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<title>Variance-based sensitivity analysis</title>

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Variance-based sensitivity analysis

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Variance-based sensitivity analysis (often referred to as the Sobol’ method or Sobol’ indices, after <a href="Ilya_M._Sobol" class="mw-redirect" title="Ilya M. Sobol">Ilya M. Sobol’</a>) is a form of global <a href="Sensitivity_analysis" title="Sensitivity analysis">sensitivity analysis</a>.<a href="#cite_note-Sobol2001-1">[1]</a><a href="#cite_note-Primer-2">[2]</a> Working within a <a href="Probability" title="Probability">probabilistic</a> framework, it decomposes the <a href="Variance" title="Variance">variance</a> of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. For example, given a model with two inputs and one output, one might find that 70% of the output variance is caused by the variance in the first input, 20% by the variance in the second, and 10% due to <a href="Interaction_(statistics)" title="Interaction (statistics)">interactions</a> between the two. These percentages are directly interpreted as measures of sensitivity. Variance-based measures of sensitivity are attractive because they measure sensitivity across the whole input space (i.e. it is a global method), they can deal with <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a> responses, and they can measure the effect of interactions in non-<a href="Additive_map" title="Additive map">additive</a> systems.<a href="#cite_note-OAT-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Decomposition_of_variance">Decomposition of variance</h2></div>

From a <a href="Black_box" title="Black box">black box</a> perspective, any <a href="Mathematical_model" title="Mathematical model">model</a> may be viewed as a function Y=f(X), where X is a vector of d uncertain model inputs {X1, X2, ... Xd}, and Y is a chosen <a href="Univariate" title="Univariate">univariate</a> model output (note that this approach examines scalar model outputs, but multiple outputs can be analysed by multiple independent sensitivity analyses). Furthermore, it will be assumed that the inputs are independently and uniformly distributed within the unit hypercube, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}\in [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}\in [0,1]}</annotation>

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</math><img src="./10589c4dfecffbd9da44b08aa9124b7df7f25a5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.217ex; height:2.843ex;" alt="{\displaystyle X_{i}\in [0,1]}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,...,d}">

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<annotation encoding="application/x-tex">{\displaystyle i=1,2,...,d}</annotation>

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</math><img src="./666d3b3d893ab8b642f8c29edbfd5a196d52880a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.645ex; height:2.509ex;" alt="{\displaystyle i=1,2,...,d}" loading="lazy">. This incurs no loss of generality because any input space can be transformed onto this unit hypercube. f(X) may be decomposed in the following way,<a href="#cite_note-Sob1-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle Y=f_{0}+\sum _{i=1}^{d}f_{i}(X_{i})+\sum _{i<j}^{d}f_{ij}(X_{i},X_{j})+\cdots +f_{1,2,\dots ,d}(X_{1},X_{2},\dots ,X_{d})}</annotation>

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</math><img src="./1872a8c63fa96555325bcba287cb9943254e5ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:70.292ex; height:7.676ex;" alt="{\displaystyle Y=f_{0}+\sum _{i=1}^{d}f_{i}(X_{i})+\sum _{i<j}^{d}f_{ij}(X_{i},X_{j})+\cdots +f_{1,2,\dots ,d}(X_{1},X_{2},\dots ,X_{d})}" loading="lazy"></dd></dl>

where f0 is a constant and fi is a function of Xi, fij a function of Xi and Xj, etc. A condition of this decomposition is that,

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}f_{i_{1}i_{2}\dots i_{s}}(X_{i_{1}},X_{i_{2}},\dots ,X_{i_{s}})dX_{k}=0,{\text{ for }}k=i_{1},...,i_{s}}</annotation>

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</math><img src="./4b8c4b59da17bd129eae14fb3c4d8da2271ba250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.618ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}f_{i_{1}i_{2}\dots i_{s}}(X_{i_{1}},X_{i_{2}},\dots ,X_{i_{s}})dX_{k}=0,{\text{ for }}k=i_{1},...,i_{s}}" loading="lazy"></dd></dl>

i.e. all the terms in the <a href="Functional_decomposition" title="Functional decomposition">functional decomposition</a> are <a href="Orthogonality" title="Orthogonality">orthogonal</a>. This leads to definitions of the terms of the functional decomposition in terms of conditional expected values,

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<annotation encoding="application/x-tex">{\displaystyle f_{0}=E(Y)}</annotation>

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</math><img src="./3c6e50a0fcbbdf31ea2c510b8da095a506370df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.65ex; height:2.843ex;" alt="{\displaystyle f_{0}=E(Y)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f_{i}(X_{i})=E(Y|X_{i})-f_{0}}</annotation>

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</math><img src="./54850dc61749479f9c24fd2947bc70abcd81ec90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.333ex; height:2.843ex;" alt="{\displaystyle f_{i}(X_{i})=E(Y|X_{i})-f_{0}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f_{ij}(X_{i},X_{j})=E(Y|X_{i},X_{j})-f_{0}-f_{i}-f_{j}}</annotation>

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</math><img src="./12ed4231cf1d55440f33439441071001cf6d1caa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.415ex; height:3.009ex;" alt="{\displaystyle f_{ij}(X_{i},X_{j})=E(Y|X_{i},X_{j})-f_{0}-f_{i}-f_{j}}" loading="lazy"></dd></dl>

From which it can be seen that fi is the effect of varying Xi alone (known as the <a href="Main_effect" title="Main effect">main effect</a> of Xi), and fij is the effect of varying Xi and Xj simultaneously, additional to the effect of their individual variations. This is known as a second-order <a href="Interaction_(statistics)" title="Interaction (statistics)">interaction</a>. Higher-order terms have analogous definitions.

Now, further assuming that the f(X) is <a href="Square-integrable_function" title="Square-integrable function">square-integrable</a>, the functional decomposition may be squared and integrated to give,

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<annotation encoding="application/x-tex">{\displaystyle \int f^{2}(\mathbf {X} )d\mathbf {X} -f_{0}^{2}=\sum _{s=1}^{d}\sum _{i_{1}<\dots <i_{s}}^{d}\int f_{i_{1}\dots i_{s}}^{2}dX_{i_{1}}\dots dX_{i_{s}}}</annotation>

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</math><img src="./39b28105f0c8dfa9bdf8fc549dec663f0409342b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.335ex; height:7.676ex;" alt="{\displaystyle \int f^{2}(\mathbf {X} )d\mathbf {X} -f_{0}^{2}=\sum _{s=1}^{d}\sum _{i_{1}<\dots <i_{s}}^{d}\int f_{i_{1}\dots i_{s}}^{2}dX_{i_{1}}\dots dX_{i_{s}}}" loading="lazy"></dd></dl>

Notice that the left hand side is equal to the variance of Y, and the terms of the right hand side are variance terms, now decomposed with respect to sets of the Xi. This finally leads to the decomposition of variance expression,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (Y)=\sum _{i=1}^{d}V_{i}+\sum _{i<j}^{d}V_{ij}+\cdots +V_{12\dots d}}</annotation>

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</math><img src="./867953ec5c27b2b0aa6d9086df4281fa84396949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.231ex; height:7.676ex;" alt="{\displaystyle \operatorname {Var} (Y)=\sum _{i=1}^{d}V_{i}+\sum _{i<j}^{d}V_{ij}+\cdots +V_{12\dots d}}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle V_{i}=\operatorname {Var} _{X_{i}}\left(E_{{\textbf {X}}_{\sim i}}(Y\mid X_{i})\right)}</annotation>

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</math><img src="./c793a853f71c1e5d6015aee1a4d873182b5434d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.38ex; height:3.009ex;" alt="{\displaystyle V_{i}=\operatorname {Var} _{X_{i}}\left(E_{{\textbf {X}}_{\sim i}}(Y\mid X_{i})\right)}" loading="lazy">,</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle V_{ij}=\operatorname {Var} _{X_{ij}}\left(E_{{\textbf {X}}_{\sim ij}}\left(Y\mid X_{i},X_{j}\right)\right)-V_{i}-V_{j}}</annotation>

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</math><img src="./d4e687db03066aec9e9e0a6f0f122a1aa67287ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.833ex; height:3.343ex;" alt="{\displaystyle V_{ij}=\operatorname {Var} _{X_{ij}}\left(E_{{\textbf {X}}_{\sim ij}}\left(Y\mid X_{i},X_{j}\right)\right)-V_{i}-V_{j}}" loading="lazy"></dd></dl>

and so on. The X~i notation indicates the set of all variables except Xi. The above variance decomposition shows how the variance of the model output can be decomposed into terms attributable to each input, as well as the interaction effects between them. Together, all terms sum to the total variance of the model output.

<div class="mw-heading mw-heading2"><h2 id="First-order_indices">First-order indices</h2></div>

A direct variance-based measure of sensitivity Si, called the "first-order <a href="Sensitivity_index" title="Sensitivity index">sensitivity index</a>", or "main effect index" is stated as follows,<a href="#cite_note-Sob1-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle S_{i}={\frac {V_{i}}{\operatorname {Var} (Y)}}}</annotation>

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</math><img src="./2c3aaf51997536786a3a4bdf35d095d61be17004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.559ex; height:6.009ex;" alt="{\displaystyle S_{i}={\frac {V_{i}}{\operatorname {Var} (Y)}}}" loading="lazy"></dd></dl>

This is the contribution to the output variance of the main effect of Xi, therefore it measures the effect of varying Xi alone, but averaged over variations in other input parameters. It is standardised by the total variance to provide a fractional contribution. Higher-order interaction indices Sij, Sijk and so on can be formed by dividing other terms in the variance decomposition by Var(Y). Note that this has the implication that,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{d}S_{i}+\sum _{i<j}^{d}S_{ij}+\cdots +S_{12\dots d}=1}</annotation>

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</math><img src="./5170f43d7cd0683af3a9b24b3fb272b1e59d9c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.203ex; height:7.676ex;" alt="{\displaystyle \sum _{i=1}^{d}S_{i}+\sum _{i<j}^{d}S_{ij}+\cdots +S_{12\dots d}=1}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Total-effect_index">Total-effect index</h2></div>

Using the Si, Sij and higher-order indices given above, one can build a picture of the importance of each variable in determining the output variance. However, when the number of variables is large, this requires the evaluation of 2d-1 indices, which can be too computationally demanding. For this reason, a measure known as the "Total-effect index" or "Total-order index", STi, is used.<a href="#cite_note-5">[5]</a> This measures the contribution to the output variance of Xi, including all variance caused by its interactions, of any order, with any other input variables. It is given as,

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<annotation encoding="application/x-tex">{\displaystyle S_{Ti}={\frac {E_{{\textbf {X}}_{\sim i}}\left(\operatorname {Var} _{X_{i}}(Y\mid \mathbf {X} _{\sim i})\right)}{\operatorname {Var} (Y)}}=1-{\frac {\operatorname {Var} _{{\textbf {X}}_{\sim i}}\left(E_{X_{i}}(Y\mid \mathbf {X} _{\sim i})\right)}{\operatorname {Var} (Y)}}}</annotation>

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</math><img src="./b10a1d4e523376e04f7da21b90e1493b26a5800f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.641ex; height:6.509ex;" alt="{\displaystyle S_{Ti}={\frac {E_{{\textbf {X}}_{\sim i}}\left(\operatorname {Var} _{X_{i}}(Y\mid \mathbf {X} _{\sim i})\right)}{\operatorname {Var} (Y)}}=1-{\frac {\operatorname {Var} _{{\textbf {X}}_{\sim i}}\left(E_{X_{i}}(Y\mid \mathbf {X} _{\sim i})\right)}{\operatorname {Var} (Y)}}}" loading="lazy"></dd></dl>

Note that unlike the Si,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{d}S_{Ti}\geq 1}</annotation>

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</math><img src="./2fac829dae973493a80cf2abd592d91e3aac6e64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.385ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{d}S_{Ti}\geq 1}" loading="lazy"></dd></dl>

due to the fact that the interaction effect between e.g. Xi and Xj is counted in both STi and STj. In fact, the sum of the STi will only be equal to 1 when the model is purely <a href="Additive_map" title="Additive map">additive</a>.

<div class="mw-heading mw-heading2"><h2 id="Calculation_of_indices">Calculation of indices</h2></div>

For analytically tractable functions, the indices above may be calculated analytically by evaluating the integrals in the decomposition. However, in the vast majority of cases they are estimated – this is usually done by the <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a>.

<div class="mw-heading mw-heading3"><h3 id="Sampling_sequences">Sampling sequences</h3></div>

The Monte Carlo approach involves generating a sequence of randomly distributed points inside the unit hypercube (strictly speaking these will be <a href="Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom</a>). In practice, it is common to substitute random sequences with <a href="Low-discrepancy_sequence" title="Low-discrepancy sequence">low-discrepancy sequences</a> to improve the efficiency of the estimators. This is then known as the <a href="Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">quasi-Monte Carlo method</a>. Some low-discrepancy sequences commonly used in sensitivity analysis include the <a href="Sobol_sequence" title="Sobol sequence">Sobol’ sequence</a> and the <a href="Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube</a> design.

<div class="mw-heading mw-heading3"><h3 id="Procedure">Procedure</h3></div>

To calculate the indices using the (quasi) Monte Carlo method, the following steps are used:<a href="#cite_note-Sobol2001-1">[1]</a><a href="#cite_note-Primer-2">[2]</a>

<ol><li>Generate an N×2d sample matrix, i.e. each row is a sample point in the hyperspace of 2d dimensions. This should be done with respect to the probability distributions of the input variables.</li>

<li>Use the first d columns of the matrix as matrix A, and the remaining d columns as matrix B. This effectively gives two independent samples of N points in the d-dimensional unit hypercube.</li>

<li>Build d further N×d matrices ABi, for i = 1,2,...,d, such that the ith column of ABi is equal to the ith column of B, and the remaining columns are from A.</li>

<li>The A, B, and the d ABi matrices in total specify N(d+2) points in the input space (one for each row). Run the model at each design point in the A, B, and ABi matrices, giving a total of N(d+2) model evaluations – the corresponding f(A), f(B) and f(ABi) values.</li>

<li>Calculate the sensitivity indices using the estimators below.</li></ol>

The accuracy of the estimators is of course dependent on N. The value of N can be chosen by sequentially adding points and calculating the indices until the estimated values reach some acceptable convergence. For this reason, when using low-discrepancy sequences, it can be advantageous to use those that allow sequential addition of points (such as the Sobol’ sequence), as compared to those that do not (such as Latin hypercube sequences).

<div class="mw-heading mw-heading3"><h3 id="Estimators">Estimators</h3></div>

There are a number of possible Monte Carlo estimators available for both indices. Two that are currently in general use are,<a href="#cite_note-Sobol2001-1">[1]</a><a href="#cite_note-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{X_{i}}(E_{\mathbf {X} _{\sim i}}(Y|X_{i}))\approx {{\frac {1}{N}}\sum _{j=1}^{N}f\left(\mathbf {B} \right)_{j}\left(f\left(\mathbf {A} _{B}^{i}\right)_{j}-f\left(\mathbf {A} \right)_{j}\right)}}</annotation>

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</math><img src="./ee4d26df10e62df6b434c4534c28fd1ac01012ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.699ex; height:7.676ex;" alt="{\displaystyle \operatorname {Var} _{X_{i}}(E_{\mathbf {X} _{\sim i}}(Y|X_{i}))\approx {{\frac {1}{N}}\sum _{j=1}^{N}f\left(\mathbf {B} \right)_{j}\left(f\left(\mathbf {A} _{B}^{i}\right)_{j}-f\left(\mathbf {A} \right)_{j}\right)}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle E_{\mathbf {X} _{\sim i}}\left(\operatorname {Var} _{X_{i}}\left(Y\mid \mathbf {X} _{\sim i}\right)\right)\approx {{\frac {1}{2N}}\sum _{j=1}^{N}\left(f\left(\mathbf {A} \right)_{j}-f\left(\mathbf {A} _{B}^{i}\right)_{j}\right)^{2}}}</annotation>

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</math><img src="./2b3507f76c9d308b731de08c44d5611c7096fce9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.682ex; height:7.676ex;" alt="{\displaystyle E_{\mathbf {X} _{\sim i}}\left(\operatorname {Var} _{X_{i}}\left(Y\mid \mathbf {X} _{\sim i}\right)\right)\approx {{\frac {1}{2N}}\sum _{j=1}^{N}\left(f\left(\mathbf {A} \right)_{j}-f\left(\mathbf {A} _{B}^{i}\right)_{j}\right)^{2}}}" loading="lazy"></dd></dl>

for the estimation of the Si and the STi respectively.

<div class="mw-heading mw-heading3"><h3 id="Computational_expense">Computational expense</h3></div>

For the estimation of the Si and the STi for all input variables, N(d+2) model runs are required. Since N is often of the order of hundreds or thousands of runs, computational expense can quickly become a problem when the model takes a significant amount of time for a single run. In such cases, there are a number of techniques available to reduce the computational cost of estimating sensitivity indices, such as <a href="Sensitivity_analysis#Emulators" title="Sensitivity analysis">emulators</a>, <a href="Sensitivity_analysis#High-dimensional_model_representations_(HDMR)" title="Sensitivity analysis">HDMR</a> and <a href="Sensitivity_analysis#Fourier_amplitude_sensitivity_test_(FAST)" title="Sensitivity analysis">FAST</a>.

<ul><li><a href="Sensitivity_analysis" title="Sensitivity analysis">Sensitivity analysis</a></li>

<li><a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a></li>

<li><a href="Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">Quasi-Monte Carlo method</a></li>

<li><a href="Sobol_sequence" title="Sobol sequence">Sobol’ sequence</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Sobol2001-1">^ <a href="#cite_ref-Sobol2001_1-0">a</a> <a href="#cite_ref-Sobol2001_1-1">b</a> <a href="#cite_ref-Sobol2001_1-2">c</a> Sobol, I.M. (2001), Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. MATH COMPUT SIMULAT,55(1–3),271-280, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0378-4754%2800%2900270-6">10.1016/S0378-4754(00)00270-6</a>

</li>

<li id="cite_note-Primer-2">^ <a href="#cite_ref-Primer_2-0">a</a> <a href="#cite_ref-Primer_2-1">b</a> Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D. Saisana, M., and Tarantola, S., 2008, Global Sensitivity Analysis. The Primer, John Wiley & Sons.

</li>

<li id="cite_note-OAT-3"><a href="#cite_ref-OAT_3-0">^</a> Saltelli, A., Annoni, P., 2010, How to avoid a perfunctory sensitivity analysis, Environmental Modeling and Software 25, 1508–1517.

</li>

<li id="cite_note-Sob1-4">^ <a href="#cite_ref-Sob1_4-0">a</a> <a href="#cite_ref-Sob1_4-1">b</a> Sobol’, I. (1990). Sensitivity estimates for nonlinear mathematical models. Matematicheskoe Modelirovanie 2, 112–118. in Russian, translated in English in Sobol’ , I. (1993). Sensitivity analysis for non-linear mathematical models. Mathematical Modeling & Computational Experiment (Engl. Transl.), 1993, 1, 407–414.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Homma, T. and A. Saltelli (1996). Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering and System Safety, 52, 1–17.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Andrea Saltelli, Paola Annoni, Ivano Azzini, Francesca Campolongo, Marco Ratto, and Stefano Tarantola. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181(2):259{270, 2010

</li>

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