Deriving a Simplified Formula from 2D Data Using Curve Fitting

I like tables, So I took a 2D table and made a formula...

Introduction

In the realm of data analysis, the ability to model relationships between variables is crucial for prediction and understanding underlying processes. This article outlines my method for analyzing a two-dimensional table of data concerning the relationship between ratio and wall thickness, ultimately resulting in a simplified formula that captures the observed trends.

Data Overview

The dataset comprises a matrix with two variables: Ratio (ranging from 1.0 to 7.0) and Wall Thickness (ranging from 50mm to 300mm). The data points represent a dependent variable that varies according to these two parameters. Here’s a snapshot of the data:

Methodology

Step 1: Curve Fitting for Individual Rows

The first step in the analysis involved fitting each row of the data table to an exponential curve. The exponential function was chosen due to its flexibility in modeling growth patterns observed in the data. For each ratio, the following exponential model was fitted:

Using a suitable regression method, I calculated the parameters for each row to obtain individual formulas.

Step 2: Analyzing the Parameters

Once the row-specific exponential formulas were established, the next phase involved fitting the parameters (derived from the first step) to a higher-level curve. In this case, I identified that a Michaelis-Menten curve and a Power curve effectively represented the parameters across different ratios. This process involved:

Fitting a Michaelis-Menten Curve: Often used in enzyme kinetics, this curve helped in understanding the saturation point of the parameters.

Fitting a Power Curve: This curve provided insight into the power-law relationships among the parameters.

Step 3: Deriving the Simplified Formula

After establishing the higher-level curves for the parameters, I integrated them back into the original exponential functions. The resulting simplified formula to describe the relationship between the ratio and wall thickness is:

Results and Conclusion

The derived formula effectively captures the relationship between the ratio and wall thickness across the data set. This method not only streamlined the data analysis process but also provided a predictive tool for evaluating wall thickness effects based on different ratios.

The results illustrate the importance of curve fitting and model selection in data analysis. By systematically approaching the problem and validating each step, I was able to derive a useful formula that can aid further investigations or practical applications in material science and engineering fields.