Archive for the ‘Statistics’ Category
A gentle introduction to support vector machines using R
Introduction
Most machine learning algorithms involve minimising an error measure of some kind (this measure is often called an objective function or loss function). For example, the error measure in linear regression problems is the famous mean squared error – i.e. the averaged sum of the squared differences between the predicted and actual values. Like the mean squared error, most objective functions depend on all points in the training dataset. In this post, I describe the support vector machine (SVM) approach which focuses instead on finding the optimal separation boundary between datapoints that have different classifications. I’ll elaborate on what this means in the next section.
Here’s the plan in brief. I’ll begin with the rationale behind SVMs using a simple case of a binary (two class) dataset with a simple separation boundary (I’ll clarify what “simple” means in a minute). Following that, I’ll describe how this can be generalised to datasets with more complex boundaries. Finally, I’ll work through a couple of examples in R, illustrating the principles behind SVMs. In line with the general philosophy of my “Gentle Introduction to Data Science Using R” series, the focus is on developing an intuitive understanding of the algorithm along with a practical demonstration of its use through a toy example.
The rationale
The basic idea behind SVMs is best illustrated by considering a simple case: a set of data points that belong to one of two classes, red and blue, as illustrated in figure 1 below. To make things simpler still, I have assumed that the boundary separating the two classes is a straight line, represented by the solid green line in the diagram. In the technical literature, such datasets are called linearly separable.
In the linearly separable case, there is usually a fair amount of freedom in the way a separating line can be drawn. Figure 2 illustrates this point: the two broken green lines are also valid separation boundaries. Indeed, because there is a non-zero distance between the two closest points between categories, there are an infinite number of possible separation lines. This, quite naturally, raises the question as to whether it is possible to choose a separation boundary that is optimal.
The short answer is, yes there is. One way to do this is to select a boundary line that maximises the margin, i.e. the distance between the separation boundary and the points that are closest to it. Such an optimal boundary is illustrated by the black brace in Figure 3. The really cool thing about this criterion is that the location of the separation boundary depends only on the points that are closest to it. This means, unlike other classification methods, the classifier does not depend on any other points in dataset. The directed lines between the boundary and the closest points on either side are called support vectors (these are the solid black lines in figure 3). A direct implication of this is that the fewer the support vectors, the better the generalizability of the boundary.
Although the above sounds great, it is of limited practical value because real data sets are seldom (if ever) linearly separable.
So, what can we do when dealing with real (i.e. non linearly separable) data sets?
A simple approach to tackle small deviations from linear separability is to allow a small number of points (those that are close to the boundary) to be misclassified. The number of possible misclassifications is governed by a free parameter C, which is called the cost. The cost is essentially the penalty associated with making an error: the higher the value of C, the less likely it is that the algorithm will misclassify a point.
This approach – which is called soft margin classification – is illustrated in Figure 4. Note the points on the wrong side of the separation boundary. We will demonstrate soft margin SVMs in the next section. (Note: At the risk of belabouring the obvious, the purely linearly separable case discussed in the previous para is simply is a special case of the soft margin classifier.)
Real life situations are much more complex and cannot be dealt with using soft margin classifiers. For example, as shown in Figure 5, one could have widely separated clusters of points that belong to the same classes. Such situations, which require the use of multiple (and nonlinear) boundaries, can sometimes be dealt with using a clever approach called the kernel trick.
The kernel trick
Recall that in the linearly separable (or soft margin) case, the SVM algorithm works by finding a separation boundary that maximises the margin, which is the distance between the boundary and the points closest to it. The distance here is the usual straight line distance between the boundary and the closest point(s). This is called the Euclidean distance in honour of the great geometer of antiquity. The point to note is that this process results in a separation boundary that is a straight line, which as Figure 5 illustrates, does not always work. In fact in most cases it won’t.
So what can we do? To answer this question, we have to take a bit of a detour…
What if we were able to generalize the notion of distance in a way that generates nonlinear separation boundaries? It turns out that this is possible. To see how, one has to first understand how the notion of distance can be generalized.
The key properties that any measure of distance must satisfy are:
- Non-negativity – a distance cannot be negative, a point that needs no further explanation I reckon 🙂
- Symmetry – that is, the distance between point A and point B is the same as the distance between point B and point A.
- Identity– the distance between a point and itself is zero.
- Triangle inequality – that is the sum of distances between point A and B and points B and C must be less than or equal to the distance between A and C (equality holds only if all three points lie along the same line).
Any mathematical object that displays the above properties is akin to a distance. Such generalized distances are called metrics and the mathematical space in which they live is called a metric space. Metrics are defined using special mathematical functions designed to satisfy the above conditions. These functions are known as kernels.
The essence of the kernel trick lies in mapping the classification problem to a metric space in which the problem is rendered separable via a separation boundary that is simple in the new space, but complex – as it has to be – in the original one. Generally, the transformed space has a higher dimensionality, with each of the dimensions being (possibly complex) combinations of the original problem variables. However, this is not necessarily a problem because in practice one doesn’t actually mess around with transformations, one just tries different kernels (the transformation being implicit in the kernel) and sees which one does the job. The check is simple: we simply test the predictions resulting from using different kernels against a held out subset of the data (as one would for any machine learning algorithm).
It turns out that a particular function – called the radial basis function kernel (RBF kernel) – is very effective in many cases. The RBF kernel is essentially a Gaussian (or Normal) function with the Euclidean distance between pairs of points as the variable (see equation 1 below). The basic rationale behind the RBF kernel is that it creates separation boundaries that it tends to classify points close together (in the Euclidean sense) in the original space in the same way. This is reflected in the fact that the kernel decays (i.e. drops off to zero) as the Euclidean distance between points increases.
The rate at which a kernel decays is governed by the parameter – the higher the value of , the more rapid the decay. This serves to illustrate that the RBF kernel is extremely flexible….but the flexibility comes at a price – the danger of overfitting for large values of . One should choose appropriate values of C and so as to ensure that the resulting kernel represents the best possible balance between flexibility and accuracy. We’ll discuss how this is done in practice later in this article.
Finally, though it is probably obvious, it is worth mentioning that the separation boundaries for arbitrary kernels are also defined through support vectors as in Figure 3. To reiterate a point made earlier, this means that a solution that has fewer support vectors is likely to be more robust than one with many. Why? Because the data points defining support vectors are ones that are most sensitive to noise- therefore the fewer, the better.
There are many other types of kernels, each with their own pros and cons. However, I’ll leave these for adventurous readers to explore by themselves. Finally, for a much more detailed….and dare I say, better… explanation of the kernel trick, I highly recommend this article by Eric Kim.
Support vector machines in R
In this demo we’ll use the svm interface that is implemented in the e1071 R package. This interface provides R programmers access to the comprehensive libsvm library written by Chang and Lin. I’ll use two toy datasets: the famous iris dataset available with the base R package and the sonar dataset from the mlbench package. I won’t describe details of the datasets as they are discussed at length in the documentation that I have linked to. However, it is worth mentioning the reasons why I chose these datasets:
- As mentioned earlier, no real life dataset is linearly separable, but the iris dataset is almost so. Consequently, it is a good illustration of using linear SVMs. Although one almost never uses these in practice, I have illustrated their use primarily for pedagogical reasons.
- The sonar dataset is a good illustration of the benefits of using RBF kernels in cases where the dataset is hard to visualise (60 variables in this case!). In general, one would almost always use RBF (or other nonlinear) kernels in practice.
With that said, let’s get right to it. I assume you have R and RStudio installed. For instructions on how to do this, have a look at the first article in this series. The processing preliminaries – loading libraries, data and creating training and test datasets are much the same as in my previous articles so I won’t dwell on these here. For completeness, however, I’ll list all the code so you can run it directly in R or R studio (a complete listing of the code can be found here):
The output from the SVM model show that there are 24 support vectors. If desired, these can be examined using the SV variable in the model – i.e via svm_model$SV.
The test prediction accuracy indicates that the linear performs quite well on this dataset, confirming that it is indeed near linearly separable. To check performance by class, one can create a confusion matrix as described in my post on random forests. I’ll leave this as an exercise for you. Another point is that we have used a soft-margin classification scheme with a cost C=1. You can experiment with this by explicitly changing the value of C. Again, I’ll leave this for you an exercise.
Before proceeding to the RBF kernel, I should mention a point that an alert reader may have noticed. The predicted variable, Species, can take on 3 values (setosa, versicolor and virginica). However, our discussion above dealt with a binary (2 valued) classification problem. This brings up the question as to how the algorithm deals multiclass classification problems – i.e those involving datasets with more than two classes. The libsvm algorithm (which svm uses) does this using a one-against-one classification strategy. Here’s how it works:
- Divide the dataset (assumed to have N classes) into N(N-1)/2 datasets that have two classes each.
- Solve the binary classification problem for each of these subsets
- Use a simple voting mechanism to assign a class to each data point.
Basically, each data point is assigned the most frequent classification it receives from all the binary classification problems it figures in.
With that said for the unrealistic linear classifier, let’s move to the real world. In the code below, I build SVM models using three different kernels
- Linear kernel (this is for comparison with the following 2 kernels).
- RBF kernel with default values for the parameters and .
- RBF kernel with optimal values for and . The optimal values are obtained using the tune.svm function (also available in e1071), which essentially builds models for multiple combinations of parameter values and selects the best.
OK, lets go:
I’ll leave you to examine the contents of the model. The important point to note here is that the performance of the model with the test set is quite dismal compared to the previous case. This simply indicates that the linear kernel is not appropriate here. Let’s take a look at what happens if we use the RBF kernel with default values for the parameters:
That’s a pretty decent improvement from the linear kernel. Let’s see if we can do better by doing some parameter tuning. To do this we first invoke tune.svm and use the parameters it gives us in the call to svm:
Which is fairly decent improvement on the un-optimised case.
Wrapping up
This bring us to the end of this introductory exploration of SVMs in R. To recap, the distinguishing feature of SVMs in contrast to most other techniques is that they attempt to construct optimal separation boundaries between different categories.
SVMs are quite versatile and have been applied to a wide variety of domains ranging from chemistry to pattern recognition. They are best used in binary classification scenarios. This brings up a question as to where SVMs are to be preferred to other binary classification techniques such as logistic regression. The honest response is, “it depends” – but here are some points to keep in mind when choosing between the two. A general point to keep in mind is that SVM algorithms tend to be expensive both in terms of memory and computation, issues that can start to hurt as the size of the dataset increases.
Given all the above caveats and considerations, the best way to figure out whether an SVM approach will work for your problem may be to do what most machine learning practitioners do: try it out!
A gentle introduction to random forests using R
Introduction
In a previous post, I described how decision tree algorithms work and demonstrated their use via the rpart library in R. Decision trees work by splitting a dataset recursively. That is, subsets arising from a split are further split until a predetermined termination criterion is reached. At each step, a split is made based on the independent variable that results in the largest possible reduction in heterogeneity of the dependent variable.
(Note: readers unfamiliar with decision trees may want to read that post before proceeding)
The main drawback of decision trees is that they are prone to overfitting. The reason for this is that trees, if grown deep, are able to fit all kinds of variations in the data, including noise. Although it is possible to address this partially by pruning, the result often remains less than satisfactory. This is because makes a locally optimal choice at each split without any regard to whether the choice made is the best one overall. A poor split made in the initial stages can thus doom the model, a problem that cannot be fixed by post-hoc pruning.
In this post I describe random forests, a tree-based algorithm that addresses the above shortcoming of decision trees. I’ll first describe the intuition behind the algorithm via an analogy and then do a demo using the R randomForest library.
Motivating random forests
One of the reasons for the popularity of decision trees is that they reflect the way humans make decisions: by weighing up options at each stage and choosing the best one available. The analogy is particularly useful because it also suggests how decision trees can be improved.
One of the lifelines in the game show, Who Wants to be A Millionaire, is “Ask The Audience” wherein a contestant can ask the audience to vote on the answer to a question. The rationale here is that the majority response from a large number of independent decision makers is more likely to yield a correct answer than one from a randomly chosen person. There are two factors at play here:
- People have different experiences and will therefore draw upon different “data” to answer the question.
- People have different knowledge bases and preferences and will therefore draw upon different “variables” to make their choices at each stage in their decision process.
Taking a cue from the above, it seems reasonable to build many decision trees using:
- Different sets of training data.
- Randomly selected subsets of variables at each split of every decision tree.
Predictions can then made by taking the majority vote over all trees (for classification problems) or averaging results over all trees (for regression problems). This is essentially how the random forest algorithm works.
The net effect of the two strategies is to reduce overfitting by a) averaging over trees created from different samples of the dataset and b) decreasing the likelihood of a small set of strong predictors dominating the splits. The price paid is reduced interpretability as well as increased computational complexity. But then, there is no such thing as a free lunch.
The mechanics of the algorithm
Although we will not delve into the mathematical details of the algorithm, it is important to understand how two points made above are implemented in the algorithm.
Bootstrap aggregating… and a (rather cool) error estimate
A key feature of the algorithm is the use of multiple datasets for training individual decision trees. This is done via a neat statistical trick called bootstrap aggregating (also called bagging).
Here’s how bagging works:
Assume you have a dataset of size N. From this you create a sample (i.e. a subset) of size n (n less than or equal to N) by choosing n data points randomly with replacement. “Randomly” means every point in the dataset is equally likely to be chosen and “with replacement” means that a specific data point can appear more than once in the subset. Do this M times to create M equally-sized samples of size n each. It can be shown that this procedure, which statisticians call bootstrapping, is legit when samples are created from large datasets – that is, when N is large.
Because a bagged sample is created by selection with replacement, there will generally be some points that are not selected. In fact, it can be shown that, on the average, each sample will use about two-thirds of the available data points. This gives us a clever way to estimate the error as part of the process of model building.
Here’s how:
For every data point, obtain predictions for trees in which the point was out of bag. From the result mentioned above, this will yield approximately M/3 predictions per data point (because a third of the data points are out of bag). Take the majority vote of these M/3 predictions as the predicted value for the data point. One can do this for the entire dataset. From these out of bag predictions for the whole dataset, we can estimate the overall error by computing a classification error (Count of correct predictions divided by N) for classification problems or the root mean squared error for regression problems. This means there is no need to have a separate test data set, which is kind of cool. However, if you have enough data, it is worth holding out some data for use as an independent test set. This is what we’ll do in the demo later.
Using subsets of predictor variables
Although bagging reduces overfitting somewhat, it does not address the issue completely. The reason is that in most datasets a small number of predictors tend to dominate the others. These predictors tend to be selected in early splits and thus influence the shapes and sizes of a significant fraction of trees in the forest. That is, strong predictors enhance correlations between trees which tends to come in the way of variance reduction.
A simple way to get around this problem is to use a random subset of variables at each split. This avoids over-representation of dominant variables and thus creates a more diverse forest. This is precisely what the random forest algorithm does.
Random forests in R
In what follows, I use the famous Glass dataset from the mlbench library. The dataset has 214 data points of six types of glass with varying metal oxide content and refractive indexes. I’ll first build a decision tree model based on the data using the rpart library (recursive partitioning) that I covered in an earlier article and then use then show how one can build a random forest model using the randomForest library. The rationale behind this is to compare the two models – single decision tree vs random forest. In the interests of space, I won’t explain details of the rpart here as I’ve covered it at length in the previous article. However, for completeness, I’ll list the demo code for it before getting into random forests.
Decision trees using rpart
Here’s the code listing for building a decision tree using rpart on the Glass dataset (please see my previous article for a full explanation of each step). Note that I have not used pruning as there is little benefit to be gained from it (Exercise for the reader: try this for yourself!).
Now, we know that decision tree algorithms tend to display high variance so the hit rate from any one tree is likely to be misleading. To address this we’ll generate a bunch of trees using different training sets (via random sampling) and calculate an average hit rate and spread (or standard deviation).
The decision tree algorithm gets it right about 69% of the time with a variation of about 5%. The variation isn’t too bad here, but the accuracy has hardly improved at all (Exercise for the reader: why?). Let’s see if we can do better using random forests.
Random forests
As discussed earlier, a random forest algorithm works by averaging over multiple trees using bootstrapped samples. Also, it reduces the correlation between trees by splitting on a random subset of predictors at each node in tree construction. The key parameters for randomForest algorithm are the number of trees (ntree) and the number of variables to be considered for splitting (mtry). The algorithm sets a default of 500 for ntree and sets mtry to one-third the total number of predictors for classification problems and square root of the the number of predictors for regression. These defaults can be overridden by explicitly providing values for these variables.
The preliminary stuff – the creation of training and test datasets etc. – is much the same as for decision trees but I’ll list the code for completeness.
randomForest(formula = Type ~ ., data = trainGlass, importance = TRUE, xtest = testGlass[, -typeColNum], ntree = 1001)
1 | 2 | 3 | 5 | 6 | 7 | class.error | |
1 | 40 | 7 | 2 | 0 | 0 | 0 | 0.1836735 |
2 | 8 | 49 | 1 | 2 | 2 | 1 | 0.2222222 |
3 | 6 | 3 | 6 | 0 | 0 | 0 | 0.6000000 |
5 | 0 | 1 | 0 | 11 | 0 | 1 | 0.1538462 |
6 | 1 | 2 | 0 | 1 | 6 | 0 | 0.5000000 |
7 | 1 | 2 | 0 | 1 | 0 | 21 | 0.1600000 |
The first thing to note is the out of bag error estimate is ~ 24%. Equivalently the hit rate is 76%, which is better than the 69% for decision trees. Secondly, you’ll note that the algorithm does a terrible job identifying type 3 and 6 glasses correctly. This could possibly be improved by a technique called boosting, which works by iteratively improving poor predictions made in earlier stages. I plan to look at boosting in a future post, but if you’re curious, check out the gbm package in R.
Finally, for completeness, let’s see how the test set does:
1 | 2 | 3 | 5 | 6 | 7 | |
1 | 19 | 2 | 0 | 0 | 0 | 0 |
2 | 1 | 9 | 1 | 0 | 0 | 0 |
3 | 1 | 1 | 1 | 0 | 0 | 0 |
5 | 0 | 1 | 0 | 0 | 0 | 0 |
6 | 0 | 0 | 0 | 0 | 3 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 4 |
The test accuracy is better than the out of bag accuracy and there are some differences in the class errors as well. However, overall the two compare quite well and are significantly better than the results of the decision tree algorithm.
Variable importance
Random forest algorithms also give measures of variable importance. Computation of these is enabled by setting importance, a boolean parameter, to TRUE. The algorithm computes two measures of variable importance: mean decrease in Gini and mean decrease in accuracy. Brief explanations of these follow.
Mean decrease in Gini
When determining splits in individual trees, the algorithm looks for the largest class (in terms of population) and attempts to isolate it first. If this is not possible, it tries to do the best it can, always focusing on isolating the largest remaining class in every split.This is called the Gini splitting rule (see this article for a good explanation of the rule).
The “goodness of split” is measured by the Gini Impurity, . For a set containing K categories this is given by:
where is the fraction of the set that belongs to the ith category. Clearly, is 0 when the set is homogeneous or pure (1 class only) and is maximum when classes are equiprobable (for example, in a two class set the maximum occurs when and are 0.5). At each stage the algorithm chooses to split on the predictor that leads to the largest decrease in . The algorithm tracks this decrease for each predictor for all splits and all trees in the forest. The average is reported as the mean decrease in Gini.
Mean decrease in accuracy
The mean decrease in accuracy is calculated using the out of bag data points for each tree. The procedure goes as follows: when a particular tree is grown, the out of bag points are passed down the tree and the prediction accuracy (based on all out of bag points) recorded . The predictors are then randomly permuted and the out of bag prediction accuracy recalculated. The decrease in accuracy for a given predictor is the difference between the accuracy of the original (unpermuted) tree and the those obtained from the permuted trees in which the predictor was excluded. As in the previous case, the decrease in accuracy for each predictor can be computed and tracked as the algorithm progresses. These can then be averaged by predictor to yield a mean decrease in accuracy.
Variable importance plot
From the above, it would seem that the mean decrease in accuracy is a more global measure as it uses fully constructed trees in contrast to the Gini measure which is based on individual splits. In practice, however, there could be other reasons for choosing one over the other…but that is neither here nor there, if you set importance to TRUE, you’ll get both. The numerical measures of importance are returned in the randomForest object (Glass.rf in our case), but I won’t list them here. Instead, I’ll just print out the variable importance plots for the two measures as these give a good visual overview of the relative importance of variables. The code is a simple one-liner:
The plot is shown in Figure 1 below.
In this case the two measures are pretty consistent so it doesn’t really matter which one you choose.
Wrapping up
Random forests are an example of a general class of techniques called ensemble methods. These techniques are based on the principle that averaging over a large number of not-so-good models yields a more reliable prediction than a single model. This is true only if models in the group are independent of each other, which is precisely what bootstrap aggregation and predictor subsetting are intended to achieve.
Although considerably more complex than decision trees, the logic behind random forests is not hard to understand. Indeed, the intuitiveness of the algorithm together with its ease of use and accuracy have made it very popular in the machine learning community.
A gentle introduction to decision trees using R
Introduction
Most techniques of predictive analytics have their origins in probability or statistical theory (see my post on Naïve Bayes, for example). In this post I’ll look at one that has more a commonplace origin: the way in which humans make decisions. When making decisions, we typically identify the options available and then evaluate them based on criteria that are important to us. The intuitive appeal of such a procedure is in no small measure due to the fact that it can be easily explained through a visual. Consider the following graphic, for example:
(Original image: https://www.flickr.com/photos/dullhunk/7214525854, Credit: Duncan Hull)
The tree structure depicted here provides a neat, easy-to-follow description of the issue under consideration and its resolution. The decision procedure is based on asking a series of questions, each of which serve to further reduce the domain of possibilities. The predictive technique I discuss in this post,classification and regression trees (CART), works in much the same fashion. It was invented by Leo Breiman and his colleagues in the 1970s.
In what follows, I will use the open source software, R. If you are new to R, you may want to follow this link for more on the basics of setting up and installing it. Note that the R implementation of the CART algorithm is called RPART (Recursive Partitioning And Regression Trees). This is essentially because Breiman and Co. trademarked the term CART. As some others have pointed out, it is somewhat ironical that the algorithm is now commonly referred to as RPART rather than by the term coined by its inventors.
A bit about the algorithm
The rpart algorithm works by splitting the dataset recursively, which means that the subsets that arise from a split are further split until a predetermined termination criterion is reached. At each step, the split is made based on the independent variable that results in the largest possible reduction in heterogeneity of the dependent (predicted) variable.
Splitting rules can be constructed in many different ways, all of which are based on the notion of impurity- a measure of the degree of heterogeneity of the leaf nodes. Put another way, a leaf node that contains a single class is homogeneous and has impurity=0. There are three popular impurity quantification methods: Entropy (aka information gain), Gini Index and Classification Error. Check out this article for a simple explanation of the three methods.
The rpart algorithm offers the entropy and Gini index methods as choices. There is a fair amount of fact and opinion on the Web about which method is better. Here are some of the better articles I’ve come across:
https://www.garysieling.com/blog/sklearn-gini-vs-entropy-criteria
http://www.salford-systems.com/resources/whitepapers/114-do-splitting-rules-really-matter
The answer as to which method is the best is: it depends. Given this, it may be prudent to try out a couple of methods and pick the one that works best for your problem.
Regardless of the method chosen, the splitting rules partition the decision space (a fancy word for the entire dataset) into rectangular regions each of which correspond to a split. Consider the following simple example with two predictors x1 and x2. The first split is at x1=1 (which splits the decision space into two regions x1<1 and x1>1), the second at x2=2, which splits the (x1>1) region into 2 sub-regions, and finally x1=1.5 which splits the (x1>1,x2>2) sub-region further.
It is important to note that the algorithm works by making the best possible choice at each particular stage, without any consideration of whether those choices remain optimal in future stages. That is, the algorithm makes a locally optimal decision at each stage. It is thus quite possible that such a choice at one stage turns out to be sub-optimal in the overall scheme of things. In other words, the algorithm does not find a globally optimal tree.
Another important point relates to well-known bias-variance tradeoff in machine learning, which in simple terms is a tradeoff between the degree to which a model fits the training data and its predictive accuracy. This refers to the general rule that beyond a point, it is counterproductive to improve the fit of a model to the training data as this increases the likelihood of overfitting. It is easy to see that deep trees are more likely to overfit the data than shallow ones. One obvious way to control such overfitting is to construct shallower trees by stopping the algorithm at an appropriate point based on whether a split significantly improves the fit. Another is to grow a tree unrestricted and then prune it back using an appropriate criterion. The rpart algorithm takes the latter approach.
Here is how it works in brief:
Essentially one minimises the cost, , a quantity that is a linear combination of the error (essentially, the fraction of misclassified instances, or variance in the case of a continuous variable), and the number of leaf nodes in the tree, :
First, we note that when , this simply returns the original fully grown tree. As increases, we incur a penalty that is proportional to the number of leaf nodes. This tends to cause the minimum cost to occur for a tree that is a subtree of the original one (since a subtree will have a smaller number of leaf nodes). In practice we vary and pick the value that gives the subtree that results in the smallest cross-validated prediction error. One does not have to worry about programming this because the rpart algorithm actually computes the errors for different values of for us. All we need to do is pick the value of the coefficient that gives the lowest cross-validated error. I will illustrate this in detail in the next section.
An implication of their tendency to overfit data is that decision trees tend to be sensitive to relatively minor changes in the training datasets. Indeed, small differences can lead to radically different looking trees. Pruning addresses this to an extent, but does not resolve it completely. A better resolution is offered by the so-called ensemble methods that average over many differently constructed trees. I’ll discuss one such method at length in a future post.
Finally, I should also mention that decision trees can be used for both classification and regression problems (i.e. those in which the predicted variable is discrete and continuous respectively). I’ll demonstrate both types of problems in the next two sections.
Classification trees using rpart
To demonstrate classification trees, we’ll use the Ionosphere dataset available in the mlbench package in R. I have chosen this dataset because it nicely illustrates the points I wish to make in this post. In general, you will almost always find that algorithms that work fine on classroom datasets do not work so well in the real world…but of course, you know that already!
We begin by setting the working directory, loading the required packages (rpart and mlbench) and then loading the Ionosphere dataset.
Next we separate the data into training and test sets. We’ll use the former to build the model and the latter to test it. To do this, I use a simple scheme wherein I randomly select 80% of the data for the training set and assign the remainder to the test data set. This is easily done in a single R statement that invokes the uniform distribution (runif) and the vectorised function, ifelse. Before invoking runif, I set a seed integer to my favourite integer in order to ensure reproducibility of results.
In the above, I have also removed the training flag from the training and test datasets.
Next we invoke rpart. I strongly recommend you take some time to go through the documentation and understand the parameters and their defaults values. Note that we need to remove the predicted variable from the dataset before passing the latter on to the algorithm, which is why we need to find the column index of the predicted variable (first line below). Also note that we set the method parameter to “class“, which simply tells the algorithm that the predicted variable is discrete. Finally, rpart uses Gini rule for splitting by default, and we’ll stick with this option.
The resulting plot is shown in Figure 3 below. It is quite self-explanatory so I won’t dwell on it here.
Next we see how good the model is by seeing how it fares against the test data.
pred true | bad | good |
bad | 17 | 2 |
good | 9 | 43 |
Note that we need to verify the above results by doing multiple runs, each using different training and test sets. I will do this later, after discussing pruning.
Next, we prune the tree using the cost complexity criterion. Basically, the intent is to see if a shallower subtree can give us comparable results. If so, we’d be better of choosing the shallower tree because it reduces the likelihood of overfitting.
As described earlier, we choose the appropriate pruning parameter (aka cost-complexity parameter) by picking the value that results in the lowest prediction error. Note that all relevant computations have already been carried out by R when we built the original tree (the call to rpart in the code above). All that remains now is to pick the value of :
CP | nsplit | rel error | xerror | xstd | |
1 | 0.57 | 0 | 1.00 | 1.00 | 0.080178 |
2 | 0.20 | 1 | 0.43 | 0.46 | 0.062002 |
3 | 0.02 | 2 | 0.23 | 0.26 | 0.048565 |
4 | 0.01 | 4 | 0.19 | 0.35 |
It is clear from the above, that the lowest cross-validation error (xerror in the table) occurs for (this is CP in the table above). One can find CP programatically like so:
Next, we prune the tree based on this value of CP:
Note that rpart will use a default CP value of 0.01 if you don’t specify one in prune.
The pruned tree is shown in Figure 4 below.
Let’s see how this tree stacks up against the fully grown one shown in Fig 3.
This seems like an improvement over the unpruned tree, but one swallow does not a summer make. We need to check that this holds up for different training and test sets. This is easily done by creating multiple random partitions of the dataset and checking the efficacy of pruning for each. To do this efficiently, I’ll create a function that takes the training fraction, number of runs (partitions) and the name of the dataset as inputs and outputs the proportion of correct predictions for each run. It also optionally prunes the tree. Here’s the code:
Note that in the above, I have set the default value of the prune_tree to FALSE, so the function will execute the first branch of the if statement unless the default is overridden.
OK, so let’s do 50 runs with and without pruning, and check the mean and variance of the results for both sets of runs.
So we see that there is an improvement of about 3% with pruning. Also, if you were to plot the trees as we did earlier, you would see that this improvement is achieved with shallower trees. Again, I point out that this is not always the case. In fact, it often happens that pruning results in worse predictions, albeit with better reliability – a classic illustration of the bias-variance tradeoff.
Regression trees using rpart
In the previous section we saw how one can build decision trees for situations in which the predicted variable is discrete. Let’s now look at the case in which the predicted variable is continuous. We’ll use the Boston Housing dataset from the mlbench package. Much of the discussion of the earlier section applies here, so I’ll just display the code, explaining only the differences.
Next we invoke rpart, noting that the predicted variable is medv (median value of owner-occupied homes in $1000 units) and that we need to set the method parameter to “anova“. The latter tells rpart that the predicted variable is continuous (i.e that this is a regression problem).
The plot of the tree is shown in Figure 5 below.
Next, we need to see how good the predictions are. Since the dependent variable is continuous, we cannot compare the predictions directly against the test set. Instead, we calculate the root mean square (RMS) error. To do this, we request rpart to output the predictions as a vector – one prediction per record in the test dataset. The RMS error can then easily be calculated by comparing this vector with the medv column in the test dataset.
Here is the relevant code:
Again, we need to do multiple runs to check on the reliability of the predictions. However, you already know how to do that so I will leave it to you.
Moving on, we prune the tree using the cost complexity criterion as before. The code is exactly the same as in the classification problem.
The tree is unchanged so I won’t show it here. This means, as far as the cost complexity pruning is concerned, the optimal subtree is the same as the original tree. To confirm this, we’d need to do multiple runs as before – something that I’ve already left as as an exercise for you :). Basically, you’ll need to write a function analogous to the one above, that computes the root mean square error instead of the proportion of correct predictions.
Wrapping up
This brings us to the end of my introduction to classification and regression trees using R. Unlike some articles on the topic I have attempted to describe each of the steps in detail and provide at least some kind of a rationale for them. I hope you’ve found the description and code snippets useful.
I’ll end by reiterating a couple points I made early in this piece. The nice thing about decision trees is that they are easy to explain to the users of our predictions. This is primarily because they reflect the way we think about how decisions are made in real life – via a set of binary choices based on appropriate criteria. That said, in many practical situations decision trees turn out to be unstable: small changes in the dataset can lead to wildly different trees. It turns out that this limitation can be addressed by building a variety of trees using different starting points and then averaging over them. This is the domain of the so-called random forest algorithm.We’ll make the journey from decision trees to random forests in a future post.
Postscript, 20th September 2016: I finally got around to finishing my article on random forests.
A gentle introduction to network graphs using R and Gephi
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 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.
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.
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:
where the two vectors are and , and 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 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 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 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:
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, ““”)
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”)
“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”)
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.
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.
Once that is done, click on overview (top left of the screen). You should end up with something like Figure 4.
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).
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.
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).
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.
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.
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.
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.
…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.
A gentle introduction to Naïve Bayes classification using R
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)
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:
The plots are shown in Figures 1 through 3.
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, 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,
(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,
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:
Basically, the second term on the left hand side, , 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:
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,
and
The quantity of interest, the numerator of equation (1) can then be written as:
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 and 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).
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))}
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.
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.
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:
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:
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:
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:
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
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:
[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
0.8519 0.9074 0.9170 0.9177 0.9310 0.9643
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 topic modeling using R
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:
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, ““”)
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”)
“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”)
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 500^{th} 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).
iter <- 2000
thin <- 500
seed <-list(2003,5,63,100001,765)
nstart <- 5
best <- TRUE
That done, we can now do the actual work – run the topic modelling algorithm on our corpus. Here is the code:
write.csv(ldaOut.topics,file=paste(“LDAGibbs”,k,”DocsToTopics.csv”))
write.csv(ldaOut.terms,file=paste(“LDAGibbs”,k,”TopicsToTerms.csv”))
write.csv(topicProbabilities,file=paste(“LDAGibbs”,k,”TopicProbabilities.csv”))
sort(topicProbabilities[x,])[k]/sort(topicProbabilities[x,])[k-1])
sort(topicProbabilities[x,])[k-1]/sort(topicProbabilities[x,])[k-2])
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:
- 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).
- Each topic contains all terms (words) in the corpus, albeit with different probabilities. We list only the top 6 terms in the second file.
- 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.
A gentle introduction to cluster analysis using R
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):
[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.
docs <- Corpus(DirSource(“C:/Users/Kailash/Documents/TextMining”))
#inspect a particular document
writeLines(as.character(docs[[30]]))
…<output not shown>
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):
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.
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.”:
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 = “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):
+ “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:
#print a summary
dtm
<<DocumentTermMatrix (documents: 30, terms: 4131)>>
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).
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 )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.
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) and , then one can define the straight line distance (also called Euclidean distance) between them as:
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:
- Assign each document to its own (single member) cluster
- Find the pair of clusters that are closest to each other and merge them. So you now have one cluster less than before.
- Compute distances between the new cluster and each of the old clusters.
- 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:
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)
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:
rect.hclust(groups,2)
The result is shown in the figure below.
The figures 4 and 5 below show the grouping for 3, and 5 clusters.
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:
- Assign the documents randomly to k bins
- Compute the location of the centroid of each bin.
- Compute the distance between each document and each centroid
- Assign each document to the bin corresponding to the centroid closest to it.
- 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)
kfit <- kmeans(d, 2, nstart=100)
#plot – need library cluster
library(cluster)
clusplot(m, kfit$cluster, color=T, shade=T, labels=2, lines=0)
The plot is shown in Figure 6.
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.
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):
#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.
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.