Hi,

      i would like to recommend you Linear regression Model for it because values are constant but steadily depends on time [which is not constant]

Regards

Muddsair

thank you!

i used a black box model. my question is about tools/test/indices that helps to mesure the goodness of estimation nowing that i dont have obeserved data to compar them with the estimated value. for instance Bravais-Pearson test 

You asked how to evaluate extrapolated data. I would go with the 95% confidence interval about the extrapolated data and assume that the real value is anything in that range. If that does not fit your idea of how things should work then you either need to get data that is closer to what you are trying to measure, or change your idea to fit the data.

In your background for the question it sounds like you are wanting to take extrapolated data and use it in place of observed data. This is generally not done except in a case where you develop a model, predict some result that is beyond the scope of the data used to build the model, and then collect new observations in this region that confirm/refute the model that you developed. Repeat these steps as needed.

Goodness of fit tests are inappropriate to evaluate the fit of anything with extrapolated data. The problem is that you are evaluating the fit of data generated from the model you are trying to fit.

I can think of at least one exception to the above rule. Say I have five models, and each one results in a probability density function for some outcome. I now want to combine these models and generate several estimates for points not covered by the data in any of the models. I then want to see how these estimates fit with each of the five models individually. If this matches your situation, then try running thousands of simulations and using computer intensive methods of data analysis. Rather than comparing means, compare a goodness of fit statistics.

How to assess model predictions is a good question.  It's a challenge and for this reason model predictions are usually considered hypotheses generating.  One option is to consider the confidence intervals.  Often these are rather broad.  However, confidence intervals for extrapolated values depend on the assumed model being the correct model.  To address this issue, you can use techniques used in data mining.  Typically, the available data is partitioned to provide one or more hold out samples.  The predictions made based on the model developed for the first data set are then compared with the data from the hold out sample(s).

The Jackknife method (leave-one-out, cross validation) is a good and almost universal tool to quantify an estimation error. It can be applied to many cases like distribution models or regression analysis. But autocorrelation can disturb this method.

It might help to have more context to the question. Are we talking about a biological test where a drug was administered to adult people weighing less than 90 kilograms and we are estimating the effect on people weighing 180 kilograms? Are we talking about the corrosion rate of metal resting in a salt solution of 0.5 molar and 2 molar with estimation of the effect of a 1 molar solution?

Are you building a new model using data that were estimated using an old model, or is this really a question about prediction outside the range of data? Are there real data on either side of the predicted value, or is the prediction way outside the observed data?

Thank you.

Exactly it is about prediction outside the range of data

I would combine some of the previous answers. I would build the simplest linear regression model possible. I would make sure that the data conform to the assumptions of the model. I would then plot the 95% confidence interval.

The next step assumes that you have an intermediate sized data set. If your data consist of four observations, a Jackknife method will be pointless, and it may be of limited value if you have thousands of observations because each observation will have limited impact on the final outcome if your data are well behaved. Assuming your data sets falls between: I would then use a Jackknife procedure. Build a model for all datasets that can be made by leaving out one observation. Then consider the cost of making a mistake. If the cost of making a mistake is large, then you want wide confidence intervals to maximize the chance that you have captured the true value. In this case, take the 95% CI from the model with the widest CI at any point. If the cost is small, then you might take an average value, or median value.

The next step depends on what you want to do with the estimated values. The best thing to do is to take some new measurements that are closer to the estimated values and see how the model predicts the results.

I guess another factor to consider is if you are using a well known model. A pesticide dose-response curve is a well known model. People may have less trouble with prediction outside the data range in this case relative to something like modeling the effect of CO2 with data at 350 ppm or less and then predicting the effect at 400 ppm.

I would recommend the suggestion made by Dena L Schanzer  above. You fit your model excluding some data for a fixed period of time at the end. Now use your model to extrapolate the data that you did not included in your model. If the model is good, the extrapolated values should be very close to the excluded data. You can produce a scatter plot of actual value vs extrapolated value to see positive linear relationship. The stronger the relationship is, the better your model extrapolates. Also, you can apply a chi-square test to check how close these two sets of data are to diagnose your model fit.