Curve Fitting: Linear Regression and Polynomial Fitting in MATLAB

Curve fitting is a fundamental technique in data analysis and scientific research. It involves finding a mathematical function that best describes the relationship between a set of data points. This allows us to understand trends, make predictions, and simplify complex data. In this module, we'll explore two common curve fitting methods: linear regression and polynomial fitting, using MATLAB.

Understanding Curve Fitting

The goal of curve fitting is to find a model that minimizes the difference between the observed data and the values predicted by the model. This difference is often quantified using a 'cost function' or 'loss function', such as the sum of squared errors (SSE).

Curve fitting finds a mathematical function to represent data trends.

We aim to find a function that closely matches our data points, allowing us to understand relationships and make predictions.

The process involves selecting a type of function (e.g., linear, polynomial, exponential) and then determining the parameters of that function that best fit the given data. The 'best fit' is typically achieved by minimizing a measure of error, such as the sum of the squared differences between the actual data points and the values predicted by the fitted curve.

Linear Regression

Linear regression is used when we suspect a linear relationship between two variables. It finds the best-fitting straight line through a set of data points. The equation of a straight line is typically represented as $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

What is the standard equation for a straight line used in linear regression?

$y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

In MATLAB, the

code

polyfit

function can be used for linear regression by specifying a polynomial of degree 1. Alternatively, the

code

fitlm

function provides a more comprehensive linear model object.

Linear regression finds the line of best fit for a set of data points. The function $y = mx + c$ is fitted to the data by minimizing the sum of squared errors between the observed $y$ values and the predicted $y$ values. The polyfit([x1, x2, ...], [y1, y2, ...], 1) function in MATLAB returns the coefficients [m, c] for this linear model.

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Polynomial Fitting

Polynomial fitting extends linear regression to fit curves that are not straight lines. A polynomial of degree $n$ has the form $y = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ . By increasing the degree $n$ , we can fit more complex relationships in the data.

Choosing the correct polynomial degree is crucial. Too low a degree may underfit the data, while too high a degree can lead to overfitting, where the model fits the noise in the data rather than the underlying trend.

MATLAB's

code

polyfit

function is ideal for polynomial fitting. You simply provide the x and y data, and the desired degree of the polynomial. The function returns the coefficients of the polynomial in descending order of power.

What is the primary risk associated with using a very high degree polynomial for curve fitting?

Overfitting, where the model fits the noise in the data rather than the underlying trend.

Once the coefficients are obtained, you can use the

code

polyval

function to evaluate the polynomial at specific x-values to generate the fitted curve for plotting.

Practical Application in MATLAB

Let's consider an example. Suppose you have experimental data points (x, y) and you want to fit a quadratic curve (degree 2 polynomial) to it.

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The

code

polyfit

function in MATLAB is versatile. For linear regression, you use

code

polyfit(x, y, 1)

. For a quadratic fit, you use

code

polyfit(x, y, 2)

, and so on. The output coefficients can then be used with

code

polyval

to generate the fitted curve for visualization and further analysis.

Learning Resources

Curve Fitting - MATLAB & Simulink(documentation)

Official MathWorks documentation providing an overview of curve fitting techniques and functions available in MATLAB.

Polynomial Fitting - MATLAB & Simulink(documentation)

Detailed documentation for the `polyfit` function, explaining its syntax, usage, and return values for polynomial fitting.

Linear Regression - MATLAB & Simulink(documentation)

Documentation for the `fitlm` function, a powerful tool for fitting linear models, including linear regression, in MATLAB.

Introduction to Curve Fitting in MATLAB(video)

A video tutorial demonstrating how to perform curve fitting, including linear and polynomial regression, using MATLAB.

MATLAB Curve Fitting Tutorial(video)

Another practical video guide showcasing curve fitting techniques and their implementation in MATLAB for engineering applications.

Understanding Overfitting and Underfitting(blog)

An explanation of the concepts of overfitting and underfitting, crucial for selecting appropriate curve fitting models.

Least Squares Regression(wikipedia)

Wikipedia article detailing the mathematical principles behind least squares regression, the foundation for many curve fitting methods.

Polynomial Regression - A Complete Guide(blog)

A comprehensive guide to understanding and applying polynomial regression, including its statistical aspects.

MATLAB Central - File Exchange(documentation)

A repository of user-contributed MATLAB code, often including examples and tools for curve fitting and data analysis.

Data Fitting in Engineering(paper)

A scientific overview of data fitting methodologies commonly employed in various engineering disciplines.

Curve Fitting: Linear Regression, Polynomial Fitting

Curve Fitting: Linear Regression and Polynomial Fitting in MATLAB

Understanding Curve Fitting

Curve fitting finds a mathematical function to represent data trends.

Linear Regression

Polynomial Fitting

Practical Application in MATLAB

Learning Resources