Dear Jurandi,

I depends on the pattern of the data. As a rule of thumb, you can choose a value for A close to the max SC (observed or theoretical).

B depends on how low is the SC (at minimum SC) - the lower the SC (at t=0) the higher B should be.

k will reflect how steep is the change of predicted SC. B and k also depend on the "t" range.

If you're willing to share an example dataset I'll recomend some initial values.

Best regards

Luis

Dear Jurandy,

When you consider a mathematical model for a phenomenon, you have already assumed a behavior for that phenomenon similar to your model.

Therefore, the best way to estimate the values of parameters in your model is to fit some real data on the it.

So, try to generate a set of data (Ai, Bi, ki, SCi ; i=1,2,...,n) and then fit them on your model using Least Square Fit Methods or any other fitting algorithm.

Kind regards,

Omid

What I've always done is plot the data I am trying to fit in a program like Excel, and then add a line to the plot which is the model I am trying to fit.  Have the parameters for the model in cells so that if you change the number, the line on the graph will automatically change.  By trial-and-error, you will find good starting values pretty easily.

NLIN also has a grid search option for finding starting parameters.  Use best = 5 (or some other value) in the main nlin statement.  And then parms a = -5 to 5 by 1, etc.

Also, you might try the method=marquardt option.  It seems to do better not getting lost if the starting values aren't great.

When I first replied to Jurandi I specifically took into account the model he described [SC = A/[1 + B exp(-kt)]]. Theerfore "A" will be the max observed value (if it is already a plateau) or the max theoretical one. B and k relate to the shape of the sigmoid and are not difficult to figure out.

This is a quite simple model and I do not foresee any trouble in getting suitable initial estimates. I also follow the approach depicted by Salvatore (i.e. plot the experimental data as well as the fitted model) using different parameter values in order to guess an adequate set of parameter values. One of the issues (in more complex models) is when you have cross-correlation between model variables or when you do have several optimal sets (suitable solutions) of values for the model parameters. The fitting/optimization process does not always converge to the same solution. But again it depends on the available data.

Best regards

In your example, you have 3 parameters in your model, so you can pick 3 data points in your data pool and plug these 3 data points into your model to solve the parameter values. Although your model is nonlinear, the equations you need to solve may be linear. In the case that these equations are nonlinear equations, root-finding problem is still easier than your original problem. The solution for these 3 parameters can be used as a reasonable initial guess for nonlinear least squares fitting. More details of this approach (e.g., how to choose 3 data points from the data pool) can be found in the following paper:

Article Taking the Guess Work Out of the Initial Guess: A Solution I...

If you want to find the guaranteed optimal parameter estimators that correspond to the global minimum of the squared error of the fit, you can use the focused regions identification method introduced in the following paper:

Article Reducing the Search Space for Global Minimum: A Focused Regi...

Thanks for all your valuable comments. However, Sal Mangiafico the NLIN statement you mentioned is it in Excell or R?

Model correlation: p = A/[1 + Bexp(-kt)]

i) Note that for |t| = 0; p0 ≡ A/(1 + B)

ii) Note that for |t| → +∞, 0; p∞ ≡ lim|t| → +∞{A/[1 + Bexp(-kt)]} = A

iii) From above, by examining the intercept and the asymptotic behaviour after the available data, we may possibly be able to estimate both A and B. At least other data point (|t| > 0) would be required to find a first guess estimate for the other parameter, k. First guess estimates can be denoted (A0, B0, k0).

iv) More generally, we can simply take just some 3 data points {(t1,p1), (t2,p2), (t3,p3)} to solve the corresponding system of (3) equations and find first guess estimates for the parameters (A, B, k), denoted (A0, B0, k0).

{p1 = A0/[1 + B0·exp(-k0·t1)], p2 = A0/[1 + B0·exp(-k0·t2)], p3 = A0/[1 + B0·exp(-k0·t3)]}

or with q ≡ (t1-t2)/(t2-t3):

{A0 = p1·{1 + (A0/p2 - 1)q+1/(A0/p3 - 1)q}, k0 = - {1/(t2-t3)}·ln{(A0/p2 - 1)/(A0/p3 - 1)}, B0 = (A0/p3 - 1)/exp(-k0·t3)}

where first equation can be iteratively solved.

v) The whole set of data can then be depicted in a linearized plot as exp(-k0·t) vs. 1/p, correlated as exp(-k0·t) = -1/B1 + (A1/B1)/p, from which improved parameters {A1, B1} can be fitted by least-squares linear regression (after intercept and slope).

vi) Last straight line correlation rewrites as: k0 = - {ln{-1/B1 + (A1/B1)/p}}/t. A better estimate for k can be found after the arithmetic mean for the whole set of data points k1 = - {∑{ln{-1/B1 + (A1/B1)/pi}}/ti}/n, i = 1, 2, ... , n.

vii) Both last steps can be iteratively repeated to improve the fitted parameters {A, B, k} until some suitable convergence criterion is met. These are unbiased estimates for the linearized correlation exp(-k·t) = -1/B + (A/B)/p; but not for the original correlation, p = A/[1 + Bexp(-kt)]. The corresponding linearized plot is quite convenient to linearly display the set of data points as well as scattering around the trendline.

viii) Last estimates of the parameters can still be refined, by iterative nonlinear least-squares regression, to find unbiased estimates for the original correlation.

This easy to use program will do that for you!

www.lerenisplezant.be/fitting.htm

Jamilu Yunusa Falgore , I'm only a couple of years late to respond, but, anyway, my comments on NLIN, best =, parms, and method= , refer to PROC NLIN in SAS, which the original post was asking about.

estimating initial values in nonlinear models is crucial for effective optimization. Several approaches include utilizing theoretical knowledge, graphical methods, parameter transformations, literature review, trial and error, parameter estimation software, sensitivity analysis, Monte Carlo methods, global optimization algorithms, and considering constraints. Employing these approaches can help researchers effectively estimate initial values for parameters in nonlinear models, facilitating the optimization process and improving the accuracy of model fitting.

This is discussed in my new book, which will appear later in 2024, but I can already ship preprints.

www.lerenisplezant.be/fitting-book.htm

My software does this automatically:

www.lerenisplezant.be/fitting.htm