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A gentle introduction to network graphs using R and Gephi

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Introduction

Graph theory is the an area of mathematics that analyses relationships between pairs of objects. Typically graphs consist of nodes (points representing objects) and edges (lines depicting relationships between objects). As one might imagine, graphs are extremely useful in visualizing relationships between objects. In this post, I provide a detailed introduction to network graphs using  R, the premier open source tool statistics package for calculations and the excellent Gephi software for visualization.

The article is organised as follows: I begin by defining the problem and then spend some time developing the concepts used in constructing the graph  Following this,  I do the data preparation in R  and then finally build the network graph using Gephi.

The problem

In an introductory article on cluster analysis, I provided an in-depth introduction to a couple of algorithms that can be used to categorise documents automatically.  Although these techniques are useful, they do not provide a feel for the relationships between different documents in the collection of interest.  In the present piece I show network graphs can be used to to visualise similarity-based relationships within a corpus.

Document similarity

There are many ways to quantify similarity between documents. A popular method is to use the notion of distance between documents. The basic idea is simple: documents that have many words in common are “closer” to each other than those that share fewer words. The problem with distance, however, is that it can be skewed by word count: documents that have an unusually high word  count will show up as outliers even though they may be similar (in terms of words used) to other documents in the corpus. For this reason, we will use another related measure of similarity that does not suffer from this problem – more about this in a minute.

Representing documents mathematically

As I explained in my article on cluster analysis, a document can be represented as a point in a conceptual space that has dimensionality equal to the number of distinct words in the collection of documents. I revisit and build on that explanation below.

Say one has a simple document consisting of the words “five plus six”, one can represent it mathematically in a 3 dimensional space in which the individual words are represented by the three axis (See Figure 1). Here each word is a coordinate axis (or dimension).  Now, if one connects the point representing the document (point A in the figure) to the origin of the word-space, one has a vector, which in this case is a directed line connecting the point in question to the origin.  Specifically, the point A can be represented by the coordinates (1, 1, 1) in this space. This is a nice quantitative representation of the fact that the words five, plus and one appear in the document exactly once. Note, however, that we’ve assumed the order of words does not matter. This is a reasonable assumption in some cases, but not always so.

Figure 1

Figure 1

As another example consider document, B, which consists of only two words: “five plus” (see Fig 2). Clearly this document shares some similarity with document but it is not identical.  Indeed, this becomes evident when we note that document (or point) B is simply the point $latex(1, 1, 0)$ in this space, which tells us that it has two coordinates (words/frequencies) in common with document (or point) A.

Figure 2

Figure 2

To be sure, in a realistic collection of documents we would have a large number of distinct words, so we’d have to work in a very high dimensional space. Nevertheless, the same principle holds: every document in the corpus can be represented as a vector consisting of a directed line from the origin to the point to which the document corresponds.

Cosine similarity

Now it is easy to see that two documents are identical if they correspond to the same point. In other words, if their vectors coincide. On the other hand, if they are completely dissimilar (no words in common), their vectors will be at right angles to each other.  What we need, therefore, is a quantity that varies from 0 to 1 depending on whether two documents (vectors) are dissimilar(at right angles to each other) or similar (coincide, or are parallel to each other).

Now here’s the ultra-cool thing, from your high school maths class, you know there is a trigonometric ratio which has exactly this property – the cosine!

What’s even cooler is that the cosine of the angle between two vectors is simply the dot product  of the two vectors, which is sum of the products of the individual elements of the vector,  divided by the product of the  lengths of the two vectors. In three dimensions this can be expressed mathematically as:

\cos(\theta)= \displaystyle \frac{x_1 x_2+y_1 y_2+z_1 z_2}{\sqrt{x_1^2+y_1^2+z_1^2}\sqrt{x_2^2+y_2^2+z_2^2}}...(1)

where the two vectors are (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}) , and \theta is the angle between the two vectors (see Fig 2).

The upshot of the above is that the cosine of the angle between the vector representation of two documents is a reasonable measure of similarity between them. This quantity, sometimes referred to as cosine similarity, is what we’ll take as our similarity measure in the rest of this article.

The adjacency matrix

If we have a collection of N documents, we can calculate the similarity between every pair of documents as we did for A and B in the previous section. This would give us a set of N^2 numbers between 0 and 1, which can be conveniently represented as a matrix.  This is sometimes called the adjacency matrix. Beware, though, this term has many different meanings in the math literature. I use it in the sense specified above.

Since every document is identical to itself, the diagonal elements of the matrix will all be 1. These similarities are trivial (we know that every document is identical to itself!)  so we’ll set the diagonal elements to zero.

Another important practical point is that visualizing every relationship is going to make  a very messy graph. There would be N(N-1) edges in such a graph, which would make it impossible to make sense of if we have more than a handful of documents. For this reason, it is normal practice to choose a cutoff value of similarity below which it is set to zero.

Building the adjacency matrix using R

We now have enough background to get down to the main point of this article – visualizing relationships between documents.

The first step is to build the adjacency matrix.  In order to do this, we have to build the document term matrix (DTM) for the collection of documents,  a process which I have dealt with at length in my  introductory pieces on text mining and topic modeling. In fact, the steps are actually identical to those detailed in the second piece. I will therefore avoid lengthy explanations here. However,  I’ve listed all the code below with brief comments (for those who are interested in trying this out, the document corpus can be downloaded here and a pdf listing of the R code can be obtained here.)

OK, so here’s the code listing:

#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)

The  rows of a DTM are document vectors akin to the vector representations of documents A and B discussed earlier. The DTM therefore contains all the information we need to calculate the cosine similarity between every pair of documents in the corpus (via equation 1). The R code below implements this, after taking care of a few preliminaries.

#convert dtm to matrix
m<-as.matrix(dtm)
#write as csv file
write.csv(m,file=”dtmEight2Late.csv”)
#Map filenames to matrix row numbers
#these numbers will be used to reference
#files in the network graph
filekey <- cbind(rownames(m),filenames)
write.csv(filekey,”filekey.csv”)
#compute cosine similarity between document vectors
#converting to distance matrix sets diagonal elements to 0
cosineSim <- function(x){
as.dist(x%*%t(x)/(sqrt(rowSums(x^2) %*% t(rowSums(x^2)))))
}
cs <- cosineSim(m)
write.csv(as.matrix(cs),file=”csEight2Late.csv”)
#adjacency matrix: set entries below a certain threshold to 0.
#We choose half the magnitude of the largest element of the matrix
#as the cutoff. This is an arbitrary choice
cs[cs < max(cs)/2] <- 0
cs <- round(cs,3)
write.csv(as.matrix(cs),file=”AdjacencyMatrix.csv”)

A few lines need a brief explanation:

First up, although the DTM is a matrix, it is internally stored in a special form suitable for sparse matrices. We therefore have to explicitly convert it into a proper matrix before using it to calculate similarity.

Second, the names I have given the documents are way too long to use as labels in the network diagram. I have therefore mapped the document names to the row numbers which we’ll use in our network graph later. The mapping back to the original document names is stored in filekey.csv. For future reference, the mapping is shown in Table 1 below.

File number Name
1 BeyondEntitiesAndRelationships.txt
2 bigdata.txt
3 ConditionsOverCauses.txt
4 EmergentDesignInEnterpriseIT.txt
5 FromInformationToKnowledge.txt
6 FromTheCoalface.txt
7 HeraclitusAndParmenides.txt
8 IroniesOfEnterpriseIT.txt
9 MakingSenseOfOrganizationalChange.txt
10 MakingSenseOfSensemaking.txt
11 ObjectivityAndTheEthicalDimensionOfDecisionMaking.txt
12 OnTheInherentAmbiguitiesOfManagingProjects.txt
13 OrganisationalSurprise.txt
14 ProfessionalsOrPoliticians.txt
15 RitualsInInformationSystemDesign.txt
16 RoutinesAndReality.txt
17 ScapegoatsAndSystems.txt
18 SherlockHolmesFailedProjects.txt
19 sherlockHolmesMgmtFetis.txt
20 SixHeresiesForBI.txt
21 SixHeresiesForEnterpriseArchitecture.txt
22 TheArchitectAndTheApparition.txt
23 TheCloudAndTheGrass.txt
24 TheConsultantsDilemma.txt
25 TheDangerWithin.txt
26 TheDilemmasOfEnterpriseIT.txt
27 TheEssenceOfEntrepreneurship.txt
28 ThreeTypesOfUncertainty.txt
29 TOGAFOrNotTOGAF.txt
30 UnderstandingFlexibility.txt

Table 1: File mappings

Finally, the distance function (as.dist) in the cosine similarity function sets the diagonal elements to zero  because the distance between a document and itself is zero…which is just a complicated way of saying that a document is identical to itself 🙂

The last three lines of code above simply implement the cutoff that I mentioned in the previous section. The comments explain the details so I need say no more about it.

…which finally brings us to Gephi.

Visualizing document similarity using Gephi

Gephi is an open source, Java based network analysis and visualisation tool. Before going any further, you may want to download and install it. While you’re at it you may also want to download this excellent quick start tutorial.

Go on, I’ll wait for you…

To begin with, there’s a little formatting quirk that we need to deal with. Gephi expects separators in csv files to be semicolons (;) . So, your first step is to open up the adjacency matrix that you created in the previous section (AdjacencyMatrix.csv) in a text editor and replace commas with semicolons.

Once you’ve done that, fire up Gephi, go to File > Open,  navigate to where your Adjacency matrix is stored and load the file. If it loads successfully, you should see a feedback panel as shown in Figure 3.  By default Gephi creates a directed graph (i.e one in which the edges have arrows pointing from one node to another). Change this to undirected and click OK.

Figure 3: Gephi import feedback

Figure 3: Gephi import feedback

Once that is done, click on overview (top left of the screen). You should end up with something like Figure 4.

Figure 4: Initial overview after loading adjacency matrix

Figure 4: Initial overview after loading adjacency matrix

Gephi has sketched out an initial network diagram which depicts the relationships between documents…but it needs a bit of work to make it look nicer and more informative. The quickstart tutorial mentioned earlier describes various features that can be used to manipulate and prettify the graph. In the remainder of this section, I list some that I found useful. Gephi offers many more. Do explore, there’s much more than  I can cover in an introductory post.

First some basics. You can:

  • Zoom and pan using mouse wheel and right button.
  • Adjust edge thicknesses using the slider next to text formatting options on bottom left of main panel.
  • Re-center graph via the magnifying glass icon on left of display panel (just above size adjuster).
  • Toggle node labels on/off by clicking on grey T symbol on bottom left panel.

Figure 5 shows the state of the diagram after labels have been added and edge thickness adjusted (note that your graph may vary in appearance).

Figure 5: graph with node labels and adjusted edge thicknesses

Figure 5: graph with node labels and adjusted edge thicknesses

The default layout of the graph is ugly and hard to interpret. Let’s work on fixing it up. To do this, go over to the layout panel on the left. Experiment with different layouts to see what they do. After some messing around, I found the Fruchtermann-Reingold and Force Atlas options to be good for this graph. In the end I used Force Atlas with a Repulsion Strength of 2000 (up from the default of 200) and an Attraction Strength of 1 (down from the default of 10). I also adjusted the figure size and node label font size from the graph panel in the center. The result is shown in Figure 6.

Figure 6: Graph after using Force Atlas layout

Figure 6: Graph after using Force Atlas layout

This is much better. For example, it is now evident that document 9 is the most connected one (which table 9 tells us is a transcript of a conversation with Neil Preston on organisational change).

It would be nice if we could colour code edges/nodes and size nodes by their degree of connectivity. This can be done via the ranking panel above the layout area where you’ve just been working.

In the Nodes tab select Degree as  the rank parameter (this is the degree of connectivity of the node) and hit apply. Select your preferred colours via the small icon just above the colour slider. Use the colour slider to adjust the degree of connectivity at which colour transitions occur.

Do the same for edges, selecting weight as the rank parameter(this is the degree of similarity between the two douments connected by the edge). With a bit of playing around, I got the graph shown in the screenshot below (Figure 7).

Figure 7: Connectivity-based colouring of edges and nodes.

Figure 5: Connectivity-based colouring of edges and nodes.

If you want to see numerical values for the rankings, hit the results list icon on the bottom left of the ranking panel. You can see numerical ranking values for both nodes and edges as shown in Figures 8 and 9.

Figure 8: Node ranking

Figure 8: Node ranking (see left of figure)

Figure 9: Edge ranking

Figure 9: Edge ranking

It is easy to see from the figure that documents 21 and 29 are the most similar in terms of cosine ranking. This makes sense, they are pieces in which I have ranted about the current state of enterprise architecture – the first article is about EA in general and the other about the TOGAF framework. If you have a quick skim through, you’ll see that they have a fair bit in common.

Finally, it would be nice if we could adjust node size to reflect the connectedness of the associated document. You can do this via the “gem” symbol on the top right of the ranking panel. Select appropriate min and max sizes (I chose defaults) and hit apply. The node size is now reflective of the connectivity of the node – i.e. the number of other documents to which it is cosine similar to varying degrees. The thickness of the edges reflect the degree of similarity. See Figure 10.

Figure 10: Node sizes reflecting connectedness

Figure 10: Node sizes reflecting connectedness

Now that looks good enough to export. To do this, hit the preview tab on main panel and make following adjustments to the default settings:

Under Node Labels:
1. Check Show Labels
2. Uncheck proportional size
3. Adjust font to required size

Under Edges:
1. Change thickness to 10
2. Check rescale weight

Hit refresh after making the above adjustments. You should get something like Fig 11.

Figure 11: Export preview

Figure 11: Export preview

All that remains now is to do the deed: hit export SVG/PDF/PNG to export the diagram. My output is displayed in Figure 12. It clearly shows the relationships between the different documents (nodes) in the corpus. The nodes with the highest connectivity are indicated via node size and colour  (purple for high, green for low) and strength of similarity is indicated by edge thickness.

Figure 12: Gephi network graph

Figure 12: Gephi network graph of document corpus

…which brings us to the end of this journey.

Wrapping up

The techniques of text analysis enable us to quantify relationships between documents. Document similarity is one such relationship. Numerical measures are good, but the comprehensibility of these can be further enhanced through meaningful visualisations.  Indeed, although my stated objective in this article was to provide an introduction to creating network graphs using Gephi and R (which I hope I’ve succeeded in doing), a secondary aim was to show how document similarity can be quantified and visualised. I sincerely hope you’ve found the discussion interesting and useful.

Many thanks for reading! As always, your feedback would be greatly appreciated.

Written by K

December 2, 2015 at 7:20 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

A gentle introduction to cluster analysis using R

with 27 comments

Introduction

Welcome to the second part of my introductory series on text analysis using R (the first article can be accessed here).  My aim in the present piece is to provide a  practical introduction to cluster analysis. I’ll begin with some background before moving on to the nuts and bolts of clustering. We have a fair bit to cover, so let’s get right to it.

A common problem when analysing large collections of documents is to categorize them in some meaningful way. This is easy enough if one has a predefined classification scheme that is known to fit the collection (and if the collection is small enough to be browsed manually). One can then simply scan the documents, looking for keywords appropriate to each category and classify the documents based on the results. More often than not, however, such a classification scheme is not available and the collection too large. One then needs to use algorithms that can classify documents automatically based on their structure and content.

The present post is a practical introduction to a couple of automatic text categorization techniques, often referred to as clustering algorithms.  As the Wikipedia article on clustering tells us:

Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in other groups (clusters).

As one might guess from the above, the results of clustering depend rather critically on the method one uses to group objects. Again, quoting from the Wikipedia piece:

Cluster analysis itself is not one specific algorithm, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their notion of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances [Note: we’ll use distance-based methods] among the cluster members, dense areas of the data space, intervals or particular statistical distributions [i.e. distributions of words within documents and the entire collection].

…and a bit later:

…the notion of a “cluster” cannot be precisely defined, which is one of the reasons why there are so many clustering algorithms. There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties. Understanding these “cluster models” is key to understanding the differences between the various algorithms.

An upshot of the above is that it is not always straightforward to interpret the output of clustering algorithms. Indeed, we will see this in the example discussed below.

With that said for an introduction, let’s move on to the nut and bolts of clustering.

Preprocessing the corpus

In this section I cover the steps required to create the R objects necessary in order to do clustering. It goes over territory that I’ve covered in detail in the first article in this series – albeit with a few tweaks, so you may want to skim through even if you’ve read my previous piece.

To begin with I’ll assume you have R and RStudio (a free development environment for R) installed on your computer and are familiar with the basic functionality in the text mining ™ package.  If you need help with this, please look at the instructions in my previous article on text mining.

As in the first part of this series,  I will use 30 posts from my blog as the example collection (or corpus, in text mining-speak). The corpus can be downloaded here. For completeness, I will run through the entire sequence of steps – right from loading the corpus into R, to running the two clustering algorithms.

Ready? Let’s go…

The first step is to fire up RStudio and navigate to the directory in which you have unpacked the example corpus. Once this is done, load the text mining package, tm.  Here’s the relevant code (Note: a complete listing of the code in this article can be accessed here):

getwd()
[1] “C:/Users/Kailash/Documents”

#set working directory – fix path as needed!
setwd(“C:/Users/Kailash/Documents/TextMining”)
#load tm library
library(tm)

Loading required package: NLP

Note: R commands are in blue, output in black or red; lines that start with # are comments.

If you get an error here, you probably need to download and install the tm package. You can do this in RStudio by going to Tools > Install Packages and entering “tm”. When installing a new package, R automatically checks for and installs any dependent packages.

The next step is to load the collection of documents into an object that can be manipulated by functions in the tm package.

#Create Corpus
docs <- Corpus(DirSource("C:/Users/Kailash/Documents/TextMining"))
#inspect a particular document
writeLines(as.character(docs[[30]]))

The next step is to clean up the corpus. This includes things such as transforming to a consistent case, removing non-standard symbols & punctuation, and removing numbers (assuming that numbers do not contain useful information, which is the case here):

#Transform to lower case
docs <- tm_map(docs,content_transformer(tolower))
#remove potentiallyy 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, " -")
docs <- tm_map(docs, toSpace, "“")
docs <- tm_map(docs, toSpace, "”")
#remove punctuation
docs <- tm_map(docs, removePunctuation)
#Strip digits
docs <- tm_map(docs, removeNumbers)

Note: please see my previous article for more on content_transformer and the toSpace function defined above.

Next we remove stopwords – common words (like “a” “and” “the”, for example) and eliminate extraneous whitespaces.

#remove stopwords
docs <- tm_map(docs, removeWords, stopwords("english"))
#remove whitespace
docs <- tm_map(docs, stripWhitespace)
writeLines(as.character(docs[[30]]))

flexibility eye beholder action increase organisational flexibility say redeploying employees likely seen affected move constrains individual flexibility dual meaning characteristic many organizational platitudes excellence synergy andgovernance interesting exercise analyse platitudes expose difference espoused actual meanings sign wishing many hours platitude deconstructing fun

At this point it is critical to inspect the corpus because  stopword removal in tm can be flaky. Yes, this is annoying but not a showstopper because one can remove problematic words manually once one has identified them – more about this in a minute.

Next, we stem the document – i.e. truncate words to their base form. For example, “education”, “educate” and “educative” are stemmed to “educat.”:

docs <- tm_map(docs,stemDocument)

Stemming works well enough, but there are some fixes that need to be done due to my inconsistent use of British/Aussie and US English. Also, we’ll take this opportunity to fix up some concatenations like “andgovernance” (see paragraph printed out above). Here’s the code:

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")

The next step is to remove the stopwords that were missed by R. The best way to do this  for a small corpus is to go through it and compile a list of words to be eliminated. One can then create a custom vector containing words to be removed and use the removeWords transformation to do the needful. Here is the code (Note:  + indicates a continuation of a statement from the previous line):

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 ")
#remove custom stopwords
docs <- tm_map(docs, removeWords, myStopwords)

Again, it is a good idea to check that the offending words have really been eliminated.

The final preprocessing step is to create a document-term matrix (DTM) – a matrix that lists all occurrences of words in the corpus.  In a DTM, documents are represented by rows and the terms (or words) by columns.  If a word occurs in a particular document n times, then the matrix entry for corresponding to that row and column is n, if it doesn’t occur at all, the entry is 0.

Creating a DTM is straightforward– one simply uses the built-in DocumentTermMatrix function provided by the tm package like so:

dtm <- DocumentTermMatrix(docs)
#print a summary
dtm
Non-/sparse entries: 13312/110618
Sparsity           : 89%
Maximal term length: 48
Weighting          : term frequency (tf)

This brings us to the end of the preprocessing phase. Next, I’ll briefly explain how distance-based algorithms work before going on to the actual work of clustering.

An intuitive introduction to the algorithms

As mentioned in the introduction, the basic idea behind document or text clustering is to categorise documents into groups based on likeness. Let’s take a brief look at how the algorithms work their magic.

Consider the structure of the DTM. Very briefly, it is a matrix in which the documents are represented as rows and words as columns. In our case, the corpus has 30 documents and 4131 words, so the DTM is a 30 x 4131 matrix.  Mathematically, one can think of this matrix as describing a 4131 dimensional space in which each of the words represents a coordinate axis and each document is represented as a point in this space. This is hard to visualise of course, so it may help to illustrate this via a two-document corpus with only three words in total.

Consider the following corpus:

Document A: “five plus five”

Document B: “five plus six”

These two  documents can be represented as points in a 3 dimensional space that has the words “five” “plus” and “six” as the three coordinate axes (see figure 1).

Figure 1: Documents A and B as points in a 3-word space

Figure 1: Documents A and B as points in a 3-word space

Now, if each of the documents can be thought of as a point in a space, it is easy enough to take the next logical step which is to define the notion of a distance between two points (i.e. two documents). In figure 1 the distance between A and B  (which I denote as D(A,B))is the length of the line connecting the two points, which is simply, the sum of the squares of the differences between the coordinates of the two points representing the documents.

D(A,B) = \sqrt{(2-1)^2 + (1-1)^2+(0-1)^2} = \sqrt 2

Generalising the above to the 4131 dimensional space at hand, the distance between two documents (let’s call them X and Y) have coordinates (word frequencies)  (x_1,x_2,...x_{4131}) and (y_1,y_2,...y_{4131}), then one can define the straight line distance (also called Euclidean distance)  D(X,Y) between them as:

D(X,Y) = \sqrt{(x_1 - y_1)^2+(x_2 - y_2)^2+...+(x_{4131} - y_{4131})^2}

It should be noted that the Euclidean distance that I have described is above is not the only possible way to define distance mathematically. There are many others but it would take me too far afield to discuss them here – see this article for more  (and don’t be put off by the term metric,  a metric  in this context is merely a distance)

What’s important here is the idea that one can define a numerical distance between documents. Once this is grasped, it is easy to understand the basic idea behind how (some) clustering algorithms work – they group documents based on distance-related criteria.  To be sure, this explanation is simplistic and glosses over some of the complicated details in the algorithms. Nevertheless it is a reasonable, approximate explanation for what goes on under the hood. I hope purists reading this will agree!

Finally, for completeness I should mention that there are many clustering algorithms out there, and not all of them are distance-based.

Hierarchical clustering

The first algorithm we’ll look at is hierarchical clustering. As the Wikipedia article on the topic tells us, strategies for hierarchical clustering fall into two types:

Agglomerative: where we start out with each document in its own cluster. The algorithm  iteratively merges documents or clusters that are closest to each other until the entire corpus forms a single cluster. Each merge happens at a different (increasing) distance.

Divisive:  where we start out with the entire set of documents in a single cluster. At each step  the algorithm splits the cluster recursively until each document is in its own cluster. This is basically the inverse of an agglomerative strategy.

The algorithm we’ll use is hclust which does agglomerative hierarchical clustering. Here’s a simplified description of how it works:

  1. Assign each document to its own (single member) cluster
  2. Find the pair of clusters that are closest to each other and merge them. So you now have one cluster less than before.
  3. Compute distances between the new cluster and each of the old clusters.
  4. Repeat steps 2 and 3 until you have a single cluster containing all documents.

We’ll need to do a few things before running the algorithm. Firstly, we need to convert the DTM into a standard matrix which can be used by dist, the distance computation function in R (the DTM is not stored as a standard matrix). We’ll also shorten the document names so that they display nicely in the graph that we will use to display results of hclust (the names I have given the documents are just way too long). Here’s the relevant code:

#convert dtm to matrix
m <- as.matrix(dtm)
#write as csv file (optional)
write.csv(m,file="dtmEight2Late.csv")
#shorten rownames for display purposes
rownames(m) <- paste(substring(rownames(m),1,3),rep("..",nrow(m)),
+                      substring(rownames(m), nchar(rownames(m))-12,nchar(rownames(m))-4))
#compute distance between document vectors
d <- dist(m)

Next we run hclust. The algorithm offers several options check out the documentation for details. I use a popular option called Ward’s method – there are others, and I suggest you experiment with them  as each of them gives slightly different results making interpretation somewhat tricky (did I mention that clustering is as much an art as a science??). Finally, we visualise the results in a dendogram (see Figure 2 below).


#run hierarchical clustering using Ward’s method
groups <- hclust(d,method="ward.D")
#plot dendogram, use hang to ensure that labels fall below tree
plot(groups, hang=-1)
Figure 2: Dendogram from hierarchical clustering of corpus

Figure 2: Dendogram from hierarchical clustering of corpus

A few words on interpreting dendrograms for hierarchical clusters: as you work your way down the tree in figure 2, each branch point you encounter is the distance at which a cluster merge occurred. Clearly, the most well-defined clusters are those that have the largest separation; many closely spaced branch points indicate a lack of dissimilarity (i.e. distance, in this case) between clusters. Based on this, the figure reveals that there are 2 well-defined clusters – the first one consisting of the three documents at the right end of the cluster and the second containing all other documents. We can display the clusters on the graph using the rect.hclust function like so:

#cut into 2 subtrees – try 3 and 5
rect.hclust(groups,2)

The result is shown in the figure below.

Figure 3: 2 cluster solution

Figure 3: 2 cluster grouping

The figures 4 and 5 below show the grouping for 3,  and 5 clusters.

Figure 4: 3 cluster solution

Figure 4: 3 cluster grouping

Figure 5: 5 cluster solution

Figure 5: 5 cluster grouping

I’ll make just one point here: the 2 cluster grouping seems the most robust one as it happens at large distance, and is cleanly separated (distance-wise) from the 3 and 5 cluster grouping. That said, I’ll leave you to explore the ins and outs of hclust on your own and move on to our next algorithm.

K means clustering

In hierarchical clustering we did not specify the number of clusters upfront. These were determined by looking at the dendogram after the algorithm had done its work.  In contrast, our next algorithm – K means –   requires us to define the number of clusters upfront (this number being the “k” in the name). The algorithm then generates k document clusters in a way that ensures the within-cluster distances from each cluster member to the centroid (or geometric mean) of the cluster is minimised.

Here’s a simplified description of the algorithm:

  1. Assign the documents randomly to k bins
  2. Compute the location of the centroid of each bin.
  3. Compute the distance between each document and each centroid
  4. Assign each document to the bin corresponding to the centroid closest to it.
  5. Stop if no document is moved to a new bin, else go to step 2.

An important limitation of the k means method is that the solution found by the algorithm corresponds to a local rather than global minimum (this figure from Wikipedia explains the difference between the two in a nice succinct way). As a consequence it is important to run the algorithm a number of times (each time with a different starting configuration) and then select the result that gives the overall lowest sum of within-cluster distances for all documents.  A simple check that a solution is robust is to run the algorithm for an increasing number of initial configurations until the result does not change significantly. That said, this procedure does not guarantee a globally optimal solution.

I reckon that’s enough said about the algorithm, let’s get on with it using it. The relevant function, as you might well have guessed is kmeans. As always, I urge you to check the documentation to understand the available options. We’ll use the default options for all parameters excepting nstart which we set to 100. We also plot the result using the clusplot function from the cluster library (which you may need to install. Reminder you can install packages via the Tools>Install Packages menu in RStudio)

#k means algorithm, 2 clusters, 100 starting configurations
kfit <- kmeans(d, 2, nstart=100)
#plot – need library cluster
library(cluster)
clusplot(as.matrix(d), kfit$cluster, color=T, shade=T, labels=2, lines=0)

The plot is shown in Figure 6.

Figure 6: principal component plot (k=2)

Figure 6: principal component plot (k=2)

The cluster plot shown in the figure above needs a bit of explanation. As mentioned earlier, the clustering algorithms work in a mathematical space whose dimensionality equals the number of words in the corpus (4131 in our case). Clearly, this is impossible to visualize.  To handle this, mathematicians have invented a dimensionality reduction technique called Principal Component Analysis which reduces the number of dimensions to 2 (in this case) in such a way that the reduced dimensions capture as much of the variability between the clusters as possible (and hence the comment, “these two components explain 69.42% of the point variability” at the bottom of the plot in figure 6)

(Aside  Yes I realize the figures are hard to read because of the overly long names, I leave it to you to fix that. No excuses, you know how…:-))

Running the algorithm and plotting the results for k=3 and 5 yields the figures below.

Figure 7: Principal component plot (k=3)

Figure 7: Principal component plot (k=3)

Figure 8: Principal component plot (k=5)

Figure 8: Principal component plot (k=5)

Choosing k

Recall that the k means algorithm requires us to specify k upfront. A natural question then is: what is the best choice of k? In truth there is no one-size-fits-all answer to this question, but there are some heuristics that might sometimes help guide the choice. For completeness I’ll describe one below even though it is not much help in our clustering problem.

In my simplified description of the k means algorithm I mentioned that the technique attempts to minimise the sum of the distances between the points in a cluster and the cluster’s centroid. Actually, the quantity that is minimised is the total of the within-cluster sum of squares (WSS) between each point and the mean. Intuitively one might expect this quantity to be maximum when k=1 and then decrease as k increases, sharply at first and then less sharply as k reaches its optimal value.

The problem with this reasoning is that it often happens that the within cluster sum of squares never shows a slowing down in decrease of the summed WSS. Unfortunately this is exactly what happens in the case at hand.

I reckon a picture might help make the above clearer. Below is the R code to draw a plot of summed WSS as a function of k for k=2 all the way to 29 (1-total number of documents):

#kmeans – determine the optimum number of clusters (elbow method)
#look for “elbow” in plot of summed intra-cluster distances (withinss) as fn of k
wss <- 2:29
for (i in 2:29) wss[i] <- sum(kmeans(d,centers=i,nstart=25)$withinss)
plot(2:29, wss[2:29], type="b", xlab="Number of Clusters",ylab="Within groups sum of squares")

…and the figure below shows the resulting plot.

Figure 10: WSS as a function of k (“elbow plot”)

The plot clearly shows that there is no k for which the summed WSS flattens out (no distinct “elbow”).  As a result this method does not help. Fortunately, in this case  one can get a sensible answer using common sense rather than computation:  a choice of 2 clusters seems optimal because both algorithms yield exactly the same clusters and show the clearest cluster separation at this point (review the dendogram and cluster plots for k=2).

The meaning of it all

Now I must acknowledge an elephant in the room that I have steadfastly ignored thus far. The odds are good that you’ve seen it already….

It is this: what topics or themes do the (two) clusters correspond to?

Unfortunately this question does not have a straightforward answer. Although the algorithms suggest a 2-cluster grouping, they are silent on the topics or themes related to these.   Moreover,  as you will see if you experiment, the results of clustering depend on:

  • The criteria for the construction of the DTM  (see the documentation for DocumentTermMatrix for options).
  • The clustering algorithm itself.

Indeed, insofar as clustering is concerned, subject matter and corpus knowledge is the best way to figure out cluster themes. This serves to reinforce (yet again!) that clustering is as much an art as it is a science.

In the case at hand, article length seems to be an important differentiator between the 2 clusters found by both algorithms. The three articles in the smaller cluster are in the top 4 longest pieces in the corpus.  Additionally, the three pieces are related to sensemaking and dialogue mapping. There are probably other factors as well, but none that stand out as being significant. I should mention, however, that the fact that article length seems to play a significant role here suggests that it may be worth checking out the effect of scaling distances by word counts or using other measures such a cosine similarity – but that’s a topic for another post! (Note added on Dec 3 2015: check out my article on visualizing relationships between documents using network graphs for a detailed discussion on cosine similarity)

The take home lesson is that  is that the results of clustering are often hard to interpret. This should not be surprising – the algorithms cannot interpret meaning, they simply chug through a mathematical optimisation problem. The onus is on the analyst to figure out what it means…or if it means anything at all.

Conclusion

This brings us to the end of a long ramble through clustering.  We’ve explored the two most common methods:  hierarchical and k means clustering (there are many others available in R, and I urge you to explore them). Apart from providing the detailed steps to do clustering, I have attempted to provide an intuitive explanation of how the algorithms work.  I hope I have succeeded in doing so. As always your feedback would be very welcome.

Finally, I’d like to reiterate an important point:  the results of our clustering exercise do not have a straightforward interpretation, and this is often the case in cluster analysis. Fortunately I can close on an optimistic note. There are other text mining techniques that do a better job in grouping documents based on topics and themes rather than word frequencies alone.   I’ll discuss this in the next article in this series.  Until then, I wish you many enjoyable hours exploring the ins and outs of clustering.

Note added on September 29th 2015:

If you liked this article, you might want to check out its sequel – an introduction to topic modeling.

Written by K

July 22, 2015 at 8:53 pm

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