Curve Fitting

Curve fitting is a numerical analysis method that enables a dataset to be represented by a specific mathematical model. It is commonly applied to establish a relationship between discrete data points obtained through measurement or experimentation. This relationship is expressed through a functional structure, and the resulting model is expected to produce values as close as possible to the observed data.

The curve fitting process involves a series of data pairs $(x_i,y_i)$

A lightly curved, simple model. It generally fits the data but may fail to accurately represent certain points.

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Higher-degree polynomials are more flexible. Although they may appear to fit the data better, they can cause excessive oscillations (carrying a risk of overfitting).

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Linear model (least squares): Assumes a constant trend throughout the entire dataset.

Exponential model: More suitable when growth or decline is accelerating. In this example, it shows excessive growth and is therefore not appropriate for the data.

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Performed using piecewise cubic polynomials. It ensures a smooth and continuous transition between data points, providing a flexible and precise fit to the data.

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The suitability of a model is assessed using various statistical indicators. The most common is the coefficient of determination $(R^2)$

$R^2=1-\frac{\sum (y_i-f(x_i){)}^2}{\sum (y_i-\bar{y}{)}^2}$

Here, $\overline{y}$ represents the mean of the observed values. A value of $R^2$ approaching 1 indicates a high degree of fit between the model and the data.

Curve fitting is applied across diverse fields such as engineering, physics, biology, economics, and data science. It is used for modeling experimental results, constructing prediction systems, and determining system parameters.

• Engineering: Temperature-time relationships are fitted from sensor data. For example, the behavior of engine temperature over time is modeled.
• Physics: The function $s=f(t)$ can be derived from experimental data in free-fall experiments.
• Biology: Cell growth or bacterial population increase can be modeled using exponential functions.
• Economics: Curves are fitted between inflation rates and consumer price indices to make predictions.
• Medicine: Drug release curves are fitted in pharmacokinetic studies to estimate distribution within the body.
• Artificial Intelligence / Data Science: Regression models learn by fitting curves to training data.

As model complexity increases, the risk of overfitting also rises. This results in a model that fits the existing data well but performs poorly on new data. Therefore, when selecting a model, its generalization capacity must also be considered.

• Overfitting: If the model learns noise along with the underlying pattern, it will fail on new data.
• Underfitting: If the model is too simple, it will fail to capture the trends in the data.
• Sensitivity: Small changes in the data can lead to large variations in high-degree polynomials.
• Data Distribution: Curve fitting can produce biased estimates if the data is unevenly distributed.
• Multiple Variables: Curve fitting is typically applied to single variables; for multiple variables, surface fitting or multivariate regression is required.