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A gentle introduction to Naïve Bayes classification using R

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Preamble

One of the key problems of predictive analytics is to classify entities or events based on a knowledge of their attributes.  An example: one might want to classify customers into two categories, say, ‘High Value’ or ‘Low Value,’ based on a knowledge of their buying patterns.  Another example: to figure out the party allegiances of  representatives based on their voting records.  And yet another:  to predict the species a particular plant or animal specimen based on a list of its characteristics. Incidentally, if you haven’t been there already, it is worth having a look at Kaggle to get an idea of some of the real world classification problems that people tackle using techniques of predictive analytics.

Given the importance of classification-related problems, it is no surprise that analytics tools offer a range of options. My favourite (free!) tool, R, is no exception: it has a plethora of state of the art packages designed to handle a wide range of problems. One of the problems with this diversity of choice is that it is often confusing for beginners to figure out which one to use in a particular situation. Over the next several months, I intend to write up tutorial articles covering many of the common algorithms, with a particular focus on their strengths and weaknesses; explaining where they work well and where they don’t. I’ll kick-off this undertaking with a simple yet surprisingly effective algorithm – the Naïve Bayes classifier.

Just enough theory

I’m going to assume you have R and RStudio installed on your computer. If you need help with this, please follow the instructions here.

To introduce the Naive Bayes algorithm, I will use the HouseVotes84 dataset, which contains US congressional voting records for 1984. The data set is in the mlbench package which is not part of the base R installation. You will therefore need to install it if you don’t have it already.  Package installation is a breeze in RStudio – just go to Tools > Install Packages and follow the prompts.

The HouseVotes84 dataset describes how 435 representatives voted – yes (y), no (n) or unknown (NA) – on 16 key issues presented to Congress.  The dataset also provides the party affiliation of each representative – democrat or republican.

Let’s begin by exploring the dataset. To do this, we load mlbench, fetch the dataset and get some summary stats on it. (Note: a complete listing of the code in this article can be found here)

#load mlbench library
library(mlbench)
#set working directory if needed (modify path as needed)
setwd(“C:/Users/Kailash/Documents/NaiveBayes”)
#load HouseVotes84 dataset
data(“HouseVotes84”)

It is good to begin by exploring the data visually.  To this end, let’s do some bar plots using the basic graphic capabilities of R:

#barplots for specific issue
plot(as.factor(HouseVotes84[,2]))
title(main=”Votes cast for issue”, xlab=”vote”, ylab=”# reps”)
#by party
plot(as.factor(HouseVotes84[HouseVotes84$Class==’republican’,2]))
title(main=”Republican votes cast for issue 1″, xlab=”vote”, ylab=”# reps”)
plot(as.factor(HouseVotes84[HouseVotes84$Class==’democrat’,2]))
title(main=”Democrat votes cast for issue 1″, xlab=”vote”, ylab=”# reps”)

The plots are shown in Figures 1 through 3.

Fig 1: y and n votes for issue 1

Fig 1: y and n votes for issue 1

Fig 2: Republican votes for issue 1.

Fig 2: Republican votes for issue 1.

Fig 3: Democrat votes for issue 1.

Fig 3: Democrat votes for issue 1.

Among other things, such plots give us a feel for the probabilities associated with how representatives from parties tend to vote on specific issues.

The classification problem at hand is to figure out the party affiliation from a knowledge of voting patterns. For simplicity let us assume that there are only 3 issues voted on instead of the 16 in the actual dataset. In concrete terms we wish to answer the question, “what is the probability that a representative is, say, a democrat (D) given that he or she has voted, say,  (v1 = y, v2=n,v3 = y) on the three issues?” To keep things simple I’m assuming there are no NA values.

In the notation of conditional probability this can be written as,

P(D|v1=y, v2=n,v3=y)

(Note:  If you need a refresher on conditional probability, check out this post for a simple explanation.)

By Bayes theorem, which I’ve explained at length in this post, this can be recast as,

P(D|v1=y, v2=n,v3=y) = \displaystyle \frac{p(D) p(v1=y, v2=n,v3=y|D)}{p(v1=y, v2=n,v3=y)}......(1)

We’re interested only in relative probabilities of the representative being a democrat or republican because the predicted party affiliation depends only on which of the two probabilities is larger (the actual value of the probability is not important). This being the case, we can factor out any terms that are constant.  As it happens, the denominator of the above equation – the probability of a particular voting pattern – is a constant because it depends on the total number of representatives (from both parties)  who voted a particular way.

Now, using the chain rule of conditional probability, we can rewrite the numerator as:

p(D) p(v1=y, v2=n,v3=y|D)

= p(D)p(v1=y|D) p(v2=n,v3=y|D,v1=y)

Basically, the second term on the left hand side, p(v1=y, v2=n,v3=y|D), is the probability of getting a particular voting pattern  (y,n,y) assuming the rep is a Democrat (D). The definition of conditional probability allows us to rewrite this as the probability of getting a n vote for issue v2 and a y vote for issue v3 given that the rep is a Democrat who has voted y on issue v1. Again, this is simply a consequence of the definition of conditional probability.

Another application of the chain rule gives:

p(D) p(v1=y, v2=n,v3=y|D)

= p(D)p(v1=y|d) p(v2=n|D,v1=y) p(v3=y|D,v1=y,v2=n)

Where we have now factored out the n vote on the second issue.

The key assumption of Naïve Bayes is that the conditional probability of each feature given the class is independent of all other features. In mathematical terms this means that,

p(v2=n|D,v1=y) = p(v2=n|D)

and

p(v3=y|D,v1=y,v2=n) = p(v3=y|D)

The quantity of interest, the numerator of equation (1) can then be written as:

p(D) p(v1=y, v2=n,v3=y|D)

= p(D)p(v1=y|D)p(v2=n|D)p(v3=y|D).......(2)

The assumption of independent conditional probabilities is a drastic one.  What it is saying is that the features are completely independent of each other. This is clearly not the case in the situation above: how representatives vote on a particular issue is coloured by their beliefs and values. For example, the conditional probability of voting patterns  on socially progressive issues are definitely not independent of each other. However, as we shall see  in the next section, the Naïve Bayes assumption works well for this problem as it does in many other situations where we know upfront that it is grossly incorrect.

Another good example of the unreasonable efficacy of Naive Bayes is in spam filtering.  In the case of spam, the features are individual words in an email.  It is clear that certain word combinations tend to show up consistently in spam – for example, “online”, “meds”, “Viagra” and “pharmacy.” In other words, we know upfront that their occurrences are definitely not independent of each other. Nevertheless, Naïve Bayes based spam detectors which assume mutual independence of features do remarkably well in distinguishing spam from ham.

Why is this so?

To explain why, I return to a point I mentioned earlier: to figure out  the affiliation associated with a particular voting pattern (say, v1=y, v2=n,v3=y) one only needs to know which of the two probabilities p(R| v1=y, v2=n,v3=y) and p(D| V1=y, V2=n, V3=y) is greater than the other.  That is, the values of these probabilities are not important in determining the party affiliations.

This hints as to why the independence assumption might not be so quite so idiotic. Since the prediction depends only the on the maximum, the algorithm will get it right even if there are dependencies between feature providing the dependencies do not change which class has the maximum probability (once again, note that only the maximal class is important here, not  the value of the maximum).

Yet another reason for the surprising success of Naïve Bayes is that dependencies often cancel out across a large set of features. But, of course, there is no guarantee that this will always happen.

In general, Naïve Bayes algorithms work better for problems in which the dependent (predicted) variable is discrete, even when there are dependencies between features (spam detection is a good example).  They work less well for regression problems – i.e those in  which predicted variables are continuous.

I hope the above has given you an intuitive feel for how Naïve Bayes algorithms work. I don’t know about you, but my head’s definitely spinning after writing out all that mathematical notation.

It’s time to clear our heads by doing some computation.

Naïve Bayes in action

There are a couple of well-known implementations of Naïve Bayes in R. One of them is the naiveBayes method in the e1071 package and the other is NaiveBayes method in  the klaR package.  I’ll use the former for no other reason than it seems to be more popular. That said, I have used the latter too and can confirm that it works just as well.

We’ve already loaded and explored the HouseVotes84 dataset. One of the things you may have noticed when summarising the data is  that there are a fair number of NA values. Naïve Bayes algorithms typically handle NA values either by ignoring records that contain any NA values or by ignoring just the NA values. These choices are indicated by the value of the variable na.action in the naiveBayes algorithm, which is set to na.omit (to ignore the record) or na.pass (to ignore the value).

Just for fun, we’ll take a different approach. We’ll impute NA values for a given issue and party by looking at how other representatives from the same party voted on the issue. This is very much in keeping with the Bayesian spirit: we infer unknowns based on a justifiable belief – that is, belief based on the evidence.

To do this I write two functions: one to  compute the number of NA values for a given issue (vote) and class (party affiliation), and the other to calculate the fraction of yes votes for a given issue (column) and class (party affiliation).

#Functions needed for imputation
#function to return number of NAs by vote and class (democrat or republican)
na_by_col_class <- function (col,cls){return(sum(is.na(HouseVotes84[,col]) & HouseVotes84$Class==cls))}
#function to compute the conditional probability that a member of a party will cast a ‘yes’ vote for
#a particular issue. The probability is based on all members of the party who #actually cast a vote on the issue (ignores NAs).
p_y_col_class <- function(col,cls){
sum_y<-sum(HouseVotes84[,col]==’y’ & HouseVotes84$Class==cls,na.rm = TRUE)
sum_n<-sum(HouseVotes84[,col]==’n’ & HouseVotes84$Class==cls,na.rm = TRUE)
return(sum_y/(sum_y+sum_n))}
#Check that functions work!
> p_y_col_class(2,’democrat’)
[1] 0.6046512
> p_y_col_class(2,’republican’)
[1] 0.1878788
> na_by_col_class(2,’democrat’)
[1] 9
> na_by_col_class(2,’republican’)
> [1] 3

Before proceeding,  you might want to go back to the data and convince yourself that these values are sensible.

We can now impute the NA values based on the above. We do this by randomly assigning values ( y or n) to NAs, based on the proportion of members of a party who have voted y or n. In practice, we do this by invoking the uniform distribution and setting an NA value to y if the random number returned is less than the probability of a yes vote and to n otherwise. This is not as complicated as it sounds; you should be able to figure the logic out from the code below.

#impute missing values.
for (i in 2:ncol(HouseVotes84)) {
if(sum(is.na(HouseVotes84[,i])>0)) {
c1 <- which(is.na(HouseVotes84[,i])& HouseVotes84$Class==’democrat’,arr.ind = TRUE)
c2 <- which(is.na(HouseVotes84[,i])& HouseVotes84$Class==’republican’,arr.ind = TRUE)
HouseVotes84[c1,i] <-
ifelse(runif(na_by_col_class(i,’democrat’))<p_y_col_class(i,’democrat’),’y’,’n’)
HouseVotes84[c2,i] <-
ifelse(runif(na_by_col_class(i,’republican’))<p_y_col_class(i,’republican’),’y’,’n’)}
}

Note that the which function filters  indices by the criteria specified in the arguments and ifelse is a vectorised conditional function which enables us to apply logical criteria to multiple elements of a vector.

At this point it is a good idea to check that the NAs in each column have been set according to the voting patterns of non-NAs for a given party. You can use the p_y_col_class() function to check that the new probabilities are close to the old ones. You might want to do this before you proceed any further.

The next step is to divide the available data into training and test datasets. The former will be used to train the algorithm and produce a predictive model. The effectiveness of the model will then be tested using the test dataset. There is a great deal of science and art behind the creation of training and testing datasets. An important consideration is that both sets must contain records that are representative of the entire dataset. This can be difficult to do, especially when data is scarce and there are predictors that do not vary too much…or vary wildly for that matter. On the other hand, problems can also arise when there are redundant predictors. Indeed, the much of the art of successful prediction lies in figuring out which predictors are likely to lead to better predictions, an area known as feature selection. However, that’s a topic for another time. Our current dataset does not suffer from any of these complications so we’ll simply divide the it in an 80/20 proportion, assigning the larger number of records to the training set.

#divide into test and training sets
#create new col “train” and assign 1 or 0 in 80/20 proportion via random uniform dist
HouseVotes84[,”train”] <- ifelse(runif(nrow(HouseVotes84))<0.80,1,0)
#get col number of train / test indicator column (needed later)
trainColNum <- grep(“train”,names(HouseVotes84))
#separate training and test sets and remove training column before modeling
trainHouseVotes84 <- HouseVotes84[HouseVotes84$train==1,-trainColNum]
testHouseVotes84 <- HouseVotes84[HouseVotes84$train==0,-trainColNum]

Now we’re finally good to build our Naive Bayes model (machine learning folks call this model training rather than model building – and I have to admit, it does sound a lot cooler).

The code to train the model is anticlimactically simple:

#load e1071 library and invoke naiveBayes method
library(e1071)
nb_model <- naiveBayes(Class~.,data = trainHouseVotes84)

Here we’ve invokedthe naiveBayes method from the e1071 package. The first argument uses R’s formula notation.In this notation, the dependent variable (to be predicted) appears on the left hand side of the ~ and the independent variables (predictors or features) are on the right hand side. The dot (.) is simply shorthand for “all variable other than the dependent one.” The second argument is the dataframe that contains the training data. Check out the documentation for the other arguments of naiveBayes; it will take me too far afield to cover them here. Incidentally, you can take a look at the model using the summary() or str() functions, or even just entering the model name in the R console:

nb_model
summary(nb_model)
str(nb_model)

Note that I’ve suppressed the output above.

Now that we have a model, we can do some predicting. We do this by feeding our test data into our model and comparing the predicted party affiliations with the known ones. The latter is done via the wonderfully named confusion matrix – a table in which true and predicted values for each of the predicted classes are displayed in a matrix format. This again is just a couple of lines of code:

#…and the moment of reckoning
nb_test_predict <- predict(nb_model,testHouseVotes84[,-1])
#confusion matrix
table(pred=nb_test_predict,true=testHouseVotes84$Class)
pred true democrat republican
democrat 38 3
republican 5 22

The numbers you get will be different because your training/test sets are almost certainly different from mine.

In the confusion matrix (as defined above), the true values are in columns and the predicted values in rows. So, the algorithm has correctly classified 38 out of 43 (i.e. 38+5) Democrats and 22 out of 25 Republicans (i.e. 22+3). That’s pretty decent. However, we need to keep in mind that this could well be quirk of the choice of dataset. To address this, we should get a numerical measure of the efficacy of the algorithm and for different training and testing datasets. A simple measure of efficacy would be the fraction of predictions that the algorithm gets right. For the training/testing set above, this is simply 60/68 (see the confusion matrix above). The simplest way to calculate this in R is:

#fraction of correct predictions
mean(nb_test_predict==testHouseVotes84$Class)
[1] 0.8823529

A natural question to ask at this point is: how good is this prediction. This question cannot be answered with only a single run of the model; we need to do many runs and look at the spread of the results. To do this, we’ll create a function which takes the number of times the model should be run and the training fraction as inputs and spits out a vector containing the proportion of correct predictions for each run. Here’s the function

#function to create, run and record model results
nb_multiple_runs <- function(train_fraction,n){
fraction_correct <- rep(NA,n)
for (i in 1:n){
HouseVotes84[,”train”] <- ifelse(runif(nrow(HouseVotes84))<train_fraction,1,0)
trainColNum <- grep(“train”,names(HouseVotes84))
trainHouseVotes84 <- HouseVotes84[HouseVotes84$train==1,-trainColNum]
testHouseVotes84 <- HouseVotes84[HouseVotes84$train==0,-trainColNum]
nb_model <- naiveBayes(Class~.,data = trainHouseVotes84)
nb_test_predict <- predict(nb_model,testHouseVotes84[,-1])
fraction_correct[i] <- mean(nb_test_predict==testHouseVotes84$Class)
}
return(fraction_correct)
}

I’ve not commented the above code as it is essentially a repeat of the steps described earlier. Also, note that I have not made any effort to make the code generic or efficient.

Let’s do 20 runs with the same training fraction (0.8) as before:

#20 runs, 80% of data randomly selected for training set in each run
fraction_correct_predictions <- nb_multiple_runs(0.8,20)
fraction_correct_predictions
[1] 0.9417476 0.9036145 0.9294118 0.9302326 0.9213483 0.9404762 0.8777778 0.9102564
[9] 0.9102564 0.9080460 0.9139785 0.9200000 0.9090909 0.9239130 0.9605263 0.9333333
[17] 0.9052632 0.8977273 0.9642857 0.8518519
#summary of results
summary(fraction_correct_predictions)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.8519 0.9074 0.9170 0.9177 0.9310 0.9643
#standard deviation
sd(fraction_correct_predictions)
[1] 0.02582419

We see that the outcome of the runs are quite close together, in the 0.85 to 0.95 range with a standard deviation of 0.025. This tells us that Naive Bayes does a pretty decent job with this data.

Wrapping up

I originally intended to cover a few more case studies in this post, a couple of which highlight the shortcomings of the Naive Bayes algorithm. However, I realize that doing so would make this post unreasonably long, so I’ll stop here with a few closing remarks, and a promise to write up the rest of the story in a subsequent post.

To sum up: I have illustrated the use of a popular Naive Bayes implementation in R and attempted to convey an intuition for how the algorithm works.  As we have seen, the algorithm works quite well in the example case, despite the violation of the assumption of independent conditional probabilities.

The reason for the unreasonable effectiveness of the algorithm is two-fold. Firstly, the algorithm picks the predicted class based on the largest predicted probability, so ordering is more important than the actual value of the probability. Secondly, in many cases, a bias one way for a particular vote may well be counteracted by a bias the other way for another vote. That is, biases tend to cancel out, particularly if there are a large number of features.

That said, there are many cases in which the algorithm fails miserably – and we’ll look at some of these in a future post.  However, despite its well known shortcomings, Naive Bayes is often the first port of call in prediction problems simply because it is easy to set up and is fast compared to many of the iterative algorithms we will explore later in this series of articles.

Endnote

Thanks for reading! If you liked this piece, you might enjoy the other articles in my “Gentle introduction to analytics using R” series. Here are the links:

A gentle introduction to text mining using R

A gentle introduction to cluster analysis using R

A gentle introduction to topic modeling using R

Written by K

November 6, 2015 at 7:33 am

A gentle introduction to topic modeling using R

with 49 comments

Introduction

The standard way to search for documents on the internet is via keywords or keyphrases. This is pretty much what Google and other search engines do routinely…and they do it well.  However, as useful as this is, it has its limitations. Consider, for example, a situation in which you are confronted with a large collection of documents but have no idea what they are about. One of the first things you might want to do is to classify these documents into topics or themes. Among other things this would help you figure out if there’s anything interest while also directing you to the relevant subset(s) of the corpus. For small collections, one could do this by simply going through each document but this is clearly infeasible for corpuses containing thousands of documents.

Topic modeling – the theme of this post – deals with the problem of automatically classifying sets of documents into themes

The article is organised as follows: I first provide some background on topic modelling. The algorithm that I use, Latent Dirichlet Allocation (LDA), involves some pretty heavy maths which I’ll avoid altogether. However, I will provide an intuitive explanation of how LDA works before moving on to a practical example which uses the topicmodels library in R. As in my previous articles in this series (see this post and this one), I will discuss the steps in detail along with explanations and provide accessible references for concepts that cannot be covered in the space of a blog post.

(Aside: Beware, LDA is also an abbreviation for Linear Discriminant Analysis a classification technique that I hope to cover later in my ongoing series on text and data analytics).

Latent Dirichlet Allocation – a math-free introduction

In essence, LDA is a technique that facilitates the automatic discovery of themes in a collection of documents.

The basic assumption behind LDA is that each of the documents in a collection consist of a mixture of collection-wide topics. However, in reality we observe only documents and words, not topics – the latter are part of the hidden (or latent) structure of documents. The aim is to infer the latent topic structure given the words and document.  LDA does this by recreating the documents in the corpus by adjusting the relative importance of topics in documents and words in topics iteratively.

Here’s a brief explanation of how the algorithm works, quoted directly from this answer by Edwin Chen on Quora:

  • Go through each document, and randomly assign each word in the document to one of the K topics. (Note: One of the shortcomings of LDA is that one has to specify the number of topics, denoted by K, upfront. More about this later.)
  • This assignment already gives you both topic representations of all the documents and word distributions of all the topics (albeit not very good ones).
  • So to improve on them, for each document d…
  • ….Go through each word w in d…
  • ……..And for each topic t, compute two things: 1) p(topic t | document d) = the proportion of words in document d that are currently assigned to topic t, and 2) p(word w | topic t) = the proportion of assignments to topic t over all documents that come from this word w. Reassign w a new topic, where you choose topic t with probability p(topic t | document d) * p(word w | topic t) (according to our generative model, this is essentially the probability that topic t generated word w, so it makes sense that we resample the current word’s topic with this probability).  (Note: p(a|b) is the conditional probability of a given that b has already occurred – see this post for more on conditional probabilities)
  • ……..In other words, in this step, we’re assuming that all topic assignments except for the current word in question are correct, and then updating the assignment of the current word using our model of how documents are generated.
  • After repeating the previous step a large number of times, you’ll eventually reach a roughly steady state where your assignments are pretty good. So use these assignments to estimate the topic mixtures of each document (by counting the proportion of words assigned to each topic within that document) and the words associated to each topic (by counting the proportion of words assigned to each topic overall).

For another simple explanation of how LDA works in, check out  this article by Matthew Jockers. For a more technical exposition, take a look at this video by David Blei, one of the inventors of the algorithm.

The iterative process described in the last point above is implemented using a technique called Gibbs sampling.  I’ll say a bit more about Gibbs sampling later, but you may want to have a look at this paper by Philip Resnick and Eric Hardesty that explains the nitty-gritty of the algorithm (Warning: it involves a fair bit of math, but has some good intuitive explanations as  well).

As a general point, I should also emphasise that you do not need to understand the ins and outs of an algorithm to use it but it does help to understand, at least at a high level, what the algorithm is doing. One needs to develop a feel for algorithms even if one doesn’t understand the details. Indeed, most people working in analytics do not know the details of the algorithms they use, but that doesn’t stop them from using algorithms intelligently. Purists may disagree. I think they are wrong.

Finally – because you’re no doubt wondering  🙂 – the term “Dirichlet” in LDA refers to the fact that topics and words are assumed to follow Dirichlet distributions. There is no “good” reason for this apart from convenience – Dirichlet distributions provide good approximations to word distributions in documents and, perhaps more important, are computationally convenient.

Preprocessing

As in my previous articles on text mining, I will use a collection of 30 posts from this blog as an example corpus. The corpus can be downloaded here. I will assume that you have R and RStudio installed. Follow this link if you need help with that.

The preprocessing steps are much the same as described in my previous articles.  Nevertheless, I’ll risk boring you with a detailed listing so that you can reproduce my results yourself:

 

#load text mining library
library(tm)

 

#set working directory (modify path as needed)
setwd(“C:\\Users\\Kailash\\Documents\\TextMining”)

 

#load files into corpus
#get listing of .txt files in directory
filenames <- list.files(getwd(),pattern=”*.txt”)

 

#read files into a character vector
files <- lapply(filenames,readLines)

 

#create corpus from vector
docs <- Corpus(VectorSource(files))

 

#inspect a particular document in corpus
writeLines(as.character(docs[[30]]))

 

#start preprocessing
#Transform to lower case
docs <-tm_map(docs,content_transformer(tolower))

 

#remove potentially problematic symbols
toSpace <- content_transformer(function(x, pattern) { return (gsub(pattern, ” “, x))})
docs <- tm_map(docs, toSpace, “-“)
docs <- tm_map(docs, toSpace, “’”)
docs <- tm_map(docs, toSpace, “‘”)
docs <- tm_map(docs, toSpace, “•”)
docs <- tm_map(docs, toSpace, “””)
docs <- tm_map(docs, toSpace, ““”)

 

#remove punctuation
docs <- tm_map(docs, removePunctuation)
#Strip digits
docs <- tm_map(docs, removeNumbers)
#remove stopwords
docs <- tm_map(docs, removeWords, stopwords(“english”))
#remove whitespace
docs <- tm_map(docs, stripWhitespace)
#Good practice to check every now and then
writeLines(as.character(docs[[30]]))
#Stem document
docs <- tm_map(docs,stemDocument)

 

#fix up 1) differences between us and aussie english 2) general errors
docs <- tm_map(docs, content_transformer(gsub),
pattern = “organiz”, replacement = “organ”)
docs <- tm_map(docs, content_transformer(gsub),
pattern = “organis”, replacement = “organ”)
docs <- tm_map(docs, content_transformer(gsub),
pattern = “andgovern”, replacement = “govern”)
docs <- tm_map(docs, content_transformer(gsub),
pattern = “inenterpris”, replacement = “enterpris”)
docs <- tm_map(docs, content_transformer(gsub),
pattern = “team-“, replacement = “team”)
#define and eliminate all custom stopwords
myStopwords <- c(“can”, “say”,”one”,”way”,”use”,
“also”,”howev”,”tell”,”will”,
“much”,”need”,”take”,”tend”,”even”,
“like”,”particular”,”rather”,”said”,
“get”,”well”,”make”,”ask”,”come”,”end”,
“first”,”two”,”help”,”often”,”may”,
“might”,”see”,”someth”,”thing”,”point”,
“post”,”look”,”right”,”now”,”think”,”‘ve “,
“‘re “,”anoth”,”put”,”set”,”new”,”good”,
“want”,”sure”,”kind”,”larg”,”yes,”,”day”,”etc”,
“quit”,”sinc”,”attempt”,”lack”,”seen”,”awar”,
“littl”,”ever”,”moreov”,”though”,”found”,”abl”,
“enough”,”far”,”earli”,”away”,”achiev”,”draw”,
“last”,”never”,”brief”,”bit”,”entir”,”brief”,
“great”,”lot”)
docs <- tm_map(docs, removeWords, myStopwords)
#inspect a document as a check
writeLines(as.character(docs[[30]]))

 

#Create document-term matrix
dtm <- DocumentTermMatrix(docs)
#convert rownames to filenames
rownames(dtm) <- filenames
#collapse matrix by summing over columns
freq <- colSums(as.matrix(dtm))
#length should be total number of terms
length(freq)
#create sort order (descending)
ord <- order(freq,decreasing=TRUE)
#List all terms in decreasing order of freq and write to disk
freq[ord]
write.csv(freq[ord],”word_freq.csv”)

Check out the  preprocessing section in either this article or this one for detailed explanations of the code. The document term matrix (DTM) produced by the above code will be the main input into the LDA algorithm of the next section.

Topic modelling using LDA

We are now ready to do some topic modelling. We’ll use the topicmodels package written by Bettina Gruen and Kurt Hornik. Specifically, we’ll use the LDA function with the Gibbs sampling option mentioned earlier, and I’ll say  more about it in a second. The LDA function has a fairly large number of parameters. I’ll describe these briefly below. For more, please check out this vignette by Gruen and Hornik.

For the most part, we’ll use the default parameter values supplied by the LDA function,custom setting only the parameters that are required by the Gibbs sampling algorithm.

Gibbs sampling works by performing a random walk in such a way that reflects the characteristics of a desired distribution. Because the starting point of the walk is chosen at random, it is necessary to discard the first few steps of the walk (as these do not correctly reflect the properties of distribution). This is referred to as the burn-in period. We set the burn-in parameter to  4000. Following the burn-in period, we perform 2000 iterations, taking every 500th  iteration for further use.  The reason we do this is to avoid correlations between samples. We use 5 different starting points (nstart=5) – that is, five independent runs. Each starting point requires a seed integer (this also ensures reproducibility),  so I have provided 5 random integers in my seed list. Finally I’ve set best to TRUE (actually a default setting), which instructs the algorithm to return results of the run with the highest posterior probability.

Some words of caution are in order here. It should be emphasised that the settings above do not guarantee  the convergence of the algorithm to a globally optimal solution. Indeed, Gibbs sampling will, at best, find only a locally optimal solution, and even this is hard to prove mathematically in specific practical problems such as the one we are dealing with here. The upshot of this is that it is best to do lots of runs with different settings of parameters to check the stability of your results. The bottom line is that our interest is purely practical so it is good enough if the results make sense. We’ll leave issues  of mathematical rigour to those better qualified to deal with them 🙂

As mentioned earlier,  there is an important parameter that must be specified upfront: k, the number of topics that the algorithm should use to classify documents. There are mathematical approaches to this, but they often do not yield semantically meaningful choices of k (see this post on stackoverflow for an example). From a practical point of view, one can simply run the algorithm for different values of k and make a choice based by inspecting the results. This is what we’ll do.

OK, so the first step is to set these parameters in R… and while we’re at it, let’s also load the topicmodels library (Note: you might need to install this package as it is not a part of the base R installation).

#load topic models library
library(topicmodels)

 

#Set parameters for Gibbs sampling
burnin <- 4000
iter <- 2000
thin <- 500
seed <-list(2003,5,63,100001,765)
nstart <- 5
best <- TRUE

 

#Number of topics
k <- 5

That done, we can now do the actual work – run the topic modelling algorithm on our corpus. Here is the code:

#Run LDA using Gibbs sampling
ldaOut <-LDA(dtm,k, method=”Gibbs”, control=list(nstart=nstart, seed = seed, best=best, burnin = burnin, iter = iter, thin=thin))

 

#write out results
#docs to topics
ldaOut.topics <- as.matrix(topics(ldaOut))
write.csv(ldaOut.topics,file=paste(“LDAGibbs”,k,”DocsToTopics.csv”))

 

#top 6 terms in each topic
ldaOut.terms <- as.matrix(terms(ldaOut,6))
write.csv(ldaOut.terms,file=paste(“LDAGibbs”,k,”TopicsToTerms.csv”))

 

#probabilities associated with each topic assignment
topicProbabilities <- as.data.frame(ldaOut@gamma)
write.csv(topicProbabilities,file=paste(“LDAGibbs”,k,”TopicProbabilities.csv”))

 

#Find relative importance of top 2 topics
topic1ToTopic2 <- lapply(1:nrow(dtm),function(x)
sort(topicProbabilities[x,])[k]/sort(topicProbabilities[x,])[k-1])

 

#Find relative importance of second and third most important topics
topic2ToTopic3 <- lapply(1:nrow(dtm),function(x)
sort(topicProbabilities[x,])[k-1]/sort(topicProbabilities[x,])[k-2])

 

#write to file
write.csv(topic1ToTopic2,file=paste(“LDAGibbs”,k,”Topic1ToTopic2.csv”))
write.csv(topic2ToTopic3,file=paste(“LDAGibbs”,k,”Topic2ToTopic3.csv”))

The LDA algorithm returns an object that contains a lot of information. Of particular interest to us are the document to topic assignments, the top terms in each topic and the probabilities associated with each of those terms. These are printed out in the first three calls to write.csv above. There are a few important points to note here:

  1. Each document is considered to be a mixture of all topics (5 in this case). The assignments in the first file list the top topic – that is, the one with the highest probability (more about this in point 3 below).
  2. Each topic contains all terms (words) in the corpus, albeit with different probabilities. We list only the top  6 terms in the second file.
  3. The last file lists the probabilities with  which each topic is assigned to a document. This is therefore a 30 x 5 matrix – 30 docs and 5 topics. As one might expect, the highest probability in each row corresponds to the topic assigned to that document.  The “goodness” of the primary assignment (as discussed in point 1) can be assessed by taking the ratio of the highest to second-highest probability and the second-highest to the third-highest probability and so on. This is what I’ve done in the last nine lines of the code above.

Take some time to examine the output and confirm for yourself that that the primary topic assignments are best when the ratios of probabilities discussed in point 3 are highest. You should also experiment with different values of k to see if you can find better topic distributions. In the interests of space I will restrict myself to k = 5.

The table below lists the top 6 terms in topics 1 through 5.

Topic 1 Topic 2 Topic 3 Topic 4 Topic 5
1 work question chang system project
2 practic map organ data manag
3 mani time consult model approach
4 flexibl ibi manag design organ
5 differ issu work process decis
6 best plan problem busi problem

The table below lists the document to (primary) topic assignments:

 

Document Topic
BeyondEntitiesAndRelationships.txt 4
bigdata.txt 4
ConditionsOverCauses.txt 5
EmergentDesignInEnterpriseIT.txt 4
FromInformationToKnowledge.txt 2
FromTheCoalface.txt 1
HeraclitusAndParmenides.txt 3
IroniesOfEnterpriseIT.txt 3
MakingSenseOfOrganizationalChange.txt 5
MakingSenseOfSensemaking.txt 2
ObjectivityAndTheEthicalDimensionOfDecisionMaking.txt 5
OnTheInherentAmbiguitiesOfManagingProjects.txt 5
OrganisationalSurprise.txt 5
ProfessionalsOrPoliticians.txt 3
RitualsInInformationSystemDesign.txt 4
RoutinesAndReality.txt 4
ScapegoatsAndSystems.txt 5
SherlockHolmesFailedProjects.txt 3
sherlockHolmesMgmtFetis.txt 3
SixHeresiesForBI.txt 4
SixHeresiesForEnterpriseArchitecture.txt 3
TheArchitectAndTheApparition.txt 3
TheCloudAndTheGrass.txt 2
TheConsultantsDilemma.txt 3
TheDangerWithin.txt 5
TheDilemmasOfEnterpriseIT.txt 3
TheEssenceOfEntrepreneurship.txt 1
ThreeTypesOfUncertainty.txt 5
TOGAFOrNotTOGAF.txt 3
UnderstandingFlexibility.txt 1

From a quick perusal of the two tables it appears that the algorithm has done a pretty decent job. For example,topic 4 is about data and system design, and the documents assigned to it are on topic. However, it is far from perfect – for example, the interview I did with Neil Preston on organisational change (MakingSenseOfOrganizationalChange.txt) has been assigned to topic 5, which seems to be about project management. It ought to be associated with Topic 3, which is about change. Let’s see if we can resolve this by looking at probabilities associated with topics.

The table below lists the topic probabilities by document:

Topic 1 Topic 2 Topic 3 Topic 4 Topic 5
BeyondEn 0.071 0.064 0.024 0.741 0.1
bigdata. 0.182 0.221 0.182 0.26 0.156
Conditio 0.144 0.109 0.048 0.205 0.494
Emergent 0.121 0.226 0.204 0.236 0.213
FromInfo 0.096 0.643 0.026 0.169 0.066
FromTheC 0.636 0.082 0.058 0.086 0.138
Heraclit 0.137 0.091 0.503 0.162 0.107
IroniesO 0.101 0.088 0.388 0.26 0.162
MakingSe 0.13 0.206 0.262 0.089 0.313
MakingSe 0.09 0.715 0.055 0.067 0.074
Objectiv 0.216 0.078 0.086 0.242 0.378
OnTheInh 0.18 0.234 0.102 0.12 0.364
Organisa 0.089 0.095 0.07 0.092 0.655
Professi 0.155 0.064 0.509 0.128 0.144
RitualsI 0.103 0.064 0.044 0.676 0.112
Routines 0.108 0.042 0.033 0.69 0.127
Scapegoa 0.135 0.088 0.043 0.185 0.549
Sherlock 0.093 0.082 0.398 0.195 0.232
sherlock 0.108 0.136 0.453 0.123 0.18
SixHeres 0.159 0.11 0.078 0.516 0.138
SixHeres 0.104 0.111 0.366 0.212 0.207
TheArchi 0.111 0.221 0.522 0.088 0.058
TheCloud 0.185 0.333 0.198 0.136 0.148
TheConsu 0.105 0.184 0.518 0.096 0.096
TheDange 0.114 0.079 0.037 0.079 0.69
TheDilem 0.125 0.128 0.389 0.261 0.098
TheEssen 0.713 0.059 0.031 0.113 0.084
ThreeTyp 0.09 0.076 0.042 0.083 0.708
TOGAFOrN 0.158 0.232 0.352 0.151 0.107
Understa 0.658 0.065 0.072 0.101 0.105

In the table, the highest probability in each row is in bold. Also, in cases where the maximum and the second/third largest probabilities are close, I have highlighted the second (and third) highest probabilities in red.   It is clear that Neil’s interview (9th document in the above table) has 3  topics with comparable probabilities – topic 5 (project management), topic 3 (change) and topic 2 (issue mapping / ibis), in decreasing order of probabilities. In general, if a document has multiple topics with comparable probabilities, it simply means that the document speaks to all those topics in proportions indicated by the probabilities. A reading of Neil’s interview will convince you that our conversation did indeed range over all those topics.

That said, the algorithm is far from perfect. You might have already noticed a few poor assignments. Here is one – my post on Sherlock Holmes and the case of the failed project has been assigned to topic 3; I reckon it belongs in topic 5. There are a number of others, but I won’t belabor the point, except to reiterate that this precisely why you definitely want to experiment with different settings of the iteration parameters (to check for stability) and, more important, try a range of different values of k to find the optimal number of topics.

To conclude

Topic modelling provides a quick and convenient way to perform unsupervised classification of a corpus of documents.  As always, though, one needs to examine the results carefully to check that they make sense.

I’d like to end with a general observation. Classifying documents is an age-old concern that cuts across disciplines. So it is no surprise that topic modelling has got a look-in from diverse communities. Indeed, when I was reading up and learning about LDA, I found that some of the best introductory articles in the area have been written by academics working in English departments! This is one of the things I love about working in text analysis, there is a wealth of material on the web written from diverse perspectives. The term cross-disciplinary often tends to be a platitude , but in this case it is simply a statement of fact.

I hope that I have been able to convince you to explore this rapidly evolving field. Exciting times ahead, come join the fun.

Written by K

September 29, 2015 at 7:18 pm

Setting up an internal data analytics practice – some thoughts from a wayfarer

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Introduction

This year has been hugely exciting so far: I’ve been exploring and playing with various techniques that fall under the general categories of data mining and text analytics. What’s been particularly satisfying is that I’ve been fortunate to find meaningful applications for these techniques within my organization.

Although I have a fair way to travel yet, I’ve learnt that common wisdom about data analytics – especially the stuff that comes from software vendors and high-end consultancies – can be misleading, even plain wrong. Hence this post in which I dispatch some myths and share a few pointers on establishing data analytics capabilities within an organization.

Busting a few myths

Let’s get right to it by taking a critical look at a few myths about setting up an internal data analytics practice.

  1. Requires high-end technology and a big budget: this myth is easy to bust because I can speak from recent experience. No, you do not need cutting-edge technology or an oversized budget.   You can get started for with an outlay of 0$ – yes, that’s right, for free!  All you need to is the open-source statistical package R (check out this section of my article on text mining for more on installing and using R) and the willingness to roll-up your sleeves and learn (more about this  later).  No worries if you prefer to stick with familiar tools – you can even begin with Excel.
  2. Needs specialist skills: another myth floating around is that you need Phd level knowledge in statistics or applied mathematics to do practical work in analytics. Sorry, but that’s plain wrong. You do need a PhD to do research in the analytics and develop your own algorithms, but not if you want to apply algorithms written by others.Yes, you will need to develop an understanding of the algorithms you plan to use, a feel for how they work and the ability to tell whether the results make sense. There are many good resources that can help you develop these skills – see, for example, the outstanding books by James, Witten, Hastie and Tibshirani and Kuhn and Johnson.
  3. Must have sponsorship from the top: this one is possibly a little more controversial than the previous two. It could be argued that it is impossible to gain buy in for a new capability without sponsorship from top management. However, in my experience, it is OK to start small by finding potential internal “customers” for analytics services through informal conversations with folks in different functions.I started by having informal conversations with managers in two different areas: IT infrastructure and sales / marketing.  I picked these two areas because I knew that they had several gigabytes of under-exploited data – a good bit of it unstructured – and a lot of open questions that could potentially be answered (at least partially) via methods of data and text analytics.  It turned out I was right. I’m currently doing a number of proofs of concept and small projects in both these areas.  So you don’t need sponsorship from the top as long as you can get buy in from people who have problems they believe you can solve. If you deliver, they may even advocate your cause to their managers.

A caveat is in order at this point:  my organization is not the same as yours, so you may well need to follow a different path from mine. Nevertheless, I do believe that it is always possible to find a way to start without needing permission or incurring official wrath.  In that spirit, I now offer some suggestions to help kick-start your efforts

Getting started

As the truism goes, the hardest part of any new effort is getting started.  The first thing to keep in mind is to start small. This is true even if you have official sponsorship and a king-sized budget. It is very tempting to spend a lot of time garnering management support for investing in high-end technology.  Don’t do it!  Do the following instead:

  1. Develop an understanding of the problems faced by people you plan to approach: The best way to do this is to talk to analysts or frontline managers. In my case, I was fortunate to have access to some very savvy analysts in IT service management and marketing who gave me a slew of interesting ideas to pursue. A word of advice: it is best not to talk to senior managers until you have a few concrete results that you can quantify in terms of dollar values.
  2. Invest time and effort in understanding analytics algorithms and gaining practical experience with them: As mentioned earlier, I started with R – and I believe it is the best choice. Not just because it is free but also because there are a host of packages available to tackle just about any analytics problem you might encounter.  There are some excellent free resources available to get you started with R (check out this listing on the r-statistics blog, for example).It is important that you start cutting code as you learn. This will help you build a repertoire of techniques and approaches as you progress. If you get stuck when coding, chances are you will find a solution on the wonderful stackoverflow site.
  3. Evangelise, evangelise, evangelise: You are, in effect, trying to sell an idea to people within your organization. You therefore have to identify people who might be able to help you and then convince them that your idea has merit. The best way to do the latter is to have concrete examples of problems that you have tackled. This is a chicken-and-egg situation in that you can’t have any examples until you gain support.  I got support by approaching people I know well. I found that most – no, all – of them were happy to provide me with interesting ideas and access to their data.
  4. Begin with small (but real) problems: It is important to start with the “low-hanging fruit” – the problems that would take the least effort to solve. However, it is equally important to address real problems, i.e. those that matter to someone.
  5. Leverage your organisation’s existing data infrastructure: From what I’ve written thus far, I may have given you the impression that the tools of data analytics stand separate from your existing data infrastructure. Nothing could be further from the truth. In reality, I often do the initial work  (basic preprocessing and exploratory analysis) using my organisation’s relational database infrastructure. Relational databases have sophisticated analytical extensions to SQL as well as efficient bulk data cleansing and transport facilities. Using these make good sense, particularly if your R installation is on a desktop or laptop computer as it is in my case. Moreover, many enterprise database vendors now offer add-on options that integrate R with their products. This gives you the best of both worlds – relational and analytical capabilities on an enterprise-class platform.
  6. Build relationships with the data management team: Remember the work you are doing falls under the ambit of the group that is officially responsible for managing data in your organization. It is therefore important that you keep them informed of what you’re doing.  Sooner or later your paths will cross, and you want to be sure that there are no nasty surprises (for either side!) at that point. Moreover, if you build connections with them early, you may even find that the data management team supports your efforts.

Having waxed verbose, I should mention that my effort is work in progress and I do not know where it will lead. Nevertheless, I offer these suggestions as a wayfarer who is considerably further down the road from where he started.

Parting thoughts

You may have noticed that I’ve refrained from using the overused and over-hyped term “Big Data” in this piece. This is deliberate. Indeed, the techniques I have been using have nothing to do with the size of the datasets. To be honest, I’ve applied them to datasets ranging from a few thousand to a few hundred thousand records, both of which qualify as Very Small Data in today’s world.

Your vendor will be only too happy to sell you Big Data infrastructure that will set you back a good many dollars. However, the chances are good that you do not need it right now.  You’ll be much better off going back to them after you hit the limits of your current data processing infrastructure. Moreover, you’ll also be better informed about your needs then.

You may also be wondering why I haven’t said much about the composition of the analytics team (barring the point about not needing PhD statisticians) and how it should be organized.  The reason I haven’t done so is that I believe the right composition and organizational structure will emerge from the initial projects done and feedback received from internal customers. The resulting structure will be better suited to the organization than one that is imposed upfront.  Long time readers of this blog might recognize this as a tenet of emergent design.

Finally, I should reiterate that my efforts are still very much in progress and I know not where they will lead. However, even if they go nowhere, I would have learnt something about my organization and picked up a useful, practical skill. And that is good enough for me.

Written by K

September 3, 2015 at 8:28 pm

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