@ Saad Sharjeel 

Let me ask a question about the topic: Are you asking how to verify that the chosen differential stochatic equation (DSE) and its parameters are appropriate for describing the observed Time Series Emission Data?

I am not familiar with the AIC, however I am interested in the structure of the criteria. One can easily guess that choosing  a model a comparisson to the data is crucial. Therefore I wonder that you are seeking a "fit test in absolute terms (not model to model comparison for selection of best model)". Does it mean that the data are not taken into account? Which models are compared in the "model to model" way? If this is true, then probably the admitted models are somehow correlated to the data. So what then  can be better with absolute terms? Is this a question of neglecting some implicitely assumed information involved, say, by a bayesian procedure?

Any hint allowing catching the structure is appreciated. Perhaps then we can jointly come to the key point of your problem :) In any case I'll got a chance to increase my knowledge about modern statistical methods.

Sincerely yours, Joachim

@Joachim

Sir,

AIC (Akaike Information Criteria) provides relative quality of statistical/  stochastic models like in my case i have compared 4 proposed/ developed models for given set of data (as attached) . Given AIC estimates suitability of each model relative to other proposed models is carried out. Thus for the given data, the preferred model is the one with the lowest AIC value.

But there is problem with this method i.e. it does not provide a test of a model in the sense of testing a null hypothesis, so it does not provide any idea of quality of the model in absolute sense. AIC discourages overfitting, as increasing the number of parameters in the model  improves the goodness of fit.

Quote

"Let me ask a question about the topic: Are you asking how to verify that the chosen differential stochatic equation (DSE) and its parameters are appropriate for describing the observed Time Series Emission Data?"

Yes Sir, i want to check the validity of my model (in absolute sense and not just selecting best model out of four models). As pointed out earlier, i have used AIC but this gives quality of model in relative sense i.e. in comparison to other proposed models. Other relevant calculated/ simulated values like var-covariance matrix values for parameters (estimated through pseudo-maximum likelihood estimation) and model mean plot, actual  Time-Series data plot, simulated trajectories plot and 95 % confidence interval plot are given in my first post as an attachment. 

As in case of linear models, generalized linear models etc we have the option of calculating RSS (Residual Sum of Squares), conduct Chi-sq testing, R-squared value and adjusted R-squared value etc to check efficacy of model. What needs to be done in case of Stochastic model (apart from the work which i've already done) to check the validity against given dataset. Kindly do see my plots as well. I would be obliged if you can also provide any valuable comments, suggestions or point out any shortcomings.

Best Regards.

Saad Sharjeel

@ Saad Sharjeel

Dear Sir,

Thak you very much for your exhaustive answer. I like the fact, that the models are still  sought among other possibilities, and that you are not passing easily to hints which are not sufficiently justified. On my side, I am not fmiliar with many of your statistical tools, mainly due to my age, and the entire history :( , but I hope some remarks can be given; especially with respect to the equation to be considered for the process you are interested in. The main way of obtaining such a model preferred by me is to start with a QUALITATIVE model ofchanges to the quantity of observation. Obviously, particularly with respect to the emission [of pollution ? ],  this requires a study of the potential causes 9which are physical, economcal, political and . . .) thus some time i required; even for getting the simplest possible model. What I mean, can be said briefly as an example: the changes are caused not only by the current state but a little bit at least from the past (thus, Markovianity of say order 2 might be required). But this is pointed out only as an illustration of my way I would like to follow.

At the first glance, my (perhaps naive) proposal is to analyse THEORETICALLY, what is the covariance function for the models of your current choice, just to get involved into the core of the problem. This is my intrisic need. Hopefully I shall get some insight into the subject soon.

Best regards

Joachim Domsta

I'm rather familiar with modelling with sde.

There are many issues in what you write that should be taken into account.

1) you should have a qualitative motivation to propose a specific model to fit a dataset, you do not just try models by chance. The models you propose possess a stationary distribution, while your dataset seem to suggest an illimitated growth. Have you considered this fact.

2) Some of your models could have problems if zero is reached, you should specify parameters range in which this does not happen and verify that you estimates are compatible

3) If you have a reason to model the data with one of the sde you propose you'd better observe that for all your models the transition density is know in an explicit form. That means that you can do maximum likelihood (without PSEUDO!), and that you can also apply goodness of fit tests (if you really like). However things are a bit long to discuss here and it seem that you need some guidance.

All the best, Enrico

@ Joachim @ Enrico

Thanks for the guidance!

Best Regards.

Saad Sharjeel

Notes:

-- the third model on the list by Saad (in his second attachment) is the same as the one  presented in the original question;

-- referring to the remark by Enrico, seemingly, none of the three models ensures  avoiding negative values by the proces X_{\cdot} ;

-- the simplest way to stay within positive processes is to model the logarithmY_t := \log X_t, t\ge 0;

-- I support the remarks by Enrico, in particular, the researcher should have an idea of the sources of randomness first! In the simplest cases, the log-processes lead to log-normally distributed models (comp. the sde  dY_t = a Y_t dW_t  for the constant-sign  martingale 

Y_t = \exp[a W_t -  ( t  a^2 /2) ], whose constant mean-value equals 1 ;

-- Despite the need of a more careful analysis with taking into account the natural causes of randomness, still staying within quesses, perhaps the 2-parameter sde .dY_t =  Y_t  ( a  dW_t  +  b dt )  is suffient for fitting at least the best approximation of exponential mean of the form

y(t) := E\{ Y_t \} to be compared to  A \exp( b t ),  t \ge 0.

Then for good choice of  parameter  a,   a theoretical formula for - say the  total sum of squares of the random process devitions from its expectation should be compared to the same for the data with respect to the exponential function. Another way is to pass to logarithms of  Y  and the given data. THIS IS JUST A PROPOSAL! Probably far not consistent with the data, but it seems to be a good way for getting experienced in changing scale etc. For instance, since the data correspond to discrete time, and the model is time-continuous, some other algorithms/procedures should be learned up to the very detail :(  :)

Sincerely,

Joachim Domsta

--

@ Joachim

Sir,

I just tried to avoid a long post however i think i need to clarify few things:

All models have a qualitative background. e.g. the simplest model case

dX = (theta1).dt+(theta2).dW

is the one with the assumption that rate of change of Carbon dioxide emissions from cement industry with time is proportional to some parameter say theta1(drift part). Furthermore, it is assumed that uncertainty/ randomness  is associated with the emission process represented by parameter theta2(diffusion part). This randomness(diffusion part) is result of uncertainty due methodology used for estimating the emissions. Other factors which contribute to the uncertainty are confidentiality of production data at industry level, type of cement produced, CKD (Cement Kiln dust) estimations, electricity utilized by the plant, transportation related emissions etc. it has been assumed that diffusion process is purely Gaussian in nature (White noise). 

Another model that i.e. last model in my second post has been derived from  probability distribution in continuous time (letting Δt → 0) satisfying the Forward Kolmogorov equation.

Quote:

"referring to the remark by Enrico, seemingly, none of the three models ensures avoiding negative values by the process X_{\cdot} "

Sir, up till now, i have not faced this problem with models and available datasets.

Quote:

"Despite the need of a more careful analysis with taking into account the natural causes of randomness, still staying within quesses, perhaps the 2-parameter sde .dY_t =  Y_t  ( a  dW_t  +  b dt )  is suffient for fitting at least the best approximation of exponential mean of the form

y(t) := E\{ Y_t \} to be compared to  A \exp( b t ),  t \ge 0.

Then for good choice of  parameter  a,   a theoretical formula for - say the  total sum of squares of the random process devitions from its expectation should be compared to the same for the data with respect to the exponential function. Another way is to pass to logarithms of  Y  and the given data. THIS IS JUST A PROPOSAL! Probably far not consistent with the data, but it seems to be a good way for getting experienced in changing scale etc."

Sir noted!

Quote:

"For instance, since the data correspond to discrete time, and the model is time-continuous, some other algorithms/procedures should be learned up to the very detail"

Sir! this problem can be tackled by various means/ R software.

Sir! just an idea... Is it right to maybe plot the mean of the simulated model and extract the values out at each point. Then compare those simulated stochastic model mean values against available Time Series data for analysis like chi square testing, variance analysis, error bar plot etc.

Thanking you in anticipation and best regards.

Saad Sharjeel

@ Saad Sharjeel

Thank you for your kind reply. Let me just two "buts":

-- I am suggesting to perform the theoretical expectation analysis from the assumed model sde, and then the relevant parameters get from the best fitting to the data. At the next step only the parameters influencing the variance can be found (e.g. by substracting or some other means).

-- Despite the fact that you didn't get negative sumulations, it is not appropriate to consider models which do not exclude such a "catastroph", same with infinities, unless they appear as a predicted event.

Best regards,

Joachim Domsta

@ Joachim

Sir, your suggestions are very valuable. Definitely i need to put in some more effort to get the things right.

I am very grateful to you for providing guidance.

Best Regards.

Saad Sharjeel