Global Sensitivity Analysis
A Julia package which implements global sensitivity analysis methods. It is also created to work in concert with the sensitivity analysis functionality of [Mimi.jl][https://www.mimiframework.org].
Much of this package is based on SALib (Herman and Usher, 2017) which implements several global sensitivity analysis measures in Python. The package seeks to implement several of these same algorithms in Julia using a clear, userfriendly API.
The package currently includes the following methods:

Sobol Sensitivity Analysis (Sobol 2001, Saltelli 2002, Saltelli et al. 2010)

Delta MomentIndependent Measure (Borgonovo 2007, Plischke et al. 2013)
The API
The API contains two primary functions: sample
and analyze
. These two functions call methods based on the typeparameterization of their data
argument, which is either of type SobolData
or DeltaData
.
Note: For now the sample
function will call the most used sampling protocol for the particular method, (Sobol sequence for Sobol method and Latin Hypercube sampling for Delta method), but in the future this will be rearranged and generalized since, for example, the Delta method can also just as well use Sobol sequence sampling and other methods.
Sobol Sensitivity Analyis
Sampling with sample
is the first of the two main steps in an analysis, generating the model inputs to be run through a model of choice and produce the outputs analyzed in the analyze
function. The signature for this function is as follows.
"""
sample(data::SobolData)
Generate a matrix containing the model inputs for Sobol sensitivity analysis with
the information in `data`. In this function we apply Saltelli's
extension of the Sobol sequence. Saltelli's scheme extends the Sobol sequence in
a way to reduce the error rates in the resulting sensitivity index calculations.
The resulting matrix has `N` * (`D` + 2) rows, where `D` is the number of parameters
and `N` is the number of samples.
"""
The single argument to this function is of type SobolData
, a custom type designed to hold all information needed for sampling and analysis. A SobolData
struct is parameterized by a params
dictionary (NOTE that this must be an OrderedDict
, not a Dict
) which maps parameter names to their Distributions, calc_second_order
determining whether or not to calculate secondorder sensitivity indices, and the desired number of runs N
.
"""
SobolData
A struct which holds all information needed for the sampling and analysis of a
specific problem using Sobol Analysis:
`params::Union{OrderedDict{Symbol, <:Any}, Nothing} = nothing`: a dictionary mapping parameter names to their Distribution
`calc_second_order::Bool = true`: whether or not to calculate second order sensitivity indices
`N::Int = 1000`: the number of runs
"""
After sampling with sample
, use the resulting matrix of parameter combinations to run your model, producing a vector of results. The next and final step is to analyze the results with your model_output
using the analyze
function with the signature below. This function takes the same SobolData
as sample
, as well as the model_output
vector and produces a dictionary of results. This dictionary will include the :firstorder
, :totalorder
indices and (optionally) confidence intervals for each parameter.
"""
function analyze(data::SobolData, model_output::AbstractArray{<:Number, S}; num_resamples::Union{Nothing, Int} = 1_000, conf_level::Union{Nothing, Number} = 0.95, progress_meter::Bool = true, N_override::Union{Nothing, Integer}=nothing)
Performs a Sobol Analysis on the `model_output` produced with the problem defined by the information in `data` and returns the a dictionary of results with the sensitivity indices and respective confidence intervals for each of the
parameters defined using the `num_resamples` and `conf_level` keyword args. If these are `nothing` than no confidence intervals will be calculated. The `progress_meter` keyword argument indicates whether a progress meter will be displayed and defaults to true. The `N_override` keyword argument allows users to override the `N` used in a specific `analyze` call to analyze just a subset (useful for convergence graphs).
"""
An example of the basic flow can be found in src/main.jl
using the Ishigami test function in src/test_functions/ishigami.jl
, and is copied and commented below for convenience.
using Distributions
using DataStructures
include("src/sample_sobol.jl")
include("src/analyze_sobol.jl")
include("src/test_functions/ishigami.jl")
# define the data
data = SobolData(
params = OrderedDict(:x1 => Uniform(3.14159265359, 3.14159265359),
:x2 => Uniform(3.14159265359, 3.14159265359),
:x3 => Uniform(3.14159265359, 3.14159265359)),
N = 1000
)
# generate samples using Sobol sequence
samples = sample(data)
# run model (example)
Y = ishigami(samples)
# perform Sobol Analysis
analyze(data, Y)
Delta MomentIndependent Measure
Sampling with sample
is the first of the two main steps in an analysis, generating the model inputs to be run through a model of choice and produce the outputs analyzed in the analyze
function. The signature for this function is as follows.
"""
sample(data::DeltaData)
Generate a matrix containing the model inputs for Delta MomentIndependent Measure sensitivity analysis with the information in the `data`. In this function we apply Latin Hypercube Sampling. The resulting matrix has `N` columns * `D` rows, where `D` is the number of parameters and `N` is the number of samples.
"""
The single argument to this function is of type DeltaData
, a custom type designed to hold all information needed for sampling and analysis. A DeltaData
struct is parameterized by a params
dictionary (NOTE that this must be an OrderedDict
, not a Dict
) which maps parameter names to their Distributions and the desired number of runs N
.
"""
DeltaData
A struct which holds all information needed for the sampling and analysis of a
specific problem using Sobol Analysis:
`params::Union{OrderedDict{Symbol, <:Any}, Nothing} = nothing`: a dictionary mapping parameter names to their Distribution
`N::Int = 1000`: the number of runs
"""
After sampling with sample
, use the resulting matrix of parameter combinations to run your model, producing a vector of results. The next and final step is to analyze the results with your model_output
using the analyze
function with the signature below. This function takes the same DeltaData
as sample
, as well as the model_output
vector and produces a dictionary of results. This dictionary will include the :firstorder
, :delta
indices and confidence intervals for each parameter.
function analyze(data::DeltaData, model_input::AbstractArray{<:Number, S1}, model_output::AbstractArray{<:Number, S2}; num_resamples::Int = 1_000, conf_level::Number = 0.95, progress_meter::Bool = true, N_override::Union{Nothing, Integer}=nothing)
Performs a Delta MomentIndependent Analysis on the `model_output` produced with the problem defined by the information in `data` and `model_input` and returns a dictionary of results with the sensitivity indices and respective confidence intervals for each of the parameters defined using the `num_resamples` and `conf_level` keyword args. The `progress_meter` keyword argument indicates whether a progress meter will be displayed and defaults to true. The `N_override` keyword argument allows users to override the `N` used in a specific `analyze` call to analyze just a subset (useful for convergence graphs).
References
References from the peerreviewed literature include:
Borgonovo, E. (2007). A new uncertainty importance measure. Reliability Engineering
& System Safety, 92(6), 771784. Chicago.
Herman, J. and Usher, W. (2017) SALib: An opensource Python library for sensitivity
analysis. Journal of Open Source Software, 2(9).
Plischke, E., Borgonovo, E., & Smith, C. L. (2013). Global sensitivity measures
from given data. European Journal of Operational Research, 226(3), 536550.
Saltelli, A. (2002). "Making best use of model evaluations to compute sensitivity indices."
Computer Physics Communications,145(2):280297, doi:10.1016/S00104655(02)002801.
Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010).
"Variance based sensitivity analysis of model output. Design and estimator
for the total sensitivity index." Computer Physics Communications, 181(2):259270,
doi:10.1016/j.cpc.2009.09.018.
Sobol, I. M. (2001). "Global sensitivity indices for nonlinear mathematical
models and their Monte Carlo estimates." Mathematics and Computers in Simulation,
55(13):271280, doi:10.1016/S03784754(00)002706.
Copyright Information
Some of the code in this package is derivative code based on the work of John Herman, Will Usher, and others:
The MIT License (MIT)
Copyright (c) 20132017 Jon Herman, Will Usher, and others.
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