General interactive user guide for GSUA-CSB Toolbox

Global Sensitivity and Uncertainty Analysis - Confidence Subcontour Box (GSUA-UCI) Toolbox is a product developed by Universidad EAFIT for command-line mathematical model validation in both of Simulink or Symbolic Math Toolbox environment . At present time, the toolbox allows to perform the following functions: To apply and visualize several variance-based sensitivity (SA) and uncertainty (UA) analysis, to estimate model parameters (PE) and to to estimate confidence subcontour boxes (CSB) for estimated parameters.. This toolbox is based on the previous work of Carlos Mario Vélez: GSUA of dynamical systems using variance-based methods, published in this mathworks file exchange link.

Developers

Daniel Rojas Díaz - drojasd@eafit.edu.co
Carlos Mario Vélez Sánchez - cmvelez@eafit.edu.co

Addtitonal resources

Toolbox paper, Slides on UA/SA, model identifiability paper.

Table of Contents

General interactive user guide for GSUA-CSB Toolbox Developers Addtitonal resources Initial configuration Working with Simulink models Working with Symbolic math models Working with User-defined models Toolbox routines for selected model 1. Simulating the model 1.1 gsua_eval function 2. Sample the space of factors 2.1 gsua_dmatrix function 3. Uncertainty analysis 3.1 gsua_ua function 4. Sensitivity analysis (SA) 4.1 gsua_sa function 4.2 SA results visualization and interpretation 5. Parameter estimation 5.1 gsua_pe function 5.2 Identifiability analysis 6. Uncertainty Interval Calculus 6.1 gsua_oatr function 6.2 gsua_ucis function Interactive user guide for Toolbox initial configuration (Simulink) 1. Model preparation 2. Environment configuration and additional features 2.1 Additional features: 2.1.1 Range definition 2.1.2 Nominal factor family 2.1.3 Model kind (default: Dynamic) 2.1.4 Fixed step (Default: []) Interactive user guide for Toolbox initial configuration (Symbolic Matlab) 1. Model creation 2. Environment configuration and additional features 2.1 Additional features 2.1.1 Range definition 2.1.2 Nominal factor family 2.1.3 Specific model output (Default: 1:model_order) 2.1.4 Specific ode solver (Default: ode45) 2.1.5 Ode solver configuration (Default: none) 2.1.6 Static model simulation Interactive user guide for Toolbox initial configuration (User-Defined functions) 1. Time-dependent functions 1.1 Model creation 1.1.1 Example 1.2 Model implementation 1.3 Additional features 1.3.1 Range definition 1.3.2 Nominal factor family 1.3.3 Specific model output (Default: 1:model_order) 1.3.4 Specific names for model factors (Default: {}) 1.3.5 Specific names for model outputs (Default: {}) 1.3.6 Run as vectorized function (Default: false) 1.3.7 Any additional user-provided configuration (Default: {}) 2. Time-free functions 2.1 Model creation 1.1.1 Example 2.2 Model implementation 2.3 Additional features

IMPORTANT NOTE: If you lost any of the results in this guide, you can recover it easy with the following command:

load userguidesession.mat

Initial configuration

Working with Simulink models

It is necessary to create and configure a simulink model as indicated in GSUA Simulink userguide. Also, check the guide for additional configurations.

[T,tout]=gsua_dataprep(model_name,Ranges,Parameter_names)

Working with Symbolic math models

It is necessary to create a system of equations as indicated in GSUA Symbolic Math userguide. Also, check the guide for additional configurations.

[T,solver]=gsua_dataprep(odes,vars,domain,model_name)

Working with User-defined models

It is necessary to create a system of equations as indicated in GSUA User-defined functions userguide. Also, check the guide for additional configurations.

[T,sol]=gsua_dataprep(func,ranges,'domain',domain)

Toolbox routines for selected model

For the following examples we will use a symbolic math model (same as constructed in GSUA Symbolic Math userguide). You can execute your own model or the Simulink model example following the respective userguide steps.

1. Simulating the model

%load default example model

default_example

Setting environment to work with symbolic Matlab Introduce ranges in the following order:

ans(t) =

T%check the summary table

T = 4×2 table 

	Range		Nominal
1 S0	100	3000	1550
2 I0	1	500	250.5000
3 beta	0	1	0.5000
4 gamma	0	1	0.5000

1.1 gsua_eval function

help gsua_eval%check features
  Function for few model evaluations
  
  Y=gsua_eval(values,Table)
  Parameters:
  values <-- array of NpxN (number of factors x number of simulations)
  Table  <-- summary table from gsua_dataprep function
  Outputs:
  Y <-- array with model output
  Additional features:
  If you provide an array (xdata) that belongs to the model domain, then
  the model output is interpolated to match xdata. Also you can provide a
  previous model output or experimental data as an array (ydata) that must
  coincide with the xdata array:
  Y=gsua_eval(values,Table,xdata,ydata)

Execute the following line to modify the number of model outputs (valid only for symbolic math and user-defined models)

T.Properties.CustomProperties.output=1:3;%set to visualize the three states of the model

y=gsua_eval(T.Nominal,T);% simple simulation with factor values stored in T.Nominal

%now, we are going to use the additional features of the function

xdata=linspace(0,264,265);%create 265 points between 0 and 264

noise= normrnd(0,0.2,size(xdata));%create noise

%simulate the model with user defined parameters and return values in xdata

y=gsua_eval([1000 200 0.1 0.06]',T,xdata);

yexp=y+y.*noise;%create a noisy output

%compare results and simulate another factors family

y=gsua_eval([1000 200 0.1 0.06; 2000 200 0.1 0.06]',T,xdata,yexp);

2. Sample the space of factors

2.1 gsua_dmatrix function

help gsua_dmatrix

Function for design of experiments (factor space sampling) [M,T2]=gsua_dmatrix(T,N) Parameters: T <-- summary table from gsua_dataprep N <-- number of samples Outpus: M <-- design matrix of NxNp for later routines T2 <-- summary table with fixed parameters actualized Additional features: You can choose a method for factor space sampling between uniform distribution design and latin hypercube design (default). To switch between methods use the paired feature 'Method' and 'Uniform', 'Sobol', or 'LatinHypercube'. Also you can visualize the sampling result using the paired feature 'Show', 'on' M=gsua_dmatrix(T,N,'Method','Sobol','Show','on')

M=gsua_dmatrix(T,500,'Show','on');%design matrix with 500 samples

%we ignore T2 because there is no new fixed parameters

3. Uncertainty analysis

3.1 gsua_ua function

help gsua_ua

Function for uncertainty analysis Y=gsua_ua(M,T) Parameters: M <-- design matrix from gsua_dmatrix function T <-- summary table from gsua_dataprep function Output: Y <-- result of Monte-Carlo simulation Additional features: The function automatically apply a Monte-Carlo filtering over results and present it as figures (gsua_MCF function). It is possible to perform UA over multiple model outputs. To obtain extrapolated results, use the paired feature 'xdata'. To compare results with a specific output use the paired feature 'ynom' (ynom length must coindice with xdata length). To avoid parallel computing (no speed up), use the paired feature 'parallel',false. Y=gsua_ua(M,T,'xdata',xdata,'ynom',ynom,'parallel',false)

M=gsua_dmatrix(T,500);%design matrix

xdata=linspace(0,264,265);%265 points between 0 and 264

Y=gsua_ua(M,T,'xdata',xdata);

Progress: 100% Estimated processing time (h:m:s): 0:0:1 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:0:1 Estimated stop time (h:m:s): 2:31:45 Number of simulations: 500

Since we do not specify a nominal output, the function will take as nominal the output associated with T.Nominal values (red line in first figure above).

The Monte-Carlo filtering interpretation is as follows:

The bigger the difference between low values (blue) and high values (red) lines, the bigger is the factor relevance.
If low values (blue) is the bottom line, then high values for the correspondent factor generates low values for model output and low values for the correspondent factor generates high values for model output..
If high values (red) is the botton line, then high values for the correspondent factor generates hich values for model output and low values for the correspondent factor generates low values for model output.

%if you change ynom, then Monte-Carlo filtering results could vary

ynom=gsua_eval([1000 200 0.1 0.06]',T,xdata);

Y=gsua_ua(M,T,'xdata',xdata,'ynom',ynom);

Progress: 100% Estimated processing time (h:m:s): 0:0:1 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:0:1 Estimated stop time (h:m:s): 2:31:50 Number of simulations: 500

Now, results interpretation is much more tricky. It is better to perform a SA (next section).

4. Sensitivity analysis (SA)

4.1 gsua_sa function

Sensitivity indices estimators implemented in this toolbox are based on the following works:

[1]: Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., and Tarantola, S. (2010). Variance based sensitivity analysis of model output. design and estimator for the total sensitivity index. Computer Physics Communications, 181(2):259–270.

[2]: Xiao, S., Lu, Z., and Wang, P. (2018). Multivariate global sensitivity analysis based on distance components decomposition. Risk Analysis, 38(12):2703–2721.

Saltelli method has been configured to work with Minimun Least-Squares outputs, if you do not provide a specific 'ynom' the function assume that 'ynom' is given by the nominal Factors in T.

Xiao method is especially good for time-dependent model responses.

help gsua_sa
  Function for sensitivity analysis
 
  T=gsua_sa(M,T)
  Parameters:
  M <-- design matrix from gsua_dmatrix function
  T <-- summary table from gsua_dataprep function
  Output:
  T <-- summary table with sensitivity indices.
  Additional features:
  It is possible to request 2 extra positional outputs:
  J <-- vector of model outputs in scalar representation (MSE)
  Y <-- model output matrix (same as gsua_ua output)
  It is possible to choose between 6 SA methods using the paired feature
  'SensMethod',method. Where method is one of the following:
  'Saltelli'    <-- requires N*(Np/2+1) simulations
  'Jansen'      <-- requires N*(Np/2+1) simulations
  'Xiao'        <-- newest method (Default), requires N*(Np/2+1)
                    simulations
  'Sobol'       <-- classic method, requires N*(Np+1) simulations
  'brute-force' <-- explores all possible combinations, N + Np2*N^2
                    simulations
  'OAT'         <-- non-global method, requires N*(Np+1) simulations
  Also, it is possible to specify an output for indices estimation using
  the paired feature 'ynom'. Finally, you can avoid parallel speed up 
  for simulation process using the paired feature 'parallel', false.
  [ParT,J,Y] = gsua_sa(M,T,'SensMethod','Xiao','ynom',ynom)
  Note: ynom size must coincide with xdata size, where xdata is the next
  xdata=linspace(Domain(1),Domain(2),Domain(2)-Domain(1)+1) and Domain is
  the array of the model domain

Let's use the same M matrix as in previous section, without a specific ynom. Hence, the toolbox will take as ynom the output associated to T.Nominal values. Sensitivity indices are not affected by ynom selection. Additional outputs J and Y will be used for posterior analysis.

[T,J,Y]=gsua_sa(M,T,'SensMethod','Xiao');
Progress: 17%
Estimated processing time (h:m:s): 0:0:5
Remaining time (h:m:s): 0:0:4
Elapsed time (h:m:s): 0:0:0
Estimated stop time (h:m:s): 2:32:11
Number of simulations: 1500
Progress: 34%
Estimated processing time (h:m:s): 0:0:6
Remaining time (h:m:s): 0:0:4
Elapsed time (h:m:s): 0:0:2
Estimated stop time (h:m:s): 2:32:12
Number of simulations: 1500
Progress: 50%
Estimated processing time (h:m:s): 0:0:8
Remaining time (h:m:s): 0:0:4
Elapsed time (h:m:s): 0:0:4
Estimated stop time (h:m:s): 2:32:14
Number of simulations: 1500
Progress: 67%
Estimated processing time (h:m:s): 0:0:9
Remaining time (h:m:s): 0:0:3
Elapsed time (h:m:s): 0:0:6
Estimated stop time (h:m:s): 2:32:15
Number of simulations: 1500
Progress: 84%
Estimated processing time (h:m:s): 0:0:9
Remaining time (h:m:s): 0:0:1
Elapsed time (h:m:s): 0:0:8
Estimated stop time (h:m:s): 2:32:15
Number of simulations: 1500
Progress: 100%
Estimated processing time (h:m:s): 0:0:9
Remaining time (h:m:s): 0:0:0
Elapsed time (h:m:s): 0:0:9
Estimated stop time (h:m:s): 2:32:15
Number of simulations: 1500

4.2 SA results visualization and interpretation

gsua_plot('UncertaintyAnalysis',T,Y,xdata,ynom); <- Uncertainty analysis graph
gsua_plot('ScatterOutput',T,Y,M,xdata,xref); <- Uncertainty analysis for output in time=xref
gsua_plot('ScatterOutput',T,J,M,xdata); <- Uncertainty analysis for scalar output (MSE of the output with respecto to ynom)
gsua_plot('Pie',T,T.STi/Si); <- Pie graph of sensitivity indices
gsua_plot('Bar',T,T.STi/Si); <- Bar graph of sensitivity indices
gsua_plot('FractionalSensitivityArea',T,T.STi_vec/Si_vec,xdata); <- Area graph of vectorial sensitivity indices (a index for each point in xdata)
gsua_plot('TotalSensitivityArea',T,T.STi_vec/Si_vec,xdata); <- Normalized area graph of vectorial sensitivity indices (a index for each point in xdata)
gsua_plot('FractionalSensitivityPlots',T,T.STi_vec/Si_vec,xdata); <- Figure of subgraphs with the sensitivity indices plot for each factor, and for each point in xdata.
gsua_plot('Pie',T,T.STi_vec/Si_vec,xdata,[t1,t2,t3,t4]); <- Figure of pie subgraphs. As many subgraphs as points in [t1,t2,...tn]. Each pie represent the sensitivity indices for a specific time [t1,t2,...tn].

The most relevant index to assess parameter relevance is STi. However, it is neccesary to point out a relevant feature about the indices: The summatory of STi is always greater or equal to one, while the summatory of Si is always lesser or equal to one. If one of the sumatories is equal to one, the other one must be equal too, and hence, the model is purely additive.

A general advice is to perform a global SA over the model when the interaction among factors it is unknown. If the summatory of Si is greater than 0.65, then you can state that the model has weak interactions among factors and said interactions are negligible. Thereby, for subsequent analysis, it is better to perform local SA analysis or analytical SA methods.

IMPORTANT NOTE: Choosing an appropriate N for SA is a challenging task. However, Saltelli and Distance method allow an aproximation: A good N has been chosen when there are no negative Si. Thereby, if you get negative Si values, you should consider to probe with a greater N.

T.Si

ans = 4×1

sum(T.Si)

ans = 0.9270

sum(T.STi)

ans = 1.2884

clf

gsua_plot('Pie',T,T.STi)

colormap jet

clf

gsua_plot('TotalSensitivityArea',T,T.STi_vec,xdata);

colormap jet

clf

gsua_plot('ScatterOutput',T,J,M,xdata);

5. Parameter estimation

5.1 gsua_pe function

help gsua_pe

Parameter estimation function [T,res] = gsua_pe(T,xdata,ydata,N) Parameters: T <-- summary table from gsua_dataprep function xdata <-- array of points where the model will be evaluated ydata <-- array with expected model output Outputs: T <-- summary table with parameter estimation results res <-- cost functions for each estimation Additional paired features: 'N',N <-- number of parameter estimations 'Multistart',k <-- activate multistart feature for lsqcurvefit optimizer, perform k parameter estimations in each cycle 'solver',{'lsqc','lsqn','ga','particle','psearch','surrogate','annealing'} <-- allows to choose among several matlab optimizers. Default: lsqc 'opt',optimoptions(optimizer,...) <-- allows to configure the respective matlab optimizer, optimizer must match with the real matlab optimizer name. 'ipoint',point <-- allows to give to the optimizer the initial points for estimations. point must have the same number of columns as factors to estimate and the same number of rows as N. 'Show',{'off','on'}. If activated, the function plot optimizer results Default: off. [T,res] = gsua_pe(T,xdata,ydata,'Show','on','solver','particle','N',3)

solver='lsqc';%this is the default solver

opt=optimoptions('lsqcurvefit','UseParallel',true);%configure optimizer

[T,res]=gsua_pe(T,xdata,yexp,'solver',solver,'Show','on','opt',opt,'N',10);

Generating a valid matrix for estimations

Estimation 1 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 2 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 3 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 4 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 5 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 6 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 7 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 8 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 9 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details> Estimation 10 Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. <stopping criteria details>

T%look the actual table, there is a lot of information
T = 4×7 table 
 RangeNominalSiSTiSTi_vecSi_vecEstlsqc
1 S0100300015500.16990.133900.02320.06560.09810.11920.13500.14890.16190.17410.18510.19460.20220.20770.21110.21290.21330.21260.21120.20910.20660.20380.20070.19760.19440.19120.18810.18510.18220.17930.17640.17360.17080.16800.16520.16250.15970.15690.15420.15140.14870.14600.14330.14060.13790.13520.13260.13000.12740.12490.12240.11990.11740.11500.11260.11030.10800.10570.10340.10120.09910.09690.09480.09280.09070.08880.08680.08490.08300.08110.07930.07750.07580.07410.07240.07070.06910.06750.06600.06440.06290.06150.06000.05860.05720.05590.05460.05330.05200.05070.04950.04830.04720.04600.04490.04380.04270.04170.04060.03960.03870.03770.03680.03590.03500.03410.03320.03240.03160.03080.03000.02920.02850.02780.02710.02640.02570.02500.02440.02380.02320.02260.02200.02140.02090.02030.01980.01930.01880.01830.01780.01730.01690.01640.01600.01560.01520.01480.01440.01400.01360.01330.01290.01260.01220.01190.01160.01130.01100.01070.01040.01010.00980.00960.00930.00910.00880.00860.00840.00810.00790.00770.00750.00730.00710.00690.00670.00650.00630.00620.00600.00580.00570.00550.00540.00520.00510.00490.00480.00470.00450.00440.00430.00420.00410.00400.00380.00370.00360.00350.00340.00330.00320.00320.00310.00300.00290.00280.00270.00270.00260.00250.00240.00240.00230.00220.00220.00210.00210.00200.00190.00190.00180.00180.00170.00170.00160.00160.00160.00150.00150.00140.00140.00130.00130.00130.00120.00120.00120.00110.00110.00110.00100.00100.00100.00100.00090.00090.00090.00080.00080.00080.00080.00080.00070.00070.00070.00070.00070.00060.00060.00060.00060.00060.00050.00050.00050.00050.00050.00050.00050.00040.00040.00040.00040.000400.00380.01300.01480.01090.00640.00400.00510.00920.01520.02130.02610.02870.02930.02830.02610.02330.02010.01690.01370.01080.00800.00550.00330.0013-0.0005-0.0020-0.0034-0.0046-0.0057-0.0066-0.0074-0.0081-0.0088-0.0093-0.0098-0.0101-0.0104-0.0107-0.0109-0.0110-0.0111-0.0111-0.0111-0.0111-0.0110-0.0109-0.0108-0.0107-0.0105-0.0103-0.0101-0.0099-0.0097-0.0095-0.0093-0.0090-0.0088-0.0086-0.0083-0.0081-0.0079-0.0076-0.0074-0.0072-0.0069-0.0067-0.0065-0.0063-0.0061-0.0059-0.0057-0.0055-0.0053-0.0051-0.0049-0.0048-0.0046-0.0044-0.0043-0.0041-0.0040-0.0038-0.0037-0.0035-0.0034-0.0033-0.0032-0.0030-0.0029-0.0028-0.0027-0.0026-0.0025-0.0024-0.0023-0.0022-0.0021-0.0021-0.0020-0.0019-0.0018-0.0018-0.0017-0.0016-0.0016-0.0015-0.0014-0.0014-0.0013-0.0013-0.0012-0.0012-0.0011-0.0011-0.0010-0.0010-0.0009-0.0009-0.0009-0.0008-0.0008-0.0008-0.0007-0.0007-0.0007-0.0006-0.0006-0.0006-0.0006-0.0005-0.0005-0.0005-0.0005-0.0005-0.0004-0.0004-0.0004-0.0004-0.0004-0.0004-0.0003-0.0003-0.0003-0.0003-0.0003-0.0003-0.0003-0.0003-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0002-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0001-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000-0.0000336.3700336.3739336.3758336.3773336.3791336.3828336.3869336.3871336.3959336.4517
2 I01500250.50000.02200.02140.92160.62370.34400.20810.13460.09150.06630.05170.04220.03440.02750.02160.01690.01320.01030.00800.00620.00480.00380.00310.00270.00250.00250.00260.00280.00290.00300.00300.00300.00290.00280.00260.00240.00230.00210.00190.00170.00160.00150.00130.00120.00110.00100.00090.00090.00080.00070.00070.00060.00060.00060.00050.00050.00050.00040.00040.00040.00040.00040.00030.00030.00030.00030.00030.00030.00020.00020.00020.00020.00020.00020.00020.00020.00020.00020.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.80610.52250.23010.10960.05840.03290.01800.00830.0019-0.0020-0.0040-0.0046-0.0044-0.0036-0.0026-0.0015-0.00040.00070.00170.00260.00340.00420.00470.00520.00550.00570.00580.00580.00570.00550.00530.00510.00480.00450.00430.00400.00380.00350.00330.00310.00280.00270.00250.00230.00220.00200.00190.00180.00170.00160.00150.00140.00130.00120.00120.00110.00100.00100.00090.00090.00080.00080.00070.00070.00070.00060.00060.00060.00050.00050.00050.00040.00040.00040.00040.00030.00030.00030.00030.00030.00030.00020.00020.00020.00020.00020.00020.00020.00020.00020.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000171.4599171.4604171.4609171.4610171.4612171.4618171.4625171.4625171.4637171.4726
3 beta010.50000.04560.063500.12040.20810.22780.22040.20440.18600.16710.14920.13340.11970.10800.09810.09000.08350.07850.07460.07150.06910.06710.06540.06390.06250.06120.05990.05850.05710.05550.05390.05220.05050.04870.04690.04510.04330.04160.03990.03820.03660.03500.03350.03210.03070.02940.02810.02690.02570.02460.02350.02240.02150.02050.01960.01870.01790.01710.01630.01560.01490.01420.01360.01300.01240.01180.01130.01080.01030.00980.00930.00890.00850.00810.00770.00740.00700.00670.00640.00610.00580.00560.00530.00510.00480.00460.00440.00420.00400.00380.00370.00350.00340.00320.00310.00300.00280.00270.00260.00250.00240.00240.00230.00220.00210.00210.00200.00200.00190.00190.00180.00180.00180.00170.00170.00170.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00160.00170.00170.00170.00170.00170.00170.00180.00180.00180.00180.00180.00190.00190.00190.00190.00200.00200.00200.00200.00210.00210.00210.00220.00220.00220.00220.00230.00230.00230.00230.00240.00240.00240.00250.00250.00250.00250.00260.00260.00260.00260.00270.00270.00270.00270.00280.00280.00280.00280.00280.00290.00290.00290.00290.00290.00300.00300.00300.00300.00300.00310.00310.00310.00310.00310.00310.00310.00320.00320.00320.00320.00320.00320.00320.00320.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00340.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00330.00320.00320.00320.00320.00320.00320.00320.003200.23270.24770.19260.14390.10750.07970.05830.04220.03040.02200.01620.01230.00970.00810.00700.00630.00590.00550.00520.00490.00460.00440.00410.00390.00370.00350.00340.00320.00310.00300.00280.00270.00260.00250.00240.00220.00210.00200.00190.00180.00170.00160.00140.00130.00130.00120.00110.00100.00090.00090.00080.00070.00070.00060.00060.00060.00050.00050.00040.00040.00040.00040.00030.00030.00030.00030.00030.00030.00020.00020.00020.00020.00020.00020.00020.00020.00020.00020.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.12910.12910.12910.12910.12910.12910.12910.12910.12910.1290
4 gamma010.50000.68951.069600.25180.46440.53460.57350.61510.66270.71140.75630.79580.83050.86100.88790.91150.93190.94950.96460.97760.98870.99811.00611.01281.01851.02371.02851.03331.03821.04341.04901.05511.06161.06861.07601.08371.09171.10011.10861.11731.12621.13511.14411.15321.16221.17131.18021.18921.19801.20681.21551.22411.23251.24091.24911.25731.26531.27311.28091.28851.29611.30351.31071.31791.32501.33191.33871.34551.35211.35861.36501.37131.37751.38371.38971.39561.40151.40731.41301.41861.42411.42961.43501.44031.44561.45081.45591.46091.46601.47091.47581.48071.48551.49021.49491.49961.50421.50881.51331.51781.52231.52681.53121.53561.53991.54431.54861.55291.55721.56151.56571.56991.57421.57841.58261.58681.59101.59521.59941.60361.60781.61201.61611.62031.62451.62881.63301.63721.64141.64571.65001.65421.65851.66281.66721.67151.67591.68031.68471.68911.69351.69801.70251.70701.71161.71611.72071.72541.73001.73471.73941.74421.74901.75381.75861.76351.76841.77331.77831.78331.78841.79351.79861.80381.80901.81421.81951.82481.83021.83561.84101.84651.85201.85761.86321.86881.87451.88031.88611.89191.89781.90371.90961.91561.92171.92781.93391.94011.94631.95261.95901.96531.97181.97821.98481.99131.99802.00462.01132.01812.02492.03182.03872.04572.05272.05982.06692.07412.08132.08862.09592.10332.11072.11822.12572.13332.14102.14872.15642.16422.17212.18002.18802.19602.20412.21222.22042.22862.23692.24532.25372.26212.27072.27922.28792.29662.30532.31412.32302.33192.34082.34992.35902.36812.37732.38662.39592.40522.41472.42422.43372.44332.45302.46272.47252.48242.49232.50222.51232.52232.53252.54272.55302.56332.57372.58412.594600.31940.56880.68270.74170.78040.81220.84190.87110.90060.93080.96140.99171.02141.05011.07761.10361.12801.15061.17151.19071.20831.22451.23971.25411.26801.28171.29521.30891.32261.33651.35071.36501.37941.39401.40871.42341.43821.45291.46771.48241.49711.51161.52611.54041.55461.56871.58261.59631.60991.62321.63641.64941.66231.67491.68731.69961.71161.72351.73521.74671.75801.76911.78011.79091.80151.81191.82221.83221.84211.85191.86151.87091.88011.88921.89821.90701.91561.92411.93241.94061.94871.95661.96441.97201.97951.98691.99412.00122.00822.01512.02182.02842.03492.04132.04762.05382.05982.06582.07162.07742.08302.08852.09402.09942.10462.10982.11492.11992.12482.12972.13442.13912.14372.14832.15272.15712.16152.16572.16992.17412.17822.18222.18612.19012.19392.19772.20152.20522.20892.21252.21612.21962.22312.22662.23002.23342.23672.24002.24332.24662.24982.25302.25622.25932.26252.26562.26862.27172.27472.27772.28072.28372.28672.28962.29262.29552.29842.30132.30422.30712.31002.31282.31572.31852.32142.32422.32712.32992.33272.33552.33842.34122.34402.34682.34972.35252.35532.35822.36102.36382.36672.36952.37242.37522.37812.38102.38392.38672.38962.39252.39542.39842.40132.40422.40722.41012.41312.41612.41912.42212.42512.42812.43122.43422.43732.44042.44352.44662.44972.45282.45602.45922.46232.46552.46882.47202.47522.47852.48182.48512.48842.49172.49512.49852.50192.50532.50872.51212.51562.51912.52262.52612.52962.53322.53682.54042.54402.54762.55132.55502.55872.56242.56612.56992.57372.57752.58132.58512.58902.59292.59682.60082.60472.60872.61272.61672.62082.62482.62892.63302.63722.64132.64552.64970.03260.03260.03260.03260.03260.03260.03260.03260.03260.0326

	Range	Nominal	Si	STi	STi_vec	Si_vec	Estlsqc
1 S0	100	3000	1550	0.1699	0.1339	0	0.0232	0.0656	0.0981	0.1192	0.1350	0.1489	0.1619	0.1741	0.1851	0.1946	0.2022	0.2077	0.2111	0.2129	0.2133	0.2126	0.2112	0.2091	0.2066	0.2038	0.2007	0.1976	0.1944	0.1912	0.1881	0.1851	0.1822	0.1793	0.1764	0.1736	0.1708	0.1680	0.1652	0.1625	0.1597	0.1569	0.1542	0.1514	0.1487	0.1460	0.1433	0.1406	0.1379	0.1352	0.1326	0.1300	0.1274	0.1249	0.1224	0.1199	0.1174	0.1150	0.1126	0.1103	0.1080	0.1057	0.1034
2 I0	1	500	250.5000	0.0220	0.0214	0.9216	0.6237	0.3440	0.2081	0.1346	0.0915	0.0663	0.0517	0.0422	0.0344	0.0275	0.0216	0.0169	0.0132	0.0103	0.0080	0.0062	0.0048	0.0038	0.0031	0.0027	0.0025	0.0025	0.0026	0.0028	0.0029	0.0030	0.0030	0.0030	0.0029	0.0028	0.0026	0.0024	0.0023	0.0021	0.0019	0.0017	0.0016	0.0015	0.0013	0.0012	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004
3 beta	0	1	0.5000	0.0456	0.0635	0	0.1204	0.2081	0.2278	0.2204	0.2044	0.1860	0.1671	0.1492	0.1334	0.1197	0.1080	0.0981	0.0900	0.0835	0.0785	0.0746	0.0715	0.0691	0.0671	0.0654	0.0639	0.0625	0.0612	0.0599	0.0585	0.0571	0.0555	0.0539	0.0522	0.0505	0.0487	0.0469	0.0451	0.0433	0.0416	0.0399	0.0382	0.0366	0.0350	0.0335	0.0321	0.0307	0.0294	0.0281	0.0269	0.0257	0.0246	0.0235	0.0224	0.0215	0.0205	0.0196	0.0187	0.0179	0.0171	0.0163	0.0156
4 gamma	0	1	0.5000	0.6895	1.0696	0	0.2518	0.4644	0.5346	0.5735	0.6151	0.6627	0.7114	0.7563	0.7958	0.8305	0.8610	0.8879	0.9115	0.9319	0.9495	0.9646	0.9776	0.9887	0.9981	1.0061	1.0128	1.0185	1.0237	1.0285	1.0333	1.0382	1.0434	1.0490	1.0551	1.0616	1.0686	1.0760	1.0837	1.0917	1.1001	1.1086	1.1173	1.1262	1.1351	1.1441	1.1532	1.1622	1.1713	1.1802	1.1892	1.1980	1.2068	1.2155	1.2241	1.2325	1.2409	1.2491	1.2573	1.2653	1.2731	1.2809	1.2885

5.2 Identifiability analysis

T_newRange=gsua_ia(T,T.Estlsqc)

T_newRange = 4×7 table 

	Range		Nominal	Si	STi	STi_vec																																																																Si_vec																																																																Estlsqc
1 S0	336.3700	336.4517	336.3700	0.1699	0.1339	0	0.0232	0.0656	0.0981	0.1192	0.1350	0.1489	0.1619	0.1741	0.1851	0.1946	0.2022	0.2077	0.2111	0.2129	0.2133	0.2126	0.2112	0.2091	0.2066	0.2038	0.2007	0.1976	0.1944	0.1912	0.1881	0.1851	0.1822	0.1793	0.1764	0.1736	0.1708	0.1680	0.1652	0.1625	0.1597	0.1569	0.1542	0.1514	0.1487	0.1460	0.1433	0.1406	0.1379	0.1352	0.1326	0.1300	0.1274	0.1249	0.1224	0.1199	0.1174	0.1150	0.1126	0.1103	0.1080	0.1057	0.1034	0.1012	0.0991	0.0969	0.0948	0.0928	0.0907	0.0888	0.0868	0.0849	0.0830	0.0811	0.0793	0.0775	0.0758	0.0741	0.0724	0.0707	0.0691	0.0675	0.0660	0.0644	0.0629	0.0615	0.0600	0.0586	0.0572	0.0559	0.0546	0.0533	0.0520	0.0507	0.0495	0.0483	0.0472	0.0460	0.0449	0.0438	0.0427	0.0417	0.0406	0.0396	0.0387	0.0377	0.0368	0.0359	0.0350	0.0341	0.0332	0.0324	0.0316	0.0308	0.0300	0.0292	0.0285	0.0278	0.0271	0.0264	0.0257	0.0250	0.0244	0.0238	0.0232	0.0226	0.0220	0.0214	0.0209	0.0203	0.0198	0.0193	0.0188	0.0183	0.0178	0.0173	0.0169	0.0164	0.0160	0.0156	0.0152	0.0148	0.0144	0.0140	0.0136	0.0133	0.0129	0.0126	0.0122	0.0119	0.0116	0.0113	0.0110	0.0107	0.0104	0.0101	0.0098	0.0096	0.0093	0.0091	0.0088	0.0086	0.0084	0.0081	0.0079	0.0077	0.0075	0.0073	0.0071	0.0069	0.0067	0.0065	0.0063	0.0062	0.0060	0.0058	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048	0.0047	0.0045	0.0044	0.0043	0.0042	0.0041	0.0040	0.0038	0.0037	0.0036	0.0035	0.0034	0.0033	0.0032	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0027	0.0026	0.0025	0.0024	0.0024	0.0023	0.0022	0.0022	0.0021	0.0021	0.0020	0.0019	0.0019	0.0018	0.0018	0.0017	0.0017	0.0016	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014	0.0013	0.0013	0.0013	0.0012	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010	0.0010	0.0010	0.0009	0.0009	0.0009	0.0008	0.0008	0.0008	0.0008	0.0008	0.0007	0.0007	0.0007	0.0007	0.0007	0.0006	0.0006	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0004	0	0.0038	0.0130	0.0148	0.0109	0.0064	0.0040	0.0051	0.0092	0.0152	0.0213	0.0261	0.0287	0.0293	0.0283	0.0261	0.0233	0.0201	0.0169	0.0137	0.0108	0.0080	0.0055	0.0033	0.0013	-0.0005	-0.0020	-0.0034	-0.0046	-0.0057	-0.0066	-0.0074	-0.0081	-0.0088	-0.0093	-0.0098	-0.0101	-0.0104	-0.0107	-0.0109	-0.0110	-0.0111	-0.0111	-0.0111	-0.0111	-0.0110	-0.0109	-0.0108	-0.0107	-0.0105	-0.0103	-0.0101	-0.0099	-0.0097	-0.0095	-0.0093	-0.0090	-0.0088	-0.0086	-0.0083	-0.0081	-0.0079	-0.0076	-0.0074	-0.0072	-0.0069	-0.0067	-0.0065	-0.0063	-0.0061	-0.0059	-0.0057	-0.0055	-0.0053	-0.0051	-0.0049	-0.0048	-0.0046	-0.0044	-0.0043	-0.0041	-0.0040	-0.0038	-0.0037	-0.0035	-0.0034	-0.0033	-0.0032	-0.0030	-0.0029	-0.0028	-0.0027	-0.0026	-0.0025	-0.0024	-0.0023	-0.0022	-0.0021	-0.0021	-0.0020	-0.0019	-0.0018	-0.0018	-0.0017	-0.0016	-0.0016	-0.0015	-0.0014	-0.0014	-0.0013	-0.0013	-0.0012	-0.0012	-0.0011	-0.0011	-0.0010	-0.0010	-0.0009	-0.0009	-0.0009	-0.0008	-0.0008	-0.0008	-0.0007	-0.0007	-0.0007	-0.0006	-0.0006	-0.0006	-0.0006	-0.0005	-0.0005	-0.0005	-0.0005	-0.0005	-0.0004	-0.0004	-0.0004	-0.0004	-0.0004	-0.0004	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	336.3700	336.3739	336.3758	336.3773	336.3791	336.3828	336.3869	336.3871	336.3959	336.4517
2 I0	171.4599	171.4726	171.4599	0.0220	0.0214	0.9216	0.6237	0.3440	0.2081	0.1346	0.0915	0.0663	0.0517	0.0422	0.0344	0.0275	0.0216	0.0169	0.0132	0.0103	0.0080	0.0062	0.0048	0.0038	0.0031	0.0027	0.0025	0.0025	0.0026	0.0028	0.0029	0.0030	0.0030	0.0030	0.0029	0.0028	0.0026	0.0024	0.0023	0.0021	0.0019	0.0017	0.0016	0.0015	0.0013	0.0012	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.8061	0.5225	0.2301	0.1096	0.0584	0.0329	0.0180	0.0083	0.0019	-0.0020	-0.0040	-0.0046	-0.0044	-0.0036	-0.0026	-0.0015	-0.0004	0.0007	0.0017	0.0026	0.0034	0.0042	0.0047	0.0052	0.0055	0.0057	0.0058	0.0058	0.0057	0.0055	0.0053	0.0051	0.0048	0.0045	0.0043	0.0040	0.0038	0.0035	0.0033	0.0031	0.0028	0.0027	0.0025	0.0023	0.0022	0.0020	0.0019	0.0018	0.0017	0.0016	0.0015	0.0014	0.0013	0.0012	0.0012	0.0011	0.0010	0.0010	0.0009	0.0009	0.0008	0.0008	0.0007	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	171.4599	171.4604	171.4609	171.4610	171.4612	171.4618	171.4625	171.4625	171.4637	171.4726
3 beta	0.1290	0.1291	0.1291	0.0456	0.0635	0	0.1204	0.2081	0.2278	0.2204	0.2044	0.1860	0.1671	0.1492	0.1334	0.1197	0.1080	0.0981	0.0900	0.0835	0.0785	0.0746	0.0715	0.0691	0.0671	0.0654	0.0639	0.0625	0.0612	0.0599	0.0585	0.0571	0.0555	0.0539	0.0522	0.0505	0.0487	0.0469	0.0451	0.0433	0.0416	0.0399	0.0382	0.0366	0.0350	0.0335	0.0321	0.0307	0.0294	0.0281	0.0269	0.0257	0.0246	0.0235	0.0224	0.0215	0.0205	0.0196	0.0187	0.0179	0.0171	0.0163	0.0156	0.0149	0.0142	0.0136	0.0130	0.0124	0.0118	0.0113	0.0108	0.0103	0.0098	0.0093	0.0089	0.0085	0.0081	0.0077	0.0074	0.0070	0.0067	0.0064	0.0061	0.0058	0.0056	0.0053	0.0051	0.0048	0.0046	0.0044	0.0042	0.0040	0.0038	0.0037	0.0035	0.0034	0.0032	0.0031	0.0030	0.0028	0.0027	0.0026	0.0025	0.0024	0.0024	0.0023	0.0022	0.0021	0.0021	0.0020	0.0020	0.0019	0.0019	0.0018	0.0018	0.0018	0.0017	0.0017	0.0017	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0017	0.0017	0.0017	0.0017	0.0017	0.0017	0.0018	0.0018	0.0018	0.0018	0.0018	0.0019	0.0019	0.0019	0.0019	0.0020	0.0020	0.0020	0.0020	0.0021	0.0021	0.0021	0.0022	0.0022	0.0022	0.0022	0.0023	0.0023	0.0023	0.0023	0.0024	0.0024	0.0024	0.0025	0.0025	0.0025	0.0025	0.0026	0.0026	0.0026	0.0026	0.0027	0.0027	0.0027	0.0027	0.0028	0.0028	0.0028	0.0028	0.0028	0.0029	0.0029	0.0029	0.0029	0.0029	0.0030	0.0030	0.0030	0.0030	0.0030	0.0031	0.0031	0.0031	0.0031	0.0031	0.0031	0.0031	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0	0.2327	0.2477	0.1926	0.1439	0.1075	0.0797	0.0583	0.0422	0.0304	0.0220	0.0162	0.0123	0.0097	0.0081	0.0070	0.0063	0.0059	0.0055	0.0052	0.0049	0.0046	0.0044	0.0041	0.0039	0.0037	0.0035	0.0034	0.0032	0.0031	0.0030	0.0028	0.0027	0.0026	0.0025	0.0024	0.0022	0.0021	0.0020	0.0019	0.0018	0.0017	0.0016	0.0014	0.0013	0.0013	0.0012	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1290
4 gamma	0.0326	0.0326	0.0326	0.6895	1.0696	0	0.2518	0.4644	0.5346	0.5735	0.6151	0.6627	0.7114	0.7563	0.7958	0.8305	0.8610	0.8879	0.9115	0.9319	0.9495	0.9646	0.9776	0.9887	0.9981	1.0061	1.0128	1.0185	1.0237	1.0285	1.0333	1.0382	1.0434	1.0490	1.0551	1.0616	1.0686	1.0760	1.0837	1.0917	1.1001	1.1086	1.1173	1.1262	1.1351	1.1441	1.1532	1.1622	1.1713	1.1802	1.1892	1.1980	1.2068	1.2155	1.2241	1.2325	1.2409	1.2491	1.2573	1.2653	1.2731	1.2809	1.2885	1.2961	1.3035	1.3107	1.3179	1.3250	1.3319	1.3387	1.3455	1.3521	1.3586	1.3650	1.3713	1.3775	1.3837	1.3897	1.3956	1.4015	1.4073	1.4130	1.4186	1.4241	1.4296	1.4350	1.4403	1.4456	1.4508	1.4559	1.4609	1.4660	1.4709	1.4758	1.4807	1.4855	1.4902	1.4949	1.4996	1.5042	1.5088	1.5133	1.5178	1.5223	1.5268	1.5312	1.5356	1.5399	1.5443	1.5486	1.5529	1.5572	1.5615	1.5657	1.5699	1.5742	1.5784	1.5826	1.5868	1.5910	1.5952	1.5994	1.6036	1.6078	1.6120	1.6161	1.6203	1.6245	1.6288	1.6330	1.6372	1.6414	1.6457	1.6500	1.6542	1.6585	1.6628	1.6672	1.6715	1.6759	1.6803	1.6847	1.6891	1.6935	1.6980	1.7025	1.7070	1.7116	1.7161	1.7207	1.7254	1.7300	1.7347	1.7394	1.7442	1.7490	1.7538	1.7586	1.7635	1.7684	1.7733	1.7783	1.7833	1.7884	1.7935	1.7986	1.8038	1.8090	1.8142	1.8195	1.8248	1.8302	1.8356	1.8410	1.8465	1.8520	1.8576	1.8632	1.8688	1.8745	1.8803	1.8861	1.8919	1.8978	1.9037	1.9096	1.9156	1.9217	1.9278	1.9339	1.9401	1.9463	1.9526	1.9590	1.9653	1.9718	1.9782	1.9848	1.9913	1.9980	2.0046	2.0113	2.0181	2.0249	2.0318	2.0387	2.0457	2.0527	2.0598	2.0669	2.0741	2.0813	2.0886	2.0959	2.1033	2.1107	2.1182	2.1257	2.1333	2.1410	2.1487	2.1564	2.1642	2.1721	2.1800	2.1880	2.1960	2.2041	2.2122	2.2204	2.2286	2.2369	2.2453	2.2537	2.2621	2.2707	2.2792	2.2879	2.2966	2.3053	2.3141	2.3230	2.3319	2.3408	2.3499	2.3590	2.3681	2.3773	2.3866	2.3959	2.4052	2.4147	2.4242	2.4337	2.4433	2.4530	2.4627	2.4725	2.4824	2.4923	2.5022	2.5123	2.5223	2.5325	2.5427	2.5530	2.5633	2.5737	2.5841	2.5946	0	0.3194	0.5688	0.6827	0.7417	0.7804	0.8122	0.8419	0.8711	0.9006	0.9308	0.9614	0.9917	1.0214	1.0501	1.0776	1.1036	1.1280	1.1506	1.1715	1.1907	1.2083	1.2245	1.2397	1.2541	1.2680	1.2817	1.2952	1.3089	1.3226	1.3365	1.3507	1.3650	1.3794	1.3940	1.4087	1.4234	1.4382	1.4529	1.4677	1.4824	1.4971	1.5116	1.5261	1.5404	1.5546	1.5687	1.5826	1.5963	1.6099	1.6232	1.6364	1.6494	1.6623	1.6749	1.6873	1.6996	1.7116	1.7235	1.7352	1.7467	1.7580	1.7691	1.7801	1.7909	1.8015	1.8119	1.8222	1.8322	1.8421	1.8519	1.8615	1.8709	1.8801	1.8892	1.8982	1.9070	1.9156	1.9241	1.9324	1.9406	1.9487	1.9566	1.9644	1.9720	1.9795	1.9869	1.9941	2.0012	2.0082	2.0151	2.0218	2.0284	2.0349	2.0413	2.0476	2.0538	2.0598	2.0658	2.0716	2.0774	2.0830	2.0885	2.0940	2.0994	2.1046	2.1098	2.1149	2.1199	2.1248	2.1297	2.1344	2.1391	2.1437	2.1483	2.1527	2.1571	2.1615	2.1657	2.1699	2.1741	2.1782	2.1822	2.1861	2.1901	2.1939	2.1977	2.2015	2.2052	2.2089	2.2125	2.2161	2.2196	2.2231	2.2266	2.2300	2.2334	2.2367	2.2400	2.2433	2.2466	2.2498	2.2530	2.2562	2.2593	2.2625	2.2656	2.2686	2.2717	2.2747	2.2777	2.2807	2.2837	2.2867	2.2896	2.2926	2.2955	2.2984	2.3013	2.3042	2.3071	2.3100	2.3128	2.3157	2.3185	2.3214	2.3242	2.3271	2.3299	2.3327	2.3355	2.3384	2.3412	2.3440	2.3468	2.3497	2.3525	2.3553	2.3582	2.3610	2.3638	2.3667	2.3695	2.3724	2.3752	2.3781	2.3810	2.3839	2.3867	2.3896	2.3925	2.3954	2.3984	2.4013	2.4042	2.4072	2.4101	2.4131	2.4161	2.4191	2.4221	2.4251	2.4281	2.4312	2.4342	2.4373	2.4404	2.4435	2.4466	2.4497	2.4528	2.4560	2.4592	2.4623	2.4655	2.4688	2.4720	2.4752	2.4785	2.4818	2.4851	2.4884	2.4917	2.4951	2.4985	2.5019	2.5053	2.5087	2.5121	2.5156	2.5191	2.5226	2.5261	2.5296	2.5332	2.5368	2.5404	2.5440	2.5476	2.5513	2.5550	2.5587	2.5624	2.5661	2.5699	2.5737	2.5775	2.5813	2.5851	2.5890	2.5929	2.5968	2.6008	2.6047	2.6087	2.6127	2.6167	2.6208	2.6248	2.6289	2.6330	2.6372	2.6413	2.6455	2.6497	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326

6. Uncertainty Interval Calculus

6.1 gsua_oatr function

help gsua_oatr

Function for once-at-time ranges expantion T=gsua_oatr(T) Parameters: T <-- summary table from gsua_dataprep function. The objective output is given by T.Nominal. Outputs: T <-- summary table with new factor intervals in T.Range Additional features: To define limit for range expantion use the paired feature 'lim'. Default:0.3. To speed up range calculus, use the paired feature 'parallel',true. T=gsua_oatr(T,'lim',0.2,'parallel',true)

T.Nominal=[1000 200 0.1 0.06]';

T_oat=gsua_oatr(T)

Expansion-reduction Method OAT is being launched ------------------------------------ Calculus of gamma Starting Activating intern_counter for gamma Activating intern_counter for gamma Range for gamma Done! Initial range of gamma(4)--> 0 1 Actual range --> 0 0.135 Calculus of beta Starting Activating intern_counter for beta Activating intern_counter for beta Range for beta Done! Initial range of beta(3)--> 0 1 Actual range --> 0 0.225 Calculus of I0 Starting Range for I0 Done! Initial range of I0(2)--> 1 500 Actual range --> 137.375 265.5931 Calculus of S0 Starting Activating intern_counter for S0 Activating intern_counter for S0 Range for S0 Done! Initial range of S0(1)--> 100 3000 Actual range --> 0 2250

T_oat = 4×7 table 

	Range		Nominal	Si	STi	STi_vec																																																																Si_vec																																																																Estlsqc
1 S0	0	2250	1000	0.1699	0.1339	0	0.0232	0.0656	0.0981	0.1192	0.1350	0.1489	0.1619	0.1741	0.1851	0.1946	0.2022	0.2077	0.2111	0.2129	0.2133	0.2126	0.2112	0.2091	0.2066	0.2038	0.2007	0.1976	0.1944	0.1912	0.1881	0.1851	0.1822	0.1793	0.1764	0.1736	0.1708	0.1680	0.1652	0.1625	0.1597	0.1569	0.1542	0.1514	0.1487	0.1460	0.1433	0.1406	0.1379	0.1352	0.1326	0.1300	0.1274	0.1249	0.1224	0.1199	0.1174	0.1150	0.1126	0.1103	0.1080	0.1057	0.1034	0.1012	0.0991	0.0969	0.0948	0.0928	0.0907	0.0888	0.0868	0.0849	0.0830	0.0811	0.0793	0.0775	0.0758	0.0741	0.0724	0.0707	0.0691	0.0675	0.0660	0.0644	0.0629	0.0615	0.0600	0.0586	0.0572	0.0559	0.0546	0.0533	0.0520	0.0507	0.0495	0.0483	0.0472	0.0460	0.0449	0.0438	0.0427	0.0417	0.0406	0.0396	0.0387	0.0377	0.0368	0.0359	0.0350	0.0341	0.0332	0.0324	0.0316	0.0308	0.0300	0.0292	0.0285	0.0278	0.0271	0.0264	0.0257	0.0250	0.0244	0.0238	0.0232	0.0226	0.0220	0.0214	0.0209	0.0203	0.0198	0.0193	0.0188	0.0183	0.0178	0.0173	0.0169	0.0164	0.0160	0.0156	0.0152	0.0148	0.0144	0.0140	0.0136	0.0133	0.0129	0.0126	0.0122	0.0119	0.0116	0.0113	0.0110	0.0107	0.0104	0.0101	0.0098	0.0096	0.0093	0.0091	0.0088	0.0086	0.0084	0.0081	0.0079	0.0077	0.0075	0.0073	0.0071	0.0069	0.0067	0.0065	0.0063	0.0062	0.0060	0.0058	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048	0.0047	0.0045	0.0044	0.0043	0.0042	0.0041	0.0040	0.0038	0.0037	0.0036	0.0035	0.0034	0.0033	0.0032	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0027	0.0026	0.0025	0.0024	0.0024	0.0023	0.0022	0.0022	0.0021	0.0021	0.0020	0.0019	0.0019	0.0018	0.0018	0.0017	0.0017	0.0016	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014	0.0013	0.0013	0.0013	0.0012	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010	0.0010	0.0010	0.0009	0.0009	0.0009	0.0008	0.0008	0.0008	0.0008	0.0008	0.0007	0.0007	0.0007	0.0007	0.0007	0.0006	0.0006	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0004	0	0.0038	0.0130	0.0148	0.0109	0.0064	0.0040	0.0051	0.0092	0.0152	0.0213	0.0261	0.0287	0.0293	0.0283	0.0261	0.0233	0.0201	0.0169	0.0137	0.0108	0.0080	0.0055	0.0033	0.0013	-0.0005	-0.0020	-0.0034	-0.0046	-0.0057	-0.0066	-0.0074	-0.0081	-0.0088	-0.0093	-0.0098	-0.0101	-0.0104	-0.0107	-0.0109	-0.0110	-0.0111	-0.0111	-0.0111	-0.0111	-0.0110	-0.0109	-0.0108	-0.0107	-0.0105	-0.0103	-0.0101	-0.0099	-0.0097	-0.0095	-0.0093	-0.0090	-0.0088	-0.0086	-0.0083	-0.0081	-0.0079	-0.0076	-0.0074	-0.0072	-0.0069	-0.0067	-0.0065	-0.0063	-0.0061	-0.0059	-0.0057	-0.0055	-0.0053	-0.0051	-0.0049	-0.0048	-0.0046	-0.0044	-0.0043	-0.0041	-0.0040	-0.0038	-0.0037	-0.0035	-0.0034	-0.0033	-0.0032	-0.0030	-0.0029	-0.0028	-0.0027	-0.0026	-0.0025	-0.0024	-0.0023	-0.0022	-0.0021	-0.0021	-0.0020	-0.0019	-0.0018	-0.0018	-0.0017	-0.0016	-0.0016	-0.0015	-0.0014	-0.0014	-0.0013	-0.0013	-0.0012	-0.0012	-0.0011	-0.0011	-0.0010	-0.0010	-0.0009	-0.0009	-0.0009	-0.0008	-0.0008	-0.0008	-0.0007	-0.0007	-0.0007	-0.0006	-0.0006	-0.0006	-0.0006	-0.0005	-0.0005	-0.0005	-0.0005	-0.0005	-0.0004	-0.0004	-0.0004	-0.0004	-0.0004	-0.0004	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0003	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0002	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0001	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	336.3700	336.3739	336.3758	336.3773	336.3791	336.3828	336.3869	336.3871	336.3959	336.4517
2 I0	137.3750	265.5931	200	0.0220	0.0214	0.9216	0.6237	0.3440	0.2081	0.1346	0.0915	0.0663	0.0517	0.0422	0.0344	0.0275	0.0216	0.0169	0.0132	0.0103	0.0080	0.0062	0.0048	0.0038	0.0031	0.0027	0.0025	0.0025	0.0026	0.0028	0.0029	0.0030	0.0030	0.0030	0.0029	0.0028	0.0026	0.0024	0.0023	0.0021	0.0019	0.0017	0.0016	0.0015	0.0013	0.0012	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.8061	0.5225	0.2301	0.1096	0.0584	0.0329	0.0180	0.0083	0.0019	-0.0020	-0.0040	-0.0046	-0.0044	-0.0036	-0.0026	-0.0015	-0.0004	0.0007	0.0017	0.0026	0.0034	0.0042	0.0047	0.0052	0.0055	0.0057	0.0058	0.0058	0.0057	0.0055	0.0053	0.0051	0.0048	0.0045	0.0043	0.0040	0.0038	0.0035	0.0033	0.0031	0.0028	0.0027	0.0025	0.0023	0.0022	0.0020	0.0019	0.0018	0.0017	0.0016	0.0015	0.0014	0.0013	0.0012	0.0012	0.0011	0.0010	0.0010	0.0009	0.0009	0.0008	0.0008	0.0007	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	171.4599	171.4604	171.4609	171.4610	171.4612	171.4618	171.4625	171.4625	171.4637	171.4726
3 beta	0	0.2250	0.1000	0.0456	0.0635	0	0.1204	0.2081	0.2278	0.2204	0.2044	0.1860	0.1671	0.1492	0.1334	0.1197	0.1080	0.0981	0.0900	0.0835	0.0785	0.0746	0.0715	0.0691	0.0671	0.0654	0.0639	0.0625	0.0612	0.0599	0.0585	0.0571	0.0555	0.0539	0.0522	0.0505	0.0487	0.0469	0.0451	0.0433	0.0416	0.0399	0.0382	0.0366	0.0350	0.0335	0.0321	0.0307	0.0294	0.0281	0.0269	0.0257	0.0246	0.0235	0.0224	0.0215	0.0205	0.0196	0.0187	0.0179	0.0171	0.0163	0.0156	0.0149	0.0142	0.0136	0.0130	0.0124	0.0118	0.0113	0.0108	0.0103	0.0098	0.0093	0.0089	0.0085	0.0081	0.0077	0.0074	0.0070	0.0067	0.0064	0.0061	0.0058	0.0056	0.0053	0.0051	0.0048	0.0046	0.0044	0.0042	0.0040	0.0038	0.0037	0.0035	0.0034	0.0032	0.0031	0.0030	0.0028	0.0027	0.0026	0.0025	0.0024	0.0024	0.0023	0.0022	0.0021	0.0021	0.0020	0.0020	0.0019	0.0019	0.0018	0.0018	0.0018	0.0017	0.0017	0.0017	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0016	0.0017	0.0017	0.0017	0.0017	0.0017	0.0017	0.0018	0.0018	0.0018	0.0018	0.0018	0.0019	0.0019	0.0019	0.0019	0.0020	0.0020	0.0020	0.0020	0.0021	0.0021	0.0021	0.0022	0.0022	0.0022	0.0022	0.0023	0.0023	0.0023	0.0023	0.0024	0.0024	0.0024	0.0025	0.0025	0.0025	0.0025	0.0026	0.0026	0.0026	0.0026	0.0027	0.0027	0.0027	0.0027	0.0028	0.0028	0.0028	0.0028	0.0028	0.0029	0.0029	0.0029	0.0029	0.0029	0.0030	0.0030	0.0030	0.0030	0.0030	0.0031	0.0031	0.0031	0.0031	0.0031	0.0031	0.0031	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0034	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0033	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0.0032	0	0.2327	0.2477	0.1926	0.1439	0.1075	0.0797	0.0583	0.0422	0.0304	0.0220	0.0162	0.0123	0.0097	0.0081	0.0070	0.0063	0.0059	0.0055	0.0052	0.0049	0.0046	0.0044	0.0041	0.0039	0.0037	0.0035	0.0034	0.0032	0.0031	0.0030	0.0028	0.0027	0.0026	0.0025	0.0024	0.0022	0.0021	0.0020	0.0019	0.0018	0.0017	0.0016	0.0014	0.0013	0.0013	0.0012	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0006	0.0006	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1291	0.1290
4 gamma	0	0.1350	0.0600	0.6895	1.0696	0	0.2518	0.4644	0.5346	0.5735	0.6151	0.6627	0.7114	0.7563	0.7958	0.8305	0.8610	0.8879	0.9115	0.9319	0.9495	0.9646	0.9776	0.9887	0.9981	1.0061	1.0128	1.0185	1.0237	1.0285	1.0333	1.0382	1.0434	1.0490	1.0551	1.0616	1.0686	1.0760	1.0837	1.0917	1.1001	1.1086	1.1173	1.1262	1.1351	1.1441	1.1532	1.1622	1.1713	1.1802	1.1892	1.1980	1.2068	1.2155	1.2241	1.2325	1.2409	1.2491	1.2573	1.2653	1.2731	1.2809	1.2885	1.2961	1.3035	1.3107	1.3179	1.3250	1.3319	1.3387	1.3455	1.3521	1.3586	1.3650	1.3713	1.3775	1.3837	1.3897	1.3956	1.4015	1.4073	1.4130	1.4186	1.4241	1.4296	1.4350	1.4403	1.4456	1.4508	1.4559	1.4609	1.4660	1.4709	1.4758	1.4807	1.4855	1.4902	1.4949	1.4996	1.5042	1.5088	1.5133	1.5178	1.5223	1.5268	1.5312	1.5356	1.5399	1.5443	1.5486	1.5529	1.5572	1.5615	1.5657	1.5699	1.5742	1.5784	1.5826	1.5868	1.5910	1.5952	1.5994	1.6036	1.6078	1.6120	1.6161	1.6203	1.6245	1.6288	1.6330	1.6372	1.6414	1.6457	1.6500	1.6542	1.6585	1.6628	1.6672	1.6715	1.6759	1.6803	1.6847	1.6891	1.6935	1.6980	1.7025	1.7070	1.7116	1.7161	1.7207	1.7254	1.7300	1.7347	1.7394	1.7442	1.7490	1.7538	1.7586	1.7635	1.7684	1.7733	1.7783	1.7833	1.7884	1.7935	1.7986	1.8038	1.8090	1.8142	1.8195	1.8248	1.8302	1.8356	1.8410	1.8465	1.8520	1.8576	1.8632	1.8688	1.8745	1.8803	1.8861	1.8919	1.8978	1.9037	1.9096	1.9156	1.9217	1.9278	1.9339	1.9401	1.9463	1.9526	1.9590	1.9653	1.9718	1.9782	1.9848	1.9913	1.9980	2.0046	2.0113	2.0181	2.0249	2.0318	2.0387	2.0457	2.0527	2.0598	2.0669	2.0741	2.0813	2.0886	2.0959	2.1033	2.1107	2.1182	2.1257	2.1333	2.1410	2.1487	2.1564	2.1642	2.1721	2.1800	2.1880	2.1960	2.2041	2.2122	2.2204	2.2286	2.2369	2.2453	2.2537	2.2621	2.2707	2.2792	2.2879	2.2966	2.3053	2.3141	2.3230	2.3319	2.3408	2.3499	2.3590	2.3681	2.3773	2.3866	2.3959	2.4052	2.4147	2.4242	2.4337	2.4433	2.4530	2.4627	2.4725	2.4824	2.4923	2.5022	2.5123	2.5223	2.5325	2.5427	2.5530	2.5633	2.5737	2.5841	2.5946	0	0.3194	0.5688	0.6827	0.7417	0.7804	0.8122	0.8419	0.8711	0.9006	0.9308	0.9614	0.9917	1.0214	1.0501	1.0776	1.1036	1.1280	1.1506	1.1715	1.1907	1.2083	1.2245	1.2397	1.2541	1.2680	1.2817	1.2952	1.3089	1.3226	1.3365	1.3507	1.3650	1.3794	1.3940	1.4087	1.4234	1.4382	1.4529	1.4677	1.4824	1.4971	1.5116	1.5261	1.5404	1.5546	1.5687	1.5826	1.5963	1.6099	1.6232	1.6364	1.6494	1.6623	1.6749	1.6873	1.6996	1.7116	1.7235	1.7352	1.7467	1.7580	1.7691	1.7801	1.7909	1.8015	1.8119	1.8222	1.8322	1.8421	1.8519	1.8615	1.8709	1.8801	1.8892	1.8982	1.9070	1.9156	1.9241	1.9324	1.9406	1.9487	1.9566	1.9644	1.9720	1.9795	1.9869	1.9941	2.0012	2.0082	2.0151	2.0218	2.0284	2.0349	2.0413	2.0476	2.0538	2.0598	2.0658	2.0716	2.0774	2.0830	2.0885	2.0940	2.0994	2.1046	2.1098	2.1149	2.1199	2.1248	2.1297	2.1344	2.1391	2.1437	2.1483	2.1527	2.1571	2.1615	2.1657	2.1699	2.1741	2.1782	2.1822	2.1861	2.1901	2.1939	2.1977	2.2015	2.2052	2.2089	2.2125	2.2161	2.2196	2.2231	2.2266	2.2300	2.2334	2.2367	2.2400	2.2433	2.2466	2.2498	2.2530	2.2562	2.2593	2.2625	2.2656	2.2686	2.2717	2.2747	2.2777	2.2807	2.2837	2.2867	2.2896	2.2926	2.2955	2.2984	2.3013	2.3042	2.3071	2.3100	2.3128	2.3157	2.3185	2.3214	2.3242	2.3271	2.3299	2.3327	2.3355	2.3384	2.3412	2.3440	2.3468	2.3497	2.3525	2.3553	2.3582	2.3610	2.3638	2.3667	2.3695	2.3724	2.3752	2.3781	2.3810	2.3839	2.3867	2.3896	2.3925	2.3954	2.3984	2.4013	2.4042	2.4072	2.4101	2.4131	2.4161	2.4191	2.4221	2.4251	2.4281	2.4312	2.4342	2.4373	2.4404	2.4435	2.4466	2.4497	2.4528	2.4560	2.4592	2.4623	2.4655	2.4688	2.4720	2.4752	2.4785	2.4818	2.4851	2.4884	2.4917	2.4951	2.4985	2.5019	2.5053	2.5087	2.5121	2.5156	2.5191	2.5226	2.5261	2.5296	2.5332	2.5368	2.5404	2.5440	2.5476	2.5513	2.5550	2.5587	2.5624	2.5661	2.5699	2.5737	2.5775	2.5813	2.5851	2.5890	2.5929	2.5968	2.6008	2.6047	2.6087	2.6127	2.6167	2.6208	2.6248	2.6289	2.6330	2.6372	2.6413	2.6455	2.6497	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326	0.0326

6.2 gsua_ucis function

Please, do not execute gsua_csb function inside live script environment.

help gsua_csb
  Function for uncertainty-based confidence intervals
 
  T= gsua_csb(T,N)
  Parameters:
  T <-- summary table from gsua_oatr function
  N <-- Number of samples per cycle
  Outputs:
  T <-- summary table with new confidence intervals in T.Range
  Additional features:
  It is possible to request the following additional positional outputs
  New_range <-- array with range modification record.
  J_test <-- MSE output of last iteration
  Y_test <-- output of last iteration
  sup    <-- record of good scalars
  Also, it is possible to apply the next paired features
  'ynom',ynom <-- to especify a new output objective
  'reps',k <--to perform k cycles. Default: 100.
  'recort',r <-- to set another recort criteria (r). Default: 0.5.
  'select',s <-- to set another select criteria (s). Default: 0.5.
  'lim',l <-- to set another distance criteria (l). Default: 0.3.
  'stop',p <-- to set the confidence of the interval in (p). Default: 0.95.
  'parallel',false <-- to avoid the speed up of parallel computing toolbox.
  [T,New_range,J_test,Y_test,sup]= gsua_ucis(T,N,'lim',0.5,'select',0.4)

Execute the following in matlab command window

T_uci=gsua_csb(T_oat,100)

Interactive user guide for Toolbox initial configuration (Simulink)

We will use a preconfigured predator - prey model (order: 2, factors: 5) as example throughout the present guide. To acces the model, type:

open predatorprey.slx

1. Model preparation

It is advisable to create a mask for factor (input, parameter or initial condition) value assignation. However, it is not necessary.

It is fundamental to assign the value of each factor as a reference to some variable in the Model workspace, it is necessary because the toolbox will identify and assign values for the factors by name matching. If you want to exclude a factor for the analysis, you must to assign it a constant value insted of a reference name. As you can see in the previous figures, the predator-prey model has 5 factors, and we will use all of them but D0 for analysis. If you have preferred a configuration, apply it to the model (use the Model configuration Parameters option, redbox in next figure), otherwise, the toolbox will apply the Simulink current configuration.

At the moment, it is possible to work only with a single model output, hence, you must to connect an output block to the objective output and a terminator block for any other output, as shown in following figure.

For this example, we will apply analysis over prey population (P).

Resources:

Simulink masks - MathWorks

Espacio de estado y simulación - spanish slides

2. Environment configuration and additional features

[ParT,tOut] = sens_dataprep(model,Ranges,ParIn)

Parameters:

model: file name of the simulink model
Ranges: Npx2 array or cell array with the range of Np factors (see additional features)
ParIn: cell array with the factor names of the model

Outputs:

T: table array with all required system information
tout: array of output time for the model

model='predatorprey';
ParIn={'beta','delta','ro','P0'};
Ranges=[0.4 0.6; 0.4 0.6; 2.2 2.8; 2.7 3.2];
[T,time] = gsua_dataprep(model,Ranges,ParIn)
Setting environment to work with simulink
Setting nominal values
All done!
T = 4×2 table 
 RangeNominal
1 beta0.40000.60000.5000
2 delta0.40000.60000.5000
3 ro2.20002.80002.5000
4 P02.70003.20002.9500
time = 391×1
    1.0000
    1.1000
    1.2000
    1.3000
    1.4000
    1.5000
    1.6000
    1.7000
    1.8000
    1.9000

	Range	Nominal
1 beta	0.4000	0.6000	0.5000
2 delta	0.4000	0.6000	0.5000
3 ro	2.2000	2.8000	2.5000
4 P0	2.7000	3.2000	2.9500

2.1 Additional features:

2.1.1 Range definition

Paired feature 'rMethod' (default: range)

You can choose between 4 methods to define the ranges of the model factors, each method requires a different definition of varible Ranges

std <-- {Nominal value, Distance to interval limits}
percent <-- {Nominal value, percent of distance to interval limits}
range <-- {Inferior limit, superior limit}
normal <-- {Nominal value, standart deviation}

Examples:

%use of std

Ranges={0.5 0.1; 0.5 0.1; 2.5 0.3; 3 0.2}

[T,time] = gsua_dataprep(model,Ranges,ParIn,'rMethod','std');

%use of percent

s=10;

Ranges={0.5 s; 0.5 s; 2.5 s; 3 s}%you can define a different value for each s

[T,time] = gsua_dataprep(model,Ranges,ParIn,'rMethod','percent');

%use of normal

Ranges={0.5 0.05; 0.5 0.01; 2.5 0.1; 3 0.1}

[T,time] = gsua_dataprep(model,Ranges,ParIn,'rMethod','normal');

Note: The use of normal distribution affects the generation of samples from the space of factors when using the gsua_dmatrix function.

A 'rMethod' for each single range (do not use normal rMethod)

You can define a different rMethod for each factor range. To do this, just add a third column to Range array and specify the rMethod there.

Example:

Ranges={0.5 0.1 'std'; 0.5 0.1 'percent'; 2.5 0.3 'range'; 3 0.2 'percent'};

[T,time] = gsua_dataprep(model,Ranges,ParIn);

2.1.2 Nominal factor family

You can specify a nominal family of factor values using the paired feature 'nominal'. It could be useful for some toolbox functions.

Example:

nominal=[0.6,0.4,2.5,3.2]%array with nominal values for factors

[T,time] = gsua_dataprep(model,Ranges,ParIn,'nominal',nominal);

2.1.3 Model kind (default: Dynamic)

It is possible to work with static models in simulink (initial time = final time = 0). To active the configuration for static models use the paired feature 'modelkind'

Example:

[T,time] = gsua_dataprep(model,Ranges,ParIn,'modelkind','Static');

2.1.4 Fixed step (Default: [])

It is possible to specify from the toolbox a fixed-step magnitude for the model using the paired feature 'Step'

Example:

step=0.1;

[T,time] = gsua_dataprep(model,Ranges,ParIn,'Step',step);

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Interactive user guide for Toolbox initial configuration (Symbolic Matlab)

We will create and configure a simple SIR model with Symbolic Math Toolbox

1. Model creation

syms S(t) I(t) R(t) P(t) beta gamma P %symbolic state variables and parameters

P=S+I+R;

%Defining the system of differential equations

ode1 = diff(S) == -S*I*beta/P;

ode2 = diff(I) == S*I*beta/P - gamma*I;

ode3 = diff(R) == gamma*I;

%Array with the system

odes=[ode1; ode2; ode3]

odes(t) =

%choose an order for state variables

vars = [R S I];

If your model is in the form

then use the the following command

[odes, vars]=reduceDifferentialOrder(ode,var)% ode=f(), var=y^n;

2. Environment configuration and additional features

[T,solver] = gsua_dataprep(odes,vars,domain,modelName)

Parameters:

odes: array with the system of differential equations
vars: array with state variables
domain: 1x2 array with the range of the domain
modelName: the function creates a file with the model for later simulations, modelName is the name for said output model file

Outputs:

Table: table array with all required system information
solver: function handle for simulations (not necessary)

Use:

Run the following to obtain the correct order for the factors

domain=[0 264];

modelName='SIR';

[T,solver]=gsua_dataprep(odes,vars,domain,modelName);

Setting environment to work with symbolic Matlab Introduce ranges in the following order:

ans(t) =

Now, you can manually add information to the Table T or use the following additional features:

2.1 Additional features

2.1.1 Range definition

Paired feature 'range' and 'rMethod' (Default rMethod: range)

You can directly provide the range of model factors.

You can choose between 4 methods to define the ranges of the model factors, each method requires a different definition of varible Ranges

std <-- {Nominal value, Distance to interval limits}
percent <-- {Nominal value, percent of distance to interval limits}
range <-- {Inferior limit, superior limit}
normal <-- {Nominal value, standart deviation}

Examples:

%use of std

Ranges=[0 0; 1500 1000; 300 200; 0.5 0.5; 0.5 0.5]

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'range',Ranges,'rMethod','std');

%use of percent

s=10;

Ranges=[0 s; 100 s; 200 s; 0.5 s; 0.5 s]%you can define a different value for each s

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'range',Ranges,'rMethod','percent');

%use of normal

Ranges=[0 0; 2000 0.1; 300 0.1; 0.5 0.05; 0.5 0.05]

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'range',Ranges,'rMethod','normal');

Note: The use of normal distribution affects the generation of samples from the space of factors when using the gsua_dmatrix function.

Ranges=[0 0; 100 3000; 1 500; 0 1; 0 1];

[T,solver] = gsua_dataprep(odes,vars,domain,modelName,'range',Ranges)

Setting environment to work with symbolic Matlab Introduce ranges in the following order:

ans(t) =

T = 4×2 table 

	Range		Nominal
1 S0	100	3000	1550
2 I0	1	500	250.5000
3 beta	0	1	0.5000
4 gamma	0	1	0.5000

solver = function_handle with value:

@(init,pars)ode45(@(t,Y)odefun(t,Y,pars),domain,[Table.Range(fixvars,1)',init],opt)

Note: To fix a factor, give to the function the respective range as a degenerated one (min=max). As you can see, fixed factors will not appear in table T (see previous table for R0 factor).

2.1.2 Nominal factor family

You can specify a nominal family of factors values using the paired feature 'nominal'. It could be useful for some toolbox functions.

Example:

nominal=[0,2000,50,0.1,0.06]%array with nominal values for factors

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'nominal',nominal);

2.1.3 Specific model output (Default: 1:model_order)

You can specify the output(s) of interest for the model as an array (it should be only one output for SA, and UCI) using the paired feature 'output'.

Example:

%output=[1,3]

output=3;% selected output

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'output',output);

2.1.4 Specific ode solver (Default: ode45)

You can choose between a fixed step solver (ode4) and an adaptative ode solver (ode45) to solve your model using the paired feature 'solver'.

Example:

solver='ode4';%change the default solver

domain=[0 264 0.1]%add a third column with fixed step size to domain variable

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'solver',solver);

2.1.5 Ode solver configuration (Default: none)

You can configure the ode45 solver with all possible options (check Matlab odeset) using the paired feature 'opt'.

Example:

opt = odeset('NonNegative',1:3);%nonnegative feature is the only supported for ode4 configuration

[T,sol] = gsua_dataprep(odes,vars,domain,modelName,'opt',opt);

2.1.6 Static model simulation

You can simulate static models defining the domain variable as a single number.

Example:

domain=1;

[T,sol] = gsua_dataprep(odes,vars,domain,modelName);

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Interactive user guide for Toolbox initial configuration (User-Defined functions)

User defined functions must be stored as .m files.

1. Time-dependent functions

1.1 Model creation

Time-dependent functions must be configured in the form output=user_function(factors,domain). Where:

Parameters:

factors: is a vector of parameters of NpxN, being Np the number of parameters and N the number of simulations to run (if vectorized, otherwise, N=1).
domain: is a vector of 1x2 that represents the time domain for the simulation in the form [initial_time, final_time]

Output:

output is a struct() object with the following fields at least:

output.x: simulation time.
output.y: simulation outputs, as a matrix of nOutputsx|output.x|, where nOutputs is the number of outputs of the model and |output.x| is the length of the simulation time.

In this way, the outputs from matlab ode and dde solvers meet the requirements.

1.1.1 Example

function sol = user_dependent(params,domain)

% This problem is an epidemic model due to Cooke et alia, more information

% can be found in 'Interaction of maturation delay and nonlinear birth in

% population and epidemic models' J. Math. Biol., 39 (1999) 332-352.

% (This is reference 3 of the tutorial).

% You can Download the tutorial from:

% https://www.mathworks.com/matlabcentral/fileexchange/3899-tutorial-on-solving-ddes-with-dde23

T = params(1,:);

lambda = params(2,:);

sol = dde23(@prob4f,T,[2; 3.5],domain,[],lambda,T);%the output from dde23 meet the requirements to be the output of the function.

%-----------------------------------------------------------------------

function yp = prob4f(t,y,Z,lambda,T)

%PROB4F The derivative function for Problem 4 of the DDE Tutorial.

a = 1;

b = 80;

d = 1;

d1 = 1;

e = 10;

gamma = 0.5;

I = y(1);

N = y(2);

Nlag = Z(2,1);

dIdt = lambda*(N - I)*(I/N) - ( d + e + gamma)*I;

dNdt = b*exp(-a*Nlag)*Nlag*exp(-d1*T) - d*N - e*I;

yp = [ dIdt; dNdt];

1.2 Model implementation

To prepare the environment to work with time-dependent user-defined functions, you must run the following code, replacing input parameters as indicated

[Table,solver]=gsua_dataprep(func,ranges,'domain',domain)

Parameters:

func: name of the .m file as char
ranges: Npx2 array with the range of the factors
domain: 1x2 array with the range of the domain

Outputs:

Table: table array with all required system information
solver: function handle for simulations (not essential)

Use:

We will expose an example implementing the example function user_defined.m (defined above), you can use the command

open user_defined.m

to access the function

func='user_dependent';
ranges=[0.1 0.3; 10 14];
domain=[0 25];
[T,sol]=gsua_dataprep(func,ranges,'domain',domain)
Setting environment to work with user-defined function
T = 2×2 table 
 RangeNominal
1 10.10000.30000.2000
2 210.000014.000012.0000
sol = function_handle with value:
    @(pars)user_func(pars',domain)

	Range	Nominal
1 1	0.1000	0.3000	0.2000
2 2	10.0000	14.0000	12.0000

Test if the model is actually working by running a simmulation with the values stored at T.Nominal

gsua_eval(T.Nominal,T);

Change the number of outputs (max 2) with the following command

T.Properties.CustomProperties.output=1:2;

Provide a specific xdata to achieve the model behaviour at those points

xdata=0:0.1:25;

Evaluate the model again

gsua_eval(T.Nominal,T,xdata);

1.3 Additional features

1.3.1 Range definition

Paired feature 'rMethod' (Default rMethod: range)

You can choose between 4 methods to define the ranges of the model factors, each method requires a different definition of varible ranges

std <-- {Nominal value, Distance to interval limits}
percent <-- {Nominal value, percent of distance to interval limits}
range <-- {Inferior limit, superior limit}
normal <-- {Nominal value, standart deviation}

Examples:

%use of std

ranges=[1 0.1; 12 2]

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'rMethod','std');

%use of percent

s=10;

ranges=[0.2 s; 12 s]%you can define a different value for each s

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'rMethod','percent');

%use of normal

ranges=[0.3 0.001; 12 0.1]

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'rMethod','normal');

Note: The use of normal distribution affects the generation of samples from the space of factors when using the gsua_dmatrix function.

Note: To fix a factor, give to the function the respective range as a degenerated one (min=max). As you can see, fixed factors will not appear in table T (see previous table for R0 factor).

1.3.2 Nominal factor family

You can specify a nominal family of factors values using the paired feature 'nominal'. It could be useful for some toolbox functions.

Example:

nominal=[0.2,11]%array with nominal values for factors

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'nominal',nominal);

1.3.3 Specific model output (Default: 1:model_order)

You can specify the output(s) of interest for the model as an array (it should be only one output for SA, and UCI) using the paired feature 'output'.

Example:

%output=[1,2]

output=2;% selected output

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'output',output);

1.3.4 Specific names for model factors (Default: {})

You can specify the names of the model factors as an array using the paired feature 'names'. The names of the factors are displayed when plotting some graphs.

Example:

names={'\tau','\lambda'};% selected output

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'names',names);

1.3.5 Specific names for model outputs (Default: {})

You can specify the names of the model output(s) as an array using the paired feature 'out_names'. The names of the outputs are displayed when plotting some graphs.

Example:

out_names={'I(t)','N(t)'};% selected output

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'out_names',out_names);

1.3.6 Run as vectorized function (Default: false)

If you have a vectorized function, then you can exploit its capabilities using the paired feature 'vectorized'

Example:

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'vectorized',true);

1.3.7 Any additional user-provided configuration (Default: {})

If your function requires on three inputs, then the third input must be for additional configuration. You can provide anything you want for this function input using the paired feature 'opt'

Example:

opt=struct();%anything you want

[T,sol] = gsua_dataprep(func,ranges,'domain',domain,'opt',opt);

2. Time-free functions

2.1 Model creation

Time-free functions must be configured in the form output=user_function(factors). Where:

Parameters:

factors: a vector that contains all the parameters needed to run the function

Output:

A numerical array (if vectorized) or a single number.

Tip: If your model is time-free but its output is a vector, treat it as a time-dependent model. Just ignore the domain input inside the function.

1.1.1 Example

function out = user_free(params)

%R_0

% ODES = R_0(BETA_M,BETA_H,THETA_M,THETA_H,ALPHA,MU_M,GAMMA_H,MU_H)

% This function was generated by the Symbolic Math Toolbox version 8.3.

% 04-Aug-2019 17:48:42

beta_m=params(1,:);

beta_h=params(2,:);

theta_m=params(3,:);

theta_h=params(3,:);

alpha=params(3,:);

mu_m=params(3,:);

gamma_h=params(3,:);

mu_h=params(3,:);

t2 = alpha.*mu_m;

out = (beta_h.*beta_m.*theta_h.*theta_m)./(t2.*(gamma_h+mu_h).*(mu_h+theta_h).*(t2+theta_m));

2.2 Model implementation

To prepare the environment to work with time-free user-defined functions, you must run the following code, replacing input parameters as indicated

[T,solver] = gsua_dataprep(Function,Ranges)

Parameters:

Function: Is the name of the user-defined function as char ('my_function')
Ranges: Numerical array of Npx2

Outputs:

T: table array with all required system information
solver: function handle for simulations (not necessary)

func='user_free';
range=[0 4; 0 4; 0 1; 0 1; 1 2; 0 0.3; 0 1; 0 0.1];
[T,sol]=gsua_dataprep(func,range)
Setting environment to work with user-defined function
T = 8×2 table 
 RangeNominal
1 104.00002.0000
2 204.00002.0000
3 301.00000.5000
4 401.00000.5000
5 512.00001.5000
6 600.30000.1500
7 701.00000.5000
8 800.10000.0500
sol = function_handle with value:
    @(pars)user_func(pars')

	Range	Nominal
1 1	0	4.0000	2.0000
2 2	0	4.0000	2.0000
3 3	0	1.0000	0.5000
4 4	0	1.0000	0.5000
5 5	1	2.0000	1.5000
6 6	0	0.3000	0.1500
7 7	0	1.0000	0.5000
8 8	0	0.1000	0.0500

2.3 Additional features

For additional features see Time-dependent features

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