Double exponential (DE) is a popular non linear model for calculating kinetic degradation of organic molecules/pollutants. In DE model, the two distinctive phase of degradation (fast and slow) is observed and for these two phases have two different coefficient (K) value and partial portion (P) value.
But, I am unable to find out the K1, K2; P1, P2 values from this equation. Some reference said it is possible using solver add in Excel or Matlab. But, I do not understand that, how they do it?
How they do it? - They find the parameter values for which the sum of the squared residuals is minimal. The Excel solver uses a simplex search.
How do you do it? - You have your data and some guess for the parameters. According to the guesses you can calculate the model values that correspond to each observed value. The difference between an observed value and a model value is called "residual". You square these residuals and sum up the squares. You tell the solver to minimize this value by changing the parameter values. You will find more detailed instructions in the internet if you dare to search.
How they do it? - They find the parameter values for which the sum of the squared residuals is minimal. The Excel solver uses a simplex search.
How do you do it? - You have your data and some guess for the parameters. According to the guesses you can calculate the model values that correspond to each observed value. The difference between an observed value and a model value is called "residual". You square these residuals and sum up the squares. You tell the solver to minimize this value by changing the parameter values. You will find more detailed instructions in the internet if you dare to search.
Thanks Dr. Jochen Wilhelm for your kind reply regarding this query. I understand the minimization of square error/square residual for best fit expected model value with the dummy values from the solver tutorials. But, I wonder that, the equation for DE, C(t) = C0 - (P1 (1-e^-k1 t) + P2 (1-e^k2 t). Here, P1, P2; K1, K2 are unkown. So, these 4 values may be dummy values as you said based on guess.
Let me try with this. Thanks again.
I think "dummy variables" is a wrong term. These are the coefficients of the model. And yes, you need some educated guess from which you can start the minimization of the SSE. Depending on the problem and on the guess, it may happen that some local minimum far away from the global minimum is found, giving a bad fit. It is thus important to always check the result graphically.
Dear Aniruddha, if you attach/share your data in excel format (or any other "general" format, e.g. tab, csv, ...) I can try to send you back the non-linear best fit for your model using solver. Remove any sensitive information from the file but keep proper and clear labels for the results. Regards Luis
Thanks Dr. Luis F. Gouveia for your kind interest. I already get the fit curve from one of my colleague. But, I like to know how they do it? Any such link/tutorial for this non-linear fit? I need to do it ownself for further such activities.
Regard,
Aniruddha
Dear Aniruddha, did your colleague performed the fit, using Excel? If yes, kindly ask him/her to explain to you how he/she did it. If not, send the data and I will share an excel workbook with the formulas and clear comments.
Alternatively and, as suggested by Jochen Wilhelm, you can search the internet and find lots of resources available online.
Search for "non-linear fit using excel solver" and you will get a lot of interesting, useful and clear resources.
Regards, Luis
Here is an example (not exactly your model, as C0 is missing, but I needed some example data - I took it from a paper using a double exp. model without C0; you should be able to modify this according to your needs).
What I don't like is that the solver does not provide any uncertainty measures (standard errors, confidence intervals), and that it will be hard in Excel to get confidence bands for predictions or for the expectations. The paper I used is a bad example either, as they don't show any uncertainty associated with their fits. Additionally, I think the data there show a bi-phasic saturation, what is not well fitted by the proposed double-exponentials. One should have used this to actually reject their model... but tha's off topic.
I would prefer fitting a non-linear model using a proper statistical software (like with nlm in R, jut to give one example), where one would get the var-covar-matrix of the parameter estimates and where one can bootstrap appropriate confidence bands.
Jochen Wilhelm : I downloaded your excel sheet. I have what is likely an embarrassing question. What determines whether you wish to minimize or maximize the value of SSE? Solver has the option of doing both. If there is an introductory explanation you could refer me to I would be extremely appreciative.
In an unrelated connection: Congratulations on your MONSTER ResearchGate impact score.
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