sbiosobol

Perform global sensitivity analysis by computing first- and total-order Sobol indices (requires Statistics and Machine Learning Toolbox)

Syntax

sobolResults = sbiosobol(modelObj,params,observables)

sobolResults = sbiosobol(modelObj,params,observables,Name,Value)

Description

sobolResults = sbiosobol(modelObj,params,observables) performs global sensitivity analysis [1] on a SimBiology model modelObj by decomposing the variances of observables with respect to the sensitivity inputs params.

sobolResults = sbiosobol(modelObj,params,observables,Name,Value) uses additional options specified by one or more name-value pair arguments.

Examples

Perform Global Sensitivity Analysis by Computing First- and Total-Order Sobol Indices

Load the Tumor Growth Model.

sbioloadproject tumor_growth_vpop_sa.sbproj

Get a variant with the estimated parameters and the dose to apply to the model.

v = getvariant(m1);
d = getdose(m1,'interval_dose');

Get the active configset and set the tumor weight as the response.

cs = getconfigset(m1);
cs.RuntimeOptions.StatesToLog = 'tumor_weight';

Simulate the model and plot the tumor growth profile.

sbioplot(sbiosimulate(m1,cs,v,d));

Perform global sensitivity analysis (GSA) on the model to find the model parameters that the tumor growth is sensitive to.

First, retrieve model parameters of interest that are involved in the pharmacodynamics of the tumor growth. Define the model response as the tumor weight.

modelParamNames = {'L0','L1','w0','k1','k2'};
outputName = 'tumor_weight';

Then perform GSA by computing the first- and total-order Sobol indices using sbiosobol. Set 'ShowWaitBar' to true to show the simulation progress. By default, the function uses 1000 parameter samples to compute the Sobol indices [1].

sobolResults = sbiosobol(m1,modelParamNames,outputName,'Variants',v,'Doses',d,'ShowWaitBar',true)

sobolResults = 
  Sobol with properties:

                Time: [444x1 double]
        SobolIndices: [5x1 struct]
            Variance: [444x1 table]
         Observables: {'[Tumor Growth Model].tumor_weight'}
    ParameterSamples: [1000x5 table]
      SimulationInfo: [1x1 struct]

You can change the number of samples by specifying the 'NumberSamples' name-value pair argument. The function requires a total of (number of input parameters + 2) * NumberSamples model simulations.

Show the mean model response, the simulation results, and a shaded region covering 90% of the simulation results.

plotData(sobolResults);

You can adjust the quantile region to a different percentage by specifying 'Alphas' for the lower and upper quantiles of all model responses. For instance, an alpha value of 0.1 plots a shaded region between the 100 * alpha and 100 * (1 - alpha) quantiles of all simulated model responses.

plotData(sobolResults,'Alphas',0.1);

Plot the time course of the first- and total-order Sobol indices.

h = plot(sobolResults);
% Resize the figure.
h.Position(:) = [100 100 1280 800];

The first-order Sobol index of an input parameter gives the fraction of the overall response variance that can be attributed to variations in the input parameter alone. The total-order index gives the fraction of the overall response variance that can be attributed to any joint parameter variations that include variations of the input parameter.

From the Sobol indices plots, parameters L1 and w0 seem to be the most sensitive parameters to the tumor weight before the dose was applied at t = 7. But after the dose is applied, k1 and k2 become more sensitive parameters and contribute most to the after-dosing stage of the tumor weight. The total variance plot also shows a larger variance for the after-dose stage at t > 35 than for the before-dose stage of the tumor growth, indicating that k1 and k2 might be more important parameters to investigate further. The fraction of unexplained variance shows some variance at around t = 33, but the total variance plot shows little variance at t = 33, meaning the unexplained variance could be insignificant. The fraction of unexplained variance is calculated as 1 - (sum of all the first-order Sobol indices), and the total variance is calculated using var(response), where response is the model response at every time point.

You can also display the magnitudes of the sensitivities in a bar plot.

bar(sobolResults)

You can specify more samples to increase the accuracy of the Sobol indices, but the simulation can take longer to finish. Use addsamples to add more samples. For example, if you specify 1500 samples, the function performs 1500 * (2 + number of input parameters) simulations.

gsaMoreSamples = addsamples(gsaResults,1500)

The SimulationInfo property of the result object contains various information for computing the Sobol indices. For instance, the model simulation data (SimData) for each simulation using a set of parameter samples is stored in the SimData field of the property. This field is an array of SimData objects.

sobolResults.SimulationInfo.SimData

 
   SimBiology SimData Array : 1000-by-7
 
   Index:    Name:         ModelName:         DataCount: 
   1           -           Tumor Growth Model 1          
   2           -           Tumor Growth Model 1          
   3           -           Tumor Growth Model 1          
   ...                                                   
   7000        -           Tumor Growth Model 1          
 

You can find out if any model simulation failed during the computation by checking the ValidSample field of SimulationInfo. In this example, the field shows no failed simulation runs.

all(sobolResults.SimulationInfo.ValidSample)

ans = 1x7 logical array

   1   1   1   1   1   1   1

SimulationInfo.ValidSample is a table of logical values. It has the same size as SimulationInfo.SimData. If ValidSample indicates that any simulations failed, you can get more information about those simulation runs and the samples used for those runs by extracting information from the corresponding column of SimulationInfo.SimData. Suppose that the fourth column contains one or more failed simulation runs. Get the simulation data and sample values used for that simulation using getSimulationResults.

[samplesUsed,sd,validruns] = getSimulationResults(sobolResults,4);

You can add custom expressions as observables and compute Sobol indices for the added observables. For example, you can compute the Sobol indices for the maximum tumor weight by defining a custom expression as follows.

% Suppress an information warning that is issued during simulation.
warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON');
% Add the observable expression.
sobolObs = addobservable(sobolResults,'Maximum tumor_weight','max(tumor_weight)','Units','gram');

Plot the computed simulation results showing the 90% quantile region.

h2 = plotData(sobolObs);
h2.Position(:) = [100 100 1280 800];

You can also remove the observable by specifying its name.

gsaNoObs = removeobservable(sobolObs,'Maximum tumor_weight');

Restore the warning settings.

warning(warnSettings);

Input Arguments

modelObj — SimBiology model

SimBiology model object

SimBiology model, specified as a SimBiology model object.

params — Names of model parameters, species, or compartments

character vector | string | string vector | cell array of character vectors

Names of model parameters, species, or compartments, specified as a character vector, string, string vector, or cell array of character vectors.

Example: ["k1","k2"]

Data Types: char | string | cell

observables — Model responses

character vector | string | string vector | cell array of character vectors

Model responses, specified as a character vector, string, string vector, or cell array of character vectors. Specify the names of species, parameters, compartments, or observables.

Example: ["tumor_growth"]

Data Types: char | string | cell

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: sobolResults = sbiosobol(modelObj,params,observables,'ShowWaitbar',true) specifies to show a simulation progress bar.

'Bounds' — Parameter bounds

numeric matrix

Parameter bounds, specified as the comma-separated pair consisting of 'Bounds' and a numeric matrix with two columns. The first column contains the lower bounds and the second column contains the upper bounds. The number of rows must be equal to the number of parameters in params.

If a parameter has a nonzero value, the default bounds are ±10% of the value. If the parameter value is zero, the default bounds are [0 1].

Example: [0.5 5]

Data Types: double

'Doses' — Doses to use during simulations

ScheduleDose object | RepeatDose object | vector of dose objects

Doses to use during model simulations, specified as the comma-separated pair consisting of 'Doses' and a ScheduleDose or RepeatDose object or a vector of dose objects.

'Variants' — Variants to apply before simulations

variant object | vector of variant objects

Variants to apply before model simulations, specified as the comma-separated pair consisting of 'Variants' and a variant object or vector of variant objects.

When there are multiple variants and if there are duplicate specifications for a property's value, the last occurrence for the property value in the array of variants is used during simulation.

'NumberSamples' — Number of samples to compute Sobol indices

1000 (default) | positive integer

Number of samples to compute Sobol indices, specified as the comma-separated pair consisting of 'NumberSamples' and a positive integer. The function requires (number of input params + 2) * NumberSamples model simulations to compute the first- and total-order Sobol indices.

Data Types: double

'SamplingMethod' — Method to generate parameter samples

'Sobol' (default) | character vector | string

Method to generate parameter samples, specified as the comma-separated pair consisting of 'SamplingMethod' and a character vector or string. Valid options are:

• 'Sobol' — Use the low-discrepency Sobol sequence to generate samples.

• 'Halton' — Use the low-discrepency Halton sequence to generate samples.

• 'lhs' — Use the low-discrepency Latin hypercube samples.

• 'random' — Use uniformly distributed random samples.

Data Types: char | string

'StopTime' — Simulation stop time

nonnegative scalar

Simulation stop time, specified as the comma-separated pair consisting of 'StopTime' and a nonnegative scalar. If you specify neither StopTime nor OutputTimes, the function uses the stop time from the active configuration set of the model. You cannot specify both StopTime and OutputTimes.

Data Types: double

'OutputTimes' — Simulation output times

numeric vector

Simulation output times, specified as the comma-separated pair consisting of 'OutputTimes' and a numeric vector. The function computes the Sobol indices at these output time points. You cannot specify both StopTime and OutputTimes. By default, the function uses the output times of the first model simulation.

Example: [0 1 2 3.5 4 5 5.5]

Data Types: double

'UseParallel' — Flag to run model simulations in parallel

false (default) | true

Flag to run model simulations in parallel, specified as the comma-separated pair consisting of 'UseParallel' and true or false.

Data Types: logical

'Accelerate' — Flag to turn on model acceleration

true (default) | false

Flag to turn on model acceleration, specified as the comma-separated pair consisting of 'Accelerate' and true or false.

Data Types: logical

'InterpolationMethod' — Method for interpolation of model simulations

"interp1q" (default) | character vector | string

Method for interpolation of model simulations to new output times, specified as the comma-separated pair consisting of 'InterpolationMethod' and a character vector or string. The valid options follow.

• "interp1q" — Use the interp1q function.

• Use the interp1 function by specifying one of the following methods:

  • "nearest"

  • "linear"

  • "spline"

  • "pchip"

  • "v5cubic"

• "zoh" — Specify the zero-order hold.

Data Types: char | string

'ShowWaitbar' — Flag to show progress of model simulations

false (default) | true

Flag to show the progress of model simulations by displaying a progress bar, specified as the comma-separated pair consisting of 'ShowWaitbar' and true or false. By default, no wait bar is displayed.

Data Types: logical

Output Arguments

sobolResults — Results containing Sobol indices

SimBiology.gsa.Sobol object

Results containing the first- and total-order Sobol indices, returned as a SimBiology.gsa.Sobol object. The object also contains the parameter sample values and model simulation data used to compute the Sobol indices.

The results object can contain a significant amount of simulation data (SimData). The size of the object exceeds (1 + number of observables) * number of output time points * (2 + number of parameters) * number of samples * 8 bytes. For example, if you have one observable, 500 output time points, 8 parameters, and 100,000 samples, the object size is (1 + 1) * 500 * (2 + 8) * 100000 * 8 bytes = 8 GB. If you need to save such large objects, use this syntax:

save(fileName,variableName,'-v7.3');

For details, see MAT-file version.

More About

Saltelli Method to Compute Sobol Indices

sbiosobol implements the Saltelli method [1] to compute Sobol indices.

Consider a SimBiology model response Y expressed as a mathematical model $$Y=f\left(X_1,X_2,X_3,\mathrm{...},X_k\right)$$, where X_i is a model parameter and i = 1,…,k.

The first-order Sobol index (S_i) gives the fraction of the overall response variance V(Y) that can be attributed to variations in X_i alone. S_i is defined as follows.

$$S_i=\frac{V_{X_i}\left(E_{X_{\sim i}}\left(Y|X_i\right)\right)}{V\left(Y\right)}$$

The total-order Sobol index (S_Ti) gives the fraction of the overall response variance V(Y) that can be attributed to any joint parameter variations that include variations of X_i. S_Ti is defined as follows.

$$S_{Ti}=1-\frac{V_{X_{\sim i}}\left(E_{X_i}\left(Y|X_{\sim i}\right)\right)}{V\left(Y\right)}=\frac{E_{X_{\sim i}}\left(V_{X_i}\left(Y|X_{\sim i}\right)\right)}{V\left(Y\right)}$$

To compute individual values for Y corresponding to samples of parameters X₁, X₁, …, X_k, consider two independent sampling matrices A and B.

$$A=\left(\begin{array}{cccc}{X}_{11}& X_{12}& \mathrm{...}& X_{1k}\\ X_{21}& X_{22}& \mathrm{...}& X_{2k}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ X_{n1}& X_{n2}& \mathrm{...}& X_{nk}\end{array}\right)$$

$$B=\left(\begin{array}{cccc}{X}_{11}^{\text{'}}& X_{12}^{\text{'}}& \mathrm{...}& X_{1k}^{\text{'}}\\ X_{21}^{\text{'}}& X_{22}^{\text{'}}& \mathrm{...}& X_{2k}^{\text{'}}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ X_{n1}^{\text{'}}& X_{n2}^{\text{'}}& \mathrm{...}& X_{nk}^{\text{'}}\end{array}\right)$$

n is the sample size. Each row of the matrices A and B corresponds to one parameter sample set, which is a single realization of model parameter values.

Estimates for S_i and S_Ti are obtained from model simulation results using sample values from the matrices A, B, and $${{\displaystyle A}}_B^i$$, which is a matrix where all columns are from A except the ith column, which is from B for i = 1, 2, …, params.

$${{\displaystyle A}}_B^i=\left(\begin{array}{cccccc}{X}_{11}& X_{12}& \mathrm{...}& X_{1i}^{\text{'}}& \mathrm{...}& X_{1k}\\ X_{21}& X_{22}& \mathrm{...}& X_{2i}^{\text{'}}& \mathrm{...}& X_{2k}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ X_{n1}& X_{n2}& \mathrm{...}& X_{ni}^{\text{'}}& \mathrm{...}& X_{nk}\end{array}\right)$$

The formulas to approximate the first- and total-order Sobol indices are as follows.

$${\widehat{S}}_i=\frac{\frac{1}{n}{\displaystyle \sum _{j=1}^{n}f{(B)}_j\left(f{\left({{\displaystyle A}}_B^i\right)}_j-f{(A)}_j\right)}}{V(Y)}$$

$${\widehat{S}}_{Ti}=\frac{\frac{1}{2n}{\displaystyle \sum _{j=1}^n{\left(f{(A)}_j-f{\left({{\displaystyle A}}_B^i\right)}_j\right)}^2}}{V(Y)}$$

f(A), f(B), and $$f{\left({{\displaystyle A}}_B^i\right)}_j$$ are the model simulation results using the parameter sample values from matrices A, B, and $${{\displaystyle A}}_B^i$$.

The matrix A corresponds to the ParameterSamples property of the Sobol results object (resultsObj.ParameterSamples). The matrix B corresponds to the SupportSamples property (resultsObj.SimulationInfo.SupportSamples).

The $${{\displaystyle A}}_B^i$$ matrices are stored in the SimData structure of the SimulationInfo property (resultsObj.SimulationInfo.SimData). The size of SimulationInfo.SimData is NumberSamples-by-params + 2, where NumberSamples is the number of samples and param is the number of input parameters. The number of columns is 2 + params because the first column of SimulationInfo.SimData contains the model simulation results using the sample matrix A. The second column contains simulation results using SupportSamples, which is another sample matrix B. The rest of the columns contain simulation results using $${{\displaystyle A}}_B^1$$, $${{\displaystyle A}}_B^2$$, …, $${{\displaystyle A}}_B^i$$, …, $${{\displaystyle A}}_B^{params}$$. See getSimulationResults to retrieve the model simulation results and samples for a specified ith index ($${{\displaystyle A}}_B^i$$) from the SimulationInfo.SimData array.

References

[1] Saltelli, Andrea, 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, no. 2 (February 2010): 259–70. https://doi.org/10.1016/j.cpc.2009.09.018.