Math CentralQuandaries & Queries


Question from aika:

Could one show me the complete algorithm and formula for finding the coefficients (a,b, and c) in exponential regression model


Firstly, it is important to realize that a formula like this, specifying a fit line, does not specify a regression model. A regression model is specified by a parametrized family of models for the conditional probability distribution of Y given X.

In some cases (such as ordinary least squares regression) there is a "best fit line" that corresponds to most probable values of Y(X). In other cases, such as logistic regression, there is no such line. [Logistic regression fits models in which Y is always 0 or 1, with P(Y=1) a function of X. For instance, it could model the probability of sinking a basketball penalty shot from X feet away. But the ball is always either in or out.]

Anyhow - a best fit curve is never enough on its own. An error model is also needed, and the choice of error model will affect the choice of best fit curve. For ordinary least squares, the error model is Gaussian with standard deviation independent of X ("homoskedastic"). That is to say, an observation Y(X) has an equal probability of being some distance off the best fit line whatever X is.

For exponential models the error model is usually NOT homoskedastic, but it is assumed that the error is proportional to y-c.
Fitting the full model that you have given is (as I recall) rather complicated, and would -I think- be done by a repeated approximation process, not a closed-form formula. In this case you are out of your depth and need a statistician.

However, the model you give is also more general than is usually required. Usually the asymptotic lower limit c is equal to 0 or some other value that is given by theory. If this is the case, then you may make the transformation Z = log(Y-c) [decimal or natural log, it does not matter]. This transforms the parametric family of best-fit curves into Z = [log a] + bX, AND makes the proportional error model mentioend above into a homoskedastic model. Ordinary least squares can then be used on the transformed data. If you are not sure whether this applies, you are out of your depth and need a statistician.

If the errors follow any other model (in particular if the observed Y can possibly be less than c) this simple method will not work and you will have to fall back on iterative methods. In this case - or if you are not sure - you are out of your depth and need a statistician.

Good Hunting!


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