How to calculate errors in pearson linear fit

Hello everyone,

I have done straight line fitting by using polyfit command where i gave polynomial = 1 and found the values of m and c of the equation mx + c = y

Now i want to find errors associated with m and c i.e I want finally

m +/- error c +/- error and also the r value

It would be nice if anyone could help me with it. I am badly stuck :(

Thanks

Answer 1

[RANGA BHARATH]

Hi @aditi. Here is the solution and code for your question.

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)]);

    "Slope (m): "    "2"

disp(["Slope Error: ", num2str(m_std_error)]);

    "Slope Error: "    "9.9301e-17"

disp(["Y-Intercept (c): ", num2str(c)]);

    "Y-Intercept (c): "    "5.1884e-17"

disp(["Y-Intercept Error: ", num2str(c_std_error)]);

    "Y-Intercept Error: "    "6.5869e-16"

disp(["Coefficient of Determination (r): ", num2str(r)]);

    "Coefficient of Determination (r): "    "1"

• 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)

[aditi]

Thank you so much.

I will try it out. I quickly ran the code and it is working great.

Thanks for your help :-)

[aditi]

Hi

Could you please help me understand the maths behind the above code.. like how to derive the formulae of slope and intercept errors? And how to find the residuals?

Thanks

[Francesco Sarnari]

Hi @RANGA BHARATH,

thanks so much for this very useful piece of code. My question to you is the following:

how would this script change if I were willing to calculate the errors associated with a

quadratic, rather than linear fitting? How did you determine the analytical expressions for m_std_error and c_std_error?

Cheers,

Francesco