reslr: Statistical Models for examining Relative Sea Level Change in R

Maeve Upton, Andrew Parnell & Niamh Cahill

2023-06-14

Introduction

If you require fast instructions, check out the reslr: Quick start.

The reslr package is specifically developed for Bayesian modeling of relative sea-level data. It offers a diverse selection of statistical models, including linear regression, change-point regression, integrated Gaussian process regression, splines, and generalized additive models. One notable feature is the incorporation of measurement uncertainty in multiple dimensions, which is crucial when analyzing relative sea-level data. The package provides a unified framework for data loading, model fitting, and summarising changes in relative sea level (RSL) over time and space. The generated plots depict sea level curves and corresponding rates of change, taking into account the associated uncertainty.

There are a number of modelling options available to the user:

Statistical Model Model Information model_type code
Errors in variables simple linear regression A straight line of best fit taking account of any age and measurement errors in the RSL values using the method of Cahill et al (2015) “eiv_slr_t”
Errors in variables change point model An extension of the linear regression modelling process. It uses piece-wise linear sections and estimates where/when trend changes occur in the data (Cahill et al. 2015). “eiv_cp_t”
Errors in variables integrated Gaussian Process A non linear fit that utilities a Gaussian process prior on the rate of sea-level change that is then integrated (Cahill et al. 2015). “eiv_igp_t”
Noisy Input spline in time A non-linear fit using regression splines using the method of Upton et al (2023). “ni_spline_t”
Noisy Input spline in space and time A non-linear fit for a set of sites across a region using the method of Upton et al (2023). “ni_spline_st”
Noisy Input Generalised Additive model for the decomposition of the RSL signal A non-linear fit for a set of sites across a region and provides a decomposition of the signal into regional, local-linear (commonly GIA) and local non-linear components. Again this full model is as described in Upton et al (2023). “ni_gam_decomp”

For all of the above models the user is able to quantify and visualise changes of RSL and rates of change with associated uncertainties. Indeed a full posterior distribution ensemble of values is available in the output of the functions. For the decomposed full model, “ni_gam_decomp”, the user is able to access the posterior probability distributions of the individual components.

Installation of the reslr package

The reslr package uses the JAGS (Just Another Gibbs Sampler) software to run the models. Before installing reslr, visit the JAGS website and download and install JAGS for your operating system.

Next, start Rstudio and find the window with the command prompt (the symbol >). Type

# install.packages("reslr")
# library(devtools)
# devtools::install()
#devtools::install_github("maeveupton/reslr")
install_github("maeveupton/reslr")

It may ask you to pick your nearest CRAN mirror (the nearest site which hosts R packages). You will then see some activity on the screen as the reslr package and the other packages it uses are downloaded. The final line should then read:

package 'reslr' successfully unpacked and MD5 sums checked

You then need to load the package.

library(reslr)

This will load the reslr package and all the associated packages. You’ll need to type the library(reslr) command every time you start R. If you have problems, visit the Issues page and leave a message to tell us what went wrong.

Considerations before running reslr

Prior to running the reslr package, there are a few points to consider.

Installating JAGS software

In this package, the models are written using Just Another Gibbs Sample (JAGS) which uses Gibbs sampling and Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the unknown parameters. To download the JAGS package use this link.

Working with scripts

The best way to use the reslr package is by creating scripts. A script can be created in Rstudio by clicking File > New File > Rscript. This opens a text window which allows commands to be typed in order and saved. The command can be sent to the command prompt (which Rstudio calls the Console) by highlighting the command and clicking Run (or going to Code > Run Lines). There are also keyboard shortcuts to speed up the process. We strongly recommend you learn to run R via scripts.

Inputting User’s data

reslr can handle three different types of data structure. It is important to note that varying the number of data sites will require different statistical modelling strategy:

The user must ensure that the input data is a dataframe. For a single site or multiple sites only one dataframe should be given to the package, i.e. combined all sites into one dataframe, with the following columns names:

Site Region Age Age_err RSL RSL_err Longitude Latitude linear_rate linear_rate_err
“Leeds Point” “New Jersey” 1000 8 0.5 0.01 39.5 - 74.4 1.69 0.03
“Leeds Point” “New Jersey” 1050 11 0.6 0.01 39.5 - 74.4 1.69 0.03
… … … … … … … … … …
… … … … … … … … … …
… … … … … … … … … …
“Cedar Island” “North Carolina” 1700 12 0.8 0.06 -76.4 35 0.74 0.01
… … … … … … … … … …
… … … … … … … … … …

Tide Gauge Data

There is an option in the reslr package to include tide gauge data as an additional source of data which we recommend when using the model_type = "ni_gam_decomp". The package will extract tide gauge data from the PSMSL website. The data is downloaded from this website and stored in a temporary directory.

The tide gauges undergo a number of processing steps within the package. Firstly, certain tide gauges have been flagged by the PSMSL website and we remove these locations. Secondly, the tide gauge data in the PSMSL database is given in millimetres relative to a revised local reference datum (a coordinate system which defines the zero level for sea level measurements (Pugh et al., 2014)). We transform the data by removing 7000 mm to revert the tide gauge data into the observed reference frame and convert the RSL to metres following the PSMSL guidance as described in Aarup et al. 2006 . Lastly, the tide gauge data is averaged over a decade to make it comparable with sedimentation rates associated proxy records sedimentation rates. The user can alter the size of the average if required when accumulation rates for the sediment in the proxy record is estimated to have a higher or lower accumulation rate, e.g. longer sediment accumulation rate result in a larger window average of 20 years.

Within the reslr_load function, the user has three options to choose from:

  1. Provide a list of the preferred tide gauges from the PSMSL website, ensuring spelling, capitalisation and spacing is exactly the same as the website. Note, the package will not work if error in spelling occurs. In addition, certain tide gauges have been flagged by the PSMSL website and are not included in this package and will return an error if selected. This is done by giving a list to the list_preferred_TGs = c("ARGENTIA","MAYPORT") option in the reslr_mcmc function

  2. The nearest tide gauge to proxy site based on minimum distance in kilometers, which is done by setting TG_minimum_dist_proxy = TRUE.

  3. Any tide gauge within 1 degree from the proxy site, which is done by setting all_TG_1deg = TRUE.

The user can select a combination of option 1 and option 2 or option 1 and option 3 which allows for additional tide gauge data to be included. The final output is a data frame which contains an additional column, called data_type_id, identifying the data source “ProxyRecord” or “TideGaugeData” depending on the observation in question.

Glacial Isostatic Adjustment (GIA)

For the NI GAM decomposition, the statistical model requires an estimate for the local linear rate arising from processes such as GIA and associated uncertainty for this rate each site. According to Whitehouse (2018), GIA represents the Earth’s reaction to the growth or melting of ice sheets, including the gravitational field and ocean. GIA can be approximated as a linear contribution over a short timescale, but with variable effects along the coast (Engelhart et al., 2009). Earth-ice models, which incorporate the physical structure of the Earth to predict GIA changes due to ice loading and unloading, can provide estimates of GIA rates. There are a range of Earth-ice models with one such example being the ICE5G VM2-90 (Peltier, 2004). It should be noted that other processes, such as tectonic vertical land motion, can mimic the linear trend of GIA. Therefore, the linear local component within the NI GAM decomposition may account for contributions from processes other than GIA that drive changes in relative sea level. These are included as additional columns, linear_rate and linear_rate_err, in the input dataframe provided by the user.

If the GIA rate for the proxy site is not provided then package will automatically calculate these rates using the data provided and we do not estimates the rates from any Earth-ice physical model. The user can source their own rate estimates as previously mentioned. Two examples of GIA rate sources (not limited to) include Prof. Peltier’s webpage and the associated publication (Peltier, 2004) or the Caron et al. 2018 publication and data.

Important to note, the tide gauge data require values for the linear_rate and linear_rate_err columns. This is calculated using ICE-5G (VM2 L90) Earth ice model (Peltier et al. 2004) with an uncertainty value of 0.3 mm/year from Engelhart et al. 2009.

Example Data Set

The reslr package possesses a large dataset used as an example called NAACproxydata. This dataset contains proxy records from the Atlantic coast of North America as used in Upton et al 2023 along with tide gauge data which will be discussed below. The 21 different proxy data sites and the references for each data source can be found in the following table:

Site Name Reference
Barn Island, Connecticut Donnelly et al (2004), Gehrels et al (2020)
Big River Marsh, Newfoundland Kemp et al (2018)
Cape May Courthouse, New Jersey Kemp et al (2013), Cahill et al (2016)
Cedar Island, North Carolina Kemp et al (2011), Kemp et al (2017)
Cheesequake, New Jersey Walker et al (2021)
Chezzetcook Inlet, Nova Scotia Gehrels et al (2020)
East River Marsh, Connecticut Kemp et al (2015), Stearns et al (2023)
Fox Hill Marsh, Rhode Island Stearns et al (2023)
Leeds Point, New Jersey Kemp et al (2013), Cahill et al (2016)
Les Sillons, Magdelen Islands Barnett et al (2017)
Little Manatee River, Florida Gerlach et al (2017)
Nassau, Florida Kemp et al (2014)
Pelham Bay, New York Kemp et al (2017), Stearns et al (2017)
Placentia, Newfoundland Kemp et al (2018)
Revere, Massachusetts Donnelly et al (2006)
Saint Simeon, Quebec Barnett et al (2017)
Sanborn Cove, Maine Gehrels et al (2020)
Sand Point, North Carolina Kemp et al (2011), Kemp et al (2017)
Snipe Key, Florida Khan et al (2022)
Swan Key, Florida Khan et al (2022)
Wood Island, Massachusetts Kemp et al (2011)

The NAACproxydata is a data frame with 1715 rows and 8 columns which include:

If you are interested in a specific site or multiple sites from the example dataset, then filter for that site prior to running the package, using the following method:

# For 1 site
data_1site <- reslr::NAACproxydata %>% dplyr::filter(Site == "Cedar Island")
# For multiple sites
data_multisite <- reslr::NAACproxydata %>% dplyr::filter(Site %in% c(
  "Snipe Key", "Cheesequake",
  "Placentia", "Leeds Point"
))

How to run reslr

The general structure for running reslr is as follows:

Step 1. Load in the data using reslr_load. If tide gauge data is required update the argument include_tide_gauge = TRUE, from this the user has three options as described above. First, provide the list of names for the tide gauges from PSMSL website that the wish to use in the list_preferred_TGs option. Second, the package find the tide gauge closest to the proxy site TG_minimum_dist_proxy = TRUE. Third, the package uses all tide gauges within 1 degree of the proxy site all_TG_1deg = TRUE. If sedimentation accumulation rates for the proxy records are less than or greater than a decade the user can alter this size using sediment_average_TG = 10 which has a default of 10 years. If linear_rate is of interest to the user update the argument include_linear_rate = TRUE. The user can select the resolution of the output by changing the value of prediction_grid_res = 50 with the default of 50 years. The input_age_type argument is associated with the type of input age where the default is in Common Era. The package can model Before Present by updating this setting to “BP” and more information is in the advanced vignette.

Step 1a. The print function provides a brief insight into the inputted data.

Step 2. Plot the raw data using plot and select whether to include tide gauges in the output plot. The user can update the title (plot_title) and axis labels (xlab,ylab). The captions (plot_captions) can be included on the plots which provides a summary of the number of proxy sites and tide gauge sites.

Step 3. Choose your preferred model type from the available list above and use the reslr_mcmc function to obtain the parameter estimates and the dataframes required for plotting the outputs. This function has a number of settings which allow the user to improve model diagnostics. In addition, this function allows the user to chose their preferred credible interval size, the default setting is CI = 0.95.

Step 3a. The print function provides a brief insight into the output of the reslr_mcmc function.

Step 4. Check the model converged and examine the results of the parameters with the summary function

Step 5. Visualise the results with plot and access the dataframes used to create the plots. The plot_type option allows users to print individual plots, for example the model fits (“model_fit_plot”) and the rates (“rate_plot”) separately. The captions (plot_captions) can be included on the plots which provides a summary of the model type, the number of proxy sites and tide gauge sites. The user can select to include the tide gauge (plot_tide_gauges) in the output plots.

Errors-in-Variables Simple Linear Regression (“eiv_slr_t”)

The simplest model the reslr package can fit is a simple linear regression using the Errors-in-Variables method to account for the uncertainty associated with the proxy records, i.e. uncertainty associated with input (age) and the output (RSL). We would not recommend any model simpler than this (e.g. lm) as it will ignore some of the key uncertainties in the data.

This technique focuses on 1 site and is not recommended for multiple proxy sites together. Tide gauge data can be included to gain insight into recent changes in RSL, however, the user must investigate which tide gauge is suitable. As an example, we will filter the example dataset NAACproxydata to select one site to demonstrate the process:

# For 1 site
CedarIslandNC <- NAACproxydata %>% dplyr::filter(Site == "Cedar Island")

Step 1: Load in the data using the reslr_load function:

CedarIslandNC_input <- reslr_load(
  data = CedarIslandNC,
  include_tide_gauge = FALSE,
  include_linear_rate = FALSE,
  TG_minimum_dist_proxy = FALSE,
  list_preferred_TGs = NULL,
  all_TG_1deg = FALSE,
  prediction_grid_res = 50,
  input_age_type = "CE",
  sediment_average_TG = 10
)

In this function, the user can select to add tide gauge data and estimates for linear_rate, by changing include_tide_gauge = TRUE and include_linear_rate = TRUE respectfully. If include_tide_gauge = TRUE the user must decide if they require the closest tide gauge i.e. TG_minimum_dist_proxy = TRUE, or select specific tide gauge i.e. list_preferred_TGs = c("ARGENTIA"), or all tide gauges within 1 degree of the proxy site i.e. all_TG_1deg = TRUE. The default setting is rolling_window_average = 10 which corresponds to sediment accumulation rates of the proxy records, yet the user has the ability to alter this sediment accumulation rate. Note that for a simple linear regression we recommend using the default settings as demonstrated in the above code chunk. The user can alter the resolution of the output plots using prediction_grid_res with the default set at 50 years.

The output of this function is a list of two dataframes called data and data_grid.

data <- CedarIslandNC_input$data
data_grid <- CedarIslandNC_input$data_grid

Step 1a: A brief insight into the outputs of the reslr_input function can be obtained using:

print(CedarIslandNC_input)
#> This is a valid reslr input object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> Decadally averaged tide gauge data was not included. It is recommended for the ni_gam_decomp model 
#> The linear_rate or linear_rate_err was not included. It is required for the ni_gam_decomp model

Step 2: Plotting the data the raw data with:

plot(
  x = CedarIslandNC_input,
  title = "Plot of the raw data",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_tide_gauges = FALSE,
  plot_proxy_records = TRUE,
  plot_caption = TRUE
)

This will produce a plot of Age on the x-axis and Relative Sea Level on the y-axis in meters. Grey boxes represent the uncertainty associated with the vertical and horizontal uncertainty. The black data points are the midpoints of these uncertainty boxes. The following extra arguments can be used which allows the user to updated the titles and axis labels. The caption plot_caption, included by default, provides the number of proxy sites and tide gauge sites that will be used in the model and can be removed if required plot_caption = FALSE. In addition, the user can select to plot the additional tide gauge data, plot_tide_gauge = TRUE.

Step 3: To run the the model the following code is used:

res_eiv_slr_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_slr_t",
  CI = 0.95
)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 208
#>    Unobserved stochastic nodes: 107
#>    Total graph size: 1003
#> 
#> Initializing model

This command takes the input data and the user specifies the statistical model, i.e. a simple linear regression using the EIV uncertainty method (“eiv_slr_t”). The CI setting allows the user to set the credible intervals, the current default is CI = 0.95. The function tells reslr to store the output of the model run in an object called res_eiv_slr_t.

Step 3a: A brief insight into the outputs of the reslr_output function can be obtained using:

print(res_eiv_slr_t)
#> This is a valid reslr output object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> The model used was the Errors-in-Variables Simple Linear Regression model.
#> The input data has been run via reslr_mcmc and has produced 3000 iterations over 3 MCMC chains.

Step 4: The convergence of the algorithm is examined and he parameter estimates from the model can be investigated using the following:

summary(res_eiv_slr_t)
#> No convergence issues detected.
#> # A tibble: 3 × 7
#>   variable    mean      sd     mad      q5     q95  rhat
#>   <chr>      <num>   <num>   <num>   <num>   <num> <num>
#> 1 alpha    -1.99   0.0166  0.0162  -2.02   -1.96    1.00
#> 2 beta      0.823  0.0128  0.0126   0.802   0.844   1.00
#> 3 sigma_y   0.0664 0.00956 0.00970  0.0508  0.0824  1.00

If the model run has the package will print: “No convergence issues detected”. If the package prints: “Convergence issues detected, a longer run is necessary”. The user is required to update the reslr_mcmc function with additional iterations in the following manner:

res_eiv_slr_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_slr_t",
  # Update these values
  n_iterations = 6000, # Number of iterations
  n_burnin = 1000, # Number of iterations to discard at the beginning
  n_thin = 4, # Reduces number of output samples to save memory and computation time
  n_chains = 3 # Number of Markov chains
)

The output of this function allows to user to examine the parameter estimates. For the eiv_slr_t model, the parameters of interest are the intercept (“alpha”), the slope (“beta”) and the residual standard deviation of the model (“sigma_y”). When using the eiv_slr_t model, an estimate of the of the rate of sea-level change can be obtained by examining the value of the slope, i.e. “beta”.

Step 5: The results from the eiv_slr_t model can be visualised using the following function:

plot(res_eiv_slr_t,
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)"
)

The output of this function is a graph of the input data, i.e. Age and RSL and associated uncertainty boxes, and the model fit with 95 % credible interval. The caption provides the model type used and number of proxy sites and tide gauge sites used and can be removed if necessary with plot_caption = FALSE.

To examine the data creating these plots the user types the following:

output_dataframes <- res_eiv_slr_t$output_dataframes
head(output_dataframes)
#>   Longitude Latitude                       SiteName data_type_id  Age      pred
#> 1    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -800 -2.648602
#> 2    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -750 -2.607433
#> 3    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -700 -2.566265
#> 4    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -650 -2.525096
#> 5    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -600 -2.483927
#> 6    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -550 -2.442758
#>         upr       lwr  CI
#> 1 -2.699503 -2.597972 95%
#> 2 -2.657669 -2.558027 95%
#> 3 -2.615300 -2.517924 95%
#> 4 -2.572879 -2.477893 95%
#> 5 -2.530531 -2.438004 95%
#> 6 -2.488174 -2.398186 95%

Errors-in-Variable Change Point Model (“eiv_cp_t”)

The Errors-in-Variable Change Point model is an extension of the linear regression and allows the user to specify the number of change points required.

This technique focuses on 1 site and the maximum number of change points available to the user is 3. We do not recommended for multiple proxy sites together. Tide gauge data can be included to gain insight into recent changes in RSL, however, the user must investigate which tide gauge is most suitable. It is important to note that certain data sites will not work with 2 or 3 change points as there is no distinct changing points in the data. In this case, we recommend testing different number of change points and reviewing the resulting plots to confirm the correct number of change points is selected.

As an example, we will filter the example dataset NAACproxydata to select one site to demonstrate the process:

# For 1 site
CedarIslandNC <- reslr::NAACproxydata %>% dplyr::filter(Site == "Cedar Island")

Step 1: Load in the data using the reslr_load function:

CedarIslandNC_input <- reslr_load(
  data = CedarIslandNC,
  include_tide_gauge = FALSE,
  include_linear_rate = FALSE,
  TG_minimum_dist_proxy = FALSE,
  list_preferred_TGs = NULL,
  all_TG_1deg = FALSE,
  prediction_grid_res = 50,
  sediment_average_TG = 10
)

In this function, the user can select to add tide gauge data and estimates for linear_rate, by changing include_tide_gauge = TRUE and include_linear_rate = TRUE respectfully. If include_tide_gauge = TRUE the user must decide if they require the closest tide gauge i.e. TG_minimum_dist_proxy = TRUE, or select specific tide gauge i.e. list_preferred_TGs = c("ARGENTIA"), or all tide gauges within 1 degree of the proxy site i.e. all_TG_1deg = TRUE. The default setting is sediment_average_TG = 10 which corresponds to sediment accumulation rates of the proxy records, yet the user has the ability to alter this sediment accumulation rate by changing the size of the rolling window average.

Note that for a change point model, we recommend using the default settings as demonstrated in the above code chunk. The user can alter the resolution of the output plots using prediction_grid_res with the default set at 50 years. The output of this function is a list of two dataframes called data and data_grid. - The data dataframe is the inputted data with additional column for the data_type_id which will contain, “ProxyRecord”. It can be accessed by:

data <- CedarIslandNC_input$data
head(data)
#>           Region         Site Latitude Longitude   RSL  Age Age_err RSL_err
#> 1 North Carolina Cedar Island   34.971    -76.38 -0.12 2005    2.25    0.06
#> 2 North Carolina Cedar Island   34.971    -76.38 -0.14 1996    2.00    0.06
#> 3 North Carolina Cedar Island   34.971    -76.38 -0.16 1988    5.00    0.06
#> 4 North Carolina Cedar Island   34.971    -76.38 -0.18 1979    5.75    0.06
#> 5 North Carolina Cedar Island   34.971    -76.38 -0.19 1974    5.50    0.06
#> 6 North Carolina Cedar Island   34.971    -76.38 -0.21 1963    5.50    0.06
#>                         SiteName data_type_id
#> 1 Cedar Island,\n North Carolina  ProxyRecord
#> 2 Cedar Island,\n North Carolina  ProxyRecord
#> 3 Cedar Island,\n North Carolina  ProxyRecord
#> 4 Cedar Island,\n North Carolina  ProxyRecord
#> 5 Cedar Island,\n North Carolina  ProxyRecord
#> 6 Cedar Island,\n North Carolina  ProxyRecord
data_grid <- CedarIslandNC_input$data_grid
head(data_grid)
#> # A tibble: 6 × 5
#> # Groups:   SiteName [1]
#>   Longitude Latitude SiteName                         data_type_id   Age
#>       <dbl>    <dbl> <fct>                            <fct>        <dbl>
#> 1     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -800
#> 2     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -750
#> 3     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -700
#> 4     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -650
#> 5     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -600
#> 6     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -550

Step 1a: A brief insight into the outputs of the reslr_input function can be obtained using:

print(CedarIslandNC_input)
#> This is a valid reslr input object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> Decadally averaged tide gauge data was not included. It is recommended for the ni_gam_decomp model 
#> The linear_rate or linear_rate_err was not included. It is required for the ni_gam_decomp model

Step 2: Plotting the data the raw data with:

plot(
  x = CedarIslandNC_input,
  title = "Plot of the raw data",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_proxy_records = TRUE,
  plot_tide_gauges = FALSE
)

This will produce a plot of Age on the x-axis and Relative Sea Level on the y-axis in meters. Grey boxes represent the uncertainty associated with the vertical and horizontal uncertainty. The black data points are the midpoints of these uncertainty boxes. The extra arguments can be used which allows the user to updated the titles and axis labels. The caption plot_caption, included by default, provides the number of proxy sites and tide gauge sites that will be used in the model and can be removed if required plot_caption = FALSE. The user can select to plot the additional tide gauge data, plot_tide_gauge = TRUE.

Step 3: Run the model using the following code and select the number of change points you require:

res_eiv_cp1_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_cp_t",
  n_cp = 1,
  CI = 0.95
)
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 208
#>    Unobserved stochastic nodes: 109
#>    Total graph size: 1651
#> 
#> Initializing model

If the user is interested in running 2 change points use method:

res_eiv_cp2_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_cp_t",
  n_cp = 2, # Updating the default setting to include an additional change point.
  CI = 0.95
)

The CI setting allows the user to set the credible intervals, the current default is CI = 0.95. Similar to the earlier model, the output object res_eiv_cp1_t stores the JAGS model run and should take a second to run.

Step 3a: A brief insight into the outputs of the reslr_output function can be obtained using:

print(res_eiv_cp1_t)
#> This is a valid reslr output object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> The model used was the Errors-in-Variables Change Point model with 1 change point.
#> The input data has been run via reslr_mcmc and has produced 3000 iterations over 3 MCMC chains.

Step 4: The convergence of the algorithm is examined and the parameter estimates from the model can be investigated using the following:

summary(res_eiv_cp1_t)
#> No convergence issues detected.
#> # A tibble: 5 × 7
#>   variable                 mean     sd    mad        q5     q95  rhat
#>   <chr>                   <num>  <num>  <num>     <num>   <num> <num>
#> 1 alpha                 -1.05   0.618  0.0920 -2.04     -0.557   1.00
#> 2 beta[1]                0.651  0.146  0.0272  0.361     0.766   1.00
#> 3 beta[2]                1.92   0.808  0.913   0.882     3.07    1.00
#> 4 Change Point in CE: 1268      0.753  0.0924  0.0457    1.85    1.00
#> 5 sigma_y                0.0156 0.0140 0.0108  0.000966  0.0444  1.00

If the model run has the package will print: “No convergence issues detected”. If the package prints: “Convergence issues detected, a longer run is necessary”. The user is required to update the reslr_mcmc function with additional iterations in the following manner:

res_eiv_cp1_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_cp_t",
  # Update these values
  n_iterations = 6000, # Number of iterations
  n_burnin = 1000, # Number of iterations to discard at the beginning
  n_thin = 4, # Reduces number of output samples to save memory and computation time
  n_chains = 3 # Number of Markov chains
)

For the eiv_cp_t model, the parameters of interest are the intercept (alpha), the slopes before the change point (“beta[1]”) and after the change point (“beta[2]”), the year of the change point (Change Point) and “sigma_y” the variance of the model.

Step 5: The results from the EIV Change Point model can be illustrated using:

plot(res_eiv_cp1_t,
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
)

The output of this function is a graph of the input data, i.e. Age and RSL and associated uncertainty boxes, and the model fit with 95 % credible interval. The caption provides the model type used and number of proxy sites and tide gauge sites used and can be removed if necessary with plot_caption = FALSE.

To examine the data creating these plots the user types the following:

output_dataframes <- res_eiv_cp1_t$output_dataframes
head(output_dataframes)
#>   Longitude Latitude                       SiteName data_type_id  Age      pred
#> 1    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -800 -2.498389
#> 2    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -750 -2.465848
#> 3    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -700 -2.433308
#> 4    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -650 -2.400767
#> 5    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -600 -2.368226
#> 6    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -550 -2.335686
#>         upr       lwr  CI
#> 1 -2.593383 -2.322540 95%
#> 2 -2.554933 -2.303383 95%
#> 3 -2.516741 -2.286490 95%
#> 4 -2.478136 -2.269123 95%
#> 5 -2.439818 -2.251211 95%
#> 6 -2.401164 -2.232406 95%

Errors-in-Variable Integrated Gaussian Process Model (“eiv_igp_t”)

The EIV Integrated Gaussian Process model provides the underlying rate of the process directly from the model. Further reading on this modeling approach can be found here.

This technique focuses on 1 site and we do not recommended for multiple proxy sites together. Tide gauge data can be included to gain insight into recent changes in RSL, however, the user must investigate which tide gauge is suitable. As an example, we will filter the example dataset NAACproxydata to select one site to demonstrate the process:

# For 1 site
CedarIslandNC <- reslr::NAACproxydata %>% dplyr::filter(Site == "Cedar Island")

Step 1: Load in the data using the reslr_load function:

CedarIslandNC_input <- reslr_load(
  data = CedarIslandNC,
  include_tide_gauge = FALSE,
  include_linear_rate = FALSE,
  TG_minimum_dist_proxy = FALSE,
  list_preferred_TGs = NULL,
  all_TG_1deg = FALSE,
  prediction_grid_res = 50,
  sediment_average_TG = 10
)

In this function, the user can select to add tide gauge data and estimates for linear_rate, by changing include_tide_gauge = TRUE and include_linear_rate = TRUE respectfully. If include_tide_gauge = TRUE the user must decide if they require the closest tide gauge i.e. TG_minimum_dist_proxy = TRUE, or select specific tide gauge i.e. list_preferred_TGs = c("ARGENTIA"), or all tide gauges within 1 degree of the proxy site i.e. all_TG_1deg = TRUE. The default setting is sediment_average_TG = 10 which corresponds to sediment accumulation rates of the proxy records, yet the user has the ability to alter this sediment accumulation rate.

Note that for an IGP we recommend using the default settings as demonstrated in the above code chunk. The user can alter the resolution of the output plots using prediction_grid_res with the default set at 50 years. The output of this function is a list of two dataframes called data and data_grid. - The data dataframe is the inputted data with additional columns for the data_type_id which will contain “ProxyRecord”. It can be accessed by:

data <- CedarIslandNC_input$data
data_grid <- CedarIslandNC_input$data_grid

Step 1a: A brief insight into the outputs of the reslr_input function can be obtained using:

print(CedarIslandNC_input)
#> This is a valid reslr input object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> Decadally averaged tide gauge data was not included. It is recommended for the ni_gam_decomp model 
#> The linear_rate or linear_rate_err was not included. It is required for the ni_gam_decomp model

Step 2: Plotting the data the raw data with:

plot(
  x = CedarIslandNC_input,
  title = "Plot of the raw data",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_proxy_records = TRUE,
  plot_tide_gauges = FALSE
)

This will produce a plot of Age on the x-axis and Relative Sea Level on the y-axis in meters. Grey boxes represent the uncertainty associated with the vertical and horizontal uncertainty. The black data points are the midpoints of these uncertainty boxes. The extra arguments can be used which allows the user to updated the titles and axis labels. The caption plot_caption, included by default, provides the number of proxy sites and tide gauge sites that will be used in the model and can be removed if required plot_caption = FALSE. In addition, the user can select to plot the additional tide gauge data, plot_tide_gauge = TRUE.

Step 3: To run the eiv_igp_t model the following function should be used:

res_eiv_igp_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_igp_t",
  CI = 0.95,
  
)

This command takes the input data and the user specifies the statistical model, i.e. an integrated Gaussian process using the EIV uncertainty method (“eiv_slr_t”). It tells reslr to store the output of the model run in an object called res_eiv_igp_t. The CI setting allows the user to set the credible intervals, the current default is CI = 0.95. The computational run time for this model is approximately 14 minutes.

Step 3a: A brief insight into the outputs of the reslr_output function can be obtained using:

print(res_eiv_igp_t)

Step 4: The convergence of the algorithm is examined and he parameter estimates from the model can be investigated using the following:

summary(res_eiv_igp_t)

If the model run has the package will print: “No convergence issues detected”. If the package prints: “Convergence issues detected, a longer run is necessary”. The user is required to update the reslr_mcmc function with additional iterations in the following manner:

res_eiv_igp_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "eiv_igp_t",
  # Update these values
  n_iterations = 6000, # Number of iterations
  n_burnin = 1000, # Number of iterations to discard at the beginning
  n_thin = 4, # Reduces number of output samples to save memory and computation time
  n_chains = 3 # Number of Markov chains
)

For the parameter estimates, the length scale parameter, “rho” is the correlation parameter and “nu” is the standard deviation of the rate process. “sigma_y” is the variation of the model.

Step 5: The results from the EIV IGP model can be illustrated using:

plot(res_eiv_igp_t,
  plot_type = "model_fit_plot",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_proxy_records = TRUE,
  plot_tide_gauges = FALSE
)

The output of this function is a graph of the input data, i.e. Age and RSL and associated uncertainty boxes, and the model fit with 95 % credible interval. The caption provides the model type used and number of proxy sites and tide gauge sites used and can be removed if necessary with plot_caption = FALSE. In order to view the rate of change plot, the following setting should be used:

plot(res_eiv_igp_t,
  plot_type = "rate_plot",
  xlab = "Year (CE)",
  y_rate_lab = "Rate of Change (mm per year)"
)

This prints the plot of the rate of change with 95 % credible intervals. The caption provides the model type, the number of proxy sites and tide gauge sites that were used.

To examine the data creating these plots the user types the following:

output_dataframes <- res_eiv_igp_t$output_dataframes

Noisy input spline in time (“ni_spline_t”)

An alternative method to examine how the response variable varies in time is using the Noisy input spline in time (ni_spline_t). It model can obtain results in more efficient computational run times when compared with the eiv_igp_t model.

This technique focuses on 1 site and we do not recommended for multiple proxy sites together. Tide gauge data can be used to gain insight into recent RSL changes. As an example, we will filter the example dataset NAACproxydata to select one site to demonstrate the process:

# For 1 site
CedarIslandNC <- reslr::NAACproxydata %>% dplyr::filter(Site == "Cedar Island")

Step 1: Load in the data using the reslr_load function:

CedarIslandNC_input <- reslr_load(
  data = CedarIslandNC,
  include_tide_gauge = FALSE,
  include_linear_rate = FALSE,
  TG_minimum_dist_proxy = FALSE,
  list_preferred_TGs = NULL,
  all_TG_1deg = FALSE,
  prediction_grid_res = 50,
  sediment_average_TG = 10
)

In this function, the user can select to add tide gauge data and estimates for linear_rate, by changing include_tide_gauge = TRUE and include_linear_rate = TRUE respectfully. If include_tide_gauge = TRUE the user must decide if they require the closest tide gauge i.e. TG_minimum_dist_proxy = TRUE, or select specific tide gauge i.e. list_preferred_TGs = c("ARGENTIA"), or all tide gauges within 1 degree of the proxy site i.e. all_TG_1deg = TRUE. The default setting is sediment_average_TG = 10 which corresponds to sediment accumulation rates of the proxy records, yet the user has the ability to alter this sediment accumulation rate by changing the size of the rolling window average.

Note that for a spline in time, we recommend using the default settings as demonstrated in the above code chunk. The user can alter the resolution of the output plots using prediction_grid_res with the default set at 50 years. The output of this function is a list of two dataframes called data and data_grid. - The data dataframe is the inputted data with additional columns for the data_type_id which will contain “ProxyRecord”. It can be accessed by:

data <- CedarIslandNC_input$data
data_grid <- CedarIslandNC_input$data_grid

Step 1a: A brief insight into the outputs of the reslr_input function can be obtained using:

print(CedarIslandNC_input)
#> This is a valid reslr input object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> Decadally averaged tide gauge data was not included. It is recommended for the ni_gam_decomp model 
#> The linear_rate or linear_rate_err was not included. It is required for the ni_gam_decomp model

Step 2: Plotting the data the raw data with:

plot(
  x = CedarIslandNC_input,
  title = "Plot of the raw data",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_proxy_records = TRUE,
  plot_tide_gauges = FALSE
)

This will produce a plot of Age on the x-axis and Relative Sea Level on the y-axis in meters. Grey boxes represent the uncertainty associated with the vertical and horizontal uncertainty. The black data points are the midpoints of these uncertainty boxes. The following extra arguments can be used which allows the user to updated the titles and axis labels. The caption plot_caption, included by default, provides the number of proxy sites and tide gauge sites that will be used in the model and can be removed if required plot_caption = FALSE. In addition, the user can select to plot the additional tide gauge data, plot_tide_gauge = TRUE.

Step 3: To run this model type use the following:

res_ni_spline_t <- reslr_mcmc(
  input_data = CedarIslandNC_input,
  model_type = "ni_spline_t",
  CI = 0.95
)

The output object res_ni_spline_t stores the JAGS model run. The CI setting allows the user to set the credible intervals, the current default is CI = 0.95. Note that there will be two model runs printed in the console here but the output will be the same format as earlier models.

Step 3a: A brief insight into the outputs of the reslr_output function can be obtained using:

print(res_ni_spline_t)
#> This is a valid reslr output object with 104 observations and  1 site(s).
#> There are  1  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> The model used was the Noisy Input Spline in time model.
#> The input data has been run via reslr_mcmc and has produced 3000 iterations over 3 MCMC chains.

Step 4: The convergence of the algorithm is examined and he parameter estimates from the model can be investigated using the following:

summary(res_ni_spline_t)
#> No convergence issues detected.
#> # A tibble: 2 × 7
#>   variable      mean      sd     mad       q5    q95  rhat
#>   <chr>        <num>   <num>   <num>    <num>  <num> <num>
#> 1 sigma_beta 2.07    0.661   0.530   1.29     3.35    1.00
#> 2 sigma_y    0.00622 0.00466 0.00466 0.000500 0.0154  1.00

If the model run has the package will print: “No convergence issues detected”. If the package prints: “Convergence issues detected, a longer run is necessary”. The user is required to update the reslr_mcmc function with additional iterations in the following manner:

res_ni_spline_t <- reslr_mcmc(
  input_data = CedarIslandNC,
  model_type = "ni_spline_t",
  # Update these values
  n_iterations = 6000, # Number of iterations
  n_burnin = 1000, # Number of iterations to discard at the beginning
  n_thin = 4, # Reduces number of output samples to save memory and computation time
  n_chains = 3 # Number of Markov chains
)

For the parameter estimates, we can present the standard deviation associated with the NI spline time model. Where “sigma_beta” highlights the variation associated with the spline coefficient for the spline in time and “sigma_y” presenting the overall variation of the model.

Step 5: the results from the ni_spline_t model can be illustrated using:

plot(res_ni_spline_t,
  plot_type = "model_fit_plot",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)"
)

The output of this function is a graph of the input data, i.e. Age and RSL and associated uncertainty boxes, and the model fit with 95 % credible interval. The caption provides the model type used and number of proxy sites and tide gauge sites used and can be removed if necessary with plot_caption = FALSE. In order to view the rate of change plot, the following setting should be used:

plot(res_ni_spline_t,
  plot_type = "rate_plot",
  xlab = "Year (CE)",
  y_rate_lab = "Rate of Change (mm per year)"
)

This prints the plot of the rate of change with 95 % credible intervals. Again, the caption provides the model type, number of proxy sites and tide gauge sites that were used.

To examine the data creating these plots the user types the following:

output_dataframes <- res_ni_spline_t$output_dataframes
head(output_dataframes)
#>   Longitude Latitude                       SiteName data_type_id  Age      pred
#> 1    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -800 -2.311456
#> 2    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -750 -2.316224
#> 3    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -700 -2.316846
#> 4    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -650 -2.313524
#> 5    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -600 -2.306459
#> 6    -76.38   34.971 Cedar Island,\n North Carolina  ProxyRecord -550 -2.295854
#>         upr       lwr   rate_pred    rate_upr  rate_lwr  CI
#> 1 -2.395822 -2.224850 -0.13819005 -0.66088582 0.3964452 95%
#> 2 -2.382671 -2.248344 -0.05323343 -0.49746281 0.3934967 95%
#> 3 -2.372799 -2.260280  0.02767858 -0.34356103 0.4017406 95%
#> 4 -2.363248 -2.262135  0.10454578 -0.20120408 0.4124406 95%
#> 5 -2.354248 -2.258640  0.17736817 -0.07123049 0.4231023 95%
#> 6 -2.344469 -2.247720  0.24614575  0.05076784 0.4398068 95%

Noisy input spline in space time (“ni_spline_st”)

The Noisy input spline in space time examines changes in RSL over multiple locations and throughout time. For this model, a minimum of 2 proxy sites should be used and tide gauge data provides insight into recent changes in RSL if the user is requires_ As an example, we will filter the example dataset NAACproxydata to select two sites to demonstrate the process:

# For 2 site
multi_site <- reslr::NAACproxydata %>%
  dplyr::filter(Site %in% c("Cedar Island", "Nassau"))

Step 1: Load in the data using the reslr_load function:

multi_site_input <- reslr_load(
  data = multi_site,
  include_tide_gauge = FALSE,
  include_linear_rate = FALSE,
  TG_minimum_dist_proxy = FALSE,
  list_preferred_TGs = NULL,
  all_TG_1deg = FALSE,
  prediction_grid_res = 50,
  sediment_average_TG = 10
)

In this function, the user can select to add tide gauge data and estimates for linear_rate, by changing include_tide_gauge = TRUE and include_linear_rate = TRUE respectfully. If include_tide_gauge = TRUE the user must decide if they require the closest tide gauge i.e. TG_minimum_dist_proxy = TRUE, or select specific tide gauge i.e. list_preferred_TGs = c("ARGENTIA"), or all tide gauges within 1 degree of the proxy site i.e. all_TG_1deg = TRUE. The default setting is rolling_window_average = 10 which corresponds to sediment accumulation rates of the proxy records, yet the user has the ability to alter this sediment accumulation rate.

Note that for a spline in space time, we recommend using the default settings as demonstrated in the above code chunk or investigating the resulting plots if additional tide gauge data could provide insight into recent changes. The user can alter the resolution of the output plots using prediction_grid_res with the default set at 50 years. The output of this function is a list of two dataframes called data and data_grid. - The data dataframe is the inputted data with additional columns for the linear_rate, linear_rate_err and data_type_id which will contain two options, “ProxyRecord” or “TideGaugeData”. It can be accessed by:

data <- multi_site_input$data
head(data)
#>    Region   Site Latitude Longitude   RSL  Age Age_err RSL_err
#> 1 Florida Nassau   30.587   -81.666  0.05 2002    4.25    0.07
#> 2 Florida Nassau   30.587   -81.666  0.03 1990    5.50    0.07
#> 3 Florida Nassau   30.587   -81.666  0.01 1980    4.25    0.07
#> 4 Florida Nassau   30.587   -81.666 -0.01 1974    4.50    0.07
#> 5 Florida Nassau   30.587   -81.666 -0.03 1964    9.50    0.07
#> 6 Florida Nassau   30.587   -81.666 -0.05 1936   10.75    0.07
#>            SiteName data_type_id
#> 1 Nassau,\n Florida  ProxyRecord
#> 2 Nassau,\n Florida  ProxyRecord
#> 3 Nassau,\n Florida  ProxyRecord
#> 4 Nassau,\n Florida  ProxyRecord
#> 5 Nassau,\n Florida  ProxyRecord
#> 6 Nassau,\n Florida  ProxyRecord
data_grid <- multi_site_input$data_grid
head(data_grid)
#> # A tibble: 6 × 5
#> # Groups:   SiteName [1]
#>   Longitude Latitude SiteName                         data_type_id   Age
#>       <dbl>    <dbl> <fct>                            <fct>        <dbl>
#> 1     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -800
#> 2     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -750
#> 3     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -700
#> 4     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -650
#> 5     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -600
#> 6     -76.4     35.0 "Cedar Island,\n North Carolina" ProxyRecord   -550

Step 1a: A brief insight into the outputs of the reslr_input function can be obtained using:

print(multi_site_input)
#> This is a valid reslr input object with 169 observations and  2 site(s).
#> There are  2  proxy site(s) and  0  tide gauge site(s).
#> The age units are; Common Era. 
#> Decadally averaged tide gauge data was not included. It is recommended for the ni_gam_decomp model 
#> The linear_rate or linear_rate_err was not included. It is required for the ni_gam_decomp model

Step 2: Plotting the data the raw data with:

plot(
  x = multi_site_input,
  title = "Plot of the raw data",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)",
  plot_proxy_records = TRUE,
  plot_tide_gauges = FALSE
)

This will produce a plot of Age on the x-axis and Relative Sea Level on the y-axis in meters. Grey boxes represent the uncertainty associated with the vertical and horizontal uncertainty. The black data points are the midpoints of these uncertainty boxes. The separate sites will appear in separate windows on the plot. The extra arguments can be used which allows the user to updated the titles and axis labels. The caption plot_caption, included by default, provides the number of proxy sites and tide gauge sites that will be used in the model and can be removed if required plot_caption = FALSE. In addition, the user can select to plot the additional tide gauge data, plot_tide_gauge = TRUE.

Step 3: Run the model for the two sites.

res_ni_spline_st <- reslr_mcmc(
  input_data = multi_site_input,
  model_type = "ni_spline_st",
  CI = 0.95
)

The output object jags_output.ni_spline_st stores the JAGS model run. The CI setting allows the user to set the credible intervals, the current default is CI = 0.95. Note that additional computational run time is required for this model compared with the ni_spline_t.

Step 3a: A brief insight into the outputs of the reslr_output function can be obtained using:

print(res_ni_spline_st)

Step 4: The convergence of the algorithm is examined and he parameter estimates from the model can be investigated using the following:

summary(res_ni_spline_st)

If the model run has the package will print: “No convergence issues detected”. If the package prints: “Convergence issues detected, a longer run is necessary”. The user is required to update the reslr_mcmc function with additional iterations in the following manner:

res_ni_spline_st <- reslr::reslr_mcmc(
  input_data = multi_site_input,
  model_type = "ni_spline_st",
  # Update these values
  n_iterations = 6000, # Number of iterations
  n_burnin = 1000, # Number of iterations to discard at the beginning
  n_thin = 4, # Reduces number of output samples to save memory and computation time
  n_chains = 3 # Number of Markov chains
)

For the parameter estimates, we can present the standard deviation associated with the NI spline space time model. Where “sigma_beta” highlights the variation associated with the spline coefficient of the spline in time and “sigma_y” presenting the overall variation.

Step 5: the results from the ni_spline_st model can be illustrated using:

plot(res_ni_spline_st,
  plot_type = "model_fit_plot",
  xlab = "Year (CE)",
  ylab = "Relative Sea Level (m)"
)