How Can You Fit Exponential and Logarithmic Curves in Python Without Built-in Functions?

Curve Fitting: Beyond Polynomials in Python

When analyzing data, it's often necessary to determine the best mathematical expression to describe the relationship between variables. While polynomial fitting is a common choice, exponential and logarithmic curves can also provide valuable insights.

Performing Exponential and Logarithmic Fitting Without Existing Functions

Despite the absence of dedicated functions for exponential and logarithmic fitting in Python's standard library, there are ways to accomplish this task using transformations.

Logarithmic Curve Fitting (y = A B log x)

To fit a logarithmic curve, simply plot y against (log x). The resulting coefficients from the linear regression will give the parameters of the logarithmic equation (y ≈ A B log x).

Exponential Curve Fitting (y = Ae^Bx)

Fitting an exponential curve is slightly more involved. Take the logarithm of both sides of the equation (log y = log A Bx) and plot (log y) against x. The resulting linear regression coefficients provide the parameters for the exponential equation (y ≈ Ae^Bx).

Note on Bias in Weighted Least Squares:

When fitting exponential curves, it's important to consider that polyfit's default weighted-least-squares method can bias the results towards small values of y. To alleviate this, specify weights proportionate to y using the w keyword argument.

Using Scipy's Curve_Fit for Flexibility

Scipy's curve_fit function offers a more versatile approach to curve fitting, allowing you to specify any model without transformations.

Logarithmic Curve Fitting using Scipy:

Curve_fit returns identical results to the transformation method for the logarithmic curve model.

Exponential Curve Fitting using Scipy:

For exponential curve fitting, curve_fit provides a more accurate fit by calculating Δ(log y) directly. However, it requires an initial guess to reach the desired local minimum.

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