Basics of simulating sawn timber strength with WoodSimulatR"

In WoodSimulatR: Generate Simulated Sawn Timber Strength Grading Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 5
)

library(WoodSimulatR)
library(magrittr)
library(ggplot2)
pander::panderOptions('knitr.auto.asis', FALSE);

Introduction

The WoodSimulatR package provides functions for generating artificial datasets of sawn wood properties obtained by destructive and non-destructive testing.

An existing dataset containing some of these properties can be enriched by adding simulated values for the missing properties.

Aim of this document

This document aims to provide an overview of the capabilities of the WoodSimulatR package.

On the one hand, we will simulate a dataset with varying parameters, highlighting both the capabilities of the WoodSimulatR functions and the direction where it should go.

On the other hand, we will illustrate the capabilities of WoodSimulatR with respect to simulating grade determining properties for a dataset with different pre-existing variables.

Simulate a whole dataset

Preliminaries

As a quick summary of each dataset, we will show mean and CoV for all variables, split by country and subsample, and we will show the matrix of correlations.

For this, we define the following function:

summ_fun <- function(ds, grp = c('country', 'subsample', 'loadtype')) {
  grp <- intersect(grp, names(ds));
  v <- setdiff(names(ds), grp);

  r <- cor(ds[v]);

  ds <- tibble::add_column(ds, n = 1);
  v <- c('n', v);
  ds <- tidyr::gather(ds, 'property', 'value', !!! rlang::syms(v));
  ds <- dplyr::mutate(
    ds,
    property = factor(
      property,
      levels=v,
      labels=ifelse(v=='n', v, paste0(v, '_mean')),
      ordered = TRUE
    )
  );

  grp <- c(grp, 'property');
  ds <- dplyr::group_by(ds, !!! rlang::syms(grp));

  summ <- dplyr::summarise(
    ds,
    res = if (property[1] == 'n') sprintf('%.0f', sum(value)) else
      sprintf(
      if(property[1] %in% c('f_mean', 'ip_f_mean')) '%.1f (%.0f)' else '%.0f (%.0f)',
      mean(value), 100*sd(value)/mean(value)),
    .groups = 'drop_last'
  );
  pander::pander(
    tidyr::spread(summ, property, res),
    split.tables = Inf
  );

  pander::pander(r)

  invisible(summ);
}

compare_with_def <- function(ds, ssd, target = c('mean', 'cov')) {
  target <- match.arg(target);

  ds <- dplyr::group_by(ds, country);
  summ <- dplyr::summarise(
    ds,
    f_mean.ach = mean(f),
    f_cov.ach = sd(f) / f_mean.ach,
    E_mean.ach = mean(E),
    E_cov.ach = sd(E) / E_mean.ach,
    rho_mean.ach = mean(rho),
    rho_cov.ach = sd(rho) / rho_mean.ach,
    .groups = 'drop_last'
  );

  stopifnot(!anyDuplicated(ssd$country));
  summ <- dplyr::left_join(
    summ,
    dplyr::select(
      dplyr::mutate(ssd, f_cov = f_sd / f_mean, E_cov = E_sd / E_mean, rho_cov = rho_sd / rho_mean), 
      country, f_mean, f_cov, E_mean, E_cov, rho_mean, rho_cov
    ),
    by = 'country'
  );

  summ <- tidyr::pivot_longer(
    summ,
    -country,
    names_to = c('gdpname', '.value'),
    names_sep = '_'
  );
  summ <- dplyr::mutate(
    summ,
    gdpname = factor(gdpname, levels = c('f', 'E', 'rho'), ordered = TRUE)
  );

  if (target == 'mean') {
    ggplot(data = summ, aes(mean.ach, mean)) +
      geom_abline(slope = 1, intercept = 0) +
      geom_text(aes(label = country)) +
      geom_point(alpha = 0.5) +
      facet_wrap(vars(gdpname), scales = 'free') +
      theme(axis.text.x = element_text(angle = 90));
  } else {
    ggplot(data = summ, aes(cov.ach, cov)) +
      geom_abline(slope = 1, intercept = 0) +
      geom_text(aes(label = country)) +
      geom_point(alpha = 0.5) +
      facet_wrap(vars(gdpname), scales = 'free') +
      theme(axis.text.x = element_text(angle = 90));
  }
}

Default dataset

The main function for dataset simulation is simulate_dataset(). It can be called without any further arguments to yield a "default" dataset.

For reproducibility, we will call it with the extra argument random_seed = 12345. This means that we will always generate the same random numbers.

dataset_0 <- simulate_dataset(random_seed = 2345);

summ_fun(dataset_0);

The meaning of the properties in dataset_0 is as follows:

• country, subsample: when analysing properties of sawn wood, the timber typically are grouped by country of origin and possibly by further criteria into subsamples for each country. In a call without further specifications, simulate_dataset() returns generic country names "C1", "C2" etc., and sets the subsample names equal to the country names. Further options for countries and subsamples are explained below.
• f: strength of the sawn timber in N/mm² as obtained by destructive testing (tensile strength or bending strength)
• E: Modulus of elasticity in N/mm² as obtained by destructive testing in tension or in bending
• rho (as from the Greek symbol $\rho$): density of a small clear sample cut from the sawn timber, in kg/m³
• E_dyn_u: dynamic modulus of the green sawn timber in N/mm²
• ip_f: an "indicating property" (IP) for the strength f in N/mm², obtained non-destructively as a function of E_dyn, ip_rho and the total knot area ratio (tKAR) of a knot cluster of length 150mm (knot_tc -- not included in the dataset).
  • IP for tension strength: $ip_f = 11.98 + 0.003913 E_{dyn} - 0.04822 ip_\rho - 34.72 * knot_{tc}$
  • IP for bending strength: $ip_f = 21.67 + 0.004302 E_{dyn} - 0.05366 * ip_\rho - 38.43 knot_{tc}$
• E_dyn: dynamic modulus of the dry sawn timber in N/mm²
• ip_rho: density of the whole board in kg/m³, calculated by weighing the board and dividing the weight by the product of the measured dimensions.

All properties except E_dyn_u are to be taken as measured on the dry timber and corrected to a moisture content of 12%.

Customising options

The default dataset created above relies on the following assumptions:

• It's a dataset for tensile strength of spruce (Picea abies).
• The correlations are taken from the "full" sample of Holzforschung Austria's research project SiOSiP ("SImulation-based Optimisation of Sawn tImber Production", 2014-2017) on Austrian spruce wood.
• After log-transforming $f$, for all variables a normal distribution is assumed.
• We create data for 5000 boards, split equally between four subsamples.
• The means and standard deviations of $f$, $E$ and $rho$ are based on random values chosen from the range of available reference values.
• The reference values for mean and standard deviation of transformed variables ($f$ in our case) are currently enforced exactly instead of statistically -- this has to be improved yet.

All of these assumptions can be modified more or less freely.

Available subsample definitions

For convenience, the WoodSimulatR package contains tables with means and standard deviations for f, E and rho for different countries, obtained in the research projects SiOSiP and GradeWood [@Ranta_Maunus_et_al_2011_] or reported in scientific papers [@Stapel_et_al_2014_; @Rohanova_2014_]. Data from both destructive tension and bending tests are available. Currently, the data is restricted to European spruce (Picea abies, PCAB).

Tensile tests

get_subsample_definitions(loadtype = 't') %>% 
  dplyr::select(-species, -loadtype) %>%
  dplyr::arrange(country) %>%
  pander::pander(split.table = Inf);

Bending tests

get_subsample_definitions(loadtype = 'be') %>% 
  dplyr::select(-species, -loadtype) %>%
  dplyr::arrange(country) %>%
  pander::pander(split.table = Inf);

Simulated dataset with data from specific countries

ssd_c <- get_subsample_definitions(
  country = c('at', 'de', 'fi', 'pl', 'se', 'si', 'sk'),
  loadtype = 't'
);

dataset_c <- simulate_dataset(
  random_seed = 12345,
  n = 5000,
  subsets = ssd_c
);

summ_fun(dataset_c);

Compare achieved means with the defined values. It can be seen that the means of $f$ are met exactly, while the means of $E$ and $rho$ are only met approximately, which is the desideratum when we are dealing with simulation.

compare_with_def(dataset_c, ssd_c, 'm')

Compare achieved coefficients of variation with the defined values. Again, we have undesirable exact values for $f$.

compare_with_def(dataset_c, ssd_c, 'cov')

Different subsample sizes

ssd_cn <- get_subsample_definitions(
  country = c(at = 1, de = 3, fi = 1.5, pl = 2, se = 3, si = 1, sk = 1),
  loadtype = 't'
);

dataset_cn <- simulate_dataset(
  random_seed = 12345,
  n = 5000,
  subsets = ssd_cn
);

summ_fun(dataset_cn);

Own specification of means and standard deviations

In a similar manner to the predefined country specifications, we can also define our own. Since Version 0.6.0, we can also used different sample identifier columns (instead of the standard "country" and "subsample") -- for details, check the help on simulate_dataset().

As an example, we define different properties for boards with different cross sections (width and thickness, given in mm).

ssd_custom <- tibble::tribble(
  ~width, ~thickness, ~f_mean, ~f_sd,
      80,     40,      27.5,    9.0,
     140,     40,      29.4,    9.7,
     160,     60,      31.6,    9.3,
     200,     50,      30.2,   11.4, 
     240,     95,      25.5,    4.8,
     250,     40,      25.3,   11.2
);

dataset_custom <- simulate_dataset(
  random_seed = 12345,
  n = 5000,
  subsets = ssd_custom
);

summ_fun(dataset_custom, grp = c('width', 'thickness', 'loadtype'));

Further available options

• bending strength simulation
• without log-transform
• from own data using simbase_covar()

Add simulated values to a dataset

For adding simulated values to a dataset, we first need to establish the relationship between these values and some variables in the dataset.

In WoodSimulatR, relationships are established in the following way:

1. We determine the covariance matrix and the means of a set of variables, based on some kind of learning dataset. This is done using the function simbase_covar(); the resulting simbase has class "simbase_covar".
2. We also have the option to establish different relationships for different subsets of the data, e.g. for different countries of origin. This is done by grouping the dataset accordingly before calling simbase_covar(); the resulting simbase has class "simbase_list".

For both these options, it is possible to transform the involved variables.

To visualise the result of the simulation, we use scatterplots and define them in the following function:

plot_sim_gdp <- function(ds, simb, simulated_vars, ...) {
  extra_aes <- rlang::enexprs(...);
  ds <- dplyr::rename(ds, f_ref = f, E_ref = E, rho_ref = rho);
  if (!any(simulated_vars %in% names(ds))) ds <- simulate_conditionally(data = ds, simbase = simb);
  ds <- tidyr::pivot_longer(
    data = ds,
    cols = tidyselect::any_of(c('f_ref', 'E_ref', 'rho_ref', simulated_vars)),
    names_to = c('name', '.value'),
    names_sep = '_'
  );
  ds <- dplyr::mutate(
    ds,
    name = factor(name, levels = c('f', 'E', 'rho'), ordered = TRUE)
  );
  simname <- names(ds);
  simname <- simname[dplyr::cumany(simname == 'name')];
  simname <- setdiff(simname, c('name', 'ref'));
  stopifnot(length(simname) == 1);
  ggplot(data = ds, mapping = aes(.data[[simname]], ref, !!!extra_aes)) +
    geom_point(alpha = .2, shape = 20) +
    geom_abline(slope = 1, intercept = 0, alpha = .5, linetype = 'twodash') +
    facet_wrap(vars(name), scales = 'free') +
    theme(axis.text.x = element_text(angle = 90));
} # undebug(plot_sim_gdp)

simbase_covar without transformation

The main approach in WoodSimulatR is to conditionally simulate based on the means and the covariance matrix. As a start, we create basis data for the simulation without applying any transformation.

As we later want to add simulated GDP values to a dataset which already contains GDP values, we rename the GDP values for the simbase_covar to some other names not yet present in the target dataset, by suffixing with _siml (for SIMulation with Linear relationships)

sb_untransf <- dataset_0 %>%
  dplyr::rename(f_siml = f, E_siml = E, rho_siml = rho) %>%
  simbase_covar(
    variables = c('f_siml', 'E_siml', 'rho_siml', 'ip_f', 'E_dyn', 'ip_rho')
  );

sb_untransf;

Adding the simulated GDP values to a dataset is done by calling simulate_conditionally().

dataset_c_sim <- simulate_conditionally(dataset_c, sb_untransf);
names(dataset_c_sim) %>% pander::pander();

For a visual comparison:

plot_sim_gdp(dataset_c_sim, sb_untransf, c('f_siml', 'E_siml', 'rho_siml'));

This looks good for $E$ and $\rho$, but wrong in the $f$ simulation.

simbase_covar with log-transformed $f$

We might try using transforms to improve the result. For this, we have to pass a list with named entries corresponding to the GDP we want to transform.

The entry itself must be an object of class "trans" (from the package scales). As we want to use a log-transform, the required entry is scales::log_trans().

sb_transf <- dataset_0 %>%
  dplyr::rename(f_simt = f, E_simt = E, rho_simt = rho) %>%
  simbase_covar(
    variables = c('f_simt', 'E_simt', 'rho_simt', 'ip_f', 'E_dyn', 'ip_rho'),
    transforms = list(f_simt = scales::log_trans())
  );
dataset_c_sim <- simulate_conditionally(dataset_c_sim, sb_transf);
plot_sim_gdp(dataset_c_sim, sb_transf, c('f_simt', 'E_simt', 'rho_simt'));

Now, this looks much better (which is no surprise, as dataset_c itself has been simulated with lognormal $f$).

simbase_covar with log-transformed $f$ and derived on a grouped dataset

If we group the reference dataset (dataset_0), e.g. by country, we get an object of class "simbase_list" with separate simbases for each group (technically, this is a tibble with the grouping variables and an extra column .simbase which contains several objects of class "simbase_covar").

sb_group <- dataset_0 %>%
  dplyr::group_by(country) %>%
  dplyr::rename(f_simg = f, E_simg = E, rho_simg = rho) %>%
  simbase_covar(
    variables = c('f_simg', 'E_simg', 'rho_simg', 'ip_f', 'E_dyn', 'ip_rho'),
    transforms = list(f_simg = scales::log_trans())
  );

sb_group

If we add variables to a dataset using such a "simbase_list", it is required that all grouping variables stored in the "simbase_list" object are also available in this dataset.

In our case: the dataset must contain the variable "country". Values of "country" which do not also exist in our "simbase" object will result in NA values for the variables to be simulated.

Therefore, we add the variables in this case not to the dataset dataset_c (which has different values for "country") but to the dataset_0 itself.

dataset_0_sim <- simulate_conditionally(dataset_0, sb_group);
plot_sim_gdp(dataset_0_sim, sb_group, c('f_simg', 'E_simg', 'rho_simg'), colour=country);

Simulate a whole dataset, based on a simbase_list object

Simbase objects of class "simbase_list" can also be used for simulating an entire dataset, as long as the "simbase_list" only has the grouping variable(s) "country" and/or "subsample", and as long as the value combinations in "country"/"subsample" match those given in the "subsets" argument to the function simulate_dataset.

To demonstrate, we calculate a "simbase_list" based on the dataset_c created above. Here, we do not rename any of the variables.

sb_group_c <- dataset_c %>%
  dplyr::group_by(country) %>%
  simbase_covar(
    variables = c('f', 'E', 'rho', 'ip_f', 'E_dyn', 'ip_rho'),
    transforms = list(f = scales::log_trans())
  );

sb_group_c

This "simbase_list" is now used as input to simulate_dataset with the subset definitions used previously (ssd_cn).

dataset_cn2 <- simulate_dataset(
  random_seed = 12345,
  n = 5000,
  subsets = ssd_cn,
  simbase = sb_group_c
);

summ_fun(dataset_cn2);

Conclusions

The package WoodSimulatR has functions for simulating entire datasets of sawn timber properties, both based on internal definitions and on externally supplied base data.

WoodSimulatR also has functions for adding simulated grade determining properties (or other properties) to a given dataset, based on a covariance matrix approach.

The functions for adding simulated variables are suitable for all kinds of datasets, if one calculates an appropriate simbase_covar object oneself, by a call to simbase_covar using reference data.

The simulation methods also support variable transformations to accommodate non-normally distributed variables.

References

WoodSimulatR documentation built on June 20, 2022, 9:05 a.m.