Weighted Exponential Fit Calculator

Mathematical Formulation of Weighted Exponential Regression

In physics laboratory experiments, exponential processes govern phenomena such as radioactive decay, capacitor discharge, thermal cooling, and optical absorption. The general functional model describing exponential behavior is:

y = a · e^bx

To perform weighted linear fitting, we apply a logarithmic transformation to the exponential expression to express it as a linear model:

ln(y) = ln(a) + bx

Defining $Y' = \ln(y)$ and $A = \ln(a)$, we obtain a standard linear equation $Y' = A + bx$. The weighted sum of squared residuals $S$ to minimize is defined as:

S = ∑ w_i (Y'_i - (A + b x_i))²

Minimizing $S$ with respect to parameters $A$ and $b$ yields the following system of normal equations:

b = [ (∑ w) (∑ w x Y') - (∑ w x) (∑ w Y') ] / [ (∑ w) (∑ w x²) - (∑ w x)² ]
A = [ (∑ w Y') (∑ w x²) - (∑ w x) (∑ w x Y') ] / [ (∑ w) (∑ w x²) - (∑ w x)² ]

Finally, parameter $a$ is recovered using the exponential inverse transformation $a = e^{A}$.

How to Use This Calculator

Input X Data: Enter independent variables such as time, distance, or voltage in the first column.
Input Y Data: Enter corresponding measured dependent variables. Ensure all Y values are positive.
Specify Weights: Optional weight factors (e.g., inverse variance $1/\sigma^2$) account for experimental uncertainty.
Compute Results: Press the calculate button to instantly compute amplitude, exponent, and fit accuracy metrics.

Frequently Asked Questions

Why are positive Y values mandatory?

Logarithmic transformations require positive inputs because the natural logarithm function is undefined for zero or negative real numbers.

How should I choose weights for my physics experiment?

Weights should ideally reflect measurement accuracy. Set $w_i = 1/\sigma_i^2$, where $\sigma_i$ is the uncertainty of each individual data point.

What does the coefficient of determination ($R^2$) mean?

The $R^2$ value measures model accuracy, ranging from 0 to 1, where higher values indicate closer agreement with experimental observations.