Approach

The methods underpinning serosolver are motivated by the following base assumptions: (i) antibody titres may be measured using serum samples taken at some point in time; (ii) these antibody titres are an incomplete observation of true, underlying antibody levels that undergo a dynamical process following infection; (iii) these underlying antibody kinetics arise from the culmination of repeated exposures to antigenically related or identical pathogens. The aim is infer the combination of infections at different times or with different strains that are most consistent with observed antibody titres. In the main text, we describe how this system is implemented specifically for the subsequent case studies on influenza. S1 Text describes the framework in a more generalised form as a reference for future development of serosolver to other disease systems.

We frame the overall inference challenge as obtaining estimates for the joint posterior distribution of antibody kinetics parameters (Θ), individual infection histories (Z) for all n individuals in the sample, and the time-varying probability of infection in the population (Φ) across m possible discrete infection periods given an observed serological dataset (Y, which may include assay measurements against one or more strains). This distribution, P(Z, Φ, Θ|Y), is comprised of three components:

1. The combined observation and antibody kinetics models f(Y_i|Z_i, Θ), which give the likelihood of observing a set of titres Y_i for each individual i at discrete serum sampling times (t_i) given infection history Z_i and the antibody kinetics parameters Θ;
2. The infection history model P(Z_i,j|Φ_j), which gives the probability of individual i having been infected with strains circulating in each discrete time period j when infection might have occurred (between time j_min and j_max), conditional on the time-varying population infection probabilities Φ;
3. The prior probabilities of the antibody kinetics parameters, P(Θ), and the prior probability of any infection in each discrete time period j, P(Φ_j).

(1)

In each discrete time period, j, we assume that there is only one strain that circulates. Reference to j therefore refers to both the time period itself and the index of the strain that circulated during that time. Treating time as discrete differs to some previous approaches which model infection times as continuous variables [13, 35]. The time resolution of j can be set when running serosolver depending on the amount of data; using only one possible infection period (m = 1) is conceptually similar to an analysis of seropositivity, whereas choosing m to represent many small intervals of time (e.g. months) becomes conceptually similar to continuous time.

Antibody kinetics model

For a given individual infection history and set of biological parameters, the antibody kinetics model generates a set of expected log titres for that individual against all possible test strains. These antibody titres are observed at only a subset of times for which serum samples are available, and the model-predicted antibody titres across all times are therefore referred to as latent antibody titres. Although other functions for f(Y_i,t|Z_i, Θ) may be implemented and used with only minor modifications to the code, the model used here follows previous work [27]. The expected log titre individual i has against the strain that circulated during discrete time period j when observed at time t (X_i,j,t) is defined as a linear, deterministic combination of contributing antibody responses from each prior infection: (2)

The model components are defined by:

1. Long-term boosting defined by a parameter μ_l, giving the expected persistent rise in titre against a homologous strain following primary infection.
2. Short-term boosting. The transient component of the antibody kinetics model defined by μ_sw(t, k) = μ_s max{0, 1 − ω(t − k)}, where μ_s is the boost in homologous titre, ω is a waning parameter to be fitted, and t − k is the time since infection with strain k.
3. Cross-reactive antibody responses from related strains. We assume the level of cross-reaction between a test strain j and infecting strain k ∈ Z_i decreases linearly with antigenic distance (defined below) [24]. The cross-reaction functions are d_l(j, k) = max{0, 1 − σ_l δ_j,k} and d_s(j, k) = max{0, 1 − σ_s δ_j,k} for the long-term and the short-term boosts respectively. δ_j,k is the antigenic distance between strains j and k, and σ_l and σ_s are fitted parameters.
4. Antigenic seniority from boosting suppression. This results in lower titre boosts from later compared to earlier infections. In the model, the contribution of an exposure is scaled by s(Z_i, k) = max{0, 1 − τ(N_k − 1)}, where N_k is the infection number (i.e. primary infection is 1, secondary is 2) and τ is a fitted parameter.

In the model, the antigenic distance δ_j,k between strains j and k is defined by a matrix of pairwise distances in an antigenic map. δ_j,k = 1 if serum from a naive ferret infected with strain k gives a titre against strain j one log unit lower than against the homologous strain k [24]. The serosolver model can accommodate antigenically varying strains (all δ_j,k are specified) or a single homologous strain (all δ_j,k = 0). The extent to which strains are antigenically distinct or similar can be specified by the distance matrix.

The antibody kinetics model can be reduced to simpler models by setting certain parameter values equal to 0. These nested sub-models allow hypothesis testing to distinguish between immunological mechanisms. For example, a model without antigenic seniority can be created by setting τ = 0 or a model with only waning responses by setting μ_l = 0. In addition, serosolver can be extended to include more complex antibody kinetics (e.g. boosting that scales as a function of pre-existing titre), as described in S2 Text. We note that the additional immunological phenomena described in S2 Text are not exhaustive, and additional mechanisms may be easily implemented by making minor modifications to the package code.

Observation model

The expected titre X_i,j,t defined in Eq 2 feeds into the observation model. For HI and NT titres, this necessitates conversion of continuous latent titres into discrete observations. The distribution of the observed titre consists of a normally distributed random variable g(s) with mean X_i,j,t and variance ε, which is then censored to account for integer-valued log titres in the assay. Hence the probability of observing an empirical titre at time t within the limits of a particular assay Y_i,j,t ∈ {0, …, Y_max} given expected titre X_i,j,t for individual i measured against strain j at serum sampling time t is: (3) There are additional options in serosolver to include strain-specific measurement bias, which may arise through strain-specific differences in assay reactivity [26, 38, 39]. Specifically, an additional observation error is added to the predicted log antibody titres; this measurement error can be different for each individual strain or can be specified for a group (or cluster) of strains. The predicted titre taking into account strain-specific measurement bias is: (4) where P_j is the measurement offset for strain j. P_j may be estimated as independent parameters for each j, or may be assumed to come from a hierarchical distribution . P_j may also be fixed for particular strains/groups e.g. fixing or .

Infection history model

Each individual’s infection history is tracked by serosolver as a vector of binary latent states indicating the presence (1) or absence (0) of infection within each unit of discrete time. Z_i = [Z_i,1, Z_i,2, …, Z_i,j≤t] represents a vector of infection states that could have occurred at or before the time a serum sample is taken during interval time t. The set of infection histories for the sample population is therefore described by a binary matrix, Z = [Z₁, Z₂, …, Z_n]. Each row of the matrix represents an individual, i, and each column represents a time, j, at which an individual could be infected once. The likelihood for the infection history model, P(Z|Φ)P(Φ), is given by: (5) where each infection event, Z_i,j, is the outcome of a single Bernoulli trial with probability . A value of z_i,j = 1 indicates that individual i was infected in discrete time period j and z_i,j = 0 indicates that they were not. The choice of the prior distribution for the probability of infection, P(Φ_j), is discussed below and in further detail in S1 Text. The time resolution of infection times may be set by the user depending on the data: frequent serum sampling times affords greater time resolutions (e.g. months), whereas less frequent sampling may be better suited to cruder time resolutions (e.g. years).

The infection history posterior can be used to calculate the population attack rate over time. Attack rates can be inferred through combining estimated infection histories post-hoc to estimate the proportion of at risk individuals that were infected in a given time period. Summing the columns of the infection history matrix gives the total number of infections for a given time period, whereas summing the rows give the total number of lifetime infections for an individual. To ensure biological plausibility, individual infection histories are constrained to prevent infections before an individual is born and after the last time at which a serum sample was taken. A key feature of the package is that the user is given control over the prior assumptions for the infection history and the probability of infection during each time period (months, years etc).

User inputs

Data.

The serosolver package requires a dataset of individual-level log titres. Each individual may have repeat titre measurements from one or more serum sampling times, t (i.e. when each serum sample was collected), against viruses that may have circulated at each discrete time period j (i.e. when the strain was originally isolated).

The initial development of serosolver focused on influenza A/H3N2, which has circulated in human populations since 1968 and has undergone substantial antigenic evolution over this time [24, 39–41]. Fig 1 illustrates how our analytical approach applies to influenza A/H3N2. In serosolver, a key but non-essential input to the framework is therefore an antigenic map: a data structure describing the two-dimensional location of viruses in antigenic space that circulated at each time point during the period of interest (specifically, an x and y coordinate for the strain that circulated in each time period j). This map is used to calculate the pairwise antigenic distance between any two viruses (i.e. δ_j,k in the antibody kinetics model, for strains j and k). Not all disease systems or epidemiological problems require an antigenic map. For example, the map may be considered equivalent to a single point when the pathogen of interest shows no antigenic variability over time. A model with only one circulating antigenic variant may therefore be specified by using a null input for the antigenic map argument.

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Fig 1. Conceptual overview of the analytical approach used in serosolver, as applied to influenza A/H3N2.

Top panel: antigenic map for influenza A/H3N2 using coordinates from [24], with different viruses coloured by year of isolation. Solid points show centroids across all strains isolated in a given calendar year, hollow points show individual strains. Dashed line shows an antigenic summary path, generated by fitting a smoothing spline through the observed isolates. Points further apart in space are less cross-reactive. Middle panel: conceptual illustration of the antibody kinetics model. An individual is infected with the orange virus, which results in boosting and waning of homologous antibody titres. In parallel, antibodies that cross react with viruses at different points in antigenic space also boost and wane (purple and blue viruses). The individual is later infected by the purple virus, which leads to further boosting and waning of antibodies. Bottom panel: HI titres measured from serum samples taken at different times capture different parts of the homologous and cross reactive antibody kinetics. Different sampling strategies will represent different subsets of these measurements e.g. a cross-sectional study might inform a single subplot, whereas a longitudinal study might inform just the orange bars from each of the three subplots. Clearly a sampling strategy with multiple serum samples and many viruses tested per sample will provide the most information.

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Implementation

In serosolver, model inputs and assumptions may be changed depending on the serological data and hypotheses under consideration. For example, in some cases the user may be most interested in short-term, fine-scale (e.g. weekly or monthly) dynamics of infection; in other situations, long-term annual dynamics may be of interest. Furthermore, although much of the development of this package came from analysis of influenza A/H3N2 dynamics, these concepts and inputs are easily adaptable to antigenically stable pathogens by specifying a null antigenic map.

The package work flow is divided into a number of distinct stages, which handle the data and parameter inputs, simulation, inference, posterior diagnostics, and analysis (Fig 2). We developed the package to rely on only a few function calls for each of these stages, but with ample room for customisation and flexibility at each stage.

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Fig 2. Inputs and outputs for the serosolver R package.

Users input the serological data to be fitted, an antigenic map if considering an antigenically variable pathogen, the infection history prior and any priors on the antibody kinetics parameters. These inputs feed into the process model that can either be used to simulate data by itself, or combined with observed data and MCMC to obtain three posterior outputs: individual-level infection histories, population probabilities of infection, and antibody kinetics parameters. Once these posteriors have been obtained, serosolver can run MCMC diagnostics and plot key immunological and epidemiological processes.

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To set up the model, users only need to provide: a data frame describing the model parameters (they can also change a flag to fix or estimate any of the parameters); a data frame with the antibody titre data in long format; and (optionally) an antigenic map describing the antigenic relationship between each strain. A null argument may be specified for the antigenic map when modelling only a single circulating antigenic variant. Examples of a typical data cleaning workflow are provided in S4 Text.

Users may create their own likelihood and prior functions on top of those provided by default, requiring only that they return a vector of likelihoods (one per individual), and accept arguments for a vector of parameters (matching those defined in the general serosolver model) and the infection history matrix. Users can specify which prior assumption about infection histories is used, as specified above. In addition to the range of inbuilt options, the modular workflow of serosolver means that custom extensions tailored to particular problems should be readily achievable with only minor modifications to the code. In particular, alternative antibody kinetics models that capture pathogen-specific immunology and alternative assumptions about the infection history generating process.

Because of the complex nature of these Markov spaces, it is usually important to run multiple chains when using the Metropolis-Hastings MCMC algorithms implemented by serosolver. Multiple chains from different starting locations in the space help to detect convergence issues arising, for example, from multi-modal posterior distributions. Furthermore, model comparison and sensitivity analyses are a common output of model fitting analyses. It is simple to use serosolver with a parallel back-end, either through a computing cluster or locally with packages such as doParallel [46]. The accompanying vignettes (S3 and S4 Text) demonstrate how multiple chains may be run in parallel locally, but we note that much of our own work with serosolver is done using a computing cluster.

Infection history priors

Fig 3 compares infection history metrics simulated from models under the different infection history priors compared to the analytical probability mass functions for the total number of infections per unit time j and per lifetime of individual i. These results highlight the contribution of different prior assumptions on inferred attack rates, total number of lifetime infections and accumulation of an infection history with age. The ability of each of the different priors to recover known antibody kinetics parameter values, infection histories and attack rates is demonstrated in S1 Text using simulation-recovery experiments with different sero-survey designs under various infection history prior assumptions. Datasets may be rich in different dimensions (e.g. number of individuals vs. number of viruses tested), which leads to different inferential power for different quantities. S1 Table summarises each of these priors and the situations when each one is advised.

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Fig 3. Simulated vs. analytical infection history prior metrics (α = β = 1).

Bars show density histograms of infections from 10,000 simulated infection histories for 100 individuals across 42 infection periods. Red lines show known probability mass function. Plots A, D and G show the prior on the total number of infections per discrete time period j. Plots B, E and H show prior on the total number of lifetime infections per individual. Plots C, F and I show the prior on the cumulative number of infections across 42 time periods for one individual. Black line shows prior median, dark gray region shows 50% credible intervals and light gray region shows 95% credible intervals. Note that priors 1 and 2 are equivalent under these assumptions.

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The different shapes and widths of the distributions in Fig 3 demonstrates how different prior forms may lead to strong assumptions about an individual’s infection history. Here, a beta prior was used with α = β = 1 in all of these examples (equivalent to a uniform distribution), though priors of different shapes may be set by specifying different values for α and β as shown in S1 Fig. Under prior versions 1 and 2, this assumption leads to a highly constrained binomial prior on the total number of lifetime infections with mean mα/(α + β) (Fig 3A), but a uniform prior on the attack rate in each time period j (Fig 3B). Conversely, under prior version 3, the prior on the total number of lifetime infections follows the beta-binomial distribution, which is uniform under these values, whereas the attack rate prior follows the binomial distribution with mean nα/(α + β) (Fig 3D&3E). Both the total number of infections per unit time and total number of lifetime infections are uniformly distributed under prior version 4 (Fig 3G&3H).

Computational performance

The serosolver code uses a C++ back-end with substantial optimisation to scale the model to large datasets and high infection time resolutions with reasonable run times. Table 1 displays the mean run time of 5 MCMC chains fitting the serosolver model to serological data of different dimensions. In the most complex scenario, which involves fitting the model to 164,000 antibody titre measurements and inferring the infection state of 1000 individuals at 164 different time points (164,000 infection states), effective sample sizes >200 are achievable for both the antibody kinetics parameters and attack rate estimates in <12 hours. For smaller scale analysis (e.g. 100 individuals, <5000 titres), high effective sample sizes and well-mixed chains are easily generated within 30 minutes.

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Table 1. Comparison of run time and posterior sampling efficiency across a range of serosurvey designs.

https://doi.org/10.1371/journal.pcbi.1007840.t001

Case study results

We present two case studies to highlight the range of insights that serosolver can generate from serological samples. These cover two types of study designs commonly used to examine epidemiological and immunological dynamics using serological data, which can be thought of as subsets of the observations shown in Fig 1, bottom panel. The first is a serological survey testing individuals against a single homologous strain, which can reveal short-term epidemic dynamics, analogous to observing each of the bars of a single colour from Fig 1. We use data from a longitudinal study conducted in Hong Kong between 2009 and 2011 to estimate short-term antibody kinetics parameters against A/H1N1pdm09 in a population with limited prior immunity [47]. The second type of study design involves testing samples against a panel of previously circulating strains, which can provide insights into historical patterns of infection, analogous to observing all of the bars within a single serum sample from Fig 1 [37, 48]. To illustrate this application, we apply the package to cross-sectional samples tested against a panel of historical A/H3N2 influenza strains to infer infection histories and antibody kinetics [37, 48].

Case study 1.

The first case study uses data from a cohort study in Hong Kong during and after the 2009 A/H1N1pdm09 outbreak [47]. With repeat serological samples tested against a given virus, serosolver can reconstruct the unobserved infection dynamics from measured titres collected several months apart. It is also possible to examine these infection dynamics stratified by available demographic variables, such as vaccination status (Fig 4A) and age (Fig 4B). Fig 4C shows example model fits to the observed antibody titres. Finally, we can estimate biological parameters shaping the short-term antibody response (Fig 4D).

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Fig 4. Influenza A/H1N1pdm09 infection dynamics in Hong Kong cohort.

A: Exposure rates in unvaccinated and vaccinated individuals. Shaded regions show 80% (dark) and 95% (light) credible intervals (CI). Solid lines shows posterior medians. X-axis gives midpoint for that quarter. B: Age-specific exposure rates in unvaccinated individuals. Solid lines show median estimates for each age group (pink: <19 (n = 30), green: 19-64 (n = 264), blue: >64 (n = 17)) with 80% (dark) and 95% (light) CI shaded. C: Model predicted titres and inferred infections compared to observed titres for 4 representative individuals with inferred infections. Purple diamonds show observed titres; black dashed lines indicate posterior median model predicted titres; green shading shows 95% CI on model predicted latent titres (dark) and assay observations (light); orange shading indicates posterior probability of infection. Grey region shows titres outside the limit of detection. X-axis gives midpoint for that quarter. D: Posterior densities of antibody kinetics parameters and total number of infections (∑Z_i). Vertical lines represent 2.5th, 50th, and 97.5th percentiles.

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We estimated quarterly exposure rates, which could include either infection or vaccination. The inferred peaks in exposure rates are consistent with the observed two waves of the 2009 pandemic [47]. Constrained estimates of high incidence were obtained for Q4 2009. The following period of elevated incidence but with high uncertainty in early 2010 reflects the gap between the second and third round of serum sampling; further antibody boosts were detected in this period, but the exact quarter could not be determined.

We investigated the impact of vaccination status and age on inferred exposure rates, finding differences in exposure rates in vaccinated individuals (n = 113) compared to unvaccinated individuals (n = 307). Inferred boosting rates were almost identical between vaccinated and unvaccinated individuals up to and including Q4 2009. However, higher overall exposure rates in vaccinated individuals were inferred from Q1 2010 onward (Fig 4A). Additionally, we observed clear differences in age-stratified exposure rates of unvaccinated individuals (Fig 4B), with exposure rates highest among children (<19 years old, n = 30) and adults (19-64 years old, n = 264), and lowest among the elderly (>64 years old, n = 17), confirming previous findings of age-stratified exposure rates during the 2009 pandemic [49]. Some of the inferred infections in Q1 2009 may represent pre-existing baseline titres rather than infections with A/H1N1/pdm09.

The pandemic vaccine was available from December 2009, and this increase in inferred boosts from Q1 2010 may therefore capture pandemic vaccine-derived responses. Intuitively, we would expect infection rates to be lower in vaccinated individuals; however, the converse suggests that vaccination caused additional antibody boosting. It is unlikely that these detected boosts represent true infections, as the vaccinated cohort had a larger proportion of older individuals, who likely experienced lower infection incidence (Fig 4B). Vaccination around the first sampling round was predominantly with the older 2009/10 seasonal vaccination, which was unlikely to elicit a novel antibody response (rather than a memory-derived response) effective against the antigenically dissimilar pandemic virus.

We aimed to characterise the short-term immune response following infection by estimating long and short-term antibody kinetics parameters. We found that there is a strong short-term average boost (μ_s) of 2.59 (posterior median; 95% CI: 2.19-2.86) log titre units followed by a persistent long-term boost (μ_l) of 3.38 (posterior median; 95% CI: 3.27-3.52) log titre units. We estimated that every 3 months, 58.4% (posterior median; 95% CI: 48.5-72.6%) of the short-term boost is lost (ω) such that only the long-term boost remains after 0.428 years (posterior median; 95% CI: 0.344-0.515).

To assess whether data contain enough information to reliably estimate the infection histories and biological process parameters, serosolver can be used to run a simulation recovery study. For example, if data of the same structure as the A/H1N1pdm09 outbreak in Hong Kong are generated using plausible parameter values [27], it is possible to re-infer these parameters (Fig 5B) alongside the individual-level infection histories (Fig 5C) and overall probabilities of infection (Fig 5A). However, depending on the sampling frequency, number of tested strains and number of repeat measurements, there are varying levels of information to estimate these quantities. When antibody titre data is sparse, the priors placed on either the antibody parameters, infection histories or probability of infection parameters will have a greater effect on the estimation performance. We therefore recommend routine implementation of simulation recovery on new data to ensure that the most suitable model is being applied to the data available.

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Fig 5. Simulation-recovery of parameter and infection estimates using simulated single strain longitudinal data in same format as the Hong Kong dataset.

A: Model estimated attack rates vs. ‘true’ attack rates. Solid line shows estimated attack rate with 80% (dark) and 95% (light) credible intervals (CI); green line and points shows true attack rates. B: ‘True’ process parameters used for simulation compared to estimated posterior densities. Green vertical lines indicate true parameter values; vertical lines represent 2.5th, 50th, and 97.5th percentiles. C: Model predicted titres and inferred infections compared to observed titres and known infections. Green diamonds indicate observed titres; black dashed lines indicate posterior median model predicted titres; blue shading shows 95% CI on model predicted latent titres (dark) and assay observations (light); vertical lines indicate the timings of true infections; orange shading indicates posterior probability of infection.

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Case study 2.

The second case study considers cross-sectional serological samples collected in southern China in 2009, which were tested against nine historical influenza A/H3N2 strains that circulated between 1968 and 2008 [37, 48]. We demonstrate how serosolver can be used to reconstruct several features of the epidemiological and immunological dynamics in this cohort. First, Fig 6A shows substantial variation in the inferred historical attack rates of A/H3N2, with clear periods of high incidence interspersed by periods of very low incidence (range of posterior medians: 3.63% to 95.2%). Periods of high and low attack rates were similar to those in a previous analysis from a cohort in Ha Nam, Vietnam [27]. In these analysis, we used a weakly informative prior on the annual attack rate with a mode of 15% with prior version 2. Our posterior estimates were very similar to this, with a median inferred attack rate of 14.6%, suggesting either agreement between the data and prior or a lack of information in the data.

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Fig 6. Influenza A/H3N2 dynamics in southern China.

A: Inferred historical attack rates. Shaded regions show 80% and 95% credible intervals (CI), solid line and points shows posterior median estimate; B: Frequency of inferred antibody responses (sero-responses) by age group. Boxplots show distribution across individuals based on posterior median total number of infections per individual per 10 years alive. C: Model predicted titres and inferred infections compared to observed titres (black diamonds). Shaded regions show 95% CI on model predicted latent titres (dark) and assay observations (light).

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We also identified clear age-specific patterns of infection. Fig 6B shows the median number of infections per 10 years alive stratified by age at the time of exposure. These estimates agree with previous analyses that individuals are infected, or at least experience antibody boosting, less frequently as they get older [27]. Inference of long-term biological parameters suggested that individuals experience a long-term average antibody boost μ_l of 2.24 log units (posterior median; 95% CI: 1.95-2.51), corresponding to approximately a 4-fold boost to long-term homologous titres that wanes with antigenic distance (long-term cross reaction σ_l = 0.105 posterior median; 95% CI: 0.0962-0.113) and decreases with each successive exposure (antigenic seniority parameter, τ = 0.0310 posterior median; 95% CI: 0.0210-0.0415).

As with the first case study, simulation recovery was used to validate the ability of serosolver to correctly infer underlying processes from a given dataset (discussed in detail in S4 Text).