Team:Paris/Modeling/BOB/Akaike

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(Experiment)
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* We made a set of data out of a noised Hill function. In fact, our data set was made by using the same equations as System#2, but we introduced a normal noise for each point. Thus, System#1 is penalized because its RSS will be greater than that of System#2. Nevertheless, System#2 will be more penalized by its number of parameters.
* We made a set of data out of a noised Hill function. In fact, our data set was made by using the same equations as System#2, but we introduced a normal noise for each point. Thus, System#1 is penalized because its RSS will be greater than that of System#2. Nevertheless, System#2 will be more penalized by its number of parameters.
* With Matlab, we run a fitting simulation for each system, and we obtained the RSS. We then evaluated the different criteria for both models. The results are presented below.
* With Matlab, we run a fitting simulation for each system, and we obtained the RSS. We then evaluated the different criteria for both models. The results are presented below.
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Revision as of 13:08, 3 September 2008

(Under Construction)

Model Comparison


Short introduction to the criteria

  • Using linear equations in a biological system might seem awkards. However, we wanted to check the relevance of this approach. We have been looking for a criterium that would penalize a system that had many parameters, but that would also penalize a system which quadratic error would be too important while fitting experimental values. The goal here is to decide whether, assuming that the experimental data looks like a model based on Hill functions, the linear part of the BOB model is obsolete or not.
  • Akaike and Schwarz criteria taken from the information theory met our demands quite well :
Akaike criterion :
AIC.jpg
Hurvich and Tsai criterion :
AICc.jpg
Schwarz criterion :
BIC.jpg

where n denotes the number of experimental values, k the number of parameters and RSS the residual sum of squares. The best fitting model is the one for which those criteria are minimized.

  • It is remarquable that Akaike criterion and Hurvich and Tsai criterion are alike. AICc is therefore used for small sample size, but converges to AIC as n gets large. Since we will work with 20 points for each experiment, it seemed relevant to present both models. In addition, Schwarz criterion is meant to be more penalizing.

Experiment

  • As an experiment, we wished to compare the two models presented below :

System#1 : using the linear equations from our BOB approach :

Syste akaike 1.jpg

System#2 : using classical Hill functions :

Syste akaike 2 bis.jpg

  • We made a set of data out of a noised Hill function. In fact, our data set was made by using the same equations as System#2, but we introduced a normal noise for each point. Thus, System#1 is penalized because its RSS will be greater than that of System#2. Nevertheless, System#2 will be more penalized by its number of parameters.
  • With Matlab, we run a fitting simulation for each system, and we obtained the RSS. We then evaluated the different criteria for both models. The results are presented below.
Comparison of the systems
Criteria System#1 System#2


AIC
AICc
BIC


WHAT HAVE WE PROVED



We mostly used the definition of the criteria given in : [http://www.liebertonline.com/doi/pdf/10.1089/rej.2006.9.324 K. Kikkawa.Statistical issue of regression analysis on development of an age predictive equation. Rejuvenation research, Volume 9, n°2, 2006.]