@Anuj Prajapati well received and noted. Thanks

The same question had been asked 2 years ago. Please refer to Ahmed Said Mahmoud's asked question. There are some nice answers.

@Bestani Benaouda, thanks for the concern, I will check his moment.

Very imprecise question, but guessing what the author meant, the answer is as follows: If the function/equation is linear, we use the linear regression method. If the function/equation is nonlinear, we use nonlinear methods commonly called nonlinear regression. Therefore, comparing the results of both methods does not make sense.

@Miroslaw Grzesik, I do not disagree with your opinion but I think you've overlooked the question. For those who understand the statistical analysis done in the adsorption process they will get what I mean. Anyway thank you for your comment.

In the adsorption and many biological fields, many phenomena are non-linear in property and the simplest method before the advance of computing power is to linearize these non-linear phenomena. The best example would be the Lineweaver-Burke (LB) plot, designed to get the Km and Vmax values from the initial nonlinear Michaelis-Menten curve. Linearization not only disturbs the error structure buat also amplifies outliers most often seen at very low substrate concentration data. For example, the reciprocal values of 0.01 mM in LB plot is 100 while for 0.02 mM, it will be 20. A minor error will dramatically change the values of Km and Vmax

M. Y. Shukor thank you for good explanation and knowledge. I'm grateful.

Farid Mzee Mpatani, Everyone has the right to write what they want. However, it is a scientific forum. Therefore, verified information should be provided, and if we disagree with someone's opinion, the reason should be given. I understood your questions perfectly well and deliberately answered the second one. What I wrote is true, so if you think otherwise, write why you think so.

Yunus Shukor, I fully agree that most phenomena in nature around us, taken as a whole, are non-linear. However, what you call linearization is not linearization in a mathematical sense. Linearization of a nonlinear function concerns the local approximation with the use of two terms of the Taylor series: constant and linear. So linearization is not the L-B, H-W or E-H transformation, used since the 1930s by enzymologists - and "rediscovered" by adsorption specialists 20 years ago.

Mirosław Grzesik I understand where are you coming from. Appreciate you trying to explain to us the fundamental understanding of the linear and non-linear functions based on mathematics.

Farid Mzee Mpataniand all non-mathematicians out there who have used the term linearization of nonlinear equations liberally, the term transformation may be more correct. That aside, the following is an attempt to explain as to why we should not linearize (transform) otherwise nonlinear equations (functions) to linear ones.

Regression models are regularly utilized in numerous applied sciences for explaining the connection between an independent variable and a response variable. One of the problems of linearization of nonlinear data is that the distribution of the test statistic becomes distorted. Another issue is that data transformation results in asymmetric measurement errors which are originally symmetric. For instance, the maximum likelihood estimators (MLE) are often used in statistical inferences on the regression parameters of a nonlinear curve. Linearizing the nonlinear function around the true parameter to obtain the standard errors via MLE can produce asymptotic Wald confidence intervals and can result in incorrect confidence levels (Peddada and Haseman 2006). Another important problem is that linearization can change the error statistics and this issue is often exacerbated at low sample number (Jeričević and Kušter 2005). One of the strongest transformation tools is the logarithm function. This transformation affected strongly the error distribution with the originally homoscedastic observation vector can become strongly heteroscedastic. Parameters obtained through the use of ordinary least square (OLS) estimator are strongly affected by such transformed data. And this larger uncertainty in the ensuing parameters can lead to an error in the contingent statistical tests (Klicka and Kubácek 1997)

References

Jeričević Ž, Kušter Ž (2005) Non-linear optimization of parameters in Michaelis-Menten kinetics. Croatica chemica acta 78, 519–523.

Klicka R, Kubácek L (1997) Statistical properties of linearization of the Arrhenius equation via the logarithmic transformation. Chemometrics and intelligent laboratory systems 39, 69–75.

Peddada SD, Haseman JK (2006) Analysis of Nonlinear Regression Models: A Cautionary Note. Dose-Response 3, 342–352. doi:10.2203/dose-response.003.03.005.

Yunus Shukor, We describe various types of phenomena by means of mathematical functions or equations, therefore such a description cannot be separated from the rules of mathematics. Moreover, the transformation can be both linear and non-linear. The L-B transformation mentioned by you is a nonlinear transformation, as a result of which from an explicit hyperbolic function with respect to the dependent variable y (y=k1x/(k2+x)) we obtain an implicit function (1/y=k2/(k1x)+1/k1). Both functions are nonlinear with respect to x. If you think, like unfortunately many do, this is linearization, that's your business. However, do not mislead others.

Yunus Shukor, You are absolutely right that changing the coordinate system from y vs x to 1/y vs 1/x (L-B plot) changes the error distribution. Moreover, replacing the variable y with its inverse, obviously, changes the statistical distribution of the new variable. It is similar with the variable x and 1/x. However, these are not linear but non-linear transformations. Another example of nonlinear transformation is, as you mentioned, the use of a strongly nonlinear logarithmic function to transform power monomials (Freundlich isotherm, Ostwald-de Waele equation, criterial equations etc.) and exponential functions (Arrhenius like equations, Avrami-Kolmogorov like equations, Newton like equations, Jovanovic isotherm etc.).

George Stoica

George Stoica

I used to be passionate about mathematical statistics and in particular regression analysis. I was also interested in stochastic programming. But the time has come that I came to the conclusion that this is not the way to go. Best regards,

George Stoica

Dear @Farid Mzee Mpatani I don't know much about isotherms and kinetic studies of adsorption. However, to my opinion, non-linear regression analysis is more informative than the linear analysis.

Thanks!

Taking logarithms of pressure and concentration of solute and plotting is the best method. The slope will give an idea how nonlinear the isotherm is.

The linear regression method is used to solve the inverse problem for linear equations. The nonlinear methods, commonly called nonlinear regression, are used for inverse problem for nonlinear equations. And only this and so much.

Dear Farid Mzee Mpatani , as per literature particularly in reference to isotherms and kinetic models nonlinear models are better suited. It can be observed from the error factors employed for explaining the models, have lower value as compared to linear models.

Krishna Yadav, In the case of the description of the equilibrium data, I know only one linear equation. It is Henry's isotherm. In turn, in the case of describing kinetic data, I know only one linear equation resulting from the integration of the mass balance equation for the zero-order process. Do you really know more of such equations?

Dear Prof. Mirosław Grzesik as per I know the equations (Langmuir, Freundlich, Temkin, D-R isotherms) employed for adsorption isotherms are generally in non linear form and to solve these equations one should linearize first, while linearizing the equation there is chance of error which can not be ignored and can lead to misinformation regarding adsorption isotherms.

Thanks!

Dear Dr Krishna Yadav, All isotherms used are nonlinear functions with the exception of the Henry (Nernst) isotherm. It is impossible to transform a nonlinear function to a linear one and vice versa. So talking about linearization in this context does not make sense. By linearization, I think you mean transforming a nonlinear function from its explicit form with respect to a dependent variable, e.g. qe=KCen (Freundlich's isotherm) into its implicit form: lnqe=lnK+nlnCe. Both functions are nonlinear as is the transformation. The linear function: y=ax+b can be obtained only after introducing new variables: y=lnqe, x=lnCe, a=lnK and b=n, which is related, as you rightly noticed, with a change in the distribution of errors, but also with a change in the statistical distribution of new y and x variables.

Let me add that the term “linearization” refers to the local approximation of given function with the aid of Taylor series taking into account a constant term and a linear one. Regards,

This video demonstrates how to find error in isotherm and kinetic study. This video is Part-1 which involves linear or nonlinear regression analysis. https://youtu.be/NAgntHhAutk