When one makes kinetic models, one usually has to deal with two antagonistic facts: One needs the model to explain the phenomenon well, but at the same time one does not want the model to be too complicated.
My question is about how you have addressed this dilemma. How can one measure (objectively) that one model is better than another? Especially when the models have different number of parameters.
I believe using Akaike Information Criterion is a straightforward way to compare between models. Specially if they have different number of parameters.
https://en.wikipedia.org/wiki/Akaike_information_criterion
The answer is it depends. Since you are in the land of predictive models, I would first select some likely predictors of the rate. Then I would home in on a model using adaptive lasso variable selection. I would use adaptive lasso because it has an oracle property, i.e. the model it selects is asymptotically equivalent to the true model and it does not allow colinear terms in the model. I have attached examples for logistic and Cox regression and the theory for Cox regression and this type of model building. References for the least squares case are in the paper references. I would always plot the DV vs the potential IV to see if the IV is a serious candidate for the model. This is a fairly general approach to regression predictive model building so it should work for kinetics. Finding an exponent could require multiple plots. I hope this is of some use. By the way, the Akaike information Criterion (AIC), mentioned above, and the Bayesian I C have a place in this method, esp. if you come down to comparing models. Best wishes, David Booth
The problem you are posing does not have a simple answer. if two different (kinetic) models gives the same results in the whole range of conditions, the best choice is the simplest one. From the physical point of view these models are equivalent. But you are talking about simplified kinetic models. This means that each one have a limited range of validity. The set of conditions of validity intersect each other. Therefore if the results coincide you are in this part of the operative range. In this case you should evaluate the behaviors in the regions where the two models behave differently.
However there is another aspect to be considered. The model prediction is not the only criterion of the validity of a model. Often, simplified models only predict some parameters but not all. An example in my field is the prediction capabilities by different kinetic models of the flow properties during the atmospheric entry of space vehicles. Reduced models have very limited prediction capability in the sense that their validity is limited to a small range of operative conditions, and often predicts heat fluxes, but not the concentration of some species. Simplified models should always be used with care and knowing that they are not reproducing the full physics of the system, but only some aspects. Moreover, using simplified models not in the proper way can lead to wrong conclusions about the cause of a given behavior.
Criterion one, it is agreement with the experiment. :) Everything else is secondary. :)
Dmitry, that is not true. This agreement can be accidental. First of all the model may be non-applicable under the considered conditions. The models, at least, should be verified in your conditions. Then you may chooose the simplest.
Dear Yuriy. Physics is considered to be an experimental science. It means that every new theory is tested by some experiment. If the experiment verifies the theory it is recognized and vice versa : if it doesn't verify the theory it is rejected as being wrong. And you confirmed my words by writing - 'The models, at least, should be verified in your conditions.' )
Dear colleague, speaking dialectically, any model is only an approximation to the reality given in our sensation. Objectively, a measure of good approximation is the consistency of the results of the model within the limits of errors that you subjectively consider acceptable, i.e., with accuracy that is necessary and sufficient for practice. No matter, whether this is a physical or mathematical experiment. Therefore, better or worse – it’s all relatively! Using algebra or using Monte Carlo methods - the predictive power of the model can in principle be the same. The limits of applicability of the model are always determined by trial and error. But in terms of economy, of course, the simpler - the better. I wish you every success in this way!
Giving the primate to experiments was an idea promoted by Mach in 19th century. Where did this idea bring? To rejection of the atomistic model and kinetic theory of gases. Basing the physics on experiments first, would relegate the great theories of the 20th century in a limbo. Theory and experiments to should together to understand and explain the behavior of the "physical" world.
Now we have to ask ourself what is the purpose of a model. I think we have two choices.
1) A model can e used to reproduce the results of one experiment. It should not give any insight. Fitting functions, neural network, and so on, are also good models. However these models are not useful to understand the "physics" behind. The Rayleigh-Jeans law for black body emission is an example. Model accuracy and limits of validity are required.
2) A model can be used to verify a theory, against experimental results. This kind of models must be constructed considering the underlining theory and basic physical principles. In this case non only positive, but also negative results are important. Perfect agreement with results is not always required, but reproducing qualitative behaviors can be acceptable: There are some unknowns in the real system that are not (or cannot be) included in the model that can be responsible for the difference. An important requirement is that this model should be valid in a wider set of conditions.
These two categories are completely different, following different rules and having different purposes. However, sometime, this difference is forgotten and models of the first kind are used for the scope of the second kind. Then some problems arises. Extrapolating a physical behavior from models of the first kind, will lead to wrong results.
An example: The multi temperature (MT) and state-to-state (STS) models.
The first one assume Boltzmann distribution for internal degrees of freedom, the other considers an equation for each internal state, determining the internal distributions. Let us consider N2 molecule, for which the internal distribution is very difficult to determine after very few levels, because they are not emitting radiation.
In MT there are some parameters adjusted to reproduce different experiments. STS contains rates, hopefully determined from first principles, used for all the simulations. These rates have some errors obviously.
In the last years, more researchers are using STS. But while STS is a model of kind 2, MT is a model of kind 1. They are both useful but should be used for different purposes.
I think that this question cannot be addressed without including experimental point of view. The two antagonistic facts should be considered through a reasonable choice of hierarchy. The major requirement that a model is suppose to fulfill is to predict correctly the behavior of natural phenomena. Numerical predictions should match the the experimental observations with a good accuracy, at least for a certain range of experimental conditions. Thus, I will say that the best model is the one that leads to a correct prediction regardless than its simplicity. Let us assume that we are comparing two models that both are able to lead to a good predictions. In that case, I do believe that the best model is the one that give good predictions for the widest experimental conditions regardless than the simplicity of these models. Finally, if two models are able to give similar predictions for the same experimental conditions range, we should consider the simplicity of the models to judge which one is better. Here, the simplest one is the better.
Dear Jeanpiero! Thank you for very good point! I meant that there are two types of the model consistency: 1) relative consistency of data and results, and 2) internal consistency of hypotheses and axioms. This is a unity and struggle of non-antagonistic opposites! In every specific case of the solution of the problem, the optimal choice is a compromise (consistent optimum), in the result of which these opposites meet our expectation in the limits of errors that we allow in our model.
In our kinetic models, we use both MT and STS approaches, sometimes separately, sometimes combine them to model some molecules, for example, in excimer plasma, including both short-lived radiative states and long-lived non-radiating metastable electronically excited states. But in both case, we use quasi-Boltzmann local thermodynamic equilibrium (LTE) approximation although this is not strongly true in the case of the nonequilibrium electron energy distribution function (EEDF). And in our models of the kinetic rates of elementary reactions we use quasi-Arrhenius approximation in algebraic form, k(T)=AT^(n) exp(-E/T) although this is also not strongly true in chemical nonequilibrium conditions, especially in multicomponent reactive systems, where the kinetic mechanism includes hundreds of reactions. But this is a real state-of-the art in this field of physics and chemistry. As a reasonable compromise between model simplicity and complexity.
Dear colleague! Of course, nobody has canceled Occam's razor yet. )
This is common inverse problem. You have to start with specifying the objective function (measure of fit), e.g. in the form of the mean relative error. Then choose a method for minimizing multivariable functions. The answer will be provided by comparing the minima (preferably global) for both models. But what models you choose, it is up to you.
- Badges
- Science topic