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Combination Method - fitting two parameters to data

Dear All,

I am wondering about the principal of combination method. The combination formula is nCr to define the number of possible combinations of r objects from a set of n objects. So nCr = n!/(r!(n-r)!

But for my case, I have 2 unknowns which I want to obtain from several equations. For example, in total I have 287 equations. But I think I can not use the combination formula because in this case, r is 2 because I have 2 unknowns but n is not 287. Actually I am also looking for the best n itself.

In this attachment, a and b are 2 unknowns which I want to obtain and there are 6 equations. So I can get a and b by using those 6 equations. But I have 287 equations so I want to know the best combination, how many equations which I need to use to get the best a and b although the expected values of a and b are also unknown.

So is that possible to do so? If yes, what function should I use to solve it by using Mathematica? Thanks for your attention.

Best Regards, Intan

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POSTED BY: INTAN SUPRABA
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A Monte-Carlo simulation could proceed as follows.

(1) Select a random subset of your equations. You might try different criteria e.g. a few at a time or perhaps 30% at a time. I do not know enough about this area to have a good suggestion here.

(2) Do a fitting to get the best-fit parametervalues for {a,b}.

(3) Attempt to combine your results or at least check that they tend to give consistent results.

Having said this, it's probably overkill. It is not at all clear why one would not just do a best fit to the full set of data. One reason to use subsets, in general, is to avoid overfitting when there is a plethora of parameters. But you only have two. In such situations I think it is fairly common to either use the full set, or at worst use a large subset and then "test" the result by checking how well it "predicts" the remaining cases.

The upshot is I think there are basic questions of the science and statistics involved that you will need to consider before you can really pose this as a coding question.
POSTED BY: Daniel Lichtblau

Dear Daniel,

Thanks for your comments. There is a physical meaning for using only small range data set so that's the reason why I am trying not to use the full range data set, if possible. However, thanks a lot for your comments. I am considering your ideas.

Best Regards, Intan
POSTED BY: INTAN SUPRABA

(1) The given factors in the hyperbolic secants e.g. 128.3 and 136.6, do not correspond to actual values in the xlsx file. I don't know what is their significance in the expressions under consideration if they are not 'R' values from the data.

(2) There are different constant values being subtracted, e.g. 0.3169, 0.2162. So it's not clear how or where a "constant" value appears in the expressions.

All this is really very murky. From the broad description I would guess you are trying to do some form of resampling statistics. Can't really give any advice though, given how little I understand of your setup.
POSTED BY: Daniel Lichtblau

Dear Daniel,

Thanks for your comments and I would like to clarify as follows:

(1). Yes, there are some slight different between the raw data and the given R value in the hyperbolic secant. The real values of R is as per shown in the excel sheet. But sometimes I need to do some adjustments because the R value which I use in that basic equation is based on hourly data. Actually the purpose to show those excel data is only for showing the total number of R which is considered in the analysis. About the exact magnitude of R itself that I will use in the equation, I need to check it out one by one and need to do adjustment when necessary. So sorry for the confusion related to the discrepancies between the magnitude of R in the tanh equation and in the raw data.

(2). About the constant value for each equation, I obtained those values by doing a simulation for each rainfall event (R). So constant value for each equation varies because R is also different.

So my problem is after obtaining 250 equations where each equation contain 2 unknowns i.e. a and b, then what kind of combination method that I can do to get many pair combinations to produce a and b. I can use 1 equation to get a and b. I also can use 2 equations to get a and b. And I also can use overall 250 equations to get a and b.

I am looking for the appropriate combination method to do so and I heard about Monte-Carlo method but I am not sure how to do it in Mathematica. Hence, please kindly let me know if you know which function/command in Mathematica which adopted Monte-Carlo algorithms. Thanks for your attention.

Best Regards, Intan
POSTED BY: INTAN SUPRABA
POSTED BY: INTAN SUPRABA
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POSTED BY: INTAN SUPRABA

You really need to provide a (smallish) example that fully captures the problem at hand. From the current nb I cannot tell how your 'n' arises in the problem.
POSTED BY: Daniel Lichtblau

how many equations which I need to use to get the best a and b although the expected values of a and b are also unknown

How will one know they obtained the best a and b if even the expected values of a and b are not known?
POSTED BY: Nasser M. Abbasi
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