Mastering Text Mining with R

Probability in mathematics is built around sets. Set theory is very useful in probability; it provides a language for expressing and working with events.

The sample space of an experiment is the set of all possible outcomes of the experiment; let's call it S. An event, let's call it A, is a subset of the sample space S, and we say that A occurred if the actual outcome is in A.

Let's take an example of picking a card from a standard deck of 52 cards. The sample space S is the set of all the cards. Let's us consider an event A where the card we pick is an ace. This is a subset of the sample space. So the probability P of picking an ace is:

Probability = (number of elements in the event) / (number of elements in the sample space).

This says that the probability of the intersection of two events A and B can be computed as the product of probability of A given that B has happened times the probability of B:

The law of total probability or law of alternatives can be formulated as follows:

Probability is a way of expressing uncertainties about events. Whenever we observe new evidence or obtain data, we acquire information that may affect our uncertainties. Conditional probability is the concept that tells us how to express the probability which is affected by the newly acquired information. Conditional probability handles situations where you have some additional knowledge about the outcome of a trial or experiment.

Let's consider an event R, It will rain today, before looking at the sky. The probability P(R) will increase when we look at the sky and see dark clouds. So the new probability is P (R|C) where C is the event of dark clouds.

If A and B are events with P (B) > 0,then the conditional probability of A given B, denoted by P(A|B), is defined as:

Let us consider an example and try to perform the same using R. We are rolling two dice and the objective is to find the probability of the sum of the outcomes being greater than or equal to 8, given that the first dice has resulted in 3:

library(prob)
S <- rolldie(2, makespace = TRUE)
A <- subset(S, X1 + X2 >= 8)
B <- subset(S, X1 == 3) #Given
Prob(A, given = B) 

Bayes' formula gives us a way to test a hypothesis using conditional probabilities. A hypothesis is a suggested explanation for a specific outcome. If we see that a probability P (A | B) is high, we might hypothesize that event B is a cause of the event A. We use Bayes' formula when we know conditional probabilities of the form P (B | A) and want a conditional probability of the form P (A | B):

Two events, A and B, in the same sample space are independent if P (AB) = P (A) P (B).This formula gives us a new and simpler way to characterize independent events. Two events, A and B, are independent if the probability of both events happening together is equal to the product of the probabilities of the two events.

In probability, a random variable is a rule or function that assigns a number to each element of a sample space. In other words, a random variable gives a number for each outcome of a random experiment. In statistics, we define random variables using the letter X. There are different types of random variable.

When we toss two coins, the number of heads we can get is 0, 1, or 2 .We can define X as the number of heads that I get during this experiment. These random variable values have a probability associated with them; these variables can be represented as discrete points on a number line so they are called discrete random variables.

Once we have a random variable, we can determine the probability that the random variable will have a certain value; for example, for rolling two dice to get a sum of five outcomes, it can be (1,4) , (4,1) , (3,2) , or (2,3) so there are 4 out of 36 possible outcomes, so:

R provides a wide range of probability functions. The generic prefixes for probability functions in R are r, d, p, q, for random number generators, probability density function, cumulative density function, and quantile function, respectively.

A comprehensive list of functions available is as follows:

This frequency function gives the probabilities for each value in the range of a random variable. For a given value R of the random variable, the cumulative distribution function gives the probability of the random variable taking on a value up to and including the given value R. When R is 3, there are three outcomes, (1, 1), (1, 2), and (2, 1), so:

The cumulative distribution function is also called the CDF, or probability distribution or distribution function. The stats package in R provides the function ecdf to compute the empirical cumulative distribution function and plot it using the object created. You can also plot the ecdf object using the ggplot2 package. Let's look at an example for the same:

x <- rnorm(1000, 99.2, 1.2)
y <- rnorm(10000, 97.3, 0.85)
z <- rnorm(10000, 98.1, 0.4)

# Create a chart with all 3 Conditional distribution plots
plot(ecdf(x), col=rgb(1,0,0), main=NA)
plot(ecdf(y), col=rgb(0,1,0), add=T)
plot(ecdf(z), col=rgb(0,0,1), add=T)

# Adding legend to the chart.
legend('right', c('x', 'y', 'z'), fill=c(rgb(1,0,0), rgb(0,1,0), rgb(0,0,1)))

Using the ggplot2 package, create the CDF plot with the ecdf function:

  # Load the required packages.
  library("reshape2","plyr","ggplot2")
  
  # transform the data.
  plot_data <- melt(data.frame(x, y, z))
  plot_data <- ddply(plot_data, .(variable), transform, cd=ecdf(value)(value))
  
  # Create the CDF using ggplot.
  cdf <- ggplot(plot_data, aes(x=value)) + stat_ecdf(aes(colour=variable))
  
  # Generate the Conditional distribution plot.
  cdf

Two different random variables can be associated with the same sample space. When there are two random variables on the same sample space, we study their interaction using a joint distribution. Let's consider an example: we want to know the probability that the sum of the same dice rolled twice is 6, so S = 6, and that the lowest die is 3, so D = 3. We represent this as follows:

P{S = 6, D = 3}

P {S = 6} = (1, 5) (2, 4) (3, 3) (4, 2) (1, 5);

Of the five outcomes, only one has the lower number equal to 3 so the probability is:

If there are only two outcomes to a trial, one with probability P and the other with probability 1 – P, often one outcome is called a success and the other a failure. When this is the case, P is used as the probability of success and the probability of failure is 1 – P. Such an experiment is called a Bernoulli trial or a binomial trial, because there are only two outcomes. The random variable associated with a Bernoulli trial is the Bernoulli random variable, with value 1 for a successful outcome and value 0 for failure.

Let's take an example of flipping a coin. It gives two outcomes, heads and tails. If we assign the value 1 to heads and 0 to tails, we have a Bernoulli random variable. Let's call this random variable R and since heads and tails are equally likely to occur:

P{R = 1} = 0.5 and P{R = 0} = 0.5

If we repeat a Bernoulli trial many times over, we get a new distribution, called a binomial distribution. So in order to compute the probability of k successes in n trials we can use the following formula:

Here:

The Poisson distribution applies when occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another. The average frequency of occurrence for the time period is known. The probability of an occurrence during a small time interval is proportional to the entire length of the time interval:

Here:

When we are putting together texts, we will not know the probability distribution of a particular topic. If we consider a corpus of country's economic strategy, written by various economists, it's difficult to understand the probability of what they are emphasizing more – is it infrastructure, manufacturing, banking, and so on – without counting the members associated. One thing to be aware of is no corpus will be balanced. We need to count the occurrences of relevant words in the dataset to get some statistical information. We need to know the frequency distribution of different words. Word frequencies refer to the number of word tokens that are instances of a word type. We can perform word counts over corpora with the R tau package.

Zipf's law is an interesting phenomenon that can be applied universally in many contexts, such as social sciences, cognitive sciences, and linguistics. When we consider a variety of datasets, there will be an uneven distribution of words. Zipf's law says that the frequency of a word, f (w), appears as a nonlinearly decreasing function of the rank of the word, r (w), in a corpus. This law is a power law: the frequency is a function of the negative power of rank. C is a constant that is determined by the particulars of the corpus; it's the frequency of the most frequent word:

Given a collection of words, we can estimate the frequency of each unique word, which is nothing but the number of times the word occurs in the collection.

If we sort the words in descending order of their frequency of occurrence in the collection, and compute their rank, the product of their frequency and associated rank reveals a very interesting pattern.

Let us consider a small sample S: a a a a b b b c c d d:

Let there be a word w which has the rank r' in a document and the probability of this word to be at rank r' be defined as P(r'). The probability P(r') can be expressed as the function of frequency of occurrence of the words as follows:

P(r') = Freq(r')/N,

where N is the sample size and Freq(r') is the frequency of occurrence of r' in the corpus

Heaps' law is also known as Herdan's Law. This law was discovered by Gustav Herdan, but the law is sometimes attributed to Harold Heaps. It is an empirical law which describes the relationship between type and tokens in linguistics. In simpler terms, Heaps' law defines the relation between the count of distinct words in document and the length of the specified document.

The relation can be expressed as:

Vr(n) = C* nb

Here, Vr is the count of distinct words in document and n is the size of the document. C and b are parameters defined empirically.

The similarity between Heaps' Law and Zipf's law is attributed to the fact that type-token relation is derivable from type distribution:

library(tm)
data("acq")
termdoc <- DocumentTermMatrix(acq)
Heaps_plot(termdoc)

Quantitative analysis of lexical structure is relevant for many activities such as stylometrics, applied linguistics, computational linguistics, natural language processing, lexicology, and so on. There are different approaches to capture vocabulary richness. It can be measured by means of measure or of an index. It can be captured by means of curve, as in the case of Herdan's and Zipf's law. If we consider the empirical distribution of word types, we can derive the distribution based on combinatorial considerations or we can use consider the stochastic processes to derive the distribution.

In applied linguistics, lexical richness explains the qualified proficiency of the author in a document, in terms of language variation, width, length, and productive knowledge of vocabulary. Let's attempt to understand the multiple measures that explain the lexical richness of a text.

              Ld  = (Nlex / N) * 100

N-grams are important for a wide range of applications. They can be used to build simple language models. Let's consider a text T with W tokens. Let SW be a sliding window. If the sliding window consists of one cell then the collection of strings is called a unigram; if the sliding window consists of two cells, the output is , this is called a bigram .Using conditional probability, we can define the probability of a word having seen the previous word; this is known as bigram probability. So the conditional probability of an element, , given the previous element, is:

Extending the sliding window, we can generalize that n-gram probability as the conditional probability of an element given previous n-1 element:

The most common bigrams in any corpus are not very interesting, such as on the, can be, in it, it is. In order to get more meaningful bigrams, we can run the corpus through a part-of-speech (POS) tagger. This would filter the bigrams to more content-related pairs such as infrastructure development, agricultural subsidies, banking rates; this can be one way of filtering less meaningful bigrams.

A better way to approach this problem is to take into account collocations; a collocation is the string created when two or more words co-occur in a language more frequently. One way to do this over a corpus is pointwise mutual information (PMI).The concept behind PMI is for two words, A and B, we would like to know how much one word tells us about the other. For example, given an occurrence of A, a, and an occurrence of B, b, how much does their joint probability differ from the expected value of assuming that they are independent. This can be expressed as follows:

Applying the chain rule on n contexts can be difficult to estimate; Markov assumption is applied to handle such situations.

If predicting that a current string is independent of some word string in the past, we can drop that string to simplify the probability. Say the history consists of three words, Wi, Wi-1, Wi-2, instead of estimating the probability P(Wi+1) using P(Wi,i-1,i-2) , we can directly apply P(Wi+1 | Wi, Wi-1).

Markov chains are used to study systems that are subject to random influences. Markov chains model systems that move from one state to another in steps governed by probabilities. The same set of outcomes in a sequence of trials is called states. Knowing the probabilities of states is called state distribution. The state distribution in which the system starts is the initial state distribution. The probability of going from one state to another is called transition probability. A Markov chain consists of a collection of states along with transition probabilities. The study of Markov chains is useful to understand the long-term behavior of a system. Each arc associates to certain probability value and all arcs coming out of each node must have exhibit a probability distribution. In simple terms, there is a probability associated to every transition in states:

.

Hidden Markov models are non-deterministic Markov chains. They are an extension of Markov models in which output symbol is not the same as state. We will discuss this topic in detail in later chapters.

In order to find the similarity between documents in a corpus, we can use a document term matrix. In a document term matrix, rows represent documents, columns represent terms, and each cell value is the term frequency count for a document. It is one of the useful ways of modeling documents. Here is an example:

If we visualize this in a term-document space, each document becomes a point in it. We can then tell how similar two documents are by calculating the distance between the two points using Euclidean distance.

When a term occurs in a lot of documents, it tends to make notably less difference the terms that occur few times. For example, India Today has more to do with India than today. These frequently occurring terms can affect the similarity comparison. The term space will be biased towards these terms. In order to address this problem, we use inverse document frequency.

Inverse document frequency

A commonly used measure of a term's selective potential is calculated by its inverse document frequency (IDF). The formula for IDF is calculated as follows:

Here, N is the number of documents in the corpus and df(term) is the number of documents in which the term appears.

The weight of a term's appearance in a document is calculated by combining the term frequency (TF) in the document with its IDF:

This term–document score is known as TF*IDF, and is widely used. This is used by a lot of search platforms/APIs, such as SOLR, Elasticsearch, and lucene. TF*IDF scores are then pre-computed and stored, so that similarity comparison can be done by just a dot product.

When we look at the entries in this term–document matrix, most of the cells will be empty because only a few terms appear in each document; storing all the empty cells requires a lot of memory and it contributes no value to the dot product (similarity computation). Various sparse matrix representations are possible and these are used to for optimized query processing.

install.packages("stringdist")
library(stringdist)

Euclidean distance

Euclidean distance is the distance between two points in the term-document space; it can be calculated by using the formula for a two-dimensional space as follows:

Euclidean distance e <- sqrt((x1-x2)^2+(y1-y2)^2)

Here, (x1,y1) and (x2,y2) are the two points and e is the estimated Euclidean distance between them:

We can very easily convert the aforesaid formula into R code:

euclidean.dist <- function(x1, x2) sqrt(sum((x1 - x2) ^ 2))

Cosine similarity

Euclidean distance has its own pitfalls, documents with lots of terms are far from origin, we will find small documents relatively similar even if it's unrelated because of short distance.

To avoid length issues, we can use the angular distance and measure the similarity by the angle between the vectors; we measure the cosine of the angle. The larger the cosine value, the more similar the documents are. Since we use the cos function, this is also called cosine similarity:

The formula to calculate cosine between two points is as follows:

This kind of geometric modeling is also called vector space model:

# Create two random matrices matrixA and matrixB
ncol<-5
nrow<-5
matrixA<-matrix(runif(ncol*nrow), ncol=ncol) 
matrixB<-matrix(runif(ncol*nrow), ncol=ncol) 

# function for estimating cosine similarity in R: 
cosine_sim<-function(matrixA, matrixB){
  m=tcrossprod(matrixA, matrixB)
  c1=sqrt(apply(matrixA, 1, crossprod))
  c2=sqrt(apply(matrixB, 1, crossprod))
  m / outer(c1,c2)
}

# Estimate the cosine similarity between the two matrices initiated earlier
cosine_sim(matrixA,matrixB)

Alternately, cosine similarity can also be estimated by functions available in the packages lsa, proxy, and so on.

Library(stringdist)
stringdist('abcd', 'abdc', method='lv')
     [1] 2

stringdist('abcd', 'abdc', method='dl')
      [1] 1

stringdist('abcd', 'abdc', method='hamming')
     [1] 2

stringdist('abcd', 'abdc' , method = 'jw' , p=0.1)
      [1] 0.06666667

Gunning frog index

The Gunning fog index is one of the best-known methods that measure the level of reading difficulty of any document. The fog index level translates the number of years of education a reader needs in order to understand the given material. The ideal score is 7 or 8; anything above 12 is too hard for most people to read.

The Gunning fog index is calculated as shown in the following steps:

1. Select all the sentences in a passage of approximately 100 words.

2. We need to calculate the average sentence length by doing a simple math of dividing number of words by number of sentences.

3. Count all the words with three or more syllables. Generally, words with more than three syllables are considered to be complex.

4. Sum up the average sentence length and the percentage of complex words.

5. Multiply the result by 0.4.

   The formula is as shown here:

OpenNLP is an R package which provides an interface, Apache OpenNLP, which is a machine-learning-based toolkit written in Java for natural language processing activities. Apache OpenNLP is widely used for most common tasks in NLP, such as tokenization, POS tagging, named entity recognition (NER), chunking, parsing, and so on. It provides wrappers for Maxent entropy models using the Maxent Java package.

It provides functions for sentence annotation, word annotation, POS tag annotation, and annotation parsing using the Apache OpenNLP chunking parser. The Maxent Chunk annotator function computes the chunk annotation using the Maxent chunker provided by OpenNLP.

The Maxent entity annotator function in R package utilizes the Apache OpenNLP Maxent name finder for entity annotation. Model files can be downloaded from http://opennlp.sourceforge.net/models-1.5/. These language models can be effectively used in R packages by installing the OpenNLPmodels.language package from the repository at http://datacube.wu.ac.at.

The RWeka package in R provides an interface to Weka. Weka is an open source software developed by a machine learning group at the University of Wakaito, which provides a wide range of machine learning algorithms which can either be directly applied to a dataset or it can be called from a Java code. Different data-mining activities, such as data processing, supervised and unsupervised learning, association mining, and so on, can be performed using the RWeka package. For natural language processing, RWeka provides tokenization and stemming functions. RWeka packages provide an interface to Alphabetic, NGramTokenizers, and wordTokenizer functions, which can efficiently perform tokenization for contiguous alphabetic sequence, string-split to n-grams, or simple word tokenization, respectively.

The RcmdrPlugin.temis package in R provides a graphical integrated text-mining solution. This package can be leveraged for many text-mining tasks, such as importing and cleaning a corpus, terms and documents count, term co-occurrences, correspondence analysis, and so on. Corpora can be imported from different sources and analysed using the importCorpusDlg function. The package provides flexible data source options to import corpora from different sources, such as text files, spreadsheet files, XML, HTML files, Alceste format and Twitter search. The Import function in this package processes the corpus and generates a term-document matrix. The package provides different functions to summarize and visualize the corpus statistics. Correspondence analysis and hierarchical clustering can be performed on the corpus. The corpusDissimilarity function helps analyse and create a cross-dissimilarity table between term-documents present in the corpus.

This package provides many functions to help the users explore the corpus. For example, frequentTerms to list the most frequent terms of a corpus, specificTerms to list terms most associated with each document, subsetCorpusByTermsDlg to create a subset of the corpus. Term frequency, term co-occurrence, term dictionary, temporal evolution of occurrences or term time series, term metadata variables, and corpus temporal evolution are among the other very useful functions available in this package for text mining.

The tm package is a text-mining framework which provides some powerful functions which will aid in text-processing steps. It has methods for importing data, handling corpus, metadata management, creation of term document matrices, and preprocessing methods. For managing documents using the tm package, we create a corpus which is a collection of text documents. There are two types of implementation, volatile corpus (VCorpus) and permanent corpus (PCropus). VCorpus is completely held in memory and when the R object is destroyed the corpus is gone. PCropus is stored in the filesystem and is present even after the R object is destroyed; this corpus can be created by using the VCorpus and PCorpus functions respectively. This package provides a few predefined sources which can be used to import text, such as DirSource, VectorSource, or DataframeSource. The getSources method lists available sources, and users can create their own sources. The tm package ships with several reader options: readPlain, readPDF, and readDOC. We can execute the getReaders method for an up-to-date list of available readers. To write a corpus to the filesystem, we can use writeCorpus.

For inspecting a corpus, there are methods such as inspect and print. For transformation of text, such as stop-word removal, stemming, whitespace removal, and so on, we can use the tm_map, content_transformer, tolower, stopwords("english") functions. For metadata management, meta comes in handy. The tm package provides various quantitative function for text analysis, such as DocumentTermMatrix , findFreqTerms, findAssocs, and removeSparseTerms.

languageR provides datasets and functions for statistical analysis on text data. This package contains functions for vocabulary richness, vocabulary growth, frequency spectrum, also mixed-effects models and so on. There are simulation functions available: simple regression, quasi-F factor, and Latin-square designs. Apart from that, this package can also be used for correlation, collinearity diagnostic, diagnostic visualization of logistic models, and so on.

The koRpus package is a versatile tool for text mining which implements many functions for text readability and lexical variation. Apart from that, it can also be used for basic level functions such as tokenization and POS tagging.

The RKEA package provides an interface to KEA, which is a tool for keyword extraction from texts. RKEA requires a keyword extraction model, which can be created by manually indexing a small set of texts, using which it extracts keywords from the document.

The maxent package in R provides tools for low-memory implementation of multinomial logistic regression, which is also called the maximum entropy model. This package is quite helpful for classification processes involving sparse term-document matrices, and low memory consumption on huge datasets.

Truncated singular vector decomposition can help overcome the variability in a term-document matrix by deriving the latent features statistically. The lsa package in R provides an implementation of latent semantic analysis.