# Introduction to `chomper` package

We will demonstrate the use of the `chomper` package. This package facilitates the integration of multiple datasets through probabilistic entity resolution (record linkage, deduplication). In this vignette, we detail the process of merging the Italian Survey on Household Income and Wealth (ISHIW) data using `chomper` package, thereby illustrating various features of the package. The dataset is available for download from the Bank of Italy and is surveyed every two years. Furthermore, data collected in 2020 and 2022 have been preprocessed for the record linkage task and included in this package, made available in the `italy` dataset. Linkage structure estimation using `chomper` package proceeds in the following order.

1. Data loading and exploration
2. Hyperparameter specification
3. Posterior estimation
4. Linkage structure estimation and performance evaluation

# Example of Linkage Structure Estimation

## Step 1. Data Loading and Exploration

This vignette uses the `italy` dataset included in the package. The data consist of responses from a total of 28,000 participants surveyed by the Bank of Italy in 2020 and 2022. The original dataset contains over 300 responses, but the `chomper` package provides preprocessed data with 9 variables, including a unique identifier. The `italy` dataset is a list of two matrices, each containing 9 variables and 15,198 and 23,057 records, respectively. The data are preprocessed and stored so that categorical variables have integer values, and continuous variables are standardized to have a mean of 0 and a variance of 1. We use 5 categorical variables (SEX, ANASC, CIT, NASCREG, STUDIO) and 2 continuous variables (NETINCOME, VALABIT) for the example. We estimate the linkage structure of 952 records from participants in region 7 because the entire dataset is too large to fit at once.

    library(chomper)

    data(italy)

    # Select variables for the estimation
    #   1. Multinomial
    #     SEX: sex
    #     ANASC: year of birth
    #     CIT: whether Italian or not
    #     NASCREG: place of birth
    #     STUDIO: education level
    #   2. Gaussian (standardized)
    #     NETINCOME: net annual income
    #     VALABIT: estimated price of residence
    discrete_colnames <- c("SEX", "ANASC", "CIT", "NASCREG", "STUDIO")
    continuous_colnames <- c("NETINCOME", "VALABIT")
    common_fields <- c(discrete_colnames, continuous_colnames)

    # Set the types of variables.
    # If one type is missing, please set it to 0
    discrete_fields <- 1:5
    n_discrete_fields <- 5
    continuous_fields <- 6:7 # If no continuous fields are used, replace them with 0,
    n_continuous_fields <- 2 # and replace this argument with 0

    k <- length(italy) # number of files, 2 in this case
    n <- numeric(k) # number of records in each file

    M <- numeric(n_discrete_fields) # number of levels in each categorical variable
    for (l in seq_along(discrete_colnames)) {
      for (i in 1:k) {
        if (i == 1) {
          categories <- italy[[i]][, discrete_colnames[l]]
        } else {
          categories <- c(categories, italy[[i]][, discrete_colnames[l]])
        }
      }
      M[l] <- length(unique(categories))
    }

    dat <- list()
    for (i in 1:k) {
      # Select records from the prespecified region;
      #   IREG: current living region of a respondent
      dat[[i]] <-
        italy[[i]][italy[[i]][, "IREG"] == 7, ]
      n[i] <- nrow(dat[[i]])
    }
    N <- sum(n) # total number of records
    N_max <- N # total number of records, which is used for setting a prior

    p <- length(common_fields) # number of variables

    # Define the data to link and confirm that each data has a form of matrix
    x <- list()
    for (i in 1:k) {
      x[[i]] <- as.matrix(dat[[i]][, common_fields])
    }

## Step 2. Hyperparameter Specification

Before running the model, we must specify the necessary hyperparameters. Regardless of the inferential method, we specify the information required for the distribution of each variable and the distortion rate, \(\beta_{l}\). We start by defining the likelihood. Note that we use 5 categorical variables that follow multinomial distributions and 2 Gaussian variables. The model assumes the following likelihood: \[ x_{ijl}\mid\lambda_{ij},y_{\lambda_{ij}l},z_{ijl}\stackrel{\text{ind}}{\sim} \begin{cases} \text{MN}\left(1,\boldsymbol{\theta}_{l}\right)^{z_{ijl}}\text{MN}\left(1,\tilde{\phi}_{l}\left(x_{ijl},y_{\lambda_{ij}l},\tau_{l},\varepsilon_{l}\right)\right)^{1-z_{ijl}} & \text{for } l=1,\cdots,l_{1} \\ \text{N}\left(\eta_{l},\sigma_{l}\right)^{z_{ijl}}\text{N}\left(y_{\lambda_{ij}l},\varepsilon_{l}\right)^{1-z_{ijl}} & \text{for } l=l_{1}+1,\cdots,p \end{cases} \] where \(z_{ijl}\) is a distortion indicator and \(y_{\lambda_{ij}l}\) is a latent truth. In order to adopt locally-varying hits, we assume \[ \tilde{\phi}_{l}\left(x_{ijl},y_{\lambda_{ij}l},\tau_{l},\varepsilon_{l}\right)=\frac{\phi_{l}^{\frac{1}{\tau_{l}}\mathbb{I}\left(\lvert x_{ijl}-y_{\lambda_{ij}l}\rvert\leq\varepsilon_{l}\right)}}{\sum_{m=1}^{M_{l}}\phi_{l}^{\frac{1}{\tau_{l}}\mathbb{I}\left(\lvert x_{ijl}-m\rvert\leq\varepsilon_{l}\right)}} \] for \(\varepsilon_{l}\geq0\), \(\tau_{l}>0\), and \(\phi_{l}>1\). Thus, we have to specify \(\phi_{l}\) and \(\tau_{l}\) for \(l=1,\cdots,5\) and \(\varepsilon_{l}\) for \(l=1,\cdots,7\). We set \(\phi_{l}=2.0\) and \(\tau_{l}=0.01\). For variables with a single truth, we use \(\varepsilon_{1}=\cdots=\varepsilon_{4}=0\). However, as we assume that educational level, net income, and estimated price of residence have multiple truths, we use \(\varepsilon_{5}=1\), and \(\varepsilon_{6}=\varepsilon_{7}=0.1\).

    # hyperparameter for categorical fields (Multinomial distribution)
    hyper_phi <- rep(2.0, n_discrete_fields)
    hyper_tau <- rep(0.01, n_discrete_fields)

    hyper_epsilon_discrete <-
      c(rep(0, n_discrete_fields - 1), 1) # hitting range for categorical fields.
    # As education level is assumed to have multiple truths,
    # we set epsilon of that field as 1.
    hyper_epsilon_continuous <-
      rep(0.1, n_continuous_fields) # hitting range for continuous fields.
    # Continuous fields are assumed to have multiple truths, we set epsilon as 0.1.

`chomper` uses a non-informative Dirichlet prior for the parameter vector, \(\boldsymbol{\theta}_{l}\), of the multinomial distribution, which is defined internally within the package. The mean of the Gaussian distribution, \(\eta_{l}\), uses a diffuse prior, also defined internally within the package. However, the variance \(\sigma_{l}\) follows an inverse-gamma prior, which provides an option that can be specified by the user. Therefore, we must specify the hyperparameters of the inverse-gamma prior. In this example, both the shape and scale parameters are set to 0.01. \[ \sigma_{l}\sim\text{Inverse-Gamma}\left(0.01,0.01\right), \] where \(l\) is the index of the continuous variable. We finish specifying the hyperparameters by assuming that the average distortion rate for all variables is 1%: \[ \beta_{l}\sim\text{Beta}\left(N_{\max}\times0.01\times0.01,N_{\max}\times0.01\right), \] where \(N_{\max}=952\) for region 7.

    hyper_sigma <- matrix(
      rep(c(0.01, 0.01), n_continuous_fields),
      ncol = 2, byrow = TRUE
    ) # prior distribution for the variance of Gaussian, setting non-informative

    hyper_beta <- matrix(
      rep(c(N_max * 0.1 * 0.01, N_max * 0.1), p),
      ncol = 2, byrow = TRUE
    ) # prior distribution for distortion rate

## Step 3. Posterior Estimation

Run the model using the data loaded in Step 1 and the hyperparameters specified in Step 2. The `chomper` package provides inference using MCMC and VI, but this example demonstrates the use of `chomperMCMC`, with instructions for using `chomperEVIL` provided as comments in the code chunks. To generate posterior samples via MCMC, we must additionally define the number of samples to be used for inference. Furthermore, since `chomperMCMC` performs a split and merge steps for each Gibbs iteration, we must also define the number of split and merge steps per Gibbs iteration. We generate a total of 5,000 posterior samples, discard the first 4,000 as burn-in samples, and use the last 1,000 samples for inference. The number of split and merge procedures per Gibbs iteration is set to 1,000. The `chomper` package allows the user to set the maximum run time (in seconds), with the default being 24 hours.

    res <- chomperMCMC(
      x = x, k = k, n = n, N = N, p = p, M = M,
      discrete_fields = discrete_fields,
      continuous_fields = continuous_fields,
      hyper_beta = hyper_beta,
      hyper_phi = hyper_phi,
      hyper_tau = hyper_tau,
      hyper_epsilon_discrete = hyper_epsilon_discrete,
      hyper_epsilon_continuous = hyper_epsilon_continuous,
      hyper_sigma = hyper_sigma,
      n_burnin = 4000, # number of burn-in samples
      n_gibbs = 1000, # number of MCMC samples to store
      n_split_merge = 1000, # number of split and merge procedures in each Gibbs iteration
      max_time = 86400 # maximum time in second for MCMC
    )

    # Example code for using `chomperEVIL`
    # res <- chomperEVIL(
    #     x = x, k = k, n = n, N = N, p = p, M = M,
    #     discrete_fields = discrete_fields,
    #     continuous_fields = continuous_fields,
    #     hyper_beta = hyper_beta,
    #     hyper_phi = hyper_phi,
    #     hyper_tau = hyper_tau,
    #     hyper_delta = hyper_delta,
    #     hyper_sigma = hyper_sigma,
    #     overlap_prob = 0.5, # probability for the initialization.
    #     # This probability of records will be initially considered as linked.
    #     n_parents = 50, # number of parents, N_P
    #     n_children = 100, # number of offspring, N_O
    #     tol_cavi = 1e-5, # stopping criterion for a single CAVI
    #     max_iter_cavi = 10, # maximum number of CAVI update for a single member
    #     tol_evi = 1e-5, # stopping criterion for an evolution
    #     max_iter_evi = 50, # maximum number of evolution, N_E
    #     verbose = 1,
    #     n_threads = 7 # number of cores to be parallelized
    # )

`chomperMCMC` returns a list of posterior samples of the linkage structure and other parameters, and `chomperEVIL` returns a list of approximated parameters of the variational factors and other information. Among posterior samples, we will use \(\boldsymbol{\Lambda}\) to estimate the linkage structure, which can be accessed as `res$lambda`. If you run the model using `chomperEVIL`, you can access the approximated parameters of the variational factors corresponding to \(\boldsymbol{\Lambda}\) as `res$nu`.

## Step 4. Linkage Structure Estimation and Performance Evaluation

Using \(\boldsymbol{\Lambda}\) sampled via `chomperMCMC`, we can estimate the linkage structure and evaluate estimation performance using the unique identifiers in the `italy` dataset. While we can compute various point estimates using posterior samples, we use a Bayes estimate that minimizes Binder’s loss. Since the `chomper` package does not provide a function to compute the Bayes estimate, we use the `salso` function from the `salso` package to compute a Bayes estimate. First, we use `ID`, the unique identifier of the `italy` dataset, and the `links` function from the `blink` package to find the true linkage structure.

    library(blink)
    library(salso)

    for (i in 1:k) { # `k` represents the number of files
      if (i == 1) {
        key_true <- dat[[i]][, "ID"]
      } else {
        key_true <- c(key_true, dat[[i]][, "ID"])
      }
    }

    # Find the true linkage structure.
    # An external R package, `blink`, is used, especially, `blink::links`
    truth_binded <- matrix(key_true, nrow = 1)
    linkage_structure_true <- links(truth_binded, TRUE, TRUE)
    linkage_truth <- matrix(linkage_structure_true)

    head(linkage_truth)
    #>      [,1]
    #> [1,] 1
    #> [2,] 2
    #> [3,] 3
    #> [4,] 4
    #> [5,] numeric,2
    #> [6,] numeric,2

The `salso` function takes a posterior similarity matrix as an argument and calculates the Bayes estimate that minimizes Binder’s loss. Therefore, the process of calculating the posterior similarity matrix using `res$lambda` must precede this step. Since `chomperMCMC` returns posterior samples as a list, use the `flatten_posterior_samples` function provided by the `chomper` package to convert the posterior samples into a matrix format. Next, we compute the posterior similarity matrix using the `psm_mcmc` function. If `chomperEVIL` is used, the approximate posterior variational factors, `res$nu`, can be used directly as an argument to `psm_vi` to compute the posterior similarity matrix.

    # In order to use MCMC samples:
    #   1. slice the actual MCMC samples
    #   2. reshape a list into a matrix form
    lambda_binded <-
      flatten_posterior_samples(
        res$lambda, k, N
      )

    psm_lambda <- psm_mcmc(lambda_binded)

    # # Approximate posterior distribution is used directly
    # # to calculate a posterior similarity matrix
    # psm_lambda <- psm_vi(result$nu)

    # Calculate a Bayes estimate that minimizes Binder's loss
    salso_estimate <- salso(round(psm_lambda, nchar(N) + 1), # rounding is implemented to avoid floating-point error
      loss = binder(),
      maxZealousAttempts = 0, probSequentialAllocation = 1
    )

    # Convert a Bayes estimate to have an appropriate structure to evaluate the performance
    linkage_structure <- list()
    for (ll in seq_along(salso_estimate)) {
      linkage_structure[[ll]] <- which(salso_estimate == salso_estimate[ll])
    }
    linkage_estimation <- matrix(linkage_structure)

The package provides a function, `performance`, that evaluates performance given an estimate and truth. This function returns several performance metrics, including true positives, true negatives, false positives, false negatives, false positive rate, and false negative rate, in a list format. Using the `return_matrix=TRUE` option, the function also returns the linkage estimation results for each record pair in a matrix format.

    # Calculate the performance of the estimation
    perf <- performance(linkage_estimation, linkage_truth, N, FALSE)

    # Print the performance
    performance_matrix <-
      c(perf$fnr, perf$fpr, perf$tp, perf$fp, perf$tn, perf$fn)
    performance_matrix <-
      matrix(
        c(2 * performance_matrix[3] /
          (2 * performance_matrix[3] +
            performance_matrix[4] +
            performance_matrix[6]), performance_matrix),
        nrow = 1
      )
    colnames(performance_matrix) <- c("F1", "FNR", "FPR", "TP", "FP", "TN", "FN")
    rownames(performance_matrix) <- c("Region 7")
    print(performance_matrix)
    #>                 F1      FNR      FPR  TP FP  TN  FN
    #> Region 7 0.4471831 0.653951 0.182716 127 74 331 240

In order to diagnose convergence of MCMC, we can use a trace plot of number of unique \(\lambda_{ij}\). As only 1,000 MCMC samples were used for inference before, we re-run `chomperMCMC` without burn-in.

    res <- chomperMCMC(
      x = x, k = k, n = n, N = N, p = p, M = M,
      discrete_fields = discrete_fields,
      continuous_fields = continuous_fields,
      hyper_beta = hyper_beta,
      hyper_phi = hyper_phi,
      hyper_tau = hyper_tau,
      hyper_epsilon_discrete = hyper_epsilon_discrete,
      hyper_epsilon_continuous = hyper_epsilon_continuous,
      hyper_sigma = hyper_sigma,
      n_burnin = 0, # number of burn-in samples
      n_gibbs = 5000, # number of MCMC samples to store
      n_split_merge = 1000, # number of split and merge procedures in each Gibbs iteration
      max_time = 86400 # maximum time in second for MCMC
    )

The number of burn-in samples is determined by the trace plot of number of unique \(\lambda_{ij}\)’s in the lower left. Using the posterior samples, the number of unique entities in the data can also be estimated and visualized, as in the density plot in the lower right.

    library(ggplot2)
    library(patchwork)

    unique_entities <- numeric(5000)
    for (i in 1:5000) {
      unique_entities[i] <-
        length(unique(c(res$lambda[[i]][[1]], res$lambda[[i]][[2]])))
    }

    df_trace <- data.frame(idx = 1:5000, unique_entities = unique_entities)

    plot_trace <- ggplot(df_trace, aes(x = idx)) +
      geom_line(aes(y = unique_entities)) +
      labs(title = "Traceplot of MCMC Samples", x = NULL, y = NULL) +
      theme_bw() +
      theme(
        plot.title = element_text(hjust = 0.5),
        legend.position = "none",
        panel.border = element_blank(),
        axis.line.x = element_line(linewidth = .2),
        axis.line.y = element_line(linewidth = .2)
      )

    true_unique_entities <- length(unique(key_true))

    df_density <- data.frame(unique_entities = unique_entities[4001:5000])

    plot_density <- ggplot(df_density) +
      geom_density(aes(x = unique_entities, colour = "Estimate"),
        adjust = 2, key_glyph = "path"
      ) +
      geom_vline(aes(xintercept = length(unique(key_true)), colour = "Truth")) +
      scale_colour_manual(values = c("Estimate" = "black", "Truth" = "red")) +
      xlim(500, 1000) +
      labs(title = "Estimated Number of Unique Entities", x = NULL, y = NULL) +
      theme_bw() +
      theme(
        plot.title = element_text(hjust = 0.5),
        legend.position = "inside",
        legend.position.inside = c(0.85, 0.9),
        legend.title = element_blank(),
        legend.background = element_rect(fill = "transparent", color = NA),
        panel.border = element_blank(),
        axis.line.x = element_line(linewidth = .2),
        axis.line.y = element_line(linewidth = .2)
      )

    plot_trace + plot_density + plot_layout(ncol = 2)