Training emulators

The first step, given a dataset (here the SIRSample training set), is to train an emulator to each output. We need to provide details about the parameters (and their ranges) to the model and the outputs from the model, as well as the data itself. Then the function emulator_from_data(data, outputs, ranges, ...) produces a set of Emulator objects, as a named list.

input_ranges <- list(aSI = c(0.1, 0.8), aIR = c(0, 0.5), aSR = c(0, 0.05))
output_names <- c('nS', 'nI', 'nR')
wave1_emulators <- emulator_from_data(SIRSample$training, output_names, input_ranges)

An emulator is an R6 object, with the requisite print statement giving the core details of the emulator. It also contains a Correlator object with similar R6 properties.

wave1_emulators$nS
#> Parameters and ranges:  aSI: c(0.1, 0.8): aIR: c(0, 0.5): aSR: c(0, 0.05) 
#> Specifications: 
#>   Basis functions:  (Intercept); aSI; aIR; aSR; aSI:aIR; aIR:aSR 
#>   Active variables aSI; aIR; aSR 
#>   Regression Surface Expectation:  572.6947; -404.3915; 253.3451; 33.9916; 117.0669; -65.1342 
#>   Regression surface Variance (eigenvalues):  0; 0; 0; 0; 0; 0 
#> Correlation Structure: 
#> Bayes-adjusted emulator - prior specifications listed. 
#>   Variance (Representative):  5285.624 
#>   Expectation:  0 
#>   Correlation type: exp_sq 
#>   Hyperparameters:  theta: 0.9033 
#>   Nugget term: 0 
#> Mixed covariance:  0 0 0 0 0 0

wave1_emulators$nS$corr
#>  Correlation type: exp_sq 
#>  Hyperparameters:  theta: 0.9033 
#>  Nugget term: 0

The default training process assumes the following:

• The correlation structure is exponential-squared;
• The regression surface is no more than quadratic in the inputs;
• The specifics of regression parameters, correlation length, and overall variance is to be determined during the training process;
• The output emulators are trained using Bayes Linear adjustment before being outputted.

Any of these default behaviours can be modified; for instance, manually setting the correlation lengths requires the provision of a numeric vector of correlation lengths:

custom_emulators <- emulator_from_data(SIRSample$training, output_names, input_ranges,
specified_priors = list(hyper_p = c(0.6, 0.7, 0.85)))
custom_emulators$nR$corr
#>  Correlation type: exp_sq 
#>  Hyperparameters:  theta: 0.85 
#>  Nugget term: 0.05

One could instead specify a different correlation function, for instance Matérn:

matern_emulators <- emulator_from_data(SIRSample$training, output_names, input_ranges, corr_name = "matern")

In any event, having trained emulators it is crucial to ensure that they are accurately representing the model behaviour, at the very least over the region of potential interest. The validation_diagnostics function produces some simple plots to check this. It requires a separate hold-out or validation set of model runs. Some of the diagnostic functions also require a set of targets - values which we expect the model to match to. These can be specified as a named list, each element consisting of an observation value val and an associated uncertainty sigma; alternatively we can provide an upper and lower bound \([a,b]\) corresponding to the target range. A combination thereof can be provided in a given set of targets.

targets <- list(
  nS = c(580, 651),
  nI = list(val = 169, sigma = 8.45),
  nR = c(199, 221)
)

invalid_points <- validation_diagnostics(wave1_emulators, targets, SIRSample$validation, plt = TRUE)

Visualising Emulator Behaviour

Emulators can be used as a (statistical) surrogate for the model, and in particular can be used to investigate parameter behaviour across the space. The main function for doing so here is emulator_plot. The expectation, variance, and standard deviation can be plotted across any two-dimensional slice of the parameter space. If a list of emulators is provided, all relevant outputs are plotted; otherwise the single chosen emulator is outputted.

emulator_plot(wave1_emulators)

emulator_plot(wave1_emulators$nS, params = c('aSI', 'aIR'), fixed_vals = list(aSR = 0.03))

emulator_plot(wave1_emulators$nI, plot_type = 'var')

emulator_plot(wave1_emulators$nI, plot_type = 'sd')

Almost all the return values of plots produced by hmer are ggplot objects, and so can be augmented after the fact.

emulator_plot(wave1_emulators$nS, params = c('aIR', 'aSI')) + ggplot2::geom_point(data = SIRSample$training, ggplot2::aes(x = aSI, y = aIR))

Using these, we can plot implausibility across the parameter space, again using emulator_plot. We can also combine the implausibilities into a combined measure of point suitability by considering maximum implausibility across all outputs. While traditionally, implausibility plots are green-red, a colourblind friendly palette is available with the use of cb=TRUE.

emulator_plot(wave1_emulators, plot_type = 'imp', targets = targets)

emulator_plot(wave1_emulators, plot_type = 'nimp', targets = targets, cb = TRUE)

While these plots represent slices of parameter space, other functionality allows a rough understanding of the behaviour across the full space.In particular, the plot_lattice function provides a plot of optical depth and minimum implausibility for all parameter combinations. Since this is generating a plot that represents the full space and not a two-dimensional slice, it typically takes a little longer than the previous plots considered.

plot_lattice(wave1_emulators, targets)
#> `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
#> Warning: Removed 27 rows containing missing values or values outside the scale range
#> (`geom_smooth()`).
#> `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
#> Warning: Removed 25 rows containing missing values or values outside the scale range
#> (`geom_smooth()`).
#> `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Proposing new points for the model

In preparation for proposing new points, we can also look at the optimal ‘cutoff’, if we wish to remove a certain amount of parameter space, using the space_removed function.

space_removed(wave1_emulators, targets, ppd = 15) + geom_vline(xintercept = 3, lty = 2)

We see, for instance, that choosing a cutoff of 3 (roughly corresponding to a 3-sigma deviation from the target value) is sufficient to rule out over 95% of the space - coupled with the fact that our diagnostics were so healthy suggests that this is a very reasonable first cut. This is heuristic based on the number of points we provide to the function: the ppd (points-per-dimension) argument determines the size of the grid we evaluate over. Here we have considered \(15^3=3375\) points - if we wanted to examine the space removed in greater depth we could increase the value of this argument. A similar argument exists for plot_lattice.

Eventually, the aim of history matching is to propose new points which can enter into the model, the results of which can train new emulators, repeating the cycle; the generate_new_design function does just that. There are multiple options that can be passed to this function: the default behaviour should be adequate for most situations, with the consideration of the cutoff value mentioned above. Information about the progress of the point proposal can be suppressed by setting the argument verbose = FALSE - by default it will be TRUE if we are stepping through this by hand, but not if we run this at arm’s length (for example on a cluster).

new_points <- generate_new_design(wave1_emulators, 90, targets)
plot(rbind(rbind(SIRSample$training, SIRSample$validation)[,names(input_ranges)], new_points), pch = 16, cex = 0.8, col = rep(c('black', 'blue'), each = 90))

The plot command above is replicated in the hmer package itself, to a degree. We first create a list of ‘waves’; each element of the list is a data.frame containing all points proposed at that time. For now, we merely look at the proposed points and not the resulting model values.

wave.points <- list(SIRSample$training[,c('aSI', 'aIR', 'aSR')], new_points)

There are a number of explanatory plots that can be produced, given a set of wave data. Some require the model to have been run on the points, but one that doesn’t is wave_points:

wave_points(wave.points, c('aSI', 'aIR', 'aSR'))

This shows the evolution of the point proposals as the waves go on. In each lower-diagonal plot, we have a plot of the proposed points with respect to two of the parameters; in the diagonal is a density plot for each parameter. This allows us to consider the shape of the non-implausible space as we proceed through the history match. Related functions, with similar output, are wave_values and wave_dependencies, both of which we detail in the next section.

Multi-wave plotting

As mentioned above, functions exist to visualise the evolution of the non-implausible space and the points proposed. By definition such plots require us to have performed multiple waves of history matching - rather than repeat the commands above we will use the SIRMultiWaveData dataset provided. This contains a list of four waves of points from this model (including wave 0), along with the model evaluations. We can use this to look at dependencies of the outputs on the inputs (wave_dependencies), the evolution of the output values obtained (wave_values), and the overall behaviour of the model runs (simulator_plot).

wave_values(SIRMultiWaveData, targets)

wave_dependencies(SIRMultiWaveData, targets)

simulator_plot(SIRMultiWaveData, targets, barcol = 'white')

All of these plots, as well as wave_points, are wrapped into a single function, diagnostic_wrap. This will also produce log-transformed and normalised versions of the simulator plot, for situations where the outputs have vastly different magnitudes (for example, when matching to a proportion of infected people as well as raw case numbers).

Hands-off Emulation and History Matching

Each of the functions in constructing emulators has a number of options with which the default behaviour can be modified. However, if investigation of the emulator structure is not required and one simply wants the next set of proposed points, then the full_wave function allows this. It requires only a set of points to train the emulators on. If applying this at later waves, the emulators from previous waves are provided as a list to the old_emulators argument.

f_w <- suppressWarnings(full_wave(do.call('rbind.data.frame', SIRSample), list(aSI = c(0.1, 0.8), aIR = c(0, 0.5), aSR = c(0, 0.05)), targets))

This provides two things: a next set of points to evaluate the model on, and the emulators that gave rise to them. One should exercise caution when using this - while the function does what it can to ensure that emulators trained and points proposed are done in the most reasonable way, it is no substitute for expert judgement and consideration. Nevertheless, it can be a useful tool for a ‘first pass’ at emulating a system.