**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Polynomials, Curve Fitting, and Interpolation **Topic:** Least squares fitting for data analysis. **Overview** Least squares fitting is a fundamental technique in data analysis, allowing us to model relationships between variables by minimizing the sum of the squared residuals between observed data and predicted values. This method is widely used in various fields, including engineering, economics, and data science. In this topic, we will focus on the least squares fitting technique in MATLAB, exploring its applications, syntax, and best practices. **What is Least Squares Fitting?** Least squares fitting is a method for finding the best-fitting curve or line that minimizes the sum of the squared differences between observed data points and predicted values. This technique is used to model linear and non-linear relationships between variables. The primary goal of least squares fitting is to find the optimal parameters that minimize the sum of the squared residuals. **MATLAB's Least Squares Fitting Functions** MATLAB provides several functions for least squares fitting, including: 1. `lsqcurvefit`: A non-linear least squares fitting function that minimizes the sum of the squared residuals between observed data and predicted values. 2. `lscov`: A linear least squares fitting function that provides an estimate of the covariance matrix of the parameters. 3. `polyfit`: A polynomial least squares fitting function that fits a polynomial curve to observed data. **Least Squares Fitting Examples in MATLAB** Let's consider an example of fitting a linear curve to a set of data using `polyfit`. ```matlab % Define the x and y data points x = [1 2 3 4 5]; y = [2 4 6 8 10]; % Use polyfit to fit a linear curve p = polyfit(x, y, 1); % Plot the observed data and the fitted curve plot(x, y, 'o', x, polyval(p, x)); xlabel('x'); ylabel('y'); title('Linear Least Squares Fitting'); legend('Observed Data', 'Fitted Curve'); ``` In this example, we use `polyfit` to fit a linear curve (degree 1) to the observed data points. We then plot the observed data and the fitted curve using `plot` and `polyval`. **Non-Linear Least Squares Fitting** For non-linear least squares fitting, we use `lsqcurvefit`. Let's consider an example of fitting a sigmoidal curve to a set of data. ```matlab % Define the x and y data points x = [0 1 2 3 4 5]; y = [0 0.2 0.6 0.8 0.9 1]; % Define the sigmoidal function f = @(b, x) 1 ./ (1 + exp(-(x - b(1)) / b(2))); % Use lsqcurvefit to fit the sigmoidal curve b0 = [3 1]; % Initial guess for the parameters [b,~,~] = lsqcurvefit(f, b0, x, y); % Plot the observed data and the fitted curve plot(x, y, 'o', x, f(b, x)); xlabel('x'); ylabel('y'); title('Non-Linear Least Squares Fitting'); legend('Observed Data', 'Fitted Curve'); ``` In this example, we use `lsqcurvefit` to fit a sigmoidal curve to the observed data points. We define the sigmoidal function using an anonymous function `f` and provide an initial guess for the parameters `b0`. We then plot the observed data and the fitted curve using `plot` and `f`. **Best Practices** 1. Always check the residuals to ensure that the fitted curve is a good representation of the observed data. 2. Use a sufficient number of data points to ensure that the fitted curve is robust. 3. Use a variety of initial guesses for non-linear least squares fitting to avoid local minima. 4. Use `lsqcurvefit` for non-linear least squares fitting, as it provides more control over the fitting process. **Conclusion** Least squares fitting is a powerful technique for modeling relationships between variables. MATLAB provides several functions for least squares fitting, including `lsqcurvefit`, `lscov`, and `polyfit`. By understanding the syntax and best practices of these functions, you can effectively use least squares fitting to analyze data and make predictions. **External Resources** * MATLAB Documentation: [Least Squares Fitting](https://www.mathworks.com/help/optim/ug/least-squares.html) * MathWorks Blog: [Fitting Data with MATLAB](https://blogs.mathworks.com/loren/2009/08/04/fitting-data-with-matlab/) * Coursera Course: [Linear Algebra and Its Applications](https://www.coursera.org/specializations/linear-algebra) **Leave a comment or ask for help** If you have any questions or need further clarification on the topics covered, please leave a comment below.