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tutorial_sa_2
This tutorial illustrates the local derivative-based sensitivity analysis method
available in DAE Tools.
The problem is adopted from the section 2.1 of the following article:
- A. Saltelli, M. Ratto, S. Tarantola, F. Campolongo.
Sensitivity Analysis for Chemical Models. Chem. Rev. (2005), 105(7):2811-2828.
`doi:10.1021/cr040659d <http://dx.doi.org/10.1021/cr040659d>`_
The model is very simple and describes a simple reversible chemical reaction A <-> B,
with reaction rates k1 and k_1 for the direct and inverse reactions, respectively.
The reaction rates are uncertain and are described by continuous random variables
with known probability density functions. The standard deviation is 0.3 for k1 and
1 for k_1. The standard deviation of the concentration of the species A is
approximated using the following expression defined in the article:
.. code-block:: none
stddev(Ca)**2 = stddev(k1)**2 * (dCa/dk1)**2 + stddev(k_1)**2 * (dCa/dk_1)**2
The following derivative-based measures are used in the article:
- Derivatives dCa/dk1 and dCa/dk_1 calculated using the forward sensitivity method
- Sigma normalised derivatives:
.. code-block:: none
S(k1) = stddev(k1) / stddev(Ca) * dCa/dk1
S(k_1) = stddev(k_1)/ stddev(Ca) * dCa/dk_1
The plot of the concentrations, derivatives and sigma normalised derivatives:
.. image:: _static/tutorial_sa_2-results.png
:width: 800px
${\mathit{tutorial}}_{{\mathit{sa}}_{\mathit{2}}}$Cb$\mathit{Cb}$$${ { Cb } - \left( { { Ca0 } - { Ca } } \right) } = 0$$$${ { tutorial_sa_2.Cb } - \left( { { tutorial_sa_2.Ca0 } - { tutorial_sa_2.Ca } } \right) } = 0$$TrueTime$\mathit{Time}$$${ \left( { { \partial { { \left( { \tau } \right) } } } \over { \partial { t } } } \right) - {1 {}}} = 0$$$${ { { \partial { { tutorial_sa_2.\tau } } } \over { \partial { t } } } - {1 {}}} = 0$$False