# Introduction to tidylda

#### Tommy Jones

#### 2024-04-20

Note: for code examples, see README.md

# Introduction to the `tidylda` package

`tidylda` implements Latent Dirichlet Allocation using style conventions from the tidyverse and tidymodels. `tidylda`’s Gibbs sampler is written in C++ for performance and offers several novel features. It also has methods for sampling from the posterior of a trained model, for more traditional Bayesian analyses.

tidylda’s Gibbs sampler has several unique features, described below.

Non-uniform initialization: Most LDA Gibbs samplers initialize by assigning words to topics and topics to documents (i.e., construct the \(\boldsymbol{Cd}\) and \(\boldsymbol{Cv}\) matrices) by sampling from a uniform distribution. This ensures initialization without incorporating any prior information. tidylda incorporates the priors in its initialization. It begins by drawing \(P(\text{topic}|\text{document})\) and \(P(\text{word}|\text{topic})\) from Dirichlet distributions with parameters \(\boldsymbol\alpha\) and \(\boldsymbol\eta\), respectively. Then tidylda uses the above probabilities to construct \(P(\text{topic}|\text{word}, \text{document})\) and makes a single run of the Gibbs sampler to initialize \(\boldsymbol{Cd}\) and \(\boldsymbol{Cv}\). This non-uniform initialization powers transfer learning with LDA (tLDA), described elsewhere, by starting a Gibbs run near where the previous run left off. For initial models, it uses the user’s prior information to tune where sampling starts.

Flexible priors: tidylda has multiple options for setting LDA priors. Users may set scalar values for \(\boldsymbol\alpha\) and \(\boldsymbol\eta\) to construct symmetric priors. Users may also choose to construct vector priors for both \(\boldsymbol\alpha\) and \(\boldsymbol\eta\) for a full specification of LDA. Additionally, tidylda allows users to set a matrix prior for \(\boldsymbol\eta\), enabled by its implementation of tLDA. This lets users to set priors over word-topic relationships informed by expert input. The best practices for encoding expert input in this manner are not yet well studied. Nevertheless, this capability makes tidylda unique among LDA implementations.

Burn in iterations and posterior averaging: Most LDA Gibbs samplers construct posterior estimates of \(\boldsymbol\Theta\) and \(\boldsymbol{B}\) from \(\boldsymbol{Cd}\) and \(\boldsymbol{Cv}\)’s values of the final iteration of sampling, effectively using a single sample. This is inconsistent with best practices from Bayesian statistics, which is to average over many samples from a stable posterior. tidylda enables averaging across multiple samples of the posterior with the `burnin` argument. When `burnin` is set to a positive integer, tidylda averages the posterior across all iterations larger than `burnin`. For example, if `iterations` is 200 and `burnin` is 150, tidylda will return a posterior estimate that is an average of the last 50 sampling iterations. This ensures that posterior estimates are more likely to be representative than any single sample.

Transfer learning with tLDA: Finally, tidylda’s Gibbs sampler enables transfer learning with tLDA. The full specification of tLDA and details on its implementation in tidylda are described briefly in the tLDA vignette and more thoroughly in forthcoming research.

### Tidy Methods

tidylda’s construction follows Conventions of R Modeling Packages [@tidymodelsbook]. In particular, it contains methods for `print`, `summary`, `glance`, `tidy`, and `augment`, consistent with other “tidy” packages. These methods are briefly described below.

- `print`, `summary`, and `glance` return various summaries of the contents of a `tidylda` object, into which an LDA model trained with tidylda is stored.
- `tidy` returns the contents of \(\boldsymbol\Theta\), \(\boldsymbol{B}\), or \(\boldsymbol\Lambda\) (stored as `theta`, `beta`, and `lambda` respectively), as specified by the user, formatted as a tidy `tibble`, instead of a numeric matrix.
- `augment` appends model outputs to observational-level data. Taking the cue from tidytext, “observational-level” data is one row per word per document. Therefore, the key statistic used by `augment` is \(P(\text{topic}|\text{word}, \text{document})\). tidylda calculates this as \(\boldsymbol\Lambda \times P(\text{word}|\text{document})\), where \(P(\text{word}|\text{document}_d) = \frac{\boldsymbol{x}_d}{\sum_{v=1}^V x_{d,v}}\).

### Other Notable Features

tidylda has three evaluation metrics for topic models, two goodness-of-fit measures—\(R^2\) as implemented from mvrsquared and the log likelihood of the model given the data—and one coherence measure—probabilistic coherence. A flag set during model fitting with `calc_r2 = TRUE` 1 will return a model with an \(R^2\) statistic. Similarly, the log likelihood of the model, given the data, is calculated at each Gibbs iteration if the user selects `calc_likelihood = TRUE` during model fitting.

The coherence measure is called probabilistic coherence. (See vignette on probabilistic coherence.) Probabilistic coherence is bound between -1 and 1. Values near zero indicate that the top words in a topic are statistically independent from each other. Positive values indicate that the top words in a topic are positively correlated in a statistically-dependent manner. Negative values indicate that the words in a topic are negatively correlated with each other in a statistically-dependent manner. (In practice, negative values tend to be very near zero.)

tidylda enables traditional Bayesian uncertainty quantification by sampling from the posterior. The posterior distribution for \(\boldsymbol\theta_d\) is \(\text{Dirichlet}(\boldsymbol{Cd}_d + \boldsymbol\alpha)\) and the posterior distribution for \(\boldsymbol\beta_k\) is \(\text{Dirichlet}(\boldsymbol{Cv}_k + \boldsymbol\eta)\) (or \(\text{Dirichlet}(\boldsymbol{Cv}_k + \boldsymbol\eta_k)\) for tLDA). tidylda enables a `posterior` method for `tidylda` objects, allowing users to sample from the posterior to quantify uncertainty for estimates of estimated parameters.

tidylda uses one of two calculations for predicting topic distributions (i.e., \(\hat{\boldsymbol\theta}_d\)) for new documents. The first, and default, is to run the Gibbs sampler, constructing a new \(\boldsymbol{Cd}\) for the new documents but without updating topic-word distributions in \(\boldsymbol{B}\). The second uses a dot product as described in Appendix 1. tidylda actually uses the dot product prediction combined with the non-uniform initialization—described above—to initialize \(\boldsymbol{Cd}\) when predicting using the Gibbs sampler.

## Discussion

While many other topic modeling packages exist, tidylda is very user friendly and brings novel features. Its user friendliness comes from compatibility with the tidyverse. And tidylda includes tLDA and other methods contained in the previous chapters of this dissertation. It also has methods for sampling from the posterior of a trained model, for more traditional Bayesian analyses. tidylda’s Gibbs sampler is written in C++ for performance.

## Installation

You can install the development version from GitHub with:

    install.packages("remotes")

    remotes::install_github("tommyjones/tidylda")

## Example

    library(tidytext)
    library(dplyr)

    ##
    ## Attaching package: 'dplyr'

    ## The following objects are masked from 'package:stats':
    ##
    ##     filter, lag

    ## The following objects are masked from 'package:base':
    ##
    ##     intersect, setdiff, setequal, union

    library(ggplot2)
    library(tidyr)
    library(tidylda)
    library(Matrix)

    ##
    ## Attaching package: 'Matrix'

    ## The following objects are masked from 'package:tidyr':
    ##
    ##     expand, pack, unpack

    ### Initial set up ---
    # load some documents
    docs <- nih_sample

    # tokenize using tidytext's unnest_tokens
    tidy_docs <- docs |>
      select(APPLICATION_ID, ABSTRACT_TEXT) |>
      unnest_tokens(output = word,
                    input = ABSTRACT_TEXT,
                    stopwords = stop_words$word,
                    token = "ngrams",
                    n_min = 1, n = 2) |>
      count(APPLICATION_ID, word) |>
      filter(n>1) #Filtering for words/bigrams per document, rather than per corpus

    tidy_docs <- tidy_docs |> # filter words that are just numbers
      filter(! stringr::str_detect(tidy_docs$word, "^[0-9]+$"))

    # append observation level data
    colnames(tidy_docs)[1:2] <- c("document", "term")

    # turn a tidy tbl into a sparse dgCMatrix
    # note tidylda has support for several document term matrix formats
    d <- tidy_docs |>
      cast_sparse(document, term, n)

    # let's split the documents into two groups to demonstrate predictions and updates
    d1 <- d[1:50, ]

    d2 <- d[51:nrow(d), ]

    # make sure we have different vocabulary for each data set to simulate the "real world"
    # where you get new tokens coming in over time
    d1 <- d1[, colSums(d1) > 0]

    d2 <- d2[, colSums(d2) > 0]

    ### fit an intial model and inspect it ----
    set.seed(123)

    lda <- tidylda(
      data = d1,
      k = 10,
      iterations = 200,
      burnin = 175,
      alpha = 0.1, # also accepts vector inputs
      eta = 0.05, # also accepts vector or matrix inputs
      optimize_alpha = FALSE, # experimental
      calc_likelihood = TRUE,
      calc_r2 = TRUE, # see https://arxiv.org/abs/1911.11061
      return_data = FALSE
    )

    # did the model converge?
    # there are actual test stats for this, but should look like "yes"
    qplot(x = iteration, y = log_likelihood, data = lda$log_likelihood, geom = "line") +
        ggtitle("Checking model convergence")

    ## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
    ## This warning is displayed once every 8 hours.
    ## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    ## generated.

    # look at the model overall
    glance(lda)

    ## # A tibble: 1 × 5
    ##   num_topics num_documents num_tokens iterations burnin
    ##        <int>         <int>      <int>      <dbl>  <dbl>
    ## 1         10            50       1524        200    175

    print(lda)

    ## A Latent Dirichlet Allocation Model of  10 topics,  50  documents, and  1524  tokens:
    ## tidylda(data = d1, k = 10, iterations = 200, burnin = 175, alpha = 0.1,
    ##     eta = 0.05, optimize_alpha = FALSE, calc_likelihood = TRUE,
    ##     calc_r2 = TRUE, return_data = FALSE)
    ##
    ## The model's R-squared is  0.2503
    ## The  5  most prevalent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     4       12.5    0.0527 cdk5, cns, develop, based, lsds, ...
    ## 2     3       11.5    0.170  cells, cell, sleep, specific, memory, ...
    ## 3     1       11.4    0.114  effects, v4, signaling, stiffening, wall, ...
    ## 4     6       10.9    0.348  diabetes, numeracy, redox, extinction, health, ...
    ## 5     8       10.7    0.337  cmybp, function, mitochondrial, injury, fragment, …
    ## # ℹ 5 more rows
    ##
    ## The  5  most coherent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     6      10.9      0.348 diabetes, numeracy, redox, extinction, health, ...
    ## 2     8      10.7      0.337 cmybp, function, mitochondrial, injury, fragment, …
    ## 3     7      10.3      0.210 cancer, imaging, cells, rb, tumor, ...
    ## 4     5       9.13     0.206 program, dcis, cancer, research, disparities, ...
    ## 5    10       8.53     0.19  sud, plasticity, risk, factors, brain, ...
    ## # ℹ 5 more rows

    # it comes with its own summary matrix that's printed out with print(), above
    lda$summary

    ## # A tibble: 10 × 4
    ##    topic prevalence coherence top_terms
    ##    <dbl>      <dbl>     <dbl> <chr>
    ##  1     1      11.4     0.114  effects, v4, signaling, stiffening, wall, ...
    ##  2     2       7.01    0.0779 research, natural, antibodies, hiv, core, ...
    ##  3     3      11.5     0.170  cells, cell, sleep, specific, memory, ...
    ##  4     4      12.5     0.0527 cdk5, cns, develop, based, lsds, ...
    ##  5     5       9.13    0.206  program, dcis, cancer, research, disparities, ...
    ##  6     6      10.9     0.348  diabetes, numeracy, redox, extinction, health, ...
    ##  7     7      10.3     0.210  cancer, imaging, cells, rb, tumor, ...
    ##  8     8      10.7     0.337  cmybp, function, mitochondrial, injury, fragment,…
    ##  9     9       8       0.184  ppg, core, pd, data, imaging, ...
    ## 10    10       8.53    0.19   sud, plasticity, risk, factors, brain, ...

    # inspect the individual matrices
    tidy_theta <- tidy(lda, matrix = "theta")

    tidy_theta

    ## # A tibble: 500 × 3
    ##    document topic   theta
    ##    <chr>    <dbl>   <dbl>
    ##  1 8574224      1 0.00238
    ##  2 8574224      2 0.00524
    ##  3 8574224      3 0.00238
    ##  4 8574224      4 0.00429
    ##  5 8574224      5 0.00238
    ##  6 8574224      6 0.00238
    ##  7 8574224      7 0.00238
    ##  8 8574224      8 0.00238
    ##  9 8574224      9 0.00238
    ## 10 8574224     10 0.974
    ## # ℹ 490 more rows

    tidy_beta <- tidy(lda, matrix = "beta")

    tidy_beta

    ## # A tibble: 15,240 × 3
    ##    topic token             beta
    ##    <dbl> <chr>            <dbl>
    ##  1     1 adolescence  0.0025
    ##  2     1 age          0.0000648
    ##  3     1 application  0.0000648
    ##  4     1 depressive   0.0000648
    ##  5     1 disorder     0.0000648
    ##  6     1 emotionality 0.0000648
    ##  7     1 information  0.0025
    ##  8     1 mdd          0.0000648
    ##  9     1 onset        0.0000648
    ## 10     1 onset mdd    0.0000648
    ## # ℹ 15,230 more rows

    tidy_lambda <- tidy(lda, matrix = "lambda")

    tidy_lambda

    ## # A tibble: 15,240 × 3
    ##    topic token         lambda
    ##    <dbl> <chr>          <dbl>
    ##  1     1 adolescence  0.304
    ##  2     1 age          0.00938
    ##  3     1 application  0.00794
    ##  4     1 depressive   0.0206
    ##  5     1 disorder     0.0206
    ##  6     1 emotionality 0.0206
    ##  7     1 information  0.259
    ##  8     1 mdd          0.0115
    ##  9     1 onset        0.00795
    ## 10     1 onset mdd    0.0206
    ## # ℹ 15,230 more rows

    # append observation-level data
    augmented_docs <- augment(lda, data = tidy_docs)

    ## Joining with `by = join_by(document, term, n)`

    augmented_docs

    ## # A tibble: 4,566 × 4
    ##    document term            n topic
    ##    <chr>    <chr>       <int> <int>
    ##  1 8574224  adolescence     1    10
    ##  2 8646901  adolescence     1    10
    ##  3 8689019  adolescence     1    10
    ##  4 8705323  adolescence     1    10
    ##  5 8574224  age             1    10
    ##  6 8705323  age             1    10
    ##  7 8757072  age             1    10
    ##  8 8823186  age             1    10
    ##  9 8574224  application     1    10
    ## 10 8605875  application     1    10
    ## # ℹ 4,556 more rows

    ### predictions on held out data ---
    # two methods: gibbs is cleaner and more technically correct in the bayesian sense
    p_gibbs <- predict(lda, new_data = d2[1, ], iterations = 100, burnin = 75)

    # dot is faster, less prone to error (e.g. underflow), noisier, and frequentist
    p_dot <- predict(lda, new_data = d2[1, ], method = "dot")

    # pull both together into a plot to compare
    tibble(topic = 1:ncol(p_gibbs), gibbs = p_gibbs[1,], dot = p_dot[1, ]) |>
      pivot_longer(cols = gibbs:dot, names_to = "type") |>
      ggplot() +
      geom_bar(mapping = aes(x = topic, y = value, group = type, fill = type),
               stat = "identity", position="dodge") +
      scale_x_continuous(breaks = 1:10, labels = 1:10) +
      ggtitle("Gibbs predictions vs. dot product predictions")

    ### Augment as an implicit prediction using the 'dot' method ----
    # Aggregating over terms results in a distribution of topics over documents
    # roughly equivalent to using the "dot" method of predictions.
    augment_predict <-
      augment(lda, tidy_docs, "prob") |>
      group_by(document) |>
      select(-c(document, term)) |>
      summarise_all(function(x) sum(x, na.rm = T))

    ## Joining with `by = join_by(document, term, n)`

    ## Adding missing grouping variables: `document`

    # reformat for easy plotting
    augment_predict <-
      as_tibble(t(augment_predict[, -c(1,2)]), .name_repair = "minimal")

    colnames(augment_predict) <- unique(tidy_docs$document)

    augment_predict$topic <- 1:nrow(augment_predict) |> as.factor()

    compare_mat <-
      augment_predict |>
      select(
        topic,
        augment = matches(rownames(d2)[1])
      ) |>
      mutate(
        augment = augment / sum(augment), # normalize to sum to 1
        dot = p_dot[1, ]
      ) |>
      pivot_longer(cols = c(augment, dot))

    ggplot(compare_mat) +
      geom_bar(aes(y = value, x = topic, group = name, fill = name),
               stat = "identity", position = "dodge") +
      labs(title = "Prediction using 'augment' vs 'predict(..., method = \"dot\")'")

    # Not shown: aggregating over documents results in recovering the "tidy" lambda.

    ### updating the model ----
    # now that you have new documents, maybe you want to fold them into the model?
    lda2 <- refit(
      object = lda,
      new_data = d, # save me the trouble of manually-combining these by just using d
      iterations = 200,
      burnin = 175,
      calc_likelihood = TRUE,
      calc_r2 = TRUE
    )

    # we can do similar analyses
    # did the model converge?
    qplot(x = iteration, y = log_likelihood, data = lda2$log_likelihood, geom = "line") +
      ggtitle("Checking model convergence")

    # look at the model overall
    glance(lda2)

    ## # A tibble: 1 × 5
    ##   num_topics num_documents num_tokens iterations burnin
    ##        <int>         <int>      <int>      <dbl>  <dbl>
    ## 1         10            99       2962        200    175

    print(lda2)

    ## A Latent Dirichlet Allocation Model of  10 topics,  99  documents, and  2962  tokens:
    ## refit.tidylda(object = lda, new_data = d, iterations = 200, burnin = 175,
    ##     calc_likelihood = TRUE, calc_r2 = TRUE)
    ##
    ## The model's R-squared is  0.1389
    ## The  5  most prevalent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     5       14.5    0.107  research, program, cancer, health, disparities, ...
    ## 2     3       12.6    0.141  cell, cells, lung, sleep, specific, ...
    ## 3     1       11.9    0.0616 effects, muscle, wall, v4, signaling, ...
    ## 4    10       10.4    0.0499 risk, brain, factors, sud, plasticity, ...
    ## 5     2       10.2    0.0305 research, center, microbiome, core, hiv, ...
    ## # ℹ 5 more rows
    ##
    ## The  5  most coherent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     8       7.34     0.326 cmybp, function, mitochondrial, injury, fragment, …
    ## 2     9       7.55     0.187 core, data, ppg, studies, imaging, ...
    ## 3     7       9.9      0.159 cancer, tumor, clinical, imaging, cells, ...
    ## 4     3      12.6      0.141 cell, cells, lung, sleep, specific, ...
    ## 5     5      14.5      0.107 research, program, cancer, health, disparities, ...
    ## # ℹ 5 more rows

    # how does that compare to the old model?
    print(lda)

    ## A Latent Dirichlet Allocation Model of  10 topics,  50  documents, and  1524  tokens:
    ## tidylda(data = d1, k = 10, iterations = 200, burnin = 175, alpha = 0.1,
    ##     eta = 0.05, optimize_alpha = FALSE, calc_likelihood = TRUE,
    ##     calc_r2 = TRUE, return_data = FALSE)
    ##
    ## The model's R-squared is  0.2503
    ## The  5  most prevalent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     4       12.5    0.0527 cdk5, cns, develop, based, lsds, ...
    ## 2     3       11.5    0.170  cells, cell, sleep, specific, memory, ...
    ## 3     1       11.4    0.114  effects, v4, signaling, stiffening, wall, ...
    ## 4     6       10.9    0.348  diabetes, numeracy, redox, extinction, health, ...
    ## 5     8       10.7    0.337  cmybp, function, mitochondrial, injury, fragment, …
    ## # ℹ 5 more rows
    ##
    ## The  5  most coherent topics are:
    ## # A tibble: 10 × 4
    ##   topic prevalence coherence top_terms
    ##   <dbl>      <dbl>     <dbl> <chr>
    ## 1     6      10.9      0.348 diabetes, numeracy, redox, extinction, health, ...
    ## 2     8      10.7      0.337 cmybp, function, mitochondrial, injury, fragment, …
    ## 3     7      10.3      0.210 cancer, imaging, cells, rb, tumor, ...
    ## 4     5       9.13     0.206 program, dcis, cancer, research, disparities, ...
    ## 5    10       8.53     0.19  sud, plasticity, risk, factors, brain, ...
    ## # ℹ 5 more rows

I plan to have more analyses and a fuller accounting of the options of the various functions when I write the vignettes.

See the “Issues” tab on GitHub to see planned features as well as bug fixes.

If you have any suggestions for additional functionality, changes to functionality, changes to arguments or other aspects of the API please let me know by opening an issue on GitHub or sending me an email: jones.thos.w at gmail.com.

1. Users can calculate \(R^2\) after a model is fit by using the mvrsquared package or calling `tidylda:::calc_lda_rsquared`. `calc_lda_rsquared` is an internal function to tidylda, requiring the package name followed by three colons, as is R’s standard.↩︎