Optimization module#

This module defines functions for optimizing kinetic models and calculating residuals.

Author: alan.tabore

optimization.model(x, rate_law, experimental_args_for_rate, number_of_parameters_to_optimize_for_rate, vitrification_law, experimental_args_for_vitrification, number_of_parameters_to_optimize_for_vitrification, coupling_law, experimental_args_for_coupling, tg_law, experimental_args_for_tg, tg_args)[source]#

Calculate the overall reaction rate considering both reaction and vitrification.

Parameters:

x (array-like) – Parameter values for the kinetic model.
rate_law (function) – The rate law function for the reaction.
experimental_args_for_rate (tuple) – Experimental arguments for the rate law function.
number_of_parameters_to_optimize_for_rate (int) – Number of parameters to optimize for the rate law.
vitrification_law (function) – The vitrification law function.
experimental_args_for_vitrification (tuple) – Experimental arguments for the vitrification law function.
number_of_parameters_to_optimize_for_vitrification (int) – Number of parameters to optimize for the vitrification law.
coupling_law (function) – The coupling law for reaction and vitrification.
experimental_args_for_coupling (tuple) – Experimental arguments for the coupling law function.
tg_law (function) – The glass transition temperature (tg) law function.
experimental_args_for_tg (tuple) – Experimental arguments for the tg law function.
tg_args (tuple) – Additional arguments for the tg law function.

Returns:

global_rate – Overall global rate considering both reaction and vitrification.

Return type:

float

optimization.rss_increase_of_small_extents_impact(x, experimental_rate, rate_law, experimental_args_for_rate, number_of_parameters_to_optimize_for_rate, vitrification_law, experimental_args_for_vitrification, number_of_parameters_to_optimize_for_vitrification, coupling_law, experimental_args_for_coupling, tg_law, experimental_args_for_tg, tg_args, extent, extent_limit, amplification_factor)[source]#

Calculate the modified residual sum of squares (RSS) with an impact increase for small extents.

This function calculates the mean squared error with modifications based on the extent of the reaction. For extents smaller than a specified limit, the differences between the model rate and experimental rate are amplified by a given factor before computing the RSS.

Parameters:

x (array-like) – Vector with parameter values to optimize for the kinetic model.
experimental_rate (array-like) – Experimental reaction rate data.
rate_law (function) – The rate law function for the reaction.
experimental_args_for_rate (tuple) – Experimental arguments for the rate law function.
number_of_parameters_to_optimize_for_rate (int) – Number of parameters to optimize for the rate law.
vitrification_law (function) – The vitrification law function.
experimental_args_for_vitrification (tuple) – Experimental arguments for the vitrification law function.
tg_law (function) – The glass transition temperature (tg) law function.
experimental_args_for_tg (tuple) – Experimental arguments for the tg law function.
tg_args (tuple) – Additional arguments for the tg law function.
extent (array-like) – Extent of the reaction.
extent_limit (float) – Threshold value for the extent. Extents smaller than this limit will have an impact amplification.
amplification_factor (float) – Factor by which the differences are amplified for extents smaller than the limit.

Returns:

modified_rss – Modified mean of residual sum of squares (RSS).

Return type:

float

Notes

The modified RSS is calculated as follows:

\[\overline{RSS_{mod}} = \frac{\sum \left(\left(A*\left(y_{exp_{i}}(\alpha<\alpha_{lim})-f(x_i,\alpha<\alpha_{lim})\right)\right)^2\right)+\sum \left(\left(y_{exp_{i}}(\alpha\ge\alpha_{lim})-f(x_i,\alpha\ge\alpha_{lim})\right)^2\right)}{n}\]

where \(A\) is the amplification factor, \(y_{exp_{i}}(\alpha<\alpha_{lim})\) and \(y_{exp_{i}}(\alpha\ge\alpha_{lim})\) are the experimental rates for extents smaller and larger than or equal to the limit respectively, \(f(x_i,\alpha<\alpha_{lim})\) is the model rate for extents smaller than the limit, \(f(x_i,\alpha\ge\alpha_{lim})\) is the model rate for extents larger than or equal to the limit, and \(n\) is the number of points.

optimization.rss_increase_of_small_rates_impact_with_zones(x, experimental_rate, rate_law, experimental_args_for_rate, number_of_parameters_to_optimize_for_rate, vitrification_law, experimental_args_for_vitrification, number_of_parameters_to_optimize_for_vitrification, coupling_law, experimental_args_for_coupling, tg_law, experimental_args_for_tg, tg_args, fraction_to_amplify, amplification_factor)[source]#

Calculate the modified residual sum of squares (RSS) with an impact increase for small rates in specific zones.

This function calculates the mean squared error with modifications based on the experimental rate. For zones where the experimental rate is small compared to the maximum experimental rate, the differences between the model rate and experimental rate are amplified by a given factor before computing the RSS.

Parameters:

x (array-like) – Vector with parameter values to optimize for the kinetic model.
experimental_rate (array-like) – Experimental reaction rate data.
rate_law (function) – The rate law function for the reaction.
experimental_args_for_rate (tuple) – Experimental arguments for the rate law function.
number_of_parameters_to_optimize_for_rate (int) – Number of parameters to optimize for the rate law.
vitrification_law (function) – The vitrification law function.
experimental_args_for_vitrification (tuple) – Experimental arguments for the vitrification law function.
tg_law (function) – The glass transition temperature (tg) law function.
experimental_args_for_tg (tuple) – Experimental arguments for the tg law function.
tg_args (tuple) – Additional arguments for the tg law function.
fraction_to_amplify (float) – Fraction of the maximum experimental rate to consider as the threshold for amplification.
amplification_factor (float) – Factor by which the differences are amplified for rates below the threshold.

Returns:

modified_rss – Modified mean of residual sum of squares (RSS).

Return type:

float

Notes

The modified RSS is calculated as follows:

\[\overline{RSS_{mod}} = \frac{\left(\sum \left(\text{{modified_dif}}^2\right)\right)}{n}\]

where \(y_{exp_{i}}\) is the experimental rate and \(f(x_i)\) is the model rate, and \(n\) is the number of points.

The difference \(\text{{dif}}_i\) is modified as follows:

\[\begin{split}\text{{modified_dif}}_i = \begin{cases} \left(y_{exp_{i}}-f(x_i)\right) \times A, & \text{if } y_i < \frac{{\max(y)}}{p} \\ y_{exp_{i}}-f(x_i), & \text{otherwise} \end{cases}\end{split}\]

optimization.rss_mean(x, experimental_rate, rate_law, experimental_args_for_rate, number_of_parameters_to_optimize_for_rate, vitrification_law, experimental_args_for_vitrification, number_of_parameters_to_optimize_for_vitrification, coupling_law, experimental_args_for_coupling, tg_law, experimental_args_for_tg, tg_args)[source]#

Calculate the mean squared error.

\[\overline{RSS}= \frac{\sum (y_{exp_{i}}-f(x_i))^2}{n}\]

with \(y_{exp_{i}}\) the i-th value of experimental rate and \(f(x_i)\) the value of rate computed using the given kinetic model \(f\) and the i-th parameter vector \(x_i\) and \(n\) the number of points.

Parameters:

x (array-like) – Vector with parameter values to optimize for the kinetic model.
experimental_rate (array-like) – Experimental reaction rate data.
rate_law (function) – The rate law function for the reaction.
experimental_args_for_rate (tuple) – Experimental arguments for the rate law function.
number_of_parameters_to_optimize_for_rate (int) – Number of parameters to optimize for the rate law.
vitrification_law (function) – The vitrification law function.
experimental_args_for_vitrification (tuple) – Experimental arguments for the vitrification law function.
tg_law (function) – The glass transition temperature (tg) law function.
experimental_args_for_tg (tuple) – Experimental arguments for the tg law function.
tg_args (tuple) – Additional arguments for the tg law function.
coupling_law_for_reaction_and_vitrification (function) – The coupling law for reaction and vitrification.
coupling_law_for_reaction_and_vitrification_args (tuple) – Additional arguments for the coupling law function.

Returns:

rss – Mean of residual sum of squares (RSS).

Return type:

float

optimization.rss_standard(x, experimental_rate, rate_law, experimental_args_for_rate, number_of_parameters_to_optimize_for_rate, vitrification_law, experimental_args_for_vitrification, number_of_parameters_to_optimize_for_vitrification, coupling_law, experimental_args_for_coupling, tg_law, experimental_args_for_tg, tg_args)[source]#

Calculate the residual sum of squares (RSS).

\[RSS= \sum (y_{exp_{i}}-f(x_i))^2\]

with \(y_{exp_{i}}\) the i-th value of experimental rate and \(f(x_i)\) the value of rate computed using the given kinetic model \(f\) and the i-th parameter vector \(x_i\).

Parameters:

x (array-like) – Vector with parameter values to optimize for the kinetic model.
experimental_rate (array-like) – Experimental reaction rate data.
rate_law (function) – The rate law function for the reaction.
experimental_args_for_rate (tuple) – Experimental arguments for the rate law function.
number_of_parameters_to_optimize_for_rate (int) – Number of parameters to optimize for the rate law.
vitrification_law (function) – The vitrification law function.
experimental_args_for_vitrification (tuple) – Experimental arguments for the vitrification law function.
tg_law (function) – The glass transition temperature (tg) law function.
experimental_args_for_tg (tuple) – Experimental arguments for the tg law function.
tg_args (tuple) – Additional arguments for the tg law function.
coupling_law_for_reaction_and_vitrification (function) – The coupling law for reaction and vitrification.
coupling_law_for_reaction_and_vitrification_args (tuple) – Additional arguments for the coupling law function.

Returns:

rss – Residual sum of squares (RSS).

Return type:

float