plotData
Plot quantile summary of model simulations from global sensitivity analysis (requires Statistics and Machine Learning Toolbox)
Since R2020a
Description
plots quantiles and model responses of simulated samples and returns the figure handle
h
= plotData(resultsObj
)h
.
uses additional options specified by one or more name-value pair arguments.h
= plotData(resultsObj
,Name,Value
)
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);
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,ShowMedian=true,ShowMean=false);
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);
Perform GSA by Computing Elementary Effects
Load the tumor growth model.
sbioloadproject tumor_growth_vpop_sa.sbproj
Get a variant with 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, define model parameters of interest, which are involved in the pharmacodynamics of the tumor growth. Define the model response as the tumor weight.
modelParamNames = {'L0','L1','w0','k1'}; outputName = 'tumor_weight';
Then perform GSA by computing the elementary effects using sbioelementaryeffects
. Use 100
as the number of samples and set ShowWaitBar
to true
to show the simulation progress.
rng('default');
eeResults = sbioelementaryeffects(m1,modelParamNames,outputName,Variants=v,Doses=d,NumberSamples=100,ShowWaitbar=true);
Show the median model response, the simulation results, and a shaded region covering 90% of the simulation results.
plotData(eeResults,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(eeResults,Alphas=0.1,ShowMedian=true,ShowMean=false);
Plot the time course of the means and standard deviations of the elementary effects.
h = plot(eeResults);
% Resize the figure.
h.Position(:) = [100 100 1280 800];
The mean of effects explains whether variations in input parameter values have any effect on the tumor weight response. The standard deviation of effects explains whether the sensitivity change is dependent on the location in the parameter domain.
From the mean of effects plots, parameters L1
and w0
seem to be the most sensitive parameters to the tumor weight before the dose is applied at t = 7. But, after the dose is applied, k1
and L0
become more sensitive parameters and contribute most to the after-dosing stage of the tumor weight. The plots of standard deviation of effects show more deviations for the larger parameter values in the later stage (t > 35) than for the before-dose stage of the tumor growth.
You can also display the magnitudes of the sensitivities in a bar plot. Each color shading represents a histogram representing values at different times. Darker colors mean that those values occur more often over the whole time course.
bar(eeResults);
You can also plot the parameter grids and samples used to compute the elementary effects.
plotGrid(eeResults)
You can specify more samples to increase the accuracy of the elementary effects, but the simulation can take longer to finish. Use addsamples
to add more samples.
eeResults2 = addsamples(eeResults,200);
The SimulationInfo
property of the result object contains various information for computing the elementary effects. 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.
eeResults2.SimulationInfo.SimData
SimBiology SimData Array : 1500-by-1 Index: Name: ModelName: DataCount: 1 - Tumor Growth Model 1 2 - Tumor Growth Model 1 3 - Tumor Growth Model 1 ... 1500 - 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(eeResults2.SimulationInfo.ValidSample)
ans = logical
1
You can add custom expressions as observables and compute the elementary effects of the added observables. For example, you can compute the effects for the maximum tumor weight by defining a custom expression as follows.
% Suppress an information warning that is issued. warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON'); % Add the observable expression. eeObs = addobservable(eeResults2,'Maximum tumor_weight','max(tumor_weight)','Units','gram');
Plot the computed simulation results showing the 90% quantile region.
h2 = plotData(eeObs,ShowMedian=true,ShowMean=false); h2.Position(:) = [100 100 1500 800];
You can also remove the observable by specifying its name.
eeNoObs = removeobservable(eeObs,'Maximum tumor_weight');
Restore the warning settings.
warning(warnSettings);
Input Arguments
resultsObj
— Global sensitivity analysis results
SimBiology.gsa.Sobol
object | SimBiology.gsa.MPGSA
object | SimBiology.gsa.ElementaryEffects
Global sensitivity analysis results, specified as a SimBiology.gsa.Sobol
, SimBiology.gsa.MPGSA
, or SimBiology.gsa.ElementaryEffects
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.
Example: h = plotData(results,MedianColor=green)
plots the median
model response using the green color.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: h = plotData(results,'ResponseLeap',10)
plots every 10th
model response.
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
Alphas
— Size of shaded region
0.05 (default) | positive scalar between 0 and 1 | numeric vector
Size of the quantile region in the plot, specified as a positive scalar between 0
and 1 or numeric vector. The percentage of a region is calculated as 100 * (1
- 2 * Alpha)
. Hence, the default alpha value of 0.05 corresponds to the
90% quantile region.
You can specify multiple values as a vector to plot multiple regions. For
instance, 'Alphas',[0.05 0.1]
shows both 90% and 80%
regions.
Data Types: double
FaceColor
— Color of shaded regions
three-element row vector | hexadecimal color code | color name
Color of the shaded regions, 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. 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.
Data Types: double
MedianColor
— Color of median model response
three-element row vector | hexadecimal color code | color name
Color of the median model response, specified as a three-element row vector,
hexadecimal color code, color name, or a short name. By default, the function uses
the second MATLAB default color. 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.
Data Types: double
ShowMedian
— Flag to plot median model response
false
(default) | true
Flag to plot the median model response, specified as true
or
false
.
Data Types: logical
MeanColor
— Color of mean model response
three-element row vector | hexadecimal color code | color name
Color of the mean model response, specified as a three-element row vector,
hexadecimal color code, color name, or a short name. By default, the function uses
the second MATLAB default color. 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.
Data Types: double
ShowMean
— Flag to plot mean model response
true
(default) | false
Flag to plot the mean model response, specified as true
or
false
.
Data Types: logical
ResponseColor
— Color of model responses
[0.3 0.3 0.3]
(default) | three-element row vector | hexadecimal color code | color name
Color of model responses or simulations, specified as a three-element row vector,
hexadecimal color code, color name, or a short name. By default, the function uses
the color gray [0.3 0.3 0.3]
.
For details on valid color names and corresponding RGB triplets and hexadecimal codes, see Specify Plot Colors.
Data Types: double
ResponseLeap
— Every nth response to plot
positive integer
Every nth response to plot, specified as a positive integer. By default, the function plots 10% of all model responses.
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.
[2] 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.
[3] Morris, Max D. “Factorial Sampling Plans for Preliminary Computational Experiments.” Technometrics 33, no. 2 (May 1991): 161–74.
[4] Sohier, Henri, Jean-Loup Farges, and Helene Piet-Lahanier. “Improvement of the Representativity of the Morris Method for Air-Launch-to-Orbit Separation.” IFAC Proceedings Volumes 47, no. 3 (2014): 7954–59.
Version History
Introduced in R2020aR2022b: plotData
shows median response instead of mean response
plotData
shows the median model response instead of the mean
model response by default. Use the name-value arguments ShowMedian
or
ShowMean
to plot either the mean or median response. To preserve the
behavior prior to R2022a,
specify:
plotData(gsaResults,ShowMean=true,ShowMedian=false);
R2022a: plotData
warns and shows median response
plotData
warns and shows the median model response instead of the
mean model response.
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