclusion, the regression formula has a 0.699 prediction accuracy, and the purpose, history, and rent ranked categorical variables were statistically significant as a whole.  Therefore, the logistic regression on these eight variables shows more promise in prediction accuracy than pure chance, on who will and will not default on their loan.

Code

#

## The German credit data contains attributes and outcomes on 1,000 loan applications.

##    •   You need to use random selection for 900 cases to train the program, and then the other 100 cases will be used for testing.

##    •   Use duration, amount, installment, and age in this analysis, along with loan history, purpose, and rent.

### ———————————————————————————————————-

## Data source: https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.data

## Metadata file: https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.doc

#

#

## Reading the data from source and displaying the top six entries.

#

credits=read.csv(“https://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.data”, header = F, sep = ” “)

head(credits)

#

## Defining the variables (Taddy, n.d.)

#

default = credits$V21 – 1 # set default true when = 2

duration = credits$V2

amount = credits$V5

installment = credits$V8

age = credits$V13

history = factor(credits$V3, levels = c(“A30”, “A31”, “A32”, “A33”, “A34”))

purpose = factor(credits$V4, levels = c(“A40″,”A41″,”A42″,”A43″,”A44″,”A45″,”A46″,”A48″,”A49″,”A410”))

rent = factor(credits$V15==”A151″) # renting status only

# rent = factor(credits$V15 , levels = c(“A151″,”A152″,”153”)) # full property status

#

## Re-leveling the variables (Taddy, n.d.)

#

levels(history) = c(“great”, “good”, “ok”, “poor”, “horrible”)

levels(purpose) = c(“newcar”, “usedcar”, “furniture/equip”, “radio/TV”, “apps”, “repairs”, “edu”, “retraining”, “biz”, “other”)

# levels(rent) = c(“rent”, “own”, “free”) # full property status

#

## Create a new matrix called “cred” with the 8 defined variables (Taddy, n.d.)

#

credits$default = default

credits$duration= duration

credits$amount  = amount

credits$installment = installment

credits$age     = age

credits$history = history

credits$purpose = purpose

credits$rent    = rent

cred = credits[,c(“default”,”duration”,”amount”,”installment”,”age”,”history”,”purpose”,”rent”)]

#

##  Plotting & reading to make sure the data was transfered correctly into this dataset and present summary stats (Taddy, n.d.)

#

plot(cred)

cred[1:3,]

summary(cred[,])

#

## Create a design matrix, such that factor variables are turned into indicator variables

#

Xcred = model.matrix(default~., data=cred)[,-1]

Xcred[1:3,]

#

## Creating training and prediction datasets: Select 900 rows for esitmation and 100 for testing

#

set.seed(1)

train = sample(1:1000,900)

## Defining which x and y values in the design matrix will be for training and for testing

xtrain = Xcred[train,]

xnew = Xcred[-train,]

ytrain = cred$default[train]

ynew = cred$default[-train]

#

## logistic regresion

#

datas=data.frame(default=ytrain,xtrain)

creditglm=glm(default~., family=binomial, data=datas)

summary(creditglm)

#

## Confidence Intervals (UCLA: Statistical Consulting Group, 2007)

#

confint(creditglm)

confint.default(creditglm)

#

## Overall effect of the rank using the wald.test function from the aod library (UCLA: Statistical Consulting Group, 2007)

#

install.packages(“aod”)

library(aod)

wald.test(b=coef(creditglm), Sigma = vcov(creditglm), Terms = 6:9) # for all ranked terms for history

wald.test(b=coef(creditglm), Sigma = vcov(creditglm), Terms = 10:18) # for all ranked terms for purpose

wald.test(b=coef(creditglm), Sigma = vcov(creditglm), Terms = 19) # for the ranked term for rent

#

## Odds Ratio for model analysis (UCLA: Statistical Consulting Group, 2007)

#

exp(coef(creditglm))

exp(cbind(OR=coef(creditglm), confint(creditglm))) # odds ration next to the 95% confidence interval for odds ratios

#

## Predicting default from the test data (Alice, 2015; UCLA: Statistical Consulting Group., 2007)

#

newdatas=data.frame(default=ynew,xnew)

newestdata=newdatas[,2:19] #removing the variable default from the data matrix

newestdata$defaultPrediction = predict(creditglm, newdata=newestdata, type = “response”)

summary(newdatas)

#

## Plotting the true positive rate against the false positive rate (ROC Curve) (Alice, 2015)

#

install.packages(“ROCR”)

library(ROCR)

pr  = prediction(newestdata$defaultPrediction, newdatas$default)

prf = performance(pr, measure=”tpr”, x.measure=”fpr”)

plot(prf)

## Area under the ROC curve (Alice, 2015)

auc= performance(pr, measure = “auc”)

auc= auc@y.values[[1]]

auc # The closer this value is to 1 the better, much better than to 0.5

 

 

References