# BayesMoFo-vignette

This vignette is intended to show how an analysis of mortality data would work using the BayesMoFo R package. We start by installing and loading the package.

    # install package
    # Recommended installation
    # install.packages("BayesMoFo")

    # Development version (use only if needed)
    # install.packages("devtools")
    # devtools::install_github("jstw1g09/Rpackage-BayesMoFo")

    #load package
    library(BayesMoFo)

The package fits various mortality models to two types of data: age-period (AP) data and age-period-product (APP) data. AP data refers to data structured by individuals’ age and the calendar period (or year) in which events (such as deaths or exposures) are observed. Each observation corresponds to a specific age and period combination, summarising the number of events and the population at risk for that group. For example, an AP dataset might record the number of deaths and the exposure (population at risk) for individuals aged 50 in the year 2010, aged 51 in 2011, and so on.

APP data extends the AP framework by including an additional stratifying variable - the “product” - which in general can be any other stratifying variable (such as cause of death, country, deprivation level, gender/sex, geographical location/region, insurance product, marital status, socioeconomic group, smoking behaviour, etc.). Each observation in an APP dataset corresponds to a specific combination of age, period, and product, capturing the number of events and the population at risk for that stratum. For example, an APP dataset might record deaths and exposures for individuals aged 50 in 2010, by cause of death or by insurance product type.

We will separately consider each analysis.

# Age-Period (AP) data

## Data preparation

The package accepts several types of data formats. For AP data, users can supply a data-frame, a 3-dimensional (3D) data array, or a data matrix.

## Data supplied as a data-frame

The data needs to be formatted as a `data.frame` with columns name Age, Year, Deaths and Exposures. An example is provided below.

You can access a dataset in this format for comparison by running

    data(uk_mortalitydata)

The first few lines of the data look like the following:

    head(uk_mortalitydata, n = 20)
    #>    Age Year   Deaths Exposures
    #> 1    0 1922 74065.19  908771.5
    #> 2    1 1922 23947.07  916468.6
    #> 3    2 1922 10891.03  866680.1
    #> 4    3 1922  4507.01  727009.9
    #> 5    4 1922  2870.01  657473.8
    #> 6    5 1922  2692.41  704711.4
    #> 7    6 1922  2339.34  764959.2
    #> 8    7 1922  2009.12  811532.0
    #> 9    8 1922  1724.79  837073.8
    #> 10   9 1922  1554.36  838640.6
    #> 11  10 1922  1482.40  834416.1
    #> 12  11 1922  1414.43  836638.8
    #> 13  12 1922  1467.02  847043.2
    #> 14  13 1922  1569.20  857801.8
    #> 15  14 1922  1764.97  857463.9
    #> 16  15 1922  1855.20  846678.5
    #> 17  16 1922  2116.68  836631.3
    #> 18  17 1922  2261.27  826264.3
    #> 19  18 1922  2400.99  814563.2
    #> 20  19 1922  2582.88  805320.9

Next, we need to format the data in the format necessary for the function runBayesMoFo to work properly. To do this, we pass the dataset (separately for death and exposure) to the function preparedata_fn, which takes the arguments ages, years, and data. In the case of deaths, we pass the data using the column Age, Year, and Claim; and in the case of exposures, we use Exposure in place of Claim.

    death <- preparedata_fn(uk_mortalitydata[, c("Age", "Year", "Deaths")],
                                      ages = 30:60, years = 2000:2020)
    expo <- preparedata_fn(uk_mortalitydata[, c("Age", "Year", "Exposures")],
                                    ages = 30:60, years = 2000:2020)

## Data supplied as a 3D data array

Alternatively, users can supply the data as a 3-dimensional (3D) data array. `dxt_array_product` is a 3D array containing mortality data stratified by insurance product (see `?dxt_array_product` for details), where dim one: 4 insurance products, dim two: 83 ages, dim three: 5 years.

    data("dxt_array_product")
    data("Ext_array_product")

    # preview of death data the 1st insurance product called "ACI"
    str(dxt_array_product["ACI",,,drop = FALSE])
    #>  num [1, 1:83, 1:5] 0 1.01 0 0 2 ...
    #>  - attr(*, "dimnames")=List of 3
    #>   ..$ : chr "ACI"
    #>   ..$ : chr [1:83] "18" "19" "20" "21" ...
    #>   ..$ : chr [1:5] "2016" "2017" "2018" "2019" ...

Similarly, users can prepare the data by either by inputting the data as a 3-way array, or by specifying the name of the stratum to load using the argument `strat_name`:

    # inputting the data as a 3-way array
    death <- preparedata_fn(dxt_array_product["ACI",,,drop = FALSE], ages = 35:65, years = 2016:2020)
    expo <- preparedata_fn(Ext_array_product["ACI",,,drop = FALSE], ages = 35:65, years = 2016:2020)

    # specifying the name of the stratum to load using `strat_name`
    death <- preparedata_fn(dxt_array_product,strat_name="ACI", ages = 35:65, years = 2016:2020)
    expo <- preparedata_fn(Ext_array_product,strat_name="ACI", ages = 35:65, years = 2016:2020)

Data array types are less conventional but can be useful if data has been stored as it is. Preserving this data structure is useful for JAGS implementation later (for package development purposes).

## Data supplied as a data matrix

Suppose the data is provided in a 2-dimensional matrix format by age and year, commonly used in the literature. For example, the following is an illustration:

    # preview of death data the 1st insurance product called "ACI"
    str(dxt_array_product["ACI",,,drop = TRUE])
    #>  num [1:83, 1:5] 0 1.01 0 0 2 ...
    #>  - attr(*, "dimnames")=List of 2
    #>   ..$ : chr [1:83] "18" "19" "20" "21" ...
    #>   ..$ : chr [1:5] "2016" "2017" "2018" "2019" ...

To prepare data of type matrix, users need to specify the argument `data_matrix=TRUE`.

    death<-preparedata_fn(dxt_array_product["ACI",,,drop = TRUE],data_matrix=TRUE,ages=35:65)
    expo<-preparedata_fn(Ext_array_product["ACI",,,drop = TRUE],data_matrix=TRUE,ages=35:65)

## Running the model

For illustrative purposes, we chose the UK mortality data.

Once the data have been prepared, they can be passed to runBayesMoFo, which is the core function in the package for estimating mortality models.

As this package is built on top of the `rjags` package, it is capable of handling missing values in the death data, provided they are coded as `NA`. However, if missing values are present in the exposures data, these will be automatically replaced with a default value of 100, and predictions will be performed using that value. If the user wishes to perform prediction for a specific exposure value, they can manually set the desired value of the exposure (leaving the value of death count as `NA`, of course).

Users also have the option to perform model selection, depending on their needs. If more than one model is provided in the `models` argument, model selection is performed by default using deviance information criterion (DIC). The argument `models` can be set equal to:

- “all”, in which case, all models are fitted;
- a vector with the name of the models, in which case the models in the list are fitted;
- If the argument is left unspecified, a default set of models is fitted.

For example, the code below fit the `LC`, `CMB_M3`, and `APCI` models to the data.

    fitmodel <- runBayesMoFo(death, expo,
                             models = c("LC",
                                        "CBD_M3",
                                        "APCI")
                             )

Note that one can also run the individual functions rather than using the function runBayesMoFo. For example,

    fitmodel <- fit_LC(death, expo)

All other functions for analysing the output (see later) would work equally. That being said, users are highly recommend to use the function runBayesMoFo even when only one model is needed. That is,

    fitmodel <- runBayesMoFo(death, expo, models = "LC")

The full list of models, with the specific names, is available by checking `?runBayesMoFo`. Alternatively, one can query the model details through the documentation within the package, i.e. `?fit_LC`.

The argument `family` defines the specification for the distribution of death. A summary is as below (note that for AP data, just suppress the subscript \(p\)):

- If `family="poisson"`, then as proposed by Brouhns, Denuit, and Vermunt (2002), \[d_{x,t,p} \sim \text{Poisson}(E^c_{x,t,p} m_{x,t,p}) , \] where \(d_{x,t,p}\) represents the number of deaths at age \(x\) in year \(t\) of stratum \(p\), while \(E^c_{x,t,p}\) and \(m_{x,t,p}\) represents respectively the corresponding central exposed to risk and central mortality rate at age \(x\) in year \(t\) of stratum \(p\). The specification is used in conjunction with the log link function, i.e. \[\log(m_{x,t,p}) = \eta_{x,t,p} \] where \(\eta_{x,t,p}\) is the predictor that depends on the functional form of the mortality model, a full list of which is provided in Appendices A and B.

- Similarly, if `family="nb"` (default), then a negative binomial distribution is fitted with the log link function, i.e. \[d_{x,t,p} \sim \text{Negative-Binomial}(\phi,\frac{\phi}{\phi+E^c_{x,t,p} m_{x,t,p}}) , \] \[\log(m_{x,t,p}) = \eta_{x,t,p} , \] where \(\phi\) is the overdispersion parameter. This specification of the death distribution is standard practice for incorporating overdispersion, a phenomenon commonly observed in mortality modelling. See Wong, Forster, and Smith (2023).

- If `\code{family="binomial"}`, then \[d_{x,t,p} \sim \text{Binomial}(E^0_{x,t,p} , q_{x,t,p}) , \] where \(q_{x,t,p}\) represents the initial mortality rate at age \(x\) in year \(t\) of stratum \(p\), while \(E^0_{x,t,p}\approx E^c_{x,t,p}+\frac{1}{2}d_{x,t,p}\) is the corresponding initial exposed to risk. The binomial specification is used in conjunction with the logit link function, i.e. \[\text{logit}(q_{x,t,p})= \log(\frac{q_{x,t,p}}{1-q_{x,t,p}}) = \eta_{x,t,p}. \]

For example, the following fit the same set of models using the Poisson distribution for modelling number of death.

    fitmodel <- runBayesMoFo(death, expo,
                             models = c("LC",
                                        "CBD_M3",
                                        "APCI"),
                             family="poisson"
                             )

There are also other arguments for customising the MCMC sampling of the posterior distributions, as below:

- `n.adapt` specifies the number of iterations for the adaptation phase. See `?rjags::adapt` for more details.
- `n.chain` specifies the number of parallel chains for the posterior sampling under each model.
- `n_iter` specifies the number of iterations in the posterior sampling.
- `thin` specifies the thinning interval for the posterior sampling

## Forecast

As part of the function runBayesMoFo, forecasting can be performed by setting the argument `forecast=TRUE`, with the parameter `h` specifying the forecast horizon. For example, the code below fit the `LC`, `CMB_M3`, and `APCI` models to the data, and forecast the models for `h=6` time points into the future.

    fitmodel_forecast <- runBayesMoFo(death, expo,
                             models = c("LC",
                                        "CBD_M3",
                                        "APCI"),
                             forecast = TRUE,
                             h = 6,
                             n.chain = 2
                             )
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 651
    #>    Unobserved stochastic nodes: 248
    #>    Total graph size: 7904
    #>
    #> Initializing model
    #>
    #> Completed: LC (1/3)
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 651
    #>    Unobserved stochastic nodes: 304
    #>    Total graph size: 6383
    #>
    #> Initializing model
    #>
    #> NOTE: Stopping adaptation
    #>
    #>
    #> Completed: CBD_M3 (2/3)
    #> Compiling model graph
    #>    Resolving undeclared variables
    #>    Allocating nodes
    #> Graph information:
    #>    Observed stochastic nodes: 651
    #>    Unobserved stochastic nodes: 302
    #>    Total graph size: 8266
    #>
    #> Initializing model
    #>
    #> NOTE: Stopping adaptation
    #>
    #>
    #> Completed: APCI (3/3)

## Analyzing the output

After running the model, users can then query the best and worst models (in terms of DIC) among the competing models.

    fitmodel_forecast$best_model
    #> [1] "APCI"

    fitmodel_forecast$worst_model
    #> [1] "LC"

A table showing the DIC of all models fitted can be returned too.

    fitmodel_forecast$DIC
    #>            LC   CBD_M3   APCI
    #> [1,] 7227.823 7070.536 7064.3

One can retrieve the fitted results for the best and worst performing models, both of which are of type `fit_result`.

    fitmodel_forecast$result$best
    fitmodel_forecast$result$worst

The function `plot_param_fn` plots all the fitted parameters of the model specified, using posterior samples generated. If more than one models were specified previously when running runBayesMoFo, then only the best model will be illustrated.

    plot_param_fn(fitmodel_forecast)

As evident in the plot, all fitted and forecasted parameters will be included, with solid lines indicating the medians and dashed lines representing the credible intervals generated from the posterior samples. By default, the intervals are constructed based on 95% credibility, but can be changed using the argument `pred_int`. For instance, for 80% credible intervals,

    plot_param_fn(fitmodel_forecast, pred_int = 0.80)

The argument `legends` argument can be used to suppress the legends for better visualisation (e.g. if visibility is blocked by the legend boxes).

    plot_param_fn(fitmodel_forecast, pred_int = 0.80, legends = FALSE)

The function `plot_rates_fn` plots the fitted death rates of the model specified for specific ages and years, using posterior samples generated. Again, if more than one models were specified previously when running runBayesMoFo, then only the best model will be illustrated. By default, the (log) death rates will be plotted against age for the first nine years.

As before, both fitted and forecasted death rates will be included, with solid lines indicating the medians and dashed lines representing the credible intervals (95% by default but can be changed using the argument `pred_int`) generated from the posterior samples. Also, observed crude death rates will also be included as coloured dots.

    plot_rates_fn(fitmodel_forecast)
    #> Warning in plot_rates_fn(fitmodel_forecast): Too many years selected, only
    #> printing the first 9 years.

For better visualisation, one may customise the argument `plot_years` to plot only selected years. Note that if more than nine years have been specified, then only the first nine years will be plotted.

    plot_rates_fn(fitmodel_forecast, plot_years = c(2016,2020,2024))

The argument `plot_type` allows users to plot death rates against year instead to better visualise temporal variations in death rates. The argument `plot_ages` can be used accordingly to specify which ages to plot.

    plot_rates_fn(fitmodel_forecast, plot_type = "time", plot_ages = c(35,45,55))

The function `summary_fn` produces a summary of the model results, including posterior means, standard deviations, medians, lower and upper quantiles based on the credibility specified using `pred_int`.

    summary_fitmodel<-summary_fn(fitmodel_forecast)

To obtain the posterior means and standard deviations of all death rates,

    #posterior means
    summary_fitmodel$rates_summary$mean

    #posterior standard deviations
    summary_fitmodel$rates_summary$std

To obtain the posterior medians and lower/upper quantiles of all death rates,

    #posterior medians
    summary_fitmodel$rates_pn$median

    #lower quantiles
    summary_fitmodel$rates_pn$lower

    #upper quantiles
    summary_fitmodel$rates_pn$upper

Correspondingly, for model parameters,

    #posterior means
    summary_fitmodel$param_summary$mean

    #posterior standard deviations
    summary_fitmodel$param_summary$std

    #posterior medians
    summary_fitmodel$param_pn$median

    #lower quantiles
    summary_fitmodel$param_pn$lower

    #upper quantiles
    summary_fitmodel$param_pn$upper

## Convergence diagnostics

Users can assess if convergence has been attained by the MCMC posterior sampling procedure. The functions `diag_rates_fn` produces (by default) trace plots, density plots, as well as effective sample sizes of the posterior samples of death rates under the best model. For example,

    diagnostics_rates_result<-converge_diag_rates_fn(fitmodel_forecast)

The effective sample sizes can be viewed as:

    diagnostics_rates_result$ESS
    #>  q[1,2,12]  q[1,2,16]  q[1,2,17] q[1,16,12] q[1,16,16] q[1,16,17] q[1,17,12]
    #>  135.41161   82.48078   43.02568  128.19355   45.68017   47.54397  129.67851
    #> q[1,17,16] q[1,17,17]
    #>   65.93268   94.86059

The arguments `plot_strata`, `plot_ages`, `plot_years` can be used to specify which death rates to examine, as follows.

    converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(35,45,55), plot_years = c(2016,2020,2024))

    #> $ESS
    #>  q[1,6,17]  q[1,6,21]  q[1,6,25] q[1,16,17] q[1,16,21] q[1,16,25] q[1,26,17]
    #>   29.29179   32.68255   40.86880   47.54397   40.91059   94.21674   31.55983
    #> q[1,26,21] q[1,26,25]
    #>   32.84719  237.42925

The arguments `trace` and `density` can be used to specify only plotting one of them or none according to personal preferences.

    #for only trace plots
    converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(35,45,55), plot_years = c(2016,2020,2024), trace = TRUE, density = FALSE)

Auto-correlation plots can also displayed to check if posterior samples are too correlated.

    #for only acf plots
    converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(35,45,55), plot_years = c(2016,2020,2024), trace = FALSE, density = FALSE, acf_plot = TRUE)

    #> $ESS
    #>  q[1,6,17]  q[1,6,21]  q[1,6,25] q[1,16,17] q[1,16,21] q[1,16,25] q[1,26,17]
    #>   29.29179   32.68255   40.86880   47.54397   40.91059   94.21674   31.55983
    #> q[1,26,21] q[1,26,25]
    #>   32.84719  237.42925

Similarly, convergence can be assessed for fitted parameters using the function `converge_diag_rates_fn`.

    diagnostics_param_result<-converge_diag_param_fn(fitmodel_forecast)

    #> NOTE: Only showing three randomly selected alpha.

    #> NOTE: Only showing three randomly selected beta.

    #> NOTE: Only showing three randomly selected kappa.

    #> NOTE: Only showing three randomly selected gamma.

The effective sample sizes can be viewed as:

    diagnostics_param_result$ESS
    #>          rho    rho_gamma sigma2_kappa sigma2_gamma          phi  alpha[1,31]
    #>    969.64746     20.61668    500.96781     13.73325    569.60223   1398.90826
    #>  alpha[1,16]  alpha[1,15]   beta[1,22]   beta[1,14]    beta[1,7]  kappa[1,22]
    #>    993.10171    531.46272     11.51433     21.21420      9.20101   1707.96677
    #>  kappa[1,18]  kappa[1,13]  gamma[1,28]   gamma[1,4]  gamma[1,43]
    #>    121.54441    210.70046     13.81269     28.38499     21.55733

By default, the function examines a selection of the parameters. But the arguments `plot_params` can be used to specify which set of parameters to examine, as follows.

    converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa","gamma","rho","phi","sigma2_kappa"))

    #> NOTE: Only showing three randomly selected kappa.

    #> NOTE: Only showing three randomly selected gamma.
    #> $ESS
    #>          rho    rho_gamma          phi sigma2_kappa   kappa[1,6]   kappa[1,1]
    #>    969.64746     20.61668    569.60223    500.96781    105.06140    608.95448
    #>  kappa[1,22]  gamma[1,19]   gamma[1,1]  gamma[1,31]
    #>   1707.96677     37.41380     13.86666     12.44684

To check the names of parameters available for examining:

    fitmodel_forecast$result$best$param
    #> [1] "alpha"        "beta"         "kappa"        "gamma"        "rho"
    #> [6] "sigma2_kappa" "rho_gamma"    "sigma2_gamma" "phi"

For the rate-related parameters such as alpha, beta, kappa, gamma etc., only three of the randomly selected subset will be examined when specified. If a particular parameter is to be assessed, users need to specify clearly the indices of the parameters to be examined. For instance, the following will assess the beta parameters for the first stratum and the third age, as well as kappa for the first stratum and the fourth year.

    converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa[1,4]","gamma[1,2]"))

    #> $ESS
    #> kappa[1,4] gamma[1,2]
    #> 168.574037   9.357196

To check the full list of parameters available for examining:

    colnames(fitmodel_forecast$result$best$post_sample[[1]])[!startsWith(colnames(fitmodel_forecast$result$best$post_sample[[1]]),"q[")]
    #>   [1] "alpha[1,1]"   "alpha[1,2]"   "alpha[1,3]"   "alpha[1,4]"   "alpha[1,5]"
    #>   [6] "alpha[1,6]"   "alpha[1,7]"   "alpha[1,8]"   "alpha[1,9]"   "alpha[1,10]"
    #>  [11] "alpha[1,11]"  "alpha[1,12]"  "alpha[1,13]"  "alpha[1,14]"  "alpha[1,15]"
    #>  [16] "alpha[1,16]"  "alpha[1,17]"  "alpha[1,18]"  "alpha[1,19]"  "alpha[1,20]"
    #>  [21] "alpha[1,21]"  "alpha[1,22]"  "alpha[1,23]"  "alpha[1,24]"  "alpha[1,25]"
    #>  [26] "alpha[1,26]"  "alpha[1,27]"  "alpha[1,28]"  "alpha[1,29]"  "alpha[1,30]"
    #>  [31] "alpha[1,31]"  "beta[1,1]"    "beta[1,2]"    "beta[1,3]"    "beta[1,4]"
    #>  [36] "beta[1,5]"    "beta[1,6]"    "beta[1,7]"    "beta[1,8]"    "beta[1,9]"
    #>  [41] "beta[1,10]"   "beta[1,11]"   "beta[1,12]"   "beta[1,13]"   "beta[1,14]"
    #>  [46] "beta[1,15]"   "beta[1,16]"   "beta[1,17]"   "beta[1,18]"   "beta[1,19]"
    #>  [51] "beta[1,20]"   "beta[1,21]"   "beta[1,22]"   "beta[1,23]"   "beta[1,24]"
    #>  [56] "beta[1,25]"   "beta[1,26]"   "beta[1,27]"   "beta[1,28]"   "beta[1,29]"
    #>  [61] "beta[1,30]"   "beta[1,31]"   "gamma[1,1]"   "gamma[1,2]"   "gamma[1,3]"
    #>  [66] "gamma[1,4]"   "gamma[1,5]"   "gamma[1,6]"   "gamma[1,7]"   "gamma[1,8]"
    #>  [71] "gamma[1,9]"   "gamma[1,10]"  "gamma[1,11]"  "gamma[1,12]"  "gamma[1,13]"
    #>  [76] "gamma[1,14]"  "gamma[1,15]"  "gamma[1,16]"  "gamma[1,17]"  "gamma[1,18]"
    #>  [81] "gamma[1,19]"  "gamma[1,20]"  "gamma[1,21]"  "gamma[1,22]"  "gamma[1,23]"
    #>  [86] "gamma[1,24]"  "gamma[1,25]"  "gamma[1,26]"  "gamma[1,27]"  "gamma[1,28]"
    #>  [91] "gamma[1,29]"  "gamma[1,30]"  "gamma[1,31]"  "gamma[1,32]"  "gamma[1,33]"
    #>  [96] "gamma[1,34]"  "gamma[1,35]"  "gamma[1,36]"  "gamma[1,37]"  "gamma[1,38]"
    #> [101] "gamma[1,39]"  "gamma[1,40]"  "gamma[1,41]"  "gamma[1,42]"  "gamma[1,43]"
    #> [106] "gamma[1,44]"  "gamma[1,45]"  "gamma[1,46]"  "gamma[1,47]"  "gamma[1,48]"
    #> [111] "gamma[1,49]"  "gamma[1,50]"  "gamma[1,51]"  "gamma[1,52]"  "gamma[1,53]"
    #> [116] "gamma[1,54]"  "gamma[1,55]"  "gamma[1,56]"  "gamma[1,57]"  "kappa[1,1]"
    #> [121] "kappa[1,2]"   "kappa[1,3]"   "kappa[1,4]"   "kappa[1,5]"   "kappa[1,6]"
    #> [126] "kappa[1,7]"   "kappa[1,8]"   "kappa[1,9]"   "kappa[1,10]"  "kappa[1,11]"
    #> [131] "kappa[1,12]"  "kappa[1,13]"  "kappa[1,14]"  "kappa[1,15]"  "kappa[1,16]"
    #> [136] "kappa[1,17]"  "kappa[1,18]"  "kappa[1,19]"  "kappa[1,20]"  "kappa[1,21]"
    #> [141] "kappa[1,22]"  "kappa[1,23]"  "kappa[1,24]"  "kappa[1,25]"  "kappa[1,26]"
    #> [146] "kappa[1,27]"  "phi"          "rho"          "rho_gamma"    "sigma2_gamma"
    #> [151] "sigma2_kappa"

The arguments `trace` and `density` can be used to specify only plotting one of them or none according to personal preferences.

    #for only trace plots
    converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa[1,4]","gamma[1,2]"), trace = TRUE, density = FALSE)

Auto-correlation plots can also displayed to check if posterior samples are correlated.

    #for only acf plots
    converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa[1,4]","gamma[1,2]"), trace = FALSE, density = FALSE, acf_plot = TRUE)

    #> $ESS
    #> kappa[1,4] gamma[1,2]
    #> 168.574037   9.357196

Several other commonly used MCMC convergence diagnostics, such as Gelman’s R statistic (Gelman and Rubin (1992)), Geweke’s Z scores (Geweke (1991)), Heidel’s Stationarity and Half-width tests (see Heidelberger and Welch (1981) and Heidelberger and Welch (1983) for more details), can be computed and illustrated using the function `converge_diag_fn`.

    converge_diag_result<-converge_diag_fn(fitmodel_forecast, plot_gelman = TRUE, plot_geweke = TRUE)
    #> Note: no convergence issues identified.

    #Gelman's R
    head(converge_diag_result$gelman_diag$psrf)
    #>          Point est. Upper C.I.
    #> q[1,1,1]   1.007521   1.022611
    #> q[1,2,1]   1.122386   1.438192
    #> q[1,3,1]   1.677692   2.917502
    #> q[1,4,1]   1.003516   1.003721
    #> q[1,5,1]   1.269089   1.861320
    #> q[1,6,1]   1.543866   2.665951

    #Geweke's Z
    head(converge_diag_result$geweke_diag$z)
    #>  q[1,1,1]  q[1,2,1]  q[1,3,1]  q[1,4,1]  q[1,5,1]  q[1,6,1]
    #>  3.606717  1.164595 -5.137911 -2.043750 -1.228851 -5.200268

    #Heidel's Stationarity and Half-width tests
    head(converge_diag_result$heidel_diag)
    #>          stest start     pvalue htest         mean    halfwidth
    #> q[1,1,1]     1     1 0.18887881     1 0.0007217432 4.272638e-06
    #> q[1,2,1]     1     1 0.85854603     1 0.0007715985 5.760703e-06
    #> q[1,3,1]     1   801 0.12233304     1 0.0008153714 5.008481e-06
    #> q[1,4,1]     1     1 0.50954807     1 0.0008232145 6.828502e-06
    #> q[1,5,1]     1   601 0.07154669     1 0.0008998373 4.979771e-06
    #> q[1,6,1]     1   601 0.23798990     1 0.0009619807 8.592257e-06

# Age-period-product (APP) data

## Data preparation

Similarly to before, the data needs to be formatted in a data.frame with columns name Age, Year, Deaths,Exposures and Cause. Some examples are provided below.

## Data supplied as a data-frame

    data(uk_deathscausedata)
    head(uk_deathscausedata, n = 10)
    #> # A tibble: 10 × 5
    #>      Age  Year Deaths Exposures Cause
    #>    <dbl> <int>  <dbl>     <dbl> <chr>
    #>  1    15  2001   0     1653794. L057
    #>  2    15  2001   0.96  1653794. L108
    #>  3    15  2001   0     1653794. L110
    #>  4    15  2001   0     1653794. L115
    #>  5    15  2001   0     1653794. L132
    #>  6    20  2001   0     1577382. L057
    #>  7    20  2001   3.03  1577382. L108
    #>  8    20  2001   2.02  1577382. L110
    #>  9    20  2001   1.01  1577382. L115
    #> 10    20  2001   0     1577382. L132

    death <- preparedata_fn(uk_deathscausedata[,c("Age","Year","Deaths","Cause")])
    expo <- preparedata_fn(uk_deathscausedata[,c("Age","Year","Exposures","Cause")])

    #or if require a subset of the data
    death <- preparedata_fn(uk_deathscausedata[,c("Age","Year","Deaths","Cause")],
                                     ages = seq(45,90,by=5), years = 2001:2020)
    expo <- preparedata_fn(uk_deathscausedata[,c("Age","Year","Exposures","Cause")],
                                    ages = seq(45,90,by=5), years = 2001:2020)

    str(death)
    #> List of 7
    #>  $ data      : num [1:5, 1:10, 1:20] 272 506 588 37 39 ...
    #>   ..- attr(*, "dimnames")=List of 3
    #>   .. ..$ : chr [1:5] "L057" "L108" "L110" "L115" ...
    #>   .. ..$ : chr [1:10] "45" "50" "55" "60" ...
    #>   .. ..$ : chr [1:20] "2001" "2002" "2003" "2004" ...
    #>  $ strat_name: chr [1:5] "L057" "L108" "L110" "L115" ...
    #>  $ ages      : num [1:10] 45 50 55 60 65 70 75 80 85 90
    #>  $ years     : int [1:20] 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ...
    #>  $ n_strat   : int 5
    #>  $ n_ages    : int 10
    #>  $ n_years   : int 20

    str(expo)
    #> List of 7
    #>  $ data      : num [1:5, 1:10, 1:20] 1643774 1643774 1643774 1643774 1643774 ...
    #>   ..- attr(*, "dimnames")=List of 3
    #>   .. ..$ : chr [1:5] "L057" "L108" "L110" "L115" ...
    #>   .. ..$ : chr [1:10] "45" "50" "55" "60" ...
    #>   .. ..$ : chr [1:20] "2001" "2002" "2003" "2004" ...
    #>  $ strat_name: chr [1:5] "L057" "L108" "L110" "L115" ...
    #>  $ ages      : num [1:10] 45 50 55 60 65 70 75 80 85 90
    #>  $ years     : int [1:20] 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ...
    #>  $ n_strat   : int 5
    #>  $ n_ages    : int 10
    #>  $ n_years   : int 20

As shown above, the cause of death data consists of numbers of death and exposures for five causes of death, spanning years 2001-2020 and 5-year age groups between 45-90.

## Data supplied as a 3D data array

Alternatively, users may wish to supply data which is already sorted as a 3D array (dim 1: strata, dim 2: ages, dim 3: years).

    data("dxt_array_product");data("Ext_array_product")
    str(dxt_array_product) # 3D data array

    death<-preparedata_fn(dxt_array_product,ages=35:65)
    expo<-preparedata_fn(Ext_array_product,ages=35:65)

# Running the model

The syntax is similar to the case of the age-period data. The function automatically recognises the structure of the data after being processed by `preparedata_fn`. For illustration, we fit the model by Li and Lee (2005) on the cause of death data.

    fitmodel_forecast <- runBayesMoFo(death, expo,
                             models = "MLiLee",
                             forecast = TRUE,
                             h = 5,
                             quiet = TRUE,
                             n.chain = 2
                             )
    #> NOTE: Stopping adaptation
    #>
    #>
    #> Completed: MLiLee (1/1)

The argument `quiet=TRUE` was used to suppress messages generated during model compilation stage.

## Plot the output

    plot_param_fn(fitmodel_forecast)

    plot_rates_fn(fitmodel_forecast, plot_years = c(2005,2020,2025))

The argument `plot_type` allows users to plot death rates against year instead to better visualise temporal variations in death rates. The argument `plot_ages` can be used accordingly to specify which ages to plot.

    plot_rates_fn(fitmodel_forecast, plot_type = "time", plot_ages = c(45,65,85))

## Convergence diagnostics

    diagnostics_param_result<-converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(45,65,85), plot_years = c(2005,2020,2025))

    #for only acf plots
    converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(45,65,85), plot_years = c(2005,2020,2025), trace = FALSE, density = FALSE, acf_plot = TRUE)

    #> $ESS
    #>   q[1,1,5]  q[1,1,20]  q[1,1,25]   q[1,5,5]  q[1,5,20]  q[1,5,25]   q[1,9,5]
    #>  287.54106  189.70343  915.79553  198.28783  149.41207  911.67527  171.50621
    #>  q[1,9,20]  q[1,9,25]   q[3,1,5]  q[3,1,20]  q[3,1,25]   q[3,5,5]  q[3,5,20]
    #>   94.41890  271.40731  279.57375  107.24266 1315.06789  194.84396  169.37361
    #>  q[3,5,25]   q[3,9,5]  q[3,9,20]  q[3,9,25]   q[5,1,5]  q[5,1,20]  q[5,1,25]
    #> 1118.82091  140.48926  110.08231  962.87939  288.64842   78.80306   61.44129
    #>   q[5,5,5]  q[5,5,20]  q[5,5,25]   q[5,9,5]  q[5,9,20]  q[5,9,25]
    #>  137.38822   54.26480  110.48131  115.85330   80.25128 1310.99110

The effective sample sizes can be viewed as:

    diagnostics_param_result$ESS
    #>   q[2,1,5]  q[2,1,20]  q[2,1,25]   q[2,5,5]  q[2,5,20]  q[2,5,25]   q[2,9,5]
    #>  344.09052   75.05483  527.46728  177.55126  122.83274 1470.88709  108.79459
    #>  q[2,9,20]  q[2,9,25]   q[3,1,5]  q[3,1,20]  q[3,1,25]   q[3,5,5]  q[3,5,20]
    #>  218.11406 1654.60154  279.57375  107.24266 1315.06789  194.84396  169.37361
    #>  q[3,5,25]   q[3,9,5]  q[3,9,20]  q[3,9,25]   q[5,1,5]  q[5,1,20]  q[5,1,25]
    #> 1118.82091  140.48926  110.08231  962.87939  288.64842   78.80306   61.44129
    #>   q[5,5,5]  q[5,5,20]  q[5,5,25]   q[5,9,5]  q[5,9,20]  q[5,9,25]
    #>  137.38822   54.26480  110.48131  115.85330   80.25128 1310.99110

By default, the function examines a selection of the parameters. But the arguments `plot_params` can be used to specify which set of parameters to examine, as follows.

    converge_diag_param_fn(fitmodel_forecast, plot_params = c("beta","kappa","rho","phi","sigma2_kappa"))

    #> NOTE: Only showing three randomly selected beta.

    #> NOTE: Only showing three randomly selected kappa.
    #> $ESS
    #>    rho_Kappa    rho_kappa          phi sigma2_kappa    beta[2,9]    beta[3,5]
    #>    484.89142   1295.15052    132.26673    302.40896     37.17646     45.08807
    #>   beta[4,10]  kappa[3,19]   kappa[4,3]  kappa[2,18]
    #>     43.93540    141.73940    132.60227    127.77479

To check the names of parameters available for examining:

    fitmodel_forecast$result$best$param
    #>  [1] "alpha"        "beta"         "kappa"        "Beta"         "Kappa"
    #>  [6] "eta_kappa"    "rho_kappa"    "sigma2_kappa" "eta_Kappa"    "rho_Kappa"
    #> [11] "sigma2_Kappa" "phi"

For the predictor-related parameters such as alpha, beta, kappa, gamma etc., only three of the randomly selected subset will be examined when specified. If a particular parameter is to be assessed, users need to specify clearly the indices of the parameters to be examined. For instance, the following will assess the beta parameters for the first stratum and the third age, as well as kappa for the second stratum and the fourth year.

    converge_diag_param_fn(fitmodel_forecast, plot_params = c("beta[1,3]","kappa[2,4]"))

    #> $ESS
    #>  beta[1,3] kappa[2,4]
    #>   23.57477  151.44685

To check the full list of parameters available for examining:

    colnames(fitmodel_forecast$result$best$post_sample[[1]])[!startsWith(colnames(fitmodel_forecast$result$best$post_sample[[1]]),"q[")]
    #>   [1] "Beta[1]"      "Beta[2]"      "Beta[3]"      "Beta[4]"      "Beta[5]"
    #>   [6] "Beta[6]"      "Beta[7]"      "Beta[8]"      "Beta[9]"      "Beta[10]"
    #>  [11] "Kappa[1]"     "Kappa[2]"     "Kappa[3]"     "Kappa[4]"     "Kappa[5]"
    #>  [16] "Kappa[6]"     "Kappa[7]"     "Kappa[8]"     "Kappa[9]"     "Kappa[10]"
    #>  [21] "Kappa[11]"    "Kappa[12]"    "Kappa[13]"    "Kappa[14]"    "Kappa[15]"
    #>  [26] "Kappa[16]"    "Kappa[17]"    "Kappa[18]"    "Kappa[19]"    "Kappa[20]"
    #>  [31] "Kappa[21]"    "Kappa[22]"    "Kappa[23]"    "Kappa[24]"    "Kappa[25]"
    #>  [36] "alpha[1,1]"   "alpha[2,1]"   "alpha[3,1]"   "alpha[4,1]"   "alpha[5,1]"
    #>  [41] "alpha[1,2]"   "alpha[2,2]"   "alpha[3,2]"   "alpha[4,2]"   "alpha[5,2]"
    #>  [46] "alpha[1,3]"   "alpha[2,3]"   "alpha[3,3]"   "alpha[4,3]"   "alpha[5,3]"
    #>  [51] "alpha[1,4]"   "alpha[2,4]"   "alpha[3,4]"   "alpha[4,4]"   "alpha[5,4]"
    #>  [56] "alpha[1,5]"   "alpha[2,5]"   "alpha[3,5]"   "alpha[4,5]"   "alpha[5,5]"
    #>  [61] "alpha[1,6]"   "alpha[2,6]"   "alpha[3,6]"   "alpha[4,6]"   "alpha[5,6]"
    #>  [66] "alpha[1,7]"   "alpha[2,7]"   "alpha[3,7]"   "alpha[4,7]"   "alpha[5,7]"
    #>  [71] "alpha[1,8]"   "alpha[2,8]"   "alpha[3,8]"   "alpha[4,8]"   "alpha[5,8]"
    #>  [76] "alpha[1,9]"   "alpha[2,9]"   "alpha[3,9]"   "alpha[4,9]"   "alpha[5,9]"
    #>  [81] "alpha[1,10]"  "alpha[2,10]"  "alpha[3,10]"  "alpha[4,10]"  "alpha[5,10]"
    #>  [86] "beta[1,1]"    "beta[2,1]"    "beta[3,1]"    "beta[4,1]"    "beta[1,2]"
    #>  [91] "beta[2,2]"    "beta[3,2]"    "beta[4,2]"    "beta[1,3]"    "beta[2,3]"
    #>  [96] "beta[3,3]"    "beta[4,3]"    "beta[1,4]"    "beta[2,4]"    "beta[3,4]"
    #> [101] "beta[4,4]"    "beta[1,5]"    "beta[2,5]"    "beta[3,5]"    "beta[4,5]"
    #> [106] "beta[1,6]"    "beta[2,6]"    "beta[3,6]"    "beta[4,6]"    "beta[1,7]"
    #> [111] "beta[2,7]"    "beta[3,7]"    "beta[4,7]"    "beta[1,8]"    "beta[2,8]"
    #> [116] "beta[3,8]"    "beta[4,8]"    "beta[1,9]"    "beta[2,9]"    "beta[3,9]"
    #> [121] "beta[4,9]"    "beta[1,10]"   "beta[2,10]"   "beta[3,10]"   "beta[4,10]"
    #> [126] "eta_Kappa[1]" "eta_Kappa[2]" "eta_kappa[1]" "eta_kappa[2]" "kappa[1,1]"
    #> [131] "kappa[2,1]"   "kappa[3,1]"   "kappa[4,1]"   "kappa[1,2]"   "kappa[2,2]"
    #> [136] "kappa[3,2]"   "kappa[4,2]"   "kappa[1,3]"   "kappa[2,3]"   "kappa[3,3]"
    #> [141] "kappa[4,3]"   "kappa[1,4]"   "kappa[2,4]"   "kappa[3,4]"   "kappa[4,4]"
    #> [146] "kappa[1,5]"   "kappa[2,5]"   "kappa[3,5]"   "kappa[4,5]"   "kappa[1,6]"
    #> [151] "kappa[2,6]"   "kappa[3,6]"   "kappa[4,6]"   "kappa[1,7]"   "kappa[2,7]"
    #> [156] "kappa[3,7]"   "kappa[4,7]"   "kappa[1,8]"   "kappa[2,8]"   "kappa[3,8]"
    #> [161] "kappa[4,8]"   "kappa[1,9]"   "kappa[2,9]"   "kappa[3,9]"   "kappa[4,9]"
    #> [166] "kappa[1,10]"  "kappa[2,10]"  "kappa[3,10]"  "kappa[4,10]"  "kappa[1,11]"
    #> [171] "kappa[2,11]"  "kappa[3,11]"  "kappa[4,11]"  "kappa[1,12]"  "kappa[2,12]"
    #> [176] "kappa[3,12]"  "kappa[4,12]"  "kappa[1,13]"  "kappa[2,13]"  "kappa[3,13]"
    #> [181] "kappa[4,13]"  "kappa[1,14]"  "kappa[2,14]"  "kappa[3,14]"  "kappa[4,14]"
    #> [186] "kappa[1,15]"  "kappa[2,15]"  "kappa[3,15]"  "kappa[4,15]"  "kappa[1,16]"
    #> [191] "kappa[2,16]"  "kappa[3,16]"  "kappa[4,16]"  "kappa[1,17]"  "kappa[2,17]"
    #> [196] "kappa[3,17]"  "kappa[4,17]"  "kappa[1,18]"  "kappa[2,18]"  "kappa[3,18]"
    #> [201] "kappa[4,18]"  "kappa[1,19]"  "kappa[2,19]"  "kappa[3,19]"  "kappa[4,19]"
    #> [206] "kappa[1,20]"  "kappa[2,20]"  "kappa[3,20]"  "kappa[4,20]"  "kappa[1,21]"
    #> [211] "kappa[2,21]"  "kappa[3,21]"  "kappa[4,21]"  "kappa[1,22]"  "kappa[2,22]"
    #> [216] "kappa[3,22]"  "kappa[4,22]"  "kappa[1,23]"  "kappa[2,23]"  "kappa[3,23]"
    #> [221] "kappa[4,23]"  "kappa[1,24]"  "kappa[2,24]"  "kappa[3,24]"  "kappa[4,24]"
    #> [226] "kappa[1,25]"  "kappa[2,25]"  "kappa[3,25]"  "kappa[4,25]"  "phi"
    #> [231] "rho_Kappa"    "rho_kappa"    "sigma2_Kappa" "sigma2_kappa"

The arguments `trace` and `density` can be used to specify only plotting one of them or none according to personal preferences.

    #for only trace plots
    converge_diag_param_fn(fitmodel_forecast, plot_params = c("beta[1,3]","kappa[2,4]"), trace = TRUE, density = FALSE)

Auto-correlation plots can also displayed to check if posterior samples are correlated.

    #for only acf plots
    converge_diag_param_fn(fitmodel_forecast, plot_params = c("beta[1,3]","kappa[2,4]"), trace = FALSE, density = FALSE, acf_plot = TRUE)

    #> $ESS
    #>  beta[1,3] kappa[2,4]
    #>   23.57477  151.44685

Convergence diagnostics can be applied as usual, but are not run in the example below.

    converge_diag_result<-converge_diag_fn(fitmodel_forecast, plot_gelman = TRUE, plot_geweke = TRUE)

    #Gelman's R
    head(converge_diag_result$gelman_diag$psrf)

    #Geweke's Z
    head(converge_diag_result$geweke_diag$z)

    #Heidel's Stationarity and Half-width tests
    head(converge_diag_result$heidel_diag)

Interestingly, there is an article discussing the use of common (shared) cohort effects for modelling cause of death data as described by Cairns (2023). Thus, we can fit some of the models that incorporate shared cohort effects as below (NOT RUN).

    fitmodel_forecast <- runBayesMoFo(death, expo,
                             models = c("APCI_sharegamma",
                                        "RH_sharegamma"),
                             forecast = TRUE,
                             h = 5,
                             quiet = TRUE
                             )

    fitmodel_forecast$DIC

    fitmodel_forecast$best_model

    plot_param_fn(fitmodel_forecast)

    plot_rates_fn(fitmodel_forecast, plot_years = c(2005,2020,2025))

    plot_rates_fn(fitmodel_forecast, plot_type = "time", plot_ages = c(45,65,85))

    converge_diag_rates_fn(fitmodel_forecast, plot_ages = c(45,65,85), plot_years = c(2005,2020,2025))

    converge_diag_param_fn(fitmodel_forecast, plot_params = c("kappa","rho","phi","sigma2_kappa"))

# Appendix A: full list of age-period models considered

Model | Predictor, \(\eta_{x,t}\)
--- | ---
APCI | \(a_{x}+b_{x}(t-\bar{t})+k_{t} + \gamma_{c}\)
CBD_M3 | \(a_{x}+k_{t} + \gamma_{c}\)
CBD_M5 | \(k^1_{t} + k^2_{t}(x-\bar{x})\)
CBD_M6 | \(k^1_{t} + k^2_{t}(x-\bar{x}) +\gamma_{c}\)
CBD_M7 | \(k^1_{t} + k^2_{t}(x-\bar{x}) + k^3_{t}((x-\bar{x})^2-\hat{\sigma}_x^2) +\gamma_{c}\)
CBD_M8 | \(k^1_{t} + k^2_{t}(x-\bar{x}) +\gamma_{c}(constant-x)\)
LC | \(a_{x}+b_{x}k_{t}\)
MLiLee | \(a_{x}+B_xK_t\)
PLAT | \(a_{x}+k^1_{t} + k^2_{t}(\bar{x}-x) + k^3_{t}(\bar{x}-x)^+ +\gamma_{c}\)
RH | \(a_{x}+b_{x}k_{t} + \gamma_{c}\)

# Appendix B: full list of age-period-product models considered

Model | Predictor, \(\eta_{x,t,p}\)
--- | ---
APCI | \(a_{x,p}+b_{x,p}(t-\bar{t})+k_{t,p} + \gamma_{c,p}\)
APCI_sharealpha | \(a_{x}+b_{x,p}(t-\bar{t})+k_{t,p} + \gamma_{c,p}\)
APCI_sharebeta | \(a_{x,p}+b_{x}(t-\bar{t})+k_{t,p} + \gamma_{c,p}\)
APCI_sharegamma | \(a_{x,p}+b_{x,p}(t-\bar{t})+k_{t,p} + \gamma_{c}\)
APCI_sharealpha_sharebeta | \(a_{x}+b_{x}(t-\bar{t})+k_{t,p} + \gamma_{c,p}\)
APCI_sharealpha_sharegamma | \(a_{x}+b_{x,p}(t-\bar{t})+k_{t,p} + \gamma_{c}\)
APCI_sharebeta_sharegamma | \(a_{x,p}+b_{x}(t-\bar{t})+k_{t,p} + \gamma_{c}\)
APCI_shareall | \(a_{x}+b_{x}(t-\bar{t})+k_{t,p} + \gamma_{c}\)
CBD_M3 | \(a_{x,p}+k_{t,p} + \gamma_{c,p}\)
CBD_M3_sharealpha | \(a_{x}+k_{t,p} + \gamma_{c,p}\)
CBD_M3_sharegamma | \(a_{x,p}+k_{t,p} + \gamma_{c}\)
CBD_M3_shareall | \(a_{x}+k_{t,p} + \gamma_{c}\)
CBD_M5 | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x})\)
CBD_M6 | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) +\gamma_{c,p}\)
CBD_M6_sharegamma | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) +\gamma_{c}\)
CBD_M7 | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) + k^3_{t,p}((x-\bar{x})^2-\hat{\sigma}_x^2) +\gamma_{c,p}\)
CBD_M7_sharegamma | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) + k^3_{t,p}((x-\bar{x})^2-\hat{\sigma}_x^2) +\gamma_{c}\)
CBD_M8 | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) +\gamma_{c,p}(c_p-x)\)
CBD_M8_sharegamma | \(k^1_{t,p} + k^2_{t,p}(x-\bar{x}) +\gamma_{c}(c_p-x)\)
LC | \(a_{x,p}+b_{x,p}k_{t,p}\)
LC_sharealpha | \(a_{x}+b_{x,p}k_{t,p}\)
LC_sharebeta | \(a_{x,p}+b_{x}k_{t,p}\)
LC_shareall | \(a_{x}+b_{x}k_{t,p}\)
M1A | \(a_{x}+c_p+b_xk_t\)
M1U | \(a_{x,p}+b_xk_t\)
M1M | \(a_{x}c_p+b_xk_t\)
M2A1 | \(a_{x}+(c_p+b_x)k_t\)
M2A2 | \(a_{x}+b_{x,p}k_t\)
M2Y1 | \(a_{x}+b_x(k_t+c_p)\)
M2Y2 | \(a_{x}+b_{x}k_{t,p}\)
MLiLee | \(a_{x,p}+b_{x,p}k_{t,p}+B_xK_t\)
MLiLee_sharealpha | \(a_{x}+b_{x,p}k_{t,p}+B_xK_t\)
PLAT | \(a_{x,p}+k^1_{t,p} + k^2_{t,p}(\bar{x}-x) + k^3_{t,p}(\bar{x}-x)^+ +\gamma_{c,p}\)
PLAT_sharealpha | \(a_{x}+k^1_{t,p} + k^2_{t,p}(\bar{x}-x) + k^3_{t,p}(\bar{x}-x)^+ +\gamma_{c,p}\)
PLAT_sharegamma | \(a_{x,p}+k^1_{t,p} + k^2_{t,p}(\bar{x}-x) + k^3_{t,p}(\bar{x}-x)^+ +\gamma_{c}\)
PLAT_shareall | \(a_{x}+k^1_{t,p} + k^2_{t,p}(\bar{x}-x) + k^3_{t,p}(\bar{x}-x)^+ +\gamma_{c}\)
RH | \(a_{x,p}+b_{x,p}k_{t,p} + \gamma_{c,p}\)
RH_sharealpha | \(a_{x}+b_{x,p}k_{t,p} + \gamma_{c,p}\)
RH_sharebeta | \(a_{x,p}+b_{x}k_{t,p} + \gamma_{c,p}\)
RH_sharegamma | \(a_{x,p}+b_{x,p}k_{t,p} + \gamma_{c}\)
RH_sharealpha_sharebeta | \(a_{x}+b_{x}k_{t,p} + \gamma_{c,p}\)
RH_sharealpha_sharegamma | \(a_{x}+b_{x,p}k_{t,p} + \gamma_{c}\)
RH_sharebeta_sharegamma | \(a_{x,p}+b_{x}k_{t,p} + \gamma_{c}\)
RH_shareall | \(a_{x}+b_{x}k_{t,p} + \gamma_{c}\)

# References

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