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Latent Dirichlet allocation (LDA) model

A latent Dirichlet allocation (LDA) model is a topic model which discovers underlying topics in a collection of documents and infers word probabilities in topics. If the model was fit using a bag-of-n-grams model, then the software treats the n-grams as individual words.

Create an LDA model using the `fitlda`

function.

`NumTopics`

— Number of topicspositive integer

Number of topics in the LDA model, specified as a positive integer.

`TopicConcentration`

— Topic concentrationpositive scalar

Topic concentration, specified as a positive scalar. The function sets the
concentration per topic to `TopicConcentration/NumTopics`

.
For more information, see Latent Dirichlet Allocation.

`WordConcentration`

— Word concentration`1`

(default) | nonnegative scalarWord concentration, specified as a nonnegative scalar. The software sets
the concentration per word to `WordConcentration/numWords`

,
where `numWords`

is the vocabulary size of the input
documents. For more information, see Latent Dirichlet Allocation.

`CorpusTopicProbabilities`

— Topic probabilities of input document setvector

Topic probabilities of input document set, specified as a vector. The
corpus topic probabilities of an LDA model are the probabilities of
observing each topic in the entire data set used to fit the LDA model.
`CorpusTopicProbabilities`

is a
1-by-*K* vector where *K* is the
number of topics. The *k*th entry of
`CorpusTopicProbabilities`

corresponds to the
probability of observing topic *k*.

`DocumentTopicProbabilities`

— Topic probabilities per input documentmatrix

Topic probabilities per input document, specified as a matrix. The
document topic probabilities of an LDA model are the probabilities of
observing each topic in each document used to fit the LDA model.
`DocumentTopicProbabilities`

is a
*D*-by-*K* matrix where
*D* is the number of documents used to fit the LDA
model, and *K* is the number of topics. The
*(d,k)*th entry of
`DocumentTopicProbabilities`

corresponds to the
probability of observing topic *k* in document
*d*.

If any the topics have zero probability
(`CorpusTopicProbabilities`

contains zeros), then the
corresponding columns of `DocumentTopicProbabilities`

and
`TopicWordProbabilities`

are zeros.

The order of the rows in `DocumentTopicProbabilities`

corresponds to the order of the documents in the training data.

`TopicWordProbabilities`

— Word probabilities per topicmatrix

Word probabilities per topic, specified as a matrix. The topic word
probabilities of an LDA model are the probabilities of observing each word
in each topic of the LDA model. `TopicWordProbabilities`

is a *V*-by-*K* matrix, where
*V* is the number of words in
`Vocabulary`

and *K* is the number
of topics. The *(v,k)*th entry of
`TopicWordProbabilities`

corresponds to the
probability of observing word *v* in topic
*k*.

If any the topics have zero probability
(`CorpusTopicProbabilities`

contains zeros), then the
corresponding columns of `DocumentTopicProbabilities`

and
`TopicWordProbabilities`

are zeros.

The order of the rows in `TopicWordProbabilities`

corresponds to the order of the words in
`Vocabulary`

.

`TopicOrder`

— Topic order`'initial-fit-probability'`

(default) | `'unordered'`

Topic order, specified as one of the following:

`'initial-fit-probability'`

– Sort the topics by the corpus topic probabilities of the initial model fit. These probabilities are the`CorpusTopicProbabilities`

property of the initial`ldaModel`

object returned by`fitlda`

. The`resume`

function does not reorder the topics of the resulting`ldaModel`

objects.`'unordered'`

– Do not order topics.

`FitInfo`

— Information recorded when fitting LDA modelstruct

Information recorded when fitting LDA model, specified as a struct with the following fields:

`TerminationCode`

– Status of optimization upon exit0 – Iteration limit reached.

1 – Tolerance on log-likelihood satisfied.

`TerminationStatus`

– Explanation of the returned termination code`NumIterations`

– Number of iterations performed`NegativeLogLikelihood`

– Negative log-likelihood for the data passed to`fitlda`

`Perplexity`

– Perplexity for the data passed to`fitlda`

`Solver`

– Name of the solver used`History`

– Struct holding the optimization history`StochasticInfo`

– Struct holding information for stochastic solvers

**Data Types: **`struct`

`Vocabulary`

— List of words in the modelstring vector

List of words in the model, specified as a string vector.

**Data Types: **`string`

`logp` | Document log-probabilities and goodness of fit of LDA model |

`predict` | Predict top LDA topics of documents |

`resume` | Resume fitting LDA model |

`topkwords` | Most important words in bag-of-words model or LDA topic |

`transform` | Transform documents into lower-dimensional space |

`wordcloud` | Create word cloud chart from text, bag-of-words model, bag-of-n-grams model, or LDA model |

To reproduce the results in this example, set `rng`

to `'default'`

.

`rng('default')`

Load the example data. The file `sonnetsPreprocessed.txt`

contains preprocessed versions of Shakespeare's sonnets. The file contains one sonnet per line, with words separated by a space. Extract the text from `sonnetsPreprocessed.txt`

, split the text into documents at newline characters, and then tokenize the documents.

```
filename = "sonnetsPreprocessed.txt";
str = extractFileText(filename);
textData = split(str,newline);
documents = tokenizedDocument(textData);
```

Create a bag-of-words model using `bagOfWords`

.

bag = bagOfWords(documents)

bag = bagOfWords with properties: Counts: [154x3092 double] Vocabulary: ["fairest" "creatures" "desire" ... ] NumWords: 3092 NumDocuments: 154

Fit an LDA model with four topics.

numTopics = 4; mdl = fitlda(bag,numTopics)

Initial topic assignments sampled in 0.213222 seconds. ===================================================================================== | Iteration | Time per | Relative | Training | Topic | Topic | | | iteration | change in | perplexity | concentration | concentration | | | (seconds) | log(L) | | | iterations | ===================================================================================== | 0 | 0.49 | | 1.215e+03 | 1.000 | 0 | | 1 | 0.02 | 1.0482e-02 | 1.128e+03 | 1.000 | 0 | | 2 | 0.02 | 1.7190e-03 | 1.115e+03 | 1.000 | 0 | | 3 | 0.02 | 4.3796e-04 | 1.118e+03 | 1.000 | 0 | | 4 | 0.01 | 9.4193e-04 | 1.111e+03 | 1.000 | 0 | | 5 | 0.01 | 3.7079e-04 | 1.108e+03 | 1.000 | 0 | | 6 | 0.01 | 9.5777e-05 | 1.107e+03 | 1.000 | 0 | =====================================================================================

mdl = ldaModel with properties: NumTopics: 4 WordConcentration: 1 TopicConcentration: 1 CorpusTopicProbabilities: [0.2500 0.2500 0.2500 0.2500] DocumentTopicProbabilities: [154x4 double] TopicWordProbabilities: [3092x4 double] Vocabulary: ["fairest" "creatures" ... ] TopicOrder: 'initial-fit-probability' FitInfo: [1x1 struct]

Visualize the topics using word clouds.

figure for topicIdx = 1:4 subplot(2,2,topicIdx) wordcloud(mdl,topicIdx); title("Topic: " + topicIdx) end

Create a table of the words with highest probability of an LDA topic.

To reproduce the results, set `rng`

to `'default'`

.

`rng('default')`

Load the example data. The file `sonnetsPreprocessed.txt`

contains preprocessed versions of Shakespeare's sonnets. The file contains one sonnet per line, with words separated by a space. Extract the text from `sonnetsPreprocessed.txt`

, split the text into documents at newline characters, and then tokenize the documents.

```
filename = "sonnetsPreprocessed.txt";
str = extractFileText(filename);
textData = split(str,newline);
documents = tokenizedDocument(textData);
```

Create a bag-of-words model using `bagOfWords`

.

bag = bagOfWords(documents);

Fit an LDA model with 20 topics. To suppress verbose output, set `'Verbose'`

to 0.

```
numTopics = 20;
mdl = fitlda(bag,numTopics,'Verbose',0);
```

Find the top 20 words of the first topic.

k = 20; topicIdx = 1; tbl = topkwords(mdl,k,topicIdx)

`tbl=`*20×2 table*
Word Score
________ _________
"eyes" 0.11155
"beauty" 0.05777
"hath" 0.055778
"still" 0.049801
"true" 0.043825
"mine" 0.033865
"find" 0.031873
"black" 0.025897
"look" 0.023905
"tis" 0.023905
"kind" 0.021913
"seen" 0.021913
"found" 0.017929
"sin" 0.015937
"three" 0.013945
"golden" 0.0099608
⋮

Find the top 20 words of the first topic and use inverse mean scaling on the scores.

tbl = topkwords(mdl,k,topicIdx,'Scaling','inversemean')

`tbl=`*20×2 table*
Word Score
________ ________
"eyes" 1.2718
"beauty" 0.59022
"hath" 0.5692
"still" 0.50269
"true" 0.43719
"mine" 0.32764
"find" 0.32544
"black" 0.25931
"tis" 0.23755
"look" 0.22519
"kind" 0.21594
"seen" 0.21594
"found" 0.17326
"sin" 0.15223
"three" 0.13143
"golden" 0.090698
⋮

Create a word cloud using the scaled scores as the size data.

figure wordcloud(tbl.Word,tbl.Score);

Get the document topic probabilities (also known as topic mixtures) of the documents used to fit an LDA model.

To reproduce the results, set `rng`

to `'default'`

.

`rng('default')`

Load the example data. The file `sonnetsPreprocessed.txt`

contains preprocessed versions of Shakespeare's sonnets. The file contains one sonnet per line, with words separated by a space. Extract the text from `sonnetsPreprocessed.txt`

, split the text into documents at newline characters, and then tokenize the documents.

```
filename = "sonnetsPreprocessed.txt";
str = extractFileText(filename);
textData = split(str,newline);
documents = tokenizedDocument(textData);
```

Create a bag-of-words model using `bagOfWords`

.

bag = bagOfWords(documents);

Fit an LDA model with 20 topics. To suppress verbose output, set `'Verbose'`

to 0.

```
numTopics = 20;
mdl = fitlda(bag,numTopics,'Verbose',0)
```

mdl = ldaModel with properties: NumTopics: 20 WordConcentration: 1 TopicConcentration: 5 CorpusTopicProbabilities: [0.0500 0.0500 0.0500 0.0500 0.0500 ... ] DocumentTopicProbabilities: [154x20 double] TopicWordProbabilities: [3092x20 double] Vocabulary: ["fairest" "creatures" ... ] TopicOrder: 'initial-fit-probability' FitInfo: [1x1 struct]

View the topic probabilities of the first document in the training data.

topicMixtures = mdl.DocumentTopicProbabilities; figure bar(topicMixtures(1,:)) title("Document 1 Topic Probabilities") xlabel("Topic Index") ylabel("Probability")

To reproduce the results in this example, set `rng`

to `'default'`

.

`rng('default')`

`sonnetsPreprocessed.txt`

contains preprocessed versions of Shakespeare's sonnets. The file contains one sonnet per line, with words separated by a space. Extract the text from `sonnetsPreprocessed.txt`

, split the text into documents at newline characters, and then tokenize the documents.

```
filename = "sonnetsPreprocessed.txt";
str = extractFileText(filename);
textData = split(str,newline);
documents = tokenizedDocument(textData);
```

Create a bag-of-words model using `bagOfWords`

.

bag = bagOfWords(documents)

bag = bagOfWords with properties: Counts: [154x3092 double] Vocabulary: ["fairest" "creatures" "desire" ... ] NumWords: 3092 NumDocuments: 154

Fit an LDA model with 20 topics.

numTopics = 20; mdl = fitlda(bag,numTopics)

Initial topic assignments sampled in 0.035863 seconds. ===================================================================================== | Iteration | Time per | Relative | Training | Topic | Topic | | | iteration | change in | perplexity | concentration | concentration | | | (seconds) | log(L) | | | iterations | ===================================================================================== | 0 | 0.02 | | 1.159e+03 | 5.000 | 0 | | 1 | 0.08 | 5.4884e-02 | 8.028e+02 | 5.000 | 0 | | 2 | 0.06 | 4.7400e-03 | 7.778e+02 | 5.000 | 0 | | 3 | 0.05 | 3.4597e-03 | 7.602e+02 | 5.000 | 0 | | 4 | 0.05 | 3.4662e-03 | 7.430e+02 | 5.000 | 0 | | 5 | 0.05 | 2.9259e-03 | 7.288e+02 | 5.000 | 0 | | 6 | 0.06 | 6.4180e-05 | 7.291e+02 | 5.000 | 0 | =====================================================================================

mdl = ldaModel with properties: NumTopics: 20 WordConcentration: 1 TopicConcentration: 5 CorpusTopicProbabilities: [0.0500 0.0500 0.0500 0.0500 0.0500 ... ] DocumentTopicProbabilities: [154x20 double] TopicWordProbabilities: [3092x20 double] Vocabulary: ["fairest" "creatures" ... ] TopicOrder: 'initial-fit-probability' FitInfo: [1x1 struct]

Predict the top topics for an array of new documents.

newDocuments = tokenizedDocument([ "what's in a name? a rose by any other name would smell as sweet." "if music be the food of love, play on."]); topicIdx = predict(mdl,newDocuments)

`topicIdx = `*2×1*
19
8

Visualize the predicted topics using word clouds.

figure subplot(1,2,1) wordcloud(mdl,topicIdx(1)); title("Topic " + topicIdx(1)) subplot(1,2,2) wordcloud(mdl,topicIdx(2)); title("Topic " + topicIdx(2))

A *latent Dirichlet allocation* (LDA) model is a
document topic model which discovers underlying topics in a collection of documents and
infers word probabilities in topics. LDA models a collection of *D*
documents as topic mixtures $${\theta}_{1},\dots ,{\theta}_{D}$$, over *K* topics characterized by vectors of word
probabilities $${\phi}_{1},\dots ,{\phi}_{K}$$. The model assumes that the topic mixtures $${\theta}_{1},\dots ,{\theta}_{D}$$, and the topics $${\phi}_{1},\dots ,{\phi}_{K}$$ follow a Dirichlet distribution with concentration parameters $$\alpha $$ and $$\beta $$ respectively.

The topic mixtures $${\theta}_{1},\dots ,{\theta}_{D}$$ are probability vectors of length *K*, where
*K* is the number of topics. The entry $${\theta}_{di}$$ is the probability of topic *i* appearing in the
*d*th document. The topic mixtures correspond to the rows of the
`DocumentTopicProbabilities`

property of the `ldaModel`

object.

The topics $${\phi}_{1},\dots ,{\phi}_{K}$$ are probability vectors of length *V*, where
*V* is the number of words in the vocabulary. The entry $${\phi}_{iv}$$ corresponds to the probability of the *v*th word of the
vocabulary appearing in the *i*th topic. The topics $${\phi}_{1},\dots ,{\phi}_{K}$$ correspond to the columns of the `TopicWordProbabilities`

property of the `ldaModel`

object.

Given the topics $${\phi}_{1},\dots ,{\phi}_{K}$$ and Dirichlet prior $$\alpha $$ on the topic mixtures, LDA assumes the following generative process for a document:

Sample a topic mixture $$\theta ~\text{Dirichlet}(\alpha )$$. The random variable $$\theta $$ is a probability vector of length

*K*, where*K*is the number of topics.For each word in the document:

Sample a topic index $$z~\text{Categorical}(\theta )$$. The random variable

*z*is an integer from 1 through*K*, where*K*is the number of topics.Sample a word $$w~\text{Categorical}({\phi}_{z})$$. The random variable

*w*is an integer from 1 through*V*, where*V*is the number of words in the vocabulary, and represents the corresponding word in the vocabulary.

Under this generative process, the joint distribution of a document with words $${w}_{1},\dots ,{w}_{N}$$, with topic mixture $$\theta $$, and with topic indices $${z}_{1},\dots ,{z}_{N}$$ is given by

$$p(\theta ,z,w|\alpha ,\phi )=p(\theta |\alpha ){\displaystyle \prod _{n=1}^{N}p}({z}_{n}|\theta )p({w}_{n}|{z}_{n},\phi ),$$

where *N* is the number of words in the document.
Summing the joint distribution over *z* and then integrating over $$\theta $$ yields the marginal distribution of a document *w*:

$$p(w|\alpha ,\phi )={\displaystyle \underset{\theta}{\int}p(\theta |\alpha ){\displaystyle \prod _{n=1}^{N}{\displaystyle \sum _{{z}_{n}}p({z}_{n}|\theta )p({w}_{n}|{z}_{n},\phi )}}}d\theta .$$

The following diagram illustrates the LDA model as a probabilistic graphical model. Shaded nodes are observed variables, unshaded nodes are latent variables, nodes without outlines are the model parameters. The arrows highlight dependencies between random variables and the plates indicate repeated nodes.

The *Dirichlet distribution* is a continuous
generalization of the multinomial distribution. Given the number of categories $$K\ge 2$$, and concentration parameter $$\alpha $$, where $$\alpha $$ is a vector of positive reals of length *K*, the
probability density function of the Dirichlet distribution is given by

$$p(\theta \mid \alpha )=\frac{1}{B(\alpha )}{\displaystyle \prod}_{i=1}^{K}\text{}{\theta}_{i}^{{\alpha}_{i}-1},$$

where *B* denotes the multivariate Beta function given
by

$$B(\alpha )=\frac{{\displaystyle \prod}_{i=1}^{K}\text{}\Gamma \text{}\text{(}{\alpha}_{i})}{\Gamma \left({\displaystyle \sum}_{i=1}^{K}\text{}{\alpha}_{i}\right)}.$$

A special case of the Dirichlet distribution is the *symmetric Dirichlet
distribution*. The symmetric Dirichlet distribution is characterized by the
concentration parameter $$\alpha $$, where all the elements of $$\alpha $$ are the same.

`fitlda`

| `logp`

| `predict`

| `resume`

| `topkwords`

| `transform`

| `wordcloud`

| `bagOfWords`

| `lsaModel`

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