How to Fit Exponential and Logarithmic Curves in Python: Beyond Polynomial Fitting?

Fitting Exponential and Logarithmic Curves in Python

Background

When fitting a dataset, it is desirable to find the curve that best describes it. This process, known as curve fitting, is essential for a wide range of scientific and engineering applications. Among the different types of curves, exponential and logarithmic functions can provide insights into data trends.

Existing Function for Polynomial Fitting

In Python, the numpy.polyfit() function provides a convenient way to perform polynomial fitting. However, this function only supports polynomial models.

Fitting Exponential and Logarithmic Functions

Exponential Curves

To fit a curve of the form y = Ae^Bx, take the logarithm of both sides of the equation:

log(y) = log(A) Bx

Then, fit log(y) against x. Alternatively, you can use the scipy.optimize.curve_fit function with the lambda expression:

lambda t, a, b: a * np.exp(b * t)

Logarithmic Curves

To fit a curve of the form y = A B log x, simply fit y against log(x).

numpy.polyfit(numpy.log(x), y, 1)

Weighted Regression for Exponential Fits

When fitting exponential curves, it is important to consider the bias towards small values in the unbiased linear fitting approach. This bias can be alleviated by using weighted regression with weights proportional to y.

numpy.polyfit(x, numpy.log(y), 1, w=np.sqrt(y))

Advantages of Using scipy.optimize.curve_fit

While transformation methods can be used for fitting exponential and logarithmic functions, scipy.optimize.curve_fit offers several advantages:

• Curve fitting without transformations
• Ability to handle models that are not monotonic or linearizable
• Provides an initial guess for localization

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