SimBiology.gsa.Sobol
Description
Creation
Create a SimBiology.gsa.Sobol
object using sbiosobol
.
Properties
ParameterSamples
— Sampled parameter values
table
This property is read-only.
Sampled parameter values, specified as a table. The parameter sample values are used for approximating the Sobol indices. For details, see Saltelli Method to Compute Sobol Indices.
Data Types: table
Observables
— Names of model responses or observables
cell array of character vectors
This property is read-only.
Names of model responses or observables, specified as a cell array of character vectors.
Data Types: char
Time
— Time points
column numeric vector
This property is read-only.
Time points at which Sobol indices are computed, specified as a column numeric
vector. The property is []
if all observables are scalars.
Data Types: double
SobolIndices
— Computed sobol indices
structure array
This property is read-only.
Computed sobol indices, specified as a structure array. The size of the array is
[params,observables]
, where
params is the number of input parameters and
observables is the number of observables.
Each structure contains the following fields.
Parameter
— Name of an input parameter, specified as a character vectorObservable
— Name of an observable, specified as a character vectorFirstOrder
— First-order Sobol index, specified as a numeric vectorTotalOrder
— Total-order Sobol index, specified as a numeric vector.
If all observables are scalar, then the FirstOrder
and
TotalOrder
fields are specified as scalars. If some observables are
scalars and some are vectors, FirstOrder
and
TotalOrder
are numeric vectors of length Time
.
Scalar observables are scalar-expanded, where each time point has the same value.
Data Types: struct
Variance
— Variance values for time courses of observables
table
This property is read-only.
Variance values for time courses of observables, specified as a table. Each column of the table contains the variance values for the time courses of each observable.
If all observables are scalars, then the Variance
table has one
row. If some observables are scalars and some are vectors, then the variances for scalar
observables are scalar-expanded, where each row has the same value.
The VariableNames
property of the table
(Variance.Properties.VariableNames
) is a cell array of character
vectors containing the names of observables provided as inputs to sbiosobol
.
Names are truncated if needed. The VariableDescriptions
property
contains the untruncated observable names.
Data Types: table
SimulationInfo
— Simulation information used for computing Sobol indices
structure
This property is read-only.
Simulation information, such as simulation data and parameter samples, used for computing Sobol indices, specified as a structure. The structure contains the following fields.
SimFunction
—SimFunction
object used for simulating model responses or observables.SimData
—SimData
array of size[NumberSamples,2 + params]
, where NumberSamples is the number of samples and params is the number of input parameters.The first column contains the model simulation results from
ParameterSamples
.The second column contains simulation results from
SupportSamples
.The rest of the columns contain simulation results from combinations of parameter values from
ParameterSamples
andSupportSamples
. For information on retrieving the model simulation results and samples for a specified column (index) from thisSimData
array, seegetSimulationResults
. For details on how Sobol indices are computed, see the Saltelli Method to Compute Sobol Indices.
OutputTimes
— Numeric column vector containing the common time vector of allSimData
objects.Bounds
— Numeric matrix of size[params,2]
. params is the number of input parameters. The first column contains the lower bounds and the second column contains the upper bounds for sensitivity inputs.DoseTables
— Cell array of dose tables used for theSimFunction
evaluation.DoseTables
is the output ofgetTable(doseInput)
, where doseInput is the value specified for the'Doses'
name-value pair argument in the call tosbiosobol
,sbiompgsa
, orsbioelementaryeffects
. If no doses are applied, this field is set to[]
.ValidSample
— Logical matrix of size[NumberSamples,2 + params]
indicating whether a simulation result for a particular sample failed. Resampling of the simulation data (SimData) can result inNaN
values if the data is extrapolated. Such SimData are indicated as invalid.InterpolationMethod
— Name of the interpolation method forSimData
.SamplingMethod
— Name of the sampling method used to drawParameterSamples
.RandomState
— Structure containing the state ofrng
before drawingParameterSamples
.SupportSamples
— Table of parameter sample values used for approximating the Sobol indices. For details, see Saltelli Method to Compute Sobol Indices.
Data Types: struct
Object Functions
resample | Resample Sobol indices or elementary effects to new time vector |
addobservable | Compute Sobol indices or elementary effects for new observable expression |
removeobservable | Remove Sobol indices or elementary effects of observables |
getSimulationResults | Retrieve model simulation results and sample values used for computing Sobol indices |
addsamples | Add additional samples to increase accuracy of Sobol indices or elementary effects analysis |
plotData | Plot quantile summary of model simulations from global sensitivity analysis (requires Statistics and Machine Learning Toolbox) |
plot | Plot first- and total-order Sobol indices and variances |
bar | Create bar plot of first- and total-order Sobol indices |
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].
rng('default');
sobolResults = sbiosobol(m1,modelParamNames,outputName,Variants=v,Doses=d,ShowWaitBar=true)
sobolResults = Sobol with properties: Time: [444x1 double] SobolIndices: [5x1 struct] Variance: [444x1 table] ParameterSamples: [1000x5 table] Observables: {'[Tumor Growth].tumor_weight'} 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,ShowMedian=true,ShowMean=false);
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,ShowMedian=true,ShowMean=false);
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. Darker colors mean that those values occur more often over the whole time course.
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.SimDat
a. 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,ShowMedian=true,ShowMean=false); 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);
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 , where Xi is a model parameter
and i = 1,…,k
.
The first-order Sobol index (Si) gives the fraction
of the overall response variance V(Y)
that can be
attributed to variations in Xi alone.
Si is defined as follows.
The total-order Sobol index (STi) gives the fraction
of the overall response variance V(Y)
that can be
attributed to any joint parameter variations that include variations of
Xi.
STi is defined as follows.
To compute individual values for Y corresponding to samples of parameters X1, X1, …, Xk, consider two independent sampling matrices A and B.
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 Si and
STi are obtained from model simulation
results using sample values from the matrices A, B,
and , which is a matrix where all columns are from A except
the ith column, which is from B for i = 1, 2,
…, params
.
The formulas to approximate the first- and total-order Sobol indices are as follows.
f(A)
, f(B)
, and are the model simulation results using the parameter sample values from
matrices A, B, and .
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 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 , , …, , …, . See getSimulationResults
to retrieve the model simulation results and samples
for a specified ith index () from the SimulationInfo.SimData
array.
Tips
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');
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.
Version History
Introduced in R2020a
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