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

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

Dear Daniel,

Thanks for your comment. I really appreciate it. I will try to explain the overall problem clearly as follows:

1. The basic equation: a b Sech[b R] = constant, where R is total rainfall (mm) and a & b are unknowns which I want to obtain.
2. I have 250 nos of total rainfall so I have 250 values of R (please refer to attach."Raw Data.xlsx")
3. In this case, I will have 250 equations with 2 unknows (a and b).
4. I picked up randomly several total rainfall (R) and make few combinations based on different threshold such as: comb1 = R in the range of 50-150 mm comb2 = R in the range of 50-175 mm comb3 = R in the range of 50-200 mm comb4 = R in the range of 100-150 mm comb5 = R in the range of 100-175 mm comb6 = R in the range of 100-200 mm

Then for each combination, I got the variation values of a and b (please refer to attach."Combination.nb"). The number of R for each combination is still not based on the actual number of R for each threshold. Like for example, for comb1 I only use 6 nos of R when actually I have 105 nos of R. For trial, I picked up only 6 nos of R because I have not finish making equations for the overall 250 variations of R.

1. The summary of those 6 different combinations is as follows:

1. So I want to know what kind of combination method to be used to cover the full range of data set and if possible based on 2 conditions: 1) Depend on threshold for example 0-5 mm, 0-10 mm, 0-20 mm, 5-10 mm, 5-15 mm, .......0-550 mm. But the threshold itself is not determined. So any thresholds is to be used for making the combination. 2) Not depend on threshold, because even only by using a single equation I can get 2 unknowns.

              in= NMinimize[{Max[(a b Sech[128.3 b] - 0.3169)^2], a > 0}, {a,  b}]
              out= {1.2326*10^-32, {a -> 64.9373, b -> 0.0122389}}
   

Please kindly let me know if the above explanations are still not clear and I will try to further explain about the problem itself. Thanks in advance for your help!

Best Regards, Intan

Attachments:

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?