# sbiompgsa

Perform multiparametric global sensitivity analysis (requires Statistics and Machine Learning Toolbox)

## Syntax

## Description

performs a multiparametric global sensitivity analysis (MPGSA) [1] of
`mpgsaResults`

= sbiompgsa(`modelObj`

,`params`

,`classifiers`

)`classifiers`

with respect to model parameters
`params`

on a SimBiology model `modelObj`

.
`params`

are model quantities (sensitivity inputs) and
`classifiers`

define the expressions of model responses (model
outputs).

uses parameter `mpgsaResults`

= sbiompgsa(`modelObj`

,`samples`

,`classifiers`

)`samples`

to perform a multiparametric global sensitivity
analysis of `classifiers`

.

draws samples from `mpgsaResults`

= sbiompgsa(`modelObj`

,`scenarios`

,`classifiers`

)`scanarios`

, a `SimBiology.Scenarios`

object, to perform the analysis.

uses model simulation data `mpgsaResults`

= sbiompgsa(`simdata`

,`samples`

,`classifiers`

)`simdata`

to perform a multiparametric global
sensitivity analysis of `classifiers`

.

uses samples specified in `mpgsaResults`

= sbiompgsa(`simdata`

,`scenarios`

,`classifiers`

)`scenarios`

, a
`SimBiology.Scenarios`

object.

uses additional options specified by one or more name-value pair arguments. Available
name-value arguments differ depending on which syntax you are using.`mpgsaResults`

= sbiompgsa(___,`Name,Value`

)

## Examples

### Perform Multiparametric Global Sensitivity Analysis (MPGSA)

Load the Target-Mediated Drug Disposition (TMDD) Model.

`sbioloadproject tmdd_with_TO.sbproj`

Get the active configset and set the target occupancy (`TO`

) as the response.

```
cs = getconfigset(m1);
cs.RuntimeOptions.StatesToLog = 'TO';
```

Simulate the model and plot the `TO`

profile.

sbioplot(sbiosimulate(m1,cs));

Define an exposure (area under the curve of the TO profile) threshold for the target occupancy.

`classifier = 'trapz(time,TO) <= 0.1';`

Perform MPGSA to find sensitive parameters with respect to the TO. Vary the parameter values between predefined bounds to generate 10,000 parameter samples.

% Suppress an information warning that is issued during simulation. warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON'); rng(0,'twister'); % For reproducibility params = {'kel','ksyn','kdeg','km'}; bounds = [0.1, 1; 0.1, 1; 0.1, 1; 0.1, 1]; mpgsaResults = sbiompgsa(m1,params,classifier,'Bounds',bounds,'NumberSamples',10000)

mpgsaResults = MPGSA with properties: Classifiers: {'trapz(time,TO) <= 0.1'} KolmogorovSmirnovStatistics: [4x1 table] ECDFData: {4x4 cell} SignificanceLevel: 0.0500 PValues: [4x1 table] SupportHypothesis: [10000x1 table] ParameterSamples: [10000x4 table] Observables: {'TO'} SimulationInfo: [1x1 struct]

Plot the quantiles of the simulated model response.

plotData(mpgsaResults);

Plot the empirical cumulative distribution functions (eCDFs) of the accepted and rejected samples. Except for `km`

, none of the parameters shows a significant difference in the eCDFs for the accepted and rejected samples. The `km`

plot shows a large Kolmogorov-Smirnov (K-S) distance between the eCDFs of the accepted and rejected samples. The K-S distance is the maximum absolute distance between two eCDFs curves.

```
h = plot(mpgsaResults);
% Resize the figure.
pos = h.Position(:);
h.Position(:) = [pos(1) pos(2) pos(3)*2 pos(4)*2];
```

To compute the K-S distance between the two eCDFs, SimBiology uses a two-sided test based on the null hypothesis that the two distributions of accepted and rejected samples are equal. See `kstest2`

(Statistics and Machine Learning Toolbox) for details. If the K-S distance is large, then the two distributions are different, meaning that the classification of the samples is sensitive to variations in the input parameter. On the other hand, if the K-S distance is small, then variations in the input parameter do not affect the classification of samples. The results suggest that the classification is insensitive to the input parameter. To assess the significance of the K-S statistic rejecting the null-hypothesis, you can examine the p-values.

bar(mpgsaResults)

The bar plot shows two bars for each parameter: one for the K-S distance (K-S statistic) and another for the corresponding p-value. You reject the null hypothesis if the p-value is less than the significance level. A cross (`x`

) is shown for any p-value that is almost 0. You can see the exact p-value corresponding to each parameter.

[mpgsaResults.ParameterSamples.Properties.VariableNames',mpgsaResults.PValues]

`ans=`*4×2 table*
Var1 trapz(time,TO) <= 0.1
________ _____________________
{'kel' } 0.0021877
{'ksyn'} 1
{'kdeg'} 0.99983
{'km' } 0

The p-values of `km`

and `kel`

are less than the significance level (0.05), supporting the alternative hypothesis that the accepted and rejected samples come from different distributions. In other words, the classification of the samples is sensitive to `km`

and `kel`

but not to other parameters (`kdeg`

and `ksyn`

).

You can also plot the histograms of accepted and rejected samples. The historgrams let you see trends in the accepted and rejected samples. In this example, the histogram of `km`

shows that there are more accepted samples for larger `km`

values, while the `kel`

histogram shows that there are fewer rejected samples as `kel`

increases.

```
h2 = histogram(mpgsaResults);
% Resize the figure.
pos = h2.Position(:);
h2.Position(:) = [pos(1) pos(2) pos(3)*2 pos(4)*2];
```

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`

`simdata`

— Model simulation data

vector of `SimData`

objects

Model simulation data, specified as a vector of `SimData`

objects.

`scenarios`

— Source for drawing samples

`SimBiology.Scenarios`

object

Source for drawing samples, specified as a `SimBiology.Scenarios`

object.

**Note**

The object must not contain doses or variants as its entries.

`classifiers`

— Expressions of model responses

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

Expressions of model responses, specified as a character vector, string, string vector, or cell array of character vectors. Specify expressions referencing simulated model quantities, observables, time, or MATLAB functions that are on the path.

Each classifier must evaluate to a logical vector of the same length as the number of parameter samples. Entities, such as model quantities or observables, referenced in a classifier expression are evaluated as a matrix with columns containing time courses of the simulated quantity values. Each column represents one sample. Each row represents one output time. For details, see Multiparametric Global Sensitivity Analysis (MPGSA).

If you specify a `SimData`

object as the first input, each quantity
or `observable`

referenced by the classifier must resolve to a logged component in the
`SimData`

object.

You can formulate a classifier as a modeling question such as whether a model
parameter have an effect on the model response exceeding or falling below a target
threshold. For example, the following classifier defines an exposure (area under the
curve) threshold for the target occupancy `TO`

: ```
trapz(time,TO)
<= 0.1
```

. Another classifier, used by the authors in [1], measures the
deviation of the averaged model response from the mean model response over the parameter
domain: `mean(TO,1) < mean(TO,'all')`

. Here
`mean(TO,1)`

computes the averaged model response across time for
each sample of the sensitivity inputs, and `mean(TO,'all')`

computes
the mean value of the `TO`

response across all output times and all
samples.

**Example: **```
"max(Central.Drug(time > 1,:),[],1) <=
7"
```

**Data Types: **`char`

| `string`

| `cell`

`samples`

— Sampled values of model quantities

table

Sampled values of model quantities, specified as a table. The variable names of the table indicate the quantity names.

**Data Types: **`table`

### Name-Value Arguments

**Example:**

```
mpgsaResults = sbiompgsa(m1,{'k1','k2'},classifier,'Bounds',[0.5 5;0.1
1])
```

specifies parameter bounds for *k1*and

*k2*.

Specify one or more 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`

.

**Note**

Depending on which syntax you use, available name-value pairs differ.

**For the First Syntax**

`Bounds`

— Parameter bounds

numeric matrix

Parameter bounds, specified as 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`

`NumberSamples`

— Number of samples drawn to perform multiparametric global sensitivity analysis

`1000`

(default) | positive integer

Number of samples drawn to perform multiparametric global sensitivity analysis,
specified as the comma-separated pair consisting of
`'NumberSamples'`

and a positive integer.

**Data Types: **`double`

`SamplingMethod`

— Method to generate parameter samples

`'Sobol'`

(default) | character vector | string

Method to generate parameter samples, specified as a character vector or string. Valid options are:

`'Sobol'`

— Use the low-discrepancy Sobol sequence to generate samples.`'Halton'`

— Use the low-discrepancy Halton sequence to generate samples.`'lhs'`

— Use the low-discrepancy Latin hypercube samples.`'random'`

— Use uniformly distributed random samples.

**Data Types: **`char`

| `string`

`SamplingOptions`

— Options for sampling method

struct

Options for the sampling method, specified as a scalar struct. The options differ depending on
the sampling method: `sobol`

, `halton`

, or
`lhs`

.

For `sobol`

and `halton`

, specify each field name and value
of the structure according to each name-value argument of the `sobolset`

(Statistics and Machine Learning Toolbox) or `haltonset`

(Statistics and Machine Learning Toolbox) function. SimBiology uses the
default value of `1`

for the `Skip`

argument for both
methods. For all other name-value arguments, the software uses the same default values of
`sobolset`

or `haltonset`

. For instance, set up a
structure for the `Leap`

and `Skip`

options with
nondefault values as
follows.

s1.Leap = 50; s1.Skip = 0;

For `lhs`

, there are three samplers that support different sampling options.

If you specify a covariance matrix, SimBiology uses

`lhsnorm`

(Statistics and Machine Learning Toolbox) for sampling.`SamplingOptions`

argument is not allowed.Otherwise, use the field name

`UseLhsdesign`

to select a sampler.If the value is

`true`

, SimBiology uses`lhsdesign`

(Statistics and Machine Learning Toolbox). You can use the name-value arguments of`lhsdesign`

to specify the field names and values.If the value is

`false`

(default), SimBiology uses a nonconfigurable Latin hypercube sampler that is different from`lhsdesign`

. This sampler does not require Statistics and Machine Learning Toolbox™.`SamplingOptions`

cannot contain any other options, except`UseLhsdesign`

.

For instance, set up a structure to use `lhsdesign`

with the `Criterion`

and `Iterations`

options.

```
s2.UseLhsdesign = true;
s2.Criterion = "correlation";
s2.Iterations = 10;
```

You cannot specify this argument when a `SimBiology.Scenarios`

object is an input.

`sbiompgsa`

ignores this argument if you are also specifying
a table of samples as the second input.

`Distributions`

— Probability distributions

`prob.UniformDistribution`

(default) | `prob.ProbabilityDistribution`

object | vector of `prob.ProbabilityDistribution`

objects

Probability distributions used to draw samples, specified as a
`prob.ProbabilityDistribution`

object or vector of these objects.
Specify a scalar `prob.ProbabilityDistribution`

or vector of length
*N*, where *N* is the number of input parameters. You
can create distribution objects to sample from various distributions, such as uniform,
normal, or lognormal distributions, using `makedist`

(Statistics and Machine Learning Toolbox).

If you specify a scalar `prob.ProbabilityDistribution`

object, and there are multiple input parameters, `sbiompgsa`

uses the same distribution object to draw samples for each parameter.

You cannot specify this argument together with Bounds.

You cannot specify this argument when a `SimBiology.Scenarios`

object is an input.

`sbiompgsa`

ignores this argument when a table of samples is
an input.

**For the First and Second Syntaxes**

`Doses`

— Doses to use during simulations

`ScheduleDose`

object | `RepeatDose`

object | vector of dose objects

Doses to use during model simulations, specified as 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 a variant object or vector of variant objects.

When you specify multiple variants with duplicate specifications for a property's value, the last occurrence for the property value in the array of variants is used during simulation.

`StopTime`

— Simulation stop time

nonnegative scalar

Simulation stop time, specified as 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`

`UseParallel`

— Flag to run model simulations in parallel

`false`

(default) | `true`

Flag to run model simulations in parallel, specified as `true`

or
`false`

. When the value is `true`

and Parallel Computing Toolbox™ is available, the function runs simulations in parallel.

**Data Types: **`logical`

`Accelerate`

— Flag to turn on model acceleration

`true`

(default) | `false`

Flag to turn on model acceleration, specified as `true`

or
`false`

.

**Data Types: **`logical`

**For All Syntaxes**

`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 model
responses 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`

`SignificanceLevel`

— Significance level for Kolmogorov-Smirnov test

`0.05`

(default) | numeric scalar between 0 and 1

Significance level for Kolmogorov-Smirnov test, specified as the comma-separated
pair consisting of `'SignificanceLevel'`

and a numeric scalar between
0 and 1. For details, see Two-Sample Kolmogorov-Smirnov Test (Statistics and Machine Learning Toolbox).

**Example: **`0.1`

**Data Types: **`double`

`InterpolationMethod`

— Method for interpolation of model simulations

`"interp1q"`

(default) | character vector | string

Method for interpolation of model responses to a common set of output times, specified as a character vector or string. The valid options follow.

**Data Types: **`char`

| `string`

## Output Arguments

`mpgsaResults`

— Multiparametric global sensitivity analysis results

`SimBiology.gsa.MPGSA`

object

Multiparametric global sensitivity analysis results, returned as a
`SimBiology.gsa.MPGSA`

object. The object also contains model
simulation data used to perform MPGSA.

## More About

### Multiparametric Global Sensitivity Analysis (MPGSA)

Multiparametric global sensitivity analysis lets you study the relative
importance of parameters with respect to a classifier defined by model responses. A
classifier is an expression of model responses that evaluates to a logical vector.
`sbiompgsa`

implements the MPSA method described by Tiemann et. al.
(see supporting information text S2) [1].

`sbiompgsa`

performs the following steps.

Generate

*N*parameter samples using a sampling method.`sbiompgsa`

stores these samples as a table in a property,`mpgsaResults.ParameterSamples`

, of the returned object. The number of rows is equal to the number of samples and the number of columns is equal to the number of input parameters.**Tip**You can specify

*N*and the sampling method using the`'NumberSamples'`

and`'SamplingMethod'`

name-value pair arguments, respectively, when you call`sbiompgsa`

.Calculate the model response by simulating the model for each parameter set, which is a single realization of the model parameter values. In this case, a parameter set is equal to a row in the

`ParameterSamples`

table.Evaluate the classifier. A classifier is an expression that evaluates to a logical vector. For instance, if your model response is the AUC of plasma drug concentration, you can define a classifier with a toxicity threshold of 0.8 where the AUC of the drug concentration above that threshold is considered toxic.

Parameter sets are then separated into two different groups, such as accepted (nontoxic) and rejected (toxic) groups.

For each input parameter, compute the empirical cumulative distribution functions (

`ecdf`

(Statistics and Machine Learning Toolbox)) of accepted and rejected sample values.Compare the two eCDFs of accepted and rejected groups using the Two-Sample Kolmogorov-Smirnov Test (Statistics and Machine Learning Toolbox) to compute the Kolmogorov-Smirnov distance. The default significance level of the Kolmogorov-Smirnov test is

`0.05`

. If two eCDFs are similar, the distance is small, meaning the model response is not sensitive with respect to the input parameter. If two eCDFs are different, the distance is large, meaning the model response is sensitive to the parameter.**Note**The Kolmogorov-Smirnov test assumes that the samples follow a continuous distribution. Make sure that the eCDF plots are continuous as you increase the number of samples. If eCDFs are not continuous but step-like in the limit of infinite samples, then the results might not reflect the true sensitivities.

## References

[1] Tiemann, Christian A., Joep Vanlier, Maaike H. Oosterveer, Albert K. Groen, Peter A. J. Hilbers, and Natal A. W. van Riel. “Parameter Trajectory Analysis to Identify Treatment Effects of Pharmacological Interventions.” Edited by Scott Markel. *PLoS Computational Biology* 9, no. 8 (August 1, 2013): e1003166. https://doi.org/10.1371/journal.pcbi.1003166.

## See Also

`SimBiology.gsa.MPGSA`

| `sbiosobol`

| `sbioelementaryeffects`

| `ecdf`

(Statistics and Machine Learning Toolbox) | `kstest2`

(Statistics and Machine Learning Toolbox) | `Observable`

**Introduced in R2020a**

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