exponential

Exponential models

Exponentials are often used when the rate of change of a quantity is proportional to the initial amount of the quantity. The general exponential model is given by

\[y=\sum_{k=1}^na_k\exp{(b_kx)}\]

If the coefficient \(b_k\) for an \(\exp\) term is negative, that term represents exponential decay. If the coefficient is positive, that term represents exponential growth.

This module provides the general exponential model above for any number of terms \(k\), as well as convenience functions for the one- (\(k=1\)) and two-term (\(k=2\)) exponential models analogous to MATLAB’s exp1 and exp2, respectively.

Additionally, this module has the stretched exponential function, also known as the complementary cumulative Weibull distribution, given by

\[y=\exp{\left[-\left(\frac{x}{\alpha}\right)^\beta\right]}\]

where \(\beta\) is the stretching exponent. This expression is obtained by inserting a fractional power law into the exponential function.

This model is generally meaningful only for \(x>0\). The graph of \(\log{(y)}\) vs. \(x\) is characteristically stretched when \(0\leq\beta\leq 1\) and compressed when \(\beta>1\) (the latter case has less practical importance). When \(\beta=1\), the one-term exponential model is recovered. When \(\beta=2\), the probability density function for the normal distribution is obtained.

Functions

biexp	Bi-exponential function.
exp	General exponential model.
exp1	Convenience function for the exp1 model from MATLAB.
exp2	Convenience function for the exp2 model from MATLAB.
stretched_exp	Stretched exponential function.