# plot

Plot first- and total-order Sobol indices and variances

## Description

## 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);

## Input Arguments

`sobolObj`

— Results containing Sobol indices

`SimBiology.gsa.Sobol`

object

Results containing the first- and total-order Sobol indices, specified as a `SimBiology.gsa.Sobol`

object.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`h = plot(results,'Observables','tumor_weight')`

specifies to
plot Sobol indices corresponding to the tumor weight response.

`Parameters`

— Input parameters to plot

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

Input parameters to plot, specified as a character vector, string, string vector, cell
array of character vectors, or vector of positive integers indexing into the columns of
the `resultsObject.ParameterSamples`

table. Use this name-value
argument to select parameters and plot their corresponding GSA results. By default, all
input parameters are included in the plot.

**Data Types: **`double`

| `char`

| `string`

| `cell`

`Observables`

— Model responses or observables to plot

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

Model responses or observables to plot, specified as a character vector, string, string
vector, cell array of character vectors, or vector of positive integers indexing into
`resultsObject.Observables`

. By default, the function plots GSA
results for all model responses or observables.

**Data Types: **`double`

| `char`

| `string`

| `cell`

`Color`

— Color of first- and total-order Sobol indices

three-element row vector | hexadecimal color code | color name

Color of the first- and total-order Sobol indices, specified as a three-element
row vector, hexadecimal color code, color name, or a short name. By default, the
function uses the first MATLAB^{®} default color for the first order and the second default color for the
total order. To view the default color order, enter
`get(groot,'defaultAxesColorOrder')`

or see the ColorOrder property.

For details on valid color names and corresponding RGB triplets and hexadecimal codes, see Specify Plot Colors.

**Example: **`'Color',[0.4,0.3,0.2]`

**Data Types: **`double`

`VarianceColor`

— Color of total and unexplained variances

`[0,0,0]`

(default) | three-element row vector | hexadecimal color code | color name

Color of the total and unexplained variances, specified as a three-element row
vector, hexadecimal color code, color name, or a short name. By default, the function
uses the color black `[0,0,0]`

.

For details on valid color names and corresponding RGB triplets and hexadecimal codes, see Specify Plot Colors.

**Example: **`'VarianceColor',[0.2,0.5,0.8]`

**Data Types: **`double`

`DelimiterColor`

— Color of delimiting lines

`[0,0,0]`

(default) | three-element row vector | hexadecimal color code | color name

Color of the delimiting lines, specified as a three-element row vector,
hexadecimal color code, color name, or a short name. By default, the function uses
the color black `[0,0,0]`

.

For details on valid color names and corresponding RGB triplets and hexadecimal codes, see Specify Plot Colors.

**Example: **`'DelimiterColor',[0.2,0.5,0.8]`

**Data Types: **`double`

## Output Arguments

`h`

— Handle

figure handle

Handle to the figure, specified as a figure handle.

## 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**

## See Also

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