Question: How can I calculate the errors associated with the slope (m) and y-intercept (c) of a straight line fit using the polyfit command with polynomial degree 1? Additionally, how can I determine the coefficient of determination (r)?
Solution:
To find the errors associated with the slope (m) and y-intercept (c) in a linear regression model (polynomial = 1), as well as the coefficient of determination (r), you can use the "polyfit" function along with additional calculations.
Code:
% Sample data points
x = [1, 2, 3, 4, 5];
y = [2, 4, 6, 8, 10];
% Perform linear regression using polyfit
coefficients = polyfit(x, y, 1); % As polynomial = 1
m = coefficients(1); % Slope
c = coefficients(2); % Y-intercept
% Calculate the residuals (errors)
residuals = y - (m*x + c);
% Calculate the standard deviation of the residuals
residuals_std = std(residuals);
% Calculate the standard error of the slope and y-intercept
n = length(x); % Number of data points
x_mean = mean(x);
x_var = sum((x - x_mean).^2);
m_std_error = residuals_std/sqrt(x_var*(n - 1));
c_std_error = residuals_std*sqrt(sum(x.^2)/(n*x_var));
% Calculate the coefficient of determination (r)
y_mean = mean(y);
ss_total = sum((y - y_mean).^2);
ss_residual = sum(residuals.^2);
r_squared = 1 - (ss_residual/ss_total);
r = sqrt(r_squared);
% Display the results
disp(["Slope (m): ", num2str(m)]);
disp(["Slope Error: ", num2str(m_std_error)]);
disp(["Y-Intercept (c): ", num2str(c)]);
disp(["Y-Intercept Error: ", num2str(c_std_error)]);
disp(["Coefficient of Determination (r): ", num2str(r)]);
- In this code, we first perform linear regression using polyfit to obtain the slope (m) and y-intercept (c).
- Then, we calculate the residuals (errors) by subtracting the predicted values from the actual values.
- The standard deviation of the residuals is calculated to estimate the overall error.
- The standard error of the slope and y-intercept are then calculated using the residuals' standard deviation, the number of data points, and the variance of the x-values.
- Finally, the coefficient of determination (r) is calculated using the sum of squares of residuals and the total sum of squares.
Links to Documentation:
Here are some documentation links that you can refer to for more information:
- polyfit function: This documentation provides details about the polyfit function in MATLAB, which is used for polynomial curve fitting. Polynomial curve fitting - MATLAB polyfit (mathworks.com)
- Standard Deviation: This documentation explains how to calculate the standard deviation of a set of data points in MATLAB. Standard deviation - MATLAB std (mathworks.com)
