Getting started

Assume we have a simple model, fsens, which depends on two parameters, a and b and produces two outputs, target1 and target2.

fsens = (a,b) -> (;target1 = 10a + b -1, target2 = a + b -0.5)

Our knowledge about reasonable model parameters is encoded by a prior probability distribution. We can specify those by the kind of distribution, its mode and an upper quantile.

using SubglobalSensitivityAnalysis, Distributions
install_R_dependencies(["sensitivity"])

paramsModeUpperRows = [
    (:a, LogNormal, 0.2 , 0.5),
    (:b, LogitNormal, 0.7 , 0.9),
]

┌ Warning: RCall.jl: Loading required namespace: sensitivity
│ Registered S3 method overwritten by 'sensitivity':
│   method    from
│   print.src dplyr
└ @ RCall ~/.julia/packages/RCall/YrsKg/src/io.jl:172

The output DataFrame reports

the estimated index and confidence bounds (column value, cflower, cfupper)
for each of the parameter/index_type/output combinations

We can provide this directly to estimate_subglobal_sobol_indices below, or we estimate/specify distribution parameters directly in a DataFrame with column :dist.

df_dist = fit_distributions(paramsModeUpperRows)

2×5 DataFrame

Row	par	dType	mode	upper	dist
	Symbol	UnionAll	Float64	Float64	Distribu…
1	a	LogNormal	0.2	0.5	Distributions.LogNormal{Float64}(μ=-1.60944, σ=0.467504)
2	b	LogitNormal	0.7	0.9	Distributions.LogitNormal{Float64}(μ=0.847298, σ=0.688751)

While these distributions are reasonable for each parameter, there are probably parameter combinations that produce unreasonable results. Hence, we want to restrict our analysis to a parameter space around a central parameter vector, p0.

p0 = Dict(:a => 0.34, :b => 0.6)

By default a range around p0 is created that covers 20% of the cumulative probability range, i.e a span of 0.2.

For this range 20% of the area under the probability density function is covered.

The design matrix for the sensitivity analysis is constructed in the cumulative densities and transformed to parameter values. For each of the parameter vectors of the design matrix an output is computed. Now the Sobol indices and their confidence ranges can be computed for this output.

All this encapsulated by function estimate_subglobal_sobol_indices.

# note, for real analysis use a larger sample size
df_sobol = estimate_subglobal_sobol_indices(fsens, df_dist, p0; n_sample = 50)
df_sobol

8×6 DataFrame

Row	par	index	value	cf_lower	cf_upper	target
	Symbol	Symbol	Float64	Float64	Float64	Symbol
1	a	first_order	0.995291	0.941648	1.04893	target1
2	b	first_order	-0.0166506	-0.279313	0.246012	target1
3	a	total	1.01434	0.748023	1.28066	target1
4	b	total	0.00355108	-0.0458729	0.0529751	target1
5	a	first_order	0.649036	0.485109	0.812963	target2
6	b	first_order	0.259124	0.0952284	0.42302	target2
7	a	total	0.719884	0.501837	0.937931	target2
8	b	total	0.250996	0.164651	0.337341	target2

The resulting DataFrame reports:

the estimated Sobol indices and their confidence bounds (columns value, cflower, cfupper)
for all the combinations of parameter, which index, and output (columns par, index, target)